2009 GARP Practice (Source)

2009 FRM Practice Exams


Table of Contents 2009 FRM Level I Practice Exam ‐ Candidate Answer Sheet ............................................. 3 2009 FRM Level I Practice Exam – Questions ................................................................... 5 2009 FRM Level I Practice Exam – Answer Key ............................................................... 19 2009 FRM Level I Practice Exam – Answers & Explanations ........................................... 21 2009 FRM Full Exam FRM Practice Exam I – Candidate Answer Sheet .......................... 47 2009 FRM Full Exam FRM Practice Exam I – Questions .................................................. 49 2009 FRM Full Exam FRM Practice Exam I – Answer Key ............................................... 65 2009 FRM Full Exam FRM Practice Exam I – Answers & Explanations ........................... 67 2009 FRM Full Exam FRM Practice Exam II – Candidate Answer Sheet ......................... 99 2009 FRM Full Exam FRM Practice Exam II – Questions ............................................... 101 2009 FRM Full Exam FRM Practice Exam II – Answer Key ............................................ 117 2009 FRM Full Exam FRM Practice Exam II – Answers & Explanations ........................ 119

Introduction The FRM exam is a practice‐oriented examination. Its questions are derived from a combination of theory, as set forth in the core readings, and “real‐world” work experience. Candidates are expected to understand risk management concepts and approaches and how they would apply to a risk manager’s day‐to‐day activities. The FRM examination is also a comprehensive examination, testing a risk professional on a number of risk management concepts and approaches. It is very rare that a risk manager will be faced with an issue that can immediately be slotted into one category. In the real world, a risk manager must be able to identify any number of risk‐related issues and be able to deal with them effectively. The 2009 FRM Practice Exams have been developed to aid candidates in their preparation for the FRM Examination in November 2009. These practice exams are based on a sample of questions from the 2007 FRM Examination and are representative of the questions that will be in the 2009 FRM Examination. Wherever necessary and possible, questions, answers and references have been updated to better reflect the topics and core readings listed in the 2009 FRM Examination Study Guide. The 2009 FRM Level I Practice Exam and the FRM Full Exam Practice Exams I and II contain 40 and 50 multiple‐choice questions, respectively. Note that the 2009 FRM Level I and Full Examination will consist of a morning and afternoon session containing 50 and 70 multiple‐ choice questions, respectively. The practice exams were designed to be shorter to allow candidates to calibrate their preparedness without being overwhelming. Copyright © 2009 Global Association of Risk Professionals 1 All rights reserved.


The 2009 FRM Practice Exams do not necessarily cover all topics to be tested in the 2009 FRM Examination. For a complete list of topics and core readings, candidates should refer to the 2009 FRM Examination Study Guide. Core readings were selected by the FRM Committee to assist candidates in their review of the subjects covered by the exam. Questions for the FRM examination are derived from the “core” readings. It is strongly suggested that candidates review these readings in depth prior to sitting for the exam.

Suggested Use of Practice Exams To maximize the effectiveness of the practice exams, candidates are encouraged to follow these recommendations: •

Plan a date and time to take each practice exam. Set dates appropriately to give sufficient study/review time between each practice exam and prior to the actual exam.

•

Simulate the test environment as closely as possible. o Take each practice exam in a quiet place. o Have only the practice exam, candidate answer sheet, calculator, and writing instruments (pencils, erasers) available. o Minimize possible distractions from other people, cell phones and study material. o Allocate 90 minutes for each practice exam and set an alarm to alert you when 90 minutes have passed. Complete each exam but note the questions answered after the 90 minute mark. o Follow the FRM calculator policy. You may only use a Texas Instruments BA II Plus (including the BA II Plus Professional) calculator or a Hewlett Packard 12C (including the HP 12C Platinum) calculator.

•

After completing each practice exam, o Calculate your score by comparing your answer sheet with the practice exam answer key. Only include questions completed in the first 90 minutes. o Use the practice exam Answers & Explanations to better understand correct and incorrect answers and to identify topics that require additional review. Consult referenced core readings to prepare for exam. o Pass/fail status for the actual exam is based on the distribution of scores from all candidates, so use your scores only to gauge your own progress and preparedness.

Copyright © 2009 Global Association of Risk Professionals 2 All rights reserved.


2009 FRM Level I Practice Exam Candidate Answer Sheet 1.

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2009 FRM Level I Practice Exam Questions

1. To hedge against future, unanticipated, and significant increases in borrowing rates, which of the following alternatives offers the greatest flexibility for the borrower?

a. b. c. d.

Fixed for floating swap Interest rate collar Interest rate floor Call swaption

2. An investment bank uses the Exponentially Weighted Moving Average (EWMA) technique with lambda of 0.9 to model the daily volatility of a security. The current estimate of the daily volatility is 1.5%. The closing price of the security is USD 20 yesterday and USD 18 today. Using continuously‐ compounded returns, what is the updated estimate of the volatility? a. b. c. d.

5.44% 3.62% 2.96% 1.31%

3. Consider two stocks A and B. Assume their annual returns are jointly normally distributed, the marginal distribution of each stock has mean 2% and standard deviation 10%, and the correlation is 0.9. What is the expected annual return of stock A if the annual return of stock B is 3%?

a. b. c. d.

2.9% 2% 1.1% 4.7%


2009 FRM Practice Exams 4. In pricing a derivative using the Monte Carlo method, we need to simulate a reasonable number of paths for the price of the underlying asset. Suppose we use a simple model for the return of the underlying asset: y(t) = drift*Δt + vol * √ Δt * e(t), and e(t) is distributed ~ N(0,1),

where drift and vol are known parameters and Δt is the step size. The generation of each path requires a number of steps. Which of the following describes the correct procedure?

a. Generate a random number from a normal distribution N(0,1), use the inverse normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path. b. Generate a random number from a normal distribution N(0,1), use the cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path. c. Generate a random number from a uniform distribution defined in [0,1], use the inverse cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path. d. Generate a random number from a uniform distribution defined in [0,1], use the cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path.

5. A risk manager estimates the daily variance (ht) using a GARCH model on daily returns (rt): ht = α0 + α1r2t-1 + βht-1

Assume the model parameter values are α0 = 0.005, α = 0.04, β = 0.94. The long‐run annualized volatility is approximately: a. b. c. d.

25.00% 13.54% 72.72% 7.94%


2009 FRM Practice Exams 6. A single stock has a price of USD 10 and a current daily volatility of 2%. Using the delta‐normal method, the VaR at the 95% confidence level of a long at‐the‐money call on this stock over a 1‐day holding period is approximately:

a. b. c. d.

USD 0.23 USD 1.645 USD 0.33 USD 0.16

7. A portfolio consists of two zero coupon bonds, each with a current value of USD 10. The first bond has a modified duration of 1 year and the second has a modified duration of 9 years. The yield curve is flat and all yields are 5%. Assume all moves of the yield curve are parallel shifts. Given that the daily volatility of the yield is 1%, which of the following is the best estimate of the portfolio daily VaR at the 95% confidence level?

a. b. c. d.

USD 2.33 USD 1.65 USD 0.82 USD 1.16

8. Consider the following three methods of estimating the P&L of a bullet bond: full repricing, duration (PV01), and duration plus convexity. Ranking the estimated P&L impact of a large negative yield shock from the lowest P&L impact to the highest P&L impact, what is the ranking of the methods to estimate the P&L impact?

a. b. c. d.

Duration plus convexity, duration, full repricing Full repricing, duration plus convexity, duration Duration, duration plus convexity, full repricing Duration, full repricing, duration plus convexity

9. Consider a position in a 5‐year receive‐fixed swap that makes annual payments on a USD 100 million notional. The floating leg has just been reset. The term structure is flat at 5%, the Macaulay duration of a 5‐year par bond is 4.5 years, and the annual volatility of yield changes is 100bp. Your best estimate of the swap’s VaR with 95% confidence over the next month is: a. USD 1.6 million b. USD 2.0 million c. USD 5.5 million d. USD 7.1 million Copyright © 2009 Global Association of Risk Professionals 7 All rights reserved.

2009 FRM Practice Exams 10. If the gold lease rate is higher than the risk‐free rate, what is the market structure of the forward market for gold?

a. b. c. d.

Contango Backwardation Inversion Need more information to determine

11. The price of a 3‐year zero coupon government bond is 85.16. The price of a similar 4‐year bond is 79.81. What is the one‐year implied forward rate from year 3 to year 4?

a. b. c. d.

5.4% 5.5% 5.8% 6.7%

12. A portfolio manager has a bond position worth USD 100 million. The position has a modified duration of 8 years and a convexity of 150 years. Assume that the term structure is flat. By how much does the value of the position change if interest rates increase by 25 basis points?

a. b. c. d.

USD ‐1,953,125 USD ‐1,906,250 USD ‐2,046,875 UDS ‐2,187,500

13. A firm is going to buy 10,000 barrels of West Texas Crude Oil. It plans to hedge the purchase using the Brent Crude futures contract. The correlation between the spot and futures prices is 0.72. The volatility of the spot price is 0.35 per year. The volatility of the Brent Crude futures price is 0.27 per year. What is the hedge ratio for the firm?

a. b. c. d.

0.5554 0.9333 1.2099 0.8198


2009 FRM Practice Exams 14. It is June 2nd and a fund manager with USD 10 million invested in government bonds is concerned that interest rates will be highly volatile over the next three months. The manager decides to use the September Treasury bond futures contract to hedge the value of the portfolio. The current futures price is 95.0625. Each contract is for the delivery of USD 100,000 face value of bonds. The duration of the manager’s bond portfolio in three months will be 7.8 years. The cheapest to deliver bond in the Treasury bond futures contract is expected to have a duration of 8.4 years at maturity of the contract. At the maturity of the Treasury bond futures contract, the duration of the underlying benchmark Treasury bond is 9 years. What position should the fund manager undertake to mitigate his interest rate risk exposure? a. Short 94 contracts b. Short 98 contracts c. Short 105 contracts d. Short 113 contracts

15. A bond trader has bought a position in Treasury Bonds with a 4% annual coupon rate on February 15, 2015. The DV01 of the position is USD 80,000. The trader decides to hedge his interest rate risk with the 4.5% coupon rate Treasury Bonds maturing on May 15, 2017 which has a DV01 of .076 per USD 100 face value. To implement this hedge, approximately what face amount of the 4.5% Treasury bonds maturing on May 15, 2017 should the trader sell?

a. b. c. d.

USD 80,000 USD 10,500,000 USD 80,000,000 USD 105,000,000

16. Suppose that A and B are random variables, each follows a standard normal distribution, and the covariance between A and B is 0.35. What is the variance of (3A + 2B)?

a. b. c. d.

15.10 14.47 9.20 17.20


2009 FRM Practice Exams 17. Consider a stock price S that follows a geometric Brownian motion dS = μ S dt + β S dz, with β strictly positive and μ a fixed value. Which of the following statements is true?

a. b. c. d.

If the drift μ is negative, the price one year from now will be below today’s price. The instantaneous rate of return on the stock follows a uniform distribution. The stock price S follows a lognormal distribution. This model imposes mean reversion.

18. The joint probability distribution of random variables X and Y is given by f(x,y) = kxy for x = 1, 2, 3, y = 1, 2, 3, and k is a positive constant. What is the probability that X + Y will exceed 5?

a. b. c. d.

1/9 1/4 1/36 Cannot be determined

19. Which of the following statements regarding Hypothesis Testing is incorrect?

a. Hypothesis testing is used to make inferences about the parameters of a given population on the basis of statistics computed for a sample that is drawn from that population. b. Type II error refers to the failure to reject the null hypothesis when it is actually false. c. The p‐value decision rule is to reject the null hypothesis if the p‐value is greater than the significance level. d. All else being equal, the decrease in the chance of making a Type I error comes at the cost of increasing the probability of making a Type II error.

20. If stock returns are independently identically normally distributed and the annual volatility is 30%, then the daily VaR at the 99% confidence level of a stock market portfolio is approximately:

a. b. c. d.

2.41% 3.11% 4.40% 1.89%


2009 FRM Practice Exams 21. The current value of the S&P 500 index is 1457, and each S&P futures contract is for delivery of USD 250 times the index. A long‐only equity portfolio with market value of USD 300,100,000 has beta of 1.1. To reduce the portfolio beta to 0.75, how many S&P futures contract should you sell?

a. b. c. d.

618 contracts 288 contracts 574 contracts 906 contracts

The following information should be used for the next two questions. On January 1, a risk manager observes that the 1‐year continuously compounded interest rate is 5% and storage costs of a commodity product A is USD 0.05 per quarter (payable at each quarter end). He further observes the following forward prices for product A: March 5.35 June 5.90 September 5.30 December 5.22

22. Given the following explanation of supply and demand for commodity product A how would you best describe its forward price curve from June to December? Market description Explanation a. Backwardation Excess demand for A in early summer b. Backwardation Supply is expected to decline after summer c. Contango Excess demand for A in early summer d. Contango Supply is expected to decline after summer

23. What is the annualized rate of return earned on a cash‐and‐carry trade entered into in March and closed out in June?

a. b. c. d.

8.9% 9.8 % 35.7% 39.1%


2009 FRM Practice Exams 24. An investor sells a June 2008 call of ABC Limited with a strike price of USD 45 for USD 3 and buys a June 2008 call of ABC Limited with a strike price of USD 40 for USD 5. What is the name of this strategy and the maximum profit and loss the investor could incur? a. Bear Spread, Maximum Loss USD 2, Maximum Profit USD 3 b. Bull Spread, Maximum Loss Unlimited, Maximum Profit USD 3 c. Bear Spread, Maximum Loss USD 2, Maximum Profit Unlimited d. Bull Spread, Maximum Loss USD 2, Maximum Profit USD 3

25. Which of the following problems are NOT inherent disadvantages of the historical simulation approach to estimating VaR?

I. It gives too little weight to more recent observations II. For long‐only portfolios, it is likely to understate VaR following a recent structural increase in volatilities III. It always ignores the fat tails present in the distribution of returns on many financial assets IV. Because of the delta approximation, it inadequately measures the risk of nonlinear instruments

a. b. c. d.

I and II only II only I, III and IV only III and IV only

26. A bank holds USD 60 million worth of 10‐year 6.5% coupon bonds that are trading at a clean price of 101.82. The bank is worried by the exposure due to these bonds but cannot unwind the position for fear of upsetting the client. Therefore, it purchases a total return swap (TRS) in which it receives annual Libor + 100 bps in return for the mark‐to‐market return on the bond. For the first year, the Libor sets at 6.25% and by the end of the year the clean price of the bonds is at 99.35. The net receipt/payment for the bank in the total return swap will be:

a. b. c. d.

Receive USD 2.23 million Receive USD 1.93 million Pay USD 1.93 million Pay USD 2.23 million


2009 FRM Practice Exams 27. Which of the following trade(s) contain basis risk?

I.

Long 1,000 lots Nov 07 ICE Brent Oil contracts and short 1,000 lots Nov 07 NYMEX WTI Crude Oil contracts II. Long 1,000 lots Nov 07 ICE Brent Oil contracts and long 2,000 lots Nov 07 ICE Brent Oil at‐ the‐money put III. Long 1,000 lots Nov 07 Brent Oil contracts and short 1,000 lots Dec 07 ICE Brent Oil contracts IV. Long 1,000 lots Nov 07 ICE Brent Oil contracts and short 1,000 lots Dec 07 NYMEX WTI Crude Oil contracts

a. b. c. d.

I & III II &IV III & IV I, III & IV

28. According to put‐call parity, buying a put option on a stock is equivalent to:

a. b. c. d.

Buying a call option and buying the stock with funds borrowed at the risk‐free rate. Selling a call option and buying the stock with funds borrowed at the risk‐free rate. Buying a call option, selling the stock and investing the proceeds at the risk‐free rate. Selling a call option, selling the stock and investing the proceeds at the risk‐free rate.

29. A 3 month futures contract on an equity index is currently priced at USD 1000, the underlying index stocks are valued at USD 990 and pay dividends at a continuously‐compounded rate of 2 percent and the current continuously compounded risk‐free rate is 4 percent. The potential arbitrage profit per contract, given this set of data, is closest to

a. USD 10.00 b. USD 7.50 c. USD 5.00 d. USD 1.50


2009 FRM Practice Exams 30. Research and model projections indicate that a specific event is likely to move the CHF against the USD. While the direction of the move is highly uncertain, it is highly likely that magnitude of the move will be significant. Based on this information, which of the following strategies would provide the largest economic benefit?

a. Long a call option on USD/CHF and short a put option on USD/CHF with the same strike price and expiration date b. Long a call option on USD/CHF and long a put option on USD/CHF with the same strike price and expiration date c. Short a call option on USD/CHF and long a put option on USD/CHF with the same strike price and expiration date d. Short a call option on USD/CHF and short a put option on USD/CHF with the same strike price and expiration date

31. Initially, the call option on Big Kahuna Inc. with 90‐days to maturity trades at USD 1.40. The option has a delta of 0.5739. A dealer sells 200 call option contracts and to delta‐hedge the position, the dealer purchases 11,478 shares of the stock at the current market price of USD 100 per share. The following day, the prices of both the stock and the call option increase. Consequently, delta increases to 0.7040. To maintain the delta hedge, the dealer should:

a. b. c. d.

Purchase 2,602 shares Sell 2,602 shares Purchase 1,493 shares Sell 1,493 shares

32. Which of the following strategies creates a calendar spread? a. Sell a call option with a certain strike price and buy a longer maturity call option with the same strike price b. Buy a call option with a certain strike price and buy a longer maturity call option with the same strike price c. Sell a call option with a certain strike price and buy a shorter maturity call option with the same strike price d. Buy a call option with a certain strike price and sell a longer maturity call option with the same strike price


2009 FRM Practice Exams 33. Which of the following underlying macro‐economic conditions would leave an emerging market most vulnerable to the contagion effects of a currency crisis?

a. b. c. d.

Large current account surplus, low foreign exchange reserves, non‐convertible currency Large current account deficit, low foreign exchange reserves, fully convertible currency Small current account deficit, high foreign exchange reserves, non‐convertible currency Large current account surplus, high foreign exchange reserves, fully convertible currency

34. Consider an FRA (forward rate agreement) with the same maturity and compounding frequency as a Eurodollar futures contract. The FRA has a LIBOR underlying. Which of the following statements are true about the relationship between the forward rate and the futures rate?

a. b. c. d.

They should be exactly the same The forward rate is normally higher than the futures rate The forward rate is normally lower than the futures rate They have no fixed relationship

35. Your bank is an active player in the commodity market. The view of the economist of the bank is that inflation is expected to rise moderately in the near term and market volatility is expected to remain low. The traders are advised to undertake deals on the metals exchange to align your book to conform with the expectations of the economist of the bank. As risk manager, you are asked to monitor the positions of the traders to make sure that they have the exposures to inflation and market volatility sought by the bank. Which trader has taken an appropriate position among the traders you are monitoring?

a. Trader A bought a call and a put, both with 90‐days to expiration and with strike price equal to the existing spot level b. Trader B bought a put option with a down‐and‐in knock in feature c. Trader C bought a call option at the existing spot levels and sold a call at a higher strike price, both with 90‐days to expiration d. Trader D sold a call and bought a put at the existing levels, both with 90‐days to expiration


2009 FRM Practice Exams 36. The information ratio of the Sterole US Fund for 2006 against the S&P 500, its benchmark index, is 1. For the same time period, the fund’s Sharpe ratio is 2, the fund has a tracking error of 7% against the S&P 500, and the standard deviation of fund returns is 5%. The risk‐ free rate in the US is 4%. Calculate the return for the S&P 500 during the time period.

a. b. c. d.

3.5% 7% 11% 14%

37. A fund manager recently received a report on the performance of his portfolio over the last year. According to the report, the portfolio return is 9.3%, with a standard deviation of 13.5%, and a beta of 0.83. The risk‐free rate is 3.2%, the semi‐standard deviation σL(Rp) of the portfolio is 8.4%, and the tracking error of the portfolio to the benchmark index is 2.8%. What is the difference between the value of the fund’s Sortino ratio (computed relative to the risk‐free rate) and its Sharpe ratio?

a. b. c. d.

0.274 1.727 0.653 ‐0.378

38. Which of the following statements about the linear regression of the return of a portfolio over the return of its benchmark presented below are correct?

Portfolio parameter Value Beta 1.25 Alpha 0.26 Coefficient of determination 0.66 Standard deviation of error 2.42

I. The correlation is 0.71 II. 34% of the variation in the portfolio return is explained by variation in the benchmark return III. The portfolio is the dependent variable IV. For an estimated portfolio return of 12%, the confidence interval at 95% is [7.16%;16.84%]

a. b. c. d.

II and IV III and IV I, II and III II, III and IV



39. Your Board of Directors wants a comprehensive review of each business units’ operational risk activities. As the head of the corporate operational risk unit, you know that little has been done to implement an operational risk process at the business unit level and that you need to immediately come up with a framework. Which of the following statements offers the best strategy?

I.

The audit committee of the Board should first define its objectives to ensure that all the firm’s business units’ operational risk programs are providing required information II. The auditing department is to be charged with developing an operational risk program for each business unit, with the business unit being made clearly aware that they will be held accountable for its implementation III. That your department immediately assess the operational risk for each business unit using independent data feeds to ensure the information fed into the assessment cannot be manipulated IV. A senior manager from each profit center is to be charged with developing their own operational risk self assessment program based on guidelines you provide.

a. b. c. d.

I only I and IV only I and III only IV only

40. Which of the following risk management strategies of a firm which has principal payments to make on its debt in one year that substantially exceed the market value of its assets is most likely to be in the interest of the shareholders?

a. b. c. d.

Reduction of the overall risk level of the firm Increase of the overall risk level of the firm Keep the same risk level It is impossible to say which risk management strategy the shareholders prefer

END OF 2009 FRM Level I PRACTICE EXAM






2009 FRM Level I Practice Exam Answer Key 1.

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2009 FRM Level I Practice Exam Answers & Explanations 1. To hedge against future, unanticipated, and significant increases in borrowing rates, which of the following alternatives offers the greatest flexibility for the borrower?

a. b. c. d.


CORRECT: D

The question focuses on flexible management of borrowing expenses. While a fixed for floating swap could reduce borrowing expenses, it is a long‐term contractual commitment to exchange payments. If interest rates decline, the borrower may gross up to the agreed fixed rate. An interest rate collar is a combination of an interest rate floor and cap, i.e., it locks in the interest expenses within a tight range. Moreover, collars usually offer interest rate protection at one particular point of time unless several contracts with different maturities are exchanged. A call swaption gives the company the right to enter into a swap when the borrowing expenses exceed a certain reference rate. If the reference rate is below the borrowing expenses, the option is not exercised.

Reference: Hull, Chapter 7.

2. An investment bank uses the Exponentially Weighted Moving Average (EWMA) technique with lambda of 0.9 to model the daily volatility of a security. The current estimate of the daily volatility is 1.5%. The closing price of the security is USD 20 yesterday and USD 18 today. Using continuously‐ compounded returns, what is the updated estimate of the volatility?

a. b. c. d.

5.44% 3.62% 2.96% 1.31%

CORRECT: B The current return of the security is = ln (18/20) = ‐10.536%.


2009 FRM Practice Exams Using an EWMA model, the updated volatility is given as: V(t) = {lambda* ((V[t‐1]^2) +(1 – lambda)*(current return^2)} ^ 0.5 = {0.9 * ((0.015^2) + (1 ‐ 0.9) * ( ‐0.10536^2 )} ^ 0.5 = 3.62% INCORRECT: A – Forgets to square the volatility terms INCORRECT: C – Forgets to square the volatility terms and to take the square root of the resulting variance, then miscalculates conversion to percentage. INCORRECT: D – Forgets to take the square root of the variance, then miscalculates conversion to percentage. Reference : Hull, Chapter 21.

3. Consider two stocks A and B. Assume their annual returns are jointly normally distributed, the marginal distribution of each stock has mean 2% and standard deviation 10%, and the correlation is 0.9. What is the expected annual return of stock A if the annual return of stock B is 3%? a. 2.9% b. 2% c. 1.1% d. 4.7%

CORRECT: A E[ra | rb = x] = μa + (ρabσaσb/σ2a)(x – μb) = 0.02 + 0.9 * (0.03 – 0.02) = 0.029 Reference: Damodar Gujarati, Chap 2,3.

4. In pricing a derivative using the Monte Carlo method, we need to simulate a reasonable number of paths for the price of the underlying asset. Suppose we use a simple model for the return of the underlying asset: y(t) = drift*Δt + vol * √ Δt * e(t), and e(t) is distributed ~ N(0,1), where drift and vol are known parameters and Δt is the step size.


2009 FRM Practice Exams The generation of each path requires a number of steps. Which of the following describes the correct procedure?

a. Generate a random number from a normal distribution N(0,1), use the inverse normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path. b. Generate a random number from a normal distribution N(0,1), use the cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path. c. Generate a random number from a uniform distribution defined in [0,1], use the inverse cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path. d. Generate a random number from a uniform distribution defined in [0,1], use the cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path.

CORRECT: C This question wants to test if the candidate knows the basic steps to generate a very simple path: answering this question means that the candidate would be able to build a simple spreadsheet showing the Monte Carlo logic. The correct procedure is the one described in c); while a), b) and d) are nonsensical calculations. Reference: Philippe Jorion, Value at Risk,The New Benchmark for Managing Financial Risk, 3rd edition (New York: McGraw‐Hill, 2007), Chapter 12.

5. A risk manager estimates the daily variance (ht) using a GARCH model on daily returns (rt): ht = α0 + α1r2t‐1 + βht‐1 Assume the model parameter values are α0 = 0.005, α1 = 0.04, β = 0.94. The long‐run annualized volatility is approximately: a. 25.00% b. 13.54% c. 72.72% d. 7.94%

CORRECT: D


2009 FRM Practice Exams The long‐run variance is 0.005/(1‐0.04‐0.94) =0.005/0.02 = 0.25. The daily vol is thus the square root, or 0.5% and annual vol 7.935%. INCORRECT: A – The daily variance is indeed 0.25%, and the daily volatility 0.5% but this needs to be annualized. INCORRECT: B – Miscalculates variance as sqrt(0.04/(1 – 0.94 – 0.005)) * 15.87 = 13.54% INCORRECT: C – Miscalculates variance as 0.04/(1 – 0.94 – 0.005) = 72.72% Reference: Hull, Chapter 21.

6. A single stock has a price of USD 10 and a current daily volatility of 2%. Using the delta‐normal method, the VaR at the 95% confidence level of a long at‐the‐money call on this stock over a 1‐day holding period is approximately:

a. b. c. d.

USD 0.23 USD 1.645 USD 0.33 USD 0.16

CORRECT: D This question requires candidates to know the formula for the delta‐normal VaR approximation, and also to know that the delta of an at‐the‐money call is 0.5. VaR =| Δ | ×1.645 × σ × S = 0.5 × 1.645 × 0.02 × 10 = 0.1645 . INCORRECT: A – We get A by using 2.326 instead of 1.645. INCORRECT: B – We get B if we use 2 instead of 2% for the volatility. INCORRECT: C – We get C if we use a delta of 1. Reference: Allen et al, Chapter 3, 86‐89

7. A portfolio consists of two zero coupon bonds, each with a current value of USD 10. The first bond has a modified duration of 1 year and the second has a modified duration of 9 years. The yield curve is flat and all yields are 5%. Assume all moves of the yield curve are parallel shifts. Given that the daily volatility of the yield is 1%, which of the following is the best estimate of the portfolio daily VaR at the 95% confidence level? Copyright © 2009 Global Association of Risk Professionals 24 All rights reserved.

2009 FRM Practice Exams a. b. c. d.

USD 2.33 USD 1.65 USD 0.82 USD 1.16

CORRECT: B This question assesses candidates’ abilities to apply the duration VaR formula to two bonds simultaneously and to recall that the duration of a zero coupon bond is equal to the bond maturity. Using an obvious extension of Jorion’s equation 9.5

VaR = D1 × V1 × 1.645 × σ + D2 × V2 × 1.645 × σ = ( D1 × V1 + D2 × V2 ) × 1.645 × σ = ( D1 + D2 ) × 10 × 1.645 × σ = 10 × 10 × 1.645 × 0.01 = 1.645 INCORRECT: A – The 99% confidence level VaR INCORRECT: C – Arises if the candidate mistakenly divides the correct answer by the number of bond INCORRECT: D – Makes both mistakes Reference: Allen et al.


a. b. c. d.

Duration plus convexity, duration, full repricing Full repricing, duration plus convexity, duration Duration, duration plus convexity, full repricing Duration, full repricing, duration plus convexity

CORRECT: C The price / yield line with yield on the x axis and price on the y axis is convex to the origin. The duration at any yield level is the tangent to that curve. Therefore, except at the exact point of tangency, duration will always underestimate the price change. INCORRECT: A – Duration will always underestimate price change for negative yield shocks INCORRECT: B – Full repricing will never generate a smaller positive price change than duration because duration represents the point of tangency INCORRECT: D – Full repricing will generate a higher price for a large negative yield change than will duration plus convexity Reference: Allen, Boudoukh, Saunders, Chapter 3 Copyright © 2009 Global Association of Risk Professionals 25 All rights reserved.

2009 FRM Practice Exams 9. Consider a position in a 5‐year receive‐fixed swap that makes annual payments on a USD 100 million notional. The floating leg has just been reset. The term structure is flat at 5%, the Macaulay duration of a 5‐year par bond is 4.5 years, and the annual volatility of yield changes is 100bp. Your best estimate of the swap’s VaR with 95% confidence over the next month is a. USD 1.6 million b. USD 2.0 million c. USD 5.5 million d. USD 7.1 million CORRECT: A Because the floating‐rate leg has just been reset, its duration is 1. Net duration is 4.5‐1=3.5 year, or modified duration of 3.5/1.05=3.33. The 95% VaR of monthly changes in yields is 1.65*1%/√12 = 0.48%. Multiplying, this gives USD 100*0.48%*3.33=USD 1.588 INCORRECT: B – This uses a net duration of 4.5 years and ignores the duration of the floating‐rate leg. INCORRECT: C – This is the annual VaR, but should be translated to a monthly horizon. INCORRECT: D – This is the annual VaR computed by ignoring the duration of the floating‐rate leg. Reference: Allen et al. 10. If the gold lease rate is higher than the risk‐free rate, what is the market structure of the forward market for gold?

a. b. c. d.


CORRECT: B A lease rate higher than the risk fee rate will force a negatively sloped forward curve, i.e. backwardation INCORRECT: A – The forward price = spot*exp( risk free rate ‐ lease rate). If the lease rate is higher than the risk free rate, forwards will be lower than spot, implying contango INCORRECT: C – The term inversion is used to describe yield curves, not commodity forwards INCORRECT: D – There is enough information in the question to provide an answer Reference: MacDonald, Chapter 6 Copyright © 2009 Global Association of Risk Professionals 26 All rights reserved.

2009 FRM Practice Exams 11. The price of a 3‐year zero coupon government bond is 85.16. The price of a similar 4‐year bond is 79.81. What is the one‐year implied forward rate from year 3 to year 4?

a. b. c. d.

5.4% 5.5% 5.8% 6.7%

CORRECT: D 1 + Forward rate =

Price of three bond 85.16 = = 1.067034 Price of four year bond 79.81

Forward rate = 0.067034 or 6.7%

INCORRECT: A – This is B/C INCORRECT: B – This is the return of the 3‐year bond INCORRECT: C – This is the return of the 4‐year bond Reference: Tuckman


a. b. c. d.

USD ‐1,953,125 USD ‐1,906,250 USD ‐2,046,875 USD ‐2,187,500

CORRECT: A

ΔV = - D mod × Δy × V + 0.5 × Convexity × Δy 2 × V ΔV = - 8 × 0.0025 × 100M + 0.5 × 150 × (0.0025) 2 × 100M ΔV = - 2M + 46,875 ΔV = - 1,953,125 INCORRECT: B – Omits 0.5 from the second term INCORRECT: C – Subtracts the second term INCORRECT: D – Makes both mistakes Reference: Tuckman


2009 FRM Practice Exams 13. A firm is going to buy 10,000 barrels of West Texas Crude Oil. It plans to hedge the purchase using the Brent Crude futures contract. The correlation between the spot and futures prices is 0.72. The volatility of the spot price is 0.35 per year. The volatility of the Brent Crude futures price is 0.27 per year. What is the hedge ratio for the firm?

a. b. c. d.

0.5554 0.9333 1.2099 0.8198

CORRECT: B ⎛ 0.35 ⎞ N = 0.72 × ⎜ ⎟ ⎝ 0.27 ⎠ N= 0.9333

INCORRECT: A – Inverts the spot volatility and the futures volatility INCORRECT: C – Uses variances INCORRECT: D – Uses square roots of the volatilities Reference: Hull, Chapter 3

14. It is June 2nd and a fund manager with USD 10 million invested in government bonds is concerned that interest rates will be highly volatile over the next three months. The manager decides to use the September Treasury bond futures contract to hedge the value of the portfolio. The current futures price is 95.0625. Each contract is for the delivery of USD 100,000 face value of bonds. The duration of the manager’s bond portfolio in three months will be 7.8 years. The cheapest to deliver bond in the Treasury bond futures contract is expected to have a duration of 8.4 years at maturity of the contract. At the maturity of the Treasury bond futures contract, the duration of the underlying benchmark Treasury bond is 9 years. What position should the fund manager undertake to mitigate his interest rate risk exposure?

a. b. c. d.

Short 94 contracts Short 98 contracts Short 105 contracts Short 113 contracts

CORRECT: B.


2009 FRM Practice Exams ⎛ 10,000,000 ⎞ ⎛ 7.8 ⎞ N=⎜ ⎟×⎜ ⎟ ⎝ 95,062.50 ⎠ ⎝ 8.4 ⎠

N = 97.68 or 98 contracts

INCORRECT: A – This is made up. INCORRECT: C – This leaves out the durations INCORRECT: D – This inverts the durations Reference: Tuckman


a. b. c. d.

USD 80,000 USD 10,500,000 USD 80,000,000 USD 105,000,000

CORRECT: D USD 105,000,000x.076/100 = USD 79,800, which is pretty close to the desired DV01 of USD 80,000. To solve for the hedge, solve for F in the equation USD 80,000 = Fx.076/100, giving F = 105,263,158 INCORRECT: A – Selling this amount would offset a DV01 of only USD 80,000x.076/100 = USD 61 INCORRECT: B – USD 10,500,000x.076/100 = USD 7,980 INCORRECT: C – USD 80,000,000x.076/100 = USD 60,800 Reference: Tuckman, Chapter 5


2009 FRM Practice Exams 16. Suppose that A and B are random variables, each follows a standard normal distribution, and the covariance between A and B is 0.35. What is the variance of (3A + 2B)?

a. b. c. d.

15.10 14.47 9.20 17.20

CORRECT: D Since each variable is standardized, its variance is 1. Therefore V(3A+2B) = 9 V(A) + 4 V(B) + 2 x 3 x2 x Cov(A,B) = 9+4+4.2 = 17.2 INCORRECT: A – 9 + 4 + 6 * 0.35 = 15.1 INCORRECT: B – 9+ 4 + 12 * 0.35^2 = INCORRECT: C – 3 + 2 + 12 * 0.35 = 9.2 Reference: Damodar Gujarati

17. Consider a stock price S that follows a geometric Brownian motion dS = μ S dt + β S dz, with β strictly positive and μ a fixed value. Which of the following statements is true?

a. b. c. d.


CORRECT: C INCORRECT: A – The expected price is less than today’s price, but not the price in all the states of world. INCORRECT: B – The instantaneous rate of return on the stock follows normal distribution. INCORRECT: D – This model does not impose mean reversion. rd

Reference: Philippe Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3 ed. (New York: McGraw‐Hill, 2007). Chapter 12


2009 FRM Practice Exams 18. The joint probability distribution of random variables X and Y is given by f(x,y) = kxy for x = 1, 2, 3, y = 1, 2, 3, and k is a positive constant. What is the probability that X + Y will exceed 5?

a. b. c. d.

1/9 1/4 1/36 Cannot be determined

CORRECT: B 3

Note that

3

∑∑ f ( x, y) = 1 x =1 y =1

Substituting the various values of x and y, we get f(1,1)=k, f(1,2)=2k, f(1,3)=3k, f(2,1)=2k, f(2,2)=4k, f(2,3)=6k, f(3,1)=3k, f(3,2)=6k, and f(3,3)=9k. Therefore, k1 + 2k + 3k + 2k + 4k + 6k + 3k + 6k + 9k = 1 so that, 36k = 1 and k=1/36. P(X+Y>5)= f(3,3)= 1/36 x 3 x 3 = 1/4 Reference: Damodar Gujarati



CORRECT: C The true statement is to reject Ho if the p‐value is smaller than the significance level. INCORRECT: A – Statement A is correct regarding the primary use of Hypothesis Testing. INCORRECT: B – Statement B is correct regarding the definition of type II error. INCORRECT: D – Statement D is correct because type I error and type II error are in tradeoff. Reference: Damodar Gujarati


2009 FRM Practice Exams 20. If stock returns are independently identically normally distributed and the annual volatility is 30%, then the daily VaR at the 99% confidence level of a stock market portfolio is approximately:

a. b. c. d.

2.41% 3.11% 4.40% 1.89%

CORRECT: C The 1‐day volatility is s * (1/252)^0.5 = 0.3 * 0.629941 = 0.018898. The VaR at the 99% confidence level is then equal to 2.32635 * 0.018898 = 4.40% INCORRECT: A – One gets A if one uses 1.645 instead of 2.326 INCORRECT: B – One gets B if one uses the monthly volatility instead of the daily one INCORRECT: D – One gets D is the daily volatility Reference: Allen, Boudoukh and Saunders, 2004, chapter 1, p 6‐8

21. The current value of the S&P 500 index is 1457, and each S&P futures contract is for delivery of USD 250 times the index. A long‐only equity portfolio with market value of USD 300,100,000 has beta of 1.1. To reduce the portfolio beta to 0.75, how many S&P futures contract should you sell?

a. b. c. d.


CORRECT: B No of contracts = [0.75 – 1.1)/ 1]* [300,100,000 / {250 * 1,457}] = ‐288.36 Æ sell 288 contracts INCORRECT: A – ‐617.9135209 = ‐1*(0.75)* (300100000 / (250*1457)) INCORRECT: C – ‐561.74 = ‐1(0.75/1.1)* (300100000 / (250*1457)) INCORRECT: D – ‐906.273164 = ‐1* (1.1)* (300100000 / (250*1457)) Reference: Hull, Options, Futures and Other Derivatives, Chapter 3 and 4; Anthony Saunders, Financial Institutions Management, Chapter 10


2009 FRM Practice Exams The following information should be used for the next two questions. On January 1, a risk manager observes that the 1‐year continuously compounded interest rate is 5% and storage costs of a commodity product A is USD 0.05 per quarter (payable at each quarter end). He further observes the following forward prices for product A: March 5.35 June 5.90 September 5.30 December 5.22

22. Given the following explanation of supply and demand for commodity product A how would you best describe its forward price curve from June to December?

a. b. c. d.

Market description Explanation Backwardation Excess demand for A in early summer Backwardation Supply is expected to decline after summer Contango Excess demand for A in early summer Contango Supply is expected to decline after summer CORRECT: A A is correct ‐ when further‐term commodity forwards have lower price than near‐term forwards, the market is said to be in ‘backwardation’. Possible explanation can be seasonality of product A – excess demand in early summer causes June forwards to have higher price INCORRECT: B – Market description is correct, but explanation is not – expected decline in supply should increase further‐term commodity forward price INCORRECT: C – Wrong market description of contango INCORRECT: D – Wrong market description of contango Reference: Robert L McDonald, Derivatives Markets, Chapter 6


a. b. c. d.

8.9% 9.8% 35.7% 39.1%

CORRECT: C


2009 FRM Practice Exams By formula F0,T = S0erT + C, where F0,T = June forward price, S0 = March forward price, r = risk free interest rate, T = length of cash‐and‐carry, C = storage cost Solving 5.90 = 5.35er*3/12 + 0.05 Solution is r = 35.7% INCORRECT: A – 8.9 = LN((5.9‐0.05)/5.35) (forgets to annualize the return) INCORRECT: B – 9.8= LN((5.9)/5.35) (forgets to include the storage cost and to annualize the return) INCORRECT: D – 39.1= (12/3)LN((5.9)/5.35) – 0.05 (forgets to include the storage cost) Reference: Robert L McDonald, Derivatives Markets, Chapter 6

24. An investor sells a June 2008 call of ABC Limited with a strike price of USD 45 for USD 3 and buys a June 2008 call of ABC Limited with a strike price of USD 40 for USD 5. What is the name of this strategy and the maximum profit and loss the investor could incur?

a. b. c. d.

Bear Spread, Maximum Loss USD 2, Maximum Profit USD 3 Bull Spread, Maximum Loss Unlimited, Maximum Profit USD 3 Bear Spread, Maximum Loss USD 2, Maximum Profit Unlimited Bull Spread, Maximum Loss USD 2, Maximum Profit USD 3

CORRECT: D Buying a call option at lower stock price and selling call option at higher strike price is called as Bull Spread. Bear Spread is buying the call option at higher price and selling the call at lower strike price. The Cost of strategy will be USD 3‐USD 5 = ‐USD 2 The Payoff, when Stock price ST ≤ USD 40 will be ‐USD 2 (the cost of strategy) as none of the option will be exercised. The Payoff, when stock price ST ≥ 45, (as both options will be exercise) will be USD 5, Since the cost of strategy is ‐USD 3, hence profit will be USD 5‐USD 2 = USD 3 When Stock price is USD 40< ST > USD 45, Only the call option bought by the investor would be exercised hence the pay off will be ST – 40, since the cost of strategy is ‐USD 3, The Net profit will be ST – 43, which would always be lower than USD 3. Reference: Hull. Chapter 10‐ Trading Strategies Involving Options.


2009 FRM Practice Exams 25. Which of the following problems are NOT inherent disadvantages of the historical simulation approach to estimating VaR?


a. b. c. d.


CORRECT: C The disadvantage with the Historical Simulation Model is that it may not recognize the changes in volatility and correlation following recent structural changes. The model can be adjusted so that it gives more weight to recent observations. The other options, i.e. III & IV, are disadvantages of Monte Carlo method and Delta‐normal method. Reference: Allen et al.


a. b. c. d.

Receive USD 2.23 million. Receive USD 1.93 million. Pay USD 1.93 million. Pay USD 2.23 million.

CORRECT: B


2009 FRM Practice Exams It’s the result of this calculation: the notional amount is 60 million USD . Therefore the bank will receive the interest payment linked to the LIBOR rate: 60 million USD * (6,25%+100 bp) = 4. 35 million USD . The bank will pay the fixed coupon plus the change in the value of the bond: 60 million USD * 6.5% + 60 million *(99.35%‐101.82%) = 2.418 million USD . Hence the total net amount the bank will receive is: 4.35 million USD ‐ 2.418 million USD = 1.932 million USD . Reference: Hull, Chapter 7 27. Which of the following trade(s) contain basis risk?

I. Long 1,000 lots Nov 07 ICE Brent Oil contracts and short 1,000 lots Nov 07 NYMEX WTI Crude Oil contracts II. Long 1,000 lots Nov 07 ICE Brent Oil contracts and long 2,000 lots Nov 07 ICE Brent Oil at‐ the‐money put III. Long 1,000 lots Nov 07 ICE Brent Oil contracts and short 1,000 lots Dec 07 ICE Brent Oil contracts IV. Long 1,000 lots Nov 07 ICE Brent Oil contracts and short 1,000 lots Dec 07 NYMEX WTI Crude Oil contracts

a. b. c. d.

I & III II & IV IIII & IV I, III & IV

CORRECT: D Basis Risk is spread risk, which arise from trading the spread (long and short 2 positively correlated assets or same asset with different expiration) i is spread trade in highly correlated asset with same expiration month ii faces with gamma and vega risk iii is spread trade in trading the flattening of the forward curve iv is spread trade in trading 2 assets with different expiration date Reference: Robert L. McDonald, Derivatives Markets (Boston: Addison‐Wesley, 2003), Chapter 6.


2009 FRM Practice Exams 28. According to put‐call parity, buying a put option on a stock is equivalent to:

a. b. c. d.

Buying a call option and buying the stock with funds borrowed at the risk‐free rate. Selling a call option and buying the stock with funds borrowed at the risk‐free rate. Buying a call option, selling the stock and investing the proceeds at the risk‐free rate. Selling a call option, selling the stock and investing the proceeds at the risk‐free rate. CORRECT: C Buying a call option, selling the stock and investing the proceeds at the risk‐free rate. Put‐call parity states P = C ‐ S + X e‐RT INCORRECT: A – Buying a call option is correct, but the rest of the statement is incorrect. INCORRECT: B – The entire statement is incorrect. INCORRECT: D – Selling a call option is incorrect, but the rest of the statement is correct. Reference: Hull, Chapter 10


a. b. c. d.

USD 10.00 USD 7.50 USD 5.00 USD 1.50

CORRECT: C According to the fundamental pricing relationship between spot assets and the associated futures, the futures price, to prevent arbitrage, should equal 990 x e (0.04 – 0.02) x 0.25 or 995. Hence, the futures contract is overvalued, indicating it should be sold and the index should be purchased for an arbitrage profit of USD 1000 ‐ USD 995 = USD 5 Reference: Hull, Chapters 2,3 6.




CORRECT: B The question tests on understanding of a “straddle” strategy and its application on currency trading. A long straddle strategy involves buying (long) a call and put option with the same strike price and expiration date, and will benefit most when the underlying moves away from the current equlibrium. INCORRRECT: A – It sells a put option while it should buy one put INCORRECT: C – It sells a call option while it should buy one call INCORRECT: D – It sells both the call and put option while it should buy both Reference: Hull, Chapter 10.


a. b. c. d.

Purchase 2,602 shares. Sell 2,602 shares. Purchase 1,493 shares. Sell 1,493 shares.

CORRECT: A Number of calls = 200 contracts x 100 = 20,000 calls.


2009 FRM Practice Exams Number of shares = (Number of calls) x (New delta – Old delta) = 20,000 x (0.7040 – 0.5739) = +2,602 shares Positive sign indicates that the manager should purchase new shares. INCORRECT: B – The formula is incorrect, i.e. old delta minus new delta INCORRECT: C – The number of shares (instead of number of calls) is used in the calculation INCORRECT: D – As per explanation in ‘C’ above and sign error Reference: Hull Chapters 9 and 10 32. Which of the following strategies creates a calendar spread?

a. Sell a call option with a certain strike price and buy a longer maturity call option with the same strike price b. Buy a call option with a certain strike price and buy a longer maturity call option with the same strike price c. Sell a call option with a certain strike price and buy a shorter maturity call option with the same strike price d. Buy a call option with a certain strike price and sell a longer maturity call option with the same strike price

CORRECT: A INCORRECT: B – As buy a call option INCORRECT: C – As buy a shorter‐maturity call option INCORRECT: D – As this is a reverse calendar spread Reference: John Hull, Chapter 10.

33. Which of the following underlying macro‐economic conditions would leave an emerging market most vulnerable to the contagion effects of a currency crisis?

a. b. c. d.


CORRECT: B


2009 FRM Practice Exams INCORRECT: A – Large current account surplus and non‐convertible currency would protect the local currency INCORRECT: C – High foreign exchange reserves and non‐convertible currency would protect the local currency INCORRECT: D – Large current account surplus and high foreign exchange would protect the local currency Reference: Saunders, Chapter 15, Foreign Exchange Risk


a. b. c. d.

They should be exactly the same The forward rate is normally higher than the futures rate The forward rate is normally lower than the futures rate They have no fixed relationship

CORRECT: C As Eurodollar futures contract is marked to market and settled daily, normally forward rate is adjusted lower, so called convexity adjustment, by: Forward rate = Futures rate –

1 2 σ T1T2 2

Reference: Hull, Chapter 6. 35. Your bank is an active player in the commodity market. The view of the economist of the bank is that inflation is expected to rise moderately in the near term and market volatility is expected to remain low. The traders are advised to undertake deals on the metals exchange to align your book to conform with the expectations of the economist of the bank. As risk manager, you are asked to monitor the positions of the traders to make sure that they have the exposures to inflation and market volatility sought by the bank. Which trader has taken an appropriate position among the traders you are monitoring?

a. Trader A bought a call and a put, both with 90‐days to expiration and with strike price equal to the existing spot level b. Trader B bought a put option with a down‐and‐in knock in feature c. Trader C bought a call option at the existing spot levels and sold a call at a higher strike price, both with 90‐days to expiration d. Trader D sold a call and bought a put at the existing levels, both with 90‐days to expiration



CORRECT: C C Is correct, as the strategy popularly known as the bull spread will result in positive payoff when the spot rises. As inflation increases, spot levels in commodities are expected to rise. Selling a call at higher level will reduce the cost of the strategy. Although it may limit the upside, but that would be in line with the view as only a moderate rise is expected in spot. INCORRECT: A – Is incorrect, as the strategy popularly known as a straddle is to be used when the view is that the volatility in the market will rise, and there is no directional view on the spot INCORRECT: B – Is incorrect, as the above option will be suitable when the spot is expected to fall from the existing levels INCORRECT: D – Is incorrect, as the payoff in this case is similar to short position in spot and would make sense when the underlying is expected to fall Reference: Hull, Chapter 10.

36. The information ratio of the Sterole US Fund for 2006 against the S&P 500, its benchmark index, is 1. For the same time period, the fund’s Sharpe ratio is 2, the fund has a tracking error of 7% against the S&P 500, and the standard deviation of fund returns is 5%. The risk‐ free rate in the US is 4%. Calculate the return for the S&P 500 during the time period. a. 3.5% b. 7% c. 11% d. 14%

CORRECT: B Sharpe Ratio = 2 (Fund Return – Risk Free Rate)/SD = 2 (Fund Return – 4%)/5% = 2 Fund Return = 14% Information Ratio = 1 (Fund Return – S&P 500 Return)/ Tracking Error = 1 (14% ‐ S&P 500 Return) / 7% = 1 S&P 500 Return = 7% INCORRECT: A – Incorrectly divides S&P 500 Return by 2 INCORRECT: C – The candidate might use the Tracking Error as the Numerator in both the Ratios Copyright © 2009 Global Association of Risk Professionals 41 All rights reserved.

2009 FRM Practice Exams Sharpe Ratio = 2 (Fund Return – Risk Free Rate)/Tracking Error = 2 (Fund Return – 4%)/7% = 2 Fund Return = 18% Information Ratio = 1 (Fund Return – S&P 500 Return)/ Tracking Error = 1 (18% ‐ S&P 500 Return) / 7% = 1 S&P 500 Return = 11% INCORRECT: D – The candidate can stop with the fund return calculation, and end up with 14% Sharpe Ratio = 2 (Fund Return – Risk Free Rate)/SD = 2 (Fund Return – 4%)/5% = 2 Fund Return = 14% Reference: Amenc and Le Sourd, Portfolio Theory and Performance Analysis. Chapter 4


a. b. c. d.

0.274 1.727 0.653 ‐0.378

CORRECT: A Sharpe ratio equals to

R p − RF

σ (R p )

While Sortino ratio equals to

=

9 .3 % − 3 .2 % = 0.452 13.5%

R p − RF

σ L ( RP )

=

9.3% − 3.2% = 0.726 8.4%

Tracking error is used to calculate the value of the information ratio, which is defined as

R p − RB

σ ( RP − RB )

, The calculation of information ratio is not required in this question.

0.726 – 0.452 = 0.274 INCORRECT: B – 2.178 – 0.452 = 1.727


2009 FRM Practice Exams INCORRECT: C – 0.726 – 0.0.73 = 0.653 (0.073 = (.093 – 0.032)/0.83 INCORRECT: D – 0.73‐0.452=0.378 Reference: Amenc and Le Sourd, Portfolio Theory and Performance Analysis. Chapter 4

38. Which of the following statements about the linear regression of the return of a portfolio over the return of its benchmark presented below are correct? Portfolio parameter Value Beta 1.25 Alpha 0.26 Coefficient of determination 0.66 Standard deviation of error 2.42

I. The correlation is 0.71 II. 34% of the variation in the portfolio return is explained by variation in the benchmark return III. The portfolio is the dependent variable IV. For an estimated portfolio return of 12%, the confidence interval at 95% is [7.16%;16.84%]

a. b. c. d.

II and IV III and IV I, II and III II, III and IV CORRECT: B The portfolio return is the dependent variable and for an estimated portfolio return of 12%, the 95% confidence interval is [12% ‐ 2 * 2.42%, 12% + 2 * 2.42%] or [7.16%, 16.84%]. However, the correlation is the square root of the coefficient of determination and is therefore equal to 0.81, and 66% of the variation in the portfolio returns is explained by variation in the benchmark return. Reference: Amenc and Le Sourd, Portfolio Theory and Performance Analysis. Chapter 4


2009 FRM Practice Exams 39. Your Board of Directors wants a comprehensive review of each business units’ operational risk activities. As the head of the corporate operational risk unit, you know that little has been done to implement an operational risk process at the business unit level and that you need to immediately come up with a framework. Which of the following statements offers the best strategy?

I. The audit committee of the Board should first define its objectives to ensure that all the firm’s business units’ operational risk programs are providing required information II. The auditing department is to be charged with developing an operational risk program for each business unit, with the business unit being made clearly aware that they will be held accountable for its implementation III. That your department immediately assess the operational risk for each business unit using independent data feeds to ensure the information fed into the assessment cannot be manipulated IV. A senior manager from each profit center is to be charged with developing their own operational risk self assessment program based on guidelines you provide.

a. b. c. d.

I only I and IV only I and III only IV only

CORRECT: D The best strategy for developing an operational risk framework is to empower business units with the responsibility, accountability and authority to manage their own operational risks. The business units know their risks the best. INCORRECT: A – ‘I’ is not the responsibility of the Audit Committee of the Board INCORRECT: B – The auditing department is not the best assessor of an individual business unit’s risk, in fact many audit staff do not fully understand the risks of many of a firm’s activities INCORRECT: C – III’ is duplicative and should not come from the corporate department Reference: Risk Management and Capital Adequacy, Gallati, 2003.

40. Which of the following risk management strategies of a firm which has principal payments to make on its debt in one year that substantially exceed the market value of its assets is most likely to be in the interest of the shareholders?

a. b. c. d.

Reduction of the overall risk level of the firm Increase of the overall risk level of the firm Keep the same risk level It is impossible to say which risk management strategy the shareholders prefer


2009 FRM Practice Exams CORRECT: B Once a firm is in distress, it is not in the interests of shareholders to reduce risk. If the firm stays in distress and eventually defaults, shareholders will end up with worthless shares. In these circumstances, management intent on maximizing shareholder value will seek out new risks. Reference: Risk Management and Derivatives, Stulz, 2003

END OF 2009 FRM Level I PRACTICE EXAM Questions & Explanations






2009 FRM Full Exam Practice Exam I Candidate Answer Sheet 1.

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2009 FRM Practice Exams This page is intentionally left blank.



2009 FRM Full Exam Practice Exam I Questions 1. Given the information provided in the table below, what is the portfolio VaR, at the 99% confidence level, of the following 100 million CHF equally weighted investment portfolio?

Asset Stocks Bonds

Expected Return 24.00% 15.00%

Volatility

Correlation

18% 6%

Stocks 1 0.1

Bonds 1

a. b. c. d.

27.96 million CHF 22.77 million CHF 20.97 million CHF 13.98 million CHF

2. You are asked by your boss to estimate the exposure of a hedge fund to the S&P 500. Though the fund claims to mark to market weekly, it does not do so and marks to market once a month. The fund also does not tell investors that it simply holds an ETF which is indexed to the S&P500. Because of the claims of the hedge fund, you decide to estimate the market exposure by regressing weekly returns of the fund on the weekly return of the S&P500. Which of the following properties correctly describes a property of your regression estimates?

a. The beta of your regression will be one because the fund holds the S&P 500. b. The beta of your regression will be zero because the fund returns are not synchronous with the S&P 500 returns. c. The intercept of your regression will be positive, showing that the fund has a positive alpha when estimated using an OLS regression. d. The beta will be misestimated because hedge fund exposures are non‐linear.


2009 FRM Practice Exams 3. The following table shows the composition of the GARP Bond Fund. What are the portfolio duration and portfolio yield of the fund? GARP Bond Fund Rating Amount Duration Mn USD in years AAA Company A 600 1.5 Company B 300 4 Company C 200 2.5 AA Company D 400 4 Company E 350 0.5 A Company F 150 1.5 Total

2000

Rating valuation matrix Years 0-1 Rating AAA 6.25% AA 6.75% A 7.75%

1-2

2-3

3-4

6.75% 7.35% 8.45%

7.35% 8.05% 9.15%

8.00% 8.80% 9.85%

a. b. c. d.

14 years, 46.1% 2.3 years, 7.5% 2.3 years, 7.7% 4.4 years, 15.4%%

4. An investment bank uses the Exponentially Weighted Moving Average (EWMA) technique with lambda of 0.9 to model the daily volatility of a security. The current estimate of the daily volatility is 1.5%. The closing price of the security is USD 20 yesterday and USD 18 today. Using continuously‐ compounded returns, what is the updated estimate of the volatility? a. 5.44% b. 3.62% c. 2.96% d. 1.31%


2009 FRM Practice Exams 5. Consider two stocks A and B. Assume their annual returns are jointly normally distributed, the marginal distribution of each stock has mean 2% and standard deviation 10%, and the correlation is 0.9. What is the expected annual return of stock A if the annual return of stock B is 3%?

a. b. c. d.

2.9% 2% 1.1% 4.7%

6. In pricing a derivative using the Monte Carlo method, we need to simulate a reasonable number of paths for the price of the underlying asset. Suppose we use a simple model for the return of the underlying asset: y(t) = drift*Δt + vol * √ Δt * e(t), and e(t) is distributed ~ N(0,1), where drift and vol are known parameters and Δt is the step size. The generation of each path requires a number of steps. Which of the following describes the correct procedure? a. Generate a random number from a normal distribution N(0,1), use the inverse normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path. b. Generate a random number from a normal distribution N(0,1), use the cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path. c. Generate a random number from a uniform distribution defined in [0,1], use the inverse cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path. d. Generate a random number from a uniform distribution defined in [0,1], use the cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path.


a. b. c. d.

5.10 14.47 9.20 17.20


2009 FRM Practice Exams 8. You don’t have access to KMV’s data. Your boss wants you to tell him your estimate of the probability of default of a credit. To do so, you use the Merton Model because the credit you are considering has no systematic risk. In Merton’s Model, the distance to default (DD) and the expected default frequency (EDF) are

a. b. c. d.

positively and linearly related negatively and linearly related positively and non‐linearly related negatively and non‐linearly related

9. Suppose the rate on Company A’s one‐year zero‐coupon bond is 10.0% and the one‐year T‐bill rate is 8.0%. Assume the T‐bill is riskless and the probability of default of Company A’s bond is 10%. What is the LGD of Company A’s bond?

a. b. c. d.

18.18% 81.82% 20.01% 79.99%

10. A bank is considering ways of significantly reducing or eliminating its credit exposure to defaults on a loan portfolio so that the bank’s shareholders do not absorb the losses arising from such defaults. Ignoring institutional issues (e.g., tax, accounting, capital requirements), three of the following programs have a similar impact on the credit risk of the bank. Which alternative fails to reduce credit risk?

a. b. c. d.

Sell the loan portfolio in its entirety to another bank. Borrow to finance an additional risk reserve to supplement existing loan‐loss reserves. Securitize the loan portfolio. Buy credit protection on the loan portfolio with credit default swaps.


a. b. c. d.



2009 FRM Practice Exams 12. The joint probability distribution of random variables X and Y is given by f(x,y) = kxy for x = 1, 2, 3, y = 1, 2, 3, and k is a positive constant. What is the probability that X + Y will exceed 5? a. 1/9 b. 1/4 c. 1/36 d. Cannot be determined




a. b. c. d.

2.41% 3.11% 4.40% 1.89%


a. b. c. d.

USD 0.23 USD 1.645 USD 0.33 USD 0.16


2009 FRM Practice Exams 16. A portfolio consists of two zero coupon bonds, each with a current value of USD 10. The first bond has a modified duration of 1 year and the second has a modified duration of 9 years. The yield curve is flat and all yields are 5%. Assume all moves of the yield curve are parallel shifts. Given that the daily volatility of the yield is 1%, which of the following is the best estimate of the portfolio daily VaR at the 95% confidence level?

a. b. c. d.

USD 2.33 USD 1.65 USD 0.82 USD 1.16


a. duration plus convexity, duration, full repricing b. full repricing, duration plus convexity, duration c. duration, duration plus convexity, full repricing d. duration, full repricing, duration plus convexity 18. Consider a position in a 5‐year receive‐fixed swap that makes annual payments on a USD 100 million notional. The floating leg has just been reset. The term structure is flat at 5%, the Macaulay duration of a 5‐year par bond is 4.5 years, and the annual volatility of yield changes is 100bp. Your best estimate of the swap’s VaR with 95% confidence over the next month is

a. b. c. d.

USD 1.6 million USD 2.0 million USD 5.5 million USD 7.1 million

19. If the gold lease rate is higher than the risk‐free rate, what is the market structure of the forward market for gold?

a. b. c. d.



2009 FRM Practice Exams 20. The price of a 3‐year zero coupon government bond is 85.16. The price of a similar 4‐year bond is 79.81. What is the one‐year implied forward rate from year 3 to year 4?

a. b. c. d.

5.4% 5.5% 5.8% 6.7%


a. b. c. d.

USD ‐1,953,125 USD ‐1,906,250 USD ‐2,046,875 USD ‐2,187,500


a. b. c. d.

8.9% 9.8% 35.7% 39.1%

23. An investor sells a June 2008 call of ABC Limited with a strike price of USD 45 for USD 3 and buys a June 2008 call of ABC Limited with a strike price of USD 40 for USD 5. What is the name of this strategy and the maximum profit and loss the investor could incur?

a. b. c. d.

Bear Spread, Maximum Loss USD 2, Maximum Profit USD 3 Bull Spread, Maximum Loss Unlimited, Maximum Profit USD 3 Bear Spread, Maximum Loss USD 2, Maximum Profit Unlimited Bull Spread, Maximum Loss USD 2, Maximum Profit USD 3



I. It gives too little weight to more recent observations II. For long‐only portfolios, it is likely to understate VaR following a recent structural increase in volatilities III. It always ignores the fat tails present in the distribution of returns on many financial assets IV. Because of the delta approximation, it inadequately measures the risk of nonlinear instruments a. b. c. d.



a. b. c. d.

Receive USD 2.23 million Receive USD 1.93 million Pay USD 1.93 million Pay USD 2.23 million


2009 FRM Practice Exams 26. Which of the following trade(s) contain basis risk?

I. Long 1,000 lots Nov 07 ICE Brent Oil contracts and short 1,000 lots Nov 07 NYMEX WTI Crude Oil contracts II. Long 1,000 lots Nov 07 ICE Brent Oil contracts and long 2,000 lots Nov 07 ICE Brent Oil at‐the‐ money put III. Long 1,000 lots Nov 07 ICE Brent Oil contracts and short 1,000 lots Dec 07 ICE Brent Oil contracts IV. Long 1,000 lots Nov 07 ICE Brent Oil contracts and short 1,000 lots Dec 07 NYMEX WTI Crude Oil contracts

a. b. c. d.

I & III II & IV IIII & IV I, III & IV

27. According to put‐call parity, buying a put option on a stock is equivalent to: a. Buying a call option and buying the stock with funds borrowed at the risk‐free rate. b. Selling a call option and buying the stock with funds borrowed at the risk‐free rate. c. Buying a call option, selling the stock and investing the proceeds at the risk‐free rate. d. Selling a call option, selling the stock and investing the proceeds at the risk‐free rate.


a. b. c. d.

USD 10.00 USD 7.50 USD 5.00 USD 1.50





a. b. c. d.

Purchase 2,602 shares Sell 2,602 shares Purchase 1,493 shares Sell 1,493 shares

31. Which of the following strategies creates a calendar spread? a. Sell a call option with a certain strike price and buy a longer maturity call option with the same strike price. b. Buy a call option with a certain strike price and buy a longer maturity call option with the same strike price. c. Sell a call option with a certain strike price and buy a shorter maturity call option with the same strike price. d. Buy a call option with a certain strike price and sell a longer maturity call option with the same strike price.


2009 FRM Practice Exams 32. Which of the following underlying macro‐economic conditions would leave an emerging market most vulnerable to the contagion effects of a currency crisis?

a. b. c. d.



a. b. c. d.

They should be exactly the same. The forward rate is normally higher than the futures rate. The forward rate is normally lower than the futures rate. They have no fixed relationship.


a. Trader A bought a call and a put, both with 90‐days to expiration and with strike price equal to the existing spot level. b. Trader B bought a put option with a down‐and‐in knock in feature. c. Trader C bought a call option at the existing spot levels and sold a call at a higher strike price, both with 90‐days to expiration. d. Trader D sold a call and bought a put at the existing levels, both with 90‐days to expiration.

35. Considering options generally (i.e., not only plain vanilla calls and puts), which of the following statements about vega is correct? a. An option holder can never be vega negative. b. A deep in the money up and out call option has a negative vega. c. A deep out of the money up and out call option has a negative vega. d. A deep out of the money digital option has a negative vega. Copyright © 2009 Global Association of Risk Professionals 59 All rights reserved.

2009 FRM Practice Exams 36. To hedge against future, unanticipated, and significant increases in borrowing rates, which of the following alternatives offers the greatest flexibility for the borrower?

a. b. c. d.


37. Assuming other things constant, bonds of equal maturity will still have different DV01 per USD 100 face value. Their DV01 per USD 100 face value will be in the following sequence of highest value to lowest value: a. Zero coupon bonds, par bonds, premium bonds b. Premium bonds, par bonds, zero coupon bonds c. Premium bonds, zero coupon bonds, par bonds d. Zero coupon bonds, premium bonds, par bonds 38. The information ratio of the Sterole US Fund for 2006 against the S&P 500, its benchmark index, is 1. For the same time period, the fund’s Sharpe ratio is 2, the fund has a tracking error of 7% against the S&P 500, and the standard deviation of fund returns is 5%. The risk‐ free rate in the US is 4%. Calculate the return for the S&P 500 during the time period.

a. b. c. d.

3.5% 7% 11% 14%


a. b. c. d.

0.274 1.727 0.653 ‐0.378


2009 FRM Practice Exams 40. Your firm has no prior derivatives trades with its counterparty Super Bank. Your boss wants you to evaluate some trades she is considering. In particular, she wants to know which of the following trades will increase your firm’s credit risk exposure to Super Bank:

I. buying a put option II. selling a put option III. buying a forward contract IV. selling a forward contract

a. b. c. d.

I. and II only II and IV only III and IV only I, III and IV only

41. Consider the following one‐period transition matrix:

Initial Period State

A B

Next Period State A B 95% 5% 10% 80%

Default 0% 10%

Default

0%

100%

0%

If a company is originally in State A, what is the probability that the company will have defaulted strictly before the fourth transition period from now?

a. b. c. d.

0.875% 0.500% 1.375% 1.875%

42. As an approximation, it is true that

a. b. c. d.

Default swap spread = Return of a risky bond + Return of a risk‐free bond Default swap spread = Return of a risky bond – Return of a risk‐free bond Default swap spread = Return of a risky bond x Return of a risk‐free bond Default swap spread = Return of a risky bond x (1 – Return of a risk‐free bond)


2009 FRM Practice Exams 43. In a CDO, the SPV is typically

a. b. c. d.

AAA‐rated A‐rated BBB‐rated Not rated

44. A trader whose risk you are monitoring tells you that he wants to benefit from a credit spread widening due to a recession. Which of the following would be good trades for his strategy?

a. Go long risky bonds and short risk‐free bonds at the beginning of the recession. b. Short risky bonds and go long risk‐free bonds at the beginning of the recession. c. Sell credit default swaps on bonds with a low credit quality and buy credit default swaps on bonds with a higher credit quality at the beginning of the recession. d. Sell credit default swaps on bonds with low credit quality and go long low credit quality bonds. 45. Bank B has a EUR 100 million loan portfolio and has set aside a reserve to cover the first EUR 20 million in default‐related losses. If the bank wants to acquire protection for the remaining EUR 80 million in risk exposure, which of the following solutions would work and would expose the bank to the least amount of counterparty risk?

a. Buy credit protection in a senior subordinated CDS that covers EUR 80 million in losses above the first EUR 20 million. b. Buy credit insurance for losses up to EUR 80 million in excess of EUR 20 million on the loan portfolio. c. Issue a credit‐linked note in which interest and principal may be withheld from investors to cover up to EUR 80 million in losses above the first EUR 20 million on the loan portfolio. d. All three of the above choices work and expose the bank to the same amount of counterparty risk.

46. Mr. Rosenqvist, Asset Manager at BCD Bank, holds a portfolio of SEK 200 million. The portfolio consists of BBB‐rated bonds. Assume that the one‐year probability of default is 4%, the recovery rate is 60%, and defaults are uncorrelated over years. What is the 2‐year cumulative expected credit loss on Mr. Rosenqvist’s portfolio?

a. b. c. d.

SEK 6.35 million SEK 6.40 million SEK 9.48 million SEK 9.60 million


2009 FRM Practice Exams 47. Using the Merton model, the value of the debt increases if all other parameters are fixed and

I. The value of the firm decreases II. The riskless interest rate decreases III. Time to maturity increases IV. The volatility of the firm value decreases

a. b. c. d.

I and II only I and IV only II and III only II and IV only

48. A firm is going to buy 10,000 barrels of West Texas Crude Oil. It plans to hedge the purchase using the Brent Crude futures contract. The correlation between the spot and futures prices is 0.72. The volatility of the spot price is 0.35 per year. The volatility of the Brent Crude futures price is 0.27 per year. What is the hedge ratio for the firm?

a. b. c. d.

0.5554 0.9333 1.2099 0.8198


a. b. c. d.

Short94 contracts Short98 contracts Short105 contracts Short113 contracts


2009 FRM Practice Exams 50. A bond trader has bought a position in Treasury Bonds with a 4% annual coupon rate on February 15, 2015. The DV01 of the position is USD 80,000. The trader decides to hedge his interest rate risk with the 4.5% coupon rate Treasury Bonds maturing on May 15, 2017 which has a DV01 of .076 per USD 100 face value. To implement this hedge, approximately what face amount of the 4.5% Treasury bonds maturing on May 15, 2017 should the trader sell?

a. b. c. d.

USD 80,000 USD 10,500,000 USD 80,000,000 USD 105,000,000

END OF 2009 FRM FULL EXAM PRACTICE EXAM I



2009 FRM Full Exam Practice Exam I Answer Key

1.

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25.

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d.

50.

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c.

d.



This page is intentionally left blank. Copyright © 2009 Global Association of Risk Professionals 66 All rights reserved.


2009 FRM Full Exam Practice Exam I Answers & Explanations 1. Given the information provided in the table below, what is the portfolio VaR, at the 99% confidence level, of the following 100 million CHF equally weighted investment portfolio?

Asset Stocks Bonds

Expected Return 24.00% 15.00%

Volatility

Correlation

18% 6%

Stocks 1 0.1

Bonds 1

a. b. c. d.

27.96 million CHF 22.77 million CHF 20.97 million CHF 13.98 million CHF

CORRECT: B The variance of the equally weighted portfolio is 0.5^2 * 0.18^2 + 0.5^2 * 0.06^2 + 2 * 0.5 * 0.5 * 0.1 * 0.18 * 0.06 = 0.081 + 0.0009 + 0.0005 = 0.00954. The volatility is then 9.77%. The portfolio VaR or the risk budget is 2.33 * 9.77% * 100 million CHF = 22.77 million CHF. Reference: Allen et al. Chapters 2,3.

2. You are asked by your boss to estimate the exposure of a hedge fund to the S&P 500. Though the fund claims to mark to market weekly, it does not do so and marks to market once a month. The fund also does not tell investors that it simply holds an ETF which is indexed to the S&P500. Because of the claims of the hedge fund, you decide to estimate the market exposure by regressing weekly returns of the fund on the weekly return of the S&P500. Which of the following properties correctly describes a property of your regression estimates?

a. The beta of your regression will be one because the fund holds the S&P 500. b. The beta of your regression will be zero because the fund returns are not synchronous with the S&P 500 returns. c. The intercept of your regression will be positive, showing that the fund has a positive alpha when estimated using an OLS regression. d. The beta will be misestimated because hedge fund exposures are non‐linear. Copyright © 2009 Global Association of Risk Professionals 67 All rights reserved.

2009 FRM Practice Exams CORRECT: C The alpha is spurious and results from the fact that returns are non‐synchronous. d. is incorrect because the true exposure is linear. The beta is greater than zero and less than one because of non‐ synchroneity. Reference: Amenc and Le Sourd, Portfolio Theory and Performance Analysis. Chapter 4

3. The following table shows the composition of the GARP Bond Fund. What are the portfolio duration and portfolio yield of the fund? GARP Bond Fund Rating Amount Duration Mn USD in years AAA Company A 600 1.5 Company B 300 4 Company C 200 2.5 AA Company D 400 4 Company E 350 0.5 A Company F 150 1.5 Total

2000

Rating valuation matrix Years 0-1 Rating AAA 6.25% AA 6.75% A 7.75%

1-2

2-3

3-4

6.75% 7.35% 8.45%

7.35% 8.05% 9.15%

8.00% 8.80% 9.85%

a. b. c. d.

14 years, 46.1% 2.3 years, 7.5% 2.3 years, 7.7% 4.4 years, 15.4%%

The calculation of portfolio duration and portfolio yield is based on the proportional weightage of respective company to its duration and yield. The portfolio duration and portfolio yield after mapping the yield from rating matrix is as follows; CORRECT: B


2009 FRM Practice Exams This answer reflects the proportion of amount taken as weights to calculate the portfolio duration and portfolio yield. GARP Bond Fund Amount Proportion % Duration Yield AAA Co A 600 30% 1.5 6.75% Co B 300 15% 4 8.00% Co C 200 10% 2.5 7.35% AA Co D 400 20% 4 8.80% Co E 350 18% 0.5 6.75% A Co F 150 8% 1.5 8.45% Total

2000

100%

2.30

7.54%

INCORRECT: A – If the candidate does a simple addition of the duration and the mapped yield, he would get this answer. INCORRECT: C – Though the portfolio duration is correct, but it is arrived by taking simple average of duration. However, if the candidate would take simple average of mapped yield instead of proportion his answer would be 7.7% and not 7.5% INCORRECT: D – If the candidate averages based on rating classes (3 – AAA/AA/A) instead of the companies, he would get this answer. Reference: Tuckman; Chapter 6.

4. An investment bank uses the Exponentially Weighted Moving Average (EWMA) technique with lambda of 0.9 to model the daily volatility of a security. The current estimate of the daily volatility is 1.5%. The closing price of the security is USD 20 yesterday and USD 18 today. Using continuously‐ compounded returns, what is the updated estimate of the volatility?

a. b. c. d.

5.44% 3.62% 2.96% 1.31%

CORRECT: B The current return of the security is = ln (18/20) = ‐10.536%. Using an EWMA model, the updated volatility is given as: V(t) = {lambda* ((V[t‐1]^2) +(1 – lambda)*(current return^2)} ^ 0.5 = {0.9 * ((0.015^2) + (1 ‐ 0.9) * ( ‐0.10536^2 )} ^ 0.5 Copyright © 2009 Global Association of Risk Professionals 69 All rights reserved.

2009 FRM Practice Exams = 3.62% INCORRECT: A – Forgets to square the volatility terms INCORRECT: C – Forgets to square the volatility terms and to take the square root of the resulting variance, then miscalculates conversion to percentage. INCORRECT: D – Forgets to take the square root of the variance, then miscalculates conversion to percentage. Reference : Hull, Chapter 21.

5. Consider two stocks A and B. Assume their annual returns are jointly normally distributed, the marginal distribution of each stock has mean 2% and standard deviation 10%, and the correlation is 0.9. What is the expected annual return of stock A if the annual return of stock B is 3%?

a. b. c. d.

2.9% 2% 1.1% 4.7%

CORRECT: A

E[ra | rb = x] = μa + (ρabσaσb/σ a)(x – μb) = 0.02 + 0.9 * (0.03 – 0.02) = 0.029

Reference: Damodar Gujarati, chapter 2.

6. In pricing a derivative using the Monte Carlo method, we need to simulate a reasonable number of paths for the price of the underlying asset. Suppose we use a simple model for the return of the underlying asset:

y(t) = drift*Δt + vol * √ Δt * e(t), and e(t) is distributed ~ N(0,1), where drift and vol are known parameters and Δt is the step size. The generation of each path requires a number of steps. Which of the following describes the correct procedure? a. Generate a random number from a normal distribution N(0,1), use the inverse normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path.


2009 FRM Practice Exams b. Generate a random number from a normal distribution N(0,1), use the cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path. c. Generate a random number from a uniform distribution defined in [0,1], use the inverse cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path. d. Generate a random number from a uniform distribution defined in [0,1], use the cumulative normal function to get e(t), which will be fed into the model to get y(t). Repeat the same procedure until you get the full desired path.

CORRECT: C This question wants to test if the candidate knows the basic steps to generate a very simple path: answering this question means that the candidate would be able to build a simple spreadsheet showing the Monte Carlo logic. The correct procedure is the one described in c); while a), b) and d) are nonsensical calculations. Reference: Philippe Jorion, Value at Risk,The New Benchmark for Managing Financial Risk, 3rd edition (New York: McGraw‐Hill, 2007), Chapter 12.


a. b. c. d.

15.10 14.47 9.20 17.20 CORRECT: D Since each variable is standardized, its variance is 1. Therefore V(3A+2B) = 9 V(A) + 4 V(B) + 2 x 3 x2 x Cov(A,B) = 9+4+4.2 = 17.2 INCORRECT: A – 9 + 4 + 6 * 0.35 = 15.1 INCORRECT: B – 9+ 4 + 12 * 0.35^2 = INCORRECT: C – 3 + 2 + 12 * 0.35 = 9.2 Reference: Damodar Gujarati


2009 FRM Practice Exams 8. You don’t have access to KMV’s data. Your boss wants you to tell him your estimate of the probability of default of a credit. To do so, you use the Merton Model because the credit you are considering has no systematic risk. In Merton’s Model, the distance to default (DD) and the expected default frequency (EDF) are

a. b. c. d.

positively and linearly related negatively and linearly related positively and non‐linearly related negatively and non‐linearly related

CORRECT: D The risk neutral probability of default, EDF, in the Merton Model is 1‐ N(d2). The higher the distance

ln(

V 1 ) − σ 2V T − rT 2 De σV T , the lower the risk neutral probability of default is. On

to default, DD DD ≡ d2 = the contrary, the lower DD, the higher EDF is. The relationship is non‐linear. When the DD is low, EDF is high. If DD is imminent, EDF is high as well. Similarly, if DD is high, EDF is small and not imminent Reference: De Servigny and Renault, Measuring and Managing Credit Risk, Chapter 3.

9. Suppose the rate on Company A’s one‐year zero‐coupon bond is 10.0% and the one‐year T‐bill rate is 8.0%. Assume the T‐bill is riskless and the probability of default of Company A’s bond is 10%. What is the LGD of Company A’s bond?

a. b. c. d.

18.18% 81.82% 20.01% 79.99% CORRECT: A (1+10%)*(1‐PD)+(1+10%)*PD*(1‐LGD)=1+8% 1.1 x 0.9 + 1.1 x 0.10 x (1 – LGD) = 1.08 0.99 + 0.11 x (1 ‐ LGD) = 1.08 0.11 x (1 ‐ LGD) = 1.08 – 0.99 (1 ‐ LGD) = (1.08 – 0.99) / 0.11 LGD = 1‐ (1.08 – 0.99) / 0.11 = 18.18% or Copyright © 2009 Global Association of Risk Professionals 72 All rights reserved.

2009 FRM Practice Exams LGD = 1 – ((1+rf) – (1+r) x (1 – PD))/((1+r) x PD) Reference: De Servigny and Renault, Measuring and Managing Credit Risk, Chapter 3, 4.

10. A bank is considering ways of significantly reducing or eliminating its credit exposure to defaults on a loan portfolio so that the bank’s shareholders do not absorb the losses arising from such defaults. Ignoring institutional issues (e.g., tax, accounting, capital requirements), three of the following programs have a similar impact on the credit risk of the bank. Which alternative fails to reduce credit risk?

a. b. c. d.

Sell the loan portfolio in its entirety to another bank. Borrow to finance an additional risk reserve to supplement existing loan‐loss reserves. Securitize the loan portfolio. Buy credit protection on the loan portfolio with credit default swaps.

CORRECT: B All three of the other choices are economically equivalent. Selling loans to an external party eliminates all credit risk for the institution. Similarly, securitizing the loan portfolio removes the loans from the bank’s books and eliminates the credit risk for the institution. Buying credit protection using credit default swaps will offer protection against credit risk. This alternative implies counterparty risk. Borrowing does not work in the long run because shareholders still at some point have to take the hit for default‐related losses. Additionally, the increased borrowing to finance the loan loss reserves will increase the risk for the shareholders. Reference: Culp, Chapter 16


a. b. c. d.


CORRECT: C


2009 FRM Practice Exams INCORRECT: A – The expected price is less than today’s price, but not the price in all the states of world. INCORRECT: B – The instantaneous rate of return on the stock follows normal distribution. INCORRECT: D – This model does not impose mean reversion. rd

Reference: Philippe Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3 ed. (New York: McGraw‐Hill, 2007). Chapter 12

12. The joint probability distribution of random variables X and Y is given by f(x,y) = kxy for x = 1, 2, 3, y = 1, 2, 3, and k is a positive constant. What is the probability that X + Y will exceed 5?

a. 1/9 b. 1/4 c. 1/36 d. Cannot be determined CORRECT: B 3

Note that

3

∑∑ f ( x, y) = 1 . x =1 y =1

Substituting the various values of x and y, we get f(1,1)=k, f(1,2)=2k, f(1,3)=3k, f(2,1)=2k, f(2,2)=4k, f(2,3)=6k, f(3,1)=3k, f(3,2)=6k, and f(3,3)=9k. Therefore, k1 + 2k + 3k + 2k + 4k + 6k + 3k + 6k + 9k = 1 so that, 36k = 1 and k=1/36. P(X+Y>5) = f(3,3) = 1/36 x 3 x 3 = 1/4 Reference: Damodar Gujarati





CORRECT: C The true statement is to reject Ho if the p‐value is smaller than the significance level. INCORRECT: A – regarding the primary use of Hypothesis Testing. INCORRECT: B – regarding the definition of type II error. INCORRECT: D – type I error and type II error are in tradeoff. Reference: Damodar Gujarati


a. b. c. d.

2.41% 3.11% 4.40% 1.89%

CORRECT: C The 1‐day volatility is s * (1/252)^0.5 = 0.3 * 0.629941 = 0.018898. The VaR at the 99% confidence level is then equal to 2.32635 * 0.018898 = 4.40% INCORRECT: A – One gets A if one uses 1.645 instead of 2.326, INCORRECT: B – One gets B if one uses the monthly volatility instead of the daily one INCORRECT: D – One gets D is the daily volatility. Reference: Allen, Boudoukh and Saunders, 2004, chapter 1, p 6‐8


a. b. c. d.

USD 0.23 USD 1.645 USD 0.33 USD 0.16

CORRECT: D Copyright © 2009 Global Association of Risk Professionals 75 All rights reserved.

2009 FRM Practice Exams This question requires candidates to know the formula for the delta‐normal VaR approximation, and also to know that the delta of an at‐the‐money call is 0.5. VaR =| Δ | ×1.645 × σ × S = 0.5 × 1.645 × 0.02 × 10 = 0.1645 . INCORRECT: A – We get A by using 2.326 instead of 1.645 INCORRECT: B – We get B if we use 2 instead of 2% for the volatility INCORRECT: C – We get C if we use a delta of 1 Reference: Allen et al, Chapter 3

16. A portfolio consists of two zero coupon bonds, each with a current value of USD 10. The first bond has a modified duration of 1 year and the second has a modified duration of 9 years. The yield curve is flat and all yields are 5%. Assume all moves of the yield curve are parallel shifts. Given that the daily volatility of the yield is 1%, which of the following is the best estimate of the portfolio daily VaR at the 95% confidence level?

a. b. c. d.

USD 2.33. USD 1.65. USD 0.82. USD 1.16

CORRECT: B This question assesses candidates’ abilities to apply the duration VaR formula to two bonds simultaneously and to recall that the duration of a zero coupon bond is equal to the bond maturity. Using an obvious extension of Jorion’s equation 9.5

VaR = D1 × V1 × 1.645 × σ + D2 × V2 × 1.645 × σ = ( D1 × V1 + D2 × V2 ) × 1.645 × σ = ( D1 + D2 ) × 10 × 1.645 × σ = 10 × 10 × 1.645 × 0.01 = 1.645 INCORRECT: A – Is the 99% confidence level VaR INCORRECT: C – Arises if the candidate mistakenly divides the correct answer by the number of bonds INCORRECT: D – Makes both mistakes Reference: Tuckman


2009 FRM Practice Exams 17. Consider the following three methods of estimating the P&L of a bullet bond: full repricing, duration (PV01), and duration plus convexity. Ranking the estimated P&L impact of a large negative yield shock from the lowest P&L impact to the highest P&L impact, what is the ranking of the methods to estimate the P&L impact?

a. b. c. d.

duration plus convexity, duration, full repricing full repricing, duration plus convexity, duration duration, duration plus convexity, full repricing duration, full repricing, duration plus convexity

CORRECT: C The price / yield line with yield on the x axis and price on the y axis is convex to the origin. The duration at any yield level is the tangent to that curve. Therefore, except at the exact point of tangency, duration will always underestimate the price change. INCORRECT: A – Duration will always underestimate price change for negative yield shocks INCORRECT: B – Full repricing will never generate a smaller positive price change than duration because duration represents the point of tangency INCORRECT: D – Full repricing will generate a higher price for a large negative yield change than wil duration plus convexity Reference: Allen, Boudoukh, Saunders, Chapter 3

18. Consider a position in a 5‐year receive‐fixed swap that makes annual payments on a USD 100 million notional. The floating leg has just been reset. The term structure is flat at 5%, the Macaulay duration of a 5‐year par bond is 4.5 years, and the annual volatility of yield changes is 100bp. Your best estimate of the swap’s VaR with 95% confidence over the next month is

a. b. c. d.

USD1.6 million USD 2.0 million USD 5.5 million USD 7.1 million

CORRECT: A Because the floating‐rate leg has just been reset, its duration is 1. Net duration is 4.5‐1=3.5 year, or modified duration of 3.5/1.05=3.33. The 95% VaR of monthly changes in yields is 1.65*1%/√12 = 0.48%. Multiplying, this gives USD 100*0.48%*3.33=USD 1.588


2009 FRM Practice Exams INCORRECT: B – This uses a net duration of 4.5 years and ignores the duration of the floating‐rate leg. INCORRECT: C – This is the annual VaR, but should be translated to a monthly horizon. INCORRECT: D – This is the annual VaR computed by ignoring the duration of the floating‐rate leg. Reference: Hull, Chapter, Chapter 7 19. If the gold lease rate is higher than the risk‐free rate, what is the market structure of the forward market for gold?

a. b. c. d.


CORRECT: B A lease rate higher than the risk fee rate will force a negatively sloped forward curve, i.e. backwardation INCORRECT: A – The forward price = spot*exp( risk free rate ‐ lease rate). If the lease rate is higher than the risk free rate, forwards will be lower than spot, implying contango INCORRECT: C – The term inversion is used to describe yield curves, not commodity forwards INCORRECT: D – There is enough information in the question to provide an answer Reference: MacDonald, Chapter 6

20. The price of a 3‐year zero coupon government bond is 85.16. The price of a similar 4‐year bond is 79.81. What is the one‐year implied forward rate from year 3 to year 4?

a. b. c. d.

5.4% 5.5% 5.8% 6.7%

CORRECT: D


2009 FRM Practice Exams 1 + Forward rate =

85.16 Price of three bond = = 1.067034 Price of four year bond 79.81

Forward rate = 0.067034 or 6.7%

INCORRECT: A – Is a combination of B and C INCORRECT: B – Is the return of the 3‐year bond INCORRECT: C – Is the return of the 4‐year bond Reference: Hull, Tuckman


a. b. c. d.

USD ‐1,953,125 USD ‐1,906,250 USD ‐2,046,875 USD ‐2,187,500

CORRECT: A

ΔV = - D mod × Δy × V + 0.5 × Convexity × Δy 2 × V ΔV = - 8 × 0.0025 × 100M + 0.5 × 150 × (0.0025) 2 × 100M ΔV = - 2M + 46,875 ΔV = - 1,953,125 INCORRECT: B – Omits 0.5 from the second term INCORRECT: C – Subtracts the second term INCORRECT: D – Makes both mistakes Reference: Tuckman 22. What is the annualized rate of return earned on a cash‐and‐carry trade entered into in March and closed out in June? a. 8.9% b. 9.8% c. 35.7% d. 39.1% Copyright © 2009 Global Association of Risk Professionals 79 All rights reserved.


CORRECT: C By formula F0,T = S0erT + C, where F0,T = June forward price, S0 = March forward price, r = risk free interest rate, T = length of cash‐and‐carry, C = storage cost Solving 5.90 = 5.35er*3/12 + 0.05 Solution is r = 35.7% INCORRECT: A – 8.9 = LN((5.9‐0.05)/5.35) (forgets to annualize the return) INCORRECT: B – 9.8 = LN((5.9)/5.35) (forgets to include the storage cost and to annualize the return) INCORRECT: D – 39.1 = (12/3)LN((5.9)/5.35) – 0.05 (forgets to include the storage cost) Reference: Robert L McDonald, Derivatives Markets, Chapter 6

23. An investor sells a June 2008 call of ABC Limited with a strike price of USD 45 for USD 3 and buys a June 2008 call of ABC Limited with a strike price of USD 40 for USD 5. What is the name of this strategy and the maximum profit and loss the investor could incur? a. Bear Spread, Maximum Loss USD 2, Maximum Profit USD 3 b. Bull Spread, Maximum Loss Unlimited, Maximum Profit USD 3 c. Bear Spread, Maximum Loss USD 2, Maximum Profit Unlimited d. Bull Spread, Maximum Loss USD 2, Maximum Profit USD 3

CORRECT: D Buying a call option at lower stock price and selling call option at higher strike price is called as Bull Spread. Bear Spread is buying the call option at higher price and selling the call at lower strike price. The Cost of strategy will be USD 3‐USD 5 = ‐USD 2 The Payoff, when Stock price ST ≤ USD 40 will be ‐USD 2 (the cost of strategy) as none of the option will be exercised. The Payoff, when stock price ST ≥ 45, (as both options will be exercise) will be USD 5, Since the cost of strategy is ‐USD 3, hence profit will be USD 5‐USD 2 = USD 3 When Stock price is USD 40< ST > USD 45, Only the call option bought by the investor would be exercised hence the pay off will be ST – 40, since the cost of strategy is ‐USD 3, The Net profit will be ST – 43, which would always be lower than USD 3. Reference: Hull, Chapter 10‐ Trading Strategies Involving Options.




a. b. c. d.


CORRECT: C The disadvantage with the Historical Simulation Model is that it may not recognize the changes in volatility and correlation following recent structural changes. The model can be adjusted so that it gives more weight to recent observations. The other options, i.e. III & IV, are disadvantages of Monte Carlo method and Delta‐normal method. Reference: Allen et al. Chapters 2,3. 25. A bank holds USD 60 million worth of 10‐year 6.5% coupon bonds that are trading at a clean price of 101.82. The bank is worried by the exposure due to these bonds but cannot unwind the position for fear of upsetting the client. Therefore, it purchases a total return swap (TRS) in which it receives annual Libor + 100 bps in return for the mark‐to‐market return on the bond. For the first year, the Libor sets at 6.25% and by the end of the year the clean price of the bonds is at 99.35. The net receipt/payment for the bank in the total return swap will be:

a. Receive USD 2.23 million. b. Receive USD 1.93 million. c. Pay USD 1.93 million. d. Pay USD 2.23 million. CORRECT: B


2009 FRM Practice Exams it’s the result of this calculation: the notional amount is 60 million USD . Therefore the bank will receive the interest payment linked to the LIBOR rate: 60 million USD * (6,25%+100 bp) = 4. 35 million USD . The bank will pay the fixed coupon plus the change in the value of the bond: 60 million USD * 6.5% + 60 million *(99.35%‐101.82%) = 2.418 million USD . Hence the total net amount the bank will receive is: 4.35 million USD ‐ 2.418 million USD = 1.932 million USD . Reference: Hull Chapter 7 ‐ Swaps

26. Which of the following trade(s) contain basis risk?

I. Long 1,000 lots Nov 07 ICE Brent Oil contracts and short 1,000 lots Nov 07 NYMEX WTI Crude Oil contracts II. Long 1,000 lots Nov 07 ICE Brent Oil contracts and long 2,000 lots Nov 07 ICE Brent Oil at‐ the‐money put III. Long 1,000 lots Nov 07 ICE Brent Oil contracts and short 1,000 lots Dec 07 ICE Brent Oil contracts IV. Long 1,000 lots Nov 07 ICE Brent Oil contracts and short 1,000 lots Dec 07 NYMEX WTI Crude Oil contracts

a. b. c. d.

I & III II & IV IIII & IV I, III & IV CORRECT: D Basis Risk is spread risk, which arise from trading the spread (long and short 2 positively correlated assets or same asset with different expiration) I is spread trade in highly correlated asset with same expiration month II faces with gamma and vega risk III is spread trade in trading the flattening of the forward curve IV is spread trade in trading 2 assets with different expiration date Reference: Robert L. McDonald, Derivatives Markets (Boston: Addison‐Wesley, 2003), Chapter 6.


2009 FRM Practice Exams 27. According to put‐call parity, buying a put option on a stock is equivalent to:

a. b. c. d.

Buying a call option and buying the stock with funds borrowed at the risk‐free rate. Selling a call option and buying the stock with funds borrowed at the risk‐free rate. Buying a call option, selling the stock and investing the proceeds at the risk‐free rate. Selling a call option, selling the stock and investing the proceeds at the risk‐free rate. CORRECT: C Buying a call option, selling the stock and investing the proceeds at the risk‐free rate. INCORRECT: A – Buying a call option is correct, but the rest of the statement is incorrect. INCORRECT: B – The entire statement is incorrect. INCORRECT: D – Selling a call option is incorrect, but the rest of the statement is correct. Reference: Options, Futures, and Other Derivatives, 6th edition, by John Hull, Chapter 10.


a. b. c. d.

USD 10.00 USD 7.50 USD 5.00 USD 1.50

CORRECT: C According to the fundamental pricing relationship between spot assets and the associated futures, the futures price, to prevent arbitrage, should equal 990 x e (0.04 – 0.02) x 0.25 or 995. Hence, the futures contract is overvalued, indicating it should be sold and the index should be purchased for an arbitrage profit of USD 1000 ‐ USD 995 = USD 5 th

Reference: Hull, Options, Futures, and Other Derivatives, 6 ed. Chapter 5




CORRECT: B The question tests on understanding of a “straddle” strategy and its application on currency trading. A long straddle strategy involves buying (long) a call and put option with the same strike price and expiration date, and will benefit most when the underlying moves away from the current equlibrium. Long call and long put create a straddle. INCORRECT: A – It sells a put option while it should buy one put. . INCORRECT: C – It sells a call option while it should buy one call. INCORRECT: D – It sells both the call and put option while it should buy both. Reference: Hull.


a. b. c. d.

Purchase 2,602 shares. Sell 2,602 shares. Purchase 1,493 shares. Sell 1,493 shares. CORRECT: A Number of calls = 200 contracts x 100 = 20,000 calls. Copyright © 2009 Global Association of Risk Professionals 84 All rights reserved.

2009 FRM Practice Exams Number of shares = (Number of calls) x (New delta – Old delta) = 20,000 x (0.7040 – 0.5739) = +2,602 shares Positive sign indicates that the manager should purchase new shares. INCORRECT: B – Because the formula is incorrect, i.e. old delta minus new delta. INCORRECT: C – Because the number of shares (instead of number of calls) is used in the calculation. INCORRECT: D – As per explanation in ‘C’ above and sign error. Reference: Hull.

31. Which of the following strategies creates a calendar spread? a. Sell a call option with a certain strike price and buy a longer maturity call option with the same strike price. b. Buy a call option with a certain strike price and buy a longer maturity call option with the same strike price. c. Sell a call option with a certain strike price and buy a shorter maturity call option with the same strike price. d. Buy a call option with a certain strike price and sell a longer maturity call option with the same strike price.

CORRECT: A INCORRECT: B – As buy a call option. INCORRECT: C – As buy a shorter‐maturity call option INCORRECT: D – As this is a reverse calendar spread. Reference: Hull.

32. Which of the following underlying macro‐economic conditions would leave an emerging market most vulnerable to the contagion effects of a currency crisis?

a. b. c. d.


CORRECT: B Copyright © 2009 Global Association of Risk Professionals 85 All rights reserved.

2009 FRM Practice Exams INCORRECT: A – Large current account surplus and non‐convertible currency would protect the local currency INCORRECT: C – High foreign exchange reserves and non‐convertible currency would protect the local currency INCORRECT: D – Large current account surplus and high foreign exchange would protect the local currency Reference: Saunders, Chapter 15, Foreign Exchange Risk


a. b. c. d.

They should be exactly the same. The forward rate is normally higher than the futures rate. The forward rate is normally lower than the futures rate. They have no fixed relationship.

CORRECT: C As Eurodollar futures contract is marked to market and settled daily, normally forward rate is adjusted lower, so called convexity adjustment, by:

1 2

Forward rate = Futures rate – σ 2T1T2 Reference: Hull.


a. Trader A bought a call and a put, both with 90‐days to expiration and with strike price equal to the existing spot level. b. Trader B bought a put option with a down‐and‐in knock in feature. c. Trader C bought a call option at the existing spot levels and sold a call at a higher strike price, both with 90‐days to expiration. d. Trader D sold a call and bought a put at the existing levels, both with 90‐days to expiration. Copyright © 2009 Global Association of Risk Professionals 86 All rights reserved.

2009 FRM Practice Exams CORRECT: C As the strategy popularly known as the bull spread will result in positive payoff when the spot rises. As inflation increases, spot levels in commodities are expected to rise. Selling a call at higher level will reduce the cost of the strategy. Although it may limit the upside, but that would be in line with the view as only a moderate rise is expected in spot. INCORRECT: A – As the strategy popularly known as a straddle is to be used when the view is that the volatility in the market will rise, and there is no directional view on the spot. INCORRECT: B – As the above option will be suitable when the spot is expected to fall from the existing levels. INCORRECT: D – As the payoff in this case is similar to short position in spot and would make sense when the underlying is expected to fall. Reference: Hull, Chapter 10.

35. Considering options generally (i.e., not only plain vanilla calls and puts), which of the following statements about vega is correct?

a. b. c. d.

An option holder can never be vega negative. A deep in the money up and out call option has a negative vega. A deep out of money up and out call option has a negative vega. A deep out of money digital option has a negative vega. CORRECT: B Deep in the money Up and Out call option because an increase in the volatility of such options leads to the increasing chances of option either being knocked out (if the price increases beyond the barrier) or loosing its moneyness (if the prices falls) and hence the increasing volatility tends to have negative impact on the price of the option. INCORRECT: A – As an option holder can be Vega negative as shown above. INCORRECT: C – have positive Vega as an increase in the volatility would increase the chances of getting towards moneyness and hence positive Vega from a holder’s perspective. INCORRECT: D – have positive Vega as an increase in the volatility would increase the chances of getting towards moneyness and hence positive Vega from a holder’s perspective. Reference: Hull, Chapters 17,18 24.


2009 FRM Practice Exams 36. To hedge against future, unanticipated, and significant increases in borrowing rates, which of the following alternatives offers the greatest flexibility for the borrower?

a. b. c. d.


CORRECT: D The question focuses on flexible management of borrowing expenses. While a fixed for floating swap could reduce borrowing expenses, it is a long‐term contractual commitment to exchange payments. If interest rates decline, the borrower may gross up to the agreed fixed rate. An interest rate collar is a combination of an interest rate floor and cap, i.e., it locks in the interest expenses within a tight range. Moreover, collars usually offer interest rate protection at one particular point of time unless several contracts with different maturities are exchanged. A call swaption gives the company the right to enter into a swap when the borrowing expenses exceed a certain reference rate. If the reference rate is below the borrowing expenses, the option is not exercised. Reference: Hull.

37. Assuming other things constant, bonds of equal maturity will still have different DV01 per USD 100 face value. Their DV01 per USD 100 face value will be in the following sequence of highest value to lowest value:

a. Zero coupon bonds, par bonds, premium bonds b. premium bonds, par bonds, zero coupon bonds c. Premium bonds, zero coupon bonds, par bonds d. Zero coupon bonds, premium bonds, par bonds CORRECT: B DV01 is certain multiple of Dirty Price (which includes Coupons) and not Clean Price. Thus, it is proportional to Base Price, which is Dirty Price. Ordinarily, Premium Bond will have the highest (dirty) price followed by Par Bond and with the least price of Zero Coupon Bond. Hence, DV01 of Premium Bond is the highest while that of Zero Coupon Bonds is the lowest. INCORRECT: A – Premium Bond will have a higher Base Price and hence higher DV01 than that of Zero Coupon Bond. INCORRECT: C – Base Price of Par Bond is higher than that of Zero Coupon Bond and hence, its DV01 cannot be less than that of Zero Coupon Bond. Copyright © 2009 Global Association of Risk Professionals 88 All rights reserved.

2009 FRM Practice Exams INCORRECT: D – DV01 per USD 100 Face Value is an Absolute Amount of USD based on actual Base Price Change. Ordinarily, Base Price of a Zero Coupon Bond will be lower than that of Par & Premium Bond. Hence, DV01 of Zero Coupon Bond is less than that of Premium Bond of same maturity. Reference: Tuckman, “Fixed Income Securities”, Chapter 5.

38. The information ratio of the Sterole US Fund for 2006 against the S&P 500, its benchmark index, is 1. For the same time period, the fund’s Sharpe ratio is 2, the fund has a tracking error of 7% against the S&P 500, and the standard deviation of fund returns is 5%. The risk‐ free rate in the US is 4%. Calculate the return for the S&P 500 during the time period. a. 3.5% b. 7% c. 11% d. 14%

CORRECT: B Sharpe Ratio = 2 (Fund Return – Risk Free Rate)/SD = 2 (Fund Return – 4%)/5% = 2 Fund Return = 14% Information Ratio = 1 (Fund Return – S&P 500 Return)/ Tracking Error = 1 (14% ‐ S&P 500 Return) / 7% = 1 S&P 500 Return = 7% INCORRECT: A – Incorrectly divides S&P 500 Return by 2 INCORRECT: C – The candidate might use the Tracking Error as the Numerator in both the Ratios. Sharpe Ratio = 2 (Fund Return – Risk Free Rate)/Tracking Error = 2 (Fund Return – 4%)/7% = 2 Fund Return = 18% Information Ratio = 1 (Fund Return – S&P 500 Return)/ Tracking Error = 1 (18% ‐ S&P 500 Return) / 7% = 1 S&P 500 Return = 11% INCORRECT: D – The candidate can stop with the fund return calculation, and end up with 14%. Sharpe Ratio = 2


2009 FRM Practice Exams (Fund Return – Risk Free Rate)/SD = 2 (Fund Return – 4%)/5% = 2 Fund Return = 14% Reference: Amenc and Le Sourd, Portfolio Theory and Performance Analysis. Chapter 4


a. 0.274 b. 1.727 c. 0.653 d. ‐0.378 CORRECT: A Sharpe ratio equals to

R p − RF

σ (R p )

While Sortino ratio equals to

=

9 .3 % − 3 .2 % = 0.452 13.5%

R p − RF

σ L ( RP )

=

9.3% − 3.2% = 0.726 8.4%

Tracking error is used to calculate the value of the information ratio, which is defined as

R p − RB

σ ( RP − RB )

, The calculation of information ratio is not required in this question.

INCORRECT: B – 2.178 – 0.452 = 1.727 INCORRECT: C – 0.726 – 0.0.73 = 0.653 (0.073 = (.093 – 0.032)/0.83) INCORRECT: D – 0.073 – 0.452 = ‐0.378 Reference: Amenc and Le Sourd, Portfolio Theory and Performance Analysis. Chapter 4


2009 FRM Practice Exams 40. Your firm has no prior derivatives trades with its counterparty Super Bank. Your boss wants you to evaluate some trades she is considering. In particular, she wants to know which of the following trades will increase your firm’s credit risk exposure to Super Bank: I. buying a put option II. selling a put option III. buying a forward contract IV. selling a forward contract

a. b. c. d.

I. and II only II and IV only III and IV only I, III and IV only

CORRECT: D This tests understanding of the type of positions that create credit risk (and links to the following question). The key is to evaluate each of the component trades. Buying a put option creates credit risk. Buying or selling forward contracts creates credit risk. However, selling an option does not create credit risk you are not subject to the performance of the counterparty. Reference: Hull Chapter 22, Stulz Chapter 18.

41. Consider the following one‐period transition matrix:

Initial Period State

A B

Next Period State A B 95% 5% 10% 80%

Default 0% 10%

Default

0%

100%

0%

If a company is originally in State A, what is the probability that the company will have defaulted strictly before the fourth transition period from now?

a. 0.875% b. 0.500% c. 1.375% d. 1.875% CORRECT: C


2009 FRM Practice Exams The easiest way to determine the answer would be to make this a square matrix including default in initial state. Then self‐multiplying the matrix three times yields three‐period transition matrix. We can also manually do the calculation; After year 1 there is a 0% chance of default and 5% chance of being in state B. After year 2 there is 95%*5% + 80%*5% chance of being in state B and 5% * 10% chance of default. After year 3 there is a (95%*5% + 80%*5%)*10% additional chance of default. Answer A assumes just one year INCORRECT: A – Only considers the third year transition from B to default. INCORRECT: B – Only considers the second year transition from B to default. INCORRECT: D – Mistakenly doubles the second year transition from B to default. Reference: De Servigny and Renault, Measuring and Managing Credit Risk, Chapter 2, Appendix 2A, page 49 – 52.

42. As an approximation, it is true that

a. b. c. d.

Default swap spread = Return of a risky bond + Return of a risk‐free bond Default swap spread = Return of a risky bond – Return of a risk‐free bond Default swap spread = Return of a risky bond x Return of a risk‐free bond Default swap spread = Return of a risky bond x (1 ‐ Return of a risk‐free bond)

CORRECT: B The buyer of a risky bond can hedge the credit risk of the risky bond using a default swap. Entering into the swap trade reduces credit risk. To preclude arbitrage, the buyer of the risk bond has to receive the same return as the risk‐free asset, or: Return of a risky bond = Return of a risk‐free bond + Default swap spread Reference: Hull, Chapter 23.

43. In a CDO, the SPV is typically

a. b. c. d.

AAA‐rated A‐rated BBB‐rated Not rated

CORRECT: A


2009 FRM Practice Exams In a CDO transaction, the Special Purpose Vehicle are special entities of financial institutions and are usually AAA rated. The SPV and the institution are legally distinct and credit quality deterioration of the financial institution does not affect the SPV. In this case SPV counterparty risk is low, which is desired by the investor. Reference: Hull chapter 23, Culp Chapters 16, 17, 18.

44. A trader whose risk you are monitoring tells you that he wants to benefit from a credit spread widening due to a recession. Which of the following would be good trades for his strategy?

a. Go long risky bonds and short risk‐free bonds at the beginning of the recession. b. Short risky bonds and go long risk‐free bonds at the beginning of the recession. c. Sell credit default swaps on bonds with a low credit quality and buy credit default swaps on bonds with a higher credit quality at the beginning of the recession. d. Sell credit default swaps on bonds with low credit quality and go long low credit quality bonds.

CORRECT: B Shorting risky bonds and going long in risk‐free bonds at the beginning of the recession is the correct answer. During a recession, credit spreads typically start widening, and financing a long position in risk‐free bonds through declining credit quality risky bonds reduces the effective financing cost. INCORRECT: A – Going long in risky bonds and shorting risk‐free bonds at the beginning of the recession is incorrect. This strategy is preferable at the beginning of an economic expansion, when the credit spread typically starts tightening. INCORRECT: C – Selling credit default swaps on bonds with a certain credit quality and buying credit default swaps on bonds with a higher credit quality at the beginning of the recession would be preferable at the beginning of an economic expansion. INCORRECT: D – Selling credit default swaps on bonds with low credit quality and going long in low credit quality bonds effectively magnifies the credit risk, which under deteriorating credit conditions should be avoided. Reference: Culp, Chapter 12, 13, 16.


2009 FRM Practice Exams 45. Bank B has a EUR 100 million loan portfolio and has set aside a reserve to cover the first EUR 20 million in default‐related losses. If the bank wants to acquire protection for the remaining EUR 80 million in risk exposure, which of the following solutions would work and would expose the bank to the least amount of counterparty risk?

a. Buy credit protection in a senior subordinated CDS that covers EUR 80 million in losses above the first EUR 20 million. b. Buy credit insurance for losses up to EUR 80 million in excess of EUR 20 million on the loan portfolio. c. Issue a credit‐linked note in which interest and principal may be withheld from investors to cover up to EUR 80 million in losses above the first EUR 20 million on the loan portfolio. d. All three of the above choices work and expose the bank to the same amount of counterparty risk.

CORRECT: C Both CDS and insurance are “unfunded,” and expose the bank to the risk of non‐performance by the CDS protection seller or the insurance company offering credit insurance. When issuing a credit‐ linked note, however, the cash has been paid in up‐front by investors to the bank, eliminating counterparty risk. Reference: Culp, Chapter 16

46. Mr. Rosenqvist, Asset Manager at BCD Bank, holds a portfolio of SEK 200 million. The portfolio consists of BBB‐rated bonds. Assume that the one‐year probability of default is 4%, the recovery rate is 60%, and defaults are uncorrelated over years. What is the 2‐year cumulative expected credit loss on Mr. Rosenqvist’s portfolio?

a. b. c. d.

SEK 6.35 million SEK 6.40 million SEK 9.48 million SEK 9.60 million

CORRECT: A The Credit Loss Year 1 is SEK million [200 * 4% * (1 – 60%)] = 3.2 and the Credit Loss Year 2 is SEK million [(200 – 3.2) * 4% * (1 – 60%)] = 3.15. The cumulative expected loss over 2 years than is 3.2 + 3.15 = 6. 35. INCORRECT: B – Does not take into account the credit loss year 1 when calculating the loss for year 2.


2009 FRM Practice Exams INCORRECT: C – Wrongly interprets the recovery rate as a measure of credit loss, while credit loss equals (1 – recovery rate). In this case it equals (1 ‐ 0.60) = 40%. INCORRECT: D – Is a combination of mistake B and C. Reference: de Servigny and Renault, Measuring and Managing Credit Risk.

47. Using the Merton model, the value of the debt increases if all other parameters are fixed and

I. The value of the firm decreases II. The riskless interest rate decreases III. Time to maturity increases IV. The volatility of the firm value decreases

a. b. c. d.

I and II only I and IV only II and III only II and IV only

CORRECT: D According to the model, the value of the bond is B = V – S, where V is the value of the assets and S is the value of the equity, or Ke‐rtN(d2) + V x (1‐N(d1)). d1=ln(V/Ke‐rt)/σ√T+ σ√T/2 and d2 = 1 – d1. Value of the debt will increase if interest rates decreases and volatility of the firm decreases. INCORRECT: A – II is true but I is false. Value of the debt will increase if value of the firm increases. The value of the debt will increase if interest rate decreases. INCORRECT: B – V is true but I is false. INCORRECT: C – II is true but III is false. The value of the debt will increase of the time to maturity decreases Reference:Stulz, Risk Management & Derivatives, Chapter 18, p. 580 de Servigny and Renault, Measuring and Managing Credit Risk.

48. A firm is going to buy 10,000 barrels of West Texas Crude Oil. It plans to hedge the purchase using the Brent Crude futures contract. The correlation between the spot and futures prices is 0.72. The volatility of the spot price is 0.35 per year. The volatility of the Brent Crude futures price is 0.27 per year. What is the hedge ratio for the firm? Copyright © 2009 Global Association of Risk Professionals 95 All rights reserved.

2009 FRM Practice Exams a. b. c. d.

0.5554 0.9333 1.2099 0.8198

CORRECT: B ⎛ 0.35 ⎞ N = 0.72 × ⎜ ⎟ ⎝ 0.27 ⎠ N= 0.9333

INCORRECT: A – Inverts the spot volatility and the futures volatility. INCORRECT: C – Uses variances INCORRECT: D – Uses square roots of the volatilities. Reference: Hull, Chapter 3


a. Short 94 contracts b. Short 98 contracts c. Short 105 contracts d. Short 113 contracts CORRECT: B ⎛ 10,000,000 ⎞ ⎛ 7.8 ⎞ N=⎜ ⎟×⎜ ⎟ ⎝ 95,062.50 ⎠ ⎝ 8.4 ⎠

N = 97.68 or 98 contracts

INCORRECT: A – Is made up. Copyright © 2009 Global Association of Risk Professionals 96 All rights reserved.

2009 FRM Practice Exams INCORRECT: C – Leaves out the durations INCORRECT: D – Inverts the durations. Reference: Hull, Chapter 6.


a. b. c. d.

USD 80,000 USD 10,500,000 USD 80,000,000 USD 105,000,000

CORRECT: D USD 105,000,000x.076/100 = USD 79,800, which is pretty close to the desired DV01 of USD 80,000. To solve for the hedge, solve for F in the equation USD 80,000 = Fx.076/100, giving F = 105,263,158. INCORRECT: A – Selling this amount would offset a DV01 of only USD 80,000x.076/100 = USD 61. INCORRECT: B – USD 10,500,000x.076/100 = USD 7,980. INCORRECT: C – USD 80,000,000x.076/100 = USD 60,800. Reference: Tuckman

END OF 2009 FRM FULL EXAM PRACTICE EXAM I Questions & Explanations






2009 FRM Full Exam Practice Exam II Candidate Answer Sheet 1.

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2009 FRM Full Exam Practice Exam II Questions 1. The current value of the S&P 500 index is 1457, and each S&P futures contract is for delivery of USD 250 times the index. A long‐only equity portfolio with market value of USD 300,100,000 has beta of 1.1. To reduce the portfolio beta to 0.75, how many S&P futures contract should you sell?

a. b. c. d.


2. The risk‐free rate is 5% per year and a corporate bond yields 6% per year. Assuming a recovery rate of 75% on the corporate bond, what is the approximate market implied one‐year probability of default of the corporate bond? a. 1.33% b. 4.00% c. 8.00% d. 1.60%

3. The following table from Fitch Ratings shows the number of rated issuers migrating between two ratings categories during one year. Based on this information, what is the probability that an issue with a rating of A at the beginning of the year will be downgraded by the end of the year?

Year 0 rating

a. b. c. d.

Year 1 rating AAA AAA 45 AA 3 A 2 BBB 0 Default 0

AA 4 30 5 1 0

A 2 4 40 2 0

BBB 0 3 2 30 0

Default 0 2 3 1 0

Total 51 42 52 34 0

13.46% 13.44% 9.62% 3.85%


2009 FRM Practice Exams 4. Beta Bank owns a portfolio of 10 AA‐rated bonds with a total value of 200 million USD. The one‐year probability of default for each issuer is 5% and the recovery rate for each issue equals 40%. The one‐ year expected loss of the portfolio is:

a. b. c. d.


5. Risk Averse Bank (RAB) has made a loan of USD 100 million at 8% per annum. RAB wants to enter into a total return swap under which it will pay the interest on the loan plus the change in the mark‐ to‐market value of the loan and in exchange, RAB will get LIBOR + 30 basis points. Settlement payments are made annually. What is the cash flow for RAB on the first settlement date if the mark‐ to‐market value of the loan falls by 2% and LIBOR is 6%?

a. b. c. d.

Net inflow of USD 0.3 million Net outflow of USD 0.3 million Net inflow of USD 1.7 million Net outflow of USD 1.7 million

6. Determine the percentage of the following portfolio that is investment grade:

Moody's Rating Aa2 A3 Caa1 Baa3 Ba1 D Aaa A1 Baa1 Aa3

Percentage of Portfolio 25% 10% 2% 10% 5% 3% 10% 15% 10% 10%

a. b. c. d.

70% 80% 90% 95%


2009 FRM Practice Exams 7. As part of a currency hedging strategy, a U.S. portfolio manager entered a one‐year forward contract with a bank to deliver EUR 5,000,000 for US dollars at the end of the year. At the beginning of the year, the one‐year forward rate was 0.9216 USD/EUR. Six months into the contract the spot rate is 0.9201 USD/EUR, the U.S. interest rate is 6.5%, and the Euro interest rate is 6.25%. If the current spot rate (0.9201 USD/EUR) were to continue for the next six months, what is the credit risk that the portfolio manager would bear at maturity?

a. b. c. d.

USD 7,042 USD 7,264 USD 7,273 USD 7,500

8. Realizing the benefits of netting of the counterparty exposure may be challenging because of:

a. b. c. d.

Potential downgrade or withdrawal of the counterparty rating Differences in ratings between the rating agencies Trades being booked in different jurisdictions Cross‐product netting

9. In pricing a first‐to‐default credit basket swap, which of the following is true, all else being equal?

a. b. c. d.

The lower the correlation between the assets of the basket, the lower the premium. The lower the correlation between the assets of the basket, the higher the premium. The higher the correlation between the assets of the basket, the higher the premium. The correlation between the assets has no impact in the premium of a first‐to‐default credit basket swap.

10. The spread on a one‐year BBB rated bond relative to the risk‐free treasury of similar maturity is 2%. It is estimated that the contribution to this spread by all non‐credit factors (e.g., liquidity risk, taxes) is 0.8%. Assuming the loss given default rate for the underlying credit is 60%, what is approximately the implied default probability for this bond?

a. b. c. d.

3.33% 5.00% 3.00% 2.00%


2009 FRM Practice Exams 11. You are given the following information about firm A:

Market Value of Asset at time 0 = 1000 Market Value of Asset at time 1 = 1200 Short term Debt = 500 Long term Debt = 300 Annualized Asset Volatility = 10% According to the KMV model, what are the Default Point and the Distance to Default at time 1? Default Distance to Point Default a. 800 3.33 b. 650 7.50 c. 650 4.58 d. 500 5.83

12. Suppose the return on US treasuries is 3% and a risky bond is currently yielding 15%. A trader you supervise claims that he would be able to make an arbitrage trade earning 5% using US treasuries, the risky bond and the credit default swap. Which of the following could be the trader’s strategy and what is the credit default swap premium? Assume there are no transaction costs.

a. b. c. d.

Go long the treasury, short the risky bond, and buy the credit default swap with premium of 6%. Go long the treasury, short the risky bond, and sell the credit default swap with premium of 7%. Short the treasury, invest in the risky bond, and sell the credit default swap with premium of 6%. Short the treasury, invest in the risky bond, and buy the credit default swap with premium of 7%.

13. Bank A makes a 10 million USD five‐year loan and wants to offset the credit exposure to the obligor. A five‐year credit default swap with the loan as the reference asset trades on the market at a swap premium of 50 basis points paid quarterly. In order to hedge its credit exposure Bank A:

a. b. c. d.

Sells the 5 year CDS and receives a quarterly payment of USD 50,000. Buys the 5 year CDS and makes a quarterly payment of USD 12,500. Buys the 5 year CDS and receives a quarterly payment of USD 12,500. Sells the 5 year CDS and makes a quarterly payment of USD 50,000.


2009 FRM Practice Exams 14. A bank is considering buying (i.e. selling protection on) a AAA‐rated super senior tranche [10% ‐ 11%] of a synthetic CDO referencing an investment‐grade portfolio. The pricing of the tranche assumes a fixed recovery of 40% for all names. All else being equal, which one of the following four changes will make the principal invested more risky?

a. b. c. d.

An increase in subordination of 1%, i.e. investing in the [11% ‐ 12%] tranche An increase in the tranche thickness from 1% to 3%, i.e. investing in the [10% ‐ 13%] tranche Using a recovery rate assumption of 50% An increase in default correlation between names in the portfolio

15. Two banks enter into a five‐year first‐to‐default basket credit default swap transaction. The basket contains three uncorrelated credits, W, X and Y, each with a USUSD 25 million notional amount. The protection seller has to settle on the credit that defaults first during the transaction. After that, the protection seller has no obligation and the transaction terminates. Suppose the credits have the following 5‐year cumulative probability of defaults. 5 –Year Probabilities Credit of Default W 9.68% X 8.97% Y 8.02% Which of the following is the probability of at least one default in the basket during the 5 years? a. 8.02% b. 9.68% c. 24.38% d. 26.67%

16. Bank A has exposure to 100 million USD of debt issued by Company R. Bank A enters into a credit default swap transaction with Bank B to hedge its debt exposure to Company R. Bank B would fully compensate Bank A if Company R defaults in exchange for a premium. Assume that the defaults of Bank A, Bank B and Company R are independent and that their default probabilities are 0.3%, 0.5% and 3.6%, respectively. What is the probability that Bank A will suffer a credit loss in its exposure to Company R?

a. b. c. d.

3.6 % 4.1% 0.0180% 0.0108%


2009 FRM Practice Exams 17. A 3‐year credit‐linked note with underlying company Z has a LIBOR + 60 bps semi‐annual coupon. The face value of the CLN is USD 100. LIBOR is 5% for all maturities. Current 3‐year credit default swap (CDS) spread for company Z is 90 bps. The fair value of the CLN is closest to:

a. b. c. d.

USD 99.19 USD 100.00 USD 101.65 USD 111.05

18. A risk manager estimates the daily variance (ht) using a GARCH model on daily returns (rt): ht = α0 + α1r2t‐1 + βht‐1 Assume the model parameter values are α0 = 0.005, α = 0.04, β = 0.94. The long‐run annualized volatility is approximately: a. 25.00% b. 13.54% c. 72.72% d. 7.94%

19. A bank assigns capital to its traders using component‐VaR, which is based on the trading portfolio’s VaR estimated at the 99% confidence level. The market value of the bank’s trading portfolio is HKD 1 billion with a daily volatility of 2%. Of this portfolio, 1% is invested in a trading book with a beta of 0.6 relative to the trading portfolio. The closest estimate of the capital assigned to this trading book is:

a. b. c. d.

HKD 167,760 HKD 279,600 HKD 197,400 HKD 1,977,070


2009 FRM Practice Exams 20. Consider the following potential operational risks. Due to a rogue trader, we estimate that over a 1 year period there is a 10% chance we could lose anywhere between € 0 and € 100MM (equal probability for all points within that range and 0 probability of any losses outside that range). Due to model risk, we estimate that over a 1 year period there is a 20% chance that we will lose € 25MM normally distributed with a standard deviation of € 5MM. Which of the following statements is true?

a. The expected loss from a rogue trader is less than the expected loss from model risk. b. The expected loss from a rogue trader is greater than the expected loss from model risk. c. The maximum unexpected loss from a rogue trader at the 95% confidence level is less than the maximum unexpected loss at the 95% confidence level from model risk. d. The maximum unexpected loss at the 95% level from a rogue trader is greater than the maximum unexpected loss at the 95% level from model risk.

21. Which of the following statements about liquidity risk elasticity (LRE) is incorrect?

a. In calculating the sensitivity of a firm’s net assets to a change in its funding liquidity premium, LRE assumes a parallel shift in funding costs across all maturities. b. LRE is primarily useful for examining marginal changes in funding costs on a net asset/liability position. c. The LRE is a cash flow liquidity risk measure, not a present value liquidity risk measure. d. The LRE is only reliable for small changes in interest costs.

22. The risk of the occurrence of a significant difference between the mark‐to‐model value of a complex and/or illiquid instrument, and the price at which the same instrument is revealed to have traded in the market is referred to as:

a. b. c. d.

Dynamic Risk Liquidity Risk Mark‐to‐Market Risk Model Risk


2009 FRM Practice Exams 23. Which of the following statements regarding economic capital are true?

I. Economic capital is designed to provide a cushion against unexpected losses at a specified confidence level over a set time horizon. II. Since regulatory capital models and economic capital models have different objectives, economic capital models cannot help regulators in setting regulatory capital requirements. III. Firms whose capital exceeds their required regulatory capital are firms that employ their capital inefficiently and their shareholders would benefit if they used some of their capital to repurchase shares or increase dividends. IV. Economic capital can be used to validate a firm’s regulatory capital requirement against its own assessment of the risks it is running.

a. b. c. d.

III and IV only I, II and III only I, III and IV only I and IV only

24. You are an analyst at Bank Alpha. You were given the task to determine whether under Basel II your bank can use the simplified approach to report options exposure instead of the intermediate approach. Which of the following criteria would your bank have to satisfy in order for it to use the simplified approach?

a. The bank purchases and writes options and has significant option trading. b. The bank writes options but its options trading is insignificant in relation to its overall business activities. c. The bank purchases and writes options but its option trading is insignificant. d. The bank solely purchases options and its options trading is insignificant in relation to its overall business activities.

25. Your bank is using the internal models approach to estimate its general market risk charge. The multiplication factor ‘k’, set by the regulator, is 3 and banks are allowed to use the square root rule to scale daily VaR. The previous day’s 1‐day VaR estimate is EUR 3 million, and the average of the daily VaR over the last 60 days is EUR 2 million. Given the above information, what will be the market risk charge for your bank? a. EUR 18.97 million b. EUR 9.49 million c. EUR 6.32 million d. EUR 28.46 million


2009 FRM Practice Exams 26. Under the Basel II Capital Accord, banks that have obtained prior regulatory approval can use the internal models approach to estimate their market risk capital requirement. What approach or methodology is used under the internal models approach to compute capital requirements?

a. b. c. d.

Stress testing and backtesting. Internal rating and vendor models. VaR methodology Expected tail loss, as VaR is not a coherent measure of risk.

27. Bank Z, a medium‐size bank, uses only operational loss data from internal records to model its loss distribution from operational risk events. The bank reviewed its records and, after confirming that they were complete records of its historical losses and that its losses could be approximated by a uniform distribution, it decided against using external loss data to estimate its loss distribution. Based on that decision, which of the following statements is correct?

a. The estimated loss distribution likely overstates Bank Z’s real risk because many incidences in the past were likely “one off”. b. The estimated loss distribution likely accurately represents Bank Z’s real risk because the records are accurate and complete. c. The estimated loss distribution likely understates Bank Z’s real risk because the bank has not experienced a huge loss. d. The estimated loss distribution likely is the best estimate of Bank Z’s real risk because there is no better loss data for the bank than its own.

28. A large international bank has a trading book whose size depends on the opportunities perceived by its traders. The market risk manager estimates the one‐day VaR, at the 95% confidence level, to be USD 50 million. You are asked to evaluate how good of a job the manager is doing in estimating the one‐day VaR. Which of the following would be the most convincing evidence that the manager is doing a poor job, assuming that losses are identically independently distributed?

a. b. c. d.

Over the last 250 days, the mean loss is USD 60 million. Over the last 250 days, there is no exceedence. Over the last 250 days, there are 8 exceedences. Over the last 250 days, the largest loss is USD 500 million.


2009 FRM Practice Exams 29. To handle the financing of a large complex project, your bank is establishing a special purpose entity (SPE) for which your bank will act as trustee. Which of the following could result in liability to your bank through its role as trustee?

a. The SPE was formed to take advantage of a preferable legal jurisdiction. b. The SPE primary purpose was to allow for the deferral of income taxes. c. The SPE controls were unable to determine whether its investors used funds derived from legitimate business opportunities. d. The SPE structure provided for fewer creditors and a reduced likelihood that the project would be forced into bankruptcy.

30. Your bank is implementing the advanced IRB approach of Basel II for credit risk and the AMA approach for operational risk. The bank uses the model approach for market risk. The Chief Risk Officer (CRO) wants to estimate the bank’s total risk by adding up the regulatory capital for market risk, credit risk, and operational risk. The CRO asks you to identify the problems with using this approach to estimate the bank’s total risk. Which of the following statements about this approach is incorrect?

a. b. c. d.

It ignores the interest risk associated with the bank’s loans. It assumes market, credit, and operational risks have zero correlation. It ignores strategic risks. It uses a ten‐day horizon for market risk.

31. The bank you work for has a RAROC model. The RAROC model, computed for each specific activity, measures the ratio of the expected yearly net income to the yearly VaR risk estimate. You are asked to estimate the RAROC of its USD 500 million loan business. The average interest rate is 10%. All loans have the same Probability of Default, PD, of 2% with a Loss Given Default, LGD, of 50%. Operating costs are USD 10 million. The funding cost of the business is USD 30 million. RAROC is estimated using a credit‐VaR for loan businesses. In this case, the appropriate credit‐VaR for the loans is 7.5%. The economic capital is invested and earns 6%. The RAROC is:

a. b. c. d.

32.67% 13.33% 19.33% 46.00%


2009 FRM Practice Exams 32. Which of the following statements regarding Basel II non‐advanced approaches is incorrect?

a. The standardized approach uses data from the last three years of gross income to obtain a bank’s operational risk capital charge. b. The standardized approach makes it advantageous for a bank to book losses early if doing so reduces this year’s gross income sufficiently to make it negative. c. The standardized approach divides the bank into business lines and uses data from the last three years of a business line’s gross income and a beta factor to obtain the regulatory capital for that business line. d. Corporate finance, trading and sales, and payment and settlement are the business lines with the highest regulatory capital requirements.

33. Which one of the following statements does not apply to the Basel II Advanced Measurement Approach (AMA) for operational risk?

a. In contrast to credit risk regulatory capital for corporate loans, banks using the AMA approach may have to set aside capital for both expected and unexpected operational risk losses. b. In contrast to the credit risk IRB approaches, banks using the AMA approach may estimate the correlation between different types of operational risks if their models satisfy regulatory requirements. c. To evaluate exposure to high‐severity operational risk events, banks using the AMA approach may use either scenario analysis of expert opinion, or VaR model estimates based on internal data using extreme value theory. d. Reporting of operational risk exposure to senior management is a necessary condition for a bank’s ability to use the AMA approach.

34. Which of the following approaches for calculating operational risk capital charges leads to a higher capital charge for a given accounting income as risk increases?

a. b. c. d.

The basic indicator approach The standardized approach The advanced measurement approach (AMA) All of the above


2009 FRM Practice Exams 35. Which of the following is not included as an element in calculating operational risk capital under the Advanced Measurement Approach?

a. b. c. d.

External data Key risk indicators Factors reflecting the business environment Scenario analysis

36. Your firm’s fixed‐income portfolio has interest‐only CMOs (IO), callable corporate bonds, inverse floaters, noncallable corporate bonds. Your boss wants to know which of the following securities can lose value as yields decline.

a. b. c. d.

callable corporate only inverse floater only IO and callable corporate bond IO and noncallable corporate bond

37. A bank would like to estimate the number of operational risk events due to problems with tellers (large mistakes, fraud, and so on). The bank decides to model teller operational risk events as a Poisson Process with rate λ (number of events per year). With this model, teller operational risk events are assumed to occur independently of one another and the number of teller operational risk events in a year is Poisson distributed with mean λ. Other properties of a Poisson distribution with mean include:

Variance: Skewness: Excess kurtosis

λ λ‐0.5 1/ λ

Based on historical data regarding the number of teller operational risk events that occurred in previous years, the bank determines that the average number of events has been 5 per year and decides to set λ to 5. Which of the following is true regarding that model? a. The variance of the number of teller operational events in a year is 25. b. The corresponding exponential distribution that describes the time between two teller operational risk events has a mean value of 0.25 years. c. The model is not appropriate if a teller is more likely to have an operational risk event because his friend who is also a teller has been caught stealing. d. The number of teller operational risk events in a year cannot exceed 25. Copyright © 2009 Global Association of Risk Professionals 112 All rights reserved.

2009 FRM Practice Exams 38. All the following are operational risk loss events, except:

a. A loan officer inaccurately enters client financial information into the bank’s proprietary credit risk model b. An individual shows up at a branch presenting a check written by a customer for an amount substantially exceeding the customers low checking account balance. When the bank calls the customer to ask him for the funds, the phone is disconnected and the bank cannot recover the funds. c. During an adverse market movement, the computer network system becomes overwhelmed and only intermittent pricing information is available to the bank’s trading desk, leading to large losses as traders become unable to alter their hedges in response to falling prices d. A bank, acting as a trustee for a loan pool, receives less than the projected funds due to delayed repayment of certain loans.

39. Suppose you are holding 100 Wheelbarrow Company shares with a current price of USD 50. The daily historical mean and volatility of the return of the stock is 1% and 2%, respectively. The bid‐ask spread of the stock varies over time. The daily historical mean and volatility of the spread is 0.5% and 1%, respectively. Calculate the daily liquidity‐adjusted VaR (LVaR) at 99% confidence level: (Both the return and spread of the stock are normally distributed):

a. b. c. d.

USD 325 USD 275 USD 254 USD 229

40. Suppose you are given the following information about the operational risk losses at your bank.

Frequency distribution Probability Frequency 0.5 0 0.3 1 0.2 2

Severity Distribution Probability Severity 0.6 USD 1,000 0.3 USD 10,000 0.1 USD 100,000

What is the estimate of the VaR at the 95% confidence level, assuming that the frequency and severity distributions are independent?

a. b. c. d.

USD 200,000 USD 110,000 USD 100,000 USD 101,000


2009 FRM Practice Exams 41. To control risk taking by traders, your bank links trader compensation with their compliance with imposed VaR limits on their trading book. Why should your bank be careful in tying compensation to the VaR of each trader?

a. It encourages traders to select positions with low estimated risks, which leads to an underestimation of the VaR limits. b. It encourages traders to select positions with low estimated risks, which leads to an overestimation of the VaR limits. c. It encourages traders to select positions with high estimated risks, which leads to an underestimation of the VaR limits. d. It encourages traders to select positions with high estimated risks, which leads to an overestimation of the VaR limits.

42. The surplus of a pension fund is most important for:

a. b. c. d.

A defined contribution fund A defined benefit fund A young workforce A sponsoring company with strong financial status that operates in different industries

43. A mutual fund investing in common stocks has adopted a liquidity risk measure limiting each of its holdings to a maximum of 30% of its thirty day average value traded. If the fund size is USD 3 billion, what is the maximum weight that the fund can hold in a stock with a thirty‐day average value traded of USD 2.4 million?

a. b. c. d.

24.00% 0.08% 0.024% 80.0%

44. An investor is investigating three hedge funds as potential investments. Hedge fund A is an equity market neutral fund, B is a global macro fund with emphasis on equity markets, and C is a convertible arbitrage fund. Which answer correctly specifies the funds with the highest exposure to a worldwide value‐weighted equity index and to a credit‐default swaps index? Highest equity index exposure Highest credit‐default swap index exposure a. A B b. A C c. B A d. B C


2009 FRM Practice Exams 45. Which of the following is not an approach for detecting style drift of hedge funds?

a. b. c. d.

Performance attribution Peer group comparison Cash flow analysis Communication with fund manager

46. All of the following strategies are examples of capital structure arbitrage, except:

a. Long position in the bonds issued by the company, and short position in the company’s stock. b. Short position in the bonds issued by the company, and long position in the company’s stock. c. Long position in a credit default swap on the company and writing put options on the company’s stock. d. Short position in the preferred stock issued by the company and writing call options on the company’s stock.

47. You have been asked to evaluate the performance of two hedge funds: Global Asset Management I and International Momentum II. Both are benchmarked to MSCI EAFE. Which of the two funds had a higher relative Risk Adjusted Performance (RAP) last year, and what is the RAP?

The volatility of EAFE is 17.5% and the annualized performance is 10.6%. The risk‐free rate is 3.5%. Fund Volatility Annualized Performance Global Asset Management I 24.5% 12.5% International Momentum II 27.3% 13.6%

a. b. c. d.

Global Asset Management, 4.85% Global Asset Management, 6.16% International Momentum, 5.42% International Momentum, 1.18%

48. On January 1, 2006, a pension fund has assets of EUR 100 billion and is fully invested in the equity market. It has EUR 85 billion in liabilities. During 2006, the equity market declined by 15% and yields increased by 1.2%. If the modified duration of the liabilities is 12.5, what is the pension fund’s surplus on December 31, 2006?

a. b. c. d.

EUR 15.00 billion EUR 12.93 billion EUR 12.75 billion EUR 12.57 billion


2009 FRM Practice Exams 49. Which of the following statements regarding Extreme Value Theory (EVT) is incorrect?

a. Conventional approaches for estimating VaR that assume that the distribution of returns follow a unique distribution for the entire range of values may fail to properly account for the fat tails of the distribution of returns. b. In contrast to conventional approaches for estimating VaR, EVT only considers the tail behavior of the distribution. c. By smoothing the tail of the distribution, EVT effectively ignores extreme events and losses which can generally be labeled outliers. d. EVT attempts to find the optimal point beyond which all values belong to the tail and then models the distribution of the tail separately.

50. Consider a portfolio with 40% invested in asset X and 60% invested in asset Y. The mean and variance of return on X are 0 and 25 respectively. The mean and variance of return on Y are 1 and 121 respectively. The correlation coefficient between X and Y is 0.3. What is the nearest value for portfolio volatility? e. f. g. h.

9.51% 8.60% 13.38% 7.45%

END OF 2009 FRM FULL EXAM PRACTICE EXAM I



2009 FRM Full Exam Practice Exam II Answer Key 1.

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2009 FRM Full Exam Practice Exam II Answers & Explanations 1. The current value of the S&P 500 index is 1457, and each S&P futures contract is for delivery of USD 250 times the index. A long‐only equity portfolio with market value of USD 300,100,000 has beta of 1.1. To reduce the portfolio beta to 0.75, how many S&P futures contract should you sell?

a. b. c. d.


CORRECT: B No of contracts = [0.75 – 1.1)/ 1]* [300,100,000 / {250 * 1,457}] = ‐288.36 Hence we need to sell 288 contracts INCORRECT: A – ‐617.9135209 = ‐1*(0.75)* (300100000 / (250*1457)) INCORRECT: C – ‐561.74 = ‐1(0.75/1.1)* (300100000 / (250*1457)) INCORRECT: D – ‐906.273164 = ‐1* (1.1)* (300100000 / (250*1457)) Reference: Hull, Options, Futures and Other Derivatives, Chapter 3 and 4; Anthony Saunders, Financial Institutions Management, Chapter 10

2. The risk‐free rate is 5% per year and a corporate bond yields 6% per year. Assuming a recovery rate of 75% on the corporate bond, what is the approximate market implied one‐year probability of default of the corporate bond?

a. b. c. d.

1.33% 4.00% 8.00% 1.60%

CORRECT: B Using the approximation method, the 1‐year probability of default is (6%‐5%)/(1‐0.75) = 4% INCORRECT: A – This is calculated using (6% ‐ 5%) / 0.75 = 1.33% Copyright © 2009 Global Association of Risk Professionals 119 All rights reserved.

2009 FRM Practice Exams INCORRECT: C – This is calculated using 0.06 / 0.75 = 0.08 INCORRECT: D – This is calculated using 0.06 / (0.05 * 0.75) = 1.6 Reference: Saunders, Financial Institutions Management, 5th edition, Chapter 11, p. 313

3. The following table from Fitch Ratings shows the number of rated issuers migrating between two ratings categories during one year. Based on this information, what is the probability that an issue with a rating of A at the beginning of the year will be downgraded by the end of the year?

Year 0 rating

Year 1 rating AAA AAA 45 AA 3 A 2 BBB 0 Default 0

AA 4 30 5 1 0

A 2 4 40 2 0

BBB 0 3 2 30 0

Default 0 2 3 1 0

Total 51 42 52 34 0

a. b. c. d.

13.46% 13.44% 9.62% 3.85%

CORRECT: C Total Number of ‘A’ rated issuances = 52 Probability of ‘A’ rated issues downgraded to BBB (P1) = 2/52 = 0.0385 Probability of ‘A’ rated issues downgraded to Default (P2) = 3/52 = 0.0577 Probability of ‘A’ rated issues to be downgraded in one year = P1 + P2 = 0.0962 = 9.62%. INCORRECT: A – Is the number of upgrades from A (2+5)/52= 13.46% INCORRECT: B – Is the number of downgrades to A 2/51 + 4/42 = 3.92% + 9.52% = 13.44% INCORRECT: D – Is the number of downgrades to BBB 2/52 = 3.85% Reference: Hull, Chapter 22.


2009 FRM Practice Exams 4. Beta Bank owns a portfolio of 10 AA‐rated bonds with a total value of 200 million USD. The one‐year probability of default for each issuer is 5% and the recovery rate for each issue equals 40%. The one‐ year expected loss of the portfolio is:

a. b. c. d.


CORRECT: C Expected Loss equals exposure multiplied by the risk of default and by the recovery rate, or E(L) = Exposure * PD * (1 – Recovery Rate) E(L) = 200 million USD x 5% x 60% = 6 million USD. Correlation amongst issuers does not matter for computing expected losses. INCORRECT: A – Incorrectly set E(L) = 200 * 0.05 * 0.4 = 4.0 INCORRECT: B – Incorrectly set E(L) = 200 * 0.05 * 0.5 = 5.0 INCORRECT: D – Incorrectly set E(L) = 200 * 0.05 * 0.8 = 8.0 Reference: Hull, Chapter 22.

5. Risk Averse Bank (RAB) has made a loan of USD 100 million at 8% per annum. RAB wants to enter into a total return swap under which it will pay the interest on the loan plus the change in the mark‐ to‐market value of the loan and in exchange, RAB will get LIBOR + 30 basis points. Settlement payments are made annually. What is the cash flow for RAB on the first settlement date if the mark‐ to‐market value of the loan falls by 2% and LIBOR is 6%?

a. b. c. d.

Net inflow of USD 0.3 million Net outflow of USD 0.3 million Net inflow of USD 1.7 million Net outflow of USD 1.7 million

CORRECT: A RAB is the TROR payer and pays to the TROR receiver the interest it receives on the loan adjusted for changes in the value of the underlying loan. If the value of the underlying loan increases, RAB as the TROR payer pays to the TROR receiver the interest earned on the loan as well as the price appreciation. Conversely, if the value of the underlying loan decreases, RAB as the TROR payer pays to the TROR receiver the interest earned on the loan less the price depreciation. In return, as the TOR payer, it receives LIBOR plus spread from the TROR receiver. Copyright © 2009 Global Association of Risk Professionals 121 All rights reserved.

2009 FRM Practice Exams At the end of the first year, RAB will pay the interest earned on the loan, 8 million USD. The loan value declined by 2 million USD, thus the return earned by RAB is 6 million USD; this is the payment made by RAB to the TROR receiver. The TROR receives will pay to RAB LIBOR plus 30 basis points i.e. 6.3 % with a 6% LIBOR; 6.3% of 100 million USD = 6.3 million USD. So the payments are + 6.3 million USD – 6.0 million USD or + 0.3 million USD. Reference: Stulz, Risk Management & Derivatives. Chapter 18 – Credit Risks and Credit Derivatives 6. Determine the percentage of the following portfolio that is investment grade:

Moody's Rating Aa2 A3 Caa1 Baa3 Ba1 D Aaa A1 Baa1 Aa3

Percentage of Portfolio 25% 10% 2% 10% 5% 3% 10% 15% 10% 10%

a. b. c. d.

70% 80% 90% 95%

CORRECT: C Non‐investment grade assets are those rated below Baa3. Thus Caa1 with 2%, Ba1 with 5%, and D with 3%, or a total of 10% are non‐investment grade. Thus the investment grade part should equal 90%. Reference: Hull, Chapter 22.


2009 FRM Practice Exams 7. As part of a currency hedging strategy, a U.S. portfolio manager entered a one‐year forward contract with a bank to deliver EUR 5,000,000 for US dollars at the end of the year. At the beginning of the year, the one‐year forward rate was 0.9216 USD/EUR. Six months into the contract the spot rate is 0.9201 USD/EUR, the U.S. interest rate is 6.5%, and the Euro interest rate is 6.25%. If the current spot rate (0.9201 USD/EUR) were to continue for the next six months, what is the credit risk that the portfolio manager would bear at maturity?

a. b. c. d.

USD 7,042 USD 7,264 USD 7,273 USD 7,500

CORRECT: D Value to the manager at the settlement date = contract cash flow – spot market cash flow [both at contract maturity]. Contract cash flow = EUR 5,000,000 x 0.9216 USD/EUR = 4,608,000 USD Spot market cash flow = The amount the portfolio manager could receive if the spot rate after 6 months remained in effect until the settlement date = EUR 5,000,000 x 0.9201 USD /EUR = 4,600,500 USD Value to the manager = USD 4,608,000 ‐ USD 4,600,500 = USD 7,500 As the value is positive, the bank owes this amount to the portfolio manager. Since the question asks the amount of credit risk at maturity, we need not discount this back to six months. INCORRECT: A – This answer is arrived at by discounting the correct answer for twelve months at the USD interest rate, which is not a requirement of the question. INCORRECT: B – This answer is arrived at by discounting the correct answer for six months at the USD interest rate, which is not a requirement of the question. INCORRECT: C – The correct answer discounted at the Euro rate for six month. Reference: Stulz, Risk Management and Derivatives, Chapter 18

8. Realizing the benefits of netting of the counterparty exposure may be challenging because of:

a. b. c. d.

Potential downgrade or withdrawal of the counterparty rating Differences in ratings between the rating agencies Trades being booked in different jurisdictions Cross‐product netting

CORRECT: C Copyright © 2009 Global Association of Risk Professionals 123 All rights reserved.

2009 FRM Practice Exams This presents a legal challenge to enforcing netting agreement INCORRECT: A – These credit events do not make netting more challenging INCORRECT: B – Ratings are not required to establish netting INCORRECT: D – On contrary, cross‐product netting allows to realize more benefits in reducing exposure to counterparty Reference: Culp

9. In pricing a first‐to‐default credit basket swap, which of the following is true, all else being equal?

a. b. c. d.

The lower the correlation between the assets of the basket, the lower the premium. The lower the correlation between the assets of the basket, the higher the premium. The higher the correlation between the assets of the basket, the higher the premium. The correlation between the assets has no impact in the premium of a first‐to‐default credit basket swap.

CORRECT: B The lower the correlation between the assets of the basket, the higher the premium. In the case of a first‐to‐default swap, a credit event occurs the first time any of the entities defaults. This swap provides default protection against losses related to this first default, but not to any subsequent defaults. Thus, the question is whether the level of correlation between assets of the basket increases or decreases the likelihood of the triggering event. If the correlation between the assets in a credit basket swap is lower, the basket would be exposed to greater default risk. For example, the basket contains assets from different sectors, then the basket would be exposed to the default risk of each and every sector in the basket. If the basket only contains assets from one sector, then the correlation is higher, and the default risk is lower. Reference: Hull, 22, 23

10. The spread on a one‐year BBB rated bond relative to the risk‐free treasury of similar maturity is 2%. It is estimated that the contribution to this spread by all non‐credit factors (e.g., liquidity risk, taxes) is 0.8%. Assuming the loss given default rate for the underlying credit is 60%, what is approximately the implied default probability for this bond?



a. b. c. d.

3.33% 5.00% 3.00% 2.00%

CORRECT: D The probability of default equals the credit risk spread divided by the loss given default. PD = spread / LGD Here, the spread due to credit risk equals 2.0% ‐ 0.8% or 1.2% and the loss given default is 60%. The probability of default is then 2%. INCORRECT: A – Incorrectly sets PD = 2.0/0.6 = 3.33. INCORRECT: B – Incorrectly sets PD = 2.0/0.4 = 5.0. INCORRECT: D – Incorrectly sets PD = 1.2/0.4 = 3.00. Reference: de Servigny and Renault, Measuring and Managing Credit Risk, (New York: McGraw‐Hill, 2004) chapter 3, 4

11. You are given the following information about firm A:

• • • • •

Market Value of Asset at time 0 = 1000 Market Value of Asset at time 1 = 1200 Short term Debt = 500 Long term Debt = 300 Annualized Asset Volatility = 10%

According to the KMV model, what are the Default Point and the Distance to Default at time 1? Default Distance to Point Default a. 800 3.33 b. 650 7.50 c. 650 4.58 d. 500 5.83 CORRECT: C According to KMV, Default Point = STD + ½ LTD = 500 +1/2(300) = 650. Distance to default


2009 FRM Practice Exams = (Market value of asset at time 1 – Default Point)/Annualized Asset volatility at time 1 = (1200‐650)/ (1200*0.1) = 4.58 INCORRECT: A – Incorrectly sets Default Point as STD + LTD instead (500 + 300). INCORRECT: B – Incorrectly sets Default Point value as LTD, distance to default = (1200‐STD)/120 = 7.50 INCORRECT: D – Incorrectly sets Default Point as STD only, distance to default = (1200‐500)/120 = 5.83 Reference: de Servigny and Renault, Measuring and Managing Credit Risk, Chapter 4.

12. Suppose the return on US treasuries is 3% and a risky bond is currently yielding 15%. A trader you supervise claims that he would be able to make an arbitrage trade earning 5% using US treasuries, the risky bond and the credit default swap. Which of the following could be the trader’s strategy and what is the credit default swap premium? Assume there are no transaction costs.

a. b. c. d.

Go long the treasury, short the risky bond, and buy the credit default swap with premium of 6%. Go long the treasury, short the risky bond, and sell the credit default swap with premium of 7%. Short the treasury, invest in the risky bond, and sell the credit default swap with premium of 6%. Short the treasury, invest in the risky bond, and buy the credit default swap with premium of 7%.

CORRECT: D To prevent arbitrage, the return on the risk‐free government bond must equal the return on the risky bond less the premium on the default swap, or the risk corporate bond return must equal the return on the risk free government bond and the credit default swap premium. Risk‐free bond return = risky bond return – default swap premium. In this case, adjustment for the arbitrage profit changes the relationship to Arbitrage profit = risky bond return – default swap premium – risk‐free government bond return Risk‐free bond return = 3% Risky bond return = 15% Default swap premium = X% Arbitrage Profit = 5%. The short position in treasury costs 3%, return from the risky corporate bond is 15%, the default swap premium is 7%, ends up in an arbitrage profit of 5% (‐3%+15%‐7%). This is a valid arbitrage strategy. Reference: Hull, Chapter 23. Copyright © 2009 Global Association of Risk Professionals 126 All rights reserved.

2009 FRM Practice Exams 13. Bank A makes a 10 million USD five‐year loan and wants to offset the credit exposure to the obligor. A five‐year credit default swap with the loan as the reference asset trades on the market at a swap premium of 50 basis points paid quarterly. In order to hedge its credit exposure Bank A:

a. b. c. d.

Sells the 5 year CDS and receives a quarterly payment of USD 50,000. Buys the 5 year CDS and makes a quarterly payment of USD 12,500. Buys the 5 year CDS and receives a quarterly payment of USD 12,500. Sells the 5 year CDS and makes a quarterly payment of USD 50,000.

CORRECT: B To offset the credit risk in the loan, the bank needs to buy credit protection using the CDS with same maturity. Since the bank is buying protection, it pays the insurance premium quarterly. The annual insurance premium – swap payment ‐ on this transaction would be notional amount x swap premium or 10,000,000 * 0.005 = 50,000. The quarterly premium is USD 12,500. INCORRECT: A – Bank A needs to hedge its credit exposure and therefore has to buy the 5 Year CDS (Bank A buys credit protection) INCORRECT: C – The quarterly payment is USD 12,500 = notional x swap premium x 90/360 = 10 USD million x 0.005 x 90/360 INCORRECT: D – Bank A needs to hedge its credit exposure and therefore has to buy the 5 Year CDS (Bank A buys credit protection) Reference: Hull Chapter 23.

14. A bank is considering buying (i.e. selling protection on) a AAA‐rated super senior tranche [10% ‐ 11%] of a synthetic CDO referencing an investment‐grade portfolio. The pricing of the tranche assumes a fixed recovery of 40% for all names. All else being equal, which one of the following four changes will make the principal invested more risky?

a. b. c. d.

An increase in subordination of 1%, i.e. investing in the [11% ‐ 12%] tranche An increase in the tranche thickness from 1% to 3%, i.e. investing in the [10% ‐ 13%] tranche Using a recovery rate assumption of 50% An increase in default correlation between names in the portfolio

CORRECT: D The probability of a senior tranche being hit decreases if the subordination increases or if the fixed recovery rate increases. The expected loss of a tranche decreases if the tranche thickness increases. These all lead to the tranche being less risky. As portfolio correlation increases, the loss distribution


2009 FRM Practice Exams exhibits fatter tail and the probability of a senior tranche being hit increases, leading to a more risky tranche. Reference: Culp, Chapters 16, 17, 18.

15. Two banks enter into a five‐year first‐to‐default basket credit default swap transaction. The basket contains three uncorrelated credits, W, X and Y, each with a USUSD 25 million notional amount. The protection seller has to settle on the credit that defaults first during the transaction. After that, the protection seller has no obligation and the transaction terminates. Suppose the credits have the following 5‐year cumulative probability of defaults. 5‐Year Probabilities of Credit Default W 9.68% X 8.97% Y 8.02% Which of the following is the probability of at least one default in the basket during the 5 years? a. 8.02% b. 9.68% c. 24.38% d. 26.67%

CORRECT: C. During the 5 years, the probability of no default is (1 – 9.68%) x (1 – 8.97%) x (1 – 8.02%) = 75.62%. The probability of at least one default is (1 – 75.62%) = 24.38%. The answer is neither the maximum (B) nor minimum (A) nor the sum (D) of the default probabilities. Reference: Hull Chapter 23.

16. Bank A has exposure to 100 million USD of debt issued by Company R. Bank A enters into a credit default swap transaction with Bank B to hedge its debt exposure to Company R. Bank B would fully compensate Bank A if Company R defaults in exchange for a premium. Assume that the defaults of Bank A, Bank B and Company R are independent and that their default probabilities are 0.3%, 0.5% and 3.6%, respectively. What is the probability that Bank A will suffer a credit loss in its exposure to Company R?



a. b. c. d.

3.6 % 4.1% 0.0180% 0.0108%

CORRECT: C Bank A will only suffer a loss when Company R and Bank B jointly default. With zero correlation, this probability is equal to 0.5% x 3.6% = 0.018%. 3.6% is the maximum, 4.1% is the sum, and 0.0108% = 0.3% * 3.6% Reference: Hull, Chapters 22, 23

17. A 3‐year credit‐linked note with underlying company Z has a LIBOR + 60 bps semi‐annual coupon. The face value of the CLN is USD 100. LIBOR is 5% for all maturities. Current 3‐year credit default swap (CDS) spread for company Z is 90 bps. The fair value of the CLN is closest to:

a. b. c. d.

USD 99.19 USD 100.00 USD 101.65 USD 111.05

CORRECT: A This question requires no calculation. Because the discount factor (0.5 * (0.05 + 0.009) = 0.0295) is greater than the coupon rate, the present value has to be less than the face value – the correct answer is A. This question can be worked out by using calculator, where N=6, I/Y=2.95, PMT=2.8, FV=100 ‐> PV=99.19. INCORRECT: B – Is the face value. INCORRECT: C – Is where only Libor rate is used for discounting. N=6, I/Y=2.5, PMT=2.8, FV=100 ‐> PV=101.65. INCORRECT: D – Is where only CDS spread is used for discounting. N=6, I/Y=0.9, PMT=2.8, FV=100 ‐> PV=101.65. Reference: Hull, Culp


2009 FRM Practice Exams 18. A risk manager estimates the daily variance (ht) using a GARCH model on daily returns (rt): ht = α0 + α1r2t‐1 + βht‐1 Assume the model parameter values are α0 = 0.005, α = 0.04, β = 0.94. The long‐run annualized volatility is approximately: a. 25.00% b. 13.54% c. 72.72% d. 7.94% CORRECT: D The long‐run variance is 0.005/(1‐0.04‐0.94) =0.005/0.02 = 0.25. The daily vol is thus the square root, or 0.5% and annual vol 7.935%. INCORRECT: A – The daily variance is indeed 0.25%, and the daily volatility 0.5% but this needs to be annualized. INCORRECT: B – Miscalculates variance as sqrt(0.04/(1 – 0.94 – 0.005)) * 15.87 = 13.54% INCORRECT: C – Miscalculates variance as 0.04/(1 – 0.94 – 0.005) = 72.72% Reference: Hull

19. A bank assigns capital to its traders using component‐VaR, which is based on the trading portfolio’s VaR estimated at the 99% confidence level. The market value of the bank’s trading portfolio is HKD 1 billion with a daily volatility of 2%. Of this portfolio, 1% is invested in a trading book with a beta of 0.6 relative to the trading portfolio. The closest estimate of the capital assigned to this trading book is:

a. b. c. d.

HKD 167,760 HKD 279,600 HKD 197,400 HKD 1,977,070

CORRECT: B This question assesses candidates’ abilities to carry out CVaR calculations correctly.


2009 FRM Practice Exams Using, e.g., equation 7.29 from Jorion 2nd edition, we have CVaR = VaRwi βi , where wi is the share of the ith asset in the portfolio and βi is its beta. Using the information given, VaR = 2.33 * 0.02 * 1,000,000,000 = 46,600,000, so CVaR = 46,600,000 * 0.01 * 0.6 = 279,600. INCORRECT: A – Incorrectly multiplies the volatility by beta in the VaR calculation. INCORRECT: C – Incorrectly uses 1.645 in the VaR calculation. INCORRECT: D – Incorrectly uses the square root of the volatility in the VaR calculation. Reference: Jorion, VaR, 2nd edition, p. 160, chapter 7 (Portfolio risk: analytical methods).

20. Consider the following potential operational risks. Due to a rogue trader, we estimate that over a 1 year period there is a 10% chance we could lose anywhere between € 0 and € 100MM (equal probability for all points within that range and 0 probability of any losses outside that range). Due to model risk, we estimate that over a 1 year period there is a 20% chance that we will lose € 25MM normally distributed with a standard deviation of € 5MM. Which of the following statements is true?

a. The expected loss from a rogue trader is less than the expected loss from model risk. b. The expected loss from a rogue trader is greater than the expected loss from model risk. c. The maximum unexpected loss from a rogue trader at the 95% confidence level is less than the maximum unexpected loss at the 95% confidence level from model risk. d. The maximum unexpected loss at the 95% level from a rogue trader is greater than the maximum unexpected loss at the 95% level from model risk.

CORRECT: D This question tests understanding of expected vs. unexpected loss. The rogue trader has an expected loss (severity multiplied by probability) of €5MM while the model risk has an expected loss of €5MM. Therefore both A and B are incorrect. We therefore must examine unexpected losses. The rogue trader has a much wider distribution (Uniform) and a lower probability of occurrence than the model risk (normal distribution). Therefore the rogue trader has a greater risk of unexpected losses. Unexpected loss for rogue trader at 95% confidence level: 50MM – 5MM = 45MM The loss from model risk at the 95th percentile corresponds to the 75th percentile for the normal distribution with mean 25 and standard deviation 5, so unexpected loss for model risk at 95% confidence level: < (25MM + 1.645*5MM) – 5MM < 30MM Reference: Dowd, Chapter 16.


2009 FRM Practice Exams 21. Which of the following statements about liquidity risk elasticity (LRE) is incorrect?

a. In calculating the sensitivity of a firm’s net assets to a change in its funding liquidity premium, LRE assumes a parallel shift in funding costs across all maturities. b. LRE is primarily useful for examining marginal changes in funding costs on a net asset/liability position. c. The LRE is a cash flow liquidity risk measure, not a present value liquidity risk measure. d. The LRE is only reliable for small changes in interest costs.

CORRECT: C LRE measures liquidity risk for present values. It is not useful if a firm is concerned about per‐period cash flows. Reference: Dowd, Chapter 14 ; Saunders, Chapter 17.

22. The risk of the occurrence of a significant difference between the mark‐to‐model value of a complex and/or illiquid instrument, and the price at which the same instrument is revealed to have traded in the market is referred to as:

a. b. c. d.

Dynamic Risk Liquidity Risk Mark‐to‐Market Risk Model Risk

CORRECT: D This is how model risk is defined in the reading. INCORRECT: A – Undefined term INCORRECT: B – The risk of not being able to sell an asset quickly INCORRECT: C – Undefined term Reference: Dowd, Chapter 16.


2009 FRM Practice Exams 23. Which of the following statements regarding economic capital are true?

I. Economic capital is designed to provide a cushion against unexpected losses at a specified confidence level over a set time horizon. II. Since regulatory capital models and economic capital models have different objectives, economic capital models cannot help regulators in setting regulatory capital requirements. III. Firms whose capital exceeds their required regulatory capital are firms that employ their capital inefficiently and their shareholders would benefit if they used some of their capital to repurchase shares or increase dividends. IV. Economic capital can be used to validate a firm’s regulatory capital requirement against its own assessment of the risks it is running.

a. b. c. d.

III and IV only I, II and III only I, III and IV only I and IV only

CORRECT: C Economic capital is defined as a capital buffer that is required to absorb the impact of unexpected losses during a time horizon at a portfolio manager’s level of confidence. A firm may employ the economic capital as its own internal assessment to check the efficiency of overall amount of capital it holds. Therefore, I and III are correct. Recent regulatory changes such as the internal ratings based approach in Basel II provides encouragement for firms to develop their internal risk management models. Therefore, economic capital will become a key regulatory tool with a converging trend between economic capital and regulatory capital models. II is incorrect and IV is correct Reference: Crouhy, Galai, and Mark, Risk Management. Chapter 14 – Capital Allocation and Performance Measurement


2009 FRM Practice Exams 24. You are an analyst at Bank Alpha. You were given the task to determine whether under Basel II your bank can use the simplified approach to report options exposure instead of the intermediate approach. Which of the following criteria would your bank have to satisfy in order for it to use the simplified approach?

a. The bank purchases and writes options and has significant option trading. b. The bank writes options but its options trading is insignificant in relation to its overall business activities. c. The bank purchases and writes options but its option trading is insignificant. d. The bank solely purchases options and its options trading is insignificant in relation to its overall business activities.

CORRECT: D The bank only purchases options and its options trading is not significant. INCORRECT: A – If a bank writes options, the intermediate approach must be used. INCORRECT: B – If bank writes options, the intermediate approach must be used. INCORRECT: C – If option trading is significant, the intermediate approach must be used. Reference: Basel II: International Convergence of Capital Measurement and Capital Standards: A Revised Framework ‐ Comprehensive Version

25. Your bank is using the internal models approach to estimate its general market risk charge. The multiplication factor ‘k’, set by the regulator, is 3 and banks are allowed to use the square root rule to scale daily VaR. The previous day’s 1‐day VaR estimate is EUR 3 million, and the average of the daily VaR over the last 60 days is EUR 2 million. Given the above information, what will be the market risk charge for your bank? a. EUR 18.97 million b. EUR 9.49 million c. EUR 6.32 million d. EUR 28.46 million

CORRECT: A The required market risk charge would be square root of 10 times the maximum of previous day VaR and the average daily 60 days VaR, times k. √10 * Max(3, 3*2) = 18.97 INCORRECT: B – √10 * 3 = 9.49 INCORRECT: C – √10 * 2 = 6.32 Copyright © 2009 Global Association of Risk Professionals 134 All rights reserved.

2009 FRM Practice Exams INCORRECT: D – √10 * Max(3*3, 3*2) = 28.46 Reference: Basel II: International Convergence of Capital Measurement and Capital Standards: A Revised Framework – Comprehensive Version” (Basel Committee on Banking Supervision Publication, June 2006). 26. Under the Basel II Capital Accord, banks that have obtained prior regulatory approval can use the internal models approach to estimate their market risk capital requirement. What approach or methodology is used under the internal models approach to compute capital requirements?

a. b. c. d.

Stress testing and backtesting. Internal rating and vendor models. VaR methodology Expected tail loss, as VaR is not a coherent measure of risk.

CORRECT: C Basel II prescribes a 10‐day VaR at 99% confidence level for capital charge computation. However, banks are free to choose the method to compute VaR. INCORRECT: A – Stress testing and backtesting are used to supplement and validate the results of capital computation, and are not the method per se to be adopted for capital computation INCORRECT: B – Internal rating approach and vendor models are something associated with credit risk rather than market risk INCORRECT: D – Regulation prescribes VaR and not ETL or CVaR, even when it is known that VaR is not sub‐additive at all times. Reference: Basel II: International Convergence of Capital Measurement and Capital Standards: A Revised Framework – Comprehensive Version” (Basel Committee on Banking Supervision Publication, June 2006).

27. Bank Z, a medium‐size bank, uses only operational loss data from internal records to model its loss distribution from operational risk events. The bank reviewed its records and, after confirming that they were complete records of its historical losses and that its losses could be approximated by a uniform distribution, it decided against using external loss data to estimate its loss distribution. Based on that decision, which of the following statements is correct?


2009 FRM Practice Exams a. The estimated loss distribution likely overstates Bank Z’s real risk because many incidences in the past were likely “one off”. b. The estimated loss distribution likely accurately represents Bank Z’s real risk because the records are accurate and complete. c. The estimated loss distribution likely understates Bank Z’s real risk because the bank has not experienced a huge loss. d. The estimated loss distribution likely is the best estimate of Bank Z’s real risk because there is no better loss data for the bank than its own.

CORRECT: C One of the biggest issues of Operational Risk modeling is that there is likely not enough internal data to represent large enough loss as such loss is extremely rare. Using external loss database can help to mitigate this problem. Reference: Allen et al., Chapter 5.

28. A large international bank has a trading book whose size depends on the opportunities perceived by its traders. The market risk manager estimates the one‐day VaR, at the 95% confidence level, to be USD 50 million. You are asked to evaluate how good of a job the manager is doing in estimating the one‐day VaR. Which of the following would be the most convincing evidence that the manager is doing a poor job, assuming that losses are identically independently distributed?

a. b. c. d.

Over the last 250 days, the mean loss is USD 60 million. Over the last 250 days, there is no exceedence. Over the last 250 days, there are 8 exceedences. Over the last 250 days, the largest loss is USD 500 million.

CORRECT: B Using the Kupiec test, there is an extremely low probability of no exceedence over 250 days. INCORRECT: A – VaR tells us nothing about the mean loss over time since the size of the trading book varies over time as opportunities change. INCORRECT: C – With the same test, the probability of 8 exceedences is higher than the probability of no exceedence. INCORRECT: D – The VaR does not tell us the distribution of the maximum loss. Reference: Jorion, Chapter 6.


2009 FRM Practice Exams 29. To handle the financing of a large complex project, your bank is establishing a special purpose entity (SPE) for which your bank will act as trustee. Which of the following could result in liability to your bank through its role as trustee?

a. The SPE was formed to take advantage of a preferable legal jurisdiction. b. The SPE primary purpose was to allow for the deferral of income taxes. c. The SPE controls were unable to determine whether its investors used funds derived from legitimate business opportunities. d. The SPE structure provided for fewer creditors and a reduced likelihood that the project would be forced into bankruptcy.

CORRECT: C Reference: Culp

30. Your bank is implementing the advanced IRB approach of Basel II for credit risk and the AMA approach for operational risk. The bank uses the model approach for market risk. The Chief Risk Officer (CRO) wants to estimate the bank’s total risk by adding up the regulatory capital for market risk, credit risk, and operational risk. The CRO asks you to identify the problems with using this approach to estimate the bank’s total risk. Which of the following statements about this approach is incorrect?

a. b. c. d.

It ignores the interest risk associated with the bank’s loans. It assumes market, credit, and operational risks have zero correlation. It ignores strategic risks. It uses a ten‐day horizon for market risk.

CORRECT: B The approach assumes a correlation of one. There is no capital charge for structural interest rate risk under Basel II. Operational risk includes legal risk. It uses a ten‐day horizon for market risk. Reference: Basel II: International Convergence of Capital Measurement and Capital Standards: A Revised Framework – Comprehensive Version” (Basel Committee on Banking Supervision Publication, June 2006).

31. The bank you work for has a RAROC model. The RAROC model, computed for each specific activity, measures the ratio of the expected yearly net income to the yearly VaR risk estimate. You are asked to estimate the RAROC of its USD 500 million loan business. The average interest rate is 10%. All Copyright © 2009 Global Association of Risk Professionals 137 All rights reserved.

2009 FRM Practice Exams loans have the same Probability of Default, PD, of 2% with a Loss Given Default, LGD, of 50%. Operating costs are USD 10 million. The funding cost of the business is USD 30 million. RAROC is estimated using a credit‐VaR for loan businesses. In this case, the appropriate credit‐VaR for the loans is 7.5%. The economic capital is invested and earns 6%. The RAROC is:

a. b. c. d.

32.67% 13.33% 19.33% 46.00%

CORRECT: C (0.1*500 – 5 – 10 – 30 +0.06*37.5)/37.5 = 19.33% INCORRECT: A – (0.1*500 – 10 – 30 + 0.06*37.5)/37.5 = 32.67% INCORRECT: B – (0.1*500 – 5 – 10 – 30)/37.5 = 13.33% INCORRECT: D – (0.1*500 – 5 – 30 + 0.06*37.5)/37.5 = 46.00% Reference: Crouhy, Galai, Mark, Chapter 14.

32. Which of the following statements regarding Basel II non‐advanced approaches is incorrect?

a. The standardized approach uses data from the last three years of gross income to obtain a bank’s operational risk capital charge. b. The standardized approach makes it advantageous for a bank to book losses early if doing so reduces this year’s gross income sufficiently to make it negative. c. The standardized approach divides the bank into business lines and uses data from the last three years of a business line’s gross income and a beta factor to obtain the regulatory capital for that business line. d. Corporate finance, trading and sales, and payment and settlement are the business lines with the highest regulatory capital requirements.

CORRECT: B Only positive gross income is included in the formula. Everything else is correct. Reference: Basel II: International Convergence of Capital Measurement and Capital Standards: A Revised Framework – Comprehensive Version” (Basel Committee on Banking Supervision Publication, June 2006).



33. Which one of the following statements does not apply to the Basel II Advanced Measurement Approach (AMA) for operational risk?

a. In contrast to credit risk regulatory capital for corporate loans, banks using the AMA approach may have to set aside capital for both expected and unexpected operational risk losses. b. In contrast to the credit risk IRB approaches, banks using the AMA approach may estimate the correlation between different types of operational risks if their models satisfy regulatory requirements. c. To evaluate exposure to high‐severity operational risk events, banks using the AMA approach may use either scenario analysis of expert opinion, or VaR model estimates based on internal data using extreme value theory. d. Reporting of operational risk exposure to senior management is a necessary condition for a bank’s ability to use the AMA approach.

CORRECT: C Everything is correct using the 2006 Basel II document except for c. The Basel II document requires the use of scenarios obtained with expert advice for high severity events when the bank uses only internal data. Reference: Basel II: International Convergence of Capital Measurement and Capital Standards: A Revised Framework – Comprehensive Version” (Basel Committee on Banking Supervision Publication, June 2006).

34. Which of the following approaches for calculating operational risk capital charges leads to a higher capital charge for a given accounting income as risk increases?

a. b. c. d.

The basic indicator approach The standardized approach The advanced measurement approach (AMA) All of the above

CORRECT: C Under AMA, the capital charge is based on the banks internal measurement system. Under the 2 other approaches the capital charge is based upon the Gross Income, which has nothing to do with risk exposure. Reference: Basel II: International Convergence of Capital Measurement and Capital Standards: A Revised Framework – Comprehensive Version” (Basel Committee on Banking Supervision Publication, June 2006). Copyright © 2009 Global Association of Risk Professionals 139 All rights reserved.

2009 FRM Practice Exams 35. Which of the following is not included as an element in calculating operational risk capital under the Advanced Measurement Approach?

a. b. c. d.

External data Key risk indicators Factors reflecting the business environment Scenario analysis

CORRECT: B KRI is not included in the Basel document at all Reference: Basel II: International Convergence of Capital Measurement and Capital Standards: A Revised Framework – Comprehensive Version” (Basel Committee on Banking Supervision Publication, June 2006).

36. Your firm’s fixed‐income portfolio has interest‐only CMOs (IO), callable corporate bonds, inverse floaters, noncallable corporate bonds. Your boss wants to know which of the following securities can lose value as yields decline.

a. b. c. d.

callable corporate only inverse floater only IO and callable corporate bond IO and noncallable corporate bond

CORRECT: C The IO decreases in value because a decline in rates implies an increase in mortgage prepayments, which decreases the notional principal upon which the IO pays its interest INCORRECT: A – While the price of a callable corporate may decline as the call goes in the money, the IO also decreases in value INCORRECT: B – The IO decreases in value, but so can the callable corporate INCORRECT: D – The noncallable corporate bond increases in value as yields decline Reference: Tuckman, Chapter 21


2009 FRM Practice Exams 37. A bank would like to estimate the number of operational risk events due to problems with tellers (large mistakes, fraud, and so on). The bank decides to model teller operational risk events as a Poisson Process with rate λ (number of events per year). With this model, teller operational risk events are assumed to occur independently of one another and the number of teller operational risk events in a year is Poisson distributed with mean λ. Other properties of a Poisson distribution with mean λ include: Variance: Skewness: Excess kurtosis

λ λ-0.5 1/ λ

Based on historical data regarding the number of teller operational risk events that occurred in previous years, the bank determines that the average number of events has been 5 per year and decides to set λ to 5. Which of the following is true regarding that model?

a. The variance of the number of teller operational events in a year is 25. b. The corresponding exponential distribution that describes the time between two teller operational risk events has a mean value of 0.25 years. c. The model is not appropriate if a teller is more likely to have an operational risk event because his friend who is also a teller has been caught stealing. d. The number of teller operational risk events in a year cannot exceed 25.

CORRECT: C The Poisson model implies that teller events are independent of one another. INCORRECT: A – The teller operational risk event has a variance of 5. INCORRECT: B – The exponential probability distribution describes the distribution of the period between two teller events. If the Poisson Process has a rate of 5, then the associated exponential distribution model has a mean value of M= 1/5 = 0.2 years. INCORRECT: D – There is no bound on the number of events that can occur. Reference: Rachev, Menn, Fabozzi; Chapter 3.


2009 FRM Practice Exams 38. All the following are operational risk loss events, except:

a. A loan officer inaccurately enters client financial information into the bank’s proprietary credit risk model b. An individual shows up at a branch presenting a check written by a customer for an amount substantially exceeding the customers low checking account balance. When the bank calls the customer to ask him for the funds, the phone is disconnected and the bank cannot recover the funds. c. During an adverse market movement, the computer network system becomes overwhelmed and only intermittent pricing information is available to the bank’s trading desk, leading to large losses as traders become unable to alter their hedges in response to falling prices d. A bank, acting as a trustee for a loan pool, receives less than the projected funds due to delayed repayment of certain loans.

CORRECT: D A broad interpretation of the FDIC and Basel II rules defining operational risks would consider all, but alternative d, as operational risk events and operational risk losses. A loan officer entering inaccurate client financial information into the bank’s proprietary credit risk model is a process risk and any losses attributable to this employee error should be considered as an operational risk event. Customers writing checks exceeding the balance of a checking account and depositing the funds in a savings account is an example of check kiting. If during times of adverse market movement, the computer network becomes overwhelmed and only delayed pricing information reaches the bank’s trading desk and trades are based on the available information, the bank is subject to business disruption and system failures. When a bank, acting as a trustee for a loan pool, receives less than the projected funds due to delayed repayment of certain loans in the pool is not an operational risk loss event. Reference: Basel II: International Convergence of Capital Measurement and Capital Standards: A Revised Framework – Comprehensive Version” (Basel Committee on Banking Supervision Publication, June 2006).

39. Suppose you are holding 100 Wheelbarrow Company shares with a current price of USD 50. The daily historical mean and volatility of the return of the stock is 1% and 2%, respectively. The bid‐ask spread of the stock varies over time. The daily historical mean and volatility of the spread is 0.5% and 1%, respectively. Calculate the daily liquidity‐adjusted VaR (LVaR) at 99% confidence level: (Both the return and spread of the stock are normally distributed):

a. b. c. d.

USD 325 USD 275 USD 254 USD 229



CORRECT: C VaR=USD 50*100*(2.33*0.02‐0.01) =USD 183 Liquidity adjusted=USD 50*100*0.5*(2.33*0.01+0.005) =USD 71 LVaR=VaR+ Liquidity adjusted =USD 254 Reference: Dowd: Chapter 14.

40. Suppose you are given the following information about the operational risk losses at your bank.

Frequency distribution Probability Frequency 0.5 0 0.3 1 0.2 2

Severity Distribution Probability Severity 0.6 USD 1,000 0.3 USD 10,000 0.1 USD 100,000

What is the estimate of the VaR at the 95% confidence level, assuming that the frequency and severity distributions are independent?

a. b. c. d.

USD 200,000 USD 110,000 USD 100,000 USD 101,000

CORRECT: C The loss distribution is: Total loss 0 1,000 2,000 10,000 11,000 20,000 100,000 101,000 110,000 200,000

Probability 0.5 0.18 0.072 0.09 0.072 0.018 0.03 0.024 0.012 0.002


2009 FRM Practice Exams The 95% VaR is 100,000. The other answers are from this distribution, but not corresponding to the 95% VaR.

Reference: Allen et al., chapter 5.

41. To control risk taking by traders, your bank links trader compensation with their compliance with imposed VaR limits on their trading book. Why should your bank be careful in tying compensation to the VaR of each trader?

a. It encourages traders to select positions with low estimated risks, which leads to an underestimation of the VaR limits. b. It encourages traders to select positions with low estimated risks, which leads to an overestimation of the VaR limits. c. It encourages traders to select positions with high estimated risks, which leads to an underestimation of the VaR limits. d. It encourages traders to select positions with high estimated risks, which leads to an overestimation of the VaR limits.

CORRECT: A If a trader uses an estimated variance‐covariance matrix to select trading positions, the trader would opt for a position with low estimated risks. Selecting positions with low estimated risks biases the variance‐covariance matrix, and leads to an underestimation of the VaR position limits. Reference: Dowd, Measuring Market Risk.

42. The surplus of a pension fund is most important for:

a. A defined contribution fund b. A defined benefit fund c. A young workforce d. A sponsoring company with strong financial status that operates in different industries CORRECT: B. Under a defined contribution, beneficiaries are entitled for a percentage regardless of its value. Under a defined benefit plan, they are entitled for a value. The older the work force is and the lower the surplus value, the higher the risk of the pension plan and the more important is the surplus. Reference: Jorion, Chapter 17. Copyright © 2009 Global Association of Risk Professionals 144 All rights reserved.

2009 FRM Practice Exams 43. A mutual fund investing in common stocks has adopted a liquidity risk measure limiting each of its holdings to a maximum of 30% of its thirty day average value traded. If the fund size is USD 3 billion, what is the maximum weight that the fund can hold in a stock with a thirty‐day average value traded of USD 2.4 million?

a. b. c. d.

24.00% 0.08% 0.024% 80.0%

CORRECT: C Thirty day average value traded = USD 2.4 million 30% of thirty day average value traded = 30% x USD 2.4 = USD 0.72 million % of Portfolio = USD 0.72 million/USD 3 billion = 0.024% Reference: Grinold, Kahn, Chapters, 14,17.

44. An investor is investigating three hedge funds as potential investments. Hedge fund A is an equity market neutral fund, B is a global macro fund with emphasis on equity markets, and C is a convertible arbitrage fund. Which answer correctly specifies the funds with the highest exposure to a worldwide value‐weighted equity index and to a credit‐default swaps index? Highest equity index exposure Highest credit‐default swap index exposure a. A B b. A C c. B A d. B C

CORRECT: D Hedge fund B, global macro, has highest equity exposure. Hedge fund A (equity market neutral) has no net equity exposure as long positions are offset by short positions with regards to equity exposure. Similarly, little outright market risk for convertible arbitrage fund C as it involves arbitrage to a large extent. With regards to credit risk, convertible arbitrage fund C has highest credit risk, since it invests in interest rate sensitive instruments highly affected by changes in credit ratings. Reference: Lars Jaeger, Through the Alpha Smoke Screens: A Guide to Hedge Fund Return Sources, Chapter 5 Copyright © 2009 Global Association of Risk Professionals 145 All rights reserved.

2009 FRM Practice Exams 45. Which of the following is not an approach for detecting style drift of hedge funds? a. Performance attribution b. Peer group comparison c. Cash flow analysis d. Communication with fund manager

CORRECT: C Cash Flow analysis for the fund cannot detect style drift INCORRECT: A – Performance attribution can determine if each performance component is consistent with the per‐determined style of fund. INCORRECCT: B – Peer group can be used to determine the consistency of the fund’s risk and return profile INCORRECT: D – Discussion with fund manager can reveal change in style and strategy of the fund. Reference: Lars Jaeger, ed., The New Generation of Risk Management for Hedge Funds and Private Equity Investments, Chapter 27

46. All of the following strategies are examples of capital structure arbitrage, except:

a. Long position in the bonds issued by the company, and short position in the company’s stock. b. Short position in the bonds issued by the company, and long position in the company’s stock. c. Long position in a credit default swap on the company and writing put options on the company’s stock. d. Short position in the preferred stock issued by the company and writing call options on the company’s stock.

CORRECT: D A long position in the bonds issued by the company and short position in the company’s stock, and a short position in the bonds issued by the company and a long position in the company’s stock are both capitalizing on the different speed of adjustment to information in the bond and equities markets. A long position in a credit default swap on the company and writing put options on the company’s stock capitalizes on the differences in the volatility surface between bonds and equity. Short position in the preferred stock issued by the company and writing call options on the company’s stock in fact is a net short position which should not be sensitive to relative changes in the difference between the two instruments. Reference: Jaeger, Through the Alpha Smokescreens, Chapter 5


2009 FRM Practice Exams 47. You have been asked to evaluate the performance of two hedge funds: Global Asset Management I and International Momentum II. Both are benchmarked to MSCI EAFE. Which of the two funds had a higher relative Risk Adjusted Performance (RAP) last year, and what is the RAP?

The volatility of EAFE is 17.5% and the annualized performance is 10.6%. The risk‐free rate is 3.5%. Fund Volatility Annualized Performance Global Asset Management I 24.5% 12.5% International Momentum II 27.3% 13.6%

a. b. c. d.

Global Asset Management, 4.85% Global Asset Management, 6.16% International Momentum, 5.42% International Momentum, 1.18%

CORRECT: C The annualized risk adjusted performance is σm/ σp(Rp – Rm) + Rf = σm – benchmark volatility σp – portfolio volatility Rp– portfolio return Rm – benchmark return Rf – risk‐free return The risk adjusted performance for Global is then σm/ σp(Rp – Rm) + Rf = 17.5/24.5 x (12.5 – 10.6) + 3.5 = 4.85% The risk adjusted performance for International is then σm/ σp(Rp – Rm) + Rf = 17.5/27.3 x (13.6 – 10.6) + 3.5 = 5.42%. International has a greater RAP. Reference: Grinold, Kahn, Chapters 14, 17.

48. On January 1, 2006, a pension fund has assets of EUR 100 billion and is fully invested in the equity market. It has EUR 85 billion in liabilities. During 2006, the equity market declined by 15% and yields increased by 1.2%. If the modified duration of the liabilities is 12.5, what is the pension fund’s surplus on December 31, 2006?

a. b. c. d.

EUR 15.00 billion EUR 12.93 billion EUR 12.75 billion EUR 12.57 billion


2009 FRM Practice Exams CORRECT: C The surplus at the beginning of the year was 100 – 85 or 15 billion EUR. During the year, the equity portfolio declines 15%, or 15 billion EUR, to 85 billion EUR. Due to the increase in yields, the dollar value of the liabilities decreases by 12.5 * 1.2% * 85 billion EUR, or 12.75. Thus at the end of the year, the assets are worth (100 – 15) = 85 billion EUR and the liabilities (85 – 12.75) = 72.25 billion. The surplus is then 12.75, a decrease of 2.25 billion EUR. INCORRECT: A – 15 is the equity loss in 2006 INCORRECT: B – 12.93 = 15 * (1 – (.15‐.012)) INCORRECT: D – 12.57 = 15 * (1 – (.15+.012)) Reference: Jorion, VaR, 3rd ed, Chapter 17.

49. Which of the following statements regarding Extreme Value Theory (EVT) is incorrect?

a. Conventional approaches for estimating VaR that assume that the distribution of returns follow a unique distribution for the entire range of values may fail to properly account for the fat tails of the distribution of returns. b. In contrast to conventional approaches for estimating VaR, EVT only considers the tail behavior of the distribution. c. By smoothing the tail of the distribution, EVT effectively ignores extreme events and losses which can generally be labeled outliers. d. EVT attempts to find the optimal point beyond which all values belong to the tail and then models the distribution of the tail separately.

CORRECT: C Statements a, b, and d are valid statements. Statement c is incorrect – outliers are a big concern in risk analysis and cannot be ignored. Reference: Dowd, Chapter 7.


2009 FRM Practice Exams 50. Consider a portfolio with 40% invested in asset X and 60% invested in asset Y. The mean and variance of return on X are 0 and 25 respectively. The mean and variance of return on Y are 1 and 121 respectively. The correlation coefficient between X and Y is 0.3. What is the nearest value for portfolio volatility? a. b. c. d.

9.51% 8.60% 13.38% 7.45%

CORRECT: D The portfolio volatility is calculated as follows: portfolio variance = w x2σ x2 + w y2σ y2 + 2w x w y ρ xy σ xσ y = 0.4 2 * 25 + 0.62 * 121 + 2 * 0.4 * 0.6 * 0.3 * 5 * 11

= 55.48

portfolio volatility = (portfolio variance)0.5 = (55.48) 0.5 = 7.45 INCORRECT: A – This solution incorrectly calculates the portfolio variance using wx and wy instead of wx2 and wy2. INCORRECT: B – This solution incorrectly omits the correlation between X and Y in the portfolio variance calculation. INCORRECT: C – This solution incorrectly omits the weighting terms in the portfolio variance calculation. Reference: Philippe Jorion, Value at Risk, 3rd edition. Chapter 7.

END OF 2009 FRM FULL EXAM PRACTICE EXAM II Questions & Explanations


2009 GARP Practice (Source)

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