Financial and Actuarial Statistics an Introduction

Financial and Actuarial Statistics An Introduction

Dale S.Borowiak University of Akron Akron, Ohio, U.S.A.

MARCEL DEKKER, INC. NEW YORK • BASEL

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ISBN: 0-82474-270-2 (Print Edition) Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212–696– 9000; fax: 212–685–4540 This edition published in the Taylor & Francis e-Library, 2005. "To purchase your own copy of this or any of Taylor & Francis or Routledge's collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk." Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41–61–260–6300; fax: 41–61–260–6333 World Wide Web http://www.dekker.com/ The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.

STATISTICS: Textbooks and Monographs

D.B.Owen Founding Editor, 1972–1991

Associate Editors Statistical Computing/ Nonparametric Statistics Professor William R.Schucany Southern Methodist University Probability Professor Marcel F.Neuts University of Arizona Multivariate Analysis Professor Anant M.Kshirsagar University of Michigan Quality Control/Reliability Professor Edward G.Schilling Rochester Institute of Technology Editorial Board Applied Probability Dr. Paul R.Garvey The MITRE Corporation Economic Statistics Professor David E.A.Giles University of Victoria Experimental Designs Mr. Thomas B.Barker Rochester Institute of Technology

Multivariate Analysis Professor Subir Ghosh University of California-Riverside Statistical Distributions Professor N.Balakrishnan McMaster University Statistical Process Improvement Professor G.Geoffrey Vining Virginia Polytechnic Institute Stochastic Processes Professor V.Lakshmikantham Florida Institute of Technology Survey Sampling Professor Lynne Stokes Southern Methodist University Time Series Sastry G.Pantula North Carolina State University 1. The Generalized Jackknife Statistic, H.L.Gray and W.R.Schucany 2. Multivariate Analysis, Anant M.Kshirsagar 3. Statistics and Society, Walter T.Federer 4. Multivariate Analysis: A Selected and Abstracted Bibliography, 1957–1972, Kocherlakota Subrahmaniam and Kathleen Subrahmaniam 5. Design of Experiments: A Realistic Approach, Virgil L.Anderson and Robert A. McLean 6. Statistical and Mathematical Aspects of Pollution Problems, John W.Pratt 7. Introduction to Probability and Statistics (in two parts), Part I: Probability; Part II: Statistics, Narayan C.Giri 8. Statistical Theory of the Analysis of Experimental Designs, J.Ogawa 9. Statistical Techniques in Simulation (in two parts), Jack P.C.Kleijnen 10. Data Quality Control and Editing, Joseph I.Naus

11. Cost of Living Index Numbers: Practice, Precision, and Theory, Kali S.Banerjee 12. Weighing Designs: For Chemistry, Medicine, Economics, Operations Research, Statistics, Kali S.Banerjee 13. The Search for Oil: Some Statistical Methods and Techniques, edited by D.B.Owen 14. Sample Size Choice: Charts for Experiments with Linear Models, Robert E.Odeh and Martin Fox 15. Statistical Methods for Engineers and Scientists, Robert M.Bethea, Benjamin S. Duran, and Thomas L.Boullion 16. Statistical Quality Control Methods, Irving W.Burr 17. On the History of Statistics and Probability, edited by D.B.Owen 18. Econometrics, Peter Schmidt 19. Sufficient Statistics: Selected Contributions, Vasant S.Huzurbazar (edited by Anant M. Kshirsagar) 20. Handbook of Statistical Distributions, Jagdish K.Patel, C.H.Kapadia, and D.B.Owen 21. Case Studies in Sample Design, A.C.Rosander 22. Pocket Book of Statistical Tables, compiled by R.E.Odeh, D.B.Owen, Z.W. Bimbaum, and L.Fisher 23. The Information in Contingency Tables, D.V.Gokhale and Solomon Kullback 24. Statistical Analysis of Reliability and Life-Testing Models: Theory and Methods, Lee J. Bain 25. Elementary Statistical Quality Control, Irving W.Burr 26. An Introduction to Probability and Statistics Using BASIC, Richard A.Groeneveld 27. Basic Applied Statistics, B.L.Raktoe and J.J.Hubert 28. A Primer in Probability, Kathleen Subrahmaniam 29. Random Processes: A First Look, R.Syski 30. Regression Methods: A Tool for Data Analysis, Rudolf J.Freund and Paul D.Minton

31. Randomization Tests, Eugene S.Edgington 32. Tables for Normal Tolerance Limits, Sampling Plans and Screening, Robert E.Odeh and D.B.Owen 33. Statistical Computing, William J.Kennedy, Jr., and James E.Gentle 34. Regression Analysis and Its Application: A Data-Oriented Approach, Richard F.Gunst and Robert L.Mason 35. Scientific Strategies to Save Your Life, I.D.J.Bross 36. Statistics in the Pharmaceutical Industry, edited by C.Ralph Buncher and Jia-Yeong Tsay 37. Sampling from a Finite Population, J.Hajek 38. Statistical Modeling Techniques, S.S.Shapiro and A.J.Gross 39. Statistical Theory and Inference in Research, T.A.Bancroft and C.-P.Han 40. Handbook of the Normal Distribution, Jagdish K.Patel and Campbell B.Read 41. Recent Advances in Regression Methods, Hrishikesh D.Vinod and Aman Ullah 42. Acceptance Sampling in Quality Control, Edward G.Schilling 43. The Randomized Clinical Trial and Therapeutic Decisions, edited by Niels Tygstrup, John M Lachin, and Erik Juhl 44. Regression Analysis of Survival Data in Cancer Chemotherapy, Walter H.Carter, Jr., Galen L.Wampler, and Donald M.Stablein 45. A Course in Linear Models, Anant M.Kshirsagar 46. Clinical Trials: Issues and Approaches, edited by Stanley H.Shapiro and Thomas H. Louis 47. Statistical Analysis of DNA Sequence Data, edited by B.S.Weir 48. Nonlinear Regression Modeling: A Unified Practical Approach, David A.Ratkowsky 49. Attribute Sampling Plans, Tables of Tests and Confidence Limits for Proportions, Robert E.Odeh and D.B.Owen

50. Experimental Design, Statistical Models, and Genetic Statistics, edited by Klaus Hinkelmann 51. Statistical Methods for Cancer Studies, edited by Richard G.Comell 52. Practical Statistical Sampling for Auditors, Arthur J.Wilbum 53. Statistical Methods for Cancer Studies, edited by Edward J.Wegman and James G. Smith 54. Self-Organizing Methods in Modeling: GMDH Type Algorithms, edited by Stanley J. Farlow 55. Applied Factorial and Fractional Designs, Robert A.McLean and Virgil L.Anderson 56. Design of Experiments: Ranking and Selection, edited by Thomas J.Santner and Ajit C.Tamhane 57. Statistical Methods for Engineers and Scientists: Second Edition, Revised and Expanded, Robert M.Bethea, Benjamin S.Duran, and Thomas L.Boullion 58. Ensemble Modeling: Inference from Small-Scale Properties to Large-Scale Systems, Alan E.Gelfand and Crayton C.Walker 59. Computer Modeling for Business and Industry, Bruce L.Bowerman and Richard T. O’Connell 60. Bayesian Analysis of Linear Models, Lyle D.Broemeling 61. Methodological Issues for Health Care Surveys, Brenda Cox and Steven Cohen 62. Applied Regression Analysis and Experimental Design, Richard J.Brook and Gregory C.Arnold 63. Statpal: A Statistical Package for Microcomputers—PC-DOS Version for the IBM PC and Compatibles, Bruce J.Chalmer and David G.Whitmore 64. Statpal: A Statistical Package for Microcomputers—Apple Version for the II, II+, and lle, David G.Whitmore and Bruce J.Chalmer 65. Nonparametric Statistical Inference: Second Edition, Revised and Expanded, Jean Dickinson Gibbons 66. Design and Analysis of Experiments, Roger G.Petersen

67. Statistical Methods for Pharmaceutical Research Planning, Sten W.Bergman and John C.Gittins 68. Goodness-of-Fit Techniques, edited by Ralph B.D’Agostino and Michael A. Stephens 69. Statistical Methods in Discrimination Litigation, edited by D.H.Kaye and Mikel Aickin 70. Truncated and Censored Samples from Normal Populations, Helmut Schneider 71. Robust Inference, M.L.Tiku, W.Y.Tan, and N.Balakrishnan 72. Statistical Image Processing and Graphics, edited by Edward J.Wegman and Douglas J.DePriest 73. Assignment Methods in Combinatorial Data Analysis, Lawrence J.Hubert 74. Econometrics and Structural Change, Lyle D.Broemeling and Hiroki Tsurumi 75. Multivariate Interpretation of Clinical Laboratory Data, Adelin Albert and Eugene K. Harris 76. Statistical Tools for Simulation Practitioners, Jack P.C.Kleijnen 77. Randomization Tests: Second Edition, Eugene S.Edgington 78. A Folio of Distributions: A Collection of Theoretical Quantile-Quantile Plots, Edward B. Fowlkes 79. Applied Categorical Data Analysis, Daniel H.Freeman, Jr. 80. Seemingly Unrelated Regression Equations Models: Estimation and Inference, Virendra K.Srivastava and David E.A.Giles 81. Response Surfaces: Designs and Analyses, Andre I.Khuri and John A.Comell 82. Nonlinear Parameter Estimation: An Integrated System in BASIC, John C.Nash and Mary Walker-Smith 83. Cancer Modeling, edited by James R.Thompson and Barry W.Brown 84. Mixture Models: Inference and Applications to Clustering, Geoffrey J.McLachlan and Kaye E.Basford 85. Randomized Response: Theory and Techniques, Arijit Chaudhuri and Rahul Mukerjee

86. Biopharmaceutical Statistics for Drug Development, edited by Karl E.Peace 87. Parts per Million Values for Estimating Quality Levels, Robert E.Odeh and D.B.Owen 88. Lognormal Distributions: Theory and Applications, edited by Edwin L.Crow and Kunio Shimizu 89. Properties of Estimators for the Gamma Distribution, K.O.Bowman and L.R.Shenton 90. Spline Smoothing and Nonparametric Regression, Randall L.Eubank 91. Linear Least Squares Computations, R.W.Farebrother 92. Exploring Statistics, Damaraju Raghavarao 93. Applied Time Series Analysis for Business and Economic Forecasting, Sufi M.Nazem 94. Bayesian Analysis of Time Series and Dynamic Models, edited by James C.Spall 95. The Inverse Gaussian Distribution: Theory, Methodology, and Applications, Raj S. Chhikara and J.Leroy Folks 96. Parameter Estimation in Reliability and Life Span Models, A.Clifford Cohen and Betty Jones Whitten 97. Pooled Cross-Sectional and Time Series Data Analysis, Terry E.Dielman 98. Random Processes: A First Look, Second Edition, Revised and Expanded, R.Syski 99. Generalized Poisson Distributions: Properties and Applications, P.C.Consul 100. Nonlinear Lp-Norm Estimation, Rene Gonin and Arthur H.Money 101. Model Discrimination for Nonlinear Regression Models, Dale S.Borowiak 102. Applied Regression Analysis in Econometrics, Howard E.Doran 103. Continued Fractions in Statistical Applications, K.O.Bowman and L.R.Shenton 104. Statistical Methodology in the Pharmaceutical Sciences, Donald A.Berry 105. Experimental Design in Biotechnology, Perry D.Haaland 106. Statistical Issues in Drug Research and Development, edited by Karl E.Peace

107. Handbook of Nonlinear Regression Models, David A.Ratkowsky 108. Robust Regression: Analysis and Applications, edited by Kenneth D.Lawrence and Jeffrey L.Arthur 109. Statistical Design and Analysis of Industrial Experiments, edited by Subir Ghosh 110. U-Statistics: Theory and Practice, A.J.Lee 111. A Primer in Probability: Second Edition, Revised and Expanded, Kathleen Subrahmaniam 112. Data Quality Control: Theory and Pragmatics, edited by Gunar E.Liepins and V.R.R. Uppuluri 113. Engineering Quality by Design: Interpreting the Taguchi Approach, Thomas B.Barker 114. Survivorship Analysis for Clinical Studies, Eugene K.Harris and Adelin Albert 115. Statistical Analysts of Reliability and Life-Testing Models: Second Edition, Lee J.Bain and Max Engelhardt 116. Stochastic Models of Carcinogenesis, Wai-Yuan Tan 117. Statistics and Society: Data Collection and Interpretation, Second Edition, Revised and Expanded, Walter T.Federer 118. Handbook of Sequential Analysis, B.K.Ghosh and P.K.Sen 119. Truncated and Censored Samples: Theory and Applications, A.Clifford Cohen 120. Survey Sampling Principles, E.K.Foreman 121. Applied Engineering Statistics, Robert M.Bethea and R.Russell Rhinehart 122. Sample Size Choice: Charts for Experiments with Linear Models: Second Edition, Robert E.Odeh and Martin Fox 123. Handbook of the Logistic Distribution, edited by N.Balakrishnan 124. Fundamentals of Biostatistical Inference, Chap T.Le 125. Correspondence Analysis Handbook, J.-P.Benzécri

126. Quadratic Forms in Random Variables: Theory and Applications, A.M.Mathai and Serge B.Provost 127. Confidence Intervals on Variance Components, Richard K.Burdick and Franklin A. Graybill 128. Biopharmaceutical Sequential Statistical Applications, edited by Karl E.Peace 129. Item Response Theory: Parameter Estimation Techniques, Frank B.Baker 130. Survey Sampling: Theory and Methods, Arijit Chaudhuri and Horst Stenger 131. Nonparametric Statistical Inference: Third Edition, Revised and Expanded, Jean Dickinson Gibbons and Subhabrata Chakraborti 132. Bivariate Discrete Distribution, Subrahmaniam Kocherlakota and Kathleen Kocherlakota 133. Design and Analysis of Bioavailability and Bioequivalence Studies, Shein-Chung Chow and Jen-pei Liu 134. Multiple Comparisons, Selection, and Applications in Biometry, edited by Fred M. Hoppe 135. Cross-Over Experiments: Design, Analysis, and Application, David A.Ratkowsky, Marc A.Evans, and J.Richard Alldredge 136. Introduction to Probability and Statistics: Second Edition, Revised and Expanded, Narayan C.Giri 137. Applied Analysis of Variance in Behavioral Science, edited by Lynne K.Edwards 138. Drug Safety Assessment in Clinical Trials, edited by Gene S.Gilbert 139. Design of Experiments: A No-Name Approach, Thomas J.Lorenzen and Virgil L.Anderson 140. Statistics in the Pharmaceutical Industry: Second Edition, Revised and Expanded, edited by C.Ralph Buncher and Jia-Yeong Tsay 141. Advanced Linear Models: Theory and Applications, Song-Gui Wang and SheinChung Chow 142. Multistage Selection and Ranking Procedures: Second-Order Asymptotics, Nitis Mukhopadhyay and Tumulesh K.S.Solanky

143. Statistical Design and Analysis in Pharmaceutical Science: Validation, Process Controls, and Stability, Shein-Chung Chow and Jen-pei Liu 144. Statistical Methods for Engineers and Scientists: Third Edition, Revised and Expanded, Robert M.Bethea, Benjamin S.Duran, and Thomas L.Boullion 145. Growth Curves, Anant M.Kshirsagar and William Boyce Smith 146. Statistical Bases of Reference Values in Laboratory Medicine, Eugene K.Harris and James C.Boyd 147. Randomization Tests: Third Edition, Revised and Expanded, Eugene S.Edgington 148. Practical Sampling Techniques: Second Edition, Revised and Expanded, Ranjan K. Som 149. Multivariate Statistical Analysis, Narayan C.Giri 150. Handbook of the Normal Distribution: Second Edition, Revised and Expanded, Jagdish K.Patel and Campbell B.Read 151. Bayesian Biostatistics, edited by Donald A.Berry and Dalene K.Stangl 152. Response Surfaces: Designs and Analyses, Second Edition, Revised and Expanded, André I.Khuri and John A.Cornell 153. Statistics of Quality, edited by Subir Ghosh, William R.Schucany, and William B.Smith 154. Linear and Nonlinear Models for the Analysis of Repeated Measurements, Edward F. Vonesh and Vemon M.Chinchilli 155. Handbook of Applied Economic Statistics, Aman Ullah and David E.A.Giles 156. Improving Efficiency by Shrinkage: The James-Stein and Ridge Regression Estimators, Marvin H.J.Gruber 157. Nonparametric Regression and Spline Smoothing: Second Edition, Randall L.Eubank 158. Asymptotics, Nonparametrics, and Time Series, edited by Subir Ghosh 159. Multivariate Analysis, Design of Experiments, and Survey Sampling, edited by Subir Ghosh

160. Statistical Process Monitoring and Control, edited by Sung H.Park and G.Geoffrey Vining 161. Statistics for the 21st Century: Methodologies for Applications of the Future, edited by C.R.Rao and Gábor J.Székely 162. Probability and Statistical Inference, Nitis Mukhopadhyay 163. Handbook of Stochastic Analysis and Applications, edited by D.Kannan and V.Lakshmikantham 164. Testing for Normality, Henry C.Thode, Jr. 165. Handbook of Applied Econometrics and Statistical Inference, edited by Aman Ullah, Alan T.K.Wan, and Anoop Chaturvedi 166. Visualizing Statistical Models and Concepts, R.W.Farebrother 167. Financial and Actuarial Statistics: An Introduction, Dale S.Borowiak Additional Volumes in Preparation

For my loving wife, I love you always, thanks for the trip!

Preface

In the fields of financial and actuarial modeling modern statistical techniques are playing an increasingly prominent role. The use of statistical analysis in areas such as investment pricing models, options pricing, pension plan structuring and advanced actuarial modeling is required. After teaching two actuarial science courses I realized that both students and investigators need a strong statistical background in order to keep up with modeling advances in these fields. This book approaches both financial and actuarial modeling from a statistical point of view. The goal is to supplement the texts and writings that exist in actuarial science with statistical background and present modern statistical techniques such as saddlepoint approximations, scenario and simulation techniques and stochastic investment pricing models. The aim of this book is to provide a strong statistical background for both beginning students and experienced practitioners. Beginning students will be introduced to topics in statistics, financial and actuarial modeling from a unified point of view. A thorough introduction to financial and actuarial models such as investment pricing models, discrete and continuous insurance and annuity models, pension plan modeling and stochastic surplus models from a statistical science approach is given. Statistical techniques associated with these models, such as risk estimation, percentile estimation and prediction intervals are discussed. Advanced topics related to statistical analysis including single decrement modeling, saddlepoint approximations for aggregate models and resampling techniques are discussed and applied to financial and actuarial models. The audience for this book is made up of two sectors. Actuarial science students and financial investigators, both beginning and advanced, who desire thorough discussions on basic statistical concepts and techniques will benefit from the approach of this text that introduces statistical principles in the context of financial and actuarial topics. This approach allows the reader to develop knowledge in both areas and understand the existing connections. Advanced readers, whether students in college undergraduate or graduate programs in mathematics, economics or statistics, or professionals advancing in financial and actuarial careers will find the in-depth discussions of advanced modeling topics useful. Research discussions include approximations to aggregate sums, single decrement modeling, statistical analysis of investment pricing models and simulation approaches to stochastic status models. The approach this text takes is unique in that it presents a unified structure for both financial and actuarial modeling. This is accomplished by applying the actuarial concept of financial actions being based on the survival or failure of predefined conditions, referred to as a status. Applying either a deterministic or stochastic feature to the general

status unifies financial and actuarial models into one structure. Basic statistical topics, such as point estimation, confidence intervals and prediction intervals, are discussed and techniques are developed for these models. The deterministic setting includes basic interest, annuity, investment pricing models and aggregate insurance models. The stochastic status models include life insurance, life annuity, option pricing models and pension plans. In Chapter 1 basic statistical concepts and functions including an introduction to probability, random variables and their distributions, expectations, moment generating functions, estimation, aggregate sums of random variables, compound random variables, regression models and an introduction to autoregressive modeling are presented in the context of financial and actuarial modeling. Financial computations such as interest and annuities in both discrete and continuous modeling settings are presented in Chapter 2. The concept of deterministic status models is introduced in Chapter 3. The basic loss model along with statistical evaluation criteria are presented and applied to single risk models including investment and option pricing models, collective aggregate models and stochastic surplus models. In Chapter 4 the discrete and continuous future lifetime random variable along with the force of mortality is introduced. In particular, multiple future lifetime and decrement models are discussed. In Chapter 5, through the concept of group survivorship modeling, future lifetime random variables are used to construct life models and life tables. Ultimate, select multiple decrement and single decrement tables, along with statistical measurements are presented. Stochastic status models that include actuarial life insurance and annuity models and applications make up the material for Chapter 6. Risk and percentile premiums, reserve calculations and common notations are discussed. More advanced topics such as computational relationships between models, general time period models, multiple decrement computations, pension plan models and models that include expenses are included. In Chapter 7 modern scenario and simulation techniques, along with associated statistical inference, are introduced and applied to both deterministic and stochastic status models. In particular, collective aggregate models, investment pricing models and stochastic surplus models are evaluated using simulation techniques. In Chapter 8 introductions to advanced statistical topics, such as mortality adjustment factors for increased risk cases and mortality trend modeling, are presented. I would like to thank the people at Marcel Dekker Inc., in particular Maria Allegra, for her assistance in the production of this book. Dale S.Borowiak

Contents

Preface 1 Statistical Concepts

xv 1

2 Financial Computational Models

54

3 Deterministic Status Models

78

4 Future Lifetime Random Variable

104

5 Future Lifetime Models and Tables

135

6 Stochastic Status Models

161

7 Scenario and Simulation Testing

214

8 Further Statistical Considerations

236

Appendix: Standard Normal Tables

249

References

253

Index

257

Financial and Actuarial Statistics

1 Statistical Concepts

The modeling of financial and actuarial systems starts with the mathematical and statistical concepts of variables. There are two types of variables used in financial and actuarial statistical modeling, referred to as non-stochastic or deterministic and stochastic variables. Stochastic variables are proper random variables that have an associated probability structure. Non-stochastic variables are deterministic in nature without a probability attachment Interest and annuity calculations based on fixed time periods are examples of non-stochastic variables. Examples of stochastic variables are the prices of stocks that are bought or sold, and insurance policy and annuity computations where payments depend upon stochastic events, such as accidents or deaths. The evaluation of stochastic variables requires the use of basic probability and statistical tools. This chapter presents the basic statistical concepts and computations that are utilized in the analysis of such data. For the most part the concepts and techniques presented in this chapter are based on the frequentist approach to statistics as opposed to the Bayesian perspective and are limited to those that are required later in the analysis of financial and actuarial models. This is due in part to the lack of Bayesian model development and application in this area. The law of large numbers is relied upon to add validity to frequentist probabilistic approach. Basic theories and concepts are applied in a unifying approach to both financial and actuarial modeling. The basis of statistical evaluations and inference is probability. Therefore, we start this chapter with a brief introduction to probability and then proceed to the various statistical components. Standard statistical concepts such as discrete and continuous random variables, probability distributions, moment generating functions and estimations are discussed. Further, other topics such as approximating the aggregate sum of random variables, regression modeling and an introduction to stochastic processes through auto correlation modeling are presented. 1.1 Probability In this section we present a brief introduction into some basic ideas and concepts in probability, There are many texts in probability that give a broader background but a review is useful since the basis of statistical inference is contained in probability theory. The results discussed are used either directly in the later part of this book or give insight

Financial and actuarial statistics

2

useful to later topics. Some of these topics may be review for the reader and we refer to Larson (1995) and Ross (2002) for further background in basic probability. For a random process let the set of all possible outcomes comprise the sample space, denoted Ω. Subsets of the sample space, consisting of some or all of the possible outcomes, are called events. Primarily we are interested in assessing the likelihood of events occurring. Basic set operations are defined on the events associated with a sample space. For events A and B the union of A and B, , is comprised of all outcomes in A, B, or common to both A and B. The intersection of two events A and B is the set of all outcomes common to both A and B and is denoted A∩B. The complement of event A is the event that A does not occur and is Ac. In general, we wish to quantify the likelihood of particular events taking place. This is accomplished by defining a stochastic or probability structure over the set of events. For any event A, the probability of A, measuring the likelihood of occurrence is denoted P(A). Taking an empirical approach, if the random process is observed repeatedly then as the number of trials or samples increases the proportion of time A occurs within the trials approaches the probability of A or P(A). This is called the long run relative frequency of event A. In various settings mathematical models are developed to determine this probability function. There are certain mathematical properties that every probability function more formally referred to as a probability measure follows. A probability measure, P, is a real valued set function where (i) P(A)≥0 for all events A. (ii) P(Ω)=1 (iii) Let A1, A2,…be a collection of disjoint sets, i.e.

for i≠j.

Then (1.1.1) Conditions (i), (ii) and (iii) are called the axioms of probability and, (iii) is referred to as the countable additive property of a probability measure. These conditions form the basic structure of a probability system. In practice probability measures are constructed in two ways. The first is based on assumed functional structures derived from physical laws and are mathematically constructed. The second, more statistical in nature, relies on observed or empirical data. Both methods are used in financial and actuarial modeling. An illustrative example is now given. Ex. 1.1.1. A survey of 125 people in a particular age group, or strata, is taken. Let J denote the number of future years an individual holds a particular stock. Here J is an integer future lifetime and is the number of full years a person retains possession of a stock. From the survey data a table, Table 1.1.1, of frequencies, given by f, for values of J is constructed. The relative frequency concept is used to estimate probabilities when the choices corresponding to individual outcomes are equally likely. For example the probability a person sells the stock in less than 1 year is the proportion P(J=0) =2/125=.016. The probability a stock is held for 5 or more years is P(J≥5)= 100/125=.8.

Statistical concepts

3

The simple concepts presented in Ex. 1.1.1 introduce basic statistical ideas and notations, such as integer years, used in the development of financial and actuarial models. In later chapters model evaluation and statistical inferences are developed based on these basic ideas. Further, the concept of conditioning on observed outcomes and conditional probabilities is central to financial and actuarial calculations. For two events, A and B, the conditional probability of A given the fact B has occurred is defined by (1.1.2) provided P(B) is not zero. This probability structure satisfies the previously

Table 1.1.1 Survey of Future Holding Lifetimes of a Stock J

0

1

2

3

4

5 or more

f

2

3

4

6

10

100

stated probability axioms. The previous example is now extended to demonstrate conditioning on specified events. Ex. 1.1.2. Consider the conditions of Ex. 1.1.1. Given an individual holds a stock for the first year the conditional probability of selling the stock in subsequent years is found using (1.1.2). For J≥1 (1.1.3) For example, the conditional probability of retaining possession of the stock for at least 5 additional years is P(J≥5|J≥1)=(100/125)/(123/125)=100/123= .8130. The conditional probability concept can be utilized to compute joint probabilities corresponding to many events. For a collection of events A1, A2,…, An the probability of all Ai, 1≤i≤n, occurring is (1.1.4) The idea of independence plays a central role in many applications. A collection of events A1, A2,…, An are completely independent or just independent if (1.1.5) It is a mathematical fact that events can be “pair-wise” independent but not completely independent In practice formulas for the analysis of financial and actuarial actions are based on the ideas of conditioning and independence. A clear understanding of these concepts aids in the mastery of future statistical, financial and actuarial topics.


4

There are some properties and formulas that follow from the axioms of probability. Two such properties are used in the application and development of statistical models. For event A let the complement be Ac. Then

Fig. 1.1.1 Probability Rules (1.1.6) and (1.1.7).

(1.1.6) Also, for two events A and B the probability of the union can be written as (1.1.7) It is sometimes useful to view these probability rules in terms of graphs of the sample space and the respective events referred to as Venn Diagrams. Given in Fig. 1.1.1 are the Venn Diagrams corresponding to rules (1.1.6) and (1.1.7). In Prob 1.1 the reader is asked to verify rules (1.1.6) and (1.1.7) using probability axiom property (iii), (1.1.1) and Fig. 1.1.1, by utilizing disjoint sets. These formulas have many applications and two examples that apply these computations as well as introduce two important actuarial multiple life settings now follow. Ex. 1.1.3. Two people ages x and y takes out a financial contract that pays a benefit predicated on the survival of the two people, referred to as (x) and (y), for an additional j years. Let events be A={(x) lives past age x+j} and B={(y) lives past age y+j}. We consider two different types of status where the events A and B are considered independent. i) Joint Life Status requires both people to survive an additional n years. The probability of paying the benefit, using (1.1.5), is P(A∩B)=P(A)P(B). ii) Last Survivorship Status requires at least one person to survive an additional n years. Using (1.1.5) and (1.1.7) the probability of paying the benefit is . In particular let the frequencies presented in Table 1.1.1 hold where the two future lifetimes are given by J1 and J2. Thus, for any individual P(J≥3)=116/125=.928. From i) the probability both survive an additional 3 years is

From (1.1.7) the probability at least one of the two survive an additional 3 years is


5

These basic probabilistic concepts easily extend to more than two future lifetime variables. Ex. 1.1.4. An insurance company issues insurance policies to a group of individuals. Over a short period, such as a year, the probability of a claim for any policy is .1. The probability of no claim in the first 3 years is found assuming independence and applying (1.1.5)

Also, using (1.1.6) the probability of at least one claim in 3 years is

This insurance setting is referred to as short term insurance where the force of interest, introduced in Sec. 2.1, can be ignored. The basic formula in a variety of settings and it is helpful to understand the basic structures. For conceptual understanding and applicability it is sometimes helpful to see formulas in a variety of financial and actuarial modeling settings. We now turn our attention to topics in both applied and theoretical statistics. 1.2 Random Variables In financial and actuarial modeling there are two types of variables, stochastic and nonstochastic. Non-stochastic variables are completely deterministic lacking any stochastic structure. Examples of these variables include the fixed benefit in a life insurance policy, the length of time in a mortgage loan or the amount of a fixed interest mortgage payment Random variables are stochastic variables that possess some probabilistic or stochastic component. Random variables include the lifetime of a particular health status, the value of a stock after one year or the amount of a health insurance claim. In general notation, random variables are denoted by uppercase letters, such as X or T, and fixed constants take the form of lower case letters, like x and t. There are three types of random variables characterized by the possible values they can assume. Along with the typical discrete and continuous random variables there are combinations of discrete and continuous variables, referred to as mixed random variables. For a discussion of random variables and corresponding properties we refer to Hogg and Tanis (2001, Ch 3 and Ch 4) and Rohatgi (1976, Ch 2). In financial and actuarial modeling the time until a financial action occurs may be associated with a probability structure and therefore be stochastic. In actuarial science, conditions prior to initiation of the financial action are referred to as the holding of a status. The action is initiated when the status changes or fails to hold. We use this general concept of a status along with its change or failure to unite financial and actuarial modeling in a common framework. For example, with a life insurance policy the status is the act of the person surviving. During the person’s lifetime the status, defined as survival, is said to hold. After the death of the person the status is said to fail and an


6

insurance benefit is paid. Similarly, in finance an investor may retain a particular stock, thereby keeping ownership or the “status” of the stock the same, until the price of the stock reaches a particular level. Upon reaching the desired price the status, or ownership of the stock, changes and the status is said to fail. In general the specific conditions that dictate one or more financial actions are referred to as a status and the lifetime of a status is a random variable, which we denote by T. 1.2.1 Discrete Random Variables A discrete random variable, X, can take on a countable number of values or outcomes. Associated with each outcome is a corresponding probability. The collection of these probabilities comprise the probability density function, pdf, denoted f(x)

Fig 1.2.1 Discrete pdf.

(1.2.1) for possible outcome values x. The support of f(x), denoted by S, is the domain set on which f(x) is positive. From the association between the random variable and the probability axioms (i), (ii) and (iii) we se that f(x)≥0 for all x in S and the sum f(x) over all elements in S is one. We note that in some texts the discrete probability density function is referred to as a probability mass function. In many settings the analysis of a financial or actuarial model depends on the integer valued year a status fails denoted by J. For example, an insurance policy may pay a fixed benefit at the end of the year of death. The variable J is the year of death as measured from the date the policy was issued or T=0 and J =1, 2,….. We follow with examples in the context of life insurance that demonstrate these concepts and introduce standard probability measures and their corresponding pdfs. Ex. 1.2.1. In the case death of a insurance policy holder within five years of the initiation of the policy a fixed amount or benefit b is paid at the end of the year of death, If the policyholder survives five years amount b is immediately paid. Let J denote the year a payment is made, so that J=1, 2,…, 5 and the support is S={1, 2, 3, 4, 5}. Let the probability of an accident be q and the probability of no accident be p, so that 0≤p≤1 and q=1−p. The probability structure is contained in the pdf of J, which for demonstrational purposes takes the Geometric random variable form, given by


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(1.2.2)

The pdf (1.2.2) can be used to assess the expected cost and statistical aspects of the policy. The graph of the pdf (1.2.2) is given in Fig. 1.2.1 and is typical of a discrete pdf where the probabilities are represented as spikes at the support points of the pdf. Later in this text the expected cost is computed using the probability structure defined by the pdf along with the time value of money. Ex 1.2.2. Over a short time period a collection of m insurance policies is considered. For policy i, 1≤i≤m, let the random variable Xi=1 if a claim is made and=0 in the event of no claim. Also, for each i let P(Xi=1)=q and P(Xi=0)=p=1−q for 0
and counts the number of claims out of the m policies. Here X is discrete on support S={0, 1,…, n} with parameters m and p. The pdf gives the probability that N=n and is assumed to be (1.2.3) for n=0, 1,…, m. Here N is a binomial random variable with parameters n and p. In an infinite population setting this is a standard sampling distribution and the statistical aspects of the binomial distribution are discussed later. Ex. 1.2.3. For a set of insurance policies let N denote the number of claims over a specific time period. Let N take on a stochastic structure where the Poisson distribution is employed. Here the pdf of N is based on support S= {0, 1,…} and takes the form (1.2.4) for parameter λ>0. Hence, the probability of no claims is P(N=0)=f(0)= exp(−λ). The Poisson probability structure can be derived from a set of conditions, referred to as the Poisson postulates, that imply that the Poisson pdf is appropriate to model discrete random processes. Many classical examples of modeling random structures with a Poisson random variable exist and a detailed description is given by Helms (1997, p. 271).


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Fig. 1.2.1 Continuous pdf.

1.2.2 Continuous Random Variables For a continuous random variable, X, the stochastic structure is different from the discrete random variable and the domain consists of one or more intervals. The distribution function, df, or cumulative distribution function associated with X is always exists and is defined as probability the random variable attains at most fixed quantities and is given by (1.2.5) for constant x. We remark that the df is defined over the entire real line. In the continuous case the probability density function or pdf, if it exists, corresponding to X is a function f(x)≥0, for all x, and probability of intervals corresponds to areas under f(x). Hence, the total area under f(x) is one. The support of f(x), denoted S, designates the set where f(x) is positive. For interval (a, b) contained in S and using the Fundamental Theorem of Calculus (1.2.6) Hence, in the continuous random variable setting probabilities corresponding to intervals are represented as areas under the pdf. In Fig. 1.2.1 probability (1.2.6) is represented as the area under the curve f(x) between a and b. Thus, the df F(x) is the antiderivative of the pdf f(x) over support S. Standard continuous statistical models are introduced in the next set of examples.


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Fig. 1.2.2 Continuous Distributions.

Ex. 1.2.4. The continuous random variable X is uniform on support S=[a, b], a
The df, from definition (1.2.5), is defined by (1.2.8)

We remark that mathematically the df is bounded by one, nondecreasing and right continuous. In Fig. 1.2.2 is a graph of both the pdf and df given by (1.2.7) and (1.2.8) when b=3 and a=1. The graphs given in Fig, 1.2.2 are typical for continuous type distributions where probabilities of events correspond to areas under f(x) and cumulative probabilities are given by the df. The uniform distribution is often utilized in modeling probabilities when little or no information about the stochastic structure of a process is known. Ex. 1.2.5. Let the lifetime associated with an insurance policy be T, where T is said to follow an exponential distribution. The pdf is given by (1.2.9) for parameter constant θ>0 and the support is given by St={t: t≥0}. The probability that T exceeds constant c, called the reliability or survival, is (1.2.10) for c>0. The exponential distribution has many applications (see Walpole, Myers and Myers (1998, p. 166) and is frequently used in survival and reliability modeling. Ex. 1.2.6. Let the future lifetime of a status, T, follow a Gaussian or normal distribution with mean µ and standard deviation σ, denoted by T~n(µ, σ2). The pdf associated with T is given by


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where the support is S=(−∞, ∞). This pdf is symmetric about the mean µ and to compute probabilities the transformation to the standard normal random variable is required. The standard normal random variable, denoted Z, is a normal random variable that takes mean 0 and variance 1. The Z-random variable associated with T=t is given by the transformation Z=(T−µ)/σ. The df for T is (1.2.11) for any real valued t where Φ is the df of the standard normal random variable. The evaluation of Φ in (1.2.11) is achieved using numerical approximation methods. Tabled values of Φ(t) for fixed t, such as given in is Appendices A1 and A2, or computer packages are utilized to compute normal probabilities. For example let lifetime T be a normal random variable with parameters µ=65 and σ=10. The probability the age of an individual exceeds 80 is computed using (1.2.11) and Appendix A2 as

Further, the probability an individual dies between ages 70 and 90 is found as

We remark that the continuous nature of the above random variable, where the probability of attaining an exact value is negligible, is utilized. 1.2.3 Mixed Random Variables Mixed random variables are a combination of both discrete and continuous random variables. If X is a mixed random variable the support is partition into two disjoint parts. One part is discrete in nature while the other part of the support is continuous. Applications of mixed-type random variables are rare in many fields but this type of random variable is particularly useful in financial and actuarial modeling. Many authors atttack mixed random variable problems in the context of statistical conditioning while we present a straightforward approach. The simple example that follows demonstrates the versatility of this variable. Ex. 1.2.7. Here an insurance policy pays claims between $100 and $500. The amount of the claim, X, is defined as a mixed random variable. The discrete part defines the probability X=0 as .5 and X=$100 and $500 as .2. The continuous part is defined by a constant (or uniform) pdf over [$100, $500] with value .00025. Hence, the pdf is defined as


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The support of f(x) is decomposed into where S1={0, 100, 500} and S2= (100, 500). Probabilities are computed using the procedures for discrete and continuous random variables. For example the condition that the total probability associated with X is one implies

Also, the probability that the claim is at most $250 is the combination of discrete and continuous type calculations

This example, although simple, demonstrates the possible types of mixed discrete and continuous random variables that can be constructed. There is a variation of the mixed-type random variable that utilizes both discrete and continuous random variables in defining the pdf. This plays a part in insurance modeling and an example of this type of random variable structure follows in the next example. Ex. 1.2.8. A one-year insurance policy pays a claim or benefit denoted by B in case of an accident. The probability of a claim in the first year is given by q. Given there is a claim let B be a continuous random variable with pdf f(B). The overall claim variable can be written as X=IB where the indicator function I=1 if there is a claim and=0 if there is no claim. The pdf of X can be approached from a conditioning point of view, as introduced in (1.1.2), is (1.2.12)

The probability the claim is greater than c>0 is P(X>c)=qP(B>c). This situation of single insurance policies has many practical applications. One is extension of models of the form (1.2.12) to a set or portfolio of many policies. These are referred to as collective risk models, and discussed in Sec. 3.4. Further, over longer periods of time adjustments must be made to account for the effect or force of interest. Much statistical work concerns the estimation of collective stochastic structures. As we have seen in some of the examples the pdf, f(x) and the df, F(x) may be a function of one or more parameters. In practice the experimenter may estimate the


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unknown parameters from empirical data. Probabilistic and statistical aspects of such estimation must be accounted for in financial and actuarial models. 1.3 Expectations The propensities of a random variable or a function of a random variable to take on particular outcomes is often important to financial and actuarial modeling. The expectation is one method of predicting and assessing stochastic outcomes of random variables. The expected value of function g(x), if it exists, is denoted E{g(x)}. Often the expected values of properly selected functions are used to characterize the probability distribution associated with one or more random variables. Along with the possible types of random variables three cases, discrete, continuous and mixed random variables, produce different formulas for expected values. First, if X is discrete with support Sd and pdf f(x) (1.3.1) Second, X is continuous and the pdf f(x) has support Sc (1.3.2) In the last case if X is a mixed random variable the expected value is a combination of then (1.3.1) and (1.3.2). If the support is (1.3.3) In financial and actuarial modeling expectations play a central role. The central core of financial and actuarial risk analysis is the computation of properly chosen random variables. There are a few standard expectations that play an important role in analyzing data. Employing the identity function, g(x)=x, yields the expected value of X or the mean of X given by (1.3.4) The mean of X is a weighted average, with respect to the probability structure, over the support and is a measure of the center of the pdf. If g(x)=Xr, for positive integer r, then the expected value, E{Xr}, is referred to as the rth moment or a moment of order r of X. It is a mathematical property that if moments of order r exist then moments of order s exist for s≤r. Central moments of order r, for positive integer r, are defined by E{(X−µ)r}. The variance of X is a central moment with r=2 and is denoted by Var{X}=σ2 and after simplification the variance becomes (1.3.5)


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We note that existence of the second moment implies existence of the variance. The standard deviation of X is σ=(σ1/2)2. The variance and standard deviation of a random variable measures the dispersion or variability associated with the random variable and the associated pdf. The discrete case computation is demonstrated in the next example. Ex. 1.3.1. Let N be Poisson as described in Ex. 1.2.3. The mean of N is found using a Taylor Series (see Prob. 1.3)

In a similar manner, so that from (1.3.5), . Hence, for the Poisson random variable the mean and the variance are equivalent and completely determine the distribution. As mentioned earlier for random variable X the mean, µ, measures the “center” and the standard deviation, σ, measures the “variability” or dispersion associated with the pdf of X. Other useful moment measurements are the skewness and the kurtosis denoted by, respectively, Sk and Ku. These are defined by (1.3.6) These moments are classically used to characterize distributions in terms of shape. For an applied discussion in the usage of (1.3.6) we refer to McBean and Rovers (1998). Examples concerning moment computations in the continuous and mixed random variable cases are now given. Ex. 1.3.2. Let X be uniform on (a, b) as in Ex. 1.2.4. The mean or expected value of X is (1.3.7) Further, the second moment is

From (1.3.5), the variance of X simplifies to (1.3.8) The special case of the uniform distribution over the unit interval has many applications and takes a=0 and b=1 and from (1.3.7) and (1.3.8) produces moments µ=1/2 and σ2=1/12.


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Ex. 1.3.3. Let X have an exponential pdf given in Ex. 1.2.5. To find the mean of X we use integration by parts to find

Using integration by parts twice (see Prob. 1.6) we find

Hence, from (1.3.5)

In fact for positive integer r, the general moment formula is given by . Applying the general moment formula the skewness and kurtosis, defined by (1.3.6), can be computed (see Prob. 1.8). Ex. 1.3.4. In this example we consider the mixed variable case of Ex. 1.2.7. The supports for the discrete and continuous parts are defined by S1={0, 100, 500} and S2=(100, 500), respectively. The mean takes the form

Hence, to represent a typical value of X close to the center of the pdf we might use the mean or expectation of $150.00. Two additional general formulas are used to compute the expected value of a function of X when X≥0. Let X have pdf f(x) with support S and df F(x) and G(x) be monotone where G(x)≥0. There are two cases to consider, continuous and discrete random variables. If X is continuous we assume G(x) is differentiable with (d/dx)G(x)=g(x) and assuming E{G(X)} exists the expectation is (1.3.9) In the case X is discrete with corresponding support on the nonnegative integers then, if the expectation exists, the expected value of G(X) is (1.3.10) where δ(G(x))=G(x+1)−G(x), The proofs of these expectation formulas are outlined in Prob. 1.7.


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Ex. 1.3.5. Let the number of claims over a period of time be N so that the support is S={0, 1,…}. The pdf corresponding to X is assumed to take the form of the discrete geometric distribution introduced in Ex. 1.2.1. The general pdf is given by

for constants p, 0
and so 1−F(n)=qn +1. To find the mean of N we employ (1.3.10), noting G(n) =n, g(n)=1 and δ(G(n))=1 (1.3.13) We note that as the probability of a claim, q, increases the mean, (1.3.13), increases. The second moment can be computed by using (1.3.10) with G(x)= x2. Ex. 1.3.6. An investment of $5,000 is made in the hopes of increased value over time. The sale of the investment occurs at future time T with corresponding pdf

First, we find constant a that makes f(t) a pdf. Hence


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which implies a=.6(3)/53=.0144. We now compute the mean and variance of T. The first moment is

The second moment is

and, hence, the variance of T is

In application the value of the investment is a function of both the length of time held and the return rate on the investment. In Chapter 2 financial computations and concepts concerning the return and interest rates are presented and explored. Ex. 1.3.7. The insurance policy setting of Ex. 1.2.8 is revisited. Over a short time period let the probability of a claim be p and the benefit paid be B. The claim variable, as defined in Ex. 1.2.8, is X=IB and the pdf is given by (1.2.12) with graph given in Fig. 1.2.5. The expected value of X is (1.3.14) where S is the support of B. The second moment is (1.3.15) Hence, utilizing (1.3.5) the variance of X can be written as the combination of moments given by (1.3.16) In particular, if Y~n(1000, 3002) and p=.1 then E{X}=.1(1000)=100. Further, the probability the claim variable is more than 1,200 is computed, using Appendix A2, as

Hence, in any time period the probability of a claim is small at 2.5%. Models over short time periods where the force of interest can be ignored are referred to as short-term risk models and are discussed in Sec. 3.4.


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1.4 Moment Generating Function A special widely used expectation in both the theoretical and applied settings is the moment generating function or mgf. If g(X)=exp(tx) then the mgf is found by computing the expectation given by (1.3.1), (1.3.2) or (1.3.3) in the discrete, continuous setting or mixed random variable cases respectively. For the mgf to exist it must exist or converge for values of t in a neighborhood of zero. Hence, the mgf, when X is discrete, is defined by (1.4.1) and in the case where X is continuous X the mgf is (1.4.2) The mixed random variable case is a combination of (1.4.1) and (1.4.2) and all the formulas concerning the mgf assume the expectations exist. The mgf has many uses, such as moment computation and distribution derivation, that will be described later in this chapter. We now follow with a discussion of moment generating functions through a set of typical random variable examples. Ex. 1.4.1. From Ex. 1.2.2 let N be binomial with parameters m and p. In this discrete random variable case the mgf is computed using the binomial theorem and is given by the formula (1.4.3)

It is clear that the mgf evaluated at zero takes the value of one. One of the primary used of the mgf, as described later, is to compute to compute the moments of X. Ex. 1.4.2. Let X be continuous uniform on [c−δ, c+δ], or X~U[c− δ, c+δ], for positive constants δ
Using the Taylor Series expansion (see Prob. 1.3) we can simplify (1.4.4) as (1.4.5)


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As the value of δ decreases to zero we note that the mgf converges to Mx(t)= exp(tc) which corresponds to a random variable that takes all its probability at the single point c. The theory and the results concerning the convergence of an mgf are extensive and some of the implications are considered later, but in this case the convergence to a constant implies that the uniform random variables converges in distribution to the single point c. As mentioned earlier, in either the discrete or continuous case the mgf can be used to find the moments of X. For positive integer r the rth moment, if it exists, can be compute using (1.4.6) In the above we assume the above derivative exists in the support of the random variable. The application of (1.4.6) is now demonstrated. Ex 1.4.3. Let X be a normal random variable, introduced in Ex. 1.2.6, with mean µ and standard deviation σ. The mgf can be shown (see Hogg and Craig (1995, p. 139) to be (1.4.7) where the mgf is defined on the entire real line, Further, using (1.4.7) we show that the mean and standard deviations are, respectively, µ and σ. From (1.4.6) with r=1 the mean is calculated as

To compute the second moment using the product rule for derivatives we find

Hence, Var(X)=E{X2}−µ2=σ2. Formally, this demonstrates the fact that the parameters in the normal pdf correspond to the mean and variance. The mgf, when it exists, is unique and can be employed to find the distribution of a random variable. If a random variable under examination yields an mgf that matches a mgf of a known pdf then the pdf also matches. This is a commonly used technique when examining the distribution of sums of independent random variables. Furthermore, the Continuity Theorem states that if the limit of an mgf converges point-wise to a proper mgf then the corresponding distributions converge. For more in-depth discussions of the uses of the mgf see either Rohatgi (1976, Sec 4.6) or Hogg and Craig (1995, Sec. 4.7). An alternative to the mgf is the characteristic function defined by E{exp(itX)}. This complex valued function, similar to the mgf, determines existing moments and is unique in that it completely determines the distribution function of the associated random variable. In fact an inversion formula exists that allows derivation of the associated pdf, assuming it exists, based on the characteristic function. The characteristic function has an advantage over the mgf in that it always exists but in some cases tools and concepts from complex analysis are required. For a discussion of the characteristic function and application we refer to Laha and Rohatgi (1979).


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1.5 Survival Functions For random variable X the survival or reliability function defines the probability that X attains at least a fixed value, say x. For X the survival function associated with constant x is (1.5.1) If X is the lifetime of a status then (1.5.1) gives the probability that the lifetime is greater than constant x. Using F(x)=P(X≤x) we note the relationship with the df, S(x)=1−F(x). In the case where X is continuous the relationship between S(x) and the pdf f(x) is (1.5.2) The survival function is unique and determined the distribution of the random variable. In the balance of this section examples of survival functions and moment computations utilizing (1.5.1) are presented. Ex. 1.5.1 Let the lifetime of a status X be a continuous random variable with support S=(0, 100) where associated to survival function is given by

Using (1.5.2), the corresponding pdf is

Many measurements can be made that characterize the distribution associated with X. One classical measurement that characterizes the center of the distribution, similar to the mean, is the median. The median of the of X is the constant x1/2 where S(x1/2)=1/2. In this case the median is computed to be x1/2 =100(1−1/21/2)=29.29. If the support of the pdf is nonnegative the survival function can be used to compute moments of random variables. Letting G(X)=X in (1.3.9) or (1.3.10) alternative formulas for the mean of X can be constructed. The expected value of X can be found by (1.5.3) where X is discrete or continuous, respectively, and the summation and the integral are over the support of f(x). Other moments can also be computed using these formulas. For example, the second moment is either (1.5.4)


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depending on X being either discrete or continuous. These formulas, (1.5.3) and (1.5.4), in some cases ease computations and are discussed in Prob.1.8. Ex. 1.5.2. Let the lifetime random variable of a status X have an exponential distribution with pdf given by (1.2.9). The survival function is S(x) =exp(−x/θ) and using (1.5.3) the expected value of X is

Further, using (1.5.4), we find E{X2}=2θ2, so that Var{X}=θ2. In this case the alternate computation is easier than directly applying the definition approach of formulas (1.3.1) and (1.3.2). Ex. 1.5.3. Let X~U(0, 100) so that the pdf is f(x)=1/100 for 0≤x≤ 100. For 0≤x≤100 the survival function is

We now find the mean and variance of X. From (1.5.3) the mean is

From (1.5.4)

and hence, Var{X}=1002/3−1002/4=1002/12=833.33. The mean and variance can be used to construct approximate interval estimates for assessment statistics. 1.6 Conditional Distributions The distribution of a random variable under the condition that the variable is at least a fixed value c is central in the study of financial and actuarial science, The distribution corresponding to various statuses depend on the conditions, such as age at the time of the inception of a policy or a contract In engineering and medical statistics this type of conditioning is quite common and for a review of conditioning with various distributions we refer to Nelson (1982). Let X have pdf f(x) with support S. Based on the probability concept of conditioning given in (1.1.2) the conditional pdf of X at given X>c is


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(1.6.1) where x is in the intersection S and {x>c}. The df and moments of the conditional distribution of X given X>c can also be computed using (1.6.1) and the standard computational formulas. The relationship between conditional probabilities and conditional distributions is immediate. We follow with some simple examples. Ex. 1.6.1. Let X have an exponential distribution as presented in Ex. 1.2.5. Then the pdf of X given X≥c, from (1.6.1) is (1.6.2) From (1.6.2), if we let the future lifetime past c be T=X−c then the pdf of T is also Exponential with the identical parameter θ. This property is referred to as a lack of memory, since conditioning on {X>c} produces a pdf that is a function of only the fixture lifetime X−c and is independent of the values of c and parameter θ. Ex 1.6.2. Let X be a discrete geometric random variable with pdf given in Ex. 1.3.5. The survival function can be shown to be (1.6.3) for x=0, 1,…. For a fixed positive integer c the truncated distribution (1.6.1) becomes (1.6.4) As with the exponential distribution in the previous example the conditional distribution takes the form of the initial distribution. Hence, the geometric random variable exhibits a lack of memory property in the discrete random variable setting. In most financial and actuarial models conditioning is applied where, unlike the previous two examples, the conditioning affects the future distribution. It is common to have financial actions conditioned on statuses and their associated survival functions. For example, a stock may be sold if its price reaches or exceeds a particular value. For x>c the conditional survival function is (1.6.5) provided x>c and c is in the support of the pdf of X. The conditioning concept and related formula is central to many financial and actuarial calculations in presented in later chapters. 1.7 Joint Distributions In modeling of real data, such as that found in financial and actuarial fields, there is often more than one variable required. The situation where we have two random variables, X


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and Y, is now considered and the discussion can be extended to the general multiple variable setting. Generally, these variables can be of any type, discrete, continuous or mixed and the joint pdf is denoted f(x, y). The basic concepts and formulas relating to one random variable are now extended to the multivariable case. The initial concept in multivariable random variable modeling is that of the probability that (X, Y) falls in a defined set A. There are the following three possibilities depending on the type of random variables involved: (i) Discrete X and Y: (ii) Continuous X and Y: (iii) Discrete X and Continuous Y: Here (i), (ii) and (iii) define the probability structure of the joint random variable. Other statistical concepts, such as dependence and independence, are extended to the case of more than one random variable. Further, the distribution of single variables and relationships among the variables can be explored. The marginal distributions are the distributions of the individual variables alone. Similar to the formulas for probabilities there are three possible cases. The marginal pdfs denoted g(x) and h(y) are given by: (iv) Discrete X and Y: (v) Continuous X and Y: (vi) Discrete X and Continuous Y: Applications of these are encountered in various financial modeling and actuarial science applications. Relationships between the variables are often important in statistical modeling. To do this we need the concept of conditional distributions presented in Sec. 1.6 applied in the joint random variable setting. The conditional distributions and independence conditions follow the same pattern as those introduced in the probability measure setting of Sec. 1.1. The conditional pdf of X given Y=y explores the distribution of X while Y is held fixed at value y and is (1.7.1) Further, X and Y are called independent if either (1.7.2) or (1.7.3)


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over the support associated with the random variables. We remark that this definition of independence is analogous to the independence of sets definition given in (1.1.5). The above definitions, formulas and concepts hold for all types of random variables. Two examples are now presented. Ex. 1.7.1. Let X and Y have with support S={(x, y) where x>0, y> 0} and joint pdf given by

for positive θ1 and θ2. Using condition (v) of the joint distributional setting we have

From criteria (1.7.2) we see the X and Y are independent and the marginal distributions are both exponential in type. Ex. 1.7.2. Insurance structures are often defined separately for different groups of individuals. We consider an insurance policy where there are two risk categories or strata, J=1 or 2, for claims. The amount of the claims, in thousands of dollars, are denoted by X and the corresponding pdfs defined on their supports are for J=1

and for J=2

Now, we assess the frequencies of the risk categories are defined by P(J=1)= .6 and P(J=2)=.4. The joint pdf is a function of the two random variables J and X. Similar to the insurance modeling examples of Ex. 1.2.8 and Ex. 1.3.7 the pdf is (1.7.4)

The marginal pdfs are computed as

and (1.7.5)

Probabilities of basic events can be found using these pdfs. The overall probability a claim is at most 3, from (1.7.5),


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Common statistical measurements, such as expectations and variances can be computed in the multi-ruled distributions using the standard rules. These basic statistical concepts and formulas extend to the setting of multiple random variables. In the case of two random variables certain standard moments are useful to compute. For function g(x, y) the expected value, if it exists, is denoted E{g(x, y)}. If X and Y are discrete with corresponding pdf f(x, y) and support S the expectation is (1.7.6) In the continuous case we have (1.7.7) while in the mixed variable case the expectation takes the form (1.7.8) In these computations we assume all the moments exists. Applying (1.7.6). (1.7.7) and (1.7.8) means and variances can be computed but the relationship between variables must be considered in forming the proper summation and integral limits. A measure of the linear relationship between X and Y is the covariance between X and Y denoted by Cov{X, Y} or σxy. If the mean of X and Y are given by µx and µy then the covariance is defined as (1.7.9) We remark that if X and Y are independent then E{XY}=µxµy and σxy=0. A scaled or normed measure of variable association based on (1.7.9) is the correlation coefficient denoted by ρ. This measure or parameter is used in correlation modeling and it’s sample estimate is used as a diagnostic tool in regression modeling (see Sec. 1.13). Many applications in financial and actuarial modeling involve the sum of more than one random variable. Let two random variables X and Y have means µx and µy, variances σx2 andσy2 and covariance σxy. Since the expectation acts as a linear operator for the sum S=X+Y (see (1.7.6), (1.7.7) and (1.7.8)) the expectation is (1.7.10) Further, applying the definition of variance, (1.3.5), we write the variance of S as (1.7.11)


25

We note that if the variables are independent, (1.7.11) indicates that the variance of the sum of variables is the sum of the separate variance terms. This concept extends to the general multiple random variable case in the same manner. Expectations of functions of multiple random variables can be approached using the concept of statistical conditioning. When two variables are involved expectations are simplified by conditioning on one of the random variables inside the probability structure of the other. Let X and Y be random variables with joint pdf f(x, y)=h(y) f(x|y) and we compute the mean and variance of functions of the form g(x, y)=v(y)w(x, y). Here for any type of random variable (1.7.12) Also, the variance of g(x, y) is (1.7.13) Further, if w(x, y)=w(x) and X and Y are independent then (1.7.14) and (1.7.15) The general statistics theory required to derive the conditioning arguments is given in general theoretical statistical texts such as Mood, Graybill and Boes (1974). The derivation of these formulas is considered in Prob. 1.18. Formulas (1.7.14) and (1.7.15) can be used to derive (1.3.14) and (1.3.16). We remark that the concepts and ideas presented in this section for the two variable setting can easily be extended to more than two random variables. For a review of joint distributions and their manipulations see Hogg and Craig (1995, Chapter 2), Hogg and Tanis (2001, Chapter 5) and Bowers et al. (1997, Chapter 9). 1.8 Sampling Distributions and Estimation There are two conceptual views in statistical modeling referred to, respectively, as frequentist and Bayesian approaches. In both approaches the financial and actuarial actions are modeled utilizing one or more statistical distributions. These distributions are functions of unknowns called parameters. In frequentist statistics the parameters are considered to be fixed constants and information about the parameters, such as point estimates and bounds, is entirely a function of observed data. In Bayesian statistics the parameters themselves are modeled as random variables that are associated with specified prior probability distributions and probability estimation and statistical inference take on a more mathematical flavor. In this section, topics in parameter and statistical estimation


26

based on the frequentist approach are discussed. For a review of Bayesian methods we refer Bickel and Doksum (2001, sec. 1.2) and Hogg and Craig (1995, Sec. 8.1). Throughout this text the frequentist approach to statistical modeling is applied, Based on the information contained in sample data, such as financial or actuarial records, unknown parameters are estimated using well-chosen statistics. The computed values of the statistics depend on the samples that are observed. In this way one or more probability distributions are imposed on observed statistics. The probability distribution associated with a statistic is referred as the sampling distribution of the statistic. Much work in frequentist statistics concerns the selection and evaluation of efficient statistics through analysis of their associated sampling distributions. The topic of statistical estimation for unknown parameters, as well as useful functions of parameters, is very broad and we discuss some of the areas that are applied in financial and actuarial modeling. In this section the typical topics of point and interval estimation along with percentile measurements are presented. For a basic review of the principles of estimation we refer to Rohatgi (1976), Hogg and Tanis (2001) and Hogg and Craig (1995). 1.8.1 Point Estimation In the study of frequentist statistical theory parameters are considered fixed constants that characterize the distribution of a random variable. In this chapter some of the parameters discussed include the mean, µ, the variance and standard deviation, σ2 and σ. In financial and actuarial modeling applications, other relevant quantities such as percentile points required for model analysis are investigated. Generally, statistics are defined as functions based on sample data and are used, in part, to estimate parameters. A point estimator is a single valued statistic that is used as an estimate of a parameter. Some common statistics are now discussed A random sample is a collection of variables, X1,…, Xn, where each random variable is independent and comes from the same distribution and the sample size is fixed. This is referred to as independent and identically distributed from pdf f(x) which we denote by xi~iid. f(x) for i≥1. This is the mathematical structure conveyed in the notion of random sampling. The sample mean is given by (1.8.1) and the sample variance is (1,8.2) The sample standard deviation is S=(S2)1/2. These point estimates possess many desirable statistical properties based on their associated sampling distributions and a full discussion of these is outside of our investigation of financial and actuarial modeling. Sample moments, defined by for positive integer r, can be used to estimate central moments like the skewness and kurtosis defined by (1.3.6). The sample moments


27

are substituted for the theoretical moments, E{Xr}, in these formulas. In statistical estimation this is referred to as the method of moments procedure and is supported by the convergence in probability of these estimators. For example substitution in (1.3.6) yields the sample or estimated skewness and kurtosis, denoted by their sample or plug-in estimators and . Another common applied modeling setting involves the modeling of proportions. If Xi is a Bernoulli variable, or Xi=1 or=0 depending on if an event, say A, is realized, In this case applying (1.8.1) the sample proportion is defined as (1.8.3) and measures the proportion, out of the sample size n, that event A has occurred, The sample proportion is used to estimate the theoretical likelihood or probability of event A occurring. As the number of samples increase, (1.8.3) converges to the theoretical probability associated with event A. This is the weak law of large numbers and is the intuitive concept for the long run relative frequency approach to probability that is central to frequentist probability and statistics. There are many properties associated with the efficient statistics, such as consistency, sufficiency and unbiasedness. A statistic, W, is unbiased for parameter θ if (1.8.4) We remark that , S2 and are unbiased for µ, σ2 and P(A). The unbiased property states that the center or mean of the sampling distribution associated with the statistic matches the parameter to be estimated. To choose between unbiased estimators for a given parameter we select the estimator with the smallest variance. These estimators are referred to as minimum variance unbiased or best estimators. As an example of estimation suppose we have a random sample X1, X2,…, Xn. By this we mean Xi~iid with common pdf f(x) for i=1, 2,…, n. Let the sample mean be given by (1.8.1). Using the linear aspects of expectation we find (1.8.5) Hence, the sample mean is an unbiased estimator of the parameter mean. Further, from (1.8.3) we note that the sample proportion is an unbiased estimate for the population proportion. Ex. 1.8.1. The lifetimes of a series of people, past an initial age, are 11, 15, 21, 25, 21, 30, 32, 39 and 42. This is a sample of size n=10 and find the sums and sums of squares ΣX=278 and ΣX2=8,646. From (1.8.1) and (1.8.2) the sample mean and variance are

The sample standard deviation is s=101.91/2=10.095. Many techniques and formulas in statistics utilize these basic statistics.


28

Another useful property of point estimators is the idea of consistency. A statistic is consistent for a parameter if as the sample size increases the statistic converges to the parameter to be estimating. In this situation statistical theory supports the substitution of statistic for the unknown parameter. This is utilized in the constructing of limiting distributions for useful statistics. Examples of consistent estimators are the sample mean, variance, standard deviation and sample proportions. For a review of point estimators and related properties we refer to Hogg and Craig (1995, Ch. 6 and Ch. 7) and Bickel and Doksum (2001, Ch. 2). 1.8.2 Percentiles and Prediction Intervals Percentiles give a relative measure of data point relationships that are commonly used in many fields of study. For a random variable or statistic, X, we now define percentile values. For α, 0≤α≤1, the (1−α)100th percentile, denoted by x1−α, is defined by the relation (1.8.6) We note that the 50th percentile, x.5, is the median of a continuous distribution. An interval that covers a fixed proportion of the distribution associated with a random variable X, if the random variable is used for investigation and analysis, is referred to as a prediction interval. For α, 0≤α≤1, from (1.8.6) an example of a prediction interval is (1.8.7) Therefore, the prediction interval written as [xα/2, x1−α/2] contains probability 1 −α. We remark that one-sided prediction intervals can also be constructed. This statistical interval construction is closely related to the idea of tolerance intervals for statistics and distributions where coverage probabilities are defined Ex. 1.8.2. In this example we consider the general normal random variable conditions of Ex. 1.2.6. The lifetime of a status, T, is assumed normal with mean µ and standard deviation a. To construct a prediction interval with probability 1−α we set (1.8.8) Using (1.2.11) to standardize T we find prediction interval (1.8.9) where α=Φ(zα). For example if µ=1,000 and α=100 we find the 95th percentile t.95. Approximating from the Z tables in Appendix A2 we find z.95= 1.645 so that

We remark that intervals of this form are symmetric about the mean and in the normal random variable case posses the trait that they contain the smallest width.


29

1.8.3 Parameter Interval Estimation Point estimates lack information about their variability and reliability. To address this lack of assessment, interval estimates similar to prediction intervals, incorporating the variability and reliability, for unknown parameters are constructed. An interval estimate for parameter θ is [a, b] where we estimate a≤θ≤b. The probability the interval contains the unknown parameter is called the confidence coefficient or confidence level. Hence, a confidence interval for θ with confidence coefficient 1−α satisfies (1.8.10) We remark that the parameter is a fixed value while the interval is dependent on the selected observations and is therefore a random set. The theoretical construction of these intervals is out of the scope of this book (see Rohatgi (1976, Secs. 11.2 and 11.3)) but we follow with an illustrative example from basic statistics. Ex. 1.8.3. A random sample of size n is taken from a normal distribution with unknown mean µ. The sample mean and sample standard deviation, s, are found. Under large sample size conditions

is distributed approximately as a standard normal random variable. A confidence intervals for µ with confidence coefficient 1−α is (1.8.11) In the case where the standard deviation a is unknown and the sample size n is small the t-random variable is applied to form a confidence interval for µ. In that case the (1−α/2)100 percentile value based on a t-distribution with degrees of freedom is used in place of the standard normal percentile in (1.8.11). Confidence intervals for unknown parameters possess interpretive qualities. In statistical inference the interval estimates, such as (1.8.9) and (1.8.11), constructed for model assessment statistics are used to shed light on the structure being modeled. This is done for financial and actuarial models throughout this text. 1.9 Aggregate Sums of Independent Variables In practice we often observe a series of independent variables X1,…, Xm, where the number of variables may be fixed, denoted by m, or stochastic, represented by the random variable by N. For example, these could be the future lifetimes of a group of people or the current values of a number of stocks held in a portfolio. In financial and actuarial analysis applications involving the aggregate sum, denoted Sm, are referred to as a collective risk or aggregate modeling. If these variables hold over a short time period the force of interest, discussed in Chapter 2, can be ignored. We follow with mathematical and statistical investigations of these models starting with the two random variable setting.


30

In this section let the number of variables be fixed at m. For positive integer m we consider the distribution of the aggregate sum,

The two variable case m=2, are presented and can be extended to the multiple variable situation. To compute the distribution of the sum of two independent variables X and Y we discuss first the classical convolution method. The df for S=X+Y is defined as (1.9.1) for constant s. In the general setting, conditioning on Y=y, for the discrete case we find (1.9.2) and for continuous X and Y (1.9.3) Further, if X and Y are independent then the dfs (1.9.2) and (1.9.3) become (1.9.4)

Here (1.9.4) gives are the formulas for the convolution for the dfs Fx(x) and Fy(y) which we denote Fx*Fy. Taking the derivatives or differences yields the pdf. The corresponding pdf in the discrete case and continuous cases are, respectively, (1.9.5)

An example of the convolution method in the continuous setting is given in Prob.1.1.19. As mentioned the convolution process can be extended to the situation of n independent variables. Let Sn=X1+…+Xn where the df of X1 is F1 and the df of X1+…+Xk be F(k) for positive integer k≤n. Iteratively we find the convolution df by using (1.9.6) for i=2, 3,…, n. A computational example, analogous to the example in Bowers, et. al. (1997, p. 35), is now given.


31

Ex. 1.9.2. Let X1 and X2 be independent and S=X1+X2. The pdfs are f1(x) and the dfs are F2(x). The defining probabilities are given below: X f1(x) f2(x) F1(x) 0

.4

.3

.4

1

.4

.4

.8

2

.1

.1

.9

3

.1

.1

1.0

4

.0

.1

1.0

5

.0

.0

1.0

The distribution of S is computed using (1.9.2). Here,

Continuing the pattern, F(2)(2)=.9(.3)+.8(.4)+.4(.1)=.63, F(2)(3)=.78, F(2)(4)=.91, F(2)(5)=.97, F(2)(6)=.99 and F(2)(7)=1. The pdf of S is found by subtracting the consecutive values of F(2). In Prob. 1.20 the process is extended to the convolution for a third random variable computing df F(3). In a theoretical sense the convolution process can be used to find the distribution of the sum of independent variables as long as the number of variables is not too large. In many practical settings such as in the case of many random variables several approximations to the distribution of aggregate sums of variables are used in financial and actuarial modeling. In this chapter we present three approximations, namely, the celebrated Central Limit Theorem, Haldane Type A approximation and the statistical saddle point approximation. This section ends with theoretical distributional considerations of a sum of independent random variables. To find the distribution of the sum of random variables theoretical techniques based on transformations or dfs can be used. We discuss another procedure that utilizes the mgf. Let the sum corresponding to m random variables be Sm=ΣXi where the mgf for Xi is Mi(t) and the random variables are independent. This setting of the mgf of Sm reduces to (1.9.7)

The mgf associated with a random variable, if it exists, is unique. Thus, if the mgf of S matches the mgf of a known distribution then the distribution of S must be the


32

distribution associated with the matched mgf. If Xi, 1≤i≤n, are independent identically distributed, iid, each with mean µ, standard deviation σ and mgf M(t) then from (1.8.7) (1.9.8) This form of the mgf is useful in finding distributions of sums resulting from random samples. This matching of mgfs procedure is commonly used, in connection with the Continuity Theorem, to prove the Central Limit Theorem. It can be shown that the mgf of Sn/n, employing a Taylor Series expansion, converges to the mgf corresponding to a normal random variable with mean µ and variance σ2. The Central Limit Theorem is discussed in Sec. 1.11.1. Here the mgf (1.9.8) can be used to find the moments of Sm through the derivative formula (1.4.6). Taking the derivatives of (1.9.8) and evaluating at t= 0 we find (1.9.9) Hence, for the sample mean

the mean and variance (1.9.10)

These moments can be used to calculate or approximate relevant probabilities. In the case where the aggregate sum is modeled by a normal random variable and we standardized the sum to produce the df. (1.9.11) for constant c. The next example demonstrates the application of the normal distribution. Ex 1.9.3. Let X1,…, Xn be a random sample, or iid, from a normal distribution with mean µ and standard deviation σ. From (1.4.7) and (1.9.8) the mgf of Sm is (1.9.12) Since (1.9.12) takes the form of a “normal” mgf then Sn is a normal random variable. Further, from the mgf we see that Sm has mean and variance given in (1.9.10) and probabilities can be computed using (1.9.11). The technique employed in Ex. 1.9.3 to find the distribution of the aggregate sum works for other distributions such as Poisson, Binomial and Gamma random variables (see Prob. 21). In other distributions approximations, such as the Central Limit Theorem, are used to estimate the distribution.


33

1.10 Order Statistics In many applications there is a natural arrangement of random variables. We consider m continuous random variables given by X1, X2,…, Xm where the order statistics arise from an ordered rearrangement of the corresponding observed variables. The order statistics are denoted by X(1)0. Other combination of order statistics can be analyzed using the general theory of order statistics. Financial and actuarial contracts, and hence their models, may be a function of the first or last order statistics. Future random variables associated with statuses that are


34

functions multiple lifetimes are discussed in Sec. 4.5. Contingent models based on these random variables can also be constructed. 1.11 Approximating Aggregate Sums Applications in financial and actuarial modeling often involve a collection or portfolio of financial contracts where the aggregate sum is to be analyzed. The insurance company may have a collection of insurance policies or an investor may have a diversified collection of investments. In this section the distribution of an aggregate sum of random variables is discussed. For moderate sample sizes the convolution method becomes cumbersome at best. There are several approximations to the distribution of aggregate sums of variables used in actuarial science. One such method is the Wilson-Hilferty approximation or the Haldane Type A approximation as presented by Pentikainen (1987). For a discussion of actuarial approaches re refer to Panjer and Wilmot (1992 Chapter 6) and Bowers et. al. (1997, Chapters 2 and 12). Classical statistical approaches include the Edgeworth expansion (Feller (1971)) or the conjugate density method of Esscher (1932). In this section three methods for approximating cumulative probabilities and hence percentiles, for sums of random variables are presented. The first is the celebrated Central Limit Theorem for sums of iid. Random variables denoted by CLT while the second is the Haldane Type A approximation as discussed by Pentikainen (1987). The last is the Saddlepoint Approximation of Daniels (1954). The latter two approximations can be viewed as extensions of the CLT. 1.11.1 Central Limit Theorem Much work in theoretical and applied statistics concerns the convergence of a series of random variables. Under various restrictions limiting distributions have been demonstrated. The Lindeberg-Feller Central Limit Theorem (Feller (1946)) concerns the convergence of a set of independent, but not identically distributed, random variables. Under suitable conditions on the variance of the respective random variables, convergence in distribution of a standardized variable based on the collective sum to a standard normal random variables is achieved as the sample size increases. Under relaxed conditions the random variables are also assumed independent from identical distributions with finite mean and variance and large sample convergence to normality is assured. This common setting, referred to as the Levy Central Limit Theorem (see Levy (1954)) or simply the Central Limit Theorem, is utilized throughout this text and is denote by CLT. In this form we have m independent variables, X1, X2,…, Xm taken from the same distribution. From this random sample of size m we compute the sample mean. As the sample size n increases the sampling distribution of the sample mean approaches the normal distribution. Since

for large m the random variable


35

(1.11.1) is distributed approximately standard normal From (1.11.1) percentile, tail probabilities and prediction intervals can be approximated. The efficiency of any approximation is contingent on many factors. In this case the shape of the distribution associated with the random variables and the deviation from the assumptions play a role. In any event, the convergence of large sample approximations can be assessed by simulation techniques as discussed in the last chapter. Further, the consistency of the sample standard deviation s allows for the substitution of s for a in (1.11.1) in the case of large m. Ex. 1.11.1. From a random sample of size 100 we find X. It is claimed that the mean and standard deviation are 65 and 7, respectively. From the sample we find that

The probability that the sample mean is at least 68 is found using (1.11.1) as

From this we might conclude that the mean is not, as was assumed in the computation, 65 but somewhat larger. This type of investigation lays the groundwork for concepts in statistical hypothesis testing. In the setting of small sample sizes the Central Limit Theorem can be adjusted to approximate the distribution of the sum of random variables Sm This has many applications in financial and actuarial modeling. We follow with an introductory example. Ex. 1.11.2. A portfolio consists of 25 short term insurance policies. Over the time period for the insurance P(claim)=q=.1 and the benefit paid or claim amount, B, is a normal random variable with mean $1,000 and standard deviation $200. This type of policy was discussed in Ex. 1.2.8 and Ex. 1.3.7. For the ith policy the claim variable is Xi=IiBi and the aggregate sum of the claim variables is Sm for m=25. Using (1.3.14) and (1.3.16) we first find the mean and variance of Sm. For the ith policy the mean and variance are

and

Thus, for the aggregate sum, assuming the individual short term insurance policies lead to iid. claim random variables, the mean and variance are computed as


36

and

using the CLT percentiles for the aggregate sum can be approximated. The 95th percentile for Sm in terms of dollars is

Hence, the probability of the aggregate claims exceeding $5,021.73 is only .05. Further, using the CLT approximate, cumulative probabilities, survival probabilities and prediction intervals can be constructed in a straightforward manner. 1.11.2 Haldane Type A Approximation In this section the first alternative method to approximate tail probabilities and percentiles to the Central Limit Theorem is discussed, The Haldane Type A Approximation, denoted HAA, for a sum of independent random variables is presented as described by Pentikainen (1987). This method extends the CLT and utilizes information from the third moment or skewness measure, (1.3.6), of the variables distribution. In this section an aggregate sum corresponding to iid random variables, X1, X2,…Xm, where m is a fixed constant, is considered. An approximation for cumulative probabilities of the form P(Sm≤s) is considered where the aggregate sum is Sm and s is a constant. Let the mean and the variance for each Xi be denoted by µx and σx2 for i≥1. For the cumulative sum, using the independence among random variables, we have the mean, variance and central third moment given by (1.11.2)

The approximation uses the kurtosis of the aggregate sum, which from (1.3.6) and . To compute the kurtosis the (1.11.2) is found to be trinomial can be expanded and the moments computed term by term. This approximation is a third moment approximation and to apply the HAA we first define required constants. For a fixed value of s let

We define r=σx/(m1/2µx), h=1−kus/(3r), µ(h, r)=1−h(1−h)[1−(2−h)(1–3h)r2/4]r2/4 and σ(h, r)=hr[1−(1−h)(1−3h)r2/2]1/2. The HAA approximation takes the form (1.11.3) where Φ(a) is the standard normal distribution function. In financial and actuarial modeling the aggregate sum often consists of nonnegative values, such as claim values and the approximation applies taking h>0, µ(h, r)>0 and σ(h, r)>0.


37

Ex. 1.11.3 Consider the portfolio of 25 policies introduced in Ex. 1.11.2. The claims, denoted B, occur in 10% of the time periods, and are normal variables with mean and standard deviation given by µB=$1,000 and σB=$200 respectively. We wish to approximate the probability the aggregate sum exceeds $5,000. Using the CLT, P(S25>5,000)=1−Ф((5,000–2,500)/1532.97)=Φ(1.63082)=.05146. To apply HAA we first find a form for the 3rd central moment in this situation. Noting the claim variable is X=IB the central third moment. similar to the derivation of the variance in (1.3.16), is (1.11.4)

The normality of B and the computations of Ex. 1.11.2 imply E{(X−µx)3}= 82,800,000 and kus=.574604. Further, r=.613188, h=.6876413, µ(h, r)= .9392614 and σ(h, r)=.4082815. The desired HAA approximation is

In this example the CLT and the HAA are close in computed survival probability. The exact value obtain using a simulation resampling method is approximated with a high degree of accuracy in Chap. 7 and is computed to be .06616. Hence both approximations have a large relative error of about (.06616 −.05)/.06616=.24425. 1.11.3 Saddlepoint Approximation The previous two approximation techniques depend only on the moments associated with the iid random variables. We now turn to an approximation method that utilizes information contained in the entire distribution. Since their introduction by Daniels (1954) saddlepoint approximations, denoted SPA, have been utilized to approximate tail 0 probabilities corresponding to sums of independent random variable. For an in-depth discussion of the accuracy of saddlepoint approximations we refer to Field and Ronchetti (1990). The approximation is shown to be accurate for small sample sizes, even as small as one. Further, in the case of data from a normal distribution the SPA reduces to the CLT approximation. For this reason the SPA can be viewed as an extension to the CLT in the case of small sample sizes. Saddlepoint approximations have been applied to a variety of situations and for further references see articles by Goutis and Casella (1999), Huzurbazar (1999), Butler and Sutton (1998), Tsuchiya and Konishi (1997) and Wood, Booth and Butler (1993). We present the simplest setting where there are independent identically distributed random variables X1,…, Xm where m is fixed. Unlike other saddlepoint approximation developments that utilize the cumulants of hypothesized distributions this discussion is based on the associated moments. The moment generating function of X1 is assumed to exist and is denoted by M1(β) where E{X1}=µ and Var{X1}=σ2. The corresponding moment generating function for Z=(X−µ)/σ is


38

(1.11.5) For a fixed value of t, let β solve (1.11.6) Further, let (1.11.7) For constant s the saddlepoint approximation, denoted SPA, for percentile calculations of the form P(Sm≤s) is (1.11.8)

where t=(s−mµ)/(mσ) and Φ(x) is the standard normal distribution function. Tail probabilities are computed using the complement method (1.1.6) and approximate prediction intervals can be constructed. In application, for a chosen s the associated t value is computed and a numerical method, such as Newton’s Method or the secant method (see Stewart (1995, p. 170) or Burden and Faires (1997, Chapter 2) may be required to solve for β in (1.11.6). The saddlepoint approximation is found by substitution of (1.11.7) into (1.11.8). We remark that if the distribution of individual random variables is iid normal with any fixed mean and variance the SPA yields exact standard normal probabilities. Ex. 1.11.3. We demonstrate the SPA and compare it to the CLT and HAA in the case of the exponential random variable with pdf given by f(x)= (1/θ) exp(−x/θ) for support S=(0, ∞). The mean is µ=θ, the variance σ2=θ2 and the mgf is M1(β)=(1−βθ)−1. For a fixed value of s we find

and solving (1.11.6) for β we compute (1.11.7). Applying the SPA to the exponential distribution we have required constants that are computed with the formulas (1.11.9)

In this case the distribution of the sum is known to be Gamma (see Prob 1.21, c). For different sample sizes the exact percentile points as computed using the Gamma distribution with parameters α=m and θ=1 are found. The cumulative probabilities associated with these points using the CLT, HAA and the SPA are found and results for sample sizes of m=1 and m=2 are given in Table 1.11.1


39

From Table 1.11.1 we see that the HAA and SPA outperform the CLT. This is to be expected since these approximations use more information, specifically, information about the skewness of the random variables. The CLT is most efficient in the case of symmetric random variables. The most efficient method is the SPA yielding efficient percentile approximations for the exponential distribution even for sample sizes of one and two.

Table 1.11.1 CLT, HAA and SPA Percentile Approximations Sample Size m=1 Percentile

m=2

.99

.95

.90

.75

.99

.95

.90

.75

CLT

.9998

.9770

.9036

.6504

.9995

.9738

.9092

.6879

HAA

.9900

.9513

.9023

.7512

.9899

.9506

.9012

.7511

SPA

.9900

.9498

.8997

.7502

.9900

.9499

.8998

.7499

In general the SPA requires computation of (1.11.5), (1.11.6) and (1.11.7) which may be cumbersome. These computations can be somewhat eased by the reduction given in Prob 1.26. The SPA has applications in financial and actuarial modeling and has been extended to the case of life table data with uniform distributions within each year by Borowiak (2001). 1.12 Compound Random Variables Generally a compound random variable is a random variable that is composed of more than one random variable. We consider the structure of an aggregate sum of iid. random variables where the number of independent random variables is a random variable. In this section the statistical properties and applications of compound random variables are explored and presented using the techniques and formulas presented in previous sections. The theoretical distribution can be investigated using statistical conditioning in connection with other statistical models, such as either hierarchal or Baysian models, Compound random variables have applications in actuarial and financial modeling where examples include investment portfolio analysis and collective risk modeling (see Bowers et al. (1997, Chapter 12)). 1.12.1 Expectations of Compound Variables Let the random variables X1, X2,…, XN be independent from the same distribution and N be a discrete random variable. Let E{X1}=µ1, E{X12}=µ2 and Var{X1}=σ2. The random variable of interest is the aggregate sum


40

(1.12.1) where the pdf of N is given by P(N=n) and the support of P(N=n) is SN. The mean and variance of (1.12.1) can be found using conditioning arguments. We assume X1,…, XN and N are independent and the joint pdf is given by f(n, SN)= P(N=n) f(Sn|N=n). From this the expectation of the aggregate sum is (1.12.2) Further, using the conditioning argument on N that the variance is (1.12.3) These formulas are used to construct statistical inference such as confidence and prediction intervals. The derivations of (1.12.2) and (1.12.3) are considered in Prob 1.29. The mgf corresponding to the compound variable SN can also be found using a conditioning argument. Let the mgf of Xi be M(t) for i=1,…, N. The mgf of (1.12.1) is (1.12.4)

We can also show that (1.12.2) and (1.12.3) can be found from the mgf (1.12.4) by taking the usual derivatives (see Prob. 1.29). Two illustrative examples follow where the second describes the much-investigated compound Poisson random variable. Ex. 1.12.1. Let N be discrete geometric with pdf given by Ex. 1.3.5. where P(N=n)=pqn for n=0, 1,…. Applying the summation formula (1.3.11) on (1.12.4) the mgf (1.12.5) is computed as (1.12.5) Taking the derivative of (1.12.5) we find the mean is E{SN}=pqµ1/(1−q)2. The distribution with associated mgf, (1.12.5), is referred to as a compound geometric random variable. Ex. 1.12.2. A collection of insurance policies produces N claims where N is modeled by a Poisson random variable (see Ex. 1.2.3) with parameter . The distribution of SN is said to be a compound Poisson random variable. Since E{N}=Var{N}=λ from (1.12.2) and (1.12.3) we find (1.12.6) From (1.12.5) the mgf of SN is


41

(1.12.7) This mgf is used in the next section to validate and construct limiting distributions for the compound Poisson random variable. These distributions are employed in statistical inference methods. 1.12.2 Limiting Distributions for Compound Variables Limiting distributions exist for some compound distributions. We give two limiting distribution approximations for the compound Poisson distribution. The first utilizes the standard normal distribution and is similar to the CLT while the second applies the saddlepoint approximation approach. Other approximation approaches exist, such as the discretizing method given by Panjer (1981). We assume the number of random variables follows a Poisson distribution and we let E{Xi}=µi for i=1, 2. From (1.12.6) we form the standardized variable (1.12.8) The mgf of ZN can be written as (1.12.9) This development assumes the mgf of Xi exists and using a Taylor Series expansion (see Prob. 1.3) we have (1.12.10) Putting (1.12.7) and (1.12.10) into (1.12.9) yields (1.12.11) where o(λ) are terms that approach zero as λ approaches infinity. Hence, as λ approaches infinity Mz(t) approaches the mgf of the standard normal, distribution exp(t2/2). By the Continuity Theorem ZN defined by (1.12.8) converges to a standard normal random variables as λ approaches infinity. Hence, the limiting distribution of (1.12.8), for large λ, is standard normal. Ex. 1.12.3. Let the amounts of accident claims, Xi, be independent with mean µ=100 and variance σ2=100. Let N be Poisson with mean λ=50. Considering the sum of the claims SN, from (1.12.6) E(SN}=5,000 and Var{SN}=1,000,000. The approximate probability the sum of the claims is less than 7,000 using the limiting standard normal distribution is


42

Also, from (1.8.8) a 95% prediction interval, using z.975=1.96, for SN is

Thus, prediction limits for the aggregate sum SN are $3,040 on the low side and $6,960 on the high side. The saddlepoint approximation approach can also be applied to the compound Poisson distribution when λ is large. If the required functions are known the SPA of Sec. 1.11 can be directly applied We present a three moment SPA. Applying the approximation (1.12.10), including only the first three terms, to (1.12.7) and (1.12.9) we approximate the mgf of

by (1.12.12) The required SPA calculations are now found. For a fixed value of t=(s− λµ1)/(λµ2)1/2, from (1.11.6), we need to solve for β in t=β+ β2µ3/[2λ1/2µ23/2]. The solution is found to be (1.12.13) In addition (1.11.7) becomes (1.12.14)

The SPA, (1.11.8), can now be computed for any t=(s−λµ1)/(λµ2)1/2. This application is the topic of the next example. Ex. 1.12.4. In this example we demonstrate the SPA to the compound Poisson distribution where X is distributed as a Gamma random variable, given in Prob. 1.21c) with parameters α=β=1. and λ=10. We compute the probability the aggregate sum is at most 15. The exact probability can be found by conditioning on N=n and using the fact that, for a fixed n, Sn is distributed as a Gamma random variable with parameters α=n and β=1. We compute with the aid of a computer package the exact cumulative probability

To apply the SPA, using (1.12.13) and (1.12.14), we find β=.86631, c= 1.68308 and σ=1.2574. The SPA yields approximation .860929 which is very close to the true value with a relative error of only (.86584−.860929)/.86584= .00567. Limiting distributions exist for other compound random variables and can be sought using the mgf and the continuity theorem. Bowers et al. (1997, Chapter 11) presents the limiting distribution for compound Poisson and negative binomial distributions.


43

1.13 Regression Modeling One of the most widely used statistical techniques is that of linear regression. Linear regression applied to a collection of variables can demonstrate relationships among the variables and model predictive structures. In this section we present a basic introduction to simple linear regression that will be utilized in the modeling and analysis of financial systems, This is not meant to be a comprehensive discussion on the subject but give the flavor of the interaction of regression and financial estimation. For an introduction to the theory and application of linear regression modeling we refer to Myers (1986) and Draper and Smith (1981). In simple linear regression modeling there are two variables. The independent or predictor variable, denoted by X, impacts the dependent or response variable, denoted Y. The empirical data takes the form of ordered pairs (xj, yj), for j=1, 2,…, n, and corresponds to observed outcomes of the variables. A linear relationship between the variables is assumed and the simple linear regression model is (1.13.1) where β0 and β1 are intercept and slope parameters and ej is the error term. For model estimation we assume ej are iid from a continuous distribution that has mean zero and constant variance for j=1, 2,…, n. For inference, such as hypothesis testing and confidence intervals, the additional assumption that the errors are normally distributed with constant variance is required. The simple linear regression model is applied to observed pairs of data points and the modeling assumptions can be assessed for accuracy. A plot of the data points referred to as a scatter plot is used as an initial check for linearity. An example of a scatter plot and the graph of an estimating line is given in Fig. 1.13.1. The normality assumption can be investigated through statistical techniques such as goodness of fit tests and hazard or probability plotting. The model given by (1.13.1), containing one predictor variable, is referred to as a simple linear regression model. In general, other predictor variables may be added to the model resulting in a multiple linear regression model. In the case of multiple linear regression topics of variable selection and influence, colinearity and testing become important. In this text we apply only simple linear regression models to financial and actuarial data. The estimation of the intercept and slope parameters is investigated followed by standard statistical inference techniques. 1.13.1 Least Squares Estimation To estimate the parameters β0 and β1 the mathematical method of least squares is applied. Least squares estimators have desirable statistical properties. Under general regularity conditions Jennrich (1969) has been shown general least squares estimators to be asymptotically normal as the sample size increases. In this method the sums of squares error is used as a measure for fit of the estimated line to the data and is given by


44

(1.13.2) Taking the partial derivatives of (1.13.2) the least squares estimators are found and they take the form (1.13.3) , , and (see Prob. where 1.30). Applying the estimators (1.13.3) to (1.13.1), the estimated or fitted regression line is (1.13.4) One of the primary main applications of (1.13.4) is the prediction of y at future values of x. The accuracy of such a prediction depends on the accuracy of the model, through modeling shape, and efficiency of estimation. One of the oldest measures of the linear fit of the estimated regression model is the correlation coefficient denoted ρ. This parameter arises when where both X and Y are random variables and the random vector (X, Y) comes from the Bivariate normal distribution. The parameter ρ measures the linear association between X and Y. Theoretically, −1≤ρ≤1 with proximity to −1 or +1 indicating a linear relationship. It is a mathematical fact that ρ2=1 if and only if the probability Y is a linear funtion of X is one (see Rohatgi (1976, p. 175)). The sample estimate of this parameter is called the sample correlation coefficient and is computed as

Table 1.13.1 Stock Price Over One Year Month Price

1

2

3

4

5

6

7

8

9

10

11

12

10.5 11.31 12.75 12.63 12.17 12.56 12.17 12.56 14.69 15.13 12.75 13.44

(1.13.5) where Syy=Σyj2−(Σyj)2/n. Here, r measures the linear relationship between X and Y where −1≤r≤1 and the closer |r| is to one the closer the points lie to a straight line. Another useful diagnostic statistic connected to the correlation coefficient is the coefficient of determination defined by r2. The statistic (1.13.5) is used to asses the accuracy of the estimated regression line (1.13.4). An example is now given. Ex. 1.13.1: The price of a stock is recorded at the start of each month for a year The data is listed in Table 1.13.1 We would like to predict the stock price over time and we apply the regression of X, the month, on Y, the stock price. The least squares are found to be


45

and the estimated regression line is

The plot of the data listed in Table 1.13.1 is referred to as a scatter plot of the data. In Fig. 1.13.1 is a scatter plot of the data along with the least squares line. We see the data is increasing over time but the linear fit of the data is somewhat suspect. The estimated slope, given by .255, indicates the price of the stock is increasing over time. Further, the sample correlation coefficient is found to be r=.724 indicating a fairly linear relationship. Based on this regression model we would expect the price of the sock after 6 months to be 11.0617+6(.255)=12.59. Analysis questions about the efficiencies of these types of point estimates arise and we now consider statistical inference topics associated with regression models.

Fig. 1.13.1 Regression of Stock Prices

1.13.2 Regression Model Based Inference Desirable properties of the least squares estimators, such as unbiasedness and minimum variance, are well known and do not rely on the form of the distribution of the error term. For a review of theoretical topics in linear models we refer to Searle (1971) and Myers and Milton (1998). For inference techniques we require that the error terms are normally distributed with zero mean and common standard deviation a, i.e. in (1.13.1) we assume ei ~iid n(0, σ2) for i=1, 2,…, n. Under this assumption inference procedures rely on the resulting normality of the least squares estimators. Further, when normality of the errors is assumed the least squares estimators coincide with the maximum likelihood estimators. Normality of the error term in the linear regression model implies that many statistics used in the analysis of these models also have normal distributions. As an example, the least squares estimator of the slope and the predicted response associated with chosen point xo are both normal random variables where


46

(1.13.6)

To construct inference based on the above we require an estimate of the variance σ2 that is statistically independent of the estimated parameters. Such an estimator, based on the fitted model, is

where we denote S=(S2)1/2. Using standard distribution theory we form confidence intervals for parameters and useful functions of parameters. The (1− α)100% percentile point for a t-random variable with degrees of freedom d is denoted by t(1−α,d). From (1.13.6) a confidence interval for β with coefficient with confidence coefficient 1−α is (1.13.7) In a similar manner, a confidence interval for the predicted value of Y based on X=xo, can be constructed. Letting yo=βo+β1xo the confidence interval (1.13.8)

Other confidence intervals, such as for the regression line or for a corresponding value of x, can be constructed. In financial and actuarial applications discussed in this text we apply only the confidence intervals given in (1.13.7) and (1.13.8). The width of the confidence intervals can be used to judge the accuracy of the estimation. More accuracy is signified by tighter or shorter confidence intervals. Different factors can adversely affect the width of the interval. In addition to the choice confidence coefficient the sample size has an effect, with larger samples resulting in better accuracy. Further, in new point estimation the farther the predicted point’s dependent or x-value is from the dependent variable’s mean the wider the interval. This structure implies a penalty for estimation far in the future. Ex. 1.13.2: We consider the stock prices and regression model in Ex. 1.13.1. After running an analysis we find s2=.846332 and Sxx=143. The confidence interval with coefficient be .95 for the slope parameter, from (1.13.7), is .083971≤β≤.42679. From this we see that the price for the stock is increasing over time, but the slope of the increase is not certain as seen by large width of the confidence interval for β. Further, the estimated price in six months or at future time x=13 is estimated as 12.59. Computing a 95% confidence interval for the price from (1.13.8) we find the confidence interval


47

10.15≤y≤15.02. Again a large-width confidence interval implies that the estimated stock price after 6 months is uncertain. 1.14 Autoregressive Systems In the practice of modeling financial and actuarial systems dependent random variables play an important role. In applications we may observe a series of random variables X1, X2,…where the individual random variables are dependent. Variables such as interest and financial return rates are often modeled using dependent variable systems. Many dependent variable techniques, such as time series, moving average and auto regressive modeling, exist In this section an introduction to dependent modeling is presented where one such modeling procedure, namely autoregressive modeling of order one, is discussed For a discussion of dependent random variable models we refer to Box and Jenkins (1976). In a general autoregressive system of order j, with observed x1, x2,…, the relation between variables is defined by (1.14.1)

where and µj are constants and the error terms ej are independent random variables for j≥1. It is clear from (1.14.1) that the random variables xj are dependent. In our discussion only the autoregressive system of order one, taking j=1, is considered. An autoregressive process or system of order one, denoted AR(1), takes the form of (1.13.1) where j=1 and can be written in the form (1.14.2)

where the error terms ej are independent from the same distribution with zero mean and variance Var{ej}=σ2. From (1.14.2) we solve iteratively to find (1.14.3) From this the mean and variance of the marginal distributions are given by (1.14.4)


48

The moments given in (1.14.4) are functions of unknown parameters that need to be estimated before these formulas can be utilized. After estimation approximate inference techniques, such as confidence interval, can be applied. We remark that alternative conditions to (1.14.2) on the observed sequence of random variables lead to moving average dependent variable models. In moving average modeling conditions relating the observed variables, Xj−µj, and error terms are imposed for j≥1. A mixture of autoregressive and moving average models lead to ARMA and ARMIA models. For an introduction and exposition to these techniques see Box and Jenkins (1976, Chapter 3). In an AR(1) system we estimate the parameters and σ2 using the method of least squares. Let Zi=Xi−µi for i≥1. In matrix notation (1.14.3) is written as (1.14.5) where e=(e1,…, en)’ and P is comprised of elements pij where (1.14.6)

The inverse of P, denoted P−1, can be shown to consists of elements given by the formula (1.14.7)

In the absence of a presumed trend in the means, the means are estimated by To estimate the unknown parameters least squares is applied where

The least squares point estimate for

for 1≤j≤n.

is found to be (1.14.8)

Using (1.14.5), (1.15.7) and (1.14.8), we compute estimator of σ2 as

and write the least squares (1.14.9)

In the computation of (1.14.9) it is useful to note that


49

(1.14.10) and (1.14.11) The distributional properties of the AR(1) fitted model are not easy to assess. Based on the asymptotic normality of the least squares estimators approximate inference can be constructed for some cases. We now follow with an approximate procedure. An ad-hock approximate confidence interval for a new response can be formed using the AR(1) model when the means at the individual locations are treated as fixed. The estimators (1.14.8) and (1.14.9) are used to produce the interval estimate for a new value Xn+m (1.14.12)

We remark that to employ these AR(1) an assumed model of mean values, µj for j≥1, is required. Confidence and prediction, such as given in (1.14.12), can be evaluated for their accuracy using simulation resampling techniques as discussed in Chapter 7. An example of AR(1) modeling is now given. Ex. 1.14.1. In this example we consider the stock price data in Ex. 1.13.1. We apply the AR(1) model where we take the mean at the different locations to be fixed at µi=11+.2i for i≥1. Utilizing formulas (1.14.8) to (1.14.11) we compute the estimates

Also, the 95% confidence interval (11.14.12) at future time corresponding to 18 months computes to be 10.05408≤x18≤15.41925. Comparing these findings with the least squares analysis in Ex.1.13.2 we observe that AR(1) resulted in a slightly larger variance point estimate and wider resulting confidence interval. Autoregressive models can be used to model the aggregate sums used in collective risk models. Let the aggregate sum consisting of correlated AR(1) random variables be

for fixed constant n. We find the sum can be written in terms of the errors as (1.14.13)


50

Also, the mean and variance of Sn are computed to b (1.14.14)

We remark that the constants µj model the trend in the variables over locations or time. Utilizing (1.14.14) ad-hoc interval estimates, similar to (1.14.12), can be constructed. These interval estimates can be validated by modern simulation and resampling methods, such as those presented in Chapter 7. The section closes with a theorem that is used in conjunction with simulation methods presented in Chapter 7. Ex. 1.13.2. We consider AR(1) settings where the error terms are normally distributed with a common variance. If the errors are independent with common variance we observe that marginally, Xj is normal with mean and variance given by (1.14.4). In a more general setting if we take (X1, X2,…Xn) to be multivariate normal under (1.14.2) then the conditional distribution of Xj given Xj−1,…, X1 depends only on Xj−1. Here X1 is normal with E{X1}=µ1 and Var{X1}=σ2 and using normal theory Xj|Xj−1=Xj−1 is distributed normal where (1.14.15) This conditioning strategy is useful in certain financial settings where values at certain time intervals, such as stock prices, depend on the previous values. The formulas given in (1.14.15) are utilized in simulation procedures where a series of variable outcomes are sequentially generated. Problems 1.1. Let A and B be events and use the axioms of probability to prove by constructing disjoint

events.

a)

. 1.2. For real number a and positive integer n consider the series and associated partial sum defined by

respectively. The infinite series exists if


51

Show for j=s formula (1.3.11), . Hint: First find Sn−Sn+1. 1.3. For continuously differentiable function f(x) the general Taylor Series is f(x)=f(a)+f (1)(a)(x−a)/1!+f (2)(a)(x−a)2/2!+… a) Find the Taylor Series approximation for exp(x) with a=0. b) For the Poisson distribution given in Ex. 1.2.3 find the mean and variance. c) For the geometric distribution given in Ex. 1.2.1 find the mgf and use it to compute the mean and variance. 1.4. For random variable X find the mean and variance, µ and σ2, for pdf a) f(x)=6x(1−x) for S={x|0≤x≤1}, b) f(x)=1/5 for S={1, 2,…, 5}, c) f(x)=1/2x for S={1, 2,…} 1.5. a) Let X~U[0, 1] with pdf given in Ex. 1.2.4. Find i) F(x), ii) S(x), iii)µ and σ2, iv) P(X>1/4), v) P(X>1/2|X>1/4). b) Let X~n(100, 100). Use (1.2.11), Appendix A1 and Appendix A2 to compute i) P(X>115, ii) P(X≤ 83), iii) P(85
. Evaluate the

. b) Find and compute the variance of an integral exponential random variable. 1.7. a) For a continuous random variable with support S=[0, b] use integration by parts to show (1.3.9). b) For a discrete random variable with support S={0, 1,…, m} show (1.3.10). 1.8. a) In the setting of Ex. 1.3.3 use the general moment computation to compute Sk and Ku as defined by (1.3.6). b) Use (1.3.9) and (1.3.10) to show the alternate forms of the 1st and 2nd moments given in (1.5.3) and (1.5.4). 1.9. Let the random variable X have pdf f(x)=1/2x for S={1, 2,…Find a) F(x), b) S(x), c) Mx(t)=E{exp(tx)}, d) µ and σ2, e) the probability X is at least 2. 1.10. Consider the Poisson random variable, introduced in Ex. 1.2.3, X with pdf f(x)=exp(−λ) λx/x! for λ>0 and S={0, 1,…}. a) Find the mgf Mx(t), b) Use a) to find µ and σ2, c) If the mean is 4 what is the probability X is at least 2? at most 3? 1.11. Let X be an exponential random variable with pdf f(x)=(1/θ) exp(−x/θ) for S={x|x≥0}. a) Find S(x), b) use S(x) to find µ and σ2, c) find P(X>b| X>a) for 0

52

1.16. Let random variables X and Y have joint pdf f(x, y)=a for S={00, b) the binomial pdf, f(x)=[n!/(x!(n−x)1)]px(1−p)n−x for S ={0, 1,…, n} and 00 and β>0. 1.22. For an insurance policy the amount of the claims, denoted B, are distributed normal with mean $1,000 and standard deviation $300. The probability of a claim is .05 and the claim variable is X=IB. Find a) the mean and variance of X, b) the 99th percentile of X, and c) an interval that contains 90% of the claim variable distribution. 1.23. A random sample of claims yields 1000, 1200, 800, 750, 220, 330, 410, and 2000. Find a) , b) s, c) the median, d) an approximate 95% confidence interval for the mean using the normal random variable. 1.24. A portfolio consists of 30 insurance policies. For each policy the claim variable is X=IB. Here P(claim)=.1 and B~n(1,000, 3002). The aggregate sum of the claim variables is S30=ΣXi. a) Find the mean and variance of S30. b) Approximate the probability that S30 would not exceed 5,000 using the CLT and HAA. c) Using the CLT compute the 99th percentile and a 99% prediction interval corresponding to S30. 1.25. For an insurance policy over a short period the probability of a claim is .1. If there is a claim the benefit is distributed uniform between 500 and 1500. Let the claim variable for any policy be X=IB. a) Find the mean and variance of X. b) Let the aggregate sum of claim for m policies be denoted Sm Using (1.11.2) and (1.11.4) find the mean, variance of and skewness of Sm. 1.26. Consider the insurance portfolio setting of Ex. 1.11.3. a) Use the pdf in (1.2.12) to derive the central third moment (1.11.4). b) Consider the SPA given in Sec 1.11.3. Use (1.11.5) to rewrite formula (1.11.6) as tσ+µ= M1(1)(β/σ)/M1(β/σ). Further, Write the formulas in (1.11.7) in terms of the mgf M1(β/σ). 1.27. Demonstrate that the SPA reduces to the CLT when the distribution of the individual random variables is normal. Does the HAA reduce to a similar result? 1.28. A portfolio of stock values is denoted by Xi for i≥1. Let the aggregate sum of values be SN where due to future transactions N is a Poisson random variable with mean λ=20. The distribution of the stock values is not known but we estimate µ1=1, σ=.2 and


53

µ3=.3. a) Approximate the probability that the portfolio will be valued at more than 20 using i) normal approximation and ii) the 3 moment SPA. b) Approximate the 25th percentile for the aggregate sum. 1.29. Consider the compound random variable SN. Use the conditioning formulas (1.7.14) an (1.7.15), substituting N and X for w(x) and v(y), respectively, to show (1.12.2) and (1.12.3) hold Further, show the mgf takes the form of (1.12.4) and the moments can be computed using this mgf. 1.30. Taking the partial derivatives of Q in (1.13.2) derive the least squares estimators (1.13.3). If there is no intercept term, βo, in (1.13.1) derive the least squares estimator of the remaining parameter. 1.31. The Dow Jones Index is observed over 12 months with the following ending values in terms of 1,000 Month

1

2

3

4

5

6

7

8

9

10

11

12

Index

1.07

1.09

1.05

.96

1.08

1.10

1.06

1.05

.99

.91

.92

.98

Consider a linear regression with the month as the predictor variable and the index as the response. a) Using formulas (1.13.3) find the least squares estimators of βo and β1 and draw a scatter plot of the data and draw the least squares similar to that of Fig. 1.13.1. b) Give the point estimate of the index after the next month. c) Find r as defined by (1.13.5). d) Using (1.13.7) compute a 95% confidence interval for the slope. Do you think the slope is significant, i.e. different from zero? e) Based on (1.13.8) find a 95% confidence interval for the price index at the end of the next month. f) What assumptions are required to run the inference in c) and d)? 1.32. We model the Dow Jones Index data given in Prob. 1.30 using an AR(1) model. The true model is assumed to be given by given by µj=1.1−.01x for x =1, 2,…. a) Using (1.14.8) find the estimate of . b) Using (1.14.10) and (1.14.11) compute the estimate of σ2 given by (1.14.9). c) Approximate a 95% confidence interval for the index for the next month. Are you confident it will increase for this month?

2 Financial Computational Models

A financial or actuarial model is used to explain and quantify one or more financial actions. These actions may be contingent on many factors, such as time, price and speculation and the resulting models fall into one of two main types. In the first, the actions are deterministic and are completely defined in terms of their form and timing. This is true for monthly mortgage payments that continue for a fixed number of years. The interest rate may be either fixed or variable. In the second type, the financial action itself may be initiated or effected by a random event. For general terminology the time the action commences is defined in terms of the failure of an existing status. Examples of stochastic financial actions include the payment of a benefit associated with an insurance policy at the time of death of the policyholder or the purchase of a stock at the time its value exceeds a predefined price. In this chapter we consider financial computations dealing with financial and actuarial models where the actions are completely predetermined and are non-stochastic. These computations and procedures are generally referred to as financial computational models. The value of an investment or series of payments depends on numerous factors. Some of these include the interest rate or the return rate and the length of time of the investment. Financial strategies containing one or more monetary actions are thus a function of a future time variable which we denote by T where T≥0. These actions are evaluated in terms of some reference time. The reference time needs to be fixed before the analysis and may correspond to a future time or the present time. The present value function is an evaluation at the starting time T=0 and is a function of a financial action at future time T=t and is denoted PV(t). The future value is computed at time T=t and is given by FV(t). In this chapter non-stochastic financial actions are used to form models that include the compounding of interest, the growth of stock holdings, the evaluation of annuities where there are a series of monetary payments or investments and the computation of combination actions. Examples of combined action calculations include the computation of mortgage payments for both fixed and variable interest rates and the assessment of the future value of a series of fixed rate investments such as in pensions. In this chapter the interest or return rates may either be fixed or stochastic. There are two types of deterministic financial models that are constructed to model financial actions. In the first, a single monetary investment is analyzed. This leads to various financial and actuarial investment and interest models where the models are functions of interest and investment return rates. For simplicity and uniformity in terms

Financial computational models

55

we will refer to both interest rates and investment return rates by the single term of financial rates or simply rates, In the second there is a series of monetary payments referred to as annuity payments. The annuity payments may different in amounts or may be identical and referred to as level. These models are analyzed at both the present and general future time reference points. For a review of deterministic interest and annuity models we refer to Kellison (1991). In this chapter deterministic models are presented to describe the standard interest and annuity models in terms of future and present value computations. Both discrete and continuous interest and annuity models are discussed in this context. The chapter concludes with an extension of these models to the generalized stochastic financial rate setting. 2.1 Fixed Financial Rate Models Money can be invested in avenues such as a bank or the stock market in the hopes that its value will increase. The receiver of the investment pays the investor for the right to utilize the invested capital. An amount of money, called the principal, earns additional value, referred to as interest or investment return, over time. The additional monetary worth of an investment grows as a function of an interest or a return rate. This is often referred to as the force of interest In this section these rates are considered non-stochastic and may be modeled as either a discrete function over fixed intervals or a continuous function over future time. Formulas for both the present value and the future value of the investment are given and we start with a discussion of required basic financial computations. 2.1.1 Financial Rate Based Calculations Both financial and actuarial models contain the same structure in terms of monetary growth. To encompass both in unifying nomenclature the single term financial rate or simply rate is used. The financial rate, whether from interest or investment, is often defined in terms of yearly percentages. For a financial rate associated with one year, denoted by i, the amount of interest earned is the product iP. Thus, after one year the future value is the sum of the principal plus the interest or the annual percentage rate, APR, is (2.1.1) In the interest setting this is an example of simple yearly interest, If the time length is different from a year the financial rate is altered accordingly. If the length of the time periods are equal and correspond to fraction of a year, such as (1/m)th of a year, the period rate is computed as r=i/m. For example if for m=12 the time period is in terms of months while yearly quarters are indicated by m=4. The concept of monetary growth can be extended to the situation of arbitrary multiple time periods. In general, the rates are defined based on the partition 0=t0

56

fixed at rj. In the discrete time period model the rate corresponding to (1/m)th of a year intervals and are defined by the equal width time periods Ej/m=((j−1)/m, j/m] for j≥1. The future value of the investment after multiple time periods is found by repeated use of (2.1.1) and in the interest setting this is referred to as compounding interest. After n time periods the future time is given by tn=nm the future value of the initial invest of P is (2.1.2) This formula is general in nature and can be adapted to most common interest time frame situations. General formulas for the future value of an investment in standard settings are easily constructed. In the discrete time period model with equal width time periods often the interest rate is fixed at yearly or annual percentage rate, APR, given by i, and thus the period interest rate is r=i/m. From (2.1.2) the future value after n years is (2.1.3) In this situation the interest is said to be compounded mthly. Examples are now given that demonstrate the application of these formulas and the financial growth of an initial investment over time. Ex: 2.1.1. Let an initial investment or principal be P=$100 and the annual interest rate be i=.12. Suppose we compound interest monthly so that the periods are one month long and in yearly units the future time is denoted tj =j/12. In addition the monthly or period interest rate is r=.12/12=.01. The future value after one year, or 12 periods, using (2.1.3) is

From this we note that the APR corresponding to simple interest over one year for this investment is 12.6%. Comparing compounding interest monthly for one year to yearly simple interest an increase of .53% in the future value is realized. Ex. 2.1.2. An investment of $500 earns interest that is compounded quarterly. The annual interest rate is i=.08 so that the quarterly interest rate is r=.08/4=.02. Based on (2.1.3) the future value after one year is

The future value is slightly different if the quarterly interest rates varied. Let the quarterly interest rates be given by .01, .015, .025 and .03, noting the mean of these interest rates is .02. In this case, using (2.1.2), the future value is

We note that magnitude and not the order of the different period interest rates determines the future value of the investment. The rate of compounding, in terms of time period length, affects the future value calculation. Compounding interest or investment growth continuously over a fixed time is


57

achieved by letting the number of equal length periods approach infinity. To find the limiting future value we let m approach infinity in formula (2.1.3). If the continuous rate is denoted δ then at time T= t the future value is (2.1.4) The different rate settings, demonstrated in the example that follows, result in slightly different future value computations. Ex. 2.1.3. The future value of $100 after one year is calculated under various settings. The interest rate is constant where the annual rate is i=.12. Using (2.1.3) and (2.1.4) we find the following future values: (i) Simple Interest: FV(1)=$100(1+.12)=$112.00 (ii) Compound Monthly: FV(1)=$100(1+.01)12=$112.68 (iii) Compound Continuously : FV(1)=$100exp[(1)(. 12)]=$112.75 We note the rate of compounding is directly related with the future value of the investment with compounding continuously yielding the largest future value and simple interest producing the smallest yield. The APR for (iii) is 12.75% as compared to the simple yearly interest setting of 12% for an increase of .75/12 =6.25%. For a comparison of investment or financial growth on equal scales of measurement annual financial rates and the continuous compounding rates can be related. If we equate (1+i) with exp(δ) the resulting relations are (2.1.5) Thus, if we set either type of financial rate i or δ we can solve for the other. In the general discrete model, where the intervals correspond to (1/m)th of a year, we relate the continuous type rates for discrete rates for period Ej/m denoted by δj/m as (2.1.6) for j≥1. These relations play an important role in associations that exist among computations concerning differing financial and actuarial models. To compare interest and investment rates from different sources the annual interest or percentage rate, APR, is often used as is shown in the next example.

Table 2.1.1 Continuous Percentage Rate and Annual Rate Comparison Continuous Rate

10

12

14

16

18

20

22

APR

10.5

12.7

15.0

17.4

19.7

22.1

24.6

Ex. 2.1.4. An offer of a loan reports an interest rate of δ where the interest is compounded continuously. Using (2.1.5) the yearly interest rate or APR can be computed A comparison of these calculations is given in Table 2.1.1 where the rate δ along with the


58

associated annual rate are listed. From Table 2.1.1 we see that the APR is higher than the continuous rate and these transformations of rates are useful in making comparisons. The time frame associated with the evaluation of a financial action may be altered. Monetary sums or principles, such as investments, valued at P at T=t>0 may be analyzed at the initial or present time. This is referred to as the present value or PV. In the discrete model we find the present value by interchanging P and FV in (2.1.2) for PV and P, respectively. The present value of an investment worth P at time T=tn is (2.1.7) In an analogous manner to future value computations other present value functions are constructed. This concept is now extended to the discrete time period model consisting of equal time periods and the continuous compounding model. In the discrete setting the periods are (1/m)th of a year and the period interest rates based on the annual interest rate i, are equal. The present value associated with an investment valued at P in n years is (2.1.8) Further, if the financial growth is compounded continuously at continuous rate δ the present value of an investment of P in future time t is (2.1.9) The relationships associated with present value computations are explored in the next example. Ex. 2.1.5. The present value of an investment is desired when after 5 years the investment is valued at $1,000. Here the interest rate is constant with annual rate is i=.12. The present value after 5 years is compared under various interest schemes: (i) Simple Interest: PV(5)=$1,000 (1+.12)−5=$567.43 (ii) Compound Monthly: PV(5)=$1,000 (1+.12/12)−60=$550.45 (iii) Compound Continuously: PV(5)=$1,000 exp(−.12(5))=$548.81 As one would expect, we note that the greatest affect on the present value function occurs with continuous compounding where only an initial investment of $548.81 is required to produce the future value of $1,000. It is sometimes useful to remember these calculations in terms of the time line. For a monetary investment the future value goes forward in time and the investment increases in value. The opposite is true for a present value of an investment. The present value proceeds backward in time and the value of the investment decreases in value. We now formalize the discrete time period model.


59

2.1.2 General Period Discrete Rate Models In this section we describe general discrete rate models for both the future and present value functions. The general model presents the basic concepts and formulas for both stochastic and nonstochastic financial rate models described in later chapters. The future value of an investment of P, given by (2.1.2), can be rewritten using the technique of compounding period rates. Over time period Ej/m the interest rate, as used in (2.1.4), is δj for j≥1. If investment is compounded m times a year, then over the time period (0, tn], where tn=n/m, the cumulative financial rate and the rate function reflecting the increase in investment value are given by n follows. (2.1.10) Based on these notations the future value after n periods or at time tn is (2.1.11) This notation is useful in various discrete time period settings. The continuous time setting is defined as a limiting extension of the discrete time period model case as the time period shrinks. In the discrete rate model the continuous rates corresponding to each period can be used to construct the present value function in terms of a discount function. This is analogous to the construction of (2.1.11). The discount function defined over (0, tn] reflects the decrease in the value of the investment as time reverses is (2.1.12) where ψn is the cumulative interest rate given by (2.1.10). The present value function is (2.1.13) This formula is a generalization or formal restating of (2.1.7) where the return rates may vary from period to period. These discrete rate formulas are applied in various settings, such as when the rates are stochastic. 2.1.3 Continuous Rate Models In this section the construction of continuous return models is based on continuous compounding rate calculations. The financial rate is a function of future time T=t and is denoted δt. The future value and the present value functions are constructed to be extensions of the continuous compounding rate formulas for discrete time periods. In the simplest case the continuous interest rate δt is nonnegative and integrable for t>0 and the continuous cumulative financial rate is defined by


60

(2.1.14) Analogous to the discrete time period models the continuous rate function and the discount function reflect the time effect on monetary values and are given by (2.1.15) Based on (2.1.11) and (2.1.13) the future value and present value functions for the continuous time models are given by (2.1.16) These formulas are utilized, and sometimes combined, to analyze a variety of financial contracts and settings. A continuous time structured financial rate example that demonstrates the potential versatility associated with these models is now given. Ex. 2.1.6. A principal of $100 is invested for one year where the return rate over the year is estimated to be increasing linearly from .10 to .14. In this case δt=.04t+.10 and the cumulative interest rate, from (2.1.14), is

From (2.1.15), the rate function is RF(1)=exp(.12)=1.127497 and the future value is FV(1)=$100exp(.12)=$112.75. We note that this coincides with the compound continuous future value computed in Ex. 2.1.3. 2.2 Fixed Rate Annuities An annuity is a series of payments where the time between the payments is called the period. The value of the jth payment is denoted by πj, for j=1, 2,… The future value of the annuity is often called the amount of the annuity and is the sum of the future values of the separate payments. It is common in financial and actuarial models to consider ordinary or level payments where the payments are all equal, denoted π, and made at the end of each period. In this section the financial rates are considered non-stochastic and are fixed over future time periods. 2.2.1 Discrete Annuity Models The general discrete rate setting presented in the previous section is taken to hold. The time line is partitioned into disjoint time periods Ej/m=((j− 1)/m, j/m] where the period interest rate is rj, for j≥1. We initially consider the annuity model where payments, denoted by πj, are made at the end of each period corresponding to future time tj over some predefined time period. The amount or future value of the annuity at time tn is the


61

sum of the future values of the separate payments. Using (2.1.2) the future value of the annuity at time T =tn is (2.2.1) This formula can be adapted to many common situations where, in some cases, adjustments may be required. For example, payments may be made at the start of each period requiring an adjustment to the individual future value calculations contained in formula (2.2.1). In the most common setting, referred to as an ordinary annuity, we have the discrete rate model where the year is divided into periods of length corresponding to (1/m)th of a year. The payments are all equal and denoted by π. Further, we assume the period financial rates are all the same and are given by r=i/m. In this case the future value, (2.2.1), reduces, using the summation formula (1.3.11), to (2.2.2) This formula has many applications, such as house mortgages, loan payments, and the evaluation of future pension benefits. An annuity example is now given. Ex 2.2.1. A deposit of $150 is made at the end of each month for four years where the annual interest rate is 3.6%. The interest rate per month is r= .036/12=.003 and the future value of the annuity at t48=4 from (2.2.2) is

We remark that without interest the total future value of the payments is $7,200, which is approximately 7% less than the interest enhanced or loaded monetary sum. The formula for the present value of an annuity can be written in terms that utilize the cumulative financial rate and rate function given in (2.1.10). Letting the annuity payments be equal, for s

62

payments backward in time to T=0 to find the present value of an annuity. The present value of an annuity is the sum of the present values of the separate payments denoted by πj for j≥1. Based on (2.1.7) the present value of the annuity associated with time tn is (2.2.5) where rj is the jth period financial rate. In the common setting where equal payments, π, are made at the end of each (1/m)th of a year the present value formula can be simplified. The period financial rates are all r=i/m at time tn the present value can be written as (2.2.6) Again, this formula is a result of the summation formula (1.3.11) and an example of its application now follows. Ex. 2.2.2. A series of monthly payments of $150 is made in an account for five years where the annual interest rate is i=.036. Thus, r=.036/12= .003, π=$150 and t60=5. From (2.2.6) the present value of the annuity is

The sum of the payments without interest is $9,000 and we observe a decrease of 774.77/9,000=8.6%. The lower value of the present value calculation in the presence of interest is due to the growth rate of money over time. The formulas for the present value of an annuity can be written in terms of the discount function (2.1.15). If the payments are level or all the identical for each period the present value after payments to time T=tn, from (2.2.5), is (2.2.7) If the period payments are different the natural adjustment to (2.2.7) is easily made. In actuarial calculations the special case of payments valued at one unit made at the start of each year for k+1 years play a central role. This is a common structure used in life insurance and life annuity models. If the yearly interest rate is i the future value of the annuity called the interest accumulation at the end of year k is found using Prob. 1.2 with a=1+i and is denoted (2.2.8) where d=i/(1+i). Further, the present value of the payments made at the start of each year for k+1 years is


63

(2.2.9) Through these formulas the effect of time is evident. The notations given in (2.2.8) and (2.2.9) are given for completeness as they are commonly used in actuarial science texts and literature. 2.2.2 Continuous Annuity Models As in the case of interest models annuity models can be constructed from a continuous time point of view. Continuous annuity models are extensions of the basic discrete time period annuity models. There are two types of continuous annuity models. The first is a discrete time period model with continuous financial rate structure. The payments follow a discrete pattern, corresponding to interval time periods and the financial rate function is modeled by a continuous function. In the second, the more traditional treatment is considered where payments are considered continuous through time periods along with the rates. In both types of continuous annuity models, as in the Sec. 2.1.2, the financial rate is defined by (2.1.14) and for times between s and t, for s
The future value of the annuity after one year, utilizing (2.2.4), is computed as

In these constructions we note the potential flexibility in the structure of the general financial rate function. A variety of shapes for financial growth can be modeled.


64

The second type of continuous annuity model is considered where the payments themselves are considered to be continuous in nature. The annuity payments are a conceptual limit of the discrete time period setting as the number of periods increases. For an annuity with continuous payment, denoted by πt, the future value and present value associated with future time T=t is defined, respectively, by (2.2.11) In the case of level continuous payments, which is quite common, the payments are constant with respect to time denoted by πt=π for t≥0 and formulas (2.2.11) are adjusted. Ex. 2.2.4. We estimate the annuity structure presented in Ex. 2.2.3 where the annuity payments are level continuous payments associated with a yearly value of π=$4,000. Utilizing (2.2.11) the present value after one year becomes

We remark that the integral in the calculation is evaluated by first completing the square and then utilizing the standard normal distribution (see Prob. 2.8). Further, present value computations associated with arbitrary future time t are easily made by adjusting the integral limits. 2.3 Stochastic Rate Models In practice it is often the case that the return rates for investments or sometimes the interest rates corresponding to loans or annuities are variable or stochastic in nature. In this case the rates are modeled as random variables with associated distributions. The effect of varying interest and return rates on financial and actuarial computations has been the focus of much work. Jordan (1991) observed the change on annuity and reserve calculations with varying interest rates. The effect of interest rates on surrender features of insurance have been explored by Grosen and Jorgensen (1997). Further, examples of modeling the force of interest using stochastic processes and time series models are given in Nielsen and Sandmann (1995), Frees (1990) and Panjer and Bellhouse (1980). As in the previous sections there are two basic cases for the rate random variables, indicated by the discrete and continuous time period models. In the discrete time period model, with equal width intervals, the rate random variable δj/m over the intervals have mean and variance denoted by γj/m and βj/m2. In the continuous compounding setting the mean and variance of the random rate δu are integrable functions given by γu and βu2. To model stochastic rates in the discrete time interval setting the interest modeling approach of Borowiak (1999) is utilized. There are three types of statistical calculations that are utilized in conjunction with stochastic interest rate models. The first concerns probability questions, such as percentiles and survival probabilities, corresponding to future and present value functions. The second deals with the expected future and present value functions, defined


65

by (2.1.15), and (2.1.16). The third concerns the construction and applications of prediction intervals for the distribution of the random rates as presented in Sec. 1.8.2. In conjunction with any of these a choice for the probability distribution of the financial must be made. Following the approach of Kellison (1980) the normal distribution plays a central role in the modeling of financial return rates. To assess the applicability of modeling financial rates with the normal distribution graphical assessments, such as probability or hazard plots, may be employed. For a review of the usage of these plots in the selection of probability distributions to empirical data we refer to Nelson (1987, Ch. 3) and Tobias and Trindade (1995, Ch. 6). An example of applying probability plots to empirical data is considered in Prob. 2.17. 2.3.1 Discrete Stochastic Rate Model In the discrete time period model with equal width time periods the financial rates δj are considered random variables and are assumed to follow a specified distribution, such as the normal or uniform distribution. In the general discrete time period model the time period Ej is associated with the stochastic rate δj that have mean γj and variance βj2 for j≥1. In this section probability computations and prediction intervals for future and present value functions are demonstrated for the normal distribution. This is introduced in the following investment example. Ex. 2.3.1. An investment of P is made for n periods where the interest is compounded mthly. The period financial rates are assumed to be independent and identically distributed normal random variables. From (2.1.10), this distributional assumption implies that ψn is a normal random variable where (2.3.1) For constant c the probability that the future value exceeds c is found by first solving for ψn inside the probability function and then transforming to the standard normal random variable. The reliability associated with the future value function is given by (2.3.2)

where Φ(c) is the standard normal distribution function. Further, using the normality in (2.3.1), a (1−α)100% prediction interval for ψn is given by (µn−z1− α/2 σn, µn+z1−α/2 σn). The resulting prediction interval for the future value is (2.3.3) The prediction interval for the present value function is constructed in Prob. 2.11. For example, if P=$10,000, m=12, γj=.01 and βj=.01, for all j≥1, then after five years the probability of the present value exceeding $20,000 is, using Appendix A2,


66

From this we interpret that there is an 11.32% chance that the future value after 5 years will exceed 20,000. Also, a 95% prediction interval for the future value takes the form

resulting in interval estimate

As with any statistical interval estimate, the reliability of the future value estimate is considered in the prediction interval. Additional statistical topics in this context, such as sample size problems are left to the reader to develop. We now consider moment calculations for these financial functions. Expectations and variances for future and present value functions can be computed following basic statistical techniques. Let the mgf of δi, as discussed in Sec. 1.4, be denoted by Mi(a). In general the expected rate and discount functions, based on (2.1.15), are given by the expectations (2.3.4) Hence, the expected future value and expected present value functions corresponding to investment or principal P are (2.3.5) In special cases expectation functions can be streamlined to contain mgf values. If the return rates based on mthly spaced periods are iid random variables with corresponding mgf denoted Mm(t) then (2.3.4) becomes (2.3.6) From (2.3.6) the expected future and present values are computed using mgf values at 1 and −1 and are (2.3.7) A variety of probability distributions and corresponding moment generating functions can be applied to model the financial rates. A set of examples using the normal distribution to model the financial rates demonstrating concepts and formulas are now given. Ex. 2.3.2. The investment in Ex. 2.3.1 is considered where the return rates are assumed to be iid normal random variables with period mean and variance given by, respectively, γ and β2. As before, using (2.3.7), the expected rate function is


67

(2.3.8)

In particular if $10,000 earns interest compounded monthly the expected future value after 5 years, where in terms of each month γ=.01 and β=.01, the expected rate function is

From (2.3.5) the expected future value is

We note that, as expected, the expected future value is within the prediction bounds computed in Ex. 2.3.1. Ex. 2.3.3. In this example we apply an AR(1) system introduced in Sec. 1.14 to a collection of financial rates over time. Utilizing the notations of that section, based on n periods ψn=Sn and we have (2.3.9)

Using these formulas statistical calculations, such as survival probabilities and prediction intervals given in (2.3.2) and (2.3.3), can be computed based on hypothetical parameter values. This procedure is extended to empirical data in Ex. 2.3.5. Ex. 2.3.4. The discrete mthly period model is considered in this example. The present value of an amount P after t years is computed where the rates are assumed to be iid for all j≥1. Using normal random variables with mthly period values γj =γ and the normal mgf the expected present value associated with future time t=tm is (2.3.10) where n=tm. For example, if m=12, P=$1,000, α=σ=.01, and t=10 then n=120 and

We note that if the interest rate is fixed at .12 per year the difference is minimal with PV(10)=$301.19 In practice distribution parameters, such as means and variances are unknown and need to be estimated from observed data. Often statistical formulas are applied to observed data by treating the point estimates as the fixed constants and plugging in computed values for unknown parameters. This is theoretically sound with statistical


68

estimates that converge to their associated estimated parameters as the sample size gets large. The reliability of these plug-in estimator procedure can be evaluated using simulation resampling methods similar to the ones presented in Chapter 7. In the next example empirical data is considered and the method of using plug in estimators is demonstrated. Ex. 2.3.5. In this example we consider adapting observed data to the formulas in this section where there is assumed to be no increasing rate trend over time. At the start of each of 11 periods the value of an investment, denoted Xj, is utilized to estimate the return rates. For period Ej, the estimated return rate is computed as

Here and let we let for j≥1. These observed values are given in Table 2.3.1. The future value of the investment is to be estimated after additional 5 periods. The return rates are assumed to be normal with a constant mean and we apply, one at a time, the both the independent and AR(1) models.

Table 2.3.1 Empirical Return Rates Period

0

1

2

3

4

5

6

7

8

9

10

11

Xj

12.5

10.5

11.5

10.7

12.5

12.5

12.2

12.5

14.7

14.2

12.7

13.5

δj

−.174

.091

−.072

.156

.000

−.024

.024

.162

−.035

−.112

.062

Zj

−.181

.045

−.061

.131

.024

−.032

.011

.159

−.009

−.127

.029

A scatterplot of the return rates against months, along with a least squares line, is given in Ex. 2.3.6 in Fig. 2.3.1. From Fig. 2.3.1 we see there is, at best, a weak linear fit. In the independent rate model the point estimates of the parameters are

From (2.3.8)

and the expected present value is

Using the AR(1) approach the least squares point estimates are computed using (1.14.8) and (1.14.9). Letting

we find


69

Fig. 2.3.1 Regression of Return Rate Date.

Based on (2.3.9) we find in estimate of the expected rate function is

and

. The plug

and the expected present value estimate is

We remark that for both the iid and AR(1) models prediction intervals can be constructed using either asymptotic statistics, relying on the asymptotic normality of the least squares estimators. These inferences can be validated using simulation resampling techniques procedures discussed in Chapter 7. In the case of an increasing or decreasing tends in financial rates statistical model fitting may be applied. If the change in the financial rates are linear with time then a linear regression modeling analysis can be utilized. This is demonstrated in the next example. If the financial rate is not linear in time then transformations or other regression techniques, such as nonlinear regression, are possible. Ex. 2.3.6. The investment data in Ex. 2.3.4 is considered where the return rates are assumed to be from a normal distribution. The future value at the 12th period is to be estimated by both a point estimate and a 95% prediction interval. To do this we apply regression using δj, as the response and the period X=j as the predictor variable for 1≤j≤11 as the response. The least squares estimators are found to be

In Fig. 2.3.1 is a scatterplot of the data along with the least squares line indicating little or no linear relationship. The resulting least squares line and point estimate for the 12th period are


70

This indicates a slight increasing trend in the return rates. In fact this regression the sample correlation coefficient is found to be r=.146 indicating a weak linear fit. A 95% confidence interval for the new return rate corresponding to the 12th period is found to be −.10359≤δ11≤.17396. The resulting 95% prediction interval for the future value is

We should keep in mind that even though these intervals asses the reliability of future values they are dependent on circumstances and conditions, such as a linear trend, that may change over time. 2.3.2 Continuous Stochastic Rate Models In this section a general continuous compounding financial rate model is presented The model has many structuring options and comes from an extension of the discrete time period model For a fixed future time T=t, ψt is considered a random variable. Extending (2.3.1), the mean and variance are integrable functions γt and βt for t≥0. Here, for future time t (2.3.11) Applying a probability distribution to the cumulative financial rate ψt allows the calculation of survival probabilities and expectations for future and present value functions. Ex. 2.3.7. The investment over time discussed in Ex. 2.3.1 is altered so that the financial return rate is continuous and follows a normal distribution. Hence (2.3.12)

With this model many statistical and probabilistic computations are possible. For example the probability the future value exceeds c is (2.3.13) where the distributional properties in (2.3.12) are utilized, A (1−α)100% prediction interval for the future value is constructed using standard normal percentiles and is given by (2.3.14)


71

The counterparts of (2.3.13) and (2.3.13) in the case of the present value function are easily constructed (see Prob. 2.14). The expected present and future value functions for stochastic financial rate models are now presented. The general formulas for the expected future and present value functions given in (2.3.4) and (2.3.5) are adapted to this setting. The financial rates are assumed to be normal random variables and from (2.3.12) the expected rate function is (2.3.15) while the expected discount function is (2.3.16) Analogous to the discrete case the expected future and present value functions are (2.3.17) These formulas are adapted to form continuous time computations where the shape of the mean and variance functions can be modeled. Two examples are now given. Ex. 2.3.8. The investment example presented in Ex. 2.1.5 is modeled using the continuous approach. A principal of $100 is invested for one year. The mean of the financial rate is defined by the integrable function γu=.04u +.10 while the variance is βu2=.002(1+.001u). In this case the mean for one year is

and the variance is

Applying the normal distribution for the financial rate the expectation of the rate function is

and the expected future value follows from (2.3.17) and is computed as $186.13. Ex. 2.3.9. In this example a continuous financial rate posses a constant mean and . The statistical formulas variance function. This construction implies µt=tγ and (2.3.11) through (2.3.14) along with their present value function counterparts hold. For example the probability that the present value, associated with future time t, is at most c is (2.3.18)


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Based on this the (1−α)100% percentile for the present value function is given by (2.3.19) Further, the expected future value and present value associated with T=t are (2.3.20)

For example consider a monetary sum of $1,000 that is required in 18 months. How much money do we need to invest now to have confidence that this payment will be made? The yearly return is estimated to be between 6% and 14%. The estimated return is centered at i=10% so that the continuous return rate point estimated is taken to be δ=ln(1.1)=.0953 with a likely range of ln(1.14)−ln(1.06)=.07277. Using a 2 standard deviation interval estimate we estimate the yearly quantities γ=.0953 and β=.07277/4=.0179. Using formulas (2.3.15) through (2.3.17) we find the expected required investment

Hence, we estimate a required investment of $866.58. A more conservative estimate results from using an upper 95th percentile for the present value. From (2.3.19)

The difference in these two values is due to the skewness of the present value function and the probability requirement. 2.3.3 Discrete Stochastic Annuity Models The case of discrete time period ordinary annuities, consisting of payments made at the start of each time period, with level or constant payments is the topic for this section. Other annuities with varying payments can be modeled by adjustments to the formulas that follow. For annuities with level payments of value π made at the start of each time interval the future and present values are given by (2.3.21)


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In the annuity model setting the computation of probabilities leading to percentiles and prediction intervals is not straightforward. These statistical computations can most easily be handled by statistical approximation techniques or statistical resampling simulations. Assuming the rates are iid normal random variables and (2.3.1) holds, prediction intervals can be computed. For the future value function, defined by (2.3.21), a (1−α)100% prediction intervals is given by (2.3.22) In the case of discrete time period annuities formulas for the expected future and present value functions can be derived. For an ordinary discrete annuity the expected future value and present values are, respectively, (2.3.23)

In the case of iid rates, the formulas in (2.3.23) can be written in terms of the corresponding mgfs similar to that of (2.3.6) and is left to the reader. A computational example using the normal mgf follows. Ex. 2.3.10. An ordinary annuity continues for n periods where equal payments of value π are made at the start of each period Employing (2.3.23) the expected present value associated with future time tn is (2.3.24)

where Mm(−1)=exp(−µm+(1/2)σm2). For example, consider payments of $1,000 made at the start each month for 25 years where the interest rates are assumed to be iid normal with i=.08 and β2=.024. Hence, µ12=.006666 and σ122=.002. we find M(−1)=.994349 and EPV(25)=$1,000(143.631538)= $143,631.15. We note that if rate is non-stochastic then βj=0, M(−1)= .9933555 and EPV(25)=$1,000(129.26773)=$129,167.73. From our examples and computations we observe that the stochastic nature of the financial rates influences the annuity calculations more than the interest computations. This is due to the propagation of the stochastic effect present in formulas such as (2.3.22) and (2.3.23).


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2.3.4 Continuous Stochastic Annuity Models In this section the stochastic annuity structure is applied to continuous type annuities. This is accomplished using a limiting time period argument. For continuous annuities with constant payment π the future and present values, associated with future time t are (2.3.25) As in the case of discrete stochastic annuities percentiles and prediction intervals can be computed by resampling simulation and approximation methods. The statistical formulation of prediction intervals is now discussed. Prediction intervals for the future and present value functions in the continuous model can be found similar to the discrete setting when the rates are assumed to be normal random variables. Using (2.3.14) we find (1−α)100% prediction intervals for the future value function to take the form (2.3.26)

We remark that in the evaluation of (2.3.26) numerical integration procedures or packages make the task easier. The expected future and present values for stochastic annuities can be computed assuming an underlying distribution for financial rates. Based on the constructions in (2.3.25), for future time t these expectation formulas are (2.3.27) As mentioned before, numerical methods and approximations can be used to evaluate (2.3.27) for specific cases. Ex. 2.3.11. A continuous annuity is considered where the rates are iid and µt=tγ and for t>0. If we assume the rates are distributed as normal random variables then EDF(t)=exp(−t(γ−(1/2)β2) and the expected present value is


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For example let π=$12,000 and the corresponding yearly parameters be given by γ=.08 and β2=.024. Using (2.3.27) we compute EPV(25)= $12,000(12.01935)=$144,232.32. We remark that since the payments are made in a continuous manner the expected present value is more than the discrete time period counterpart Problems 2.1. If $950 is invested in a bank account where the annual interest rate is 6% find the future value after 3 years if the interest is a) simple, b) compounded yearly, c) compounded monthly, d) continuously. 2.2. Over a six-month period the annual financial rate is given by Month Annual Rate

1

2

3

4

5

6

.05

.06

.06

.07

.07

.09

a) Give the value of the rate and discount function over the six month period. b) If $500 were invested at the beginning of the six months what would be its value after the six months? c) If $800 were required to pay a note due at the end of the six-month period, how much money would need to be invested at the beginning? 2.3 If the annual interest or return rate is 5% what is the corresponding continuous rate? Further, if the yearly interest rate of 6% is compounded monthly what is the corresponding APR? 2.4. Consider annuity payments of π made at the start of each month in Prob. 2.2. a) If the payments are $200 what is the future value after the six months? b) If $800 were required after six months what should the monthly payments denoted by π be? 2.5 At the start of every month a deposit of $100 is made in a bank account where the annual interest rate is 6%. How much money is in the account after a) 5 years? b) 10 years? c) 20 years? 2.6. A university wants to endow a professor position where $100,000 is required annually. Money is deposited in a trust and only the interest or growth is spent. How much money is needed if it can be invested at an annual rate of 7.5%? 2.7. Over a two-year period the return rate on an investment grows from 4% to 10% in a linear fashion. a) Find the ψt function in (2.1.14). b) What is the value of an investment of $2,000 after 18 months? 2.8. Compute the present value formula for the annuity presented in Ex. 2.2.4. To do this, inside the integral complete the square and then use the form of the normal pdf given in Ex. 1.2.6, 2.9. Consider an initial investment of $2,000 where the return rates listed in Prob. 2.2 hold. A linear regression models is used to model the return rates. a) Using (1.13.3) and (1.13.4) find the estimated formula for ψt in (2.1.14). b) What is the point estimate for the future value after one year? c) Use (1.13.8), (2.1.15) and (2.1.16) to compute a 95% prediction interval for the future value after one year. 2.10. After 5 years a payment of $10,000 is due where the annual interest is i= .06. To finance this payment an annuity with level payments of π are made. Find the value of π if


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a) payments are made at the start of each month, b) payments are made continuously, from a bank account, throughout the 5 years. 2.11. Using (2.1.12), (2.1.13) and (2.3.1) construct the formula for a (1−α)100% prediction interval for the present value function associated with the discrete stochastic rate model based on mthly equal width time periods. 2.12. A sum of $5,000 is invested where the yearly return rate, δ, is considered a normal random variable with mean .08 and standard deviation .03. After 5 years find a) the expected future value, b) the probability the future value exceeds $8,000, and c) a 95% prediction interval for the future value. 2.13. For an investment, return rates are independent and random taking values between .06 and .12. Considering the rates as uniform random variables, after 5 years find a) the expected rate function (2.3.6), b) the expected discount function (2.3.6), c) the expected future value function based on an investment of $5,000. 2.14. Consider the continuous stochastic rate model given in (2.3.12). For the present value function give counterpart formulas to (2.3.13) and (1.3.14), namely a) P(PV(t)>c) for c>0, b) a (1−α)100% prediction interval for the present value function associated with future time t. 2.15. The closing price of a stock at the end of each week is given below: Week

0

1

2

3

4

5

6

7

8

9

10

11

12

13

Price

6.2

6.5

5.9

6.3

6.8

7.2

6.4

7.3

7.9

8.1

6.8

7.3

8.5

8.5

a) Find the rates δj for j=1,…, 13. b) Fit a regression line to the rates in a). What is the estimated line? c) What is the sample correlation coefficient (1.13.5)? d) Using (1.13.7) find a 95% confidence interval for the slope parameter Are the rates increasing? e) For the 15 month give the point estimate and a 95% confidence interval, (1.13.8), for the future value. 2.16. You are to receive a lump sum payment of $20,000 in 10 years. You can invest money where the return rate is distributed normal with mean .08 and standard deviation .025. For this future payment find a) the expected present value and b) a 95th percentile for the present value. What does this value mean? 2.17. In this problem we outline the procedure for probability plots for normal data. First we order the observed rates, δ(1)<δ(2)<…<δ(n) and then let the related position be

To asses the normality of δj the transformations vi=δi and wi=Φ−1(Fi), for i≥ 1, are used. Under the normality assumption the simple linear regression model (1.13.1) holds for yj=δ(j) and xj=ln(wj) for j≥1. The plot of (yj, xj) comprise the probability plot that under normality yields a straight line. a) In Prob. 2.15 give the probability plot for normal rates. b) In a) find the least squares estimates for the intercept and slope. The intercept estimates the mean while the slope estimates the standard deviation. c) Calculate r given by (1.13.5). Is the linear fit good thereby indicating normal financial rates? 2.18 An ordinary annuity making payments at the start of each month is conducted for one year. If the payments are for $1,000 and the investment rate is a random variable compute the expected present value after one year using (2.3.5) for a) iid normal


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investment rates where the yearly mean is 8% with a standard deviation of .03, b) iid uniformly distributed investment rates between 5% and 11%.

3 Deterministic Status Models

In financial and actuarial modeling we often combine financial computations, such as present and future value functions, to create models and accompanying equations to be analyzed. Theses models can be written as a function of a set of conditions referred to as a status and may contain unknown parameters or constants to be determined. A typical example includes house payments where the cost of the house is financed by a series of annuity payments. The status is defined in terms of the payments made and the status fails to hold when the loan is paid off and the payments end. The amount of the individual payments must be determined from the relevant present value functions and is a function of the amount of the loan and the interest rate. In general terms, these models are characterized by a future time component, based on the lifetime of a status, which may or may not be a random variable. In typical mortgage payments the length of the time of the loan is a fixed quantity. In this chapter models are presented that are functions of a non-random or deterministic status and are referred to as deterministic status models. The financial rates may be either fixed or random variables over the future lifetime of the model. In this chapter deterministic status models are presented in terms of a basic business loss function. The loss function is based in economic theory and in the simplest case represents the difference between expenditures and revenues. The loss function is dependent on either a fixed or stochastic rate function over time. Based on this economic modeling construction, loss criteria are defined for both fixed and stochastic rate settings that enable the statistical evaluation of these financial and actuarial models. The statistical evaluations include computations of percentiles, confidence and prediction intervals. Basic applications involving insurance, annuity and financial calculations and are presented. Also, more advanced applications such as collective risk models over short periods of time and stochastic surplus models are discussed. 3.1 Basic Loss Model To analyze financial and actuarial problems the construction of a mathematical model allows for formal investigation. From the perspective of an investor, insurance company or lending institution we construct a financial loss function or simply a loss function. At time T=t the lender or insurer may have future values of expenditures, denoted FVE(t),

Deterministic status models

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that are off set by future values of revenues, denoted FVR(t). The loss function, associated with financial action at time T=t, is the difference between the future values of expenditures and the revenues. In a like manner, at initial time, the present values of the revenue and the expenditure are denoted PVE(t) and PVR(t). Hence, the loss functions defined with respect to the two basic time frames, namely future and present values are (3.1.1) The choice of the expenditures and revenue functions in (3.1.1) depend on the context of the modeling to be done. In this construction, if LF(t)<0 at T=t then the lender or insurer makes a profit at time t. If the financial structure is completely deterministic in nature then the structure given by (3.1.1) is directly used for analysis. Specific loss models are presented that can be utilized to compute common financial computations, such as house or automobile loan payments. Loss function construction and analysis play an important role in statistical decision theory. A loss function in theoretical statistics is nonnegative and denotes a penalty in statistical actions while (3.1.1) may be negative. The expectation of the loss, referred to as the risk, is one method used in decision theoretic-based statistical inference. 3.1.1 Deterministic Loss Models We start our investigation with standard loss functions that allow for basic financial computations. In this section we assume the components in the loss function (3.1.1) are all deterministic, lacking any random variable component. In this case the analysis of these financial or actuarial models is usually accomplished in a straightforward manner. This is the case in house mortgage payments where the interest rate and the length of the contract are both fixed. In financial and actuarial models an equilibrium state holds if there is a balance between the expenditures and the revenue over future time. The initial time T=0 is used as a reference point and from (3.1.1) this balance implies (3.1.2) The equivalence principal, denoted EP, dictates that unknown constants or parameters are chosen so that the conditions and equations in (3.1.2) hold The applications of loss function construction and EP analysis are demonstrated in a series of examples. Ex. 3.1.1. A payment of P is due in t years and is offset by level annuity payments of π made m times a year. Hence, there are a total of n=mt payments defined by the contract. If the interest is compounded m times a year and the period interest rate is r the expenditure for the insurer, PVE(tn), is the present value of P associated with future time t=tm. Further, the present value of the revenue, PVR(tn), is the sum of the present values of the annuity payments. Using (2.1.8) and (2.2.6) the loss function is (3.1.3)


80

where n=tm. We again note that the financial computations in (3.1.3) take as a basis the initial time frame of t=0. Using EP, (3.1.2), the premium payment π reduces to the formula (3.1.4) For example, an amount of $10,000 is needed in five years where interest is compounded monthly and the annual interest rate is 12%. How much money does the person have to save each month to meet this requirement? Here r= .12/12=.01 and n=12(5)=60. From (3.1.3) the person needs to save

each month. In this setting the interest rate, the payment amount and the length of time involved determine the required premium payments. Ex 3.1.2 A house is purchased for $120,000 where the annual interest rate on a 30 year loan is 9%. How large a down payment, D, is required so that the monthly mortgage payments are only $900.00? The financial computations are based at the initial time and r=.09/12=.0075, n=12(30)=360, and E= $120,000. The present value of the revenue is PVR(t)=D+PVP(t) where PVP(t) is the present value of the premium payments given by (2.2.6). Using EP

In application, any cost related to the loan must be included into the expense term. Models that include various types of expenses, fixed and variable, are discussed in Sec. 6.13. Ex. 3.1.3. How many monthly payments of $225 are needed to pay off a loan of $10,000 if the annual interest rate is 8%? Here r=.08/12=.0067, PVE(t)=$10,000, π=225 and PVR(t) is given by (2.2.6). Utilizing (3.1.2) and solving for n

Hence, there are 52 full payments and one partial payment consisting of $225(.17)=$38.25. The previous examples illustrate some of the applications utilizing fully deterministic loss models and EP analysis. This modeling construction is the basis for more complex financial and actuarial analysis presented in later chapters. In practical cases the interest rate or the financial rate may be stochastic with an associated probability distribution. The case of deterministic status models associated with stochastic financial rates is considered in the next section. 3.1.2 Stochastic Rate Models In general deterministic loss models some of the components in the loss function (3.1.1) may be a function of one or more random variables. This is the case in a house mortgage where the interest rate is variable and is a function of some stochastic variable. The loss model approach is quite general and the model may include discrete, continuous or mixed


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random variables. In this section we present two examples that demonstrate the application of such deterministic loss functions. The choice of revenue and expenditure function depends on the frame of reference taken by the financial or actuarial analysis. Ex. 3.1.4. An investor wants to compare two types of investments. The first investment is a fixed rate annuity where level payments, π, are made at the start of each month for n years. In the second a one-time investment of value P is made where the return rate is considered a normal random variable in which (2.3.1) holds where µn=nγ and σn2=nβ2. A comparison of these strategies by relating the revenue to the fixed rate annuity and the expenditure to the variable rate investment at future time t=n years is conducted. Using (2.2.2), (2.1.10) and (2.1.11) we define the future values of the revenue and expenditures by (3.1.5) From the loss model construction (3.1.1), we note that the random rate investment outperforms the annuity if the loss function is negative. Based on the normality assumptions applied to (3.1.5) the probability of a negative loss, LF(n) <0, is computed as (3.1.6) This is one statistical approach to the problem of selecting the best investment and is passed on probability computations. Another uses the expectation or risk function. Using the mgf of the normal random variable, (2.3.7), the expected loss function is given by (3.1.7) In particular, for the fixed rate annuity t=10, m=12, π=$100 and r=.06/12= .005 from (2.2.2) we compute FVR(10)=$16,387.934. In the stochastic financial rate investment let P=$9,000 and for each month let the normal parameters be γ=.005 and β=.001. Using (3.1.6) we find

Hence, 58.51% of the time the variable rate investment would outperform the fixed rate competitor. Further, applying (3.1.1) we find the expectation of the loss function to be

This computation indicates that the expected difference in the future values of these investments is minimal. Both analyses indicate, marginally, that the stochastic investment should be taken over the fixed rate annuity. Ex. 3.1.5. A investor has a choice between a sum of money, S, now or a payment of P at future time T=t. If the money can be invested where the return rate is the continuous normal random variable given by the continuous model (2.1.15) and (2.1.16), which does the investor take? To answer this the present value loss function (3.1.1) is constructed where PVE(t)=S and PVR(t)=P exp(−ψt). The loss function is


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(3.1.8) There exist at least two natural criteria or strategies available to use in connection with the loss function to solve for P. One solution results if we take the expectation of (3.1.8) and apply the EP. Using the expectation of the discount function and solving for P we find (3.1.9) The second strategy utilizes the probability of a positive loss where the lump sum of S is taken at the present time over the future payment of P if the probability of a positive loss is judged to be too large. Applying the normal distribution of rates (3.1.10) In particular, if S=$1,000 and P=$1,500, the yearly parameters are α=.08 and β=.025 and t=3 using (3.1.9) we find

Hence, going with the initial sum of $1,000 over the future payment the investor incurs an expected loss of $181.05. In addition, the probability of a positive loss is almost assured since from (3.1.10)

Both the expectation and the probability of a loss indicate strongly a future payment strategy over the lump sum payment In the next section specific criteria are formally presented in connection with the stochastic loss function to aid in the selection of rival financial and actuarial strategies. These criteria, as introduced in the previous examples, utilize expectation and probability calculations. 3.2 Stochastic Loss Criterion As introduced in the previous section we consider deterministic status models where one or more financial actions is stochastic in nature. The frame of reference for any analysis is the initial time, T=0, and, hence, present value functions are utilized. There are two approaches to the analysis of financial contracts that we will consider. The first approach in the analysis of these deterministic status stochastic financial strategies or actions is to base decisions on the expected value of the loss function (3.1.1), This is broadly referred to as risk analysis. This type of analysis is quite flexible and can be used in a variety of settings. In particular, this approach is useful when the interest or return rates are


83

stochastic in nature, The second approach is to base financial and actuarial analysis on the percentile calculations for a suitable loss function. In this approach the likelihood of adverse loss realizations is kept low. These criteria are presented in a formal manner in the next two sections. Other criteria have been proposed such as the exponential utility function as presented in Bowers et al. (1997, Ch. 1). 3.2.1 Risk Criteria As in decision theoretic based statistics, in financial and actuarial modeling we define the risk as the expectation of the loss function. In many cases decision criteria is a function of the computed risk. From (3.1.1) the risk function is defined as (3.2.1) Similar to the deterministic loss model counterpart without any stochastic component given by (3.1.2), the equilibrium principle, EP implies that all estimates of unknown parameters are used so that (3.2.2) is realized. There are other criteria that have been utilized in the analysis of risks. Another method is the application of an exponential utility weight function as discussed by Gerber (1976, 1979). The risk criteria, which we denote by RC, used in conjunction with EP dictates that (3.2.2) holds and can be used as a tool for unknown parameter evaluation. For example, premiums can be found for house payments where the interest rate is a random variable. For the development of the EP we refer to Lukacs (1948). An example now follows that demonstrates the application of the risk criteria approach. Ex. 3.2.1. In this example an amount or principal P is invested for four years where the return rate is assumed to be a continuous random variable. For one year δ~n(γ, β2). If the goal is to have a sum of S after t years what amount do you need to invest? In this case the expected expenditure E{PVE(t)}=P while the present value of the revenue is PVR(t)=S exp(−δt). Computing the expectation of each part, as in (2.3.17), the EP implies (3.2.3) In particular if S=$1,000, t=4, γ=.12 and β=.06 then from (3.2.3)

Hence, $623.25 is required for the investment using an expectation approach. In other words based on many identical investments this is the mean principal investment that meets the conditions. As with all RC, estimated parameters, referred to as statistical point estimators, contain no statistical measure of their reliability. The statistical distribution of the


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stochastic loss function is utilized only by the computed mean. In the next section we present the second criteria that is associated with a probabilistic measure of reliability, 3.2.2 Percentile Criteria In this section the second criteria to assess financial and actuarial models, as an alternative to RC, is presented and is based on percentile analysis of constructed financial loss functions. The (1−α)100th percentile associated with the loss function in (3.1.1) is denoted by the constant lf(t)1−α and is defined by (3.2.4) Based on the distribution of the financial rates percentiles can be found and, hence, prediction intervals for prospective losses can be calculated. Further, unknown parameter estimates can be calculated that ensure, to a desired probability, limited loss amounts. Similar to the RC, percentiles of the loss function can be used not only to estimate unknown parameters but to select among possible financial and actuarial decisions. In a percentile criterion unknown parameters and actions are chosen so that the probability of a positive loss function is a small specified amount We define the 25th percentile criteria, denoted by PC(.25), to be the one that associates α=.25 with lf(t).75=0. Therefore, using PC(.25) parameters are chosen resulting in a positive loss function occurring only 25% of the time. Ex. 3.2.2. Consider the setting of Ex. 3.2.1 where an amount or principal P is invested for t years where for each year δ~n(γ, β2). The goal is to have a sum of S and PC(.25) is applied with E{PVE(t)}=P and PVR(t)=S exp( −δt). We note that a negative loss indicates P will be greater than the present value of the investment Thus, with (3.2.3) we have, using PC(.25),

The solution, in terms of P, is found by first forming a standard normal random variable and then equating the constructed standard normal random variable to the 75th standard normal percentile. Upon doing this we find the 25th percentile (3.2.5) In the context of Ex. 3.2.3, S=$1,000, γ=.12, and β=.06 and from (3.2.4)

where z.75=.6745. Hence, using P=$644.34 a positive loss will occur only 25% of the time under identical conditions. We note that the percentile method is more conservative than the risk method and produces a larger required investment.


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3.3 Single Risk Models We now explore and formalize the loss criterion techniques introduced in Ex. 3.2.1 and Ex. 3.2.2 for a common type of financial or actuarial model. In this section the situation where, in (3.2.1), either PVE or PVR is a constant and the other represents a deterministic financial action is investigated. This is the case in simple mortgage computations. The financial rate is a random variable of either the continuous or discrete type and the criteria used in analyses of these models is either the RC or PC(.25) as presented in the previous section. In many cases it is useful to consider an investment or future obligation that is financed by only one initial payment at time T=0. In an insurance setting an insurance company may have a future obligation, such as life insurance over a fixed time period, that is financed by one payment called a single net premium. In an investment setting, such as the stock market, the revenue depends on the stochastic return compared to a fixed monetary amount called the single net revenue. To encompass both financial and actuarial models we use the term the single net value denoted by SNV. For example, in the application of RC, using EP, implies (3.3.1) Potentially, either setting in (3.3.1) presents the same degree of calculations and the context of the problem dictates the exact notation. 3.3.1 Insurance Pricing In an insurance setting the company incurs more expenses than just the payment of the direct benefit. Other expenses, both fixed and variable, may be added to the overall cost For this reason the single net value of the direct insurance claims may be less than the expected value of the direct and indirect expenses. The difference we refer to as a loading or LD. If the present value of the claim is taken to be PVE(t), using the expectation criteria, the loaded premium is denoted by G and the loading is given by (3.3.2) In this case the policyholder because of the loaded premium may receive a dividend denoted by D. For simplicity we consider a short time period where the effect of interest can be ignored and the claim amount X may be denoted by the present value of expenditures PVE. For constant k, 0

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The concept and formulas for expected dividends are connected to insurance policies that contain deductible claim amounts. These insurance models are introduced in the next example and discussed in the setting of aggregate insurance claims presented in Sec. 3.4.3. In either model describing deductible limits or dividends, the two criteria proposed for computing unknown parameters, namely RC or PC(.25), can be applied. In the example that follows deductible insurance models are introduced and the single net value, SNV, for general insurance models discussed. Ex. 3.3.1. We consider an insurance policy that pays in the case of an accident with evaluated loss of Y, a benefit given by B introduced in Ex. 1.3.7. The loss variable is considered to be a continuous random variable with pdf f(y). We consider one time period where the effect of interest can be ignored and in the simplest case where the benefit paid is equal to the amount of the loss. Let the probability of an accident be q. The overall claim variable is X=I(claim)Y where the indicator function is defined as I(claim)=1 if there is a claim and is zero otherwise. From (1.3.14) and applying the EP the single net value is (3.3.5) The variance of the claim amount is found using (1.3.16). In the event that (3.3.5) leads to too large a SNV, inflating the resulting premium costs, the insurance policy may be altered to include a deductible amount d. In a deductible insurance policy the claim payments do not start until the loss exceeds deductible d and are decreased by the deductible amount For deductible amount d>0 the amount of the benefit paid, B, takes the form (3.3.6)

Hence, the claim payments do not start until the loss exceeds d. Let the probability of a claim be q and the overall claim variable be given by X=B if there is a claim and X=0 otherwise. The effect of the deductible amount d is demonstrated in Fig. 3.3.1 where the initial pdf corresponds to X=I(claim)Y and the deductible pdf corresponds to X=I(claim) B where (3.3.6) holds. The deductible increases the probability of a zero claim and decreases the height of the continuous pdf portion. The statistical concepts and formulas introduced in previous sections can be applied to the deductible insurance benefit defined by (3.3.6). Conditioning on a claim, the expected value of (3.3.6) is the single net value (3.3.7) For any chosen distribution of claim amounts, as signified by the pdf given by f(y) the expectation defined by (3.3.7) can be computed. In particular if the realized claim amounts follow a normal distribution, or Y~n(µ, σ2), then (3.3.7) reduces to


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(3.3.8) where q is the claim probability and Φ is the df of a standard normal variable.

Fig. 3.3.1 Deductible pdf

If the RC is followed using the EP the premium paid is associated with the SNV given in (3.3.8). If the premium is loaded, where the expectation is given by (3.3.5), then a dividend may be paid. In this case the loss function is given by L =B−SNV. Applying a percentile criteria we compute the probability of a positive loss, from (3.3.6), as q P(Y>d+SNV). Assuming the loss Y~n(µ, σ2), if .25≤q then PC(.25) yields a SNV given by (3.3.9) For example if, in terms of $1,000 units, µ=10 and σ2=9 and q=.05 then SNV from (3.3.5) is .5 units or $500. If a deductible amount of 2 units or $2,000 is imposed then from (3.3.6) and (3.3.8) we find SNV=.40017 or $400.17. We note that the SNV is changed by $100 with the addition of the deductible amount 2. Also, given there is a claim the mean claim is reduced by approximately 2 units which computes as $2,000. Ex. 3.3.2. In this example we consider a form of disability insurance. A fixed benefit, b, is paid for a stochastic number of periods. There may be an elimination period, or a length of time after which benefit payments start. Also, there may be an upper limit on the number of payments denoted by m. The length of time is considered short so that the force of interest can be ignored. The payment amounts are denoted b, in each time period, are indexed by J=1, 2,…, m with corresponding pdf f(j) for j=1, 2,…. If we let the probability of a claim be q then the present value of the total claims is X=b J I(claim) where J= 1, 2,…, m and I(claim)=1 if claim and zero otherwise. For positive integer r the rth moment corresponding to J is (3.3.10)


88

Using (1.5.3) and (1.5.4) the first and second moments of J are (3.3.11)

and the SNV using the mean of X and the variance of X, found by applying (1.7.15), are (3.3.12)

In particular suppose $3,000 is paid each month a person is disabled starting after the second month and continuing for at most 20 months. Let q=.01 and number of payments have the geometric pdf given by f(j) = .2(.8j−1) for y=1, 2,,…. In this case P(J≥j) =.8j−1 so that from (3.3.11) and (3.3.12)

The variance can be computed using (3.3.12) and a resulting approximate confidence interval estimate for the SNV can be found (see Prob. 3.5). 3.3.2 Investment Pricing Due to the prominence in the modern economic world of investment pricing these concepts and models have attracted much attention in modern financial literature. A connection between financial investment analysis and risk analysis was demonstrated in the celebrated work of Black and Scholes (1973). In this section the topic of investment pricing is approached from a statistical point of view and much of the commonly applied complicated probabilistic and mathematical structure, such as stochastic differential equations, is avoided. The basic financial formulas presented in the second chapter are used as tools for investment modeling and assessment An investment of amount P is made in the hopes of its value increasing over time and the financial rates are considered to be random variables. In this context the single risk model holds where revenue is the return on the investment and the expenditure is the amount invested. The stochastic return rate formulas introduced in Chapter 2 are assumed and the stochastic criteria introduced in this chapter are the main tools to be employed. We assume the financial rate is a normal random variable given by (2.3.12). For this model the future value after time T=t is given by (3.3.13) Using expected future value formula in (2.3.17), utilizing the form of the normal mgf, the expectation of (3.3.13) is


89

(3.3.14) As stated before, the applicability of the normal distribution to model the financial rates can be assessed through statistical techniques, such as probability plotting (see Prob. 2.17). From an investment evaluation standpoint it is useful to analyze the future value of an investment, (3.3.13), relative to a certain or guaranteed return rate at the initial time T=0. In this way the increase, over a minimum threshold, in the growth of an investment can be assessed. In this context we let a risk-free, or guaranteed, interest or return rate over the time {0, t} be denoted by r. The investment pricing formula is the difference in the present values of the investment corresponding to the stochastic financial rate and the guaranteed rate associated with future time T=t or (3.3.15) Under the normality assumption, using the EP in conjunction with RC the expected present value of the return or the single net value is (3.3.16) In particular, if the future value of the investment grows at an unknown rate then the mean γ may not be specified. In a non-informative approach we set SNV equal to zero in (3.3.16) and the resulting growth or financial rate is found to be (3.3.17) We remark that the financial rate defined by (3.3.17) is conservative in nature indicating a small growth in the investment Percentile computation and analysis can be done using the present value function in (3.3.15). Using the normality assumption in (3.3.13) the survival probability or reliability associated with constant c is (3.3.18) for c>0 where Ф(c) is the standard normal distribution function. Under the conservative approach, the future return rates are not speculated in advance, putting (3.3.17) into (3.3.18) yields (3.3.19) for c>0. Using these formulas percentile measurements and prediction intervals can be computed (see Prob. 3.7). Ex. 3.3.3. A investment of $5,000 is to be made where the guaranteed return rate is r=.02. The investor is considering a more risky investment option where the return rate is


90

a normal random variable with mean and standard deviation, per year, given by γ=.12 and β=.03, respectively. Using (3.3.14) the expected value of the investment after 2 years based on the guaranteed rate is SNV=$6,361.97. The median of this present value calculation can be found by applying (3.3.18) and takes the form

We remark that the small increase in the expectation over the median implies a slight skewness to the right associated with the distribution of the present value random variable. 3.3.3 Options Pricing An option is a contract to buy or sell an asset for a fixed price on, or possibly before, a specified data called an expiry. The fixed price is called the strike price. The option to buy is referred to as a call option while the option to sell is a put option. If the holder may exercise the option to buy only on the expiry this type of option is a European call option. If the option to buy may be exercised on a date prior to the expiry it is an American call option. In this chapter we concentrate on call options and the reader is left to adjust formulas for the put option setting. In this section we consider a European call option where the expiry is at future time T=t and the strike price is denoted k. Since the exercise to buy will only occur if the future value exceeds the strike price the present value of the option is (3.3.20) where the indicator function I(A) is 1 if A holds and is zero otherwise. Taking the analytic approach of the single risk model and RC, under a normal distributional assumption for the financial rate (3.3.13), the expectation of (3.3.20) is computed and the single net value found to be (3.3.21) where

and

The statistical analysis of the American call option is more complex and resampling simulation methods in Sec. 7.5.1 are applied to complete risk computations. The celebrated Black-Scholes formula for the pricing of options uses the conservative setting for the growth or financial rate since no future growth rate is assumed. Putting (3.3.17) in to (3.3.21) yields the option pricing formula (3.3.22)


91

where

and

Much work has been done in connection with the Black-Scholes option pricing formula, (3.3.22), that was introduced in Black and Scholes (1973). European call option pricing is often used to approximate the pricing of an American call option. This approximation is applied under the rationale that the option to buy is likely to be close to the expiry. This assumption can be checked by modern resampling simulation methods as presented in Chapter 7. In this analysis we note that no dividends are paid and no commissions are charged. We now consider the alternate to the risk criteria method of model analysis the PC(.25) method for estimating the SNV. To apply the percentile criterion first let c>0. From (3.3.18) if we assume ψt~n(tγ, tβ2) the associated tail probability reduces to (3.3.23) Using PC(.25) we set (3.3.23) equal to .25. Under the condition that P exp(α+ z.75β)>k, we solve for c and find that SNV as (3.3.24) Under the conservative value of γ given in (3.3.17), we find that (3.3.24) becomes (3.3.25) The two model evaluation approaches, namely RC and PC(.25), are demonstrated and calculations are compared in the option pricing example that follows. Ex. 3.3.4. In this example we consider a European call option as described above. Let P=100, t=1, r=.04 and β=.02. The guaranteed future value after one year is

If the strike price is k=105 we calculate SNV under both the RC and PC(.25) methods. From (3.3.22) the Black-Scholes formula using RC yields SNV= .43570. Thus, the option to buy 1000 shares is valued at $435.70 The PC(.25) method, from (3.3.25), gives SNV=.4549 and the 1000 shares of stock are valued at $454.90. We note that consistent with other comparisons the percentile approach yields a greater SNV over the mean or risk approach. The approach used to derive the option pricing formulas are different from many other approaches that use continuous stochastic processes. Further, we remark that the options pricing formula (3.3.21) can be viewed as a variation of the Black-Scholes method where the return rate is estimated to be γ. The results of this computation are explored in Prob. 3.9.


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The pricing formulas and concepts presented in this section can be modified to be applicable to the putt-option setting. In this context the option is to sell an investment for a desired price on or before a future date. This extension is left to the reader. 3.4 Short Time Period Collective Aggregate Models In this section another important application of deterministic stochastic models is presented where statistical analysis is to be done on a collection consisting of more than one financial contract. Similar to the last section these contracts may contain some stochastic components that are not connected to the survival time of the designated model status. However, the time period is considered short enough so that the force of interest as manifested by the financial rate over time, either in the interest or discount function, can be ignored. Applications of collective aggregate or risk models in insurance include works by Butler, Gardner and Gardner (1998), Butler and Worall (1991) and Cummins and Tennyson (1996). 3.4.1 Fixed Number of Variables In the collective aggregate model setting the collection consists of n random variables, such as accident claims, each denoted Xi for i=1, 2,…, n. It is assumed that the number of random variables, n, is fixed and the aggregate sum is given by (3.4.1) Over a single time period without interest, the present value of the expenditure is PVE=Sn and the single net value, SNV, can be computed. Further, we remark that either RC or PC(.25) can be utilized for financial and actuarial analysis. The computation of percentiles and prediction intervals rely on approximation techniques or resampling simulation methods as described in Sec. 1.11, Sec. 1.12 and Chapter 7, respectively. We now apply approximation techniques for aggregate sums. To find the moments of the aggregate sum the mgf technique can be applied as demonstrated in (1.9.7) and (1.9.9). In the general situation there may be contracts of different types signified by different conditions. Let there be m unique types of contracts with associated random variables Xij for i=1, 2,…, nj and j=1, 2,…, m, where n1+…+nm=n. The corresponding moments are denoted for r≥1, and the moments of (3.4.1) are computed by using statistical independence. In particular, the mean and variance of the aggregate sum are (3.4.2) The approximation methods of Sec. 1.11, namely CLT, HAA and SPA, can be applied to approximate the distribution of the aggregate sum. The central limit theorem can be applied using (3.4.2) directly. To apply HAA the first three moments must be computed


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while the SPA requires the mgfs corresponding to the different types of contracts. In the iid case the mgf of the sum is given in (1.9.8) and the SPA can be applied where the aggregate sum (3.4.1) is treated as one random variable. In this scenario the effective sample size is taken to be one. Two illustrative examples follow. Ex. 3.4.1. An aggregate sum of m=5 fire-related house insurance claims is to be assessed. The distribution of the claim amounts is skewed to the right and is modeled using an exponential distribution with mean, in thousands of dollars, of θ=100. For any claim, Xj, for i=1, 2,…so that Var{Xi}=θ2. Using (3.4.2), we find E{S5}=500 and Var{S5}=50,000. Applying the CLT the probability the sum of the claims is more than 1,000 is approximated to be

If we use the SPA as in Ex. 1.11.3, we find S=1,000 and t=(1,000− 500)/(500)=1.0. From (1.11.9), β=½=5, c=(1−1/2)exp(2/2)=1.359141 and σ2=4 and from (1.11.8) the SPA is

If we compute the true probability of the sum exceeding 1,000 as in Ex. 1.11.3, we observe that the SPA gives a better approximation than the CLT. Ex. 3.4.2. In this example there are m types of insurance contracts that, in the case of an accident, pay a benefit of B, where B is a random variable. For contract of type J=j the probability of a claim is qj where the amount is the random variable Bij, for 1≤i≤nj and 1≤j≤m. In this case (3.4.2) becomes (3.4.3)

In particular if the benefit is fixed at Bij=bj then (3.4.3) reduces to (3.4.4)

For example, consider the case of m=3 types of policies where the needed quantities are:


Policy Type

nj

qj

94

E{B1j}

Var(B1j}

1

100

.10

1.0

.09

2

150

.15

1.2

.16

3

90

.20

2.5

.25

From (3.4.4) we find SNV=82.0 and Var{Sn}=135.54. Assuming the normal distribution applies a 95% prediction interval for the aggregate sum is (3.4.5)

which computes as 59.19≤Sn≤104.82. Such prediction intervals are useful in the analysis of collective aggregate models. 3.4.2 Stochastic Number of Variables Often a portfolio of financial or actuarial items such as stocks, bonds and insurance policies may contain a stochastic number of elements. In this section the number of contracts is a random variable, denoted by N, where the associated pdf is given by P(N=n) for n≥1 where the associated mgf is assumed to exist The aggregate sum takes the form of the compound random variable discussed in Sec. 1.12. The corresponding mgf is given by (1.12.4) and the first moment or the SNV is found to be (3.4.6) Higher order moments of the aggregate sum can be computed directly or by using the proper mgf and are more complex. We consider a special compound random variable setting where the probability distribution associated with the number of random variables is Poisson. The number of random variables must be modeled by a discrete distribution. One of the most utilized modeling structures is to assume that the number of variables follows a Poisson distribution with mean λ. The development of the application of the compound Poisson distribution can be found in Seal (1969, Ch. 2). In this situation the Poisson pdf models random processes and the limiting distribution exists (see Sec. 1.12.2)). The resulting mgf is (3.4.7)

From this, applying standard differentiation techniques, we find the mean and variance take the reduced form


95

(3.4.8) Other distributions have been used to model collective risk models over short time periods. Dropkin (1959) and Simon (1960) applied the negative binomial distribution to model the stochastic claim size. An example demonstrating the compound Poisson aggregate sum follows. Ex. 3.4.3. We consider the portfolio collection of Ex. 3.4.1. There are m types of insurance policies with number of policies, claim probabilities and amounts given by nj, qj and bj=E{Bj} for 1≤j≤k. Hence, for J=j the number of claims, Nj, is binomial with parameters nj and qj. If the claim probabilities are small the Poisson approximation can be used and the number of claims made for J=j, Nj, is taken to be Poisson with mean λj=njqj. The total number of claims,

with parameter

To utilize (3.4.8) and approximations, such as the CLT, the moments µj are computed conditioning on the event of a claim. Given there is a claim and the claim amounts, bj, are all unique the probabilities for the claim amounts are (3.4.9) Here (3.4.9) is used to compute the moments by

From the data in Ex. 3.4.2 we find µ1=1.62376 and µ2=3.0673. Using (3.4.8) we compute E{SN}=82 and Vas{SN}=154.8986 and an approximate 95% prediction interval for the aggregate sum is (3.4.10) or 57.606≤SN≤106.394. These prediction intervals are useful in assessing future liabilities connected with the aggregate sum. The above approximation methods used to construct prediction intervals for the aggregate sum is based on normal approximations stemming from the Central Limit Theorem. Other approximations for the case of collective aggregate or risk models exist Borowiak (1999) applied the saddlepoint approximation method to the collective risk models in the case of heavy tailed claim distributions.


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3.4.3 Aggregate Stop—Loss Insurance and Dividends In this section the collection of random variables in the aggregate sum corresponding to insurance policy actions is considered. In either the fixed or stochastic number of insurance contracts situation the insurer may desire to limit the possible benefits that are paid out. Let the aggregate sums corresponding to insurance claims be denoted by S. A stop-loss model takes the form of an insurance policy with a deductible amount d as in Ex. 3.3.1. As in (3.3.6) the total claims paid takes the form (3.4.11)

Let the sum Sd take the form (3.4.11) with pdf given by f(s). The SNV is computed as (3.4.12) This expectation is treated as a function of the value d. An application of this concept demonstrating formula (3.4.12) in the context of the normal distribution is given in Ex. 3.4.4. We remark that in the general context of utility theory, optimal insurance for a single policy takes the form of the deductible model given by (3.4.11). The insurance is optimal if the expected claim amount is given by (3.4.12). This was proved by Arrow (1963) and we refer to Bowers, et al (1997, Sec. 1.5) for details. Ex. 3.4.4. Let the aggregate claim sum S be a normal random variable with mean µs and standard deviation σs. From (3.4.12) we leave to the reader to show that the expectation is (3.4.13)

For example, let µs=10 and σs=2. For the aggregate claims the future obligation is valued at the expectation SNV=10. To reduce this obligation we apply a deductible amount of d=3. From (3.4.13) we find the expected reduction to 7.0001. In insurance theory the expectation of the aggregate sum is referred to as a single premium or in this text SNV using RC. The concepts and applications of stop-loss insurance and resulting premiums have been explored by many authors. References include work on expectation calculation by Bohman and Esscher (1964) and Bartlett (1965) and approximations and bounds by Bowers (1969), Taylor (1977) and Goovaerts and DeVylder (1980). As in the case of single claims in Sec. 3.3.1 the concept of a dividend is explored in connection with aggregate sums. In this context the expected aggregate claim sum or the SNV plays the part of the cost or premium payment A loaded premium for the aggregate sum of insurance claims is greater than the SNV and is denoted by G. The loaded premium takes into account other fixed and variable cost not directly related to the claims. For 0

97

(3.4.14)

If S is a continuous random variable with pdf f(s) the expected dividend, under (3.4.14), is computed as (3.4.15) where the pdf of S is f(s). We note comparing (3.4.12) with d=kG and (3.4.15) a relation between E{SkG} and E{DkG} exists, We notice (3.4.16) Thus, one only needs to compute one SNV in (3.4.16) and the other can be indirectly computed. Ex. 3.4.5. In this example the normally distributed aggregate claim sum presented in Ex. 3.4.4 is discussed in terms of the dividend of the form (3.4.14). Combining (3.4.13) and (3.4.16) we have (3.4.17)

In this example for the aggregate sum, with zero deductible, the SNV is computed as 10. A loaded premium of G=15 is considered. The dividend paid is 20% of G in excess of S as defined by (3.4.14). From (3.4.17), or (3.4.16) noting k G=.2(15)=3, we have SNV=3−10+7.000113=.000113. This is the value of the resulting dividend. For further discussions of stop-loss insurance models in the context of reinsurance or deductible insurance we refer to Bowers, et. al (1997, Sec. 1.5 and Sec. 14.4), We end this chapter with an introduction to a different type of stochastic model. 3.5 Stochastic Surplus Model Another financial modeling application utilizing the loss function concept is now presented. In this section a stochastic surplus model constructed on a purely discrete time period framework is introduced. The financial actions, such as claims and premium payments, occur only at the discrete time periods denoted by tj for j=1, 2,…. Aggregate claims corresponding to time period interval Ej=(tj−1, tj] are denoted by Sj for j=1, 2,…, n,… The claims are financed by an initial amount u and a series of payments of each of value c made at time tj for j≥1. The insurers discrete surplus model is (3.5.1)


98

for j≥1. In this loss function model the aggregate sum is defined as

where Xi are iid with mean µ=E{X1}
where Mw(r) is the mgf of W1. The adjustment coefficient is defined as the value ra where 1=Mw(r), so that (3.5.5) It is interesting to note that theoretically there is exactly one positive real value r satisfying the mgf relation Mw(r)=1 (Rohatgi (1976, p. 624)). From (3.5.4) and (3.5.5) we have (3.5.6) We now compute (3.5.4) evaluated at ra by conditioning on value of Tr. For a fixed n we find (3.5.7)


99

where (3.5.8) Now in (3.5.7) for j
and we rewrite (3.5.7) as (3.5.9)

Since Uj and ΣWi are independent using (3.5.6) we have the bound (3.5.10)

. Combining (3.5.6) and (3.5.9) we have It can be shown that (3.5.8) converges to zero as n increases to infinity and we obtain the upper bound (3.5.11) Further, if the individual losses are bounded where −Uj≤b when Tr=j. In this case the lower bound is (3.5.12) Bounds given by (3.5.11) and (3.5.12) produce an interval estimator of R(u) where the length of the interval depends on the magnitude of b. Further, in some cases the adjustment coefficient can be solved explicitly. This is discussed in the next example. Ex. 3.5.1. Let the discrete surplus model hold where claim amounts Xi ~n(µ, σ2). From (3.5.5), rac=ln(exp(µra+ra2σ2/2) and the adjustment coefficient reduces to (3.5.13)


100

The resulting probability of ruin bound, (3.5.11), is computed as R(u)≤exp(− 2(c−µ)/σ2). Hence, only the moments are required to produce an upper bound on the probability of an eventual ruin. In the situation where the distribution of the claims is not specified but the moments are given by E{X}=µ and Var{X}=σ2 the adjustment coefficient can be approximated, Using the Taylor Series approximation (see Prob. 1.3) (3.5.14) and the adjustment coefficient reduces to (3.5.13). In either case if we desire the probability of ruin to be at most α, for 0<α<1, then from (3.5.11) this restriction is guaranteed to hold if exp(−rau)≤α. A suitable initial fund can be found to be (3.5.15) In particular, if ra follows (3.5.13) where µ=1, c=1.2 and σ=.2 then (3.5.13) yields ra=5. If we desire the probability of ruin to be bounded above by .05 then from (3.5.15), u=.599. Ex, 3.5.2. Claims follow an exponential distribution with mean θ. The adjustment coefficient ra, from (3.5.5), solves rac=−ln(1−θra) or θra=1− exp(−rac). For the exponential case the bound in (3.5.11) is replaced by equality (see Bowers et al. p. 414) so that, R(u)=exp(−rau). For example, if θ=1 and c =2 then solving iteratively we find ra=.797 and R(u)=exp(−.797u). For various initial amounts or values of u the ruin probability is computed where for

Fig. 3.5.1 Exponential Surplus Model

example R(1)=.451, R(2)=.203 and R(3)=.092. A graph of the probability of ruin, R(u), as a function of initial amount u is given in Fig. 3.5.1.


101

In Fig. 3.5.1 we observe that, as expected, as the initial amount u increases the probability of ruin decreases. Further, as the u increases the decrease in the probability of ruin occurs at a slower and slower rate. This example is discussed in the context of a fixed number of intervals further in Chapter 7 using resampling simulation analysis. Our investigation gives an introduction to the topic of stochastic surplus modeling. We limited our discussion to the probability of ultimate ruin as a function of the initial surplus. Authors have explored the distribution of the ruin random variable (see Seal (1978), Beekman and Bowers (1972) and Panjer and Willmot (1992, Ch. 11), There is much research being done on related topics. For example, theory has been developed for the joint distribution of ruin time and the surplus amount after ruin and for a review of the existing results we refer to Gerber and Shiu (1997). In Chapter 7 we demonstrate simulation resampling of the stochastic surplus model where some of these questions can be answered. Problems 3.1 A sum of $1,000 is required in two years and the annual interest rate is i= .05. How much money is needed to be deposited if the interest is compounded a) quarterly, b) continuously? 3.2 A couple wishes to buy a $150,000 house where their initial down payment is $20,000. They can choose between a 20-year mortgage at 11.5% annual interest and a 30year mortgage at annual interest rate at 9.5%. The payments are made at the start of each month. Which do they choose to minimize the interest paid? 3.3 A factory building is sold for $550,000. What are the monthly payments on a 30year loan where the annual interest rate is 8.6%? 3.4 A financial advisor invests $30,000 for five years where the financial rate is an approximate normal random variable with mean .08 and standard deviation .02. a) Using RC and PC(.25) compute the SNV of the investment. b) Compare the RC, SNV to a guaranteed investment at rate r=.03 over the five years. c) Form an appropriate loss function in conjunction with the comparison in b) from the point of view that assumes the guaranteed investment will outshine the speculative investment 3.5 Consider an insurance policy over a short time period that pays a benefit of B in the advent of an accident occurring with probability q. Let q=.05 and B~ n(2000, 2002). a) Using RC and PC(.25) compute the SNV. b) Consider deductible insurance of the form (3.3.6). If the deductible is 200 compute the SNV using RC. c) Find the value of d so that the resulting SNV is only 80% of that computed in a). 3.6 Disability insurance is considered where a fixed benefit of b=1 unit is paid and the growth investment rate is ignored. Let the probability of a first claim be .1 and the conditional probability for consecutive interval subsequent claims be .5. a) Find SNV using RC and Var(X) where X is the claim variable. b) Construct an approximate two standard deviation confidence limit for SNV. 3.7 In Ex. 3.3.2, a) compute the variance of the claim variable X, b) construct an approximate two standard deviation confidence interval for the SNV using RC.


102

3.8 In the investment pricing model presented in Sec. 3.3.2 give formulas for the (1−α)100% percentile points for the present value functions using both (3.3.18) and (3.3.19). How do they vary? 3.9 An investment is made where the guaranteed financial rate is r=.03. The investor considers an investment where the rate is distributed approximately normal with parameters γ=.08 and β=.03. Find a) E{FV(t)} using (3.3.14), b) SNV as defined by (3.3.16) and c) the median as in Prob. 3.8. 3.10 Consider the call option of Ex. 3.3.4. Instead of applying the Black-Scholes formula (3.3.22) utilize the SNV formula of (3.3.21) where γ=.045. How does this compare to the Black-Scholes computation? 3.11 A European call option has an expiry in 2 years and the initial investment is valued at $50. The guaranteed rate is .05 where β=.03. The strike price is 58. a) Using the Black-Scholes formula compute the SNV. b) Using PC(.25) calculate the SNV. How does this compare to a)? 3.12 A portfolio contains 20 investments where each investment is valued between $10,000 and $20,000. Without assuming any special knowledge of the distribution of values estimate the mean and variance for the aggregate value of the portfolio. Approximate a 96% lower prediction bound on the value of the investments. 3.13 A collection of 50 insurance policies is considered over a short time period where for each the probability of a claim is .08. If there is a claim the claim amount follows a distribution with survival function S(x)=(1+.01x)−2.5. a) Find E{X} and Var{X}. b) Find the mean and variance of the aggregate sum of the policies. c) Approximate a 95% prediction interval for the aggregate claim sum. 3.14 For a collection of four types of insurance policies the probability of a claim over a short time period is given by q and the claim amounts are denoted by B. The following data holds: Type n q E{B} Var{B} 1

25

.10

1.2

.31

2

30

.05

1.5

.35

3

35

.08

2.1

.45

4

40

.12

2.3

.55

a) Using RC compute the SNV for the aggregate sum. b) Find the variance of the aggregate sum. c) Find an approximate 95% prediction interval for the aggregate sum. 3.15 In a portfolio of insurance policies the number of claims over a month follows a Poisson distribution with mean equal to 25. Claims follow an exponential distribution with a mean of 1.25 units. a) Find the mean and variance of the aggregate sum of claims. b) Using the CLT approximate a 95% prediction interval for the aggregate sum. c) Using SPA estimate the probability that the aggregate sum exceeds 35 units. 3.16 For a collection of 50 policies the claims of each respective policy follows a normal distribution with mean 1.0 and standard deviation .3 units. a) What is the probability distribution of the aggregate sum of costs? b) Consider deductible policies where d=.2. What is the SNV for the aggregate collection of these policies using RC. How does this compare to a)?


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3.17 Consider the discrete stochastic surplus model where the claims have a mean of 2 and standard deviation of .5 units where c=2.5. For various values of the initial fund compute upper and lower bounds on the probability of ruin. For what initial fund are you unlikely to encounter a ruin?

4 Future Lifetime Random Variable

In this chapter we present the techniques and formulas required to model and analyze stochastic financial and actuarial contracts. Financial contracts that contain stochastic components are predicated or contingent on one or more stochastic actions. As discussed in Chapter 3, these actions are taken when conditions, called a status, change or fail Thus resulting financial and actuarial models are built around a future time random variable. The future lifetime random variable can be either discrete, continuous or a combination of these and it monitors when the status fails. The time the contract is written designates the initial time frame and is denoted by T=0. The future time random variable proceeds forward from the initial time. For example, a life insurance policy or a financial investment may be initiated for a person age 30. The initial time, T=0, corresponds to the individuals age of 30 and the future time, given as a time interval commencing at age 30, denotes the future time when the benefits are drawn or the investment is sold is given by T=t. Associated with the future time random variable are the statistical concepts and functions, such as the df, pfd and survival function, introduced in Chapter 1. Further, statistic inference techniques, such as confidence and prediction intervals, are then constructed based on these structures. In this chapter we explore the theory and formulas concerning the future lifetime random variable that are central in the study of financial and actuarial science. As discussed in Sec. 1.2, the future lifetime random variable may be either discrete, continuous or a combination of discrete and continuous and is associated with statistical distributions and measurements. An additional statistical reliability concept called the force of mortality is introduced. The force of mortality function along with its connections to the future time random variables distribution is also discussed. The concepts and formulas connected with the future lifetime random variable are extended to the multiple variable setting. Applications of the multiple random variable setting are given in the context of multiple future lifetime and multiple decrement models. 4.1 Continuous Future Lifetime In general financial and actuarial modeling the failure of a general status is designated by a random variable. Let X denote such a lifetime random variable defined in a financial or actuarial model where X is considered to be a continuous random variable. For example,

Future lifetime random variable

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X may be the lifetime of an individual taking out a life insurance policy. Let the pdf and df be, respectively, fx(x) and Fx(x) and the survival function be given by Sx(x). If a status, such as the purchase of a life insurance policy, is initiated at X=x the future lifetime random variable associated with the status is defined by T(x)=X−x. The status is often the survival of an individual which, for simplicity, is denoted by (x) with associated future lifetime random variable T(x)=T. In this context we are typically concerned with the conditional or truncated distribution based on surviving to the fixed age x that was introduced in Sec. 1.6. The pdf of T is found from truncating the pdf f(x) at X>x and using a conditioning argument is f(t)=fx(X−x|X>x). The pdf of T is denoted f(t) and the survival function corresponding to T is (4.1.1) The support of f(t) is denoted St and the form of the survival function dictates the structure of calculated mortalities and resulting formulas. We note that all concepts and formulas associated with any survival function, such as those presented in Chapter 1, hold for (4.1.1). We now introduce common actuarial notations concerning probabilities of a status, initiated at time x, with future lifetime random variable T. The probability notations and formulas in Bowers, et al (1997, Ch. 3) are followed in this development. For t>0 the df of T or the probability of status failure within time t is given by (4.1.2) and the associated survival function is (4.1.3) Here l=tpx+tqx and these quantities are often referred to as mortality and survival rates or probabilities associated with the future lifetime random variable. We remark that in terms of conditioning arguments, (4.1.2) is the probability that the status, upon surviving x years, will fail within t years or P(X≤x+t|X>x) and (4.1.3) is the probability that (x) will attain age x+t or P(X>x+t|X>x). In particular, if x=0 this notation is streamlined and the x is suppressed (4.1.4) Further, in commonly used forms if T=1 then in both (4.1.2) and (4.1.3) the l suppressed and we have (4.1.5) The practical interpretation of (4.1.5) is that the status for (x) will fail within one year from initiation and attain age x+1, respectively. Other notations have been used. For


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instance, the probability that the status for (x) will survive t years and fail within the next u years is given by (4.1.6) Following the previous formulas notations if u=1 in (4.1.6) then the u=1 is deleted and the notation is simply t|1qx=t|qx. The basic probability and statistical concepts and formulas introduced in Chapter 1 apply in this setting. Using the idea of conditional distributions presented in Sec. 1.6, other formulas naturally arise. From (1.6.5) the survival function associated with the future lifetime random variable T is (4.1.7) Other survival computations for the status can be formed. For example, the probability that the status survives t additional years but then fails within t+u additional years, using a conditioning argument on (4.1.6), can be written as (4.1.8) These formulas often have practical intuitive interpretations. In (4.1.8) the first factor is the probability of survival of the status past x+t and the second factor measures the probability of the status failing in u years past x+t. A series of illustrative example of these concepts and formulas based on common and arbitrary reliability distributions now follow. Ex. 4.1.1. Let the lifetime variable, X, have an exponential distribution with pdf given by (1.2.9). Then the survival function is

and using the lack of memory property the survival function for T, given by (4.1.7), becomes

From this formula we note that the conditional survival probability depends only on the future lifetime value t and not on the initial age or time x. This is the lack of memory property associated with the exponential distribution. Ex. 4.1.2. Let the lifetime of a status have survival function S(x)= (100−x)1/2/10 for 0≤x≤100. From (4.1.7) the survival and df corresponding to the future lifetime random variable T are

Also, the mortality computation associated with failure of the status between future life times t+u and u, from (4.1.8), is


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For illustration, the person in the above contract is age 40 and buys a life insurance policy. The status in the contract is the survival of the insured person. The probability (40) lives an additional 30 years is

Further, the probability (40) lives 20 years but dies in the next 5 years is computed as

Many other conditional reliability and mortality computations are possible. This section is concluded with an example of a status that has a stochastic future lifetime example taken from a financial investment setting. The example is simplistic in nature but introduces a basic structure that is utilized in future financial assessments. Ex. 4.1.3. A stock investment of initial value P is made where it is assumed that the future value is given by (2.1.4) and the return rate, per year, is fixed at δ. The stock is to be held for at most t years. If the future time of sale is denoted by T and is uniform (see Ex. 1.2.4) over (0, t) what is the expected future value at the time of sale? From the uniform pdf, (1.2.7), and the associated mgf, Mt(δ) given by (1.4.4) the expected future value is (4.1.9)

Thus, if P=$1,000, δ=.1 and t=5 then the expected sale prices is $1,297.44. In this example we see the combination of statistics and economics concepts and formulas that are used in financial analysis. 4.2 Discrete Future Lifetime In many applications we are interested in computations based on the number of whole years a status holds or survives. In this case a discrete random variable is appropriate to model the future lifetime and it is given special attention and nomenclature. The curtate future lifetime is the number of full years a status holds and is denoted (4.2.1) Mathematically, K(x) is the greatest integer function of the future lifetime variable T. Hence, K(x)=K is a discrete random variable with corresponding support Sks and the associated pdf of K(x) given by (4.2.2)


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Further, using a conditioning argument, the individual probabilities for the status (4.2.2), can be written as the product of the probabilities of survival to x +k and then failure during the year of x+k. Using this conditioning the pdf associated with K can be written as (4.2.3) The form of the discrete pdf given by (4.2.3) is quite general and accommodates many applications. In particular, form (4.2.3) is quite useful when the probabilities are listed in a tabular form as is done in actuarial life tables (see Sec. 5.2). As with all probability density function there is associated a distribution function. Since the random variable K(x)=K is discrete the df F(k) is a step function and is given by (4.2.4) The survival function associated with the curtate random variable K. defined in (1.5.1), is

for k≥0. An application of these formulas using a particular discrete distribution is given in the next example. Ex. 4.2.1. For a particular status let the curtate future lifetime K be a Geometric random variable as presented in Ex. 1.3.5. The form of the pdf is given by (4.2.5) Hence, the probability the status fails within one year is f(0)=.1. For positive integer k the probability of survival for at least k years is

for k=0, 1,… Thus, for an individual age x the conditional future survival probability past integer age x+j computed as (4.2.6) for j=1, 2,…. For example, an individual status age 60 with the above survival function computes the probability of surviving at least 5 years as

As before in the continuous exponential random variable case, the lack of memory property for the geometric random variable implies the survival rate or probability depends only on the number of future integer years lived and not on the conditioning age.


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4.3 Force of Mortality A standard concept in the field of reliability and actuarial life modeling is the instantaneous failure or death rate associated with the lifetime of a status. In engineering reliability modeling this is referred to as the hazard function or rate, while in healthrelated disciplines this is called the force of mortality. As in the previous discussions let the lifetime of a status be X with pdf fx(x) and df Fx(x) where we assume X is a continuous random variable with survival function Sx(x)=1−Fx(x). The instantaneous failure rate at time x is called the force of mortality and is defined as (4.3.1) The shape of the hazard function or force of mortality can be used to model mortality distributions (see Nelson (1982, Ch. 4). In Fig. 4.3.1 are the graphs of three types of force of mortality curves.

Fig. 4.3.1 Force of Mortality

In the case of infant mortality or wear-in mortality the mortality curve is decreasing signifying a decrease in the mortality rate with time. If the mortality rate, and hence the force of mortality, is increasing with time then wear-out mortality is present A combination of both infant and wear-out mortality exists for many populations such as human beings and is characterized by the bathtub shaped force of mortality curve. The force of mortality function is unique and determines the distribution associated with the future lifetime random variable. It is interesting to note that the pdf and the df of the future lifetime can be written directly as a function of the force of mortality. First, from (4.3.1) with n>0 we note (4.3.2) Thus, using (4.3.2) the conditional survival probability can be written in terms of the force of mortality as (4.3.3)


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Based on the formula for survival probabilities given by (4.3.3) we see that the force of mortality needs to satisfy certain conditions. In general these conditions imply

for a force of mortality structure to hold thereby producing a viable probability distribution. In the context and structure presented in this section formulas associated with the unconditional future lifetime random variable, X, for an individual can be derived. In this case the original pdf and df can also be written in terms of the force of mortality. If we set the initial time to 0 and take T=X then (4.3.4) Hence, the df is (4.3.5) Standard rules and associations for distributions presented in Chapter 1 hold. Taking the derivative of (4.3.4) the pdf of X is (4.3.6) Also, since the force of mortality can be written as µx=fx(x)/Sx(x)=fx(x)/xpo the pdf of X can be written as (4.3.7) From this development we note that the continuous type pdf can be written as the product of two stochastic quantities. The first being the probability of the survival to time x while the second denotes the instantaneous failure rate associated with time x. In the example that follows the strategy of using the structure of the force of mortality to establish the relevant distributions is demonstrated. Ex. 4.3.1. Let status lifetime X possess a constant force of mortality given by µ. From (4.3.4) (with x=0) the survival function is (4.3.8) which matches the survival function given in Ex. 1.5.2, Hence, if the force of mortality is constant, indicating level instantaneous failure rates, then the corresponding survival function corresponds to the exponential random variable.


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These formulas for the initial lifetime random variable are easily adapted as we move our attention to the future lifetime T=T(x) associated with a general status. Let the pdf and the df of T be given by, respectively, g(t) and G(t). Using the truncated distribution the df is (4.3.9) while the survival function for T is (4.3.10) We remark that (4.3.9) is just denoted by tqx as defined by (4.1.2). Taking the derivative of (4.3.9) with respect to t and using the fact that

the pdf of the future lifetime random variable T can be expressed as (4.3.11) Analogous to (4.3.7) the pdf of T given in (4.3.11) is a product of two functions. The first is the survival function to time t while the second is the force of mortality, or instantaneous failure rate, at time t. An example is now given. Ex. 4.3.2. Consider the lifetime of the status given in Ex. 4.1.2. The general survival function is defined by

Based on initial age of the status X=x, for x<100, from (4.3.1) and (4.3.11) the future force of mortality and pdf are found to be (4.3.12)

for 0≤t≤100−x. Also, the survival function takes the form (4.3.13) for 0≤t≤100−x. These formulas can be used to compute various mortality and survival probabilities as well as descriptive measures associated with the probability distributions. Many applications of the force of mortality exist. For example let the future lifetime be continuous where the pdf is given by (4.3.11). Two useful formulas revealing the structure of the force of mortality are now presented. The conditional probability of failure of a status within one year is


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(4.3.14) We see that the force of mortality is a weighted integral involving the survival probabilities. Also, the force of mortality relates to the conditional survival probability by (4.3.15) These formulas demonstrate basic relations and are utilized in theoretical and application developments. A classical example dealing with the force of mortality is now presented. Ex. 4.3.3. There are classical laws of mortality used in actuarial science and population growth literature. One, referred to as Gompertz Law, takes the force of mortality to be (4.3.16) for constants b>0, c>1 and age variable x≥0. From (4.3.3), for a status age x the probability of surviving an additional t years is a function of (4.3.16) and is written as (4.3.17) From (4.3.11) we compute the pdf of T as (4.3.18) A particular form of the Gompertz distribution is discussed in Prob. 4.7. In application, mortality functions, such as the force of mortality, are defined so as to fit specific populations and may need to be defined differently over disjoint intervals. For example, as represented in Fig. 4.3.1, the force of mortality may take the shape of a “bathtub curve” decreasing at the start, leveling off and then increasing as the future time increases, The decreasing structure is referred to as infant mortality and the rising part is called wear-out mortality (Nelson (1982, p. 26)), Two examples follow where the first demonstrates the multi-ruled force of mortality situation and the second involves a moment computation. Ex. 4.3.4 For a particular status let the force of mortality be defined differently in two disjoint regions and be given by the multi-rule

where a and b are positive constants. We note that the force of mortality µx is decreasing on 01. The survival function using (4.3.4) is defined in two pieces. For 0

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and for x>1

To compute reliabilities from the future lifetime random variable the condition structure holds. For example, in connection with a status age x the probability of survival past future time t, 1
We remark that if x<1 then the conditional survival rate would be a combination of the two rules for the survival rates. Ex. 4.3.5. For a status let the lifetime X have survival function S(x)= 1−x1/2/10 for 0
For example given a status is age 20 the probability the status fails before age 50 is 30q20=1–30p20=.470150. Also, statistical computations that are functions of survival and probability distributions can be made. For example, based on a status age x the expected number of future survival years is computed as

For instance if x=0 then E{X}=100/3 and when x=36 then E{T}=11.333. Other moments are computed using the basic rules presented in Chapter 1. In actuarial and financial computations the pdf and the df of the future lifetime random variable play a central role. When a financial or actuarial model is based on a status with future lifetime given by continuous random variable T the required pdf is given by (4.3.11). In the discrete setting the curtate future lifetime is a discrete random variable associated with a status lifetime and has pdf given by (4.2.3). In this section only basic concepts and formulas required for financial and actuarial modeling were presented. Many other relations among these variables exist.


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4.4 Fractional Ages In the previous sections we discussed continuous and discrete future lifetime random variables. In practice a particular mixture of these random variables is often utilized. Data on failure and survival numbers are frequently compiled and listed over intervals or periods, such as years. This type of data is discrete in nature and is commonly given in actuarial life tables where the interval boundaries correspond to consecutive years. To estimate mortality and survival rates between interval future time periods a continuous stochastic mortality structure is required Hence, reliability and mortality values and the computation of assessment statistics require the distribution of lifetimes between boundary lifetimes. The distribution of continuous future lifetimes between boundaries is often approximated by non-informative techniques. Three possible interpolation techniques have been given by Bowers, et al. (1997, Sec. 3.6) and are linear, exponential and harmonic interpolation. In our discussions we explore only linear interpolation. Let the future lifetime T be continuous and K be the curtate future lifetime. Then we have the decomposition T=K+S where S, 0≤S<1, is the fractional part of the year lived. Based on the distribution of the lifetimes within interval years specific failure and survival probabilities are constructed. If information on the failure rates within intervals is lacking then a non-informative distribution such as the uniform distribution is often applied as a default option. The procedure we present for modeling mortalities and survival rates for fractional ages applies a uniform distribution of fractional ages. Assuming the conditional failure rates within any year, [x, x+1], is distributed as a uniform random variable is referred to as the uniform distribution of death, UDD, assumption. Here, Sx(x)≤Sx(x+s)≤Sx(x+1) and Sx(x+s) is taken to be a linear function in terms of s for 0≤s≤ 1. Using linearity (4.4.1) As a result of (4.4.1) the conditional mortality corresponding to fractional age 0
The probability structure for the continuous future lifetime random variable is given by (4.4.3) Applying the linearity approximation, from (4.4.2), we can write (4.4.3) as (4.4.4) The form of the above pdf is key in financial and actuarial analysis utilizing fractional ages. Based on the form of (4.4.4) the joint pdf of K and S is given by f(k, s)=kpx qx+k where S is uniform over [0, 1], thus implying P(S≤s)=s. Further, this construction


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demonstrates that the random variables K and S are statistically independent. The relationship between independence and the fractional age assumption is well known and for a discussion see Willmot (1998). An example of fractional age modeling in terms of an assumed integer age mathematical structure follows. Ex. 4.4.1. In this example a general status is considered where the associated future lifetime random variable is continuous. The pdf of T is unknown but the probabilities at yearly intervals is given by the geometric pdf given in Ex. 4.2.1 so that

The distribution of fractional lifetimes between integer years is assumed to be uniform between 0 and 1 so that the UDD assumption holds. The probability that the status survives an additional 5−1/2 years, using (4.4.2) and (4.2.6), is

We note a decrease in the survival probability for the additional ½ year as calculated in Ex. 4.2.1 where 5p60=.59041. The structure for fractional ages applied in the previous example plays an important role in applications. In financial and actuarial modeling we commonly utilize a combination of discrete and continuous future lifetime variables. The future lifetime is decomposed as T=K+S where K is the curtate future lifetime, S is uniform on [0, 1] and K and S are independent. This decomposition allows for connections between discrete and continuous computations. In this case the df associated with the continuous future lifetime T is the combination (4.4.5) From (4.4.5) we note that the joint df of (K, S) is of a mixed type (see Sec. 1.2.3). Further, S is assumed to be a uniform random variable on 0≤s<1 then the joint pdf is obtained by taking the derivative of (4.4.4) with respect to s and is (4.4.6) for k=0, 1,…. This formula defines a mixed pdf where K is discrete and S is continuous. Distributions, other than the uniform, can be used to model S where (4.4.5) can be applied. One such distributional alternative is presented in the next example. Ex. 4.4.2. Let the force of mortality be constant. From Ex. 4.3.1 we note the survival function takes the form Sx(x)=exp(−µx), kpx=exp(−µk) and sqx+k=1−exp(−µs). Here the decomposition T=K+S yields the mixed type df (4.4.7)


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Taking the derivative of (4.4.7) with respect to S yields the joint pdf of (S, K). We remark that the modeling of the fractional age using a constant force of mortality differs from that of the uniform approach of (4.4.4). In the previous sections of this chapter the basic modeling concepts and formulas that will be utilized in later chapters associated with one the future lifetime random variable have been presented. There are situations where the general status may depend on multiple stochastic structures. In the next section we investigate the future lifetime distribution connected with two multiple stochastic lifetime structures. 4.5 Multiple Future Lifetimes The mortality or survival of a general status may depend on more than one future lifetime random variable. For example, a husband and wife may take out an insurance policy that pays benefits based on the death of one or the other spouse, In the general setting we have a set of m people with ages x1, x2,…, and xm. Analogous to the one variable case, the corresponding future lifetimes for each individual are T(xi)=Ti for i=1,…, m. The decomposition of the future lifetime for the ith individual, introduced in the previous section, is Ti=Ki+ Si where Ki is the curtate future lifetime and 0≤Si≤1. The random variables Ti are assumed independent for i≥1 and their corresponding order statistics are given by T(1)0 using the independence condition in (1.10.1) the df of T(1) is


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(4.5.2) For JLS the survival function for continuous lifetimes follows from (1.10.3) and is given by (4.5.3) To find the pdf of T(1) we follow the general rules for distributions and take the derivative of (4.5.3) with respect to t. We utilize

and the product rule for derivatives

From (4.5.3) the pdf of T(1) is given by the formula combing a survival probability and a force of mortality function. The pdf is given by (4.5.4) where the force of mortality for T(1) is the sum of the separate forces of mortality (4.5.5) A continuous example follows. Ex. 4.5.1. We consider the setting of Ex. 4.3.2 where individuals’ survival functions are given by (4.3.13). If we have JLS for two people ages x and y, the survivor function, from (4.5.3), computes as

Further, the force of mortality is found using (4.5.5) and is

We remark that the force of mortality adds in a linear fashion increasing the instantaneous mortality rate. This property is utilized in insurance adjustments to mortality rates that model increased lifestyle risks. To investigate the situation of the discrete curtate future lifetime random variable, consider the time interval (k, k+1] for positive integer k. Applying formula (4.5.3), the probability of failure within the interval is (4.5.6) Formula (4.5.6) is general and can be changed to encompass longer length of time intervals. We define the curtate future lifetime of T(1) given by K(1). For nonnegative


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integer k the probability K(1)=k is the probability either x or y will die within one year of k and is given by (4.5.6). Using probabilistic rules, such as independence, the pdf of K(1) is found to be (4.5.7) A numerical example applying the discrete probability formulas follows. Ex. 4.5.2. Consider two people ages x=55 and y=50 where the future lifetimes are independent. The pdf of K(1) is easily computed using (4.5.7). Also, the probability the first death is after 5 years but before 10 years from (4.5.6) is

The relevant survival probabilities are computed for prospective populations by defining or approximating the survival rates for T. In the case of JLS we can compute survival probabilities for fractional lifetimes as discussed in previous section. Here T(1)=K(1)+S(1) and for both variables we assume (4.4.2) holds. Hence, for fraction age s, for 00 the df of T(2) follows from statistical independence of the future lifetimes and is written as (4.5.11)


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From (4.5.11) the survival function of T(2) is (4.5.12) To find the pdf of T(m) we take the derivative of (4.5.12) using

The resulting pdf for T(2) is written as (4.5.13) where the force of mortality is given by (4.5.14) Interval probabilities can be computed. For positive integer k consider the future time interval (k, k+1]. Applying (4.5.12) we can compute probabilities corresponding to interval failure, such as (4.5.15) We note from (4.5.12) that this formulation can be written as a function of JLS and can be adapted to larger intervals. The pdf of the curtate future lifetime of T(2) denoted K(2) can be found explicitly. For nonnegative integer k, applying basic probability laws

Thus the pdf corresponding to K(2) takes the form (4.5.16)

This formula is applied in the next example. Ex. 4.5.3 Let the future lifetimes of people ages x=55 and y=50 be independent. The pdf of K(2) is given by formula (4.5.16). Further, the probability the last death is after 5 years but before 10 years using (4.5.15) and (4.5.12) is

An alternative approach to the development of such computations is to utilize basic set operations and probability rules.


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The case of fractional ages for LLS is now examined where we assume UDD holds for fractional ages. Similar to the JLS case, T(2)=K(2)+S(2) where (4.4.2) holds for both x and y. For 0
As in the JSS, when the lifetimes are independent and the uniform lifetimes with periods are applied (4.5.18) can be used to compute fractional lifetimes for LSS. A general example illustrating the basic formulas and computations for the multiple life setting concludes this section. Ex. 4.5.4. For a status let the survival function be Sx(x)=(1/(1+ .015x))2 where the support is S={x≥0}. Following basic formulas the mean lifetime is computed as

We now investigate an individual status associated with initial age x. The survival function for the future lifetime T is

From (4.3.1) we find that the force of mortality for any time x is

We observe that the force of mortality is a decreasing function of x thus we have an infant mortality structure. Multiple survival probabilities can be computed. For example, consider two statuses defined by ages x=20 and y= 25 where the corresponding future lifetimes are independent. The probability both survive at least 10 years is an example of JLS application and computes as

The probability at least one survives past 50 years deals with LSS and is computed as

It is left to the reader to compute this survival probability. In the last two sections specific forms of multiple life probabilities, based on either JLS or LSS, were presented. In practice many other combinations of conditions on future


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lifetime may be modeled. In the next section mortality rates for alternative conditioned models are introduced. 4.5.3 General Contingent Status In financial and actuarial modeling of a stochastic status based on multiple future lifetimes multiple possibilities exist Financial and actuarial contracts may be written so that financial considerations, such as the amount of a benefit or annuity payments, may be contingent on the order of mortality of the people involved. We consider the simple case of m=2 people of ages x and y where the future lifetimes are continuous and assumed to be independent Many contingent conditions are possible. For example the probability that (x) dies before (y) within n years is (4.5.19) Other common settings occur In a similar manner, the probability that (y) dies after (x) but before n years is (4.5.20) we note that in some specific cases in order to compute the integrals in (4.5.19) and (4.5.20) numerical integration methods may be utilized. Many other contingent probabilities may be required in specific contracts. These probabilities are computed with the aid of the basic laws of probability set forth in Sec. 1.1. This section ends with an example of a contingent mortality computation. Ex. 4.5.5. We consider the setting of the two people ages x and y discussed in Ex. 4.3.2. In the case where the individuals have different earning capabilities the order of death may become important. The amount of the benefits paid may be dependent on the type of status failure that occurs. The probability that (x) dies before (y) but within n years, from (4.5.19) is (4.5.21) The computation of (4.5.21) considered in Prob. 4.12, along with the benefit paid comprise a contingent insurance policy. The contingent mortality computations and resulting survival and mortality probabilities are used in the analysis of financial and actuarial models. These models are based on stochastic statuses and their analysis is discussed in Chapter 6.


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4.6 Select Future Lifetimes In practice the survival function may be dependent not only on the future lifetime random variable but also on the time of a preceding event, such as the construction of a contract or life table. This is because as time changes the force of mortality corresponding to the entire class of relevant individuals may also change at different ages. This concept is observed by noting increases in survival probabilities and mean expected future lifetimes for newborns each advancing year. Hence, a need for the adjustments in survival and failure probabilities, predicated on the date of the construction of the tables, is required. In life insurance there is thought to be lower mortality rates associated with lifetimes closer to the time the policy was written. At that time the class of individuals being insured tends to be healthier than at times away from the initial insuring. These probabilities, based on the selection of a construction point, are referred to as select probabilities. We consider financial contracts where the future lifetime random variable associated with a stochastic status is defined, in part, on the discrete variable, J, indicating the year. The required probabilities are based on a survival function that is a function of two variables. The first variable is the age at which the contract, and hence the life table, was initiated or constructed and is denoted [x]. The second is a future indicator time variable j where the status is examined at age [x]+j. The select survival probability for a status to age [x] +j is denoted Sx([x]+j). Thus, based on the conditional structure (4.1.7), select probabilities of a status with initial age [x]+j surviving i additional years or failing within i additional years are (4.6.1)

As before, if i=1 the i is suppressed in (4.6.1) leaving the notation p[x]+j. For conceptual clarity a mostly hypothetical mathematical modeling example is now presented. The survival function, while somewhat simplistic, is used to demonstrate the relevant concepts and formulas Ex. 4.6.1. The lifetime of an individual buying life insurance takes values between 0 and 100. The year the life insurance policy is written is given by [x] and the select survival function is given by (4.6.2) where 0≤100–1.2[x]−.8j≤1. The survival function (4.6.2) is a function of both [x] and j where the age is [x]+j. From (4.6.1) the survival probability corresponding to i additional years is (4.6.3)


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For example, a life insurance policy is written for a person age 50, so that [x]= [50]. After 5 years has passed the conditional probability the individual survives an additional 3 years is (4.6.4)

We note that the select probabilities are dependent on [x]. For example, in this case consider a person age 55 where the policy was written at age 52 and the likelihood of survival 3 additional years. From (4.6.3), 3p[52]+3=.96531 which, we note, is different than (4.6.4). Both survival probabilities correspond to individual age 55 and survival of an additional 3 years. The concept of select future lifetimes is useful in the construction of efficient mortality and survival rates. The select structure has many modeling possibilities. For instance the structure can also be used to model a time trend in the survival function for individual statuses. The construction of mortality and survival tables for the regular and select conditions are discussed in Sec. 5.5 and Sec. 5.6. 4.7 Multiple Decrement Lifetimes In the construction of financial and actuarial models not only the time of status failure but the particular cause of failure may be relevant. This is the case in pension and retirement plans where the benefits vary for early retirement, age retirement, disability and death. In practice a general status may fail due to one of many causes or decrements. In this case the survival and mortality probabilities are functions of two random variables. The first is the future lifetime random variable and may be either discrete or continuous. As presented in this chapter, in the continuous case this future lifetime random variable is denoted by T while in the discrete setting the future lifetime random variable is referred to as the curtate future lifetime, given by K. The second is an indexing variable that designates the type of failure or decrement that occurred. For m possible modes of decrement we define an indicator random variable J with support Sj={0, 1, 2,…, m} and J=j implies failure by mode J=j for 1≤j≤m. In engineering and theoretical statistical modeling these models are commonly referred to as competing risk models. A history of multiple decrement theory is given in Seal (1977). In the traditional setting the decrements are assumed to be independent (see Elandt-Johnson and Jonhson (1980) and Cox and Oakes (1990)). In more recent work, the dependent decrement model such as the common shock model (see Marshall and Olkin (1967) and techniques employing the copula function, as described by Genst and McKay (1986), have been proposed. The impact of dependent structures among the sources of decrement on actuarial calculations have been investigated by Gollier (1996). Ex. 4.71. A person age x enters a retirement annuity program. There may be many causes for the person to leave this program and collect benefits. In a simplistic setting there may be m=3 different types of retirement. The indicator variable J is defined by; J=1 implies retirement at standard retirement age, J=2 implies retirement due to a disability and J=3 implies retirement before retirement age. Different probabilities of


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survival and mortality due to the different modes need to be modeled since the future benefits paid may vary. The joint distribution along with associated concepts and formulas are now investigated. As in the case of any general random variable there are two major cases, discrete and continuous. A discussion of these types of random variables is now given in detail. 4.7.1 Continuous Multiple Decrements Let the future lifetime random variable T be continuous and let J be discrete with support Sj={1, 2,…, m}. The joint pdf of T and J is of the mixed variety and is denoted by f(t, j). Statistical and mortality calculations fixed at J= j are done by applying the previous concepts and formulas. For example the probability of decrement due to cause J=j on or before time T=t is given by (4.7.1) Other mortality and survival rates or probabilities can be computed using the concepts presented in earlier sections. To utilize the joint random variable setting in modeling, simplifications are needed. The standard rules and concepts for joint distributions as presented in Sec. 1.7 apply in the multiple decrement setting, The marginal dfs corresponding to T and J are given by (4.7.2) for t>0 and jεSj. The probability that the future lifetime is between a and b, a 0, due to all modes of decrement is given by


125

(4.7.5) It follows that in this case the survival probability corresponding to t additional years is (4.7.6) Also, from the marginal distributions given in (4.7.2) the overall mortality probability can be written as (4.7.7) From (4.7.7) we observe that the mortality probabilities due to separate modes of decrement sum to yield the total decrement probability. This concept is used in mortality table construction presented in the next chapter. The concept of the instantaneous mortality rate, or force of mortality, in the multiple decrement setting is now explored. 4.7.2 Forces of Mortality In this section the concept of force of mortality is extended to the multiple decrement setting. The instantaneous failure rate must be extended to the setting where all the forces of decrement are active. The force of mortality in the presence of all modes of decrement is defined by (4.7.8) To model the entire structure marginal settings need to be analyzed. If we consider just the single decrement J=j the force of mortality is given by (4.7.9) Combining (4.7.8) and (4.7.9) we note the summation (4.7.10) Hence, from (4.7.10) we observe that the separate forces of mortality sum to the overall force of mortality for the system. This is useful in integrating combinations of mortality factors that act on a general status, such as an individual’s lifetime. Insurance pricing concerns the assessment of risk factors that adversely effect mortality. As in the single decrement setting the joint distribution of T and J and the conditional distributions can be expressed using the force of mortality functions. The joint and marginal pdfs are given by


126

(4.7.11)

From these formulas using the statistical rules given in Chapter 1 standard statistical formulas are formed. The mortality probability or rates due to the jth mode of decrement associated with t additional years is (4.7.12) Further, the conditional pdf of J given T=t is given by (4.7.13) The example that follows demonstrates these concepts and formulas. The example is an extension of the two decrement example given in Bowers, et al. (1997, p. 312) to an additional decrement mode. Ex. 4.7.2. There are three forces of mortality active that are defined by the separate forces of mortality (4.7.14)

for t>0. From (4.7.10) the total force of mortality computes as the sum (4.7.15) Using the general theory the overall survival probability is (4.7.16)

The joint pdf takes the multi-rule form (4.7.17)

for t>0. The marginal distributions are now found using (7.4.2). First, for the future lifetime random variable T the pdf is


127

(4.7.18) To find the marginal pdf corresponding to J we consider each mode separately. Starting with J=2 we have the integral (4.7.19) To simplify (4.7.19) we first complete the square in the exponent and then write the integral in terms of the normal random variable with mean −3 and standard deviation of 10. We then transform to a standard normal random variable. Letting Φ(z) denote the standard normal df then (4.7.19) reduces to (4.7.20)

Similarly for j=3 we find h(3)=2 h(2)=.200. Lastly, since probabilities sum to one h(1)=1−.1−.2=.7. For the conditional pdf of J given future lifetime T =t, following the definition produces the pdf (4.7.21) A problem is concerning this construction is given in Prob. 4.13. 4.7.3 Discrete Multiple Decrements In some cases the future lifetime variable may be integer valued and discrete. As before let K be the curtate future lifetime random variable associated with a stochastic status. The fully discrete joint pdf of K and J is given by (4.7.22)

As before, we have the decomposition T=K+S and the overall survival and marginal mortality rates, respectively, are

and


128

These forms implie that (4.7.22) reduces to (4.7.23) The joint pdf formula (4.7.23) can be used to compute other basic mortality probabilities. For example the probability of decrement, due to all causes or modes, of a person age x+k within one year takes the form (4.7.24)

After simplification formula (4.7.24) reduces to (4.7.25) This formula is consistent with the additively of separate mortalities given in (4.7.7). 4.7.4 Single Decrement Probabilities In the multiple decrement model setting the different modes of decrement apply varying mortality stresses that can be modeled through the forces of mortality. In the study of the relative values of the decremental stresses the hypothetical elimination of mortality modes, leading to marginal structures, is sometimes useful. In the single or absolute decrement model all modes of decrement are eliminated except for the mode under consideration. In general, the resulting mortalities or rates are not identifiable through a unique probability distribution as many probability structures can produce an identical probability system. For a general reference see Basu and Ghosh (1980) and Langberg, Proshan and Quinzi (1978). In this section the upper and lower bounds on single decrement rates as given by Borowiak (1998) are presented. Further, we will observe that under certain situations, as noted by many sources (for example Jordan (1967)), single or absolute decrement rates can be derived. Using the relationship between the force of mortality and survival probabilities the single or absolute survival probability of a status corresponding to decrement mode J=j of age x to age x+t is defined to be (4.7.26)


129

The single decrement rate or mortality probability is (4.7.27) These single survival and mortality rates are useful in the planning and modeling of future of financial and actuarial systems where present modes of decrement may be reduced or eliminated at a future date. In general we cannot directly observe single failure rates when all forces of decrement are active, Single or absolute failure probabilities or rates are not identifiable in that they are not associated with a unique probability distribution. A lower bound on these rates is easily attained by first noting (4.7.28) It follows for all j=1, 2,…, m that a natural bound is tpxs(j)≥tpx(τ). Hence, we realize tpxs(j) µx+t(j)≥tpx(τ)µx+t(j) and thus a lower bound on the mortality rate associated with one additional year is (4.7.29) for j=1,…, m. To explore further the relationship between these mortality rates distributional assumptions are required. This is taken up in the next section. 4.7.5 Uniformly Distributed Single Decrement Rates Under certain assumptions single failure rates can be directly computed. To do this we assume each decrement has a uniform distribution of death, or UDD, within each year. For J=j and 0≤s≤1, using (4.4.2), we assume (4.7.30) Using the standard definition, the force of mortality for decrement mode J=j becomes (4.7.31) Using (4.7.30) and the survival and force of mortality relationship the single mortality rates for one additional year are computed as (4.7.32) The integral in (4.7.32) can be written as


130

After simplifying we have the direct one additional year mortality and survival formulas (4.7.33)

As noted earlier formula (4.7.33) defines an explicit relationship between single and multiple decrement quantities that is not possible without distributional assumptions. A computational example is given in the next section in Ex. 4.7.4. Reversing the roles of the probabilities is possible, Single decrement rates may be given and the multiple decrement rates are then to be determined. It may be useful to combine single decrement rates under assumption (4.7.30) to form system decrement rates. If we invert (4.7.33) using the single decrement rates as inputs we have (4.7.34) Formula (4.7.34) defines the multiple decrement probabilities in terms of the single or absolute probabilities. To apply formula (4.7.34) in practice we switch the uniform distribution of deaths within interval assumption from the multiple decrement rates to the single mortality rates. For each J=j and 0≤t<1 we assume (4.7.35) For a fixed mode J=j (4.7.36) From (4.7.35)

and we can write (7.7.36) as (4.7.37)

This formula relates the single and multiple decrement rates and is demonstrated in the next example.


131

Ex. 4.7.3. Let m=2 where a uniform distribution of mortality rates within years (4.7.35) holds. For J=j, the mortality probability for one half additional year given by (4.7.37) can be written as

We note that the technique used to derive (4.7.37) can be extended to more than two modes of decrement. 4.7.6 Single Decrement Probability Bounds In general without distributional assumptions, such as the UDD assumption, single decrement rates cannot be computed exactly. In this section bounds on the single decrement rates introduced by Borowiak (1998) are presented. There are m modes of decrement active and no distributional assumptions on the survival or mortality rates are assumed. From (4.7.28) for mode J=j we have the crude bound on one-year mortality probabilities (4.7.38) In (4.7.38) the lower bound is strict in that equality may hold in some theoretical settings, An alternative upper bound on the single decrement rates is now derived. Based on the definition of the force of mortality given in (4.7.8), for decrement J=j, the force of mortality is (4.7.39) Using a Taylor Series expansion and taking the derivative with respect to t we have (4.7.40) Combining (4.7.38) and (4.7.39) produces the form of the single survival probability for one additional year (4.7.41) If tqx(τ)<1 then for some point in (0, 1) then the integral in (4.7.41) is bounded above by (qx(τ))r−1qx(j). Hence, the single survival and mortality rate bounds corresponding to one additional year are given by (4.7.42)


132

In practice when the upper bound for the mortality rate in (4.7.42) is applied it is assumed that it is less than the default upper bound qx(τ). As expected the single decrement rates under the uniform distribution in deaths within year assumption are within the lower and upper bounds. Further, in most cases it is observed that the range for the single decrement rates defined by combining (4.7.38) and (4.7.42) is small. These concepts and formulas are demonstrated in the following example.

Table 4.7.1 Multiple Decrement Probabilities Age

qx(1)

qx(2)

qx(3)

qx(τ)

30

.001

.002

.002

.005

31

.002

.003

.004

.009

32

.002

.004

.004

.010

33

.003

.005

.006

.014

Table 4.7.2 UDD Multiple Decrement Probabilities And Upper Bounds Age

qxs(1)

ub(1)

qxs(2)

ub(2)

qxs(3)

ub(3)

30

.001002

.001005

.002003

.002008

.002003

.002008

31

.002007

.002016

.003009

.003023

.004010

.004028

32

.002008

.002018

.004012

.004032

.004012

.004032

33

.003017

.003038

.005023

.005058

.006024

.006067

Ex. 4.7.4. In this example we consider multiple decrement probabilities for a status where m=3. These decrement probabilities are listed in Table 4.7.1. Both the single decrement calculations based on UDD given by (4.7.33) and the general upper bound, ub(j), given by (4.7.42) are applied to the data in Table 4.7.1. The results are given in Table 4.7.2. First, we remark from the calculations in Table 4.7.2 that the bounds, namely (4.7.38) and (4.7.41), hold for all years. Second, there is observed a close agreement between the UDD estimate and the upper bound given by (4.7.42). In fact the narrowness of the upper and lower probability bounds leads to the application of the UDD approximation for single decrement rates. This strategy can be utilized when analyzing applications involving multiple modes of decrement. Problems 4.1 Let the lifetime of a status, X, have survival function Sx(x)=1−(x/100)2 for 0≤x≤100. Find a) µx, b) Fx(x), c) fx(x), d) P(10≤X≤40), e) For initial time x let T(x)=T be the corresponding future lifetime random variable. Give formulas for g(t) and G(t), f) Given


133

the status lasts 30 years what is the probability it will last an additional 5 years? g) Compute 5|8q20 and interpret this value. 4.2 Given µx=tan(x) for 00 and c>0 (Gompertz), b) µx=kxk for k>0 (Weibull), c) µx=a/(b+x) (Pareto). In addition, for the future lifetime random variable give the formulas for g(t) and G(t) for parts a), b), and c). 4.5 For a status let the integer future lifetime X with pdf f(k)=.9x(.1) for S= {0, 1,…}. A status at integer age x is considered. Find a) for positive integer k, kpx, b) the pdf of the curtate future lifetime K. Further, given the status lasts 10 years what is the probability it will last an additional 5 years. Given the status lasts 15 years what is the probability it will last an additional 5 years. Comment on this. 4.6 Let the pdf of an integer valued random variable J have the form f(j)=exp( −5)5j/j! for S={0, 1,…}. a) Let K=K(x) be the curtate future lifetime random variable conditioned on age x. Find the pdf of K(1). b) If the UDD assumption holds compute 1.5p2. 4.7 Consider mortality rates following the Gompertz Law given in Ex. 4.3.3. Compute the survival function and pdf where b=1 and c=e. The resulting random variable is one of the extreme valued random variables (Nelson 1982, p. 39). 4.8 For a status the future lifetime random variable is T where the notations of this chapter hold. For each of the following give the notations for an individual status age 30 a) survives 40 additional years, b) fails between the ages of 60 and 65, c) fails before age 75. 4.9 Let the lifetime of a status have a survival function given by Sx(x)=10x−2 for 0≤x≤100. a) Give formulas for fx(x) and µx. b) For the future lifetime random variable T=T(x) give the pdf and force of mortality. c) Compute the probabilities defined in Prob. 4.8 using this lifetime distribution. 4.10 The multiple future lifetime setting is considered where the individual lifetimes follow a Weibull distribution with pdf given by survival function S(t) =exp(−tβ/αi) for i=1, 2,…, m. For both JLS and LSS find the corresponding df and pdf. 4.11 Let two people ages x=30 and y=25 have independent future lifetimes where the individual pdfs are given in Prob. 4.9. a) For JLS give the survival function and pdf, b) For LSS give the survival function and pdf. c) Find the probability that the first death is i) after 20 years and ii) after 30 years but before 40 years, d) Find the probability the last death is i) after 40 years and ii) after 40 years but before 60 years. 4.12 Consider the two individual multiple life example of Ex. 4.5.5. Give the mortality formula for the probability that (y) dies after (x) but before n years. 4.13 Consider the multiple decrement model setting where m=2 and the forces of mortality are given by µt(1)=1/10 and µt(2)=t/10 for t≥0. Find a) µt(τ) and tpx(τ) b) the marginal pdf g(t) and h(j). 4.14 A multiple decrement model with m=3 independent modes of decrement follow the partial table given below. Age qx(1) qx(2) qx(3) 40

.011

.010

.015


134

41

.015

.018

.020

42

.020

.022

.031

43

.022

.025

.032

a) For each listed age find the total mortality rate qx(τ). b) Use the uniform model of Sec. 4.7.5 to estimated the single decrement rates for each mode and year listed. c) Compute the upper bound for each single decrement for each year using (4.7.42). d) Comment on the accuracy in using the uniform model to approximate the single mortality rates.

5 Future Lifetime Models and Tables

The future lifetime random variable introduced in the previous chapter is central in the development and analysis of financial and actuarial models that are built on one or more stochastic statuses. The distribution of the future lifetime random variable is required for statistical measurements, analysis and inference. Generally speaking there are two approaches used to model the distribution of the future lifetime random variable. Both techniques can be interrelated and the two methods play a part in the construction of survival and mortality rates and the analysis of stochastic status models. In the first, a theoretical statistical distribution is imposed on the future lifetime associated with a stochastic status. The statistical distribution chosen is based on past data sets or on a hypothesized mathematical structure and unknown parameters are estimated from observed data. In the second, empirical or observed survival and mortality data is utilized to construct survival and mortality rates often in the form of life mortality tables. The resulting empirical rates may be adjusted to fit lifetime modeling assumptions, such as prospective time trends or statistical consistency of mortality probabilities. Typically, the future lifetimes and associated mortality rates are defined in connection with a specified population such as a type of investment, males, females, smokers or individuals of predetermined ages. These populations can be further decomposed into specific risk categories. In practice, tables of survival and mortality rates for specified populations and risk categories are constructed and are used in statistical analysis of financial and actuarial stochastic status models. It is convenient to introduce notations concerning the population size, growth and shrinkage patterns. In this respect, to model a future lifetime random variable a hypothetical set of typical individuals, referred to as a survivorship group, is constructed. Typical survivorship groups for prospective populations are closed, allowing no new members, to new arrivals and mortality data observed using this set of individuals on a year-to-year basis. This modeling structure forms the bases of the analysis of survival and mortality statistics using in the modeling and analysis of stochastic status models. The setting where the population is dynamic and can grow in size from generation to generation is not treated here and we refer to Bowers, et al. (1997, Ch. 19) for development and applications of dynamic populations in actuarial science. In this chapter the basic theory and notations for survivorship groups and their usages are introduced and demonstrated. Specifically, these notations are utilized to construct life models and tables that form the basis of analytic modeling and analysis discussed in


136

later chapters. Applied statistical analyses, such as moment evaluation and prediction intervals, are defined on survivorship life tables. Further, survivorship theory and notations are extended to other actuarial settings. For example in the situation of multiple future lifetimes, as introduced in Sec. 4.5, select, ultimate and multiple decrement mortality tables are constructed. The notations used in this presentation are standard and are found in Bowers, et al. (1997, Sec. 3.3). 5.1 Survivorship Groups We start with a population that is defined in terms of elements with similar characteristics and develop notations to monitor the shrinkage of the population as time advances. Formally, a survivorship group is a collection of initial individuals, sometimes defined as unique statuses, where the lifetimes, X, are continuous random variables each with the same survivor function Sx(x). In this section the group is considered closed in that no new individuals, at a later time, may enter the group. The number of initial individuals is denoted lo and the number of survivors past age x is the random variable Lx. A limiting age is defined as an age w such that Sx(x)>0 for 0≤x
For a population consisting of n individuals

denotes the number of individuals that survived to age x. We remark that Lx is binomial random variables with parameters P(X≥x) and n. If lx denotes the expected number of survivors to age x then, since X is continuous, we have the relationship (5.1.1) Further, using the fact that Lx is binomial with parameters lo and S(x) the variance of Lx is given by (5.1.2)

Future lifetime models and tables

137

The expected number of individuals alive in the survivorship group at each age is used to compute the central moments, survivor and mortality rates presented in Chapter 4. For statistical analysis and inference, time interval survival and mortality rates are required. The interval of time between x and x+n is now considered. Let the random variable nDx denote the number of individuals that leave the group, or in general terms have a failure of status, in the time interval. From (5.1.1) the expected number of individual status failures between age x and x+n is denoted (5.1.3) Similar to the notations presented in the previous chapter if n=1 then the 1 is suppressed and we have the mortality notations for the random variable Dx and the expectation dx. Further, using basic probability laws (5.1.4) Life tables consist of the values of these quantities over fixed age time intervals where commonly the period length of time corresponds to one year The survivorship group statistics and probabilities can be constructed in connection with the force of mortality as defined in Sec. 4.3. If the force of mortality at age x is denoted µx then, from (4.3.1),

Since (d/dx)lx=lo(dSx(x)/dx) then the change in the size of the populations as measured by the derivative is (5.1.5) From the properties of derivatives we observe that lx is a decreasing function in age x indicating a wear-out type of mortality. Also, using the decomposition of the pdf given in (4.3.7) a relation defining the number of decrements at age x is (5.1.6) Further, using the survival probability formula defined in terms of the force of mortality (4.3.4) applied at the initial time t=0 the number of survivors from the group at time x takes the form (5.1.7) Also, the number of individual mortalities associated with future lifetimes between x and x+n is given by


138

(5.1.8) These formulas are applicable in population modeling where either the force of mortality is assumed known or extra mortality is added to a survivorship group. In insurance pricing and theory this is called excess mortality and is useful in assessing high-risk populations. The basic survival and mortality probabilities introduced in Chapter 4 can be computed using survivorship group quantities. For example, corresponding to initial age x the mortality and the survivor rates associated with n additional years can be defined in terms of the survival numbers and are given in the formulas (5.1.9) and (5.1.10) In the theoretical situation where the numbers of survivors in each year are fixed quantities or deterministic, survival and mortality probabilities, using (5.1.9) and (5.1.10), can be computed and tables comprised of these quantities can be constructed. This concept and construction forms the basis of life and mortality tables. For modeling based on observed samples that producing survival numbers the binomial distribution in (5.1.1) and (5.1.2) ensures the consistent estimation of the mortality rates as the sample sizes increase. Hence, for large samples the survival and mortality statistics are treated as deterministic quantities. These formulas can be used to construct various life distributions and life tables depending on the underlying statistical structure, In practical cases the observed or empirical data yields statistics such as Lx and Dx. In the common case these statistics are assumed to converge and the substitution method is followed and the statistics are substituted for their expected values. In this case the observed statistics are treated as fixed constants, such as lx and dx and the actuarial calculations are based on these substituted values. In this situation all subsequent computations are subject to an additional source of variation, which in the presence of a large data set, should be small in magnitude. The amount of variability in these estimators is assessed in Sec. 5.3. Modern simulation methods, such as those investigated in Chapter 7, can be used to ensure proper evaluations. Adjustments of life-tables based on observed values, such as of Lx or Dx, are often made. Mortality values of qx are often adjusted to “smooth” the life table so that the fit of these mortalities is desirable and reflects logical underlying mortality patterns. The smoothing of life-tables is called graduation of the table. The graduation process is often ad hoc in nature and produces mortality probabilities such that (5.1.11)


139

for all x≤0. Further, the difference between the smoothed mortalities and the observed mortalities is to be minimized. A discussion of life table graduation, detailing some of the issues involved, is in given in Sec. 5.6. The choice of the magnitude of graduation is a selection between smoothness and fit to the observed data. 5.2 Life Models and Tables In this section we consider the construction and utilization of life tables that contain survival and failure rates corresponding to attained ages. In practice the ages are often in terms of yearly increments. These tables are conceptually based on a group of individuals in a survivorship group and are called aggregate tables. An underlying structure for the survival rates at other than the stated times, such as fractional ages, may be either known or unknown. For expository and conceptual purposes a model for the group survival rates Sx(x) is implicitly assumed. Based on this conceptual structure life tables are constructed listing values for lx and dx by using (5.1.1) and (5.1.3). In practice, observed survival and mortality rates for various years may be utilized in the construction of the lifetables. For such life tables where quantities are based on empirical data, the observed statistics are assumed to converge to their appropriate expected values. An illustration exhibiting this process is now given. Ex. 5.2.1. For a population with initial number lo=100,000 let the limiting age, a limit by which theoretically all individuals fail, be w=101. The survival function is defined by (5.2.1) A life table consisting of the expected number survivors, lx and the number of

Table 5.2.1 Ages and Expected Survivors and Deaths x

lx

dx

x

lx

dx

x

lx

dx

0

100,000

502

50

70,710

710

70

54,772

921

1

99,495

504

51

70,000

718

71

53,851

936

2

98,994

506

52

69,282

726

72

52,915

954

3

98,488

509

53

68,556

733

73

51,961

971

4

97,979

512

54

67,823

741

74

50,990

990

5

97,467

513

55

67,082

750

75

50,000

1,010

deaths, dx, is computed and given for certain years in Table 5.2.1. It is useful to note that in practice the form of Sx(x) is implicit and is usually unknown but the life table may still be constructed using empirical data. Based on Table 5.2.1 typical probability and statistical questions can be answered, For example: i) The expected number of deaths in the first year is do=lo−l1=100,000− 99,498=502


140

ii) The expected number of survivors to age 50 is l50=70,710. iii) The probability a person age 70 survives another five years is 5p70=l75/l70 =50,000/54,772=.912875. iv) The expected number of people age 50 who will live two years and die in the following year is lo 2|1q50=100,000(3q50–2q50)=100,000(l52−l53)l50= 1,026.72. From this example we observe that the measurements in a life table allow for a wide variety of measurements based on the future lifetime random variable to be computed Another form of a life table lists survival and mortality rates for different individual years. For year x we use (5.1.9) to compute conditional failure rates qx. Utilizing such a lifetable of yearly mortality rates, in combination with a conditioning argument, we can compute some general survival and failure probabilities. For example, the probability an individual status age x survives at age x+n takes the form (5.2.2) In a like manner, for an individual status age x the probability of surviving n years and failure in the next is computed as (5.2.3) Also, the probability of failure of the status within n additional years is found by combining the separate years calculations are (5.2.4)

As we see, the mortality calculations presented in Chapter 4 can be computed using life table quantities that are based on group survivorship theory and data. The other form of a life table that presents yearly survival and mortality rates is now discussed. Ex. 5.2.2. Consider the survival function defined by (5.2.1). For selected years the mortality rates, based on formula (5.1.9), are listed in Table 5.2.2. Based on the above life table a variety of probability calculations are possible. For example: i) The probability a person age 70 survives to age 74 is 4p70= (.98319)(.98261)(.98198)(.98131)=.93095. ii) The probability the status fails for a person age 70 in the third year is 2|1q70 =(.99500) (.99494) (.00512)=.00507. We observe that this life table, since it is based on the same conceptual underlying structure as Table 5.2.1, gives the same information on mortality and survival rates as Table 5.2.1.


141

Life tables can be used in conjunction with other formulas involving the force of mortality to produce adjusted mortality and survival probabilities. The adjustments may be needed to model additional mortality stresses present for individuals in high-risk categories. This is demonstrated in the next example.

Table 5.2.2 Ages and Mortality Rates x

qx

px

x

qx

px

x

qx

px

0

.00501

.99500

50

.01005

.98995

70

.01681

.98319

1

.00506

.99494

51

.01026

.98974

71

.01739

.98261

2

.00512

.99488

52

.01047

.98953

72

.01802

.98198

3

.00517

.99483

53

.01070

.98930

73

.01869

.98131

4

.00522

.99478

54

.01093

.98907

74

.01942

.98058

Ex. 5.2.4. A person age 50 is subject to mortality rates listed in Table 5.2.2 except in year 50–51. In this year the individual is subject to an additional stress or hazard. In the 50th year there is a linear addition to the force of mortality, µx, that starts at .05 and decreases to zero. Hence, in the 50th year the total force of mortality is

for 0≤t≤1. Applying formula (4.3.3), where the survival probabilities are directly related to the force of mortality, the adjusted survival rate for the individual in year 50 is

We remark that this computed survival probability is less than the associated table value of .98995. In this way survival and mortality rates given in life tables may be adjusted for individuals in higher-risk categories who are affected by an additional stress or hazard. 5.3 Estimated Life Models and Tables Life tables, their construction, and utilization are at the center of actuarial models and studies. In this section we give an introduction to the concepts and ramifications of estimating survival and mortality rates using empirical data. There are many research opportunities in the general area of estimation and statistical inference in the context of life tables, Works in the area of actuarial life tables and measurements include textbooks by Neill (1977) and Jordan (1967). There exist other approaches to life table analysis particular to biostatistics and for a review of these we refer to Chiang (1968) and Elandt-


142

Johnson and Johnson (1980). Further, London (1988) discusses estimation topics in life tables. In this section an introduction to statistical considerations in the area of estimated life tables is presented. Basic mean and variance formulas are presented that can be used to develop new statistical measurements. For a starting point to the investigation into the estimation in life tables we consider the survivorship group theory introduced in the previous section. A deterministic survivorship group starting with lo=n individuals is under study where an underlying form of the survival function exists but is not explicitly defined, The survival function and mortality rates are estimated by the empirical or observed data. The empirical data take the form of reporting the number of status failures in disjoint years. For positive integer i the number of survivors to year i is Li. The random variable Di denotes the number of failures in the ith year so that Di=Li−1−Li. The limiting age, after which there are theoretically no survivors, is taken to be the constant m. As mentioned before these random variables are binomial random variables. Based on the structure of the observed data the discrete setting for the future lifetime random variable is considered. For any year we define the nonparametric or substitution estimators for the survival and mortality or failure function. The estimates for the survival and mortality probabilities for k additional years, kpx and qx+k are, respectively, (5.3.1) In a similar manner the estimate of the pdf of the curtate future lifetime K, given by (4.2.3), is just (5.3.2) Under the condition that the lifetime random variable X is continuous the estimate of the curtate survival function can be written as the ratio of the observed survival numbers (5.3.3) In terms of theoretical statistics these estimators are the point estimators for useful reliability and mortality functions. Statistical inference on percentiles and reliabilities under various distributional assumptions, such as confidence intervals, have been presented and for a review we refer to Nelson (1982). Also, typical estimation properties, such as consistency, for these statistics can be explored. The variances of these functions are complicated and can be approximated by large sample theory or by simulation resampling as discussed in Chapter 7. A straightforward statistical structure exists for the initial calculations. Letting x=0 then Lo=lo and the entire data set takes the form of the random vector (Dl,…, Dm). The distribution of this random vector is multinomial with parameters P(j−1≤X

143

(5.3.4)

Further, using the fact that the marginal distributions for the multinomial distribution are Binomial random variables the variance calculation are found to be (5.3.5) Based on these estimators the variance for the survival estimate is (5.3.6) Further, using general theory in statistics the covariance of these estimators, for 1≤i

144

(5.3.9)

The variance of (5.3.9) can be estimated using the presented variance and covariance structure (see Prob. 5.6). Applying these quantities approximate prediction intervals, such as two or three standard deviation intervals, for the future value can be computed. From large sample theory, based mainly in the strong and weak laws of large numbers, the convergence in probability as the sample size increases of quantities (5.3.1)—(5.3.8) is ensured. For this reason observed failure and survival probabilities are often treated as fixed constants. These values are used as basic life table values that are sometimes smoothed to give the life table a good mortality fit. We now turn our attention to statistical measures applied to life tables. 5.4 Life Models and Life Table Parameters Based on the population of individual statuses under consideration a theoretical life model or applied life table is constructed. For fixed or estimated life models and life tables there are measurements that describe population characteristics. In a statistical sense if the life model is fixed or non-random these measurements are referred to as parameters. On the other hand, if the models are estimated these calculations are statistics. We consider the quantities in the life models and tables as fixed without a stochastic component and present common lifetime parameters. 5.4.1 Population Parameters In this section population parameters, such as the mean and variance, introduced in Chapter 1 are constructed in the context of future lifetime random variables. We first consider the continuous future lifetime random variable associated with a stochastic status denoted by T. For an individual corresponding to age x two of the most basic measurements of the future lifetime random variable that yield descriptions of the “center” of the distribution are the median and mean of T. The median future lifetime is the value m(x) such that (5.4.1) Since T is assumed to be a continuous random variable, from the form of the conditional survival function given in (4.1.7), we have the equation (5.4.2)


145

Thus, to find the median we solve (5.4.2) for the constant m(x). The expected value of the future lifetime random variable T is called the complete expectation of life and can be calculated by either approach (1.3.2) or (1.5.3). The resulting two computational forms are (5.4.3) Other moments of T can be computed leading to the computation of additional statistics, such as the variance as well as the skewness and kurtosis as defined in (1.3.6). To be specific, using (1.5.4) the second moment of T can be computed as (5.4.4) and, hence, the variance of T takes the form (5.4.5) These are measurements based on the continuous random variable setting but measurements for the discrete case can also be presented. The discrete future lifetime random setting involving the curtate random variable is considered following two continuous lifetime examples. Ex. 5.4.1. For a particular status the force of mortality is modeled by µx=exp(x). From (4.3.16) we see that this is a variation of the Gompertz Law. Utilizing (4.3.3) we find that the form of the survival probability for future time t related to the force of mortality is given by (5.4.6)

We note, based on (5.4.6) that the survival rates are dependent on both the initial time x and the future time t The median, defined by (5.4.1), is found by setting (5.4.6) equal to 1/2. The median of the future lifetime random variable for an individual age x computes to be

From (5.4.3), the mean of the future lifetime random variable associated with age x computes as (5.4.7)


146

This computation is completed using numerical methods or mathematical integration tables. Ex. 5.4.2. Let the lifetime of a status have survival function given by (5.2.1), namely S(x)=(100−x)1/2/10 for 0≤x≤100. For an individual age x, from (5.1.10), the survivor probability to future time t is (5.4.8) Setting (5.4.8) equal to 1/2 the median of the future lifetime random variable can be readily found. The mean number of future lifetime years, applying (5.4.3), and the median are given by (5.4.9)

We remark that in this case the median is greater than the mean for the future lifetime random variable indicating the distribution of the future lifetimes is skewed to the left. We now consider the discrete setting where the future lifetime random variable associated with a stochastic status is the curtate random variable K. The mean curtate lifetime is found using either (1.3.1) or (1.5.3) to be (5.4.10) Also, from (1.5.4) the second moment takes the form (5.4.11) so that using (1.3.5) the variance of K is (5.4.12) Since the curtate future lifetime is a discrete random variable the computation of the median may not lead to a unique quantity. We now consider some examples that demonstrate these computations based on both an assumed underlying statistical distribution and life table data. Ex. 5.4.2. For a status the curtate future lifetime random variable has the discrete geometric distribution as discussed in Ex. 1.3.5. From (4.2.6), the survival function is given by jpx=.9j for j=0, 1,…. The mean future curtate lifetime is


147

The computation of the variance of the curtate future lifetime is straightforward and is explored in Prob. 5.8. We note that these computations rely only on the specific integer related probabilities. Ex. 5.4.3. For a stochastic status with a discrete future lifetime limited to 6 years Table 5.4.1 lists the number of survivors to specific ages. For a status age x the survival probabilities, based on (5.1.10), survival probabilities are kpx =lk+x/lx. Thus, for a status with initial age x the first and second moments of K are computed by the formulas

Table 5.4.1 Life Table Moment Example x

0

1

2

3

4

5

6

lx

1000

975

850

725

300

250

0

In particular for initial age x=2 we compute E{K}=1.5, E{K2}=3.3823 and Var{K}=3.3823−1.52=1.1323. With these moments two standard deviation, or asymptotically normal, prediction intervals can be constructed. 5.4.2 Aggregate Parameters As in previous sections, we have a deterministic survivorship group of n initial individuals or statuses where mortality and survivor numbers are measured from year to year In this section useful measurements or parameters based on the aggregate of individuals, such as functions based on the total survivors through specific years, are presented. For example, in retirement systems the number of years lived by the entire survivorship group may need to be estimated This is particularly true if the cost associated with the survival of the aggregate collection of individuals is to be assessed. If the data are in the form of a survivorship table the mean and total number of years lived over specified ages are straightforward calculations. In our discussion we consider only the continuous future lifetime random variable setting. For the entire group of individual statuses let the total number of years lived between ages x and x+1 be denoted by T(x,x+1). The expected value of T(x,x+1) is (5.4.13) Formula (5.4.13) has an intuitive interpretation. The first term is the number of years by the survivors to x+1 and the second term is the number of year as lived by those whose status fails between x and x+1. Further, using integration by parts we derive the relation


148

(5.4.14) Formula (5.4.14) relates the aggregate number of years lived to the individual quantities and other calculations can be formed. For example using the force of mortality function the total number of years lived by the group beyond age x is tx given by (5.4.15) The measurements (5.4.13)—(5.4.15) give a good idea of the shrinkage structure of the population as time increases. Other important measurements, different than the aggregate totals, on the survivorship group exist. Expectations are an important measure in the context of aggregate groups. The mean number of future lifetime years of the lx survivors of the group to age x takes the common notation µ(x) and is (5.4.16) Also, the mean number of years lived between ages x and x+1 by the members of the group whose status fails between x and x+1 is (5.4.17) These computations are more directly applicable to theoretical population models, as opposed to observed data sets. Discrete counterparts for these formulas can be constructed using the basic concepts and formulas related to future lifetime random variables and survivorship tables. In any event these computations can be used to obtain population information and we end this section with an example. Ex. 5.4.3. Let a survivorship group consist of 100,000 initial individuals. The number of survivors to age x is lx=100,000 S(x). In particular, let the survivorship function be given by (5.2.1). The number of survivors to age x is given by

Using (5.4.15) the total number of years lived by the aggregate group beyond age x is

Also, applying (5.4.16) the mean number of years lived by the group past age x is found to be


149

for 0≤x≤100. In this hypothetical case of discrete future lifetimes we observe that these measurements are direct statistical functions of the life and mortality tables. To construct continuous population measurements based on discrete life table data fractional age assumptions and adjustments are required. 5.4.3 Fractional Age Adjustments In many applications the fractional age adjustment concept introduced in Sec. 4.4 is often required in the computation of future lifetime assessment quantities. Measurement and parameter calculations associated with a continuous future lifetime random variable may need to be estimated, in part, when the only survival and mortality numbers observable are in the form of a life table. To apply discrete life table data in the continuous random variable case the distribution within periods must be assumed. We assume the common partition of the future age given by the UDD assumption, namely T=K+S where K is the curtate lifetime, S is uniform over [0, 1] and K and S are independent. From discrete life table quantities parameters can be computed for fractions of years survived. For example the probability a person age x survives an additional n+s years, for 0

150

individual lives with respective ages x and y. The following concepts and computations can be applied to more than two individuals by utilizing the general theory of order statistics. In the common setting of multiple future lifetimes, namely that of joint life status, denoted JLS, where the status fails with the first failure, and last survivor status, or LSS, where the status fails upon failure the last individual life tables similar to the ones introduced in this chapter can be constructed. For nonnegative integer k adapting the respective pdfs (4.5.4) and (4.5.13) to the discrete case produces the survival functions associated with k additional years for JLS and LSS are (5.5.1)

Table 5.4.1 Joint Life Status Mortality Rates k

kpxy

kqxy

k

kpxy

kqxy

1

.979805

.020195

20

.595900

.404100

2

.959593

.040407

21

.575697

.424303

3

.939385

.060615

22

.555492

.444508

4

.919188

.080812

23

.535283

.464717

By letting k=1 in (5.5.1) survival and mortality rates can be computed leading to the construction of life tables based on each subsequent year This is demonstrated in the next example. Ex. 5.5.1. We consider the survival function given in Ex. 5.2.1 where Table 5.2.1 holds. For two people ages x=50 and y=51 we construct part of a JLS mortality table. For JLS we use kp50 kp52=(l50+kl51+k)/(l50 l51) and produce Table 5.4.1. From Table 5.4.1 we can answer a variety of probability questions. For example we find; i) The probability that both survive at least 21 years is 21pxy=.575697 ii) The probability the first person dies within 3 years is 3qxy=.060615 iii) The probability the first person dies within 21.5 years is computed using (4.5.9). Noting x=50 and y=51 this probability is

It is left to the reader to construct, with the aid of the previous formulas and example computations, a similar multiple life table for LSS. Quantifying measurements can be applied to these life tables. We consider the most common statistical parameter measurements, namely the mean and variance, in both the discrete and continuous settings. For JLS and LSS for the case of m=2, in the continuous


151

setting the complete expectation of life is given by (5.4.3). For JLS, using the basic moment calculation (5.4.3) the mean of T(1) is found as (5.5.2) Similarly, using the survivor function for the LSS the mean or expectation of T(2) is computed as (5.5.3) As is commonly done, the variance of T(1) and T(2) can be computed by using (5.4.4) and (5.4.5). In the discrete future lifetime random variable case we apply the formula using the survivor function, namely (1.5.3), to compute the expected future lifetime. For JLS the mean of the future lifetime random variable is computed by (5.5.4) Similarly, for LSS, the expected future lifetime is found using (5.5.5) The discrete moment formulas are especially useful in conjunction with multiple life tables since these measurement computations are straightforward. In certain specific multiple life cases simplifications of computations exist. In the case of only two individual statuses with independent future lifetimes simple relationships can be realized between multiple life measurements. For example, since m=2 we have (5.5.6)

Other formulas, such as difference formulas, can be constructed. The formulas given in (5.5.6) can be utilized to compute various quantities used in the analysis of financial and actuarial models. For example, if T(x) and T(y) are independent then the covariance of T(1) and T(2) is (5.5.7) Covariance computation (5.5.7) is done while utilizing only the relevant single random variable distributions. Further, we observe that the covariance defined by (5.5.7) is positive, indicating a positive relationship between the two multiple life statuses exists.


152

Computational examples of these parametric calculations are developed in Prob. 5.9 and Prob. 5.10. 5.6 Select and Ultimate Life Tables In this section we consider the construction of life tables based on a select future lifetime random variable introduced in Sec. 4.6. The resulting life tables can be very accurate and are referred to as select and ultimate life tables. The select mortality probabilities are determined so that the tables reflect desired traits. For example, at the time a person takes out a life insurance policy his or her mortality probabilities should be at a low point. This is due, in part, to the approval process of the life insurance policy, These life table traits will be discussed in more detail later. Tables are constructed by applying the concepts and notations of select future lifetimes introduced in Sec. 4.6 on a deterministic survivorship group. Hence, the two sets of formulas are combined. For instance, at the time the life table is constructed or the contract is written the number of people alive at age [x] is given by the random variable L[x]. After j additional years the number of survivors from the initial group of L[x] individuals is denoted L[x]+j. Hence, the number of survivors at age x is given by L[x] or L[x−j]+j for j=1, 2,…, x. Further, based on the set of individuals with construction age [x] the number of deaths between ages [x]+j and [x]+j+i is denoted iD[x]+j. The notations presented in Sec 5.1 for deterministic survivorship groups are extended to select probabilities. Consider the individuals with construction age [x]. The expected number of these individuals alive at age [x] is denoted l[x] and so (5.6.1) The expected number of deaths of the survivors to age [x]+j that die within i years is (5.6.2) Further, similar to the calculation of mortality and survivor rates using the survival number given in (5.1.9) and (5.1.10), the computation, applying (5.6.1), of mortality and survival rates in this setting are given by (5.6.3) and (5.6.4) Select life tables are life tables that are comprised of the listing of pertinent computations, such as l[x]+j and q[x]+j for positive integers of [x] and j. An illustrative example follows. Ex. 5.6.1. For exposition purposes, a population at the time of construction of the table has the number of survivors for each age defined by


153

for 0≤x≤83. The select survival and failure rates are those presented in Ex. 4.6.1 where the select survival function is given by (5.6.5)

Table 5.6.1 Select Life Tables [x]

l[x]

l[x]+1

l[x]+2

q[x]

q[x]+1

q[x]+2

50

63,246

62,610

61,987

.01006

.01026

.01048

51

62,290

61,644

60,992

.01047

.01058

.01081

52

61,319

60,663

60,000

.01070

.01093

.01117

53

60,332

59,666

58,992

.01105

.01130

.01157

54

59,330

58,652

57,966

.01144

.01171

.01198

55

58,310

57,619

56,921

.01184

.01211

.01242

for appropriated values of [x] and j. A select life table for chosen values of [x] and j uses (5.6.1) and (5.6.3) to compute the relevant survival and mortality quantities. For this example the select life table is given in Table 5.6.1 that follows. From Table 5.6.1 probabilities such as the following can be evaluated: i) The probability a person age [50] survives to age 53 is

ii) The probability a person with construction age 51 dies in the third year is computed as

We remark that a wide variety of probability computations is possible. In practice the formation of select life tables allows for more accuracy in the mortality and survivor computations associated with individuals. This is observed by noting that in select tables more information is utilized. In fact, the select process can be used to model time trend for survival or mortality rates through advancing years. In practice for a table construction or selection age [x] as j increases the failure rates may converge in a uniform manner. To be precise for positive integer r and small positive constant α<0 we may have (5.6.6)


154

for all ages of selection [x] and j≥1. Thus, for all fixed initial ages [x] the future failure rates associated with subsequent years converge to ultimate failure or mortality probabilities as the future time increases. If r is the smallest positive integer such that (5.6.6) holds then the select tables can be truncated by taking (5.6.7) for i=r, r+1,…. The first r years are called the select period and the convergent rates qx+i make up the ultimate table. The combination of select

Table 5.6.2 Select—Ultimate Life Table [x]

q[x]

q[x]+1

q[x]+2

qx+3

40

.010

.013

.015

.017

41

.014

.016

.018

.019

42

.018

.019

.020

.023

43

.021

.022

.024

.027

44

.024

.026

.028

.030

probabilities and the convergent quantities make up a select-ultimate life table. An example of a select-ultimate table now follows. Ex. 5.6.2. Part of a select-ultimate table with a three-year select period is given in Table 5.6.2. The failure rates are hypothetical but the listed values converge by criteria (5.6.6) with α=.001, In a full table other columns corresponding to qx+i would be listed for i≥4. The ultimate table values are in the qx+j columns for j=3, 4,…. From Table 5.6.2 we see various decrement probabilities listed for individuals age 43, namely q[43], q[42]+1, q[41]+2 and q40+3. Also, for future lifetimes greater than 2 years past [x] these mortality rates are function of the future age and not [x]. Mortality probabilities given in select and ultimate life tables are often smoothed or graduated to create a table that reflects a reasonable mortality pattern. Guidelines for the smoothing or graduation of mortality tables have been discussed by the Society of Actuaries (see SOA (2000) and SOA (2001)). It was noted that empirical life tables are often adjusted to achieve common mortality modeling goals. For example, select life tables are often adjusted so that, except for extreme circumstances such as the very young, mortality rates should increase with age. Thus we desire life tables where (5.6.8)

Also, the mortality at an attained age should increase with the length of time, or duration, since issue or (5.6.9)


155

Table 5.6.3 Graduated Select-Ultimate Life Table [x]

q[x]

q[x]+1

q[x]+2

qx+3

40

.010

.013

.018

.020

41

.013

.017

.019

.023

42

.016

.018

.022

.026

43

.018

.021

.024

.027

44

.021

.024

.027

.030

Using the constraints given by (5.6.8) and (5.6.9) select and ultimate tables are formed using expert opinion and ad hoc methods. An example of table graduation now follows. Ex. 5.6.3. In this example we consider the select table given in Table 5.6.2. The mortality values increase within the rows and columns as the age increases so that (5.6.8) holds. However, condition (5.6.9) does not hold and the mortality rates need some adjustment. In this case mortality rates are not a minimum around initial ages. Moving along diagonal entries, according to (5.6.9), we adjust the entries about their mean to form an adjusted table given in Table 5.6.3. We remake that there are many possible adjustments that could be made and the adjustments are done so as to minimize the change in the select mortality rates. These mortality tables are considered to have a better underlying mortality pattern where inconsistent mortality computations do not arise. 5.7 Multiple Decrement Tables Another area of application of group survivorship concepts and notations involve the construction of life and mortality tables in the multiple decrement setting. In analyzing mortality or status failure data with possible competing causes the mode of decrement needs to be evaluated in terms of its likelihood. Each person in the survivorship group, initially comprised of lo individuals, is subject to decrement by one of m causes or modes. Life models and life tables are constructed for the multiple decrement model introduced in sec.4.7. A continuous future lifetime random variable and model is assumed where for an individual age x the decrement J=j the force of mortality is µx(j). The corresponding pdf is f(t, j)=tpxµx+t(j) for t > 0 and j=1,…, m. The other notations presented in Sec. 4.7, such as the overall survival probability tpx(τ) are used throughout this discussion. As in the case of a general survivorship group the number of survivors to age x is denoted by the random variable Lx. In the multiple decrement setting the expected number of survivors using the over all force of mortality is given by (5.7.1)


156

For the interval of future lifetime between x and x+n the number of people who leave the group due to decrement J=j is denoted nDx(j). The expected number of individuals who leave the group due to decrement J=j between x and x+n is given by (5.7.2) Using the fact that survival rates or probabilities can be partitioned as s+xpo(τ)= xpo(τ) spx(τ) for one year (5.7.2) becomes (5.7.3) From (5.7.3) we see that the expected number of decrements due to mode j in the year x is the product of the number of individuals alive at the start of the year times the jth decrement yearly mortality rate. These quantities are conceptually model based and can be computed when the data is table-based or a function of observed mortalities and table graduation. An example demonstrating these concepts is given in the next section. The decrement quantities related to the entire mortality system can be computed. Consider a multiple mortality system where all modes of decrement are active. The total number of decrements, due to all causes, between ages x and x+n and the corresponding expectations are (5.7.4)

If n=1 we have the total number of expected decrements to be (5.7.5) Here, qx(τ) can be viewed as the annual total decrement from x to x+1 attributed to the force of mortality µs for x≤s≤x+1. The construction of multiple decrement life tables based on these concepts and formulas is now explored. 5.7.1 Multiple Decrement Life Tables In this section the construction of multiple decrement life tables is investigated. Life tables can be constructed listing an overall decrement model as well as each separate decrement. A first type of multiple decrement table may list the number of survivors, lx, along with the number of decrements for each mode, dx(j), for j=1, 2,…, m, for ages various ages x. Based on these initial quantities a second decrement table may list the decrement probabilities for each decrement type and overall given by


157

(5.7.6) As before the overall survival probability combines failure numbers of the various modes and is computed by (5.7.7) The use of these formulas in the construction of multiple decrement life tables is demonstrated in the next example. Ex. 5.7.1. In this example a survivorship group consists initially of l50 =1,000 individuals where there are two modes of decrement, so m=2. A decrement table, Table 5.7.1, lists the number of survivors along with the number of failures by decrement per year. Based on the data given in Table 5.7.1 basic probabilities can be computed. For example: i) The probability an individual age 50 lives to age 55 is computed in a straightforward manner as

ii) The probability a person age 50 dies at age 54 by mode J=2 is

iii) The probability a person age 50 dies within two years by mode J= 1 is Many other useful probabilities concerning multiple failure mode models can be found. A second multiple decrement table listing the individual decrement rates corresponding to each competing decrement mode for each year is now constructed using the mortality and survival totals presented in Table 5.7.1. The overall and multiple decrement rates are computed using the basic concepts and are given in Table 5.7.2. Based on Table 5.7.2 basic decrement and survival probabilities can be computed. For example:

Table 5.7.1 Numbers of Survivors and Decrements for Selected Years x

dx(1)

lx

dx(2)

x

dx(1)

lx

dx(2)

50

1000

10

15

55

859

15

20

51

975

11

16

56

824

16

21

52

948

12

16

57

787

16

23

53

920

13

17

58

748

18

25

54

890

13

18

59

705

20

27


158

i) The probability a person age 55 lives more than two years is computed by using the yearly decomposition

ii) The probability a person age 52 survives two years and dies in the third year due to mode J=1 is

iii) The probability a person age 52 dies with in two years by mode J= 2 is Many possibilities exist for the computation of mortality and survival rates based on these tables. In practice Table 5.7.1 and Table 5.7.2 might be merged into a larger multiple decrement table. We note the overall decrement rate given in (5.7.6) could be listed in Table 5.7.2. 5.7.2 Single Decrement Life Tables In practice a financial or actuarial modeler might want to consider the changes in mortality patterns under the hypothetical elimination of some modes of decrement. Taking the approach of Sec. 4.7.5 and Sec. 4.7.6 single decrement mortality rates can be either bounded or, under necessary conditions, computed. In the construction of single decrement tables in this section we take both approaches. First, we assume the uniform within year distribution of

Table 5.7.2 Multiple Decrement Rates by Year qx(1)

qx(2)

px(τ)

x

qx(1)

qx(2)

px(τ)

50

.0100

.0150

.9750

55

.0175

.0233

.9592

51

.0113

.0164

.9723

56

.0194

.0255

.9551

52

.0127

.0169

.9704

57

.0203

.0292

.9505

53

.0141

.0185

.9674

58

.0241

.0334

.9425

54

.0146

.0202

.9652

59

.0284

.0383

.9333

x

Table 5.7.3 Single Decrement Rate Computations and Bounds Year

Single Decrement Rates

Upper Bounds

Lower Bounds

qxs(1)

qxs(2)

J=1

J=2

50

.0101

.0151

.0100–.0102

.0150–.0153

51

.0114

.0156

.0113–.0116

.0164–.0167

x


159

52

.0128

.0170

.0127–.0130

.0169–.0173

53

.0142

.0186

.0141–.0145

.0185–.0189

54

.0147

.0204

.0146–.0150

.0202–.0207

55

.0177

.0235

.0175–.0181

.0233–.0240

deaths, UDD, holds and thus formula (4.7.33), yielding exact mortality rates, applies. A second approach is to entertain no assumptions and apply the upper and lower bounds given in Sec.4.7.6. These procedures are demonstrated in the next example. Ex 5.7.2: For expository purposes the multiple decrement mortality calculations given in Table 5.7.2 hold. In the table that follows the calculated single mortality rates given by (4.7.33) and the upper and lower bounds given in (4.7.42) are computed. From Table 5.7.3 we remark that the difference between the lower and upper bounds is that small and as expected captures the estimated rates. This implies the uniform assumption decrement rate bounds are a good estimate of the single decrement rates as not much error is possible. If the single decrement rates are computed or treated as given quantities tables for the multiple decrement rates can be also be constructed This is accomplished by applying a fractional age assumption, such as UDD, and formulas similar to (4.7.34) and (4.7.37). Based on these tables model evaluation formulas can be computed in a hypothetical or scenario setting. Problems 5.1 Let future lifetime T have df F(t)=t/(100−x) for 0≤t<100−x.=1 for t≥ 100−x. Find a) E{T}, b) Var{T}, c) median of T. 5.2 Let the force of mortality be defined by µt=t for t>0. Find a) the pdf f(t), b) E{T}. 5.3 Let X have survival function S(x)=(1−x/w)a for 0≤x≤ w and a>0. Find a) tpx, b) E{T}, c) Var{T} 5.4 For a status over a limited time period a life table is given by: x lx dx x lx dx 0

500

3

280

1

495

4

100

2

365

5

0

a) Find dx in the table. b) Find the probability a person age 1 lives 2 years and dies in the 4th year. c) Approximate the probability a person age 2 survives at least 1.5 years. d) If x=1 find E{K} and Var{K}. Let the lifetimes corresponding to individuals ages x=1 and y=2 be independent. e) What is the probability they both survive at least one year? f) What is the probability at least one survives more than 2 years. 5.5 Using the discrete hypergeometric distribution show (5.3.4) through (5.3.8) hold (see Hogg and Craig (1995, p. 121).


160

5.6 In Ex. 5.3.1 approximate the variance of (5.3.9) utilizing (5.3.4) to (5.3.8) applying the variance structure in given (1.7.11). 5.7 Consider the Pareto distribution form of the force of mortality given by µx= a/(b+x) for positive constants a and b and x>0. a) Find tpx. b) Compute E{T} and m(T). c) What does the values of a and b indicate about the shape of the distribution? 5.8 For the discrete geometric distribution given in Ex. 5.4.2 compute the variance of the curtate future lifetime random variable. 5.9 For a closed survivorship group the number of individual statuses alive at age x, denoted by, lx, is listed below: x lx x lx x lx 0

1,000

4

750

8

225

1

995

5

635

9

110

2

931

6

495

10

51

3

860

7

350

11

0

a) Construct a mortality table listing dx, qx and px for each age. b) Find the probability a status age i) 2 fails within 2 years, ii) 4 survives at least 4 years. c) For a status age 3 find the mean and variance for the curtate future life time random variable. d) For a status age 3 compute the expectation of the continuous future lifetime random variable using UUD. 5.10 Consider a status setting where the lifetime survival data is given in Prob. 5.9. Two independent statuses are considered where the respective ages are 2 and 3. a) Construct mortality tables for both JLS and LSS. b) Find the mean of the curtate future lifetime for both JLS and LSS. 5.11 Consider the graduated select-ultimate life table given in Table 5.6.3. Find the probability a person with initial age 40 i) is now age 41 survives 2 additional years, ii) is now 4 and dies within 3 years. 5.12 For a status there are m=3 modes of failure. A multiple decrement survivorship table is listed below: x 0 1 2 3 4 5 6 7 lx

100

dx

(1)

3

4

6

8

10

6

2

1

dx

(2)

2

3

4

10

12

4

1

0

dx

(3)

1

3

2

6

8

2

1

1

a) Fill in the table and list the decrement mortality rates qx(j) for j=1, 2, 3 and qx(τ) for each year. b) Find the probability a status i) age 2 dies by age 5, ii) age 1 survives 2 years and the dies by mode 3. c) Find the expected future lifetime of an individual age x=1 5.13 Apply the UDD assumption to the multiple decrement setting given in Prob. 5.12 to compute the single decrement rates for each of the three modes, Which single decrement model has the greatest mean future lifetime? How good is the single decrement model for J=1?

6 Stochastic Status Models

In this chapter we investigate the type of financial and actuarial analysis that is required for dynamic economic structures. This is the case whether we are determining when to sell or buy a stock or how much a premium should be in connection with a whole life insurance policy for an individual in particular risk categories. In financial and actuarial modeling and analysis statistical models are developed that depend on stochastic events or actions. As described in Chapter 4 these actions are initiated by the change or failure of existing conditions, generally referred to as failure of a status. This is conceptually different than in Chapter 3 where the statuses were deterministic in structure. In this chapter the status is stochastic in nature where the lifetime of the status is defined by a future lifetime random variable. The concepts and basic formulas concerning the future lifetime random variable presented in Chapter 5 play a central role in the construction and analysis of these stochastic models. In general terms, these models are referred to as stochastic status models. In stochastic status modeling and evaluation, statistical analysis is done in conjunction with the future lifetime random variable. A best decision, whether it is a choice between types of insurance policies or the selling price of a stock, must be determined based on the possibilities, often in terms of parameter values, and their associated likelihoods. In theoretical statistics these concepts fall under the heading of decision theory where a decision criteria is constructed and utilized to judge prospective actions. In insurance modeling, utility theory (see Bowers, et al. (1997)) encompasses the theory of constructing decision making criteria. In this chapter general stochastic status model criterion are introduced and applied to practical settings and data. In connection with the proposed model decision criteria statistical concepts and techniques are presented and demonstrated that lead to useful statistical inference, such as point estimation, confidence and prediction interval estimation. These techniques when applied in an actuarial science settings lead to standard analysis of life insurance and life annuity models. Any stochastic action model has associated with it an origin or starting date. At the beginning of the contract or modeling action the initial age, X=x, is associated with the stochastic model. For example, in actuarial science the analysis of a whole life insurance policy is predicated on the age of the person at the time the contract is written. The benefit is paid upon the failure of the status, or when the person dies. In finance, a stock is bought at time T=0 and is sold when the price reaches a threshold limit. In conjunction with concepts and nomenclature previously introduced, these financial actions are


162

initiated by what is called the failure of a status and the survival lifetime of the status is measured by the future lifetime random variable discussed in Chapter 4. The modeling and analysis techniques, such as life-tables, probability distributions and fractional age developments introduced in previous chapters and section are applied to these structures. In this chapter the stochastic present value function, based on the future lifetime random variable, is defined for a variety of financial and actuarial models. Based on the present value function statistical criteria are developed, namely risk and percentile criteria in conjunction with the equivalence principle. These criteria are utilized to evaluate stochastic financial and actuarial actions Based on these methods the basic forms of life insurance and life annuities, in both the discrete and continuous settings are discussed and methods for premium calculations are presented. Standard relationships between these actuarial models are developed and demonstrated. More advanced topics in actuarial modeling, such as reserve computations, general time period models, multiple decrement policies and pension plan modeling make up the rest of the chapter. 6.1 Stochastic Present Value Functions In this section we present the basic forms that are the basis for stochastic status modeling. A financial action is initiated by the failure of a stochastic status that corresponds to a future lifetime random variable as introduced in Chapter 4. The associated continuous future lifetime random variable is denoted T. The future value of the action at time T=t, FV(t), is evaluated at initial time or T=0. The continuous interest model (2.1.9) is applied where the financial rate is either the return or interest rate. In this section we consider the simplest setting, the financial rate is consider to be fixed over the life of the contract and is denoted by δ. Thus, with fixed rate δ and taking the notation common in actuarial science V=(1+i)−1=exp(−δ) the present value is (6.1.1) We note that (6.1.1) is comprised of two pieces, the future value quantity and the discount operator, given by Vt. This model is general and can be adapted to a multitude of settings. In this continuous future lifetime setting the distribution of T is modeled by either a theoretical statistical distribution or empirical driven life table calculations. To apply the present value function (6.1.1) a breakdown of the formula is often required As we have seen in many cases of financial and actuarial modeling the stochastic structure of the future lifetime random variable can be partitioned into an integer and a continuous fractional part. The curtate future lifetime is denoted by K and the general decomposition of the future lifetime random variable T is T=K+S where the support of S is S=[0, 1]. The distribution of S models the fractional ages within years as discussed in Sec. 4.4 and depends on the structure of the financial contract. The resulting present value takes the form (6.1.2)

Stochastic status models

163

The pdf of S takes support inside unit interval and for most applications we assume S and K are independent In some settings the distribution of S is continuous and in some cases the distribution may be discrete. In financial or actuarial stochastic status models, present value functions are combined to form general loss functions similar to the loss models presented in Sec. 3.1. The difference here is that the loss function depends on the survival of a status that is a random event This additional stochastic status presents an extra source of variation to the modeling. As introduced earlier, the analysis of financial and actuarial contracts and models depends on the criteria used in the evaluation. The criteria utilized in coming sections are the risk and the percentile criteria introduced in Sec. 3.2. The chosen criteria are applied to the relevant present value functions. 6.2 Risk Evaluations The first modeling criterion we consider concerns evaluating the likelihood and associated results of stochastic actions. In this section we compute the expectation or the risk of the present value function that comprise the RC modeling method. The form of the expectation depends on the structure of the status as defined by the model. In some models the financial action takes place only at the end points of a time interval, This occurs in some discrete models where monetary payments are at one of the time period endpoints. For example insurance premium payments may be due at the start of each sixmonth time period. In others, namely continuous models, the financial actions are immediate and may occur within time periods. In any event the single risk notation of Sec. 3.3 is used where the expectations are denoted by the single net value, SNV. Higher order moment calculations follow the general rules for moment computation given in Sec. 1.3. Further, in this section the financial rates are assumed fixed over the length of the contract In the case of stochastic financial rates the modeling and analytic techniques of Sec. 2.3 can be applied. In the next two sections we introduce basic concepts and formulas for both the continuous and discrete time period models that form the basis for the analysis of stochastic status models. 6.2.1 Continuous Risk Calculations The first type of stochastic status model we consider is the type in which the lifetime of the status is continuous in nature and leads to a continuous future lifetime random variable, For these types of models the pdf of T along with unknown parameters is either assumed or fully estimated. As we have seen in previous sections the future lifetime random variable is associated with the initial age of the status X=x and the support of the pdf of T in conjunction with the insurance conditions is denoted by St. Straightforward computations of the expectation of (6.1.1) and using the form of the pdf of T given in (4.3.11) give the single net value formula (6.2.1)


164

The quantity in (6.2.1) represents the expected future value at time T=0. The integrand is a combination of three components, namely, the future value quanity, the discount operator and the pdf. This formula is general in nature and can be applied to a variety of situations. In many important specific cases (6.2.1) can be simplified to yield standard computations. This section concludes with an example that demonstrates how a stochastic status can be achieved in investment modeling. Ex. 6.2.1. An investment is to be sold at a future date for $1,000 where the return rate is estimated to be δ=.1. Due to other considerations the investment will be sold at any time at or before 4 years. The future lifetime random variable, T, is the time to sale and may follow any probability distribution. What is the expected present value of the sale? From (6.2.1)

where M(a) is the mgf of T as defined in Sec. 1.4. For demonstrational purposes we assume the distribution of the future lifetime is uniform, with pdf (1.2.7), on (0, 4), Using the uniform mgf, (1.4.4), we compute the expectation as

Hence, the value of the investment considered as one lump sum evaluated at the initial time is $824.20. 6.2.2 Discrete Risk Calculations The second basic stochastic status model is one where the time to failure of the status is modeled by a discrete random variable. In this type of model the stochastic action occurs at only one point, such as the start, end or, often as an approximation, the middle of the time interval. Risk computations involving discrete future lifetime random variables can be derived utilizing the decomposition of the present value function given in (6.1.2). In this simplest case S takes all its probability at one point which we denote by so. For example, if the financial action occurs at the end of the period then so=1 and P(S=1)= 1. Likewise if the action occurs at the start of the period, so=0 and P(S=0)=1. The pdf of K is given by (4.2.3) where the support of K is Sk and the expected future value or the single net value is (6.2.2) The risk computation (6.2.2) is completed by specifying the discrete pdf. This can be done by applying a pdf formula or life-table measurements. As in (6.2.1), the discrete type expectation is a combination of the future value quantity, discount function and likelihood associated with each sequential time period. An example of a discrete risk calculation and analysis in the financial investment setting is now given. Ex. 6.2.2. A stock investment is to be sold at the end of the first month its future value reaches a price denoted by the fixed quantity fv. The return rate is δ=.1 per year and over the next five years the probability of sale is equal or 1/60 for each month. Here, Sk={0, 1,


165

2,…, 59} and P(Sk)=1. The single net value, using the summation formula (1.3.11) with (6.2.2), is found to be

Applying V=exp( −.1/12)=.9917 we compute the single net value as SNV= .783636 fv. Thus, using the risk or expectation approach the present value of the investment is estimated at 78.36% of the sales price. We keep in mind that this calculation assumes the probability structure of a sale is discrete uniform. The computational examples given so far have demonstrated the versatility in general stochastic status models. These models are applied as parts of overall structures to analyze stochastic economic events such as financial strategies and insurance and annuity models. As an additional modeling structure we consider mixed type future lifetime random variables and their associated computations as these models form the theoretical basis of some important relations and actuarial measurements. 6.2.3. Mixed Risk Calculations Mixed type random variables play an important role in the modeling and analysis of financial and actuarial models. For example, in some contracts the financial actions take place in a continuous fashion but the pdf of T is not directly modeled. This is the case when life table data is applied in conjunction with a continuous stochastic status model. Using the decomposition T=K+S, the discrete part of T, namely K, has a pdf that is modeled by life table quantities while the pdf of S is to be assumed. For this purpose the UDD assumption is often imposed where S is uniform on [0, 1] and independent of K. From (6.2.1) using the mixed type pdf given in (4.4.4) the general single net value formula is given by (6.2.3) This expectation is general and can be applied to many modeling settings. One common modeling application occurs in the actuarial science where formula (6.2.3) is used to relate discrete and continuous types of insurance models and annuity computations. Typical reductions exist for (6.2.3). We now consider a common setting, present in life insurance policies, where the future value is a constant In particular we take FV(k+s)=fv, for all k≥0 and 0≤s<1, and consider the decomposition of T where the discrete part takes the atomic probability structure P(S=so)=1 where so=1. To do this in (6.2.3) we form Vk+1+s−1 and note the simplification


166

Applying the relation between financial rate forms given in (2.1.5) the general expectation (6.2.3) reduces to (6.2.4) In (6.2.4) the relation between discrete and continuous forms of expectations is plainly demonstrated. This is central in the development of formulas defining the relationships between discrete and continuous stochastic status models. A computational example follows. Ex. 6.2.3 Consider the stock sale setting introduced in Ex. 6.2.2 where the sale can take place at any time in the month. Here, for each month the rate is δ12=.1/12=.008333 and from (2.1.5), i12=exp(.008333)−1=.008368. Using (6.2.4) and the results of Ex. 6.2.2 we find

Thus, this stochastic status investment model has a present value expectation of 18.69% of the sales price. As we have mentioned the expectation or risk analysis basis is only one method to evaluate financial and actuarial actions. Another basic statistical measure that can be used is the application of percentiles as decision criteria. This approach was introduced in Sec. 3.2.2 and is now discussed in connection with stochastic status models. 6.3 Percentile Evaluations There exist other criteria for the evaluation of stochastic economic actions. One alternative criteria function to the risk approach used in the analysis of financial and actuarial models is based on the percentiles of the relevant stochastic quantities. The percentile is set in advance to yield a desired effect and all unknown constants are valued based on the preset percentile. In this section the 25th percentile criteria or PC(.25) is used as a discrimination criterion. For example consider the analysis on one financial action given by the single net value associated with the initial time denoted by SNV. The development follows that of Sec. 3.2.2 where the loss function approach is used and future lifetime is a random variable. The PC(.25) criteria implies that all constants are to be selected so that the resulting probability of a loss is set at .25. In the simplest setting the future value is fixed at fv for all T≥0 and financial rate is fixed at δ throughout the support of T. Percentiles are computed by the inversion of the their direct formula and, as in the expected risk discussion criteria, the relation V=(1+i)−1=exp(−δ) on the discount


167

operator is freely utilized to relate discrete and continuous models. We now consider various applications of the percentile method. First, the general case where the future lifetime of the status is a continuous random variable is explored. The pth percentile for the future lifetime random variable is denoted tp and from the definition of percentile given in (3.2.4) and using the loss function construction the single net value using PC(.25) is (6.3.1) The quantity in (6.3.1) represents the 75th percentile associated with the stochastic present value computation associated with fv. Here the distribution of the future lifetime random variable is all that is required to compute the SNV. It is clear from (6.3.1) that SNV is a decreasing function of the future lifetime percentile and denotes an upper bound to the financial evaluation of the action evaluated at the initial time. Ex. 6.3.1. The $1,000 investment discussed in Ex. 6.2.1 is evaluated using PC(.25). Since the sale made at any time in (0, 4), follows a continuous uniform distribution, then t.25=.10. The PC(.25) method using (6.3.1) with δ= .1 results in

We note that in this loss function based approach $904.84 represents an upper bound, at the 75th percentile, for the present value of the sale. The value of the utilized percentile can be changed to reflect the attitudes of the investor. We now investigate the case where the future lifetime is discrete. The decomposition of T and the UDD assumption are applied along with the definition of the discrete percentile measurement For 0<α<1, let kα be the positive integer where (6.3.2) The decomposition of T, applying the UDD assumption, is tα=kα+sα where the joint distribution (4.4.4) implies α=P(K≤kα)+sα P(K=kα+1). Solving for sα we find (6.3.3) The PC(.25) approach in conjunction with EP produces the single net value (6.3.4) We remark that even though (6.3.4) yields a percentile measurement corresponding to a continuous future lifetime random variable it can be computed using life table data in conjunction with the proper assumptions, namely UDD. A computational example of the discrete case percentile SNV follows. Ex. 6.3.2. The mixed random variable case is considered where the pdf of the curtate future lifetime is given by the geometric distribution. The survival function is given by (1.6.3) or


168

To apply PC(.25) and compute (6.3.4) we solve for k.25, the smallest k such that pk≤.75. Hence,

where gil(a) is the greatest integer less than a. From this approach the percentile calculation of (6.3.4) can be computed. For example if δ=.1 and p=.9 then k.25 =gil(3.73045)=3 and using (6.3.4) the single net value is

or 72.25% of the future value. This is a conservative estimate in that with a 75% probability the SNV associated at T=0 will produce the desired fv value or more. 6.4 Life Insurance People take out life insurance to mitigate the negative effects of possible future economic consequences associated with individual deaths, In the simplest form of life insurance a person age x takes out an insurance policy that, after death, pays a benefit to a beneficiary. The insurance policy is financed by one or more premium payments that are dependent, among other things, on the amount of the benefit and the risk category of the individual being insured. In more complex settings the benefit may be a function of several factors, such as a reliance on multiple death combinations or the exact cause of death. For this reason the general nomenclature used in connection with an insurance policy is that of setting up a general stochastic status where the benefit is paid upon failure of the status. In this section the interest rate is fixed at δ for the life of the contract. For a detailed analysis of life insurance and relevant derivations we refer to Bowers, et al. (1997, Ch. 4). Insurance policies vary in nature to meet required needs but there are policies analogous to the two types of random variables, discrete and continuous, in two major types of insurance policies. In the first the benefit is immediate and is paid at the time of failure of the status. In the second a discrete time period is considered and the benefit is paid at the end of the period in which the status fails. The conditions of the insurance policy may vary but are reflected in the support of the resulting pdf of the future lifetime random variable. In this section different types of insurance policies are analyzed and premiums are computed using RC or expected present value criteria in connection with the equivalence principle. The next example gives an indication of some of the conditions that may be included in life insurance. Ex. 6.4.1. A person age x=50 takes out a discrete life insurance policy where the expected survival and failure totals follow Table 5.2.1. Hence, the mortality and survival rates or probabilities are listed in Table 5.2.2. The benefit, paid at the end of the year, is variable and pays 3 units if death is in the first year, 2 units if death occurs in the second through fourth year and pays 1 unit upon survival to age 54. The yearly interest rate is taken to be i=.05 so that V=(1+.05)−1=.95238. The present value function corresponding to this insurance is


169

(6.4.1)

Computations used in the statistical analysis of the insurance policy, such as expectation, variances and prediction intervals, are based on the present value function (6.4.1) and life table values. Based on RC in conjunction with EP for evaluation the expected present value is computed using group survivorship quantities as demonstrated in Ex. 5.4.3. The expectation is

Further, in a similar manner the second moment is computed as

The variance of the present value function is computed to be

Using these calculations crude approximate prediction intervals can be computed For example, a two standard deviation prediction interval for SNV is given by

Hence, in the context of the insurance policy if each unit is valued at $1,000 the estimated SNV of the policy is between $599.56 and $1,067.31. We remark that the accuracy of the prediction interval depends on the distribution of the life table quantities. Typically the mean and variance calculations are employed in connection with an aggregate collection of identical policies where methods similar to the Central Limit Theorem are utilized The above example is general and demonstrates the computations based on life table measurements that are used in the analysis of life insurance contracts. There are standard types of life insurance that are common to the field. Specific forms of life insurance written for a person age x with a fixed benefit of b are now discussed. For the discrete time period insurance models the length of the time period is taken to be one year. General time period models are discussed in Sec. 6.10.


170

6.4.1 Types of Unit Benefit Life Insurance Standard insurance quantities and formulas can be derived using the notations and computations presented in previous chapters and sections. In general terms a life insurance policy pays a benefit after the failure of a defined stochastic status. For economic and statistical analysis the RC is applied in connection with EP and the resulting general expectation formula depends on the particular conditions of the insurance policy. We follow with an introduction to insurance models, listed by common nomenclature and giving the form of the present value function with constant benefit b, associated with some commonly encountered types of life insurance policies. Further, there are standard notations for the expected present values and associated second moments for some of these policies and the notations have a long history (see Actuarial Society of America (1947)). These notations correspond to unit benefit insurance and we give variations of the notations for the various types of insurance corresponding to expectation computations. For general valued benefits, denoted by the fixed quantity b, the SNV is the unit expectation calculations times b. i) Continuous Whole Life Insurance: A benefit of b is paid at the moment of death. The present value function and SNV corresponding to b=1 are (6.4.2)

ii) Discrete Whole Life Insurance: A benefit of b is paid at the end of the year of death. The present value function and the unit benefit SNV are (6.4.3)

iii) Continuous n Year Term Life Insurance: A benefit is paid at the time of death if death is before future year n. If the status survives n years nothing is paid. Here the present value function and the SNV are (6.4.4)

iv) Discrete n Year Term Life Insurance:


171

A benefit is paid at the end of the year of death if death is before future year K= n. If death occurs after year K=n−1 nothing is paid The present value and expectation are (6.4.5)

v) Continuous n Year Pure Endowment Insurance: The benefit of b is paid upon survival to future age T=n. The present value function and expectation are (6.4.6)

vi) Continuous n Year Endowment Insurance: A benefit is paid at the time of death if death or if age T=n is reached. The present value function is (6.4.7)

where the support is

. The SNV for unit benefit is (6.4.8)

vii) Continuous n Year Deferred Insurance: A benefit of b is paid at the moment of death if death occurs after future age T= n. The future value function and the SNV are (6.4.9)

The collection of insurance policies given in i) to vi) demonstrate the different types of basic insurance policies that can be formed. Of course other contractual conditions could be applied to the agreement leading to additional varieties of insurance policies. We now present a series of examples that demonstrate various types of life insurance. If the fixed benefit is different than one the expected present value function is just the unit benefit expectation times the benefit value. Ex. 6.4.2. In this example we apply the curtate future lifetime distribution and computations presented in Ex. 5.2.1 and Table 5.2.1. An individual age x=50 takes out a


172

discrete four-year term life insurance policy. The benefit paid at the end of the year of death is $100,000 and the annual interest rate is taken to be i=.05. The unit benefit expectation is computed by (6.4.5) and noting (1.05)−1 =.9524 is

Since the benefit is valued at $100,000 then

The SNV of $3,617.10 is taken to be the expected worth of this insurance policy measured at the initial time of policy issue. It can be financed by one lump payment or a series of annuity payments. Ex. 6.4.3. In this example the continuous future lifetime is exponential with mean θ where the pdf is given by (1.2.9). In the continuous setting for the computations we use the discount operator of the form V=exp(−δ). Thus, for a whole life insurance policy with unit benefit the SNV using (6.4.2) is

This again uses the RC and EP approaches. To apply PC(.25) and EP we apply (6.3.1). In the case of the exponential random variable the 25th percentile is t.25= −ln(.75)/θ and we compute the SNV as

Also, for an n year endowment policy, with expectation given in (6.4.8), the RC computation follows the mixed type construction and is given by

We remark that the RC and PC(.25) methods are both used in conjunction with the EP loss model approach and result in different SNV computations. The relative values


173

corresponding to these approaches depend on the distribution of the future lifetime random variable. Computations for other types of insurance policies, such as term life of deferred, as well as higher order moment and variance computations follow the general rules for expectations and percentiles given in Chapter 1. Moment and variance computations are taken up in Prob. 6.7. 6.5 Life Annuities Financial and actuarial stochastic status models may consist in part, of a series of payments comprising an annuity. General life annuities follow the general payment pattern introduced in Sec. 2.2 for deterministic status models. In a stochastic status life annuity a series of payments, or continuous payments through time, are made until a predefined status fails. For example premium payments that finance life insurance may be paid until a certain age is reached or death occurs and the benefit is paid. As in deterministic status annuities there are two basic types of annuities, defined as discrete and continuous. In the discrete time period model payments are made at some point in the individual time periods until failure of the status. In connection with life insurance the premium payments are typically due at the start of each surviving month. The continuous annuity setting is a limiting case of the discrete time period model and the payments are made hypothetically in a continuous manner. In this section we assume the interest rate is fixed at δ over the life of the contract and the risk or expectation criterion in conjunction with EP is the major means of evaluation. For information on the development of annuity models we refer to Bowers, et. al (1997, Ch. 5). The simplest form of a discrete life annuity oocurs when the payments are level or all equal, denoted by π, and are made at the start of each time period. The period time interval is taken to be one year with a discussion of general time period models deferred to Sec. 6.10. The annual interest rate is i and the future value of the sum of all the payments for k+1 years associated with future time T=k using the summation formula (2.2.8) is given by (6.5.1) where d=i/(1+i). We remark that applying the discount operator, Vk+1=(1+ i)−(k+1), to (6.5.1) yields the present value function given by (2.2.9)

as defined in (2.2.9). Thus, for the discrete annuity the present value is (6.5.2)


174

Using the general discrete expectation formula (6.2.2) in connection with the present value (6.5.2) the expectation is the single net value (6.5.3)

In the situation of a whole life annuity the support of the curtate future lifetime random variable is Sk={0, 1, 2,…} and the expectation given in (6.5.3), applying (1.3.10), can be written in terms of the discrete survivor function as (6.5.4) We remark that the computation of the single net value (6.5.4) is easily accomplished if the yearly survival data is readily available, such as in the form of a survivorship life table. We move to the case of a continuous annuity with level yearly payments denoted by π. The general form of the future value of the annuity is given by (2.2.11). Applying the discount function the present value of the annuity consisting of conceptual continuous payments between T=0 to T=t is the integral (6.5.5) Applying the expectation directly or utilizing (6.5.5) the expectation or SNV for the continuous annuity is computed to be (6.5.6)

In particular, in the case of a continuous whole life annuity the support is St= [0, ∞) and using integration by parts the expectation given by (6.5.6) can be rewritten as (6.5.7) Thus, to compute the expectation of a continuous whole life annuity all that is required is the form of the survivor function associated with the distribution of the stochastic status. An example of a discrete life annuity now follows. Ex. 6.5.1. Let the conditions of Ex. 5.2.1 hold where a person age 50 pays 1 unit into an account at the start of each surviving year up to, but not including, age 54. Life Table 5.2.1 applies, where the yearly rate is i=.05 so d= .05/1.05=.047619. From (6.5.2) the present value of this annuity is


175

The SNV associated with these payments is

Thus, if each unit is valued at $1,000 per year the SNV of the annuity, evaluated at the time origin, is worth $3,669.10. As in the case of life insurance a collection of typical life annuities plays a prominent role in actuarial modeling. These stochastic status annuities are used in the computation of insurance premiums and pension plan evaluations. We turn our attention to common life annuities and present basic concepts and notations. 6.5.1 Types of Unit Payment Life Annuities As in the case of life insurance, as discussed in Sec. 6.4.1, standard types of life annuities exist along with their associated notations and formulas. In this section we present standard life annuity models and common notations. We assume the annuity payments are level or all the same and valued at one per year In the discrete setting the payments are made at the start of each yearly time period and continuous annuities consist of payments made continuously through time. As with life insurance policies the different conditions of the policies dictate different formulas for the present value and associated expectations. Present value formulas along with their associated expectation and simplifications are now listed for useful stochastic status annuities. (i) Continuous Whole Life Annuity: The payment π is made in a continuous manner until the failure of the status. The present value function (6.4.4) and the expected present value corresponding to unit payments are (6.5.8)

(ii) Discrete Whole Life Annuity: The payments π are made at the start of each surviving period. The present value function and unit expectation are


176

(6.5.9)

Here ax is not to be confused with (2.2.9). iii) Continuous n Year Temporary Life Annuity: The payments are continuous up to death or till future year T=n. The present value function is (6.5.10)

The expected present value with π=1 is computed noting the multi-rule definition (6.5.10) as (6.5.11)

(iv) Discrete n Year Temporary Life Annuity: The payments are made at the start of each surviving period up to time T=n−1. The present value function is (6.5.12)

The expected present value associated with π=1 is computed as (6.5.13)

(v) Discrete n Year Deferred Whole Life Annuity: In this annuity there are no payments for the first n years or K=0, 1,…, n−1. The period payments correspond to the beginning of each period and start upon survival to future age n. The present value is


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(6.5.14) The expected present value corresponding to a unit payment is (6.5.15)

The forms of the expectations of theses annuities are easily applied to life table data. We now give two annuity computation examples that demonstrate both discrete and continuous calculations. Ex. 6.5.2. We consider the setting discussed in Ex. 6.5.1. In this example a person age 50 takes out a discrete 4-year temporary life annuity. Based on the calculation in Ex. 6.5.1 we have

If the payments were $1,000 per year the expected present value of the annuity is found to be

We remark that this is the lump sum value associated with the stochastic annuity at the initial time as measured by the risk approach. Ex. 6.5.3. In this example the continuous future lifetime random variable is an exponential random variable of the form applied in the continuous type insurance problem in Ex. 6.4.3. To compute the expectations for the continuous annuities the survival function s(t)=exp(−t/θ) is utilized and the formula (6.5.7) is applied. For a whole life annuity with unit yearly payment the expectation is (6.5.16)

For a continuous n year temporary annuity with unit payments the expectation becomes


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Following in a similar manner expectations for other types of continuous annuities can be computed. An economic complication arises from stoppage of payments in discrete annuities when the payments are made at the endpoints of the respective time periods. In this case there is either an underpayment or an over payment corresponding to the fraction of the time period between the succession of payments as dictated by the failure of a status and the end of the interval. This is the topic of the next section. 6.5.2 Apportionable Annuities A continuous stochastic status, such as an individual lifetime, may fail between time period endpoints defined in the discrete time period model. In the case of a discrete life annuity where the payments are made at the start of each time period, such as years, because a full time period is paid for there is an overpayment corresponding to the fraction of the interval between the failure of the status and the end of the time period. In an apportionable annuity there is an adjustment or repayment, to the discrete annuity to account for this fractional time period. To analyze appportionable annuities we start with a discrete annuity with level payments of π at the start of each year. For the general continuous stochastic status the future lifetime random variable T is partitioned as T=K+ S where the UDD assumption holds. As is common, K is the curtate future lifetime random variable, S is uniform on [0, 1] and K and S are independent. In an apportionable annuity a refund is paid to the insurer to account for the length of time between the failure of the status and the end of the interval. This fractional time is denoted by W, S≤W≤1, and is assumed to be a continuous uniform random variable on [0, 1]. The amount of the refund is associated with future value computations based on a continuous annuity. The future value of the refund is the ratio of

After integration this ratio can be written as the future value quantity (6.5.17) We remark that (6.5.17) is the amount of the refund at the time the status fails where s=t−k. The expectation of (6.5.17) can be written as (6.5.18) and signifies, in practice, the amount of the adjustment payment. After simplification the expectation (6.5.18) becomes a function of the expectation associated with continuous life insurance and is given by


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(6.5.19) The SNV or expectation of the present value of an apportionable annuity with unit payment is found by subtracting (6.5.19) from the annuity expectation and takes the form (6.5.20) We follow with a mathematical modeling example that demonstrates many of the calculations that are required to compute both discrete insurance and annuity quantities. Ex. 6.5.4. The curtate future lifetime of a status follows a geometric pdf where f(k)=qpk where Sk={0, 1,…}. For whole life insurance the expectation, using (6.4.3), is

For a discrete whole life annuity, from formula (6.5.9), the expectation is found to be

In particular, if δ=.1, p=.05 and q=.95 we compute Ax=.900329 and . To compute the expectation of an apportionable annuity we first find that i=e1−1=.1051709, and d=.1051709/1.1051709=.0951625. Using (6.5.20) we compute the apportionable annuity expectation as

Further, the amount of the refund, from (6.5.19), is

We remark that in this case the adjustment payment is substantial as it is about ½ a unit amount 6.6 Relating Risk Calculations Basic expectations for actuarial models, such as life insurance and life annuities, take their origin in the general present value decomposition (6.1.2) and general single net value formula (6.2.3). Using the decomposition of the continuous future lifetime random variable and the UDD assumption, life annuity and life insurance risk and general moment computations can be related. For example, in the basic present value formulas for annuities, such as (6.5.2) and (6.5.5), a relation between both continuous and discrete


180

life insurance and life annuity calculations can easily be seen. Based on these, there are formulas relating types of insurance and annuities for both the discrete and continuous setting. We restrict our discussion to the case of level benefits and level annuity payments with time. For simplicity we let these values be one, so that, b=π=1 in all policies. Further, the financial rate is fixed at δ throughout the support associated with the future lifetime random variable. In the following section relations among life insurance and life annuities expectation computations are presented. 6.6.1 Relations Among Insurance Expectations In this section the first type of connection between actuarial moments, relations among the expectations associated with life insurance models, are presented. We first consider formulas that relate life insurance SNV or expectations within either the discrete and continuous setting. For whole life and endowment life insurance, using (6.4.1) to (6.4.6), we have the relations within discrete and continuous insurance types (6.6.1) and (6.6.2) Also, a connection between deferred insurance and whole life insurance exists and is defined by (6.6.3) Understanding relationships among insurance formulas aids in overall comprehension of these types of insurance policies and an explanatory example follows. Ex. 6.6.1. In this example for a continuous type status the exponential future lifetime random variable discussed in Ex. 6.4.3 is considered. To calculate the expectation for a continuous n year term and n year endowment, both with unit benefit, insurance we use (6.6.1) and (6.6.2) to observe that (6.6.4) Applying formulas (6.6.4) in connection with the exponential pdf (see Ex. 6.4.3) we compute the expectation for the term life policy as

The reader is left to calculate the n year endowment expectation for the exponential future lifetime random variable. For the next set of moment relations for actuarial models we consider relations between the discrete and continuous insurance expectations. To do this we apply the UDD assumption and find relationships between discrete and continuous life insurance


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computations corresponding to analogous types of insurance. In Sec. 6.2.3 under the UDD assumption the general single net value formula (6.2.4) for the mixed calculations is given. This formula plays a central role in the construction of connections between continuous and discrete insurance expectations. If the benefit is b, for general continuous life insurance with support St the single net value is (6.6.5) Applying the discrete expectation notations relations between continuous and discrete whole life and endowment expectation are given by (6.6.6) We remark that based on these concepts and formulas expectations for a variety of different types of continuous insurance can be computed or estimated based on discrete mortality calculations as presented in life tables. An example relating continuous and discrete insurance follows. Ex. 6.6.2. The discrete geometric pdf is used to model the curtate future lifetime as in Ex. 6.5.4. From (6.6.6) the SNV for unit benefit insurance where the benefit payment is immediate or made at the time of death simplifies to

From this we note that the continuous life insurance expectation can be found using a discrete distribution model and the UDD assumption. This concept can be extended to life table distributions. Other relations among insurance moment computations exist. Relational formulas among life insurance risk calculations based on subsequent years can be derived. To observe this consider a fully discrete whole life insurance policy for a person age x. Using (6.4.3) we find (6.6.7) where V is the discount operator. This formula relates sequential years whole life insurance expectations and is applied in a later section in the calculation of reserves. These formulas are not exhaustive and other relations exist within insurance and annuity modeling that can be derived using the basic computational laws. 6.6.2 Relations Among Insurance and Annuity Expectations For the next investigation into the relationships among insurance and annuity models we consider similar life insurance and life annuity risk computations. There are formulas relating insurance and annuity moments where the benefit and annuity payments are level and taken to be one. Using the present value functions for specific annuities these relations can be realized. In the general discrete annuity case we use (6.5.9) and in the


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continuous case we apply (6.5.8). The resulting relations among single net values are given by (6.6.8)

The formulas in (6.6.8) can be directly applied in some actuarial settings. In the case of whole life insurance and annuities, both the discrete and continuous settings, the related expectations are (6.6.9)

Further, there exists a direct relation between expectations associated with an n year endowment insurance and an n year temporary annuity. In this case the expectations are associated by (6.6.10) In other actuarial models moment relations can be derived but the support of the associated expectations must always be considered. A computational example now follows. Ex. 6.6.3. To demonstrate the relation between life insurance and annuity expectations consider the continuous exponential future lifetime discussed in Ex. 6.4.3. Based on the expectation computation for whole life insurance the corresponding expectation for a whole life annuity follows easily from (6.6.9) as

and we remark that this computation matches the expectation found using the direct approach in Ex. 6.5.3. 6.6.3 Relations Among Annuity Expectations For the last of the expectation associations presented in this section combinations of both discrete and continuous annuities are considered We use the formulas (6.5.8) to (6.5.14) to construct relations between different varieties of life annuities. For example, the expectations for both continuous and discrete whole life annuities are related to their temporary annuity counterparts by the formulas (6.6.11)


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Similarly, for discrete and continuous deferred annuities relations with whole life annuity are given by (6.6.12)

Also, it is interesting to note that to compute the expectation for a continuous whole life annuity the discrete whole life insurance expectations can be utilized. Using (6.5.4) and (6.5.5) we find (6.6.13) These formulas can be useful in computational problems where redundant work is eliminated. Examples of such applications appear in the context of life table applications given in Sec. 6.7. A discussion of the computation of the expectation for an apportionable annuity is now presented. Using the risk association between discrete and continuous life annuities formulas for calculating the expectation for apportionable annuities, defined by (6.5.20), can be formed. Applying (6.6.6) and (6.6.9) the expectation for a unit valued whole life apportionable annuity is related to its counterpart by (6.6.14) From (6.6.14) we see that an apportionable annuity can be viewed, in terms of expectation computation, as a continuous annuity with a constant adjustment factor, namely δ/d. Using various relational formulas other apportionable annuity relations, such as the one presented in the next example, exist. Ex. 6.6.4. A whole life apportionable life annuity is considered where the curtate future lifetime distribution is assumed given. To compute the expectation we combine (6.6.9) and (6.6.13) to yield the association involving a discrete whole life insurance expectation given by (6.6.15) We remark that the adjustment payment, associated with the apportionable annuity, can easily be computed by using the expectation in either form (6.6.14) or (6.6.15) Based on these formulas annuity expectations can be computed using discrete mortality calculations, such as those found in life and mortality tables. Another annuity relational formula, used later in the calculations of reserves, relates discrete whole life annuities based on subsequent years. From (6.5.9) we can show (6.6.16) In a sequential manner, treating one year at a time, the expectation for a discrete whole life annuity can be calculated.


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Our attention now turns to expectation computations based on survival and mortality tables, such as those discussed in Chapter 5. The formulas relating models can be useful in the computation and understanding of these expectations. 6.7 Life Table Applications Survival and mortality rates or probabilities associated with stochastic status models are often are presented in terms of table values. Life and mortality tables, as introduced in Sec. 5.2 with examples, are constructed using a combination of empirical observations and table smoothing or graduation. We note that in the construction of mortality and life tables the risk category of the individuals involved can be reflected in the table quantities. The previous single net value formulas, especially the relations between discrete and continuous expectations, can be applied to life table data to calculate expectations associated with various types of life insurance and annuities, In this section an example is presented that demonstrates the application of life table data to actuarial computations. Ex. 6.7.1. We consider the mortality and survival probabilities introduced for selected years given in Ex 5.2.1 where the yearly interest rate is fixed at i=.06. For selected years x the SNV for unit benefit discrete whole life insurance and annuity are computed and are listed in Table 6.7.1. We note that if the life insurance expectation is computed first then in the calculation of the annuity expectation formula (6.6.9) can be utilized.

Table 6.7.1 SNV for Unit Whole Life Insurance and Annuities x

Ax

0

.050430

1

x

Ax

x

Ax

16.7757

50

.110048

15.7224

70

.200093

14.1317

.050979

16.7660

51

.112704

15.6756

71

.207822

13.9951

2

.051539

16.7561

52

.115485

15.6165

72

.216043

13.8500

3

.052110

16.7461

53

.118399

15.5750

73

.224796

13.6953

4

.052695

16.7357

54

.121454

15.5210

74

.234121

13.5305

In particular, using this table we can calculate various expectations or SNV. Examples of these computations are: 1) A person age 50 takes out a life insurance policy that pays a benefit of $5,000 where: a) the benefit is immediate and paid at the moment of death. Using the whole life insurance relation (6.6.6) the expectation is


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b) the benefit is paid at the end of the year of death if death occurs within 22 years. Using the expectation formula for discrete term life insurance given in (6.6.1) we compute the single net value as

where 22E50=(l72/l50)(1.06)−22=(52,915/70,710).2775051. c) the benefit is paid at the moment of death or if age 72 is reached. From (6.6.4) and (6.6.6) the single net value is computed as

2) A person takes out an annuity where the individual payments are $1,000. a) A person age 50 pays a continuous annuity for life. From (6.6.6) and (6.6.9) the expectation for the annuity is computed as

b) The person is age 3 and the payments last for at most 50 years. From (6.6.11) the expectation for the temporary annuity is

c) A person age 50 receives payments starting at year 72. The expectation for the deferred annuity is found using (6.6.12) and is

We remark that in practice other possible expectations may be required along with computations of higher moments and variances. These are accomplished using life table data. In the previous example computations corresponding to RC for some of the major insurance and annuities were aided by the relational formulas developed in this chapter. Other combinations of stochastic status calculations could be combined in financial and actuarial models but the ones chosen demonstrated typical possibilities. Computational problems are presented throughout the chapter and in the problems section.


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6.8 Insurance Premiums The most basic challenge in the field of insurance, both life and accidental, is the setting of premium payments. In a typical insurance policy future possible benefits are financed by a series of annuity payments, called premiums. The premiums must be set so as to cover all associated costs and therefore depend on many factors. To analyze this financial system the general loss function structure of (3.1.1) is imposed in connection with a stochastic status, The present values of the expenditure and the revenue correspond to the insurance benefit and the annuity premium payments, respectively. In the simplest model without a cost component the premiums are computed using either the RC or PC(.25) method in connection with the EP. In practice models are developed that account for costs associated with the insurance contract resulting in loaded premiums. The additional costs associated with insurance must be added to the expenditures side of the loss function. An introduction to these cost inclusive modeling concepts and formulas is given in Sec. 6.13. For actuarial models the interest rate plays the part of the general financial rate δ and, in this section, is considered deterministic or fixed for the future lifetime. If the interest rate is taken to be a random variable then the concepts and formulas associated with the stochastic rate models presented in Chapter 3 can be applied. In this case direct computation may not be tractable and the simulation resampling methods, such as those presented in Chapter 7, may be required. In general actuarial and financial modeling combinations of discrete and continuous types of insurance and annuity actions may be associated, thus leading to premium computations. In the first example a fully continuous model is presented where the premium annuity payments are continuous and the insurance benefit is immediate or continuous. Ex. 6.8.1. A person age x takes out a whole life insurance policy that pays a benefit of b at the moment of death. This policy is financed by whole life premium payments that are continuous in payment The expenditure and the revenue correspond to the insurance policy and the annuity payments, respectively. They are (6.8.1)

for t≥0. The resulting stochastic loss function dependent on t≥0 takes the form (6.8.2) The stochastic loss function is viewed as a function of t and is represented in Fig. 6.8.1 where it is zero at future time t*=ln[(bd+π)/π]/δ. From Fig. 6.8.1 we observe that as time increases the loss function decreases reaching zero at t*. Further, as the rate δ increases t* decreases and more of the loss function


187

Fig. 6.8.1 Loss Function

becomes negative implying a realized positive loss is less likely. Both types of stochastic loss function criteria introduced in Sec. 3.2 can be applied in connection with EP to yield premium amounts. First, using the PC(.25) method we set .25=P(LF(t)>0) resulting in .

Denoting the 25th percentile for T by t.25 the resulting premium calculation yields the premium (6.8.3) Second, applying the risk criteria, RC, approach we set E{L(t)}=0 and solve for the premium. The resulting premium is (6.8.4) We remark that the two methods for evaluating premiums yield different premium computations. This is due, in part, to the nonlinear nature of the interest function. The choice of criterion depends on the philosophy of the investigator or actuary. The most common or traditional method of calculating premiums is utilizing the risk criteria, RC, in conjunction with EP. In the next section common notations for unit valued benefit premiums based on this criterion, similar to those presented for insurance and annuity risks in Sec. 6.4.1 and Sec. 6.5.1. are presented. Using these basic expectations as a starting point complicated models can be analyzed. 6.8.1 Unit Benefit Premium Notation Various combinations of life insurance and annuity actions, yielding premium quantities, have been extensively studied and standard notations have been developed. The RC method in conjunction with the equivalence principle, EP, is used to calculate these standard premiums. Specific notations have been developed in the case of where the insurance benefit paid is one. For common combinations of life insurance and life annuity premiums a list of common premium notations associated with their particular


188

expenditure and revenue actuarial functions is given in Table 6.8.1. Further, other possibilities for the benefit structure, such as pension plan annuity payments, are possible. One such case is included in the table. This collection is not all-inclusive but demonstrates a wide variety of possible combinations of insurance and annuity payments. We remark that the notations and formulas given in Table 6.8.1 correspond to various types of unit benefit insurance. If the benefit is different from one then an adjustment is made to the unit benefit premiums. In most cases to compute a general benefit premium the unit benefit premium is multiplied by the benefit quantity. This holds when the benefit is level or constant with respect to the future lifetime of the contract

Table 6.8.1 Unit Benefit Premium Notation Expenditure

Revenue

i) Continuous Whole Life Insurance

Continuous Whole Life Annuity

ii) Discrete Whole Life Insurance

Discrete Whole Life Annuity

iii) Continuous n Year Term Insurance

Continuous h Year Temporary Annuity

iv) n Year Endowment

Discrete h Year Temporary Annuity

v) n Year Pure Endowment Endowment


vi) n Year Deferred Whole Life Annuity


EP Premiums

Computational examples now follow that demonstrate the computation of premiums and the applications of notations for unit valued actuarial contracts. It is interesting to note that both the expenditure and the revenue functions may correspond to life annuities. This is the case when the benefit to the individual is in the form of an annuity common in pension plans. Ex. 6.8.2. We again consider the life insurance setting where the life tables associated with Ex. 5.2.1 hold A person age 50 takes out a fully discrete 4-year term life insurance policy where both the life insurance and the life annuity components are discrete in nature. The expectation of the term life insurance was computed in Ex. 6.4.2 while the expected value of the temporary annuity was given in Ex. 6.5.2. These quantities are

For a unit benefit the premium, using RC and EP, is given by the common notation


189

Hence, if the death benefit is $100,000 the yearly premium computes to be π= $100,000(.0098583)=$9,858.28 and we remark that no loading to account for other incidental costs is applied to this premium. Ex. 6.8.3. We consider a fully continuous whole life insurance policy, in which both the life insurance and the premium payments are of the continuous type, where the future lifetimes are exponential with mean θ, as given in (1.2.9). The expectations of the continuous whole life insurance and annuities are given in, respectively, Ex. 6.4.3 and Ex. 6.5.3 and the RC and EP unit benefit premium is found in Table 6.8.1 as

Hence, in this case the resulting unit benefit premium is just the mean of the future lifetime random variable. After the premium is computed, using approaches such as RC or PC(.25), the resulting observed stochastic loss function becomes a function of the future lifetime random variable. The analysis of observed loss functions at future times becomes important in assessing the future financial liability that the actuarial contract has on the insuring company. We now turn to the analysis of the observed and expected loss function associated with various future lifetimes. 6.9 Reserves The economic stability and monetary value of a company, such as an insurance company, is affected by the future financial obligations it has incurred. For this reason insurance and annuity contracts must be analyzed at various times in their tenure. Thus, any policybased loss function is not only a function of the future year random variable T and the initial age x but also the time frame of the analysis. In reserve analysis the computations corresponding to a general loss function are done in association with some future date x+y. Both the future values corresponding to the expenditures and revenues are adjusted to this new age. As in typical financial and actuarial modeling there are two primary cases, discrete and continuous. In the continuous setting the new future variable is W=T−y where the discrete counterpart is denoted by the integer future lifetime random variable J=K−[y], where [y] is the greatest integer less than or equal to y. To analyze actuarial contracts at future times a general loss function is constructed based on the newly reconstituted future lifetime random variable. Loss function parameters, such as premiums, are already fixed at this point and only the future obligations are measured. The expectation of the loss function is referred to as the reserve and measures the expected future obligation associated with the financial or actuarial obligation. In general, the reserve is an increasing function in the future time variable and is represented in Fig. 6.9.1. These concepts are demonstrated in the following examples. Ex. 6.9.1. We consider the continuous whole life insurance and whole life annuity presented in Ex. 6.8.1. The analysis is done at age x+y and W=T −y represents the new


190

future lifetime random variable. The resulting present value functions analogous to (6.8.1) become (6.9.1)

for w≥0. In this time frame setting the resulting reserve loss function, based on (6.8.2), is a function of w≥0 and is given by (6.9.2) The pdf of W is formed conditioned on the status reaching age x+y and takes the form f(w)=wpx+y qx+y+w. The expectations of the quantities in (6.9.1), using the standard unit benefit and payment notations, are given by (6.9.3)

The reserve corresponding to age x+y is the expectation of the loss function (6.9.2). Utilizing the relation between insurance and annuity expectations given in (6.6.9) the reserve can be written as (6.9.4)

The EP approach implies that the reserve initially, corresponding to y=0, takes the value zero. From (6.9.4) we see that in this case as y increases, since the whole life insurance expectation increases, the reserve increases. Hence older policies are more costly to insurance companies than more recent ones based on the future time reserve analysis. Ex.6.9.2. A three-year endowment policy for an individual age x is the setting for this example. The policy is fully discrete where a benefit of $1,000 is paid at the end of the year of death and premium payments are made at the start of each surviving year. The annual interest rate is i=.15 and the mortality probabilities are qx=.1, qx+1=.1111 and qx+2=.5. Using the RC and EP the premium is π=Px:3=s$288.41. The losses each follow (6.9.2) where the integer future lifetime variable J is substituted for W and yields the alternate form as

For example

and


191

Table 6.9.1 Year-by-Year Reserve Calculations Year y

Outcome j

0

0

581.16

.1

58.12

1

216.94

.1=.9(.1111)

21.69

2

−99.76

.8=1−.1−.1

−79.81

Reserve Calculations 0

P(J=j)

E{0LF(j)}=0

LF(j) P(J=j)

and Var{0LF(j)}=46,445

0

581.16

.1111

64.57

1

216.94

.8889

192.84

Reserve Calculations 2

yLF(j)

E{1LF(j)}=257.41

0

Reserve Calculations

and Var{1LF(j)}=13,1001 581.16

E{2LF(j)}=581.16

1

581.16 and Var{2LF(j)}=0

We compute

After computing yearly loss functions measurements moment computations, such as the reserves and variances, can be done for each year separately. This is demonstrated through the information listed in Table 6.9.1 We note that as expected the EP implies that E{oLF(j)}=0 and the conditions of the endowment policy, where a benefit is paid upon survival to future year K=2, dictates Var{2LF(j)}=0. Approximate prediction interval estimates for the losses can be computed using the mean and variance calculations. Ex. 6.9.3. In this example a person age x takes out a whole life insurance policy that pays a benefit of b at the end of the year of death. The policy is financed by premium payments at the start of each surviving year for at most h years. The RC premium is

This policy is analyzed at future age x+y and the curtate future lifetime conditioned on survival to age x+y is J=K−y for positive integer y. The pdf of J takes the form jpx+y qx+y+j. Using the basic stochastic loss model structure the present value of the expenditures and revenues correspond to, respectively, the insurance benefit and the annuity payments and are (6.9.5) and (6.9.6)


192

The resulting loss function based on additional time y takes the form (6.9.7)

In this case the expectation of (6.9.7) over the separate regions gives the reserve and is (6.9.8)

We remark that reserve computations using (6.9.8) can be completed using the insurance and annuity expectations given throughout this chapter. The previous examples give the basis and insight into the computations of reserves in both the discrete and continuous settings. As in the case of premium computation common reserve calculations have special notations. In the next section some common reserve notations are presented. 6.9.1 Unit Benefit Reserves Notations As in the case of insurance premiums special notations for reserves have been developed for some common unit benefit models. The insurance benefits are taken to be one and the premium computations follow the RC and EP approach without additional costs. In Table 6.9.1 are listed some common reserves with associated notations. We remark that the unit benefit condition implies

The reserve computations and notations listed in Table 6.9.1 give insight into the structure of some of the reserve computations. In specific cases other possible combinations of insurance expenditures and annuity premium revenues can be combined to form required reserve computations

Table 6.9.1 Reserves Expenditures and Revenue i) Continuous Whole Life Insurance and Life Annuity ii) Discrete Whole Life Insurance and Life Annuity iii) Continuous n Year Term Insurance and Temporary Life Annuity

Reserve


193

iv) n Year Continuous Endowment and h h Year Temporary Annuity v) n Year Pure Endowment and h Year Discrete Temporary Annuity vi) n Year Deferred whole Life Annuity and h Year Temporary Annuity

The computation of reserves is left to applied problems that are given at the end of the chapter. In these calculations the mortality and survival structure of the future lifetime random variable may be statistically model-based or dependent on life table data. Further, the relationships between actuarial expectations presented in Sec. 6.6 are helpful in the calculation of many types of reserves. Similar to the calculations of actuarial quantities, such as life insurance, life annuities and premiums, there exist relationships between reserve computations. This is explored in the next section. In particular a sequential year-by-year approach to reserve computations based on these relationships is presented in the Sec. 6.9.3. In addition to the theoretical formulas required, a computational example dealing with reserves is given. 6.9.2 Relations Among Reserve Calculations Relations between reserve computations based on consecutive years can be derived, thereby leading to an iterative approach to the computation of reserves. For example, consider a discrete whole life insurance policy with unit benefit financed by a discrete whole life annuity written for an individual age x. Using RC and EP the premium is π=Px and the reserve formula for various years is listed in Table 6.9.1. Hence, for the first year (6.9.9) Using the formulas relating discrete insurance calculations for subsequent years, namely (6.6.7), formula (6.9.9) becomes

Solving for the reserve corresponding to year one when b=1 (6.9.10)


194

Using the same approach formulas relating consecutive reserves can be derived. Noting V−1=(1+i) for years y and y+1 we have the reserves relation (6.9.11)

Fig. 6.9.1 Reserve Function

Formula (6.9.11) defines the reserve as a function of year y. In Fig. 6.9.1 is drawn a typical graph of the reserve function as y changes. This is done by considering the survival and mortality probabilities as constant over a limited time and fixing the premium. From Fig. 6.9.1 we observe that the reserve increases as the year of analysis increases. Hence, a company with many older insurance contracts may have reserve amounts that will have a negative impact on their future financial stability. Formula (6.9.11) yields a sequential approach to the computation of reserves and is especially applicable in conjunction with survivorship group data concepts and mechanical computations. This approach is demonstrated in the next section. 6.9.3 Survivorship Group Approach to Reserve Calculations In this section we consider a closed survivorship group with definitions and notations as introduced in Sec. 5.1. In this setting there are ly individuals alive at the start of the year corresponding to the curtate future lifetime K=y. The reserves can be calculated using a cash flow approach relating each year’s reserves computations. For demonstration consider a fully discrete whole life insurance policy where we use the following notations for year y: E{PS}y is the expected premiums at the start of year y, E{FS}y denotes the expected fund at the start of year y, E(I}y is the expected interest in year y, E{B}y is the expected death benefits paid at the end of year y and the expected fund at the end of year y is E{FE}y. For year K=y in a cash flow we calculate, sequentially,


195

These quantities can be included in a sequential year-by-year table and comprise the information required for the reserve calculation. Starting in the first year or K=0 the reserves for various years are computed in a sequential manner using (6.9.11). If k=0 then E{FE}−1=0 and we have (6.9.12) Dividing (6.9.12) by l1=lo 1px and adjusting for death benefit b, if different from one, we have the reserve for year one. This expectation is (6.9.13)

For subsequent years k≥1, E{FE}k−1 may not be zero and adjusting (6.9.11) we have the reserve formula (6.9.14)

A cash flow table, containing the reserve calculations, is constructed by the addition of (6.8.14) to the other calculations. This approach is demonstrated in the following example. Ex. 6.9.3. In this example we consider the fully discrete 4-year term life insurance policy for a person age 50. The benefit is $100,000 and we computed

Table 6.9.1 Cash Flow Table for Fully Discrete Life Insurance y

l50+y

E{PS}y

E{FS}y

E{I}y

E{B}y

E{FE}y

l50+y+1

E{yLF(j)}

0

70710

69708

69708

3485

71000

2193

70000

31.3

1

70000

69008

71201

3560

71800

2961

69282

42.7


196

2

69282

68300

71261

3563

72600

2224

68556

32.4

3

68556

67585

69808

3490

73300

0000

67823

0.00

In terms of thousands of dollars the EP premium is π=.9858276. In Table 6.9.1 we give the cash flow approach where the reserve, E{yLF(j)}, is in terms of dollars. We remark that with this term life insurance example the reserve at the end of the term is fully exhausted and reaches the value zero. Another example of the cash flow approach to calculating reserves is outlined in Prob. 6.13. 6.10 General Time Period Models In the financial and actuarial models, and their associated techniques and relations, discussed to up to this point in Chapter 6 it is assumed that the time period for the discrete time period model is one year in length. In this section the length of the time period is more general. A year is divided into m equal parts so the periods are given by the intervals [(h−1)/m, h/m) for h=1, 2,… This discrete time period model is referred to as an mthly period model. The discrete interest rate is considered fixed at period rate i(m) where equating one year and m period interest we find 1+i=(1+i(m)/m)m. Thus, we have the relation (6.10.1) The financial and actuarial computations presented for the discrete one-year length time period model are extended to the discrete mthly period model, Formulas for variations in both discrete insurance and annuity models are discussed in this setting. First, we consider a general payment made at a future time. This is used in a general discrete insurance model. After h periods T=th= h/m and the present value associated with future value FV(th) is computed using the discrete time period model. If Vm=(1+i(m)/m)−1 the present value is (6.10.2) As in the case of yearly periods analytic approaches such as RC or PC(.25) can be applied along with EP to the mthly period present value functions given in (6.10.2). To finance one or more lump sum payments connected to (6.10.20) a series of premium payments is typical. Annuity payments in the mthly period model are now considered. If there are equal payments of π/m at the start of each period the present value function follows (6.5.2) and becomes (6.10.3)


197

where

and (6.10.4) As in the case of yearly period models one of the model evaluation criteria, namely RC or PC(.25), along with the EP can be applied to these general period model formulas. 6.10.1 General Period Expectation The expectation of the present value functions for life insurance and life annuities, (6.10.2) and (6.10.3), can be calculated using the basic laws of statistics. These expectations form the basis of various risk models and SNV computations. The mthly period curtate future life time variable is denoted H where the endpoints of the periods are h/m for h=0, 1,… Using the general discrete distribution structure given in (4.2.3) we observe that the pdf of h can be written as f(h)=h/mpx qx+h/m for h=0, 1,…. For discrete mthly period life insurance, where the benefit is FV(th)=b, and level life annuity, with payment π/m, the expectations are extensions of previous computational formulas. For example the net single value for discrete whole life insurance, from (6.4.5), is given by (6.10.5) In an analogous manner, adapting formula (6.5.9), for a discrete yearly unit payment whole life annuity the expectation is (6.10.6) These formulas can be directly applied to discrete mortality and survival data only if such data are recorded in terms of mthly periods. Since these data and life tables are most often kept in terms of yearly, or longer, time periods some adjustment or approximation is required. The adjustment of these mthly period formulas to account for yearly-based mortality and survival tables is the topic of the next section. 6.10.2 Relations Among General Period Expectations Similar to discrete yearly period length life insurance and life annuities, relations relating mthly period life insurance and annuity expectation formulas along with previous risk computations exist The basis in the construction of relational formulas, for fixed positive integer m, is the decomposition T=H/m+ S/m where the extension to the UDD assumptions to mthly length time periods hold. In particular it is assumed that H and S


198

are independent and S is a continuous uniform random variable defined on the unit interval. For individual contracts the support of H is defined and is denoted Sh. As mentioned, adjustments to expectation formulas are needed to utilize mortality tables in the case of mthly time periods, The required adjustments to the standard expectation formulas are presented through some basic examples. The relationship between the expectations of the present value functions corresponding to continuous and discrete mthly period life insurance, both with a unit benefit, is first considered. To derive the first relation we start with continuous whole life insurance. Following the development leading to (6.2.4), the single net value can be written as (6.10.7) First, exp(δ/m)=(1+i(m)/m) and we have

Since, V=(1+i(m)/m)m then (6.10.7) reduces to (6.10.8) Formula (6.10.8) associates the general discrete mthly period expectation, where the payment is made at the end of the decrement period, to a continuous insurance expectation. Also, using the basic relation between continuous and discrete whole life insurance expectations

formulas relating discrete insurance expectations for yearly and mthly period models can be formed. Here we have (6.10.9) where

From (6.10.9) we note that yearly-based actuarial life tables can be easily adapted to yield moment calculations, such as the mean and variance, corresponding to the mthly period setting. Annuity expectations involving mthly period payments can also be related to insurance and yearly payment annuity model expectations, These formulas relating discrete mthly period life annuities are derived in the context of specific examples. If the payments are level, adjusting the relational formula for a discrete annuity given in (6.6.8), we find


199

(6.10.10)

and to rewrite We use (6.10.8), the associations (6.10.10) into a formula that relates expectations associated with yearly and mthly period annuities, After some simplification we have (6.10.11) Hence, the mthly period discrete life annuity expectation can be written as a linear function involving the yearly-based discrete annuity expectation. Many other insurance and annuity model expectations can be rewritten into the mthly period model context. One such case relates insurances and annuities expectations for a discrete mthly period endowment and a discrete mthly period temporary annuity. In this case for future time th=h/m and for some value of h we have (6.10.12) This is an extension of the yearly period formula relating the expectations corresponding to endowment insurance and a temporary annuity. Period computations for both theoretical statistical models and life table data are demonstrated in the following two examples. Ex. 6.10.1. A benefit of b is paid upon the failure of a status. The benefit, b, is paid at the end of the month. Level premium payments are made at the start of every surviving month. Thus, using a monthly period rate, RC and EP the premium payments are given by

In particular let the geometric distribution, given in Ex. 6.5.4, be used to model the curtate future lifetime where the UDD assumption holds within time periods. From (6.10.9), (6.10.10) and noting i(m)=m(eδ/m−1) the expectations associated with the endowment insurance and temporary annuity are (6.10.13)

We remark that (6.10.13) allows for the computation of expected values based on the associated modeling parameters. Ex. 6.10.2. For an individual age 50 the life table given in Ex. 6.7.1 and the UDD within years assumption holds where i=.06. A $50,000 whole life insurance policy is purchased where the benefit is paid at the moment of death. The level premium payments


200

are made at the start of each surviving month year. Here m=12 and from (6.10.1) and (6.10.4) we compute the needed quantities

From the Table 6.9.1 and (6.6.6),

Using (6.10.9),

Also, from (6.10.10) we have

Hence, using RC and EP the monthly premium payment are found to be

We note that this premium calculation is based on the expectation of only the future benefit payments. In practice the premium would be increased, to form a loaded premium, to account for fixed and variable costs associated with the policy. 6.11 Multiple Decrement Computations Multiple decrement financial and actuarial modeling was first introduced along with the future lifetime random variable in Sec. 4.7 and extended to mortality table construction in Sec. 5.7. We now consider the case where multiple decrement models are used in connection with stochastic status structures. Following common notation, in this section we consider a stochastic status structure where the status may fail by one of J=1,…, m modes of decrement. For example an insurance policy may pay a benefit that varies based on the cause of decrements. The amount of monthly payments of a retirement annuity may depend on the conditions of a retirement. The multiple decrement modeling concepts and probability computations presented in Sec. 4.7 and Sec. 5.7 hold in their stochastic status counterparts. In the continuous model setting the future lifetime random variable is T with corresponding pdf f(t, j)=tpx(τ) µx+t(j). The benefit function depends on both the decrement time T and mode J=j and is denoted by b(t)(j). The general form of the single net value using RC and EP is the expectation (6.11.1)


201

where the support, based on the conditions of the financial contract, is denoted St. This formula is general and based on the specifications in the model, (6.11.1), can be adjusted. We now turn our attention to the discrete setting. In the discrete model setting the curtate future lifetime is K with pdf f(k, j)=kpx(τ) qx+k(j). A general benefit is a function of both J=j and K=k and is denoted by b(k)(j). The expectation is given by (6.11.2) where, as in the continuous case, the support of K is Sk. These concepts and formulas are applied with the help of the previously presented techniques. In the next example insurance benefits are defined differently corresponding to particular decrement causes. Ex. 6.11.1. A discrete whole life insurance policy contains a double indemnity provision in the case of an accident There are two modes of decrement where J=1 implies death due to accident and J=2 represents all other causes. The policy pays double if J=1 so that b(k)(1)=2 while b(k)(2)=1. From (6.11.2) the SNV for this insurance can be written as

where using discrete insurance calculations the expectation associated with mode J=j is

for j=1, 2. As shown in previous example, these computations are completed using either appropriate statistical or life table models. The computational techniques dealing with expectations in connection with discrete and continuous insurance and annuities presented in previous sections can be applied with these models. One of the major examples of the application of multiple decrement models is in constructing pension models. This topic is broad in scope with many subtopics and a brief introduction along with statistical model development is presented in the next section. 6.12 Pension Plans One of the most active actuarial science areas for both research and applications is that of pension plan analysis and construction. Individuals entered into a retirement or pension system, such as people employed by a particular company, may be viewed as a survivorship group, For the individual the stochastic status fails to hold when the group is exited. A pension plan is a financial contract where the main pension benefit, in the form of a deferred life retirement annuity, is financed by pension contributions taken from current and future salary payments. The retirement annuity is commonly a function of different factors such as age, salary and length of service. Typically the individual enters the pension plan at age x at a source of employment where the earliest and latest retirement ages are denoted by α and β, respectively. There may be other benefits related


202

to the pension plan such as a death benefit paid to a beneficiary or a life annuity paid in the event of a disability. For this reason a multiple decrement model works well for pension benefit analysis where the decrements are associated with the causes for the individual to leave the employment system. The concepts and notations presented in connection with multiple decrement modeling, presented in Sec. 4.7, are key to this development and are assumed to hold in this section. Pension benefits are financed through a series of payments referred to as contributions. The contributions may be a function of various variables such as age, salary, inflation and length of service. This is the basic structure of a general loss model, as presented in Sec. 3.1 where expenditures, namely the pension benefits, are paid for by revenues, or the individuals’ contributions. The expectation approach or RC is used along with EP to equate these two balancing factors. In the sections that follow an introduction to benefit and contribution plans is presented where many of the notations are taken from Bowers, et al. (1997, Sec. 11.5).

Table 6.12.1 Pension System Modes of Decrement Types of Decrement

Notation

Yearly Decrement Rate

Force of Mortality

(w)

µx+t(w)

General Withdrawal

w

qx

Death

d

qx(d)

µx+t(d)

Disability Retirement

s

qx(s)

µx+t(s)

Age Retirement

r

qx(r)

µx+t(r)

6.12.1 Multiple Decrement Benefits In general in a pension plan there are many different types of benefits possible and a multiple decrement theory is required for a through analysis. Often different levels of benefits are associated with the different types of employment decrement The concept of multiple decrement benefits and the associated computations is introduced in the form of an example where the types of decrements are listed in Bowers, et al. (1997, p. 350). In the example that follows the benefits discussed are typical ones and other benefits may be encountered in practice. The future lifetime random variable is considered to be a continuous random variable. Ex. 6.12.1. For expository purposes consider an employment system where there are m=4 modes of decrement for the exiting group of employed individuals. The multiple decrement system is defined in terms of types of employment along with the possible modes of decrement. The associated notations corresponding to yearly decrement rates and forces of mortality for the various decrement modes are listed in Table 6.12.1. Other causes of decrement may be encountered in practice but these represent a variety of decrement types. For a typical retirement system we now compute possible SNV based on RC approach for each mode of employment decrement presented in Table 6.12.1. A person age x enters the pension system and their retirement plan is analyzed at future age x+t. First we consider the withdrawal from the system where the benefit is a lump sum payment. The


203

withdrawal must be done prior to the lower retirement age α and is modeled by term life insurance. The unit benefit SNV takes the form of the continuous term insurance expectation (6.12.1) The computation of the above integral is often approximated using discrete approaches. As described in previous sections empirical data similar to life table data may be utilized The next benefit we consider is the death benefit The benefit is paid immediately at any time and hence, corresponds to a continuous whole life insurance policy. The unit benefit SNV is the continuous whole life insurance computation (6.12.2) As in the calculation of (6.12.1) discrete approximations may be useful. The two types of retirement annuities in the pension are now considered. For retirement due to a disability at age x+h+t must be before the upper retirement limit β. The expected present value corresponding to age x+h +t and unit payments is (6.12.3) for t<β−x−h. Retirement due to age represents a deferred annuity. This expected value is (6.12.4) where t<β−α. The computation of the SNV for the two retirement annuities at age x+h requires the integration of (6.12.3) and (6.12.5) using the proper multiple decrement pdfs. For retirement due to disability the SNV is (6.12.5)

and for retirement due to age (6.12.6) We remark that in addition to earlier computational techniques the calculations of expectation formulas (6.12.1), (6.12.2), (6.12.5) and (6.12.6) can be aided by the simplification formulas presented throughout this chapter. The expected value of the present value associated with the pension plan is a linear combination of (6.12.1), (6.12.2), (6.12.5) and (6.12.6). If the benefits and annuity


204

payments are denoted by the constants cw, cd, cs, and cr the SNV for the benefits in the pension plan is (6.12.7)

As previously indicated the preceding calculations are sometimes approximated based on yearly quantities where appropriate mortality tables are utilized. This will be demonstrated in subsequent sections, The benefits are financed through payments called contributions. The topic of pension contributions is discussed in the next section. 6.12.2 Pension Contributions Pension benefits are financed by payments connected to an individual’s salary levels and continue up to retirement or the maximum retirement age denoted by β. These payments are called pension contributions. The risk criterion, RC, in connection with the EP is used to determine the amount of pension benefits and is based on the magnitude of the pension contributions. There are two basic plans for contributions. The first is a flat rate, valued at rate c, which we denote by FR(c). The second method is a flat percentage, given by c, of the individual’s salary for designated years and it is denoted PS(c). For FR(c) contributions we assume a continuous type annuity model at a rate of c. The standard decomposition T=K+S, where the curtate future lifetime K is independent of the uniform random variable S, is utilized. At age x +t the single net value of the contribution is (6.12.8) We now simplify the above integral. Using the uniform distribution and applying a midpoint approximation an approximation for the integral part of (6.12.8) is estimated as (6.12.9) Substituting (6.12.8) into (6.12.9) yields the approximation (6.12.10) The approximating expectation (6.12.10) can be viewed as a discrete annuity with payments made in the middle of each year. Applying the uniform within year mortality assumption, the approximation

could be used in connection with formula (6.12.10).


205

Ex. 6.12.1. A pension model is considered where the contributions follow a flat rate plan, FR(c). If we use the RC and EP then equating (6.12.7) and (6.12.10) we find the general formula (6.12.11)

From (6.12.11) we see that contribution constant c is related to the benefit constants cw, cd, cs, and cr. Higher contributions, in terms of increased c values, results in more benefits. In more realistic models the individual pieces are often more complex but this model points up the general relation that exists between contributions and benefits. In practice the computations of the contributions and benefits are lengthy in nature and best handled by computer software. To help understand the underlying structure a simplified pension system example is given that is based on discrete actuarial measures. Ex. 6.12.2. For an individual age 50 let the mortality rates and actuarial computations of Table 5.2.1 and Table 6.7.1 hold where i=.06. For simplicity we consider a pension plan consisting of only two types of employment decrement, namely death and age-based retirement, that are treated as independent components. The death benefit is a discrete whole life insurance policy with a benefit of $50,000 paid at the end of the year of death. In this case the death benefit (6.12.2) is replaced by $50,000 A50=$5,502.40. Retirement takes place at ages 70, 71,…, 74 with equal probability and takes the form of a discrete deferred whole life annuity. The retirement benefit expectation given by (6.12.6) is replaced by

These benefits are financed by a flat contribution of c at the start of each surviving year up to age 70. In this case the expected contribution given by (6.12.10) is changed to

Using the EP balance in (6.12.11) the equation relating the contribution and the retirement payments become

For example if $10,000 is contributed at the start of each year the resulting pension retirement payment is $40,635.12 per year More complex components of pension models through other forms of benefits and contributions are now discussed. After these computations are made they can be combined, similar to (6.12.11), to complete the loss model and unknown parameters or constants can be computed.


206

6.12.3 Future Salary Based Benefits and Contributions For both pension benefits and the contributions may depend on the individual past and present years salaries. If the individual has attained age x+h the actual salary is known and is denoted by ASx+h. For future ages the salary amounts must be estimated. At age x+h+t the estimated salary is denoted ESx +h+t. To estimate future salaries increases due to merit, seniority and inflation, must be modeled. We assume an increase in salary reflected by continuous type financial rate γ so the relation between actual and estimated salary is given by (6.12.12) where W=(1+i*)=exp(γ). This modeling of salary growth is the same as the exponential growth used for continuous compounding interest. Contributions that are dependent on the yearly salaries are now considered. For PS(c) the contributions are a percentage, c, of the yearly salary. Using continuous annuity payments the SNV of these contributions are computed as (6.12.13) where V*=VW=exp(−δ+γ). If we apply the midpoint approximation (6.12.9) then (6.12.13) becomes (6.12.14) We remark that this computation takes the form of a discrete annuity expectation where the payment is made in the middle of the year A variety of contribution plans exists. There are some contribution plans in which contributions occur only above fixed threshold limits. For future year k contributions are payable above the fixed limit rk for k≥0. The reader is left to investigate this structure noting that formula (6.12.14) will be adjusted. We now consider benefits that are dependent on a part or all of the future years’ prospective salaries. The individual entered the pension plan at age x and is now x+h. The annual benefit rate corresponding to retirement by age at age x+h+t is denoted R(x, h, t). Based of formula (6.12.6) the expectation of this retirement benefit is (6.12.15) The standard decomposition T=K+S is used to rewrite (6.12.15) as (6.12.16)


207

where

If we apply a midpoint approximation to the integral in g(x, h, k) then (6.12.16) becomes (6.12.17) These approximate expectation formulas allow for the completion of calculations based on mortality and actuarial life table data. Ex. 6.12.3. We consider a retirement model where both the contributions, PS(c), and benefits are based on future years’ salaries, Equating only the expected contributions, given in (6.12.14), and the expectation of the retirement payments in (6.12.17) we find the expected relationship between the contribution rate c and the annual benefit rate. As expected higher values of c yields greater R(x, h, k+1/2) values. 6.12.4 Yearly Based Retirement Benefits The yearly benefits paid by the retirement due to age annuity are often, in part, a function of one or more of the yearly salaries. In this section some common concepts applied in pension models are introduced. In the simplest case the payments are a fraction, g, of the final salary. Based on (6.12.12) these payments are (6.12.18) where W=(1+i*)=eγ. This rate formula for benefits can be utilized in the single net value formula for benefits given in (6.12.16) or (6.12.17). Ex. 6.12.4. In this case an individual at age 35 makes an income of $50,000 per year. The salary growth rate is taken to be δ=.05 where contribution are 50% of the final year’s salary. In Table 6.12.1 the contributions rate function given by (6.12.18) is computed for three possible retirement ages, namely 60, 65 and 70. From Table 6.12.1 we observe that the benefit rate increase at a greater than linear rate with the number of additional years of service. This is primarily due to the exponential salary growth rate in the salary projection. Hence, the resulting pension benefits increase greatly as the number of years of service increases.

Table 6.12.1 Benefit Rate Based on Final Year’s Salary Retirement Age Benefit Rate

60

65

70

R(30, 5, t)

$87,258.57

$112.042.23

$143,865.07


208

It is a common setting in retirement annuities that the payment rate is a function of the last m years of service. If the initial age is x, the current age is x +h and if the retirement benefit rate is a fraction, g, of the final m years salary average or mean, where x+h

209

associated with actuarial calculations will not be discussed and for a more exhaustive review we refer to Bowers, et. al. (1976, Ch. 15). For life insurance common costs for the insurance company are in the broad classifications of acquisition, maintenance, settlements and general costs. Acquisition costs include selling expenses, such as advertising, risk classification, commissions, preparing of policies and recording data. Maintenance costs include premium collection and accounting and other correspondence. Claim investigation and legal defense costs fall under settlement costs. General costs include costs associated with research, actuarial services, general accounting, taxes and fees. Costs can be divided into three classifications. There are fixed costs, costs related to the amount of the benefit, and costs related to the premium payments. Further, the costs associated with these different types may change from year to year We follow with two examples of actuarial models that include different types of costs. These examples take the form of case study examples that combine different types of actuarial and financial computations. Ex. 6.13.1. A fully discrete whole life insurance policy with a benefit of $20,000 is issued to an individual of age x. Individual costs may be the same or vary from year to year. For example, the commission is of 60% of the premium the first year and 5% of the premium in subsequent years and there is a fixed maintenance cost valued at 6 units every year The costs are divided into fixed costs, denoted by F, and proportions of the benefit (per $1,000) and premium, denoted by B and P respectively. The resulting “loaded” premium is denoted by G. In Table 6.13.1 the particular costs are presented.

Table 6.13.1 Costs of Fully Discrete Whole Life Insurance First Year Classification

F

Renewal Years B

P

F

B

P

Acquisition

34.5

4.5

.85

0.0

0.0

.05

Maintenance

6.0

.5

.02

6.0

.5

.02

40.5

5.0

.87

6.0

.5

.07

Total

Using the RC and EP the loaded premium, G, can be found. Based on Table 6.13.1 we have

Solving for G we find (6.13.1)


210

Further, reserves can be computed for future ages x+y corresponding to subsequent years w=T−y for y≥1. The expected present value of the revenue and expenses is given by

and

The reserve associated with w is (6.13.2) For example, let x=50 where the SNV for whole life insurance and annuities is given in Table 6.7.1. Further, i=.06. and we find

Fig 6.13.1 Reserve Computations

Using the RC and the EP the yearly premium is π=$20,000(.110048/15.7224= $139.99. We compare this to the model with expenses as expressed in Table 6.13.1. Using Table 5.2.1 we compute the expectation of the one year deferred annuity as

and applying (6.13.1) the loaded premium is computed as

which reduces to $2497.5247/13.82178=$180.69. We remark that the loaded premium in this hypothetical example is significantly larger than the non-expense computed


211

premium. The reserves corresponding to future years defined by (6.13.2) are computed and are graphed in Fig. 6.13.1. We note from Fig. 6.13.1 the reserves, as expected, are increasing as the future time increases. The added expenses incurred in the first year cause the reserves to be negative, indicating a gain, for the first two years. Ex. 6.13.2. An individual takes out a 30-year term life insurance policy that pays $50,000 at the time of death. Premium payments are made at the start of each surviving year up to 20 years. The costs of the policy are as follows: For each year the sales commission is 7.5% and the taxes and fees are 3% of the premium. Other policy expenses are $25 the first year and $5 each renewal year and there is a claims settlement fee of $35. Using the RC in connection to the EP the loaded premium, G, satisfies the equation (6.13.3)

Solving for G we find

Using appropriate actuarial life tables or computer programs the loaded premium can be computed. Further, the reserves can be calculated corresponding to future year w. For 1≤w<20, applying the computations in (6.13.3) the reserve is

For 20≤w<30 we note there are no yearly premium payments and the reserve is computed as

In specific scenarios all these computations can be completed using appropriate life models and actuarial tables. Modern software programs are in place to compute necessary quantities. Problems 6.1 An investment is to be sold for $5,000 at a future date where the rate of return is δ=.08. The future time of sale is denoted by T. Find the expected present value where the distribution of T is a) exponential with a mean of 3 years, b) gamma with pdf g(t)=1/(Г(α)βα)tα−1e−t/β where α=1 and β=4, c) normal with mean 3 years and standard deviation given by 1 year. 6.2 In Prob. 6.1 consider c) where the future time of sale is a normal random variable normal. Construct a 95% prediction interval for the present value of the investment. 6.3 An investment is to be sold only at the end of a month for a value of P where the financial return rate is estimated to be δ. The future time of sale is the curtate lifetime K


212

which is Poisson random variable with pdf f(k)=e−λλk/k! where Sk= {0, 1,…}. Find the expected present value of the investment using (6.2.2). Also, compute the variance of the investment 6.4 Consider the future sale described in Prob.6.3. Assume the UDD assumption holds and compute the SNV using (6.2.4). Compare this with the result in Prob. 6.3. 6.5 In this problem we consider the PC(.25) method. Find the SNV for a) Prob. 6.1 part a) where the mean is 3 years, b) Prob. 6.3 where the mean is λ=3.3. 6.6 For a particular status let the future lifetime be a continuous random variable with pdf f(t)=t e−t/β/β2 where St=(0,00). Using the antiderivative formula

give formulas for a) the unit benefit insurances

and

and b) the unit premium for

and . annuities 6.7 An individual in a high risk category has a curtate future lifetime given by k f(k)

0

1

2

3

4

5

6

7

8

9

.05

.05

.07

.1

.13

.1

.1

.15

.15

.1

The interest rate is given by δ=.05. Compute the first and second moments and approximate 2 standard deviation confidence interval for the present value of a) insurance paying a benefit of $10,000 at the moment of death for i) whole life insurance, ii) 4-year term insurance, iii) 5-year endowment insurance, b) an annuity paying $1,000 at the start of each year for i) whole life annuity, ii) 4-year temporary annuity and iii) 5-year deferred annuity. 6.8 Let the future lifetime of a status be a Poisson discrete random variable with pdf given in Prob. 6.3. Give formulas for; a) the unit benefit insurance Ax, the whole life and c) the yearly premium Px using RC and EP. annuity 6.9 An individual age 50 purchases life insurance with an immediate benefit of $10,000 where Life Table 6.7.1 applies where δ=.06. Further, the UDD assumption holds where the expectation criteria are used to compute premiums. Find the level premiums for insurance where a) the premiums are paid at the start of each surviving year, b) where the premiums are paid up to each surviving year up to age 70, c) the premiums paid at the start of each surviving year comprise a whole life approtionable annuity. What is the expected pay back? 6.10 Consider the premium calculations in Prob. 6.9. Find the resulting required premiums if the period of the premium payments is monthly instead of yearly. 6.11 Consider a status where the pdf of the continuous future lifetime random variable is given in Prob. 6.6. For a fully continuous whole life insurance policy where both the benefits and premiums are continuous find the need premium using a) the expectation criteria RC, and b) the PC(.25) method using (6.8.3). 6.12 For a stochastic status let Table 6.7.1 and the UDD assumption hold where δ=.06. Calculate and give a word interpretation for the following:


213

6.13 For a particular status we have a life table with entries given by: x

0

1

2

3

4

5

6

lx

1,000

920

762

432

121

41

0

Let the interest rate be δ=.1. Consider a fully discrete whole life policy with a benefit of $20,000. a) Find the level premiums using the risk criteria. b) Use the cash flow approach, as in Table 6.91, to compute the reserves for future years. 6.14 Consider the survival data listed in Prob. 6.13 where the interest rate is i= .08. Assume where necessary that UDD holds. For an individual age 1 compute the SNV using RC where the benefits and annuity payments correspond to one unit for the following: a) discrete whole life insurance, b) 2-year discrete term life insurance, c) 2year immediate benefit endowment insurance, d) continuous whole life annuity, e) 2-year deferred discrete annuity. 6.15 A person age 70 takes out a fully discrete whole life insurance policy where the benefits are as follows: 3 units for the first year, 2 units for the second year and 1 unit for every year after the second year. a) Give a formula to compute the associated expectation. b) If Table 6.7.1 holds compute the SNV for this policy. 6.16 The pension plan problem presented in Ex. 6.12.2 is revisited where the person age 50 has a $67,000 yearly income and two types of independent employment decrement, by death and by age retirement, possible. The death benefit is $100,000 and is immediate and age based retirement takes the form of the deferred discrete annuity described in Ex. 6.12.2. Using the RC and EP find the amount of a flat rate contribution, c, required if a) the retirement annuity pays $50,000 at the start of each surviving year, b) retirement benefit rate given by (6.12.18) takes the form R(50, 0, k+1/2)=.6 AS50 Wk+1/2 where γ=.05. 6.17 An individual age 50 takes out an insurance policy that pays the beneficiary $120,000 at the moment of death. The premium payments are made at the start of each surviving month for a maximum of 20 years. The costs associated with the policy are a commission cost of 60% of the premiums the first year and 5% every subsequent year and fixed costs of $500 the first year and $50 thereafter. Give a formula for the loaded monthly premium. What assumptions are required? If Table 5.2.1 and Table 6.7.1 hold calculate the loaded premium.

7 Scenario and Simulation Testing

Many modern techniques in financial and actuarial modeling involve the exploration and analysis of mathematical and statistical models under various hypothetical settings. In fact the demonstration of the financial stability under various economic scenarios, such as changing interest and return rates, may be desired or even required for individual companies or investment plans. The models may be complex in nature and may not lend themselves to direct theoretical statistical evaluation. These techniques fall into two, sometimes related, categories. The first is scenario testing where model parameters, such as financial rates or the length of time a stock is retained, are varied over hypothetical values and the resulting effects are observed. Deterministic status models are often evaluated in this manner in terms of their risk or associated expectation. In the second, referred to as simulation testing, typical or proxy data sets are produced and the associated models and statistics are generated. Through many such data sets statistical inference on the related models, such as expectations, percentiles or prediction intervals, can be conducted, Simulation is often required to analyze stochastic status models where theoretical statistical analysis is not tractable. Further, applying simulation techniques approximate formulas, such as those resulting from asymptotic techniques, can be evaluated for accuracy. In this chapter the concepts and formulas for scenario and simulation analysis of financial and actuarial models are presented. Through examples, many of which were first introduced in previous chapters, scenario testing is presented for both stochastic and deterministic status models. In connection with simulation inference two types of simulation resampling techniques, namely bootstrap and model based simulation, are introduced. Through these resampling procedures statistical inference techniques described in earlier chapters are extended to more complex models especially in the case of stochastic status models. Expectation formulas and prediction intervals based on these resampling methods are given and applied to analytical functions such as present and future value functions for both deterministic and stochastic status models. For example the risk analysis associated with collective risk models is extended to include a stochastic force of interest and stochastic status models include further analysis on investment pricing and surplus loss modeling. Of particular note is the resampling analysis in conjunction with investment pricing models, such as American and European call options and the evaluation and validation of the Black-Scholes computations. The chapter concludes with a brief introduction to new statistical sampling techniques.

Scenario and simulation testing

215

7.1 Fixed Rate Deterministic Status Models In the simplest form of scenario testing deterministic models, containing no stochastic components are analyzed. In these models direct mathematical calculations are made and hence, no statistical measurements are appropriate. These deterministic models include examples such as house payment computations for a fixed number of years that are based on a fixed mortgage rate. An introductory example of changing settings and investigating the resulting varying calculations was given in Ex. 2.1.4, where the change in APR is explored for various continuous compounded interest settings. This type of investigation is at the heart of scenario testing. This is the case when companies demonstrate their economic soundness under adverse conditions, such as high or low interest rates or low return rates. Scenario testing in financial and actuarial modeling may involve the effect of changes in financial and economic parameters, such as financial rates or loading factors, on computations. In the two examples that follow the effects of changing financial rates on present and future value calculations in deterministic models are explored. In the first example the present value of an investment is considered and in the second the growth of an annuity is discussed. Ex. 7.1.1. The setting given in Ex. 2.1.5 is considered where the goal of an investment is to realize a monetary sum of $1,000 in five years. The present value corresponding to this future sum is dependent on the financial return rate of a hypothetical investment that is held for five years. In Table 7.1.1 the present value of the investment is computed for various return rates. From Table 7.1.1 we note that the changes in the present values related with hypothesized financial rates are not linear. Further, overestimating the financial rate will cause the present value to be badly underestimated. Hence, the present value of a

Table 7.1.1 Present Value Investment Scenario Testing Example Rate Present Value

.04

.08

.10

.12

.14

$818.73

$740.81

$670.32

$606.53

$496.58

value of a future fixed sum can be evaluated under different types of investments and scenario testing models similar to this one can be used to measure these and other changes. Ex. 7.1.2. The discrete annuity model application in Ex. 2.2.1 is explored for the effect of changing annual interest rates on future value computations. A sum of $150 is deposited at the end of each month for a total of 4 years. The future value of the sum of these deposits is computed for various annual interest rates. These computations are given in Table 7.1.2. From the computations in Table 7.1.2 we note that minor changes in the interest rate affects the future value of the discrete annuity even over a short time frame. As expected the effect of changing financial rates is more dramatic for annuities as opposed to single installment investments. The topic of general scenario testing is broad and there are many other circumstances where scenario testing is beneficial for the analysis of deterministic status models. For


216

example, house loan premiums and total interest payments can be compared for various principles, time lengths and interest rates. In fact scenario testing can be utilized even in the case where the financial rate is a random variable with a hypothesized probability structure. This situation involves some statistical distribution assumptions, such as normality of the interest or return rates, and this setting is explored in the next financial modeling example. Ex. 7.1.3. The investment setting presented in Ex. 2.3.1 is considered where $10,000 is invested for a period of five years where the financial rate is a normal random variable defined by (2.3.1) where the total number of periods is n=60. In this example two different scenario testing procedures are demonstrated. In the first, based on hypothetical monthly mean and standard deviation parameters, statistical evaluation such as the expected future value and prediction intervals can be made. Using the normal random variable mgf formula given in (2.3.8) the expected future value after 5 years, denoted EFV(5), is computed for various mean and standard deviation values, γ and β, and are listed in Table 7.1.3. From Table 7.1.3 we remark that changes in the hypothetical parameters affect the expected present value calculations. In the second scenario procedure, prediction intervals for the future value are constructed. In this scenario testing procedure the monthly mean given by γ is varied and standard deviation is fixed at β=.01. A 90% prediction interval for the future value, denoted

Table 7.1.2 Future Value Annuity Scenario Testing Example Annual Rate

.024

.036

.048

.060

.072

Future Value

$7,549

$7,731

$7,920

$8,114

$8,315

Table 7.1.3 Expected Future Value Scenario Testing Mean γ and Standard Deviation β Parameters

EFV(5)

γ=.005

γ=.008

γ=.010

γ=.012

β=008/.012

β=.008/.012

β=.008/.012

β=.008/.012

13,520/$13,550

16,190/$16,230

18,250/$18,330

20,630/$20,630

is found for various mean parameters and is listed in Table 7.1.4. Different scenarios and conceptual strategies can be utilized and developed in connection with scenario testing methods. For example, the expected future value of the investment is between $13,539 and $33,301 while if at least $16,000 is required in 5 years then a mean return rate of at least .01 is needed. Many varied applications of scenario testing exist in the field of actuarial science modeling. Here actuarial computations, such as those for single net values for insurance and annuities and premium calculations, can be made under various hypothetical settings. The effect of changes in the future interest rates or future lifetime distributions on these


217

quantities can be measured. For expository purposes a theoretical modeling example is now discussed. Ex. 7.1.4. In this example a whole life insurance policy paying a unit benefit at the moment of death is financed by a continuous whole life annuity. The future lifetime random variable is taken to be an exponential random variable with mean θ and the interest rate is given by the constant δ. Based on the exponential future lifetimes and the formulas given in Ex. 6.4.3 and Ex. 6.5.3 the expectations, namely

and resulting RC and the EP based premium

Table 7.1.4 Normal Return Rate Scenario Example Lb Return Mean

EFV(5)

Ub

.005

$11,884

$13,539

$15,333

.010

$16,041

$18,276

$20,697

.020

$29,230

$33,301

$37,712

Table 7.1.5 Exponential Scenario Testing for Ax, ax, and π Interest Rate δ .05

.08 π

.10 π

π

Mean

20

.5

10

.05

.3846

7.692

.05

.3333

6.667

.05

Future

25

.444

11.11

.04

.3333

8.333

.04

.2857

7.143

.04

Lifetime

30

.4

12

.0333

.2941

8.824.

.0333

.25

7.5

.033

are computed. For various mean lifetimes and hypothetical interest rates these quantities are calculated and are listed in Table 7.1.5. From Table 7.1.5 we note that, as expected, the insurance and annuity expectations increase with a lengthening in the mean future lifetime and decrease with an increase in the interest rate. Further, the form of the exponential distribution results in the premium being only influenced by changes in the future lifetime random variable and not by changes in the interest rate. In the evaluation of many financial and actuarial models theoretical mathematical and statistical evaluation is not tractable. This is often a result of complex stochastic structures that are imposed on the model where only incomplete statistical theory exists, In such settings only asymptotic results for statistics such as percentiles and prediction interval may exist The modeler or investigator, in this situation, may utilize simulation


218

methods to estimate and evaluate these stochastic models. The concept of simulation with the resulting formulas and statistical analysis is explored in the rest of the chapter 7.2 Simulation Methods The effect of modern computing power on modern mathematics and statistics can be seen in the number of new calculation intensive modeling estimation and evaluation techniques. The very nature of complex statistical modeling has opened up a plethora of computational techniques. Some of these methods have become quite accepted, with examples being the Quenouille-Tukey jackknife, discussed in Quenouille (1949), Tukey (1958) and Miller (1974) along with the bootstrap procedures of Efron (1979). Modern procedures, such as data mining techniques (see Hand, Blunt, Kelly and Adams (2000)), hold great promise in the modeling of random variables with less structure such as interest and financial return rates. Leo Breiman (2001 b) has suggested that two separate statistical approaches exist. The first is the standard statistical model and distribution based inference approach while the second assumes much less about the structure of the data distribution and relies on algorithmic techniques. The basis of modern algorithmic models, such as random forests (see Breiman (2001)) and support vector analysis (see Christianini and Shawe-Taylor (2000)) is modern computational techniques. For a discussion on the use of computing power in statistics we refer to Diaconis and Efron (1983). The financial and actuarial models presented and analyzed in previous chapters, by their very nature, often contain one or more stochastic components. The analysis of such compounding of stochastic actions may be mathematically complex. Further, in connection with stochastic financial or actuarial models traditional statistical inference and evaluation are not always available. Common theoretical statistical methods may not exist or may be too restrictive, in terms of their accompanying assumptions, to be employed. In this case modern simulation methods provide a useful path for the analysis for these often complex models. In this section an introduction to computational statistics is presented in the form of two types of simulation procedures that produce typical or proxy data sets. The discussion of resampling methods is not meant to be exhaustive but to give insight to resampling statistical evaluation. The first approach explored is a resampling method based on the bootstrap procedure where the observed sample serves as the basis set for the construction of future simulation samples. In the second technique typical data sets are generated from hypothetical distributions. In these resampling methods the proxy data sets are generated to mirror empirical or theoretical distributions. In this way the probability distributions associated with the observed data, though not explicitly formed, are estimated. Modern statistical analysis is applied in connection with the simulated data sets thereby extending statistical approaches to more complex models. The statistical inference methods presented in previous chapters for model evaluation, such as point estimation, parameter confidence limits and prediction intervals, are extended to the simulation data sets. The general term of resampling, denoted by RS, is used to refer to either of these two simulation techniques.


219

7.2.1 Bootstrap Resampling The first data generation method considered is a general bootstrap resampling method, denoted by BS. We have a sample of observed random variables denoted Yj for 1≤j≤n. From the set of observed values {y1, y2,…, yn} we sample at random without replacement to construct r simulated samples each of size n {y1i, y2i,…, yni} for i=1, 2,…, r. Statistics, such as future and present value functions, loss functions or reserves can be computed for each of the generated samples. For the ith sample let the associated statistics be denoted by Si for i=1, 2,…, r. In this way a proxy sampling distribution associated with these statistics is constructed which leads to statistical resampling inference techniques. An example demonstrating the BS technique in the field of stock evaluation is now given. Ex. 7.2.1. In this example the stock prices observed in Ex. 2.3.5 are analyzed using a BS procedure. Starting with an initial value of 10.5 taken after the first period, the future value after the next 10 periods is to be estimated. Using Table 2.3.1, the observed values of the rates are given by .091, −.072, .156, .000, −.024, .024, .162, −.035, −.112, and .062. The statistic to be computed based on each set of 10 simulated samples is the future value so that

for=1, 2,…, r. Using r=25 simulated samples the observed future values are given in increasing order in Table 7.2.1. Using the data produced by the BS procedure we compute the mean and standard deviation of the resampled future values to be 12.29 and 3.217 respectively. Hence, an approximate, normality based, 95% prediction interval for the future value after additional 10 periods is calculated as

or

We remark that in the case of uncertainty of the normality approximation other confidence interval approximations, such as three-standard deviation intervals, can be utilized. As in any application of statistical interval estimation the length

Table 7.2.1 RS Future Values 5.91, 7.91, 9.95, 10.45, 10.48, 10.50, 10.58, 10.84, 10.92, 11.50, 12.20, 12.33, 12.55, 12.57, 13.11, 13.45, 13.49, 13.88, 13.91, 14.71, 15.92, 17.11, 18.16, 18.20, 20.09

of the interval indicates the uncertainty in the estimation of the future value of the investment. The BS method is quite general in that no underlying distribution associated with the statistic under consideration needs to be assumed. Inference on the BS samples, such as the prediction interval of the previous example, allows the statistical evaluation of many


220

types of financial and statistical models. We now follow with the second type of RS procedure. 7.2.2 Simulation Sampling Based on past empirical data or the mathematics of the structure to be modeled a specific statistical distribution for stochastic components in the model may be assumed. In the second resampling method a theoretical model is assumed for the random variables present in the model and simulated samples are drawn. Modern random data generation methods are employed to obtain the typical data sets. In this chapter simulated sampling, referred to as SS, utilizes typical probability distributions such as the normal, exponential or uniform distribution to reproduce r replicated samples of size n. Using the notation of RS the replicated samples are given by {y1i, y2i,…, yni} with associated statistics Si, for i=1, 2,…, r. Candidates for the computed resampling statistics are any statistics that are used in financial or actuarial modeling. such as present or future value functions. The SS method requires the simulation of data from a specified distribution. To do this we first select an algorithm to generate a uniform random variable over (0, 1) which we denote U. Let the random variable of interest be Y with known pdf f(y) over support support S and df F(y). If Y is a continuous random variable then F(y) is monotone increasing with associated inverse of F denoted by F−1 over the support S. To generate observed data yi, for i≥1, with corresponding pdf and df we first generate uniform sample ui for i≥ 1 and then use the transformation (7.2.1) In the setting where the random variable is discrete the transformed method using (7.2.1) can be adjusted to yield samples taking values in the discrete type support with the correct proportions. A continuous random variable example that demonstrates SS follows. Ex. 7.2.2. For a specific type of insurance policy claims are distributed as an Exponential random variable with mean θ. The df associated with the claim amount is F(x)=1−e−x/θ for x>0. To generate exponential data we apply (7.2.1) and after solving for the inverse function use the transformation

where ui are iid uniform (0, 1) random variables for i≥1. For exposition we let θ=100 and generate 25 typical samples listing the results in Table 7.2.2. From Table 7.2.4 the mean and standard deviation of the SS data is calculated to be 106.3 and 87.1 respectively. Using the normality based confidence interval formula (1.8.11) an approximate 95% confidence interval of the mean θ is given by 72.2≤θ≤140.4. We remark that this procedure could be extended to functions of the claim amounts such as future or present value calculations. This is the topic of Prob. 7.3. Normal random variables play a central role in the statistical modeling of empirical data and this is the case in financial and actuarial modeling. To generate data from a normal distribution we use the Box-Muller transformation (see Hogg and Craig (1995, p. 177)). As in the general data generation method (7.2.1), let Ui be iid uniform random


221

variables on (0, 1) for i≥1. For pairs of uniform variables, Ui and Ui+1 the transformation generates two iid n(0, 1) random variables given by (7.2.2)

for i≥1. To generate general normal random variables with mean µ and standard deviation σ we use the linear transformation (7.2.3) Independent random variable structures are not the only ones that can be used in conjunction with resampling methods. An example

Table 7.2.2 Exponential SS Data 1, 5, 5, 6, 34, 36, 45, 47, 53, 63, 74, 76, 87, 90, 122, 123, 143, 146, 149, 156 168, 198, 205, 299, 327

of simulation in the AR(1) dependent data setting introduced in Ex. 2.3.5 utilizing the normal data SS technique follows. Statistical dependent data modeling is often theoretically complex and RS procedures are useful in the analysis of these models. Ex. 7.2.3. In this example the SS technique is applied to the observed return rates discussed in Ex. 2.3.5. The AR(1) model analysis based on the normal distribution is applied with point estimates

Here δ1~n(δ, σ2) and based on (1.14.15) for j>1,

Simulating r=25 samples of ten periods each the computed statistics are, as in Ex. 7.2.1, the future values

The ordered computed values are given in Table 7.2.3. Based on the SS data in Table 7.2.4 we find the mean is 10.504 while the variance is 3.233. Comparing this SS procedure with the BS method of Ex. 7.2.1 we see that while the variances are quite similar the observed means are different This is due to the greater mean used in the AR(1) model. In Prob. 7.4 an approximate prediction interval associated with the expected future value is constructed. Thus far in this chapter we have discussed the generation of RS samples that mirror the sampling distribution of modeling statistics. We now turn our attention to statistical


222

inference techniques used in conjunction with RS statistics. The statistical techniques used in RS evaluation take their basis in basic sampling theory.

Table 7.2.3 Future Values Based on SS AR(1) Modeling 5.52, 6.22, 7.17, 8.12, 8.21, 8.25, 8.40, 8.43, 8.69, 8.81, 9.15, 9.98, 10.19, 10.7, 10.77, 10.80, 10.85, 11.25, 11.29, 11.32, 11.85, 14.23, 16.97, 17.51, 17.85

7.3 Simulation Inference on Deterministic Status Models Deterministic status models, even though the status is fixed, may contain one or more stochastic components such as the length of time a commodity is retained or the financial rate. Simulation statistics can be generated based on the stochastic components in the model and used in inference procedures. For example each resampling technique, such as BS or SS, produces typical financial rates based on the underlying structure of the data, along with useful statistics. The present value and future value function, loss functions and surplus measurements are all examples of typical statistics to be computed. These RS data sets form the estimates of the sampling distribution of these useful statistics and measurements. In this section only deterministic status financial or actuarial models are considered and stochastic status models are considered in a subsequent section. The RS samples form a basis for estimating the sampling distribution for model assessment statistics. Statistical inference procedures discussed in this section are based on the RS samples and follow standard statistical laws and, in particular, we consider prediction intervals and percentiles based on these RS data. For RS statistics Si, for 1≤i≤r the mth moment is estimated by the empirical RS moment defined by (7.3.1) We note that the laws of large numbers imply that estimators of the form (7.3.1) are consistent and converge for large sample sizes. Using either BS or SS the mean of the statistic is estimated by taking m=1 in (7.3.1) and is denoted by E{S}rs. Similarly, the variance is computed using (7.3.1) in connection with the basic variance formula (1.3.5) and is denoted by Var{S}rs. Applying these resampled statistics approximated prediction intervals can be formed. For example using the normal distribution as a model or pivot, a (1−α)100% prediction interval for statistic S, when RS is employed, is given by (7.3.2)

The normal-type prediction interval presented in (7.3.2) is denoted NPI. The accuracy of (7.3.2) depends, in part, on the departure from normality that is present in the true distribution of S. Other proposed general confidence and prediction intervals taking a


223

nonparametric approach are not based on the normal random variable. Typically these intervals are based on RS percentiles and tail probabilities. Tail probabilities and percentiles can be found for the simulated based statistics using statistical sampling concepts. In general for each of the r samples, consisting of n observations each, observed cumulative and tail probabilities associated with statistic S are to be estimated. For a fixed s we estimate the df associated with S, namely P(S≤s), using the RS computations as a sampling distribution. For the ith sample let the computed statistic be Si and the RS the empirical distribution function is (7.3.3) where I(A)=1 if A holds and=0 otherwise. The convergence of (7.3.3) to the true df. is guaranteed by standard probabilistic laws. Percentiles associated with S can be approximated by searching for appropriate s values through relation (7.3.3). For 0≤α≤1 the RS (1−α)100% percentile, denoted by S1−α,rs, is defined by (7.3.4) We remark that in the case of continuous random variables if the resampling r size is large enough then a suitable solution to (7.3.4) can always be found. For smaller values or r interpolation methods, as discussed later, may be required. Applying (7.3.4) a symmetric RS prediction interval for S, denoted by RSPI, with coverage probability 1−α is given by (7.3.5) Alternate confidence intervals to (7.3.2) and (7.3.5) based on a histogram approach have been proposed by many authors. We consider one such approach. the bias-correcting percentile method given by Efron and Tibshirani (1986), that has been demonstrated to be useful when the statistic is not a linear function of an approximate normal random variable. To apply this method we first let zo= Φ−1[F(E{S}rs)rs] where Φ−1 is the standard normal distribution function. The interval takes tail probabilities computed as α1=Φ(2zo+z(α/2)) and αu=Ф(2zo+ z(1−α/2)). The bias correcting prediction interval, denoted BCPI, using (7.3.3) in conjunction with (7.3.4) is (7.3.6) We remark that if zo=0, or half the RS values are less than E{S}rs, than the BCPI reduced to the RSPI given in (7.3.5). This interval has been demonstrated to be effective in many circumstances. It may not be possible in some cases to compute exact prediction intervals using simulation methods. In the situation where exact solutions to the percentile equation (7.3.4), in conjunction with the empirical df (7.3.3), cannot be found due to a small number of replications interpolation formulas can be used to construct prediction intervals. In the case of either BS or SS let the corresponding order statistics be


224

S(1)
or (7.3.8) To construct a BCPI for the expected future value we first compute, using (7.3.3), F(12.8288)rs=14/15=.56. Hence zo=Φ−1(.56)=.1509693. For a 90% prediction interval we calculate the tail probabilities as

and

The lower and upper bounds are computed by applying (7.3.7). For the lower bound we find d=.089626(26)=2.330276, S2=7.91 so that

For the upper bound, d=.97423186(26)=25.33 which is greater than r=25. To avoid this problem the number of replications, r, should be increased but for now we take as a default value d=25, implying S25=20.09 and

The resulting BCPI is (7.3.9) We remark that in comparing the prediction intervals (7.3.8) and (7.3.9) the NPI is symmetric about the RS mean. The BCCI takes into account the nonsymmetric nature of the data to produce a more efficient prediction interval. Further, the RS procedure can be utilized to approximate useful percentiles. For example to estimate the 25th percentile we


225

find d=.25(26)=6.5 and [6.5]=6. From the table, S6=10.50 and S7=10.58, and using (7.3.7) we compute the RS percentile

From this RS inference we estimate there is a good chance, about 75%, that the initial value of the stock will increase from the initial price of 10.5 and the magnitude of the increased can be measured by RS prediction intervals. Ex. 7.3.2. For a population of insurance claims the center interval containing 50% of the claims is to be estimated For exposition the SS of the exponential claim amounts given in Ex. 7.2.2 are considered to be a sample of observed claims. The SS yielded E{S}rs=106.3 and Var{S}rs=7586.41. We construct prediction intervals for 50% of the claims. The NPI takes the form of

which yields the interval

The RS sample mean, 106.3, centers this interval. The RSPI is constructed where (7.3.7) is applied and the probability in each tail is taken to be .25. For the left tail d=.25(26)=6.5 and, from (7.3.7),

For the right tail d=.75(26)=19.5 which produces

Hence, the more accurate prediction interval, RSPI, is

To demonstrate the comparison in accuracy we observe that for the exponential distribution with mean θ the p100th percentile is given by xp=−θ ln(1−p). Taking the RS mean θ=106.3 produces the 50% prediction interval

which is much closer to the percentile interval RSPI. The discrepancy is due to the sampling error in the generated data. The BCPI is considered in Prob. 7.6. Ex. 7.3.3. In this example statistical inference on the SS procedure presented in Ex. 7.2.3 is discussed. This AR(1) simulation is the counterpart to the RS method of the previous example. The ordered future values given in Table 7.2.3 are used to compute percentile values. For the 25th percentile, as in Ex. 7.3.1, d=.25(26)=6.5 and

Further, the resulting 90% RSPI for the future value is computed as


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As noted in Ex. 7.2.3 this particular AR(1) utilizes a smaller mean return rate resulting in a downward shift in the future value simulation statistics corresponding to the percentile and prediction interval computations. Simulation statistical inference is an emerging field of statistics and this discussion gives a brief insight into the basic concepts and financial and actuarial modeling applications. To further demonstrate and develop these ideas we now consider useful applications of the RS techniques in the field of collective aggregate models. 7.4 Simulation Inference on Collective Aggregate Models One of the most applicable financial and actuarial modeling settings for RS analysis is the case of a collection of separate individual policies or contracts. This is the case when a portfolio of investment holdings or a collection of insurance policies is to be analyzed for value at a future date and the financial rate is modeled as a random variable. In any case the collection of models leads to a set of random variables where the properties, such as tail probabilities, prediction intervals and moments, of the resulting aggregate sum are investigated. In practice the number of random variables making up the collection may be either fixed or random. The case where the number of random variables is fixed was introduced in Sec. 1.9 and approximation methods for computing the tail probability associated with these aggregate sums were considered The RS techniques pnesented in the previous sections can be utilized as a statistical check for the accuracy of these approximations. The situation of a stochastic number of variables yields a compound random variable that was discussed in Sec. 1.12. The statistical analysis for compound random variables is more complex and a particular theoretical setting that allows direct analysis is that of the compound Poisson random variable as presented in Sec. 1.12.2. In conjunction with these models the RS procedures allow the extension of these collective aggregate models over many theoretical settings, such as those yielding non-Poisson compound random variables. In this section the application of the RS techniques for both asymptotic formula checking and modeling extensions is demonstrated. As we have seen in Sec. 3.4 actuarial collective models are often analyzed over a short time period where the effect of interest can be ignored. The first example in this section deals with RS in this short time period setting. Ex. 7.4.1. In this example the portfolio of 25 the short-term insurance policies analyzed in Ex. 1.11.3 is analyzed using an SS procedure. The probability of a claim during the time period is .1 where the claim amounts B are distributed as normal random variables with mean $1,000 and standard deviation $200. Using (7.2.2), normal claim amounts are generated condition on a probability of .1 to form SS of the aggregate of the 25 policies claims. The probability that the sum of claims exceeds $5,000 is to be estimated where in Ex. 1.11.3, using the HAA and SPA methods, this probability was estimated to be close to .05. Based on r=50,000 repetitions using (7.3.3) the SS produced a RS survival estimate of


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We observe based on the previous computations that the one percent discrepancy demonstrates the efficiency of the HAA and SPA approximations and is probably due to the skewness in the claim variable data. Simulation methods allow for the analysis of more complicated models. In the general case there is a financial effect associated with time and any modeling of a collective set of random variables should take this into account. The interest or discount functions are nonlinear and the theoretical applications of statistical approximations become more difficult In this setting RS procedures can be utilized to extend statistical inference, such as the ones introduced in Sec. 7.3, to a more complex situations. For instance assume the general discrete time period model, discussed in Sec. 2.1.2, is applied for m consecutive time periods through future time tm. The cumulative financial rate, using the notation of (2.1.10), is given by ψj for 1≤j≤m. The present value of the aggregate sum Sn over m time periods is (7.4.1) The collective model given by (7.4.1) is general in nature and statistical measurements, such as tail probability estimation, are more complex. An RS example follows that demonstrates statistical analysis of (7.4.1). Further, in an analogous method the future value function can easily be constructed and analyzed using RS techniques. Ex. 7.4.2. In this example a collection of 25 insurance policies is analyzed over a series of discrete time periods. In any time period the probability of a claim is . 1 and the claim amounts are assumed to be exponential random variables with a mean of 100. The insurance policies are tracked for one year where the financial rate is .01 per month and claims are paid at the end of the month. Here (7.4.1) holds where m=12, n=25 and ψj=.01(j) for j=1,…, 12. Using aa SS for the exponential distribution, as in Ex. 7.3.2, based on r= 1,000 repetitions the RS mean and standard deviations for the present value function are computed and found to be E{PV(1)}rs=$28,114.18 and Var{PV(1)}rs=7,156.129. Applying (7.3.2) a (1−α)100% NPI for the present value function is computed as (7.4.2)

Applying (7.4.2), based on r=1,000, an approximate 95% NPI for the present value is computed to be (7.4.3) One of the important usages of RS is to check the accuracy of a large sample or asymptotic statistics. Using the empirical distribution function given in (7.3.3) the coverage for prediction and confidence intervals can be computed. In the case of (7.4.3) using r=1,000 we find F(14088)rs=.009 and F(42140)rs=.965 and find that the RS coverage probability is computed to be .965−.009=.956, slightly larger than desired. For


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this reason a percentile prediction interval, such as (7.3.5), may be more efficient. Applying the sample df (7.3.3) with 1000 repetitions we find F(43500)rs=.975 and F(16,000)rs=.025 and a 95% RSPI is given by (7.4.4) When comparing the two interval estimates, (7.4.3) and (7.4.4), we note that the width of the RSPI is less than that of the NPI due to, in part, the overestimation of the coverage probability in (7.4.3). The examples of this section dealt with deterministic status models and give a brief introduction to the possible types of RS statistical analysis that are possible. Our attention is now focused on stochastic status models with a special exposure to investment pricing models and stochastic surplus models. 7.5 Simulation Inference on Stochastic Status Models The nature of financial and actuarial stochastic status models often leads to the construction of complex scenarios and challenging statistical model analysis. The financial actions associated with stochastic status models are initiated or predicated by a random status and, hence, any analysis of these models must include a statistical modeling component. In the analysis of such models these complexities must be considered. The future lifetime random variable associated with the status, as well as other variables such as the financial rate, can be modeled using RS techniques. An introductory example is now given. Ex. 7.5.1. In this example the status has a future lifetime random variable with survival function given by (4.3.13) or

To generate an SS we apply the df inversion formula (7.2.1), solving for the inverse function, and generate future lifetimes using the formula

For exposition purposes this hypothetical future lifetime is applied to a continuous whole life insurance policy that is paid for by a continuous whole life annuity. The benefit paid is variable with time and is given by t1/2 for t≥0. From the basic present value model (6.1.1), the present value of the benefit is

Using the stochastic present value of the investment formula (7.5.1) the SS is found and the RS expectation, (7.3.1) with m=1, yields the net single value, denoted by SNVrs. The whole life annuity is considered a continuous annuity with level payments over the future lifetime. The present value of the whole life continuous annuity, (6.5.5), is given by


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Again using the SS the expected present value of the annuity is computed and is denoted by

Based on these RS quantities the RC and EP associated premium is given as

For example the SS technique is calculated for a status age x=25 where δ=.08 and taking the replication number to be r=1,000 and the results give

and

Hence, if each unit is worth $10,000 the premium payments would be $253.15 annually. We remark that in this example added complexity arises from two sources. One is the stochastic nature of the claim amounts and the other is the force of interest over time. The application of RS techniques is broad and may be applied to the analysis techniques and models presented in Chapter 6. We remark that the distribution of the future lifetime random variable may be either known or approximated In the case of Ex. 7.5.1, the distribution of the future lifetime random variable was assumed and the expectations were approximated by RS computations. In a more common setting the distribution of the future lifetime random variable is assumed, often in conjunction with scenario testing, and the resulting statistical measurements are found using RS techniques. In our investigation of the simulation analysis of random status models we consider two previously introduced stochastic status models, namely investment pricing models and stochastic surplus models 7.5.1 Investment Pricing Models The analysis of investment scenarios is considered where the financial action, such as a buy or a sell, at a particular future time is dependent on a stochastic structure. The concepts and formulas for the investment pricing model are taken from Sec. 3.3.2. The difference between that discussion and this investigation is that in this section the sale may take place at a random future time. The simplest case is where the stochastic status defining the future time of sale is a random variable that is not related to the price of the investment An introductory example of this is considered in the first setting. Ex. 7.5.1. An in vestment of value P is made where the guaranteed return rate is r=.02. From the investment pricing formula (3.3.15) the present value of this investment is

The investment has a stochastic yearly return rate that is taken to be uniform between .06 and . 18. Hence, δt=tδ where δ is distributed as a uniform random variable on (.06, .18). The length of time the investment is held depends on outside factors and is taken to be a


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normal random variable with mean 2 years and standard deviation of .5 of a year Based on an SS procedure based on repetitions of size r=10,000 we find the mean and variance of the future value of this investment scenario. We find E{PV(t)}rs=122.4159 and Var{PV(t)}rs= 117.1523. Hence, applying (7.3.2) the 95% NPI on the pricing present value is given by

From this we see the present value of this investment is estimated to be between 100.94 and 143.63. Depending on the setting or whether the investor is buying or selling, either the lower or upper bound in the prediction interval may be important to the investor. In a more realistic and complex setting the future time of sale may be dependent on factors related to either the future price or present value of the investment. For analysis in this case a discrete pricing model is applied. Using the mthly period model setting and notation, actions, such selling or buying, may be done after j periods or at future time tj=j/m for j≥1. The present value of the investment adapted from (3.3.15), sold at future time tj is (7.5.1) A financial investment strategy is often dependent on the future sales price. For a fixed sales price, denoted k, at unspecified future time the present value of the pricing model measures the difference in the future value and the selling price. Analogous to option pricing model (3.3.20), the stochastic status investment pricing model takes the present value function (7.5.2) In the normal rate setting it is assumed that the return rate δi is distributed iid n(γ, β2) for j≥1. The application of the stochastic investment pricing model in connection with the discrete time period model is demonstrated in the following example. Ex. 7.5.2. In this example the investment scenario analyzed by the Black-Scholes method in Ex. 3.3.4 is explored. An investment of P=100 is considered where the expiry or date of sale is at one year and the guaranteed yearly rate is r=.04. We apply normal return rates where β=.02 and the period is one month long so in each period r=.04/12=.003333 and β2=.0004/12. Using the conservative estimate of (3.3.17), γ=.0033166, and we apply an SS with 10,000 repetitions. We find

which compares closely with the Black-Scholes computation of .4364. We remark that RS prediction intervals for the present value, namely NPI, RSPI or BCPI, could be found leading to a more complete picture and a reliability measure of the present value point estimate,


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The generality of the SS method has great advantage in financial investment analysis since there are so many prospective variables an investor may want to consider Applying the present value asset pricing formula in (7.5.2) in conjunction with SS to a variety of settings produces many possibilities for statistical analysis. One of these is the analysis of the more complex European call option model. We now consider two investment particular purchase strategies and their associated analyses. Both examples deal with the European call option model. Ex. 7.5.3. An American call option, where the asset may be purchased for a fixed strike price of k on or before the expiry date, may be analyzed using an SS technique. An American call option, in the setting of Ex. 7.5.2, is analyzed where the sale may be made at any future time up to one year. This model is more complex than the European call option model with an added stochastic component The sale will be made the first time period or month the future value of the investment exceeds the price k. An SS consisting of 10,000 repetitions is conducted for various choices of k and the expectation of (7.5.2) is found. The results are listed in Table 7.5.1. From Table 7.5.1 we see that the American call option, as defined in this example, yields a smaller expectation than the corresponding European call option. This is due to the strategy of buying the stock the first time the price exceeds k thereby stopping its growth. Hence, a comparison of different option strategies depends on the exact financial actions, along with time triggers, to be defined. Ex. 7.5.4. In this example a stock purchasing strategy, similar to the previous ones, with a different twist is considered An investment or purchase of a stock is contemplated. The investment strategy employed is to purchase the stock the first time, within one year, the investment present value, given by (7.5.1), exceeds k. The initial price of the investment is P=100 and the rates are taken to be independent normal random variables with yearly mean and variance given by .06 and .0004. The guaranteed rate is assumed to be .04. Based on monthly time periods γ=.06/12=.005, r=.04/12=.003333 and β2=.0004/12 =.0000333. An SS consisting of 10,000 repetitions is conducted for various selling prices k. In this way an optimum or efficient value of k can be determined. The expectation of the asset pricing expected value, (7.3.1), is

Table 7.5.1 American Call Option SS Analysis k

100

101

102

103

104

105

E{(PV(1, k)}rs

.9808

1.0266

.68049

.3214

.1184

.0422

Table 7.5.2 Asset Pricing SS Analysis At Price k E{PV(1, k)}rs

100 1.1279

101 1.5787

102 1.6278

103 1.2050

104 .7087

computed and is given in Table 7.5.2. From Table 7.5.2 we observe that the stocks asset pricing value should increase over the year. Further, the SS yields the result that the maximum value of the present value of the investment (7.5.2) occurs when the sales price k is 102. This gives the investor an efficient financial investment strategy.


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7.5.2 Stochastic Surplus Models There are additional applications of the methods introduced in this chapter Simulation methods can be used to assess the financial stability of investment and actuarial models. In this section we consider the RS analysis of the surplus model introduced in Sec. 3.5. The surplus is defined by (3.5.1), Uj= u+jc−Sj where u is the initial fund, c is the period payment, Uj is the surplus and Sj is the aggregate loss, withdrawals or claims for periods j=1, 2,…. This model is interpreted as a stochastic status model where the status is defined by the condition of a positive surplus. The future lifetime random variable is the future time of ruin Tr. The application of RS techniques allows for a broader application and analysis of the surplus model, especially in terms of the underlying distribution of losses that is assumed Simulation techniques have an advantage over asymptotic or theoretical analysis in that models can be examined for any number of periods. Theoretical analysis in the finite number of period setting is complex and is dependent on the statistical distribution of prospective losses. Aggregate data samples are drawn sequentially until either ruin is observed or the maximum number of periods, denoted n, is reached. Let the probability of ruin within n periods be

Table 7.5.2 SS Period Ruin Probability for Exponential Claims n

1

2

3

5

10

15

20

25

50

E{R(u)n}rs

.136

.165

.185

.193

.200

.206

.208

.203

.204

denoted R(u)n. An RS analysis of the stochastic surplus model is given in the following example. Ex. 7.5.5. The surplus model in Ex. 3.5.2 is considered where the claim amounts are exponential random variables with mean one and the period payment is c=2. An SS is drawn for samples sequentially until Uj<0 or n periods is reached. The SS method is used and the number of repetitions run is r =10,000. In Table 7.5.3 the observed ruin probability E{R(u)n}rs, using (7.3.1), is given of various values of n. From Table 7.5.2 we see that the observed ruin probability using SS is close to the theoretical ruin probability of .203 calculated in Ex. 3.5.2. We further note that in practice the number of repetitions is increased until convergence of the observed probabilities is achieved. From Table 7.5.2 we see that the observed ruin probability using SS is close to the theoretical ruin probability of .203 calculated in Ex. 3.5.2. We further note that in practice the number of repetitions is increased until convergence of the observed probabilities is achieved. In the analysis of stochastic surplus models other variables are of interest to the modeler. One such variable is the magnitude of the loss at the time of ruin while another is the mean surplus amount over a fixed or random time period. The analysis of general surplus models is theoretically complex and RS techniques give an avenue for economic and statistical evaluation.


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7.6 Future Directions in Resampling In this section we present an introduction to a few advanced simulation and resampling topics and discuss their impact on financial and actuarial modeling. The invention of powerful sampling and analysis algorithms allows for the analysis of large data sets that lend themselves to financial and actuarial modeling. Unlike many scientific areas there is no lack of data. Financial data as well as mortality and survival statistics are observed and published regularly. There exist many possibilities for the applications of computer intensive modeling procedures in connection with the estimation and statistical analysis of these models. We consider only possible research areas connected with the theory and application of simulation resampling. In the framework of simulation and resampling techniques Markov chain and Baysian modeling are also considered. Theses topics offer great potential for further scientific development. We first consider the broad topic of resampling techniques. One problem connected with resampling simulation methods is the realization of unstable estimators. This can happen in conjunction with the bootstrapping sampling techniques introduced in Sec 7.2.1 when the underlying distribution of the data is heavy tailed or skewed. This type of data is encountered in actuarial analysis with one example being in terms of insurance claims. In general terms an estimator is unstable if small changes in the data can produce large changes in predicted values. Unstable estimators will produce wide prediction intervals using either the normality based NPI method or the resampling RSPI technique. This trait is due to the large variation in such estimators. Further, convergence rates in connection with aggregate models will be adversely affected by unstable estimators. One solution to the unstable estimator problem is a technique introduced for jackknife resampling referred to as bagging and is based on bootstrap aggregation of the data. This procedure was first introduced by Breiman (1996) and for a general review we refer to Buhlmann and Yu (2002). Bagging, along with associated variations, can be extended with additional theory to the BS setting of Sec. 7.2.1. The BS procedure introduced in Sec 7.2.1 produces a series of associated statistics given by Si for i≥1. In the situation of unstable estimation the Si quantities are too dependent on the set of values serving as the population. In bagging, a function of Si is substituted that produces a set of more stable estimators, In the original context, that of the jackknife modeling where sequentially one element is excluded to produce samples, the expectation is suggested as a stabilizing function. Other possibilities exist such as variants based on the classical trimmed or winsorized mean estimators. In these estimation procedures the least and greatest realized values are altered. Many other possibilities exist and much work is left to be done in the area of stable simulation sampling estimation in the context of financial and actuarial modeling. Connected with general simulation techniques is the inference methods that are applied. The theory and application of bootstrap confidence intervals continues to be a source of much exploration in scientific literature. New BS confidence interval methods that improve the accuracy of interval estimates of the form presented in Sec. 7.3 have been proposed. Some of these methods reduce the bias presented by typical bootstrap estimators and have been included in this chapter. These simulation methods include the ABC, BCa, bootstrap-t and nonparametric intervals. For a review of these intervals we refer the reader to Diciccio and Efron (1996). These confidence intervals can be adapted


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to many settings in financial and actuarial modeling thereby producing efficient prediction intervals. The general bootstrap procedure can be extended to the dependent modeling structure. In the general dependent structured case the construction of BS samples is more complicated than in the independent setting. In Ex. 7.2.3 BS sampling in connection with the dependent modeling situation consisting of an AR(1) model with normal variables was presented. Other more general and advanced dependent bootstrapping methods exist. A partial list of these methods includes the block. sieve and local bootstraps. For a review and comparison of these we refer to Buhlmann (2002). Much work can be done in the dependent simulation and resampling setting. One popular dependent probabilistic modeling technique is that of Markov chain modeling. The discrete Markov chain theory concerns the migration of individuals between predefined states and has been extended to simulation methods. Markov chain simulation methods have been used extensively in various areas of scientific investigation with areas of application including statistical physics, spatial statistics and Bayesian modeling. These dependent modeling methods have yet to be widely applied to financial and actuarial models. For example, group survivorship models and tables can be constructed and analyzed using Markov chain techniques. The statistical variability of life table measurements, such as yearly mortality or survival rates, expected future lifetimes and overall reliabilities have yet to be efficiently addressed and Markov chain construction provides a possible avenue of attack. The application of Markov chain theory and modeling techniques provides a wide source of modeling possibilities. A last avenue for further developments in resampling is in terms of Bayesian applications. The utilization of Markov chain models results in greater opportunities in connection with the Baysian modeling of financial and actuarial systems. Using methods such as the Metropolis or Gibbs-Hastings algorithm a general method for the simulation of stochastic processes has been demonstrated For review articles we refer to Besag, Green, Higdon and Mengersen (1995) and Geyer (1992). In this section we have discussed only a few of the new and exciting simulation methods that are open to the modern researcher. The topics we discussed are applicable to both theoretical and applications-oriented research and will enjoy great advancement over the coming years. Problems 7.1 For a status the survival function is given by s(x)=1/(1+x) for x>0. a) Generate an SS of size r=50. b) Using a) construct an approximate 95% confidence interval on the mean using (1.8.11). c) Compute the true coverage probability for the interval constructed in b) and give any comments. 7.2 A sampling of insurance claim amounts yields the data in terms of thousands of dollars: 1.2, 2.5, .8, 1.3, 2.4, 3.5, 1.1, .7, 2.4, 1.4, 3.2, .9, 1.3, 1.4, 1.8, 3.3, 2.1 and 1.5. Generate a BS sample of size r=50 and compute an approximate 95% confidence interval for the mean claim amount using (1.8.11).


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7.3 Consider the exponential RS claim data given in Table 7.2.2. For each claim the loss is given by Si=Xi−π where Xi is the claim and π is a premium payment. Using (1.8.11) construct an approximate 95% confidence interval for the expected loss. 7.4 For the RS future values corresponding to the AR(1) modeling of the stock prices given in Table 7.2.3 find an approximate 95% prediction interval for the mean future value. 7.5 For the RS data sets produced in Prob. 7.1 and Prob. 7.2 find 95% prediction intervals using both the RSPI and BCPI methods. 7.6 For the exponential claim data produced in Ex. 7.2.2 compute a 50% BCPI for a future claim. Compare this to the RSPI constructed in Ex. 7.3.2. 7.7 In this problem the short term portfolio of 25 insurance policies where the probability of a claim is .1 discussed in Ex. 7.4.1 is to be analyzed. The survival function for the claim values is given in Prob. 7.1. a) Produce an SS of size 40 for the aggregate claim amount corresponding to the 25 policies. b) Based on a) give 75% prediction intervals for the aggregate present value using NPI, RSPI and BCPL 7.8 Consider the extension of the portfolio of 25 insurance policies discussed in Prob. 7.7 to include the force of interest over time. The policies are tracked over a 6-month time period where the financial rate is .02 per month and the present value is given by formula (7.4.1). Run an RS analysis where the SS consists of 100 runs and both the NPI and the RSPI are found for the present value function. 7.9 In this problem the continuous whole life insurance policy with future lifetime survival function given in Ex. 7.5.1 with the change that the benefit is given by the t2 is to be analyzed. Here x=25 and δ=.1. Using an SS with r= 1,000 find the single net value SNVrs, the annuity expectation and compute the resulting RC premium. 7.10 The investment scenario of Ex. 7.5.2, where the Black-Scholes method was analyzed is reexamined An investment of P=100 was made where the expiry of one year for an American call option is taken to be one year and r=.04. Using monthly time periods run an SS consisting of 1,000 repetitions where the yearly return rates are taken to be uniform between .02 and .08 and the sales price is k=104. For present value function given by (7.5.2) give the RS expectation and 95% NPI. 7.11 Revise Prob. 7.10 to fit the conditions of a European call option where the sale is made the first month the price reaches 104. Using an SS where r=1,000 find a 95% NPI for the present value function and compare the results to Prob. 7.10. 7.12 In a stochastic surplus model let the claim amounts have survival function given in Prob. 7.1. The payments for each period is given by c where u=1. For analysis SS are run where r=1,000. For various values of c compute the RS expectation for the probability of ruin within 20 periods. For what value of c is the probability of ruin at most .10?

8 Further Statistical Considerations

In the previous chapters we have presented material, both theoretic and applied, concerning the modeling of financial and actuarial systems from a statistical point of view. In financial statistics areas such as investment analysis and asset pricing were considered. In actuarial science, basic topics, such as insurance, annuities, reserves and pension plans were investigated In both areas we have applied statistic inference techniques to not only analyze possible future economic actions but also to come to a better understanding of these economic and financial systems. Statistic inference techniques, such as estimation, percentile formulations, and prediction intervals shed much light on these models. There is room for many exciting new statistical techniques, as well as some standard ones, in these models. In this chapter we consider a few statistical considerations not addressed in the previous chapters. In financial and insurance modeling there is no lack of available data. The stock prices as well as various consumer indexes are published daily. The data on human survivability that comprise the basic information in life and mortality table construction is surveyed constantly, It is the goal of many modern statistical methods to make sense of the vast supply of data available to the financial and actuarial advisor In the first section we give a brief introduction to statistical methods used in the construction of data classification techniques. Details for these statistical techniques are left to the interested reader to locate and relevant references are listed. Actuarial analysis, whether it is insurance pricing, pension planning or reserves analysis depends on the associated life and mortality tables. For this reason statuses associated with individuals are classified into various risk categories. For insurance purposes demographics, such as gender, smoking-nonsmoking, and geographic region influence mortality rates. In the past it was the job of the insurance underwriter to asses these added risks and measure their impact on survivability and insurance rates. This was often done without the aid of modem statistical techniques, In the second section in this chapter we present an introduction to the topic of statistical adjustments to the mortality rates based on risk categories. The mortality tables and resulting statistical measurements for individual statuses assume a steady-state environment. This is not the situation in practice. As time advances the mortality rates change for individuals in all risk categories. Modern advances in medicine and other areas produce a decrease in mortality rates as the future time variables increase. In the last section we present a method to adjust mortality tables for mortality

Further statistical considerations

237

trends in time based on a simple linear regression model. There are many methods present in the statistical literature to assess and make adjustments for time trends in data. The new statistical techniques presented in this chapter represent just the beginning of modern statistical techniques applied to financial and actuarial modeling. The discussion and techniques chosen represent important areas of research to financial and actuarial modeling. 8.1 Statistical Investigations In both fields of financial and actuarial modeling new data are created continuously. In financial analysis consumer indexes, stock prices and other factors that influence economic systems are continuously observed. In connection with actuarial science, advances in the accompanying statistical analysis of modern medical advances produces information on the survivability of individuals. It is the goal of many standard as well as modern statistical techniques to make sense of this great body of data. Classical methods exist in the statistical literature that concern the wide area of data reduction and the analysis of large data sets. In this section we present a brief introduction to some of the applicable statistical techniques and their associated references. In both financial and actuarial modeling the analysis and reduction of large data sets are important. In investment modeling the researcher must decide among variables, out of an ever expending collection, to form a set that is important to the future value of a portfolio under consideration. The field of insurance underwriting consists of analyzing and computing the adjustment of mortality rates, thereby affecting insurance premiums, for added risk factors. These risk factors include individual demographics such as gender, smoking-nonsmoking, general health conditions and lifestyle. The introduction of modern statistical techniques into this important field of insurance requires a thorough investigation into the nature of survival data and the efficient construction of efficient classifications. As time advances, mortality and survival data become higher and higher dimensional with the introduction of variables affecting these rates. In the future not only will standard demographics such as gender, individual health and lifestyle factors be measured, but other factors such as family health histories and specific medical measurements will be included in actuarial assessments. More survival data will be collected on these survival factors forming high-dimensional sets of data. The statistical area of high-dimension data reduction and analysis includes standard techniques such as classification, cluster, principal components and discriminant analysis. In classification and cluster analysis statistical patterns in the high dimensional data sets are explored. Graphical techniques such as proximity graphics have been introduced to aid in the analysis of these data sets. In some large data sets the search for data patterns is enhanced by the construction and application of mathematical and statistical functions, such as dissimilarity measures, that are applied to individual high-dimensional data points. For a review of these techniques we refer to Ryzin (1976). There many statistical techniques, classical and new, for the analysis of large highdimensional data sets. Two classical techniques, out of many, are now mentioned In principal components and discriminant analysis the goals are data reduction and


238

interpretation. Linear combinations or discriminant functions within the high dimensional data sets are utilized to this end. For a further look at these procedures we refer to Johnson and Wichern (1982) and Morrison (1976). 8.2 Mortality Adjustment Factors The construction and analysis of financial and actuarial models is dependent on the associated financial and mortality tables that are utilized. In many applications investment and survival data are only recorded on broad status groups. As time advances data collection and analysis techniques also advance. In actuarial modeling individual statuses are classified into separate strata based on survival characteristics. Demographics, such as gender, smoking-nonsmoking, region of the country and lifestyle analysis are used to affect the mortality measurements used in actuarial and associated statistical analysis. For some demographics, such as gender, separate mortality tables are currently being constructed and applied in the insurance industry. It is the task of the insurance underwriter to assess these survival risks and adjust the relevant computations. In the future the underwriting and mortality assessment of individual statuses will be aided, much more than it has been, by standard and new statistical techniques. In this section we present an introduction to the topic of mortality adjustment for individual statuses. The mortality adjustments are made on standard mortality table measurements and are associated with higher-risk or increased mortality statuses. We remark that decreased mortality adjustments could be made for statuses associated with lower risks. The adjustments presented are robust in that they are applied to current mortality tables based on wider characteristics and are a function of comparatively simple measures. 8.2.1 Linear Acceleration Factors The simplest form of mortality adjustment, referred to as a linear acceleration factor associates the future lifetimes of the increased risk and the standard or control groups. The concept of linear acceleration factor modeling is taken from applied engineering reliability modeling and as a reference we refer to Tobias and Trindade (1995, Ch. 7). As mentioned the lifetimes of individuals taken from two different groups are directly associated One group serves as the control group, without added mortality risk, and is associated with future lifetime random variable T. Mortality data may be available for this group in the form of a mortality table. The second group consists of statuses that are associated with higher risk leading to increased mortality values and the future lifetime random variable is denoted Ts. In a linear acceleration model we relate the two future lifetime random variables by the linear relation (8.2.1) where the linear acceleration factor is AF. Here, if AF>1 then the added stress produces a shorter future lifetime with T>Ts. The converse is true that if AF< 1 then the associated reliabilities are increased over the control group. The relationship defined by (8.2.1) can


239

be used to relate statistical functions, such as the df. or reliability, associated with the two groups. The theory and application developed in linear acceleration modeling concept can be applied to financial and actuarial models. From (8.2.1) we relate the survival functions corresponding to T=t and Ts=t by the reliability (8.2.2) The relation of the survival functions given by (8.2.2) can be reformulated utilizing the force of mortality function. If the force of mortality for the control group is µx+t then using (4.3.3) we have the reliability (8.2.3) To explore the above decreased survivor function the exponent can be rewritten. Letting w=s/AF we find s=wAF w so that ds=dwAF and (8.2.3) becomes (8.2.4) From (8.2.4) we see that the linear acceleration factor affects the reliability in two ways. The first is a direct multiplication effect and the second is an adjustment to the force of mortality scale. For specific forces of mortality the associated adjusted survival rates, and hence mortality rates, can be computed and we follow with two specific examples. Ex. 8.2.1. For a specific status let the future lifetime random variable take the form of the exponential random variable with mean θ. The survival rates are taken to be (8.2.5) From Ex. 4.3.1 we note that for the exponential random variable the force of mortality is a constant given by µx+t=µ=1/θ for t>0. Hence, for this distribution the survivor probabilities of the form (8.2.4) become (8.2.6) where θs=θ/AF. From (8.2.6) we note that the distribution of the future lifetime random variable associated with the increased stress statuses is exponential with mean given by θs. As stated earlier decreased survivor rates and increased mortality rates are associated with AF>1. Ex. 8.2.2. In this example we let the future lifetime random be Weibull with survival rate given by (8.2.7)


240

for positive constants a and β. In (8.2.7) the shape parameter is a and the scale parameter is β. Based on (8.2.2) the survival function for the stress group is written as (8.2.8) From (8.2.8) we observe that the reliability associated with the higher risk is effected through a change in the scale parameter. Clearly, as AF increases the reliabilities decrease and the mortality rates increase. As we have observed, linear acceleration factors can be used to model the effect of higher-risk statuses on mortality computations. In this area there are many open statistical questions. One area that needs to be addressed is the topic of the estimation of the acceleration factor constant AF. In a practical sense we desire robust estimates for increased risk mortalities that can be used in conjunction with constructed mortality tables. This is considered in the next two sections, The one-year survival rates associated with high risk statuses are represented by (8.2.9) In the next two sections we introduce two possible general techniques for computing or approximating increased risk probabilities of the type given in (8.2.9). These techniques are chosen due to there reliance on a minimal set of assumed information. 8.2.2. Mean Acceleration Factors As we have noted, to apply linear acceleration factors to established life or mortality tables the adjustment factor, AF, must be estimated. In this section we introduce a general strategy to adjust the mortality rates based on the mean of the future lifetime random variable, Two modeling procedures are introduced that are representative of the multitude of possibilities. We first consider the case where the future lifetime random variable for each year is modeled by an exponential random variable. In this case the AF quantity is estimated by computing the ratio of the mean future lifetime of the control group to the increased mortality risk group. For statuses at base age x let the mean of the future lifetime random variables for the control group and the increased risk group be denoted by µx and µx,s, respectively. The ratio used in the mortality adjustment factor is denoted by (8.2.10) We remark that in practice the means of the two groups, or representative approximations, may be readily available to the modeler. For example a computed life table may be used to estimate µx. The increased mortality risk mean, µx,s, may be estimated by expert testimony or based on other smaller data sets, We model the future lifetime random variable associated with any age using the exponential distribution Then using (8.2.6) we observe the mean future lifetime associated with the increased mortality group is θs=θ/AF. Further, (8.2.10) implies that


241

(8.2.11) For computed ratios r mortality adjustments to the standard mortalities can be made. Combining (8.2.11) and (8.2.6) we have the adjusted survival and mortality rates for the high stress group for selected years defined by (8.2.9) taking the form (8.2.12) The above technique suggests an easy method to adjust mortalities where only the ratio given in (8.2.10) is required. However, the exponential model assumption requires that the force of mortality associated with the future lifetime random variable be constant This model may not always fit the data in a given mortality table. Ex. 8.2.3. In this example a status has a limiting age of 10 and the survival numbers and rates are given in Table 8.2.1. Using the standard formulas the mean lifetime is computed to be µ=4.35. For a high-stress increased mortality group the mean lifetime is estimated to be µs=2.00. From (8.2.10) we find r=4.53/2=2.265. Hence, using (8.2.12) we have survival and mortality rates (8.2.13) The adjusted survival rates associated with the various ages are also listed in Table 8.2.1. We observe that the mean lifetime for the increased mortality status, based on (8.2.13), overestimates the target mean of 2.0, where we compute µs= 2.64834. This leads us to think that the exponential model for the future lifetime random variable may not be the best choice.

Table 8.2.1 Mortality and Exponential Adjusted Mortalities x

0

lx

100

1 95

2 85

3 68

4 56

5 46

6 38

7 29

8

9

20

12

10 4

px

.950

.995

.800

.824

.821

.826

.763

.690

.600

.333

0.0

px,s

.890

.777

.603

.644

.640

.649

.542

.431

.314

.083

0.0

There exists a multitude of choices for the construction of mortality adjustment functions based on the ratio of mean future lifetimes. We now present a method where the future lifetime random variable for each ages is modeled by the Weibull survival function given by (8.2.7). The theoretical mean for the Weibull distribution is given by (8.2.13)


242

In this modeling procedure we assume that both the control and the high-stress status have identical shape parameter a but differing scale parameters, denoted by β and βs, respectively. For a given ratio of the form (8.2.10) using (8.2.13) we have the relation (8.2.14) Using the df given by (8.2.7) associated with the high risk status the yearly mortality rates are defined by (8.2.9) and can be written as (8.2.15) where r(α)=rα. Hence, to use (8.2.15) in practice we need an estimate for the shape parameter α. To estimate α using a particular data set or mortality table we apply the method of least squares in conjunction with a Weibull probability plot. To do this, for i≥1 let the survival probabilities corresponding to each year be given by (8.2.16) We now define transformations ai=ln(−ln(s(i))), bi=ln(i)ln(−ln(s(i))) and ci= ln(i) for i≥1. The least squares estimator is given by (8.2.17) Using (8.2.15) in conjunction with the least squares estimator (8.2.17) adjustments to mortality tables can be made for high-risk statuses for a given set of survival or mortality data. This procedure is demonstrated in the next example.

Table 8.2.2 Mortality and Weibull Adjusted Mortalities x

0

1

2

3

4

5

6

7

8

9

10

lx

100

95

85

68

56

46

38

29

20

12

4

px

.950

.995

.800

.824

.821

.826

.763

.690

.600

.333

0.0

Px,s

.890

.777

.603

.644

.640

.649

.542

.431

.314

.083

0.0

Ex. 2.8.4. Increased mortality adjustments for the mortality table given in Ex. 2.8.5 are considered using the Weibull form outlined above. For the increased mortality group we assume a mean future lifetime of µs=2.0. We compute r=2.265 and using (8.2.17) we find the least squares estimate and function

Hence, the increased mortality group survival rates are computed as


243

(8.2.18) These decreased survival rates are given in Table 8.2.2. The computed mean future lifetime associated with rates (8.2.18) is computed to be µs=1.708. We remark that the Weibull adjusted survival rates, given by (8.2.18), are more efficient than their exponential counterparts, as defined by (8.2.13), in that they produce a mean future lifetime closer to the target 2.0 value. In this section we have presented a general mortality adjustment technique that can be applied on any given mortality table. We have demonstrated the technique using the exponential and Weibull reliability distributions. There are many other statistical reliability models that an experimenter could consider in new investigations. In the next section we introduce an additional method of constructing adjusted mortalities. 8.2.3 Survival Acceleration Factors In many survival analysis applications data are recorded on the survival rates based on a fixed number of future whole years lived. This is the case in medical studies where survival rates for various diseases, such as specific forms of cancer, are recorded and analyzed. In this section for the stress or increased risk groups the m future year survival rate is fixed at mpx,s. A mortality adjustment to a standard mortality table is presented. The adjustment procedure we discuss is based on the exponential distribution, but, as with the last section, there are many other directions an investigator could take to construct this type of mortality adjustment. To construct an adjustment to the mortality rates presented in a mortality table we use the exponential survival function of the form (8.2.5). The t year survival probability associated with the increased mortality status group is a function of the initial age x and is modeled by the relation (8.2.19) for fixed t>0 and acceleration factor AF. We fix the survival rate after m future years at mpx,s and solve (8.2.9) for AF. The solution is given by (8.2.20) We now combine (8.2.19) and (8.2.20). The adjusted mortality rates corresponding to any age x can be written as (8.2.21) where (8.2.22)


244

We remark that in this construction the experiment needs only the ratio, or an estimate of the ratio, (8.2.22) to project the mortality adjustments over the entire mortality table. This is demonstrated in the following example. Ex. 8.2.5. For expository purposes the mortality data given in Ex. 8.2.3 is revisited. The survival acceleration factor technique presented in this section is applied to this data set. From the original mortality data the 3-year survival rate is computed to be

Table 8.2.3 Mortalities and Survival-Adjusted Mortalities x

0

1

2

3

4

5

6

7

8

9

10

lx

100

95

85

68

56

46

38

29

20

12

4

px

.950

.995

.800

.824

.821

.826

.763

.690

.600

.333

0.0

px,s

.828

.867

.697

.718

.715

.720

.665

.601

.523

.290

0.0

For the increased mortality status group the 3-year survival rate is assumed to be only .5, so that from (8.2.22) the ratio r is

From (8.2.21) the adjusted mortality rates are found and are listed in Table 8.2.3. We remark that the 3-year survival rate for the high risk status is, as desired .500. The mortality adjustment procedures presented in the last two sections represent possible modeling techniques for statuses associated with increased risk or stress resulting in lower yearly mortalities. In the next section the topic of incorporating a time trend effect to mortality table construction is considered. 8.3 Mortality Trend Modeling In the mortality and life tables introduced in the previous chapters it is assumed that the mortality rates are constant over time. This is not the case as reflected in modern survival studies of humans that indicate that individuals are living longer and longer To construct accurate lifetime and mortality tables for many types of statuses the possibility of modeling for a mortality time trend is required In this section we present an introduction to time trend modeling for mortality data. Similar to the previous section, an adjustment procedure to be applied to a standard mortality table is proposed. To model for trends of mortality rates over time survival rates based on differing construction times are required. For this reason we assume that separate mortality tables are constructed at times indicated by the future lifetime random variable T. Thus, if past tables are constructed at whole years the set of possible times for T, based on the current time T=0, is given by St={−1, −2, …}. For any past time T=t the mortality tables give the set of mortality values qx(t) for x≥0. To model the time trend in the mortality rates we present a method based on simple linear regression, but various techniques are possible.


245

To model the time trend in mortality rates we apply a linear regression model to a collection of mortality tables taken at times in St. Modifying the simple linear regression model given in (1.13.1) we let the dependent or response variable be yt(x)=qx(t)/qx(0) and the independent variable be t. The resulting linear regression model is (8.3.1) We remark that the distribution of the error term and its effect on statistical inference is an open question. Using the method of least squares introduced in Sec. 1.13.1 the least squares estimator is given by (8.3.2) Based on the least squares estimator (8.3.2) the predicted mortality rate at future time T=t takes the form (8.3.3) We remark that in applying formula (8.3.3) to a given mortality table an adjustment for a time trend in mortality rates is spread throughout the entire set of age based mortalities. The procedure we have present is just one choice out of he collection of of many possible time trend modeling techniques. Standard methods as well as recent developments in statistical modeling have widened the scope of modeling choices for time trend data. Possible techniques for time trend modeling include nonlinear modeling and generalized linerar modeling techniques. For a review we refer to works by Ratkowsky (1983) and McCullagh and Nelder (1983). The choice of selection among competing models was investigated by Borowiak (1989). Ex. 8.3.1. In this example yearly based mortality rates are collected for base years represented by T=−3, T=−1 and T=0. The resulting collection of mortality tables for a given status given in Table 8.3.1. Based on the mortality data given in Table 8.3.2 the least squares estimator, given by (8.3.2), is computed for each age and is also listed in Table 8.3.2. These estimators for each year are all negative indicating a decreasing mortality rate time trend

Table 8.3.1 Mortality Trend Data and Response Mortality Rates

Response Variable

x

qx(−3)

qx(−1)

qx(0)

y−3(x)

y−1(x)

0

.016

.015

.014

1.1429

1.0714

−.0500

1

.016

.016

.015

1.0667

1.0667

−.0267

2

.017

.016

.015

1.1333

1.0667

−.0467

3

.018

.017

.016

1.1250

1.0625

−.0438

4

.019

.017

.017

1.1176

1.0000

−.0353


246

Table 8.3.2 Original and Projected Mortality Tables 1

2

3

4

5

E{K}

A10:5

Original

.014

.015

.015

.016

.017

3.937

.0593

Projected

.014

.0146

.0136

.0139

.0146

3.943

.0550

J=j

For an individual status age x efficient mortality tables can be constructed in the presence of a time trend. This is accomplished using the linear mortality time trend model given by (8.3.3). For status age x the correct mortality rates are given by qx+j(j) for j=0, 1,… The estimated mortalities are given by (8.3.4) Based on (8.3.4) an adjusted mortality table based on the mortality time trend model can be constructed. This technique is demonstrated in the next example Ex. 8.3.2. The mortality data and least squares estimators given in Ex. 8.2.1 are now utilized to form a time trend adjusted mortality table. Using (8.3.4) the adjusted yearly mortality rates are computed and are listed in Table 8.3.2. Further, the expectation for a unit benefit discrete 5-year term life insurance policy is computed for both the original and adjusted mortality values and is given in Table 8.3.2. We remark that the lower mortality values in the time trend adjusted model result in a lower expected value for the insurance policy. Hence, these adjustments are important to accurate actuarial measurements and computations such as premium computations. This section represents only an introduction to the modeling of time trends with mortality data. For the linear regression model approach presented in this section many open questions exist. The theory for the efficient modeling of the error terms and resulting prediction and confidence intervals are some of the many questions open to investigation. Problems 8.1 Let the future lifetime random variable T~n(µ, σ2). Apply the linear adjustment procedure defined by (8.2.2) What are the associated parameters associated with the stress or increased mortality group? 8.2 Consider the survival data listed below x

0

1

2

3

4

5

6

7

8

9

lx

1000

982

931

825

557

421

330

215

87

0

a) Compute px and qx for the ages listed in the table. b) Compute µ the mean of the lifetime random variable. c) Apply the mean acceleration methods presented in Sec. 8.2.1 and Sec. 8.2.2 with µs=µ/2 to find adjusted mortalities qx,s for the ages listed in the table.


247

d) Apply the survival acceleration method presented in Sec. 8.2.3 with 3po=3po,s/2 to compute adjusted mortalities qx,s for ages listed in the table. 8.3 In the context of Prob. 8.2 consider a discrete unit benefit whole life insurance policy for an individual status age x=2 where the interest rate is I= .08. Compute the expectation corresponding to the whole life insurance policy for the original table and all the adjusted mortality data values computed in parts c) and d). Is the resulting change in the expectations associated with the adjusted mortalities significant? 8.4 Using the linear regression model given by (8.3.1) use the method of least squares to derive the estimator defined by (8.3.2). 8.5 Mortality data were comprised at three different years corresponding to −2, −1 and the present time 0. The mortality data is x qx(−2) qx(−1) qx(0) 0

.014

.013

.011

1

.018

.017

.015

2

.021

.020

.018

3

.022

.021

.019

4

.024

.022

.020

For the ages listed in the mortality data set use formula (8.3.2) to compute the least squares estimates of the parameters. What do these estimators imply? 8.6 Consider the time trend data listed in Prob. 8.5. a) Use (8.3.4) to compute a time adjusted mortality table for an individual status age 0 at t=0 for the ages listed. b) Based on the mortalities corresponding to year 0 and the time trend adjusted quantities compute the expectation of a unit benefit discrete 4-year term life insurance policy where i=.06. c) Compare the expectations computed in part b).

Appendix

Appendix A1: Standard Normal Distribution Function Φ(x) z

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

−3.5

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

0.0002

0.0003

0.0003

0.0003

−3.2

0.0006

0.0007

0.0007

0.0007

0.0007

0.0008

0.0008

0.0008

0.0009

0.0009

−3

0.0013

0.0013

0.0014

0.0014

0.0015

0.0015

0.0016

0.0016

0.0017

0.0018

−2.9

0.0018

0.0019

0.0019

0.0020

0.0021

0.0021

0.0022

0.0023

0.0024

0.0024

–2.8

0.0025

0.0026

0.0027

0.0028

0.0029

0.0029

0.0030

0.0031

0.0032

0.0033

−2.7

0.0034

0.0035

0.0036

0.0037

0.0039

0.0040

0.0041

0.0042

0.0043

0.0045

−2.6

0.0046

0.0047

0.0049

0.0050

0.0052

0.0053

0.0055

0.0057

0.0058

0.0060

−2.5

0.0062

0.0063

0.0065

0.0067

0.0069

0.0071

0.0073

0.0075

0.0077

0.0079

−2.4

0.0081

0.0084

0.0086

0.0088

0.0091

0.0093

0.0096

0.0099

0.0103

0.0104

−2.3

0.0107

0.0110

0.0113

0.0116

0.0119

0.0122

0.0125

0.0128

0.0132

0.0135

–2.2

0.0139

0.0142

0.0146

0.0150

0.0153

0.0157

0.0161

0.0165

0.0170

0.0174

–2.1

0.0178

0.0183

0.0187

0.0192

0.0196

0.0201

0.0206

0.0211

0.0216

0.0222

−2

0.0227

0.0232

0.0238

0.0244

0.0249

0.0255

0.0261

0.0268

0.0274

0.0280

−1.9

0.0287

0.0293

0.0300

0.0307

0.0314

0.0321

0.0328

0.0336

0.0343

0.0351

−1.8

0.0359

0.0367

0.0375

0.0383

0.0392

0.0400

0.0409

0.0418

0.0427

0.0436

−1.7

0.0445

0.0455

0.0464

0.0474

0.0484

0.0494

0.0505

0.0515

0.0526

0.0536

−1.6

0.0547

0.0559

0.0570

0.0582

0.0593

0.0605

0.0617

0.0630

0.0642

0.0655

−1.5

0.0668

0.0681

0.0694

0.0707

0.0721

0.0735

0.0749

0.0763

0.0778

0.0792

−1.4

0.0807

0.0822

0.0837

0.0853

0.0869

0.0885

0.0901

0.0917

0.0934

0.0950

−1.3

0.0968

0.0985

0.1002

0.1020

0.1038

0.1056

0.1074

0.1093

0.1112

0.1131

−1.2

0.1150

0.1170

0.1190

0.1210

0.1230

0.1250

0.1271

0.1292

0.1313

0.1335

−1.1

0.1356

0.1378

0.1400

0.1423

0.1445

0.1468

0.1491

0.1515

0.1538

0.1562

−1

0.1586

0.1610

0.1635

0.1660

0.1685

0.1710

0.1736

0.1761

0.1787

0.1814

Appendix

250

−0.9

0.1840

0.1867

0.1894

0.1921

0.1948

0.1976

0.2004

0.2032

0.2061

0.2089

−0.8

0.2118

0.2147

0.2176

0.2206

0.2236

0.2266

0.2296

0.2326

0.2357

0.2388

−0.7

0.2419

0.2450

0.2482

0.2514

0.2546

0.2578

0.2610

0.2643

0.2676

0.2709

−0.6

0.2742

0.2775

0.2809

0.2843

0.2877

0.2911

0.2945

0.2980

0.3015

0.3050

−0.5

0.3085

0.3120

0.3156

0.3191

0.3227

0.3263

0.3299

0.3335

0.3372

0.3409

−0.4

0.3445

0.3482

0.3519

0.3556

0.3594

0.3631

0.3669

0.3707

0.3744

0.3782

−0.3

0.3820

0.3859

0.3897

0.3935

0.3974

0.4012

0.4051

0.4090

0.4129

0.4168

−0.2

0.4207

0.4246

0.4285

0.4325

0.4364

0.4403

0.4443

0.4482

0.4522

0.4562

−0.1

0.460l

0.4641

0.4681

0.4720

0.4760

0.4800

0.4840

0.4880

0.4920

0.4960

Appendix A2: Standard Normal Distribution Function Φ(x) z

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0

0.5

0.5039

0.5079

0.5119

0.5159

0.5199

0.5239

0.5279

0.5318

0.5358

0.1

0.5398

0.5437

0.5477

0.5517

0.5556

0.5596

0.5635

0.5674

0.5714

0.5753

0.2

0.5792

0.5831

0.5870

0.5909

0.5948

0.5987

0.6025

0.6064

0.6102

0.6140

0.3

0.6179

0.6217

0.6255

0.6293

0.6330

0.6368

0.6405

0.6443

0.6480

0.6517

0.4

0.6554

0.6590

0.6627

0.6664

0.6700

0.6736

0.6772

0.6808

0.6843

0.6879

0.5

0.6914

0.6949

0.6984

0.7019

0.7054

0.7088

0.7122

0.7156

0.7190

0.7224

0.6

0.7257

0.7290

0.7323

0.7356

0.7389

0.7421

0.7453

0.7485

0.7517

0.7549

0.7

0.7580

0.7611

0.7642

0.7673

0.7703

0.7733

0.7763

0.7793

0.7823

0.7852

0.8

0.7881

0.7910

0.7938

0.7967

0.7995

0.8023

0.8051

0.8078

0.8105

0.8132

0.9

0.8159

0.8185

0.8212

0.8238

0.8263

0.8289

0.8314

0.8339

0.8364

0.8389

1

0.8413

0.8437

0.8461

0.8484

0.8508

0.8531

0.8554

0.8576

0.8599

0.8621

1.1

0.8643

0.8665

0.8686

0.8707

0.8728

0.8749

0.8769

0.8789

0.8815

0.8829

1.2

0.8849

0.8868

0.8887

0.8906

0.8925

0.8943

0.8961

0.8979

0.8997

0.9014

1.3

0.9031

0.9049

0.9065

0.9082

0.9098

0.9114

0.9130

0.9146

0.9162

0.9177

1.4

0.9192

0.9207

0.9221

0.9236

0.9250

0.9264

0.9278

0.9292

0.9305

0.9318

1.5

0.9331

0.9344

0.9357

0.9369

0.9382

0.9394

0.9406

0.9417

0.9429

0.9440

1.6

0.9452

0.9463

0.9473

0.9484

0.9494

0.9505

0.9515

0.9525

0.9535

0.9544

1.7

0.9554

0.9563

0.9572

0.9581

0.9590

0.9599

0.9607

0.9616

0.9624

0.9632

1.8

0.9640

0.9648

0.9656

0.9663

0.9671

0.9678

0.9685

0.9692

0.9699

0.9706

1.9

0.9712

0.9719

0.9725

0.9731

0.9738

0.9744

0.9750

0.9755

0.9761

0.9767

Appendix

251

2

0.9772

0.9777

0.9783

0.9788

0.9793

0.9798

0.9803

0.9807

0.9812

0.9816

2.1

0.9821

0.9825

0.9829

0.9834

0.9838

0.9842

0.9846

0.9849

0.9853

0.9857

2.2

0.9860

0.9864

0.9867

0.9871

0.9874

0.9877

0.9880

0.9883

0.9886

0.9889

2.2

0.9892

0.9895

0.9898

0.9900

0.9903

0.9906

0.9908

0.9911

0.9913

0.9915

2.4

0.9918

0.9920

0.9922

0.9924

0.9926

0.9928

0.9930

0.9932

0.9934

0.9936

2.5

0.9937

0.9939

0.9941

0.9942

0.9944

0.9946

0.9947

0.9949

0.9950

0.9952

2.6

0.9953

0.9954

0.9956

0.9957

0.9958

0.9959

0.9960

0.9962

0.9963

0.9964

2.7

0.9965

0.9966

0.9967

0.9968

0.9969

0.9970

0.9971

0.9971

0.9972

0.9973

2.8

0.9974

0.9975

0.9975

0.9976

0.9977

0.9978

0.9978

0.9979

0.9980

0.9980

2.9

0.9981

0.9981

0.9982

0.9983

0.9983

0.9984

0.9984

0.9985

0.9985

0.9986

3

0.9986

0.9986

0.9987

0.9987

0.9988

0.9988

0.9988

0.9989

0.9989

0.9989

3.2

0.9993

0.9993

0.9993

0.9993

0.9994

0.9994

0.9994

0.9994

0.9994

0.9994

3.5

0.9997

0.9997

0.9997

0.9997

0.9998

0.9998

0.9998

0.9998

0.9998

0.9998

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Index

Adjustment Coefficient 127, 129 Aggregate Stop-Loss 124 American Call Option 117, 293 Annual Percentage Rate or APR 73, 76 Annuity or Annuity Model 72, 80, 83, 95, 97 Aggregate Parameters 188 Aggregate Sums of Random Variables 35, 69 approximations 44, 47, 48 mgf 41 simulation inference 287 Apportionable Annuities 227, 233 Auto Regressive Systems 61, 65, 70, 89, 281, 286 Binomial Random Variable 9, 22, 174, 182 Black-Scholes formula 114, 117, 131, 292 Central Limit Theorem 44, 50, 132 Collective Aggregate Models 119, 287 Complementary Events 5 Compounding Interest 73, 74 Compound Random Variables 51, 53, 122 Conditional Distribution 26, 27, 134 Conditional Probability 3, 134 Confidence Interval 37, 60, 100, 294 Contingent Status 154 Continuous Compounding 74, 76 Continuous Random Variable 10, 209 Continuous Rate Model 78, 83, 92 Consistency 36, 181 Convolution Method 39, 68 Correlation coefficient 31, 57 Covariance 31, 182, 194 Curtate future lifetime 137, 146, 151, 152, 186 Deductible 112, 124 Deterministic Model 102, 273, 282 Discount Function 78, 87

Index

258

Discrete Random Variable 7, 137, 210 Discrete Time Period Model 73, 77 Distribution Function 10, 18, 24, 141, 159, 279 Dividend 111, 125 Empirical Return Rates 89. 100 Equivalence Principle 104, 108, 110, 209, 237 Estimation 33, 90, 180 European Call Option 116, 131, 293 Expectations 15, 18, 22. 25, 31, 52, 216 Expected Discount Function 87, 93 Expected Future Value 87, 88, 93, 96, 97 Expected Present Value 87, 89, 90, 91, 93, 96, 97 Expected Rate Function 87, 88, 90, 93 Expense Models 265, 271 Expiry 117, 292 Exponential Random Variable 11, 67, 129, 135, 141, 148, 220, 226, 279, 304 Financial Rate 72, 73, 278 Force of Mortality 139, 140, 144, 150, 152, 160, 167, 303 Fractional Ages 145, 190 Fundamental Theorem of Calculus 10 Future Lifetime Random Variable 133, 134, 137, 142, 208 Future Value Expenditures 103 Future Value Function 73, 74, 78, 80, 81, 84, 93, 275 Future Value Revenue 103 General Time Period Models 76, 77, 249 Geometric Random Variable 8, 18, 27, 66, 138, 231 Geometric Sum 19, 66, 221 Gompertz 143, 170, 185 Graduation 197 Guaranteed Interest Rate 115 Haldane Type A Approximation 47, 50 Independent Random Variables 5, 32, 43, 146 Infinite sum 19, 66 Insurance Modeling 119, 124, 215, 288, 298 Insurance Premiums 236, 239, Notations 239, Integration by parts 66, Interest Rate 73, 75 Investment Pricing 114, 273, 291 Joint Life Status 5, 43, 149, 191 Joint Distributions 27, 158, 162 Kurtosis 16

Index

259

Last Survivor Status 5, 43, 151 Least Squares Estimation 59, 63, 89, 90, 307, 308, 311 Life Annuities 221, 251 Apportionable 227, 233 Deferred 225, 233, 269 Temporary 224, 226, 232, 253, 269 Whole Life 224, 226, 232, 234, 242, 251, 265, 269, 275 Life Insurance 215, 251 Deferred 219, 230 Endowment 218, 221, 229, 230 Pure Endowment 218, 229 Term 217, 219, 229, 230 Whole Life 217, 220, 229, 230, 231, 235, 242, 251, 265, 275 Life Table 172, 177, 180, 194, 198, 205 Limiting Distributions 53 Linear Acceleration Factors 303 Loading 111, 125, 237, 265 Loss Function 103, 104, 106, 126, 237, 242 Loss Model 103, 237, 242 Marginal Distributions 28, 68, 159, 162 Mean 15, 17, 23, 42 Mean Acceleration Factors 305, 313 Mean Number Of Future Years 185, 186, 188, Median 184, 185 Mixed Random Variables 13, 15, 18, 159, 209 Moment Generating Function 21, 49, 52 Moments 15, 18, 23 Mortality Adjustment Factors 302, 303, 305 Mortality Rates 133, 134, 158, 176, 177 Mortality Table 177, 187, 190, 195 Mortality Trend 310, 313 Multiple Decrement 157, 205, 254, 171, 198, 200, 205, 254, 257 Multiple Future Lifetimes 148, 191 Multiple Life Tables 191 Normal Distribution 12, 23, 45, 87, 115, 116, 280 Order Statistics 42 Option Pricing 116, 291 Pareto 170 Pension Benefits 257, 261, 263, 271 Pension Contributions 259, 271 Pension Plans 256, 271 Percentile Criteria 109, 212 Percentiles 36, 109, 212, 238, 283 Point estimation 34, 90, 181 Poisson Random Variable 9, 53, 67, 269

Index

260

Prediction Intervals 36, 60, 69, 92, 99, 216, 275, 282, 283, 298 Premiums 236. 239, 265 Notation 239 Present Value Function 76, 78, 81, 84, 87, 93, 207, 215, 273 Present Value of Expenditures 103, 104, 236, 241 Present Value of Revenues 103, 104, 236, 141 Probability Measure 2 Probability of Ruin 127, 294 Probability Density function 7, 10 Probability Plot 100, 307 Proportion 35 Random Variable 6, 7, 10, 13 Rate Function 77, 79, 81, 83, 87 Regression 56, 70, 310 Relations for Expectations among: Annuities 232 Insurances 229 Insurance and Annuities 231 mthly Period Expectation 251 Reserves 246 Resampling 277 Reserves 241, 243, 246, 268 Notations 244 Risk Criteria 108, 208, 210 Risk Evaluations 208 Saddlepoint Approximation 48, 50, 55, 69, 120, 132 Sample Mean 34, 36 Sample Variance 34, 36 Scenario Testing 272 Select Future Lifetimes 156 Select Life Tables 194 Short Term Insurance 119 Simulation 276, 282, 287, 289 Bias Correcting Prediction Intervals 284 Percentiles 282, 283 Prediction Intervals 282, 283 Simulation Methods 276 Bootstrap Resampling 277 Simulation Sampling 279 Single Decrement Probabilities 164, 202 Bounds 167, 202 Single Net Value 110, 112, 113, 116, 209, 210, 212, 222, 235, 255 Single Risk Models 110 Skewness 16 Standard Normal 12 Stochastic Loss Criterion 107 Stochastic Rate Model 84 Stochastic Status 206, 272, 289 Stochastic Surplus Model 126, 294

Index

261

Stop-Loss Model 124 Status Models 102, 206 Deterministic 102 Stochastic 206 Strike Price 116, 293 Sum of Random Variables 38, 44, 119 Support of a Random Variable 8, 10, 15, 209 Survival Acceleration Factors 308, 313 Survival Function 24, 25, 134, 140, 169, 149, 152 Survival Rates 134, 135, 140, 176 Survivorship Groups 173, 195, 201, 247 Taylor Series 64 Ultimate Life Tables 192 Unbiased Estimators 34 Uniform Distribution 10, 98, 144 Uniform Distribution of Deaths 146, 151, 165, 167, 190, 206, 230 Variance 15, 31, 40, 185, 187, 216 Weibull Distribution 167, 302, 304

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