HW1 2015

Homework 1
MS 4226 Risk Management Models
Due Date: in class of March 3 (Tuesday), 2015
Please submit hard-copies during the class. Late homework will NOT be accepted.

I. Short Answers
1. What is the intuitive meaning of efficient frontier? Explain in plain words.
2. What is meant by risk aggregation and risk decomposition? Which requires an in-depth understanding of individual risks? Which requires a detailed knowledge of the correlation between risks?
3. Explain in plain words the meaning of delta hedging.
4. Explain the difference between hedging, speculation, and arbitrage.
5. Suppose the true loss is heavy-tailed, while we calculate Value-at-Risk (VaR) by assuming it is normally distributed. Is the true VaR underestimated or overestimated? Why?
6. In general, does VaR satisfy risk diversification? If we assume that P&L’s follow normal distributions with mean zero, does VaR satisfy risk diversification? Why?
7. What is the difference between conditional Value-at-Risk (CVaR) and Value-at-Risk? What is the theoretical advantage of CVaR over VaR?

II. Calculation
1. A bank’s profit next year will be normally distributed with a mean of 0.6% of assets and a standard deviation of 1.5% of assets. The bank’s equity is 4% of the assets. What is the probability that the bank will have a positive equity at the end of the year?
2. A portfolio consists of 1,000 shares of HSBC stock, and 2,000 of Hang Seng stock. Current prices of HSBC and Hang Seng are 83 and 110 HKD respectively. Suppose that the percent1

age changes of HSBC and Hang Seng for the next ten days are normally distributed, with means 3% and 2%, and standard deviations 5% and 7% respectively. Let the correlation between HSBC and Hang Seng be 0.7.
(a) Calculate the ten-day 99% Value-at-Risk (VaR) of the portfolio.
(b) Explain in plain words the meaning of this VaR.
(c) Suppose further that we add to the portfolio a call option underlying 1,000 shares of
HSBC stock with strike price being 85 HKD, maturity being six months. Let the risk-free interest rate be 5%. Calculate the ten-day 99% VaR of the portfolio.
3. Suppose that the estimated 99% one-day VaR of a portfolio is 1 mil. We have the daily portfolio loss data of 500 days. We found that the daily portfolio losses of 7 days are larger than
1 mil. Backtest the VaR model. If you reject the model, conclude whether we overestimate or underestimate the true VaR.

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