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Name: .................................................... Id. Nr. .......................................................... School of Business and Economics EXAM Course : Code : Date Time Location Risk Management EBC4056 : 3 June, 2010 : 9.00 - 12.00 : MECC Westhal Electronic communications devices are not allowed - put them in your bag or on the floor, not in your pocket! (Otherwise this will be reported as possible fraud). This exam consists of: # 23 Pages (Front page included) # 10 Questions You are allowed to make use of: 1. A non-programmable calculator 2. A two-sided signed-off formula sheet Norm: : A total of 20 points can be received. The maximum points per question are provided within parentheses at the header of the question. Publication of the results: within 15 workdays Procedure for objections: After exam inspection, by e-mail to the block coordinator within 5 working days. A response will be provided within 5 working days afterwards. In case of disagreement, procedure towards exam committee to be pursued by student. Particulars: 1. Both calculation of the result and the final answer are important. Please provide both. 2. In case you miss information, clearly state what information you are missing and state what assumption you make to answer the question. 3. Be to-the-point. Inclusion of irrelevant statements will be treated as (partially) wrong. 4. All answers should be written on the exam itself. Use both sides of paper if necessary. 5. Write clearly. Text that is considered difficult to read may be treated as (partially) wrong. 6. A look-up table is available for the normal distribution in the appendix at the end of this exam. Question 1 (2 points) A financial institution has the following portfolio of over-the-counter options on sterling: Type Call Call Put Call Position -1000 -500 -2000 -500 Delta of Option 0.50 0.80 -0.40 0.70 Gamma of Option 2.2 0.6 1.3 1.8 Vega of Option 1.8 0.2 0.7 1.4 A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8. (a) What position in the traded option and in sterling would make the portfolio both gamma neutral and delta neutral? (b) What position in the traded option would make the portfolio both vega neutral and delta neutral? Answer: 2 3 Question 2 (2 points) Suppose that the zero-coupon yield curve (rates with continuously compounding) and the associated partial durations for a particular portfolio are as provided in the table below: Maturit 1 y (years) Rate 4.0 (%) Partial 2.0 Duratio n 2 3 4 5 7 10 4.5 4.8 5.0 5.1 5.2 5.3 1.6 0.6 0.2 -0.5 -1.8 -1.9 Furthermore the total duration of the portfolio is 0.2. (a) Rank order the sensitivity of the portfolio towards (A) parallel shifts, (B) to changes in shorter term interest rates and (C) to changes in longer term interest rates. (b)Using the partial durations, what would be the percentage change in the value of the portfolio if the yield curve would instantaneously shift to the following yield curve: 3.7% (1-year rate), 4.3% (2-year rate), 4.7% (3-year rate), 5.0% (4-year rate), 5.2% (5-year rate), 5.5% (7-year rate), 5.9% (10year rate). Answer: 4 5 Question 3 (2 points) Suppose that a bank has made a large number of loans of a certain type. The total amount lent is $500 million. The one-year probability of default on each loan is 1.5% and the loss when a default occurs is 70% of the amount owed. The bank uses a Gaussian copula for time to default. The correlation copula is 0.2. Estimate the loss on the portfolio that is not expected to be exceeded with a probability of 99.5%. Answer: 6 7 Question 4 (2 points) Consider an exponentially weighted moving average scheme (EWMA) and suppose that is 0.90, the volatility for day 0 is 1% per day, and during day 0 the market variable increased by 2%. (a) What is the volatility on day 1 according to the EWMA scheme. (b)Suppose that no further shocks occur during the next 60 days. What would be the volatility according to the EWMA scheme after 60 days. Answer: 8 9 Question 5 (2 points) The probability that the loss from a portfolio will be greater than $10 million in one month is estimated to be 5%. (a) What is the 1-month 99% VaR assuming the change in value of the portfolio is normally distributed? (b) What is the 1-month 99% VaR assuming that the power law applies with 3 ? Recall that the power law is of the form: Prob v x Kx Answer: 10 11 Question 6 (2 points) A risk measure used for specifying capital requirements can be thought of as the amount of cash (or capital) that must be added to a position to make its risk acceptable. Artzner et al. have proposed a number of properties that such a risk measure should have. Consider two $10 million one-year loans each of which has a 1.25% chance of defaulting. If a default occurs on one of the loans, the recovery of the loan principle is uncertain, with all recoveries between 0% and 100% being equally likely. If the loan does not default, a profit of $0.2 million is made. Suppose that if one loan defaults, it is certain that the other loan will not default. (a) Provide the 4 most important desirable properties of a risk measure and a short description thereof. (b)Determine the one-year 99% VaR for both loans separately. (c) Determine the one-year 99% VaR for the portfolio of the two loans. (d)Determine the expected shortfall from a single loan when the time horizon is one year and the confidence level is 99% (e) Determine the expected shortfall from the portfolio of the two loans when the time horizon is one year and the confidence level is 99% Answer: 12 13 Question 7 (2 points) During the period July 11, 1988 and July 10, 1998 a total of 2,256 daily return observations on the S&P 500 have been observed, ranging from -6.87% to +5.12%. Consider the left tail of the distribution and apply extreme value theory with u=0.02, there were in total 28 returns less than -2%. Furthermore use as parameters: 0.3232 , and 0.0055 . (a) Estimate the probability that the return will be less than 0.04. (b) What is the value of the one-day 99% VaR for a portfolio where $1 million is invested in the S&P 500? Answer: 14 15 Question 8 (2 points) Suppose that the Economic Capital estimates for two business units are as follows: Market Risk Credit Risk Operational Risk Business Unit 2 40 30 10 1 20 40 70 The correlations are represented in the table below. MR, CR and OR refers to Market Risk, Credit Risk and Operational Risk; 1 and 2 refer to business units MR-1 CR-1 OR-1 MR-2 CR-2 OR-2 MR-1 1.0 0.5 0.2 0.4 0.0 0.0 CR-1 0.5 1.0 0.2 0.0 0.6 0.0 OR-1 0.2 0.2 1.0 0.0 0.0 0.0 MR-2 0.4 0.0 0.0 1.0 0.5 0.2 CR-2 0.0 0.6 0.0 05 1.0 0.2 OR-2 0.0 0.0 0.0 0.2 0.2 1.0 (a) Calculate the total Economic Capital for business unit 1 and 2 separately. (b) Calculate the total Economic Capital for the two business units together. (c) What is the incremental effect of each business unit on the total Economic Capital? (d) Use the answer in (c) t0 allocate capital to each business unit. Answer: 16 17 Question 9 (2 points) The Basel Committee on Bank Supervision has identified 7 categories of operational risk. If a bank would apply the standardized approach to determine operational risk capital a bank's activities need to be divided into 8 business lines. (a) List the 7 categories of operational risk. (b)List the 8 business lines under the standardized approach. Answer: 18 19 Question 10 (2 points) (a) During the guest lecture the difference between a financial lease and an operating lease has been discussed. In terms of risk what is the most important difference between the two. (b)During the guest lecture credit risk, market risk and operational risk was discussed for a leasing company. Give an example of all three of them for DLL in particular, as discussed during the lecture. Answer: 20 21 Appendix 1 Table for N(x) when x= 0 This table shows values for N(x) for x>= 0. The table should be used with interpolation. For example, N(0.6278) = N(0.62) + 0.78[N(0.63) - N(0.62)] = 0.7324+0.78 x (0.7357 - 0.7324) = 0.7350. 23