Exam C Formulas

C Formula Sheet Probability

 f ( x)  F '( x)   F ( x)   f ( x)  F ( x)  1  S ( x)   S ( x)  1  F ( x)  S ( x)  e H ( x )   H ( x)   ln( S ( x)) h( x)  H '( x)   H ( x)   h( x) f ( x)   S '( x) f ( x) h( x )  S ( x)

E  X    x  p ( x) or k

k

x



x

k

 f ( x)dx

-

n'  E  X n  (nth Raw Moment) n n  E  X     (nth Central Moment)   1' is the mean

2  2'   2 is the variance 3  3'  32'   2 3 4  4'  43'   62'  2  3 4 Var ( X )  2   2 Standard Deviation = 

3 3  Kurtosis is  2  44 

Skewness is  1 

Coefficient of Variation is

 0 if X is symmetric

 

Skew( X )=Skew(cX )

n n! Combinations :    k  k !(n  k )!

Percentiles A 100p th percentile of a random variable X is a number  p satisfying: Pr( X   p )  p Pr( X   p )  p

Conditional Probability Pr( A  B) Pr( B) Pr( B | A)  Pr( A) Pr( A | B)  Pr( B) Pr( A | B) 

Scaling To scale a lognormal: X ~ lognormal   ,  cX ~ lognormal    ln c,  Let Y  cX y   y Then FY ( y )  Pr Y  y   Pr  cX  y   Pr  X    FX   c  c

Variance

E  XY   E  X  E Y  if Independent E  aX  bY   aE  X   bE Y 

Var  aX  b   a 2Var ( X ) Var  aX  bY   a 2Var ( X )  2abCov  X ,Y   b 2Var (Y )

 

Var ( X )  E X 2  E ( X )2 VaR  X   VaR 1  X  Cov( X , Y )  E  XY   E ( X ) E (Y ) Cov   A  B  ,  C  D    Cov ( A, C )  Cov ( A, D )  Cov ( B, C )  Cov ( B, D ) Cov( A, A)  Var ( A) Cov( A, B)  0 if Independent Cov( X , Y )  XY  (Correlation Coefficient)  X Y Bernoulli Shortcut: For any RV with only 2 values a and b: Var ( X )   a - b  pq 2

Parametric Distributions Distribution

f(x)

E(X)

Binomial n trials

n k nk  k  p (1  p)   p k (1  p)nk

np p

Bernoulli 1 trial Uniform continuous on [d, u]

Beta Exponential (memoryless) Weibull

1 u-d

ce

x    cx 1e 

x  1   cx e

Single Parameter Pareto

c

Lognormal



2 2

1    1    

2  2 1    



Gamma

Double Parameter Pareto

 u - d 2 if u  0

cx a1 (  x)b1



  1   1

x 1 c  1

 x  

 ln x    2 ce 2

x

2

e

 0.5 2

Median  e 

Var(X)

npq pq

d u 2 a ab

xd F ( x)  ud

x

 

E X2

3

 u - d 2 12  2 ab

 a  b 2 (a  b  1)

2

 2

 2  2 2 2   1  2 

e

2   2 2

 E  X 2 

  2

Frailty

Let h  x |    a( x) If a(x) is a constant then the hazard rate is exponential, otherwise Weibull  can be Gamma or Inverse Gaussian x

A( x)   a(t )dt 0

H  x |    A( x) S  x |    e A( x ) S ( x)  M    A( x) 

(Use MGF for )

Exponential a(x) leads to a Pareto distribution Weibull a(x) leads to a Burr distribution Conditional Variance Var ( X )  E Var  X | I   Var  E  X | I  E  X   E  E  X | I    P  Ai   E  X | Ai  E  X 2   E  E  X 2 | I  

Var  X   E Var  X | I 

Splices

1) Sum of functions must integrate to 1 2) To be continuous, functions must be equal at break point Shifting 3  1 If f     3e  

The mean = 1/3 + 1 (the mean of the unshifted exponential plus the shift)

Policy Limits X ^ d is the LIMITED EXPECTED VALUE X u X u

X X ^d  d

(Cost to Customer) 



0

0

Definition: E  X k    x k f ( x) dx   kx k 1S ( x)dx 



0

0

E ( X )   x  f ( x)dx 

 S ( x)dx

d

E ( X ^ d )   xf ( x)dx  d  S (d )  0 d

  S ( x)dx 0

E

 X ^ d     x k

d

k

f ( x )dx  d k  S (d ) 

0 d

  kx k 1S ( x)dx 0

d   If Y  (1  r ) X then E Y ^ d   1  r  E  X ^ 1  r  

Deductibles Ordinary Deductible of d – pays max(0,X-d) For d = 500, Loss <= 500 pays nothing, Loss of 700 pays 200 Franchise Deductible of d – pays nothing if Loss is less than d, and full amount if Loss > d

Payment per Loss with Deductible Ordinary Deductible 0 Payment from Ins. Co.   X  d 



 E  X  d  



E  X  d    ( x  d )  f ( x)dx  d

X d X d

 S (x)dx d



E  X  d  

k  

 d

k ( x  d )k  f ( x)dx  E  X k   E  X ^ d      

Payment per Payment with Deductible Ordinary Deductible Y P   X  d  | X  d FY P ( x) 

FX ( x  d )  FX (d ) 1  FX (d )

SY P ( x) 

SX (x  d ) S X (d )

e( d )  E  X  d | X  d  e(d )  E Y P    

E  X  d   S (d )



e( d ) 

  x  d  f ( x)dx d

S (d )



E X   E X ^ d  S (d )





 S ( x)dx d

S (d )

e(d )  Mean Excess Loss

E  X   E  X ^ d   E  X  d   pmt from customer

pmt from ins. co.

E  X ^ d   E  X ^ d | X  d   Pr  X  d   E  X ^ d | X  d   Pr  X  d    Average loss < d   Pr  X  d   d  Pr  X  d  Franchise Deductibles

Expected Payment per Loss  E  X  d    dS ( d ) Expected Payment per Payment  e( d )  d

Special Cases for e(d)

Distribution

e(d)

Exponential



Uniform  0,  2 Parameter Pareto

1 Parameter Pareto

 d 2

 d  1  d   1      d   d    1 

d 

d 

If X  Uniform  0,  , then  X - d  | X  d  Uniform  0,  d  If X  Pareto  ,  , then  X - d  | X  d  Pareto  ,  d 

If X  1 Parameter Pareto  ,  and d   , then  X - d  | X  d  Pareto  ,d  Loss Elimination Ratio

E X ^ d  LER(d )  E X  LER(d )  Expected % of loss not included in payment

Special Cases for LER(d)

Distribution Exponential 2 Parameter Pareto

LER(d) d

1 e   1    1    d  

1 Parameter Pareto

  d 1

 1





Properties of Risk Measures

Translation Invariance:   X  c     X   c Positive Homogeneity:  cX   c   X  Subadditivity:  X  Y     X    Y  Monotinicity:  X    Y  if X  Y A coherent risk measure satisfies all 4 properties VaR p fails subadditivity TVaR p is coherent E  X  is coherent

VaRp ( X )   p  FX1 ( p) VaR0.99 is the 99th percentile

Value-at-Risk

TVaR p ( X )  E  X | X VaR p ( X ) 

Tail-value-at-Risk







xf ( x)dx

FX1 ( p )

1 p 1

VaR ( X )dy y



p

1 p



 VaR p ( X )  e X VaR p ( X )

Distribution

Normal

Lognormal

VaR(X)

  z p

e

 



  zp  1 p

  z p

If given a mixture, use the survival function and solve for x

TVaR(X)

e 0.5 x where  ( x)  2

    z p   E  X   1  p   

2

Maximum Covered Loss For deductible 'd' and maximum covered loss 'u' 0  EX   X  d u  d 

X d d  X u X u

E  Payment per Loss   E  X ^ u   E  X ^ d  u



 S ( x)dx d

 E  X  d    E  X  u  

Policy Limit: the maximum amount the coverage will pay - If Policy Limit = 10,000 and d = 500 - Pays 10,000 for loss of 10,500 or higher - Pays Loss – 500 for losses b/w 500 and 10,500 Maximum Covered Loss: the maximum loss amount that is covered - If MCL = 10,000 and d = 500 - Pays 9,500 for loss of 10,000 or higher - Pays Loss – 500 for losses b/w 500 and 10,000

Coinsurance

E  Payment per Loss  

      E X ^ u   E X ^ d  Coinsurance  MCL  ded    



Coinsurance of 80% means that the insurance pays 80% of the costs

Inflation

u  d     E  Payment per Loss    (1  r )   E  X ^  E X ^  1  r  1  r     Variance of Payment per Loss with a deductible X  Loss RV Y L  payment per loss RV E Y L   0  Pr  X  d   E Y P  Pr  X  d 

 







  

Var Y L  E Var Y L | Case   Var E Y L | Case    2  0  Pr  X  d   Var Y P Pr  X  d    E Y P   0  Pr  X  d   Pr  X  d    



 



Bonus Pay Bonus of 50% of (500 - X) if X  500 B  0.5Max  0,500 - X   0.5Max  500  500,500 - X   0.5  500  Min  500, X    0.5  500  X ^ 500   250  0.5E  X ^ 500

Discrete Distributions The (a,b,0) class Distribution Poisson

a 0

Variance vs. Mean Variance = Mean

Negative Binomial (Geometric is NB with r = 1)

0

Variance > Mean

Binomial

0

Variance < Mean

   Pr  N  n     1  

n

Geometric Distribution (memoryless)

A sum of 'n' independent Negative Binomial random variables having the same  and parameters r1,..., rn has a Negative Binomial distribution with parameters  n

and

r

i

i 1

A sum of 'n' independent Binomial random variables having the same q and parameters m1,..., mn has a Binomial distribution with parameters q n

and

m

i

i 1

pk b a pk 1 k

k  1,2,...

ab 1 a ab Var  N   1  a 2 EN  

 x  x( x  1)  ( x  n  1) n  n!   Probability Generating Functions P ( n ) (0) pn  n! p0  P(0)

( n )  P ( n ) (1) P '(1)  E  X  P ''(1)  E  X ( X  1)  P '''(1)  E  X ( X  1)( X  2)  If given a primary and secondary pgf, substitute the secondary pgf for 'z' in the primary pgf.

The (a,b,1) class

pk b a pk 1 k

k  2,3,4,...

Zero-Truncated Distributions

p0T  0 pkT 

pk 1  p0

Zero-Modified Distributions

p0M  0



 1 p p

pkM  1  p0M 

k

0





pkM  1  p0M  pkT 1  p0M E M   E Orig   1  p0M E  Zero  Truncated  1  p0





 E  N   cm Var ( N )  c 1  c  m 2  cv c  1  p0M m is the mean of the corresponding zero-truncated distribution v is the variance of the corresponding zero-truncated distribution Sibuya ETNB with  1  r  0 and take lim as   

a 1 b  r 1 p1T  r

Poisson/Gamma The Negative Binomial is a Gamma mixture of Poissons N ~ Poisson   

 ~ Gamma  ,  Negative Binomial ( r )  Gamma( ) Negative Binomial (  )  Gamma( ) Gamma Mean   Variance   2 Negative Binomial Mean  r  Variance  r  1    Negative Binomial (r  1) is Geometric Gamma   1 is Exponential Weibull   1 is Exponential Var     Var  X   E  X 

where  ~ Gamma

Coverage Modifications Frequency Model

Original Parameters

Exposure n1

Pr  X  0   1



Poisson

m, q

Binomial

r, 

Negative Binomial

Exposure Modification

Exposure n2

Pr  X  0   1

 n2     n1   n2    m, q  n1   n2    r,   n1 

(a,b,0) and (a,b,1) adjustments

1

p0M *



 1

p0M



 1  p0*    1  p 0 

where * indicates revised parameters

Coverage Modification

Exposure n1

Pr  X  0   v

v

m, vq r , v

Aggregate Loss Models Compound Variance

S= aggregate losses N = frequency RV X =severity RV E S   E  N  E  X  Var  S  E  N Var  X   Var  N  E  X 

2

Can only be used when N and X are independent Var  S   E  X 2 

if primary distribution (# claims) is Poisson

Collective Risk vs Individual Risk

Convolution Method

pn  Pr( N  n)  f N (n) f n  Pr( X  n)  f X (n) g n  Pr( S  n)  f S (n) FS ( x) 

g  n

n x

fi1 fi2  fik

i1 ...ik  n

When given severity distributions with Pr( X  0)  0 1) Modify the frequency to eliminate 0 2) Adjust the severity probabilities after removing 0

Aggregate Deductibles

Assume severity is discrete d = stop-loss or reinsurance deductible Assume Premium  E  S ^ d  E  S  d    E  S   E  S ^ d   Net stop-loss Premium Method 1 - Definition of E  S ^ d  u

E S ^ d  

 hjg

hj

 d Pr S  d

j 0

u   d   1 (the sum of all multiples of h less than d )  h Method 2 - Integrate the Survival Function u 1

E S ^ d  

 hS (hj)  d  hu  S (hu) j 0

u   d   1 (the sum of all multiples of h less than d )  h To find E  S ^ 2.8 where x can be 2, 4 6 or 8... Method 1       E  S ^ 2.8   P  S  0   0    P  S  2   2    P  S  2   2.8         g (0) amt   amt   d     g (2)   1 g (0) g (2)             E  S ^ 4  P  S  0   0  P  S  2   2  P  S  4   4         g (0) amt   d amt   g (2) 1  g (0)  g (2)       Method 2     E  S ^ 2.8   2  Pr  S  0     0.8  Pr  S  2     dist between values of x   dist b/w highest value of x below d and d 

Aggregate Coverage Modifications If there is a per-policy deductible and you want Aggregate Payments 1) Expected Payment per Loss x Expected Number of Losses per Year

E  S   E  N   E  X  d  

OR 2) Expected Payment per Payment x Expected Number of Payments per Year

E  S   E  N '   E  X  d | X  d 

where N' is the number of positive payments  frequency  Pr( X  d )  1) Better for discrete severity distributions 2) Better for if severity is Exponential, Pareto or Uniform Exact Calculation of Aggregate Loss Distribution Two cases for which the sum of independent random variables has a simple distribution 2 1) Normal Distribution. If X i are Normal with mean  and variance  , their sum is normal.

2) Exponential or Gamma Distribution. If X i are exponential or gamma, their sum has a gamma distribution

Normal Distributions If n random variables Xi are independent and normally distributed with parameters  and  2 , their sum is normally distributed with parameters n and n 2 . Calculate Pr  S  c | n  1 using  and  2 Calculate Pr  S  c | n  2  using 2 and 2 2 ,etc. Then multiply each these probabilities by their respective p1, p2, etc.

Exponential and Gamma Distributions The sum of n exponential random variables with common mean  is a Gamma distribution with parameters   n and  . When a gamma distribution's  parameter is an integer, the gamma distribution is called an Erlang distribution. In a Poisson Process with parameter  , the number of events occurring by time t has a Poisson distribution with mean  t. In a Poisson Process with parameter  , the time between events follows an exponential distribution with parameter 1 .



In a Poisson Process with paremeter 1 , the time between events is exponential

 with mean  . Therefore, the time until the nth event occurs is Erlang with parameters n and  e

The probability of exactly n events occurring before time x is Gamma CDF n 1

FX ( x)  1 



e



x 

j

j!

j 0

If   1, F ( x)  1- e

 x 

x -   

If   2, F ( x)  1- e

x -   

x

 x  -     e    



x 

 x  n!

n

Empirical Models Bias bias    E ˆ     

 is the estimator,  is the parameter being estimated

bias is the expected value of the estimator minus its true value Estimator is unbiased if bias    0

for all 





Estimator is asymptotically unbiased if lim bias    0 x 

The sample variance (with division by n -1) is an unbiased estimator of variance Sample mean is an unbiased estimator of the true mean Consistency (weak consistency)





Definition: Estimator is consistent if lim Pr  n      1 for all   0 n

1) An estimator is consistent if it is asymptotically unbiased

 

and Var  n  0 as n   2) The MLE is always consistent 3) If MSE    0 then  is consistent Mean Square Error



 MSE    E     



2

 

 |  

MSE    Var   bias  



2

MSE is a function of the true value of the parameter

Complete Data Grouped Data f n ( x) 

nj

n  c j  c j 1 

where x is in c j , c j 1  n j = # points in the interval n = total points f n ( x)  Histogram Fn ( x)  Ogive b

To find the 2nd raw moment, calculate

 f ( x)  x dx 2

n

a

2 If there is a policy limit (say 8000), then E  X^8000   would have as its last 2 terms:   10,000

8000



f n ( x)  x dx  2

5000



f n ( x)  80002 dx

8000

Variance of Empirical Estimators

Binomial: Variance = mq(1  q) X If Y= (Binomial Proportion), m q (1  q ) Variance = m Multinomial: Variance  mqi (1  qi )  i = category  Covariance  -mqi q j If Y=

X q (1  qi ) , Variance = i m m -q q Covariance  i j m

Individual Data

Var  Sn  x    Var  Sn  x   

S  x  1  S  x  

if S is known

n Sn  x  1  Sn  x  

n nx  n  nx  Var  Sn  x    where nx is the # of survivors past time x 3 n nx  n y  n y  Var y  x p x | nx  Var y  x q x | nx  nx3









The empirical estimators of S ( x) and f ( x) are unbiased Individual Data

1) Determine the estimator 2) Determine what's random and what's not 3) Write an expression for the estimator with symbols for random variables 4) Calculate the variance of the random variables 5) Calculate the variance for the whole expression (i.e. Var ( aX  bY ), Var ( aX ), etc.)

Kaplan Meier Product Limit j 1

Sn (t ) 

 i 1

 si  1    ri 

S (t )  S  t0 

t

t0

where t 0 is the end of the study

Exponential Extrapolation

ri  risk set si  death di  entry time ui  withdrawal time xi  death time

 

S x -  Pr  X  x  Shortcut:

1 1 2   n n  1 n  0.5

Nelson Aalen Estimator j 1

Hˆ (t ) 

 i 1

si ri

ˆ Sˆ ( x)  e  H ( x )

Lives that leave at the same time as a death are in the risk set. Lives that arrive at the same time as a death are not in the risk set. Censored lives are in the risk set but are not counted as deaths k = # distinct data points Confidence Intervals For S(x), the boundaries must be between 0 and 1. For H(x), the boundaries can be anything.

Calculator Shortcuts

1) Enter

si in column 1 ri

2) Enter the formula ln(1- L1) for column 2 3) Select L1 as first variable and L2 as second e x

4) Calculate

-

e y

Nelson  Aalen Kaplan  Meier

Estimation of Related Quantities If using Kaplan-Meier or Nelson-Aalen methods, E(X^d) = area under the curve of S(x). Multiply the base x height of each of the rectangles. Bayes Theorem

P  A1 | E  

P  E | A1   P  A1 

P  E | A1   P  A1   P  E | A2   P  A2   ...  P  E | An   P  An 

Greenwood's Approximation of Variance (Kaplan-Meier)

 

Var S (t )  S (t )

2

 r r  s  sj

yj  t

 

Var S (t ) 

j

j

Sn  x  1  Sn  x   n

j

if data is complete (no censoring or truncation)

Variance of Nelson-Aalen Estimator sj Var H (t )  rj 2

 

 yj t

Linear Confidence Intervals

 S (t)  z n

0.5(1 p )

Var  Sn (t )  , Sn (t )  z0.5(1 p ) Var  S n (t ) 



Log-transformed confidence interval for S(t)  S (t ) 1U , S (t )U   n  n  

z Var  Sn (t )   0.5(1  p )  where U  exp   Sn (t )  ln Sn (t )   

Log-transformed confidence interval for H(t)  H (t )  , H (t ) U    U  

  

z  0.5(1 p ) Var H (t ) where U  exp  H (t )  

  

Kernel Smoothing

Uniform 1  k xi ( x)   2b 0  0   x   xi  b  K xi ( x)   2b  1  fˆ ( x) 

n

 i 1

Fˆ ( x) 

n

 i 1

1 n  

xi  b  x  xi  b Otherwise x  xi  b xi  b  x  xi  b x  xi  b k xi ( x)

f n  xi   probability

1 n  

K xi ( x)

f n  xi   probability

xi is a sample point x is the estimation point Kernel distribution is 1 for observation points more than one bandwidth to the left Kernel distribution is 0 for observation points more than one bandwidth to the right K 6 (13)  K10 (13)  K 25 (13) ex) To find K12 (11), linearly interpolate between K12 (7) and K12 (17) Triangular 1 b Base of triangle is 2b Height of triangle is

Expected Values EX |Y  Y

(Y is the original random variable)

The mean of the smoothed distribution is the same as the original mean  E  X   E Y  Uniform Kernel b2 Var ( X )  Var (Y )  3 Triangular Kernel b2 Var ( X )  Var (Y )  6

Approximations for Large Data Sets

d j is the # of left truncated observations in c j , c j 1 - number of new entrants





u j is the # of right censored observations in c j , c j 1 



x j is the # of losses in c j , c j 1 



r j is the risk set c j , c j 1 



q 'j is the decrement rate in c j , c j 1  q 'j 

xj rj

Pj 1  Pj  d j  u j  x j v j  # withdrawals w j  # survivors v j  wj  u j All entries/withdrawals at endpoints r j  Pj  d j All entries/withdrawals uniformly distributed



r j  Pj  0.5 d j  v j



Multiple Decrements  p   p '(1)  p '(2) ...

xt  xt 1  xt 2  '( x )  p  1   1  1   t 3 t r r  t  t 1  rt 2  '( x ) t 3 qt

 1  t 3 pt'( x)

Parametric Models Method of Moments n

m



n

xi

i 1

n

t

Distribution Exponential

2 x   i  i 1

n Formulas

Formulas

 m

ˆ 

Gamma

ˆ 

Pareto

m2 t  m2

2t  2m2 t  2m 2

ˆ  2ln(m)  0.5ln(t )

Lognormal Uniform on  0, 

2 t  m ˆ  m

ˆ 

mt t  2m 2

ˆ 2  2ln(m)  ln(t )

ˆ  2m

When they don’t specify which moment to use, use the first ‘k’ moments, where ‘k’ is the number of parameters you’re fitting. For an inverse exponential, add the reciprocals to get the mean.

Percentile Matching

ˆ p  x(n1) p

if (n  1) p is an integer

Otherwise multiply (n  1)  p and interpolate The smoothed empirical percentile is not defined if the product is less than 1 or greater than n

Maximum Likelihood Type of Data Discrete distribution, individual data Continuous distribution, individual data

Formula

px f ( x)

   

Grouped Data

F c j  F c j 1

Individual Data censored from above at u

1  F (u) for censored observations

Individual Data censored from below at d

F (d )for censored observations

Individual Data truncated from above at u

f ( x) F (u ) f ( x) 1  F (d )

Individual Data truncated from below at d

Cases where MLE = Method of Moments Estimator (if no censored or truncated data) Distribution

Result MLE = MoM MLE = MoM if fixed  MLE = MoM MLE = MoM MLE = MoM if r is known MLE = MoM if m is known

Exponential Gamma Normal Poisson Negative Binomial Binomial

If the MLE is the sample mean, the variance of the MLE is the variance of the distribution =

Var ( X ) n

Common Likelihood Functions MLE Function b a  

MLE

L     e

b a a  b a  ab



L     a eb L     a 1   

b

MLE Formulas Distribution

Formula

CT?

nc

 x  d  i

ˆ 

Exponential

Yes

i

i 1

n n

 ln x

i

ˆ 

i 1

n

Lognormal

No n

ˆ 

Inverse Exponential

 ln

2

xi   ˆ 

i 1

n

ˆ 

n n

 i 1

nc

Weibull, fixed



Uniform on Individual Data  0,  Uniform on Grouped Data  0,  OR Some observations are censored at a single point There must be at least one observation above c j

Uniform on Grouped Data  0,  All groups are bounded

ˆ 



 i 1

2

No

1 xi

xi 

nc

 di i 1

Yes

n

  max  xi   n  ˆ  c j    nj    c j  Upper bound of highest finite interval n j  Number of observations below c j

ˆ  Min of 1) UB of highest interval with data  n  2) LB of highest interval with data *    nj   

No

No

Two-parameter Pareto, fixed



ˆ  

n K

nc

nc

i 1

i 1

K   ln   di    ln   xi 

One-parameter Pareto, fixed



ˆ  

n K

nc

nc

i 1

i 1

K   ln max  , di    ln xi

aˆ   Beta, fixed



b=1

Beta, fixed

Yes

n K n

K   ln xi  n ln 

Yes

No

i 1



a=1

n bˆ   K n

K   ln   xi   n ln 

No

i 1

n = # of uncensored observations c = # of censored observations d = truncation point x = observation if uncensored or the censoring point if censored CT = formula can be used for left-truncated or right-censored data Bernoulli Technique Whenever there is one parameter and only 2 classes of observations, maximum likelihood will assign each class the observed frequency, and you can then solve for the parameter. If X can be only 2 values (a or b) # data points = a P  X  a  # data points

Reasons to Use Maximum Likelihood 1) Method of Moments and Percentile Matching only use a limited number of features from the sample. 2) Method of moments are hard to use with combined data. 3) Method of moments and percentile matching cannot always handle truncation and censoring. 4) Method of moments and percentile matching require arbitrary decisions on which moments or percentiles to use. Reasons NOT to Use Maximum Likelihood 1) There’s no guarantee that the likelihood can be maximized – it can go to infinity. 2) There may be more than one maximum. 3) There may be local maxima in addition to the global maximum; these must be avoided. 4) It may not be possible to find the maximum by setting the partial derivatives to zero; a numerical algorithm may be necessary.

Fisher’s Information 1 Parameter

 d2       E  2 l     d  1 Var     

Fisher's Information



2 Parameters  2  2 l  ,    I  ,     E      l  ,         Var   Var ˆ ,ˆ    Cov   

 

   l  ,        2  l  ,     2   Cov    1      Var   

Inverting a Matrix

a c 

b d 

1



1 d ad  bc  c

b a 

Delta Method

 dg  Var  g( X )   Var ( X )    dx 

2

1 Variable 2

 g  g g  g  Var  g( X , Y )   Var ( X )    2Cov( X , Y )  Var (Y )   x y  x   y  Var  g( X )    g     g 

General

 g g  where g   ,...,  and  is the covariance matrix  x  x k   1 Take derivative with respect to unknown variable

2

2 Variables

Fitting Discrete Distributions Distribution Poisson

Negative Binomial

Method of Moments

MLE

ˆ  x

ˆ  x

x2 rˆ  2 ˆ  x

ˆ 2  x ˆ  x

rˆ  x

ˆ 2  Sample Variance (divide by n)

qˆ 

Binomial

Choosing between (a,b,0) distributions to fit the data:

1) Compare sample variance ˆ 2 to sample mean x Poission: Variance = Mean Negative Binomial: Variance > Mean Binomial: Variance < Mean

2) Calculate

knk and observe the slope as a function of k nk -1

If ratios are increasing, then a > 0 Poisson = 0 slope Negative Binomial = positive slope Binomial = negative slope n is the # of policies/observations of k The variance of a mixture is always at least as large as the weighted average of the variances of the components and usually greater due to:

Var ( X ) variance of a mixture



E Var ( X | I )  weighted average of variances

 Var  E  X | I 

x m

Asymptotic Variance of MLE’s Distribution

 

Exponential

Var ˆ 

Uniform  0, 

Weibull

Pareto

 

Var ˆ 

fixed

Pareto fixed

fixed

Formula







 n  12 (n  2)

2 ˆ Var   2 n

 

Var ˆ  

1 Var  X   Var ( X ) n

2 n

2   2   Var ˆ  n 

 

2 n

Cov   ,   0

Var ˆ  

Poisson

n

n 2

Var  ˆ   Lognormal

2

2 2n

 Var ˆ  n

 

Hypothesis Tests – Graphic Comparison

F ( x)  F (d ) 1  F (d ) f ( x) f * ( x)  1  F (d )

F * ( x) 

D( x) plots D( x)  Fn ( x)  F *( x)  Empirical - Fitted Empirical calculation uses a denominator of n If D( x)  0, then Fn ( x)  F *( x)  had more data  x than predicted by model If D( x)  0, then Fn ( x)  F *( x)  had less data  x than predicted by model If data truncated at d , D( d )  0 Every vertical jump has distance

1 n

p  p plots On horizontal axis, one point every multiple of Domain and Range of Graph are  0,1

  

1 n 1

 

Points are Fn x j , F * x j

The first data point corresponds to the first sample value If slope is less than 45 , fitted distribution is putting too little weight in that region If slope is more than 45 , fitted distribution is putting too much weight in that region Don't plot censored values

Kolmogorov-Smirnov Test

D  max Fn ( x)  F * ( x) d  x u

Max occurs right before or after a jump If D < critical value, do not reject H 0 (null hypothesis) Having censored data lowers D and also lowers the critical value Critical values  0 as n   If data are X = 2000, 4000, 5000, 5000 and the 5000 values are right-censored





Then F4 5000  F4  5000   0.5

Anderson-Darling Test

A  nF * (u )  n 2

k

  Sn  y j   ln S *  y j   ln S *  y j 1  2

j 0

n

k

  Fn  y j   ln F *  y j 1   ln F *  y j  2

j 1

n includes censored observations

Chi-Square

Q

k



j 1



Oj  E j



2

Ej

 O j2  n Q   E  j 1  j  k



O j  # of observations in each group E j  np j (Expected # of observations in each group) n  Total # of observations Each group should have at least 5 expected observations. If not, you have to bring some groups together. Degrees of Freedom Distribution with parameters is given k  1 DoF  Distribution is fitted/estimated by MLE or using different data k  1  r DoF Parameters are fitted from the data r  # parameters k  # groups Independent Periods Q

k

O j  E j 

j 1

Vj



2

where V j is the variance

Degrees of Freeedom  k - p (p is number of estimated parameters)

KolmogorovSmirnov Individual Data

Anderson-Darling Individual Data

Continuous Fits

Chi-square Individual or Grouped Data

Loglikelihood Individual or Grouped Data

Continuous or Discrete Fits

If there is censored data    , should

If there is censored data

  ,

If there is censored data

   , no adjustment of

If there is censored data

lower critical value

should lower critical value

critical value

critical value

If parameters are fitted, critical value should be lowered

If parameters are fitted, critical value should be lowered

If parameters are fitted, critical value adjusts automatically

If parameters are fitted, critical value adjusts automatically

Larger sample size makes critical value decline

Critical value independent of sample size

Critical value independent of sample size

Critical value independent of sample size

No Discretion in grouping of data

No Discretion in grouping of data

Discretion in grouping of data

Uniform weight on all parts of distribution

Higher weight on tails of distribution

Higher weight on intervals with low fitted probability

Type I Error: Rejecting H0 when it is true Type II Error: Rejecting H1 when it is true

   , no adjustment of

Likelihood Ratio Algorithm If parameters are added to the model, the new model will have a loglikelihood at least as great. The # DoF for the likelihood ratio test is the number of free parameters in the alternative model minus the number of free parameters in the base model (null hypothesis).

Compare 2( LogL1  LogL2 ) to critical value at selected chi-square percentile and DOF LogL1  Alternative Model Loglikelihood (which will be higher) LogL2  Base Model If 2( LogL1  LogL2 ) > critical value, accept alternative hypothesis Start by comparing best 2-parameter to best 1-parameter If 2( LogL1  LogL2 )  Critical Value (it fails), compare best 3-parameter distributions to best 1-parameter If 2( LogL1  LogL2 )  Critical Value (it passes), compare 3-parameter distributions to best 2-parameter Schwarz-Bayesian Criterion

r  LogL -   ln n 2 where r is the # parameters where n is the # of data points The distribution with the highest resulting LogL is selected

Credibility eF  n0CV 2  yp  where n0     k 

2

1 p  y p  coefficient from the standard normal =  -1    2 





Given y p , P% = 100 2  percent corresponding to y p   1

k  maximum fluctuation you will accept (i.e. within 5%) Limited Fluctuation Credibility: Poisson Frequency

All  must be the same! 1  CVs2 

E  X 2  E X 

2

Credibility for Experience expressed in Exposure Units

eF Number of Claims

nF Aggregate Losses

sF

Number of Claims

Claim Size (Severity)

 

Aggregate Losses/Pure Premium

1  CV   

n0 

n0

CVs2



n0

n0

n0 CVs2



n0 1  CVs2

n0  s

n0 s CVs2





n0 s 1  CVs2



2 s





 

Pure Premium is the expected aggregate loss per policyholder per time period.

Limited Fluctuation Credibility: Non-Poisson Frequency

eF  n0CVs2 nF  eF  f Credibility for Experience expressed in

Number of Claims

Exposure Units

eF Number of Claims

nF Aggregate Losses

sF

Claim Size (Severity)

  2f n0  2   f

  

  s2 n0  2     s f

  2f n0   f 

  

  s2  n0  2   s 

  2f  s2  n0     f  2  s  

  s2  n0  s  2   s 

  2f  s2  n0  s     f  2  s  

  2f n0  s   f 

  

# of Insureds is Exposure Partial Credibility PC  ZX  1  Z  M PC  M  Z  X  M  PC  Credibility Premium M  Manual Premium Z  Credibility X  Observed Mean Z

Aggregate Losses/Pure Premium

n nF

n  Expected Claims nF  Number of Expected Claims needed for Full Credibility

  

  2f  s2 n0  2  2   f s   f

  

Bayesian Credibility Bayesian Methods – Discrete Prior Class 1 1) Prior Probabilities 2) Likelihood of Experience 3) Joint Probabilities 4) Posterior Probabilities 5) Hypothetical Means 6) Bayesian Premium

Class 2

Product of rows above Quotients of row 3 over row 3 sum Product of rows 4 and 5

Product of rows above Quotients of row 3 over row 3 sum Product of rows 4 and 5

Bayesian Premium is the predicted expected value of the next trial # claims is Bernoulli means at most 1 can occur Bayesian Methods – Continuous Prior

  | x1 ,..., xn  

    f  x1 ,..., xn |        f  x1,..., xn |   d

Posterior Density

Limits of Integration are according to prior distribution f  xn 1 | x1 ,..., xn    f  xn 1 |     | x1 ,..., xn  d

Predictive Density

   is the prior density   | x1 ,..., xn  is the posterior density f  x1 ,..., xn |   is the likelihood function of the data given   conditional  n

f  x1 ,..., xn |     f  xi |   i 1

f  x1 ,..., xn  is the unconditional joint density function f  x1 ,..., xn    f  x1 ,..., xn |     d



n  t t  e dt  0

n!

 n1

Bayesian Credibility: Poisson/Gamma N ~ Poisson    1   ~ Gamma   ,      *   + claims  *    exposures  Pc  * *  1 Posterior: Gamma  * ,*   *   Posterior mean is the avg. # claims/policy Predictive: Negative Binomial  r  * ,   * 

Bayesian Credibility: Normal/Normal

X ~ Normal  , v 

 ~ Normal   , a  x  Observed Average n  Exposure v  anx Posterior Mean v  an va a*  Posterior Variance v  an

* 

Predictive Mean  * Predictive Variance  v  a*

Bayesian Credibility: Lognormal/Normal

X

Lognormal  , v 



Normal   , a 

Find

 ln xi  x

n v  anx *  Posterior Mean v  an va a*  Posterior Variance v  an E  X |    E e  0.5v   E e  e0.5v  e

*  0.5 a*  0.5v

e

Bayesian Credibility: Bernoulli/Beta Probability of a claim = q q Unif  0,1 Uniform is a special case of Beta distribution with a  b  1 and   1 k  # claims n = exposure Beta  Cx a 1 (1  x)b 1 a*  a  claims

  Plug into Posterior Distribution b*  b  exposures - claims  a E  | x   * a*  b* If m  1, treat as a series of 'm' Bernoullis If exposure is 2 years, treat as '2m' Bernoullis 1) n  2m and 2)

 a  a*  m *  a*  b*  a*  b* 

( x  1)  x( x)

Bayesian Credibility: Exponential/Inverse Gamma

1 x f  x |   e  

Exponential



 e       Inverse Gamma      1 *    n *    nx E (next loss) 

* *  1

If f ( x) is Gamma instead of exponential x 1  1   f  x |   x e    

*     n *    nx

Loss Functions

For the loss function minimizing MSE

  

l ˆ,  ˆ  



2

Bayesian Point Estimate is the mean of the posterior distribution For the loss function minimizing Absolute Value of the Error

 

l ˆ,  ˆ   Bayesian Point Estimate is the median of the posterior distribution For the zero-one loss function Bayesian Point Estimate is the mode of the posterior distribution

Buhlmann Credibility Buhlmann Credibility: Basics

  E      

Overall Mean  Expected Value of the Hypothetical Mean 

v  E v    

Expected Value of the Process Variances

a  Var      

Variance of the Hypothetical Mean

a  v  Overall Variance For Poisson frequency HM = PV Buhlmann's k v a Buhlmann's Credibility Z k 

Z 

n na  n  k na  v

n  # periods when studying frequency or aggregate losses n  # claims when studying severity If given 2 classes and there are multiple groups within each class, you must find the mean and variance of each group separately.

Buhlmann Credibility: Continuous

Given a distribution function and the prior function: 1) Use the distribution function to get the HM and PV 2) Find v and a using the prior distribution

Model

Prior

Posterior

Predictive

Poisson

Gamma

Gamma

Negative Binomial

 



1   Bernoulli (q)

*    claims  * =  exposures Beta

Beta

a b

a*  a  claims

Normal

Normal

 ,v 

 a

v  anx v  an va a*  v  an

 

Inverse Gamma

 

1   * = *

* 

Inverse Gamma

*    n *    nx

Buhlmann a



 2

Bernoulli

q  E  | x  

b* =b  exposures - claims

Normal

Exponential

r  *

Buhlmann v

a* a*  b*

ab ab (a  b)(a  b  1) (a  b)2 (a  b  1)

Normal

  *

v

a

 2  a*  v Pareto

  *   *

2 (  1)(  2)

2 (  1)2 (  2)

Exact Credibility If you have conjugate pairs and they ask for a Buhlmann estimate, use the Bayesian estimate. Buhlmann as Least Squares Estimates of Bayes E  Initial probabilities x Outcomes  = E  Initial probabilities x Bayesian Estimates  Yˆi     X i



Cov  X , Y  Z Var ( X )

  (1  Z ) E  X  Var ( X )   pi X i2  E  X 

2

Cov  X , Y    pi X iYi  E  X  E Y 

where X are the initial outcomes and Y are the Bayesian observations Buhlmann Predictions Pc (

0

)  (1  Z ) E  X 

first observation = 0

Pc (2)  (1  Z ) E  X   2Z Pc (8)  (1  Z ) E  X   8Z Graphics Questions 1) The Bayesian prediction must be within the range of the hypothetical means -within range of the prior distribution 2) The Buhlmann predictions must lie on a straight line 3) There should be Bayesian predictions both above and below the Buhlmann line 4) The Buhlmann prediction must be between the overall mean and the observation

Cov  X i X j  Cov  X i X j   a Var  X i   v  a

Empirical Bayes Non-Parametric Methods

ˆ

Uniform Exposures

Non-Uniform Exposures

x

x

Mean of all data

r

2 m ( x  x )  ij ij i

r n 1 ( xij  xi )2  r (n  1) i 1 j 1 avg/class

vˆ

i 1 j 1

avg/cell

( i 1

r

 vˆ 1  r 2   ( xi  x )   r  1  i 1 avg/class overall avg  n  

avg/class

avg/cell r

Mean of sample variances of the rows

aˆ

n



 1)

ni per class

mi ( xi 

i 1 exp/class avg/class

x

) 2  vˆ(r  1)

overall avg r

 m  overall

mi2

i 1 exp/class

m overall

n

Years of Experience

r = # groups m = # exposures To calculate individual variances

v1

 Xi  X  

2

n 1

In the PC formula, M = the average of ALL claims

# Policyholders

Empirical Bayes Semi-Parametric Methods

Poisson Model ˆ  x vˆ  x aˆ  s 2  vˆ s2 



 Xi  X 

2

r 1 r  # Policyholders aˆ regardless of # of years (but if non-uniform exposures, use n = # exposures aˆ  vˆ for the group you are looking at) If non-uniform exposures, aˆ must be calculated using Non-parametric formula Z

For PC 1) X= total # observed claims (but if non-uniform exposures, use the average) 2) If exposure is 5 years, divide PC by 5 to get estimate for 1 year (next year) Non-Poisson Model 1) Negative Binomial with fixed  E  N | r   r Var  N | r   r  (1   )

ˆ  x vˆ  x (1   ) aˆ  s 2  vˆ 2) Gamma with fixed  E  X |     Var  X |     2

ˆ  x vˆ  x aˆ  s 2  vˆ

Simulation Inversion Method

1) Get u  F ( x) 2) Solve for x 3) Plug in 'u' to get simulated value If F (2 )  .25 F (2)  .75 Then .25  u  .75 is mapped to x  2 If F(x) = 'a' (constant) in range  x1 , x2  , then map a  x 2 If given a graph with (x, F(x)) 1) Start on the y-axis with the u values 2) Move right until you hit the line If the line is horizontal, keep going right until it starts going up 3) Go vertically down to x 4) x is the simulated value

Number of Data Values to Generate

F ( x) 1  F ( x)  Var Fˆ ( x)  n eF  n0CV 2





Estimated Item Mean Confidence Interval:

s  x  z  n   n sn is the square root of the unbiased sample variance after n runs Number of Runs: Calculates number of runs needed for the sample mean to be within 100k% of the true mean.

n  n0CV 2

Must use unbiased Variance in CV 2

1 Var X  Var  X    Remember that n

F(x) Confidence Interval: F ( x)  z VaR  F ( x)  F ( x)  z

F ( x) 1  F ( x)  n

Number of Runs:  Pn 1 n  n  Pn  n  n0    n0  P  Pn   n  n Pn  # runs below x

    

Percentiles

q

Confidence Interval:

Y , Y  a

b

  a   nq  0.5  z1 p nq 1  q     2   b   nq  0.5  z1 p nq 1  q     2

Risk Measures

TVaR p ( X )  E  L | L  VaR p   VaR p ( X )  eX VaR p ( X )  

E  L   E  L ^  p  1 p







xf ( x)dx

FX1 ( p )

1 p 1

VaR ( X )dy y



p

1 p

TVaRq ( X ) is the mean of the upper tail of the distribution TVaRq ( X )  Conditional Tail Expectation n

TVaRq ( X )  sq2 

Y

j

j k

n  k 1





2 n     E TVaR ( X )  2  E TVaR ( X ) q q      n  1  





Var TVaR q ( X ) 



sq2  q TVaR q ( X )  VaR q ( X )



2

n  k 1



Confidence Interval =TVaR q ( X )  z Var TVaR q ( X )



p

Estimate of VaRq ( X ) is Yk where k   nq   1 So if simulation has 1000 runs and you're estimating 95th percentile, Then use Y951 Bootstrap Approximation

 ( F ) is the parameter g  x1 ,..., xn  is an estimator based on a sample of 'n' items 2  MSE g  x1 ,..., xn   ( F )   EFn  g  x1 ,..., xn    ( Fn )     n

2 

2 x  x    i i 1

n

Estimating the mean with the Sample Mean n

ˆ 2 MSE x      n

2 x  x    i i 1

n2

Sums of Distributions Single Bernoulli Binomial Poisson Geometric Negative Binomial Normal Exponential Gamma Chi-Square

Multiple Binomial Binomial Poisson Negative Binomial Negative Binomial Normal Gamma Gammas Chi-Square