F537-1.s2024.HW04

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Finance 537 Financial Risk Management Spring 2024 Neil D. Pearson 1 Homework 4 Due Sunday, February 18, 2024 at 10:00 p.m. US central time Total 10 points Instructions . This is a group assignment. Groups may include up to 4 people. Please submit a Word or .pdf document with your solutions via the Canvas site prior to 10:00 p.m. on Sunday, February 18. You should also upload the computer codes and data files you use. 1. (2 points) Find Goldman Sachs’ Annual Report for the year ended December 31, 2022. Page 104 of the Annual Report indicates that, “Net losses incurred on a single day for such [trading] positions exceeded our 95% one-day VaR (i.e., a VaR exception) on two occasions during 2022 and on one occasion during 2021.” (a) (1/2 point) If a VaR model is accurate, what is the expected number of times that the daily trading loss exceeds the VaR during a year? (Assume 252 trading days per year.) (b) (1 point) Consider 252 independent Bernoulli trials (i.e., 252 trials of a binomial random variable) where on each trial the probability of a “success” is 0.95 and the probability of a “failure” is 0.05. What is the probability that the 252 trials result in two or fewer failures? What is the probability that 252 Bernoulli trials result in one or fewer failures? (You might want to look at your statistics book to refresh your memory about the binomial distribution and the distribution of Bernoulli trials.) Hint . You might find the R functions dbinom() and/or pbinom() to be useful. Alternatively, you might find the Excel function binom.dist() to be helpful. (c) (1/2 point) What does this analysis of Bernoulli trials tell you about whether Goldman’s VaR model is or is not biased? 2. (1 point) You own a portfolio of CUB stock and options on CUB stock that has a current value of $10 million. The current price of CUB stock is $100 per share. Simple returns on CUB stock are described by the normal distribution with a mean of 0.5% per month and a standard deviation of 3.6538% per month. The figure below shows the value of your portfolio after one month. If you use the Monte Carlo method and a probability of 1%, what is the VaR of your portfolio? Remark . The horizon of the VaR estimate is one month, so do not assume the expected return is zero. Do not do the simulation  rather, based on the figure, figure out what the outcome of the simulation would be.

Finance 537 Financial Risk Management Spring 2024 Neil D. Pearson 2 3. (1 point) At the close of trading on Friday, December 16, 2022, the prices of XLE, XLF, and XLY are $84.36, $33.70, and $133.90 per share. The implied volatilities of ATM options with 91 days to expiration are 0.317964, 0.227176, and 0.290478, and the continuously compounded 3- month interest rate is 0.0485 per year. XLE, XLF, and XLY will pay dividends of $0.85963, $0.20962, and $0.36581, respectively, on December 22, 2022. The ex-dividend dates are Monday, December 19, 2022, the same for all three ETFs. Using these inputs, compute the values of a 91-day call option on XLE with a strike price of 84, a 91-day call option on XLF with a strike price of 34, and 91-day call option on XLY with a strike price of 134. Remark. You might find the R function am_call_bin_propdiv in the R library AmericanCallOpt to be useful in computing the option prices. For this function, the dividend is entered as a fraction of the stock price. For the three ETFs, the fractions are $0.85963/$84.36, $0.20962/$33.70, and $0.36581/$133.90, respectively. The 91-day call options expire on March 17, 2023. In using the R function, compute the fraction of a year using trading days. December 16, 2022 to March 17, 2023 is 91 calendar days but only 13 × 5 – 4 = 61 trading days, due to the four holidays (Christmas, New Year, MLK, and President’s Day). Also, the ex-dividend date of December 19 is one trading day after December 16. Use 1,000 timesteps for the binomial model. 4. (2 points) At the close of trading on December 16, 2022, you hold 30,000 shares of XLE, 60,000 shares of XLF, and 10,000 shares of XLY. In addition, you have written call options on 30,000 shares of XLE, 60,000 shares of XLF, and 10,000 shares of XLY. The call options all expire on March 17, 2022, that is, in 91 days. The strike prices of the options on XLE, XLF, and 8,500,000 9,000,000 9,500,000 10,000,000 10,500,000 11,000,000 11,500,000 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 Portfolio value after one month

Finance 537 Financial Risk Management Spring 2024 Neil D. Pearson 3 XLY are 84, 34, and 134, respectively. The continuously compounded interest rate is r = 0.0485. XLE, XLF, and XLY will pay dividends of $0.85963, $0.20962, and $0.36581, respectively, on December 22, 2022. The ex-dividend dates are Monday, December 19, 2022, the same for all three ETFs. Please use the simple historical simulation method to compute the VaR of this portfolio as of the close of trading on December 16, 2022. Use a probability of 5% and a holding period of one day. Use six market factors, these being the log returns on XLE, XLF, and XLY, and the log changes in the three ATM implied volatilities. Relevant data are in the three .csv files ETFprices.csv, ETFreturns.csv, and ETFATMvols.csv. For this question, use only 100 timesteps in the binomial model you use to compute the simulated options prices. Use the most recent 500 returns, including the return on December 16, in the historical simulation method. 5. (1 point) Use the most recent 500 daily log returns (including the return on December 16, 2022) for the three ETFs and most recent 500 daily log changes in the implied volatilities to compute the exponentially-weighted covariance matrix of the six market factors. Use λ = 0.94, and the expected log returns or log changes are zero when you compute the exponentially- weighted covariance matrix. Remark . Compute the covariance matrix using the log returns and log changes, not the simple return and percentage changes. Note that while the question tells you to use 500 past returns, you do not actually need to use this many past returns. In thinking about how many past days should you use to compute the exponentially-weighted covariance matrix, you should think about how large is λ τ for various values of τ . 6. (2 points) Consider the portfolio from Question 4. Please use the Monte Carlo method with holding period of one day, a probability of 5%, and the exponentially-weighted covariance matrix with λ = 0.94 from Question 5 to compute the Monte Carlo VaR of your portfolio as of the close of trading on December 16, 2022. Use 1,000 Monte Carlo trials to estimate the VaR. Remark. You will make it easier for our TA to grade your homeworks if you set the seed equal to 137. Specifically, letting mu and cov be the mean vector and covariance matrix, generate the simulated log returns using the code n <- 1000 set.seed(137) simreturns <- mvrnorm(n, mu = mu, Sigma = cov)

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