Risk Management Midterm
Solutions to Exam 1 1. (Chapter 1, slides, Q3) Suppose there are two investments. The expected returns from the investments are 10% and 15%, the standard deviation of the returns are 16% and 24%, and the correlation between returns is 0. 2. Let w1 be the proportion of wealth put into the first investment. (a). Calculate the expected return and the standard deviation for portfolio w1=0,0. 2,0. 4,0. 6. (b). Draw a picture of these risk and returns for w1. (c). What is it called? (d). Draw the picture of (c) when there is a riskless asset. (e). How should an investor choose the optimal investment?
Solutions: (a) (b) Page 1 of 8 (c) It is called Efficient Frontier. An efficient frontier represents the limit of how far we can move in a northwest direction and is illustrated in the following figure. There is no investment that dominates a point on the efficient frontier, in the sense that it has both a higher expected return and a lower standard deviation of return. (d) (e) Every possible combination of the risky assets, without including any holdings of the risk-free asset, can be plotted in risk-expected return space, and the collection of all such possible portfolios defines a region in this space.
The left boundary of this region is a hyperbola, and the upper edge of this region is the efficient frontier in the absence of a risk-free asset. 2. (Chapter 1, Practice Questions 14) The return from the market last year was 10% and the risk-free rate was 5%. A hedge fund manager with a beta of 0. 6 has an alpha of 4%. What return did the hedge fund manager earn? Solutions: 3. (Chapter 2, Q3) Page 2 of 8 (a). How many banks are there in the United States? (b). What is the size of the assets held by these banks? Solutions: (a)&(b) 4. Chapter 2, slides, Q8) Suppose that there is a severe recession and as a result the bank’s loan losses rise by 3. 2%of assets, to 4% next year. We assume that other items on the income statement in Table 2. 3 are unaffected. (a) Calculate the pre-tax net operating loss. (b) Assuming a tax rate of 30%. Calculate the after-tax loss of about 1. 8% of assets. (c) Calculate the reduction of the equity capital. (d) Determine whether the bank can survive. (e) What is the lesson of this exercise? Solutions: (a) The pretax net operating loss is pre-tax operating income 0. 6 increase in loan loss . 2 2. 6 (b) The after-tax loss is after tax payment 0. 7 *( 2. 6) 1. 8 (c) The reduction in equity capital is Page 3 of 8 (d) An arter-tax loss equal to 1. 8% of assets can be absorbed by the equity capital. It would result in a reduction of the equity capital to 3. 2% of assets. Even a second bad year similar to the first would not totally wipe out the equity. (e) Equity capital is important. 5. (Chapter 3, slides, Q7) Consider a mortality table. Assume that interest rates for all maturities are 4% and premiums are paid once a year at the beginning of the year. (a).
What is an insurance company’s break-even premium for $100,000 of oneyear term life insurance for a man of average health aged 90? (b). What is the premium payment? Solutions: (a) & (b) If the term insurance lasts one year, the expected payout is 0. 181789 x 100,000, or $18,179. Assume that the payout occurs halfway through the year. The premium is $18,179 discounted for six months at 4%. Assuming semiannual compounding, this is 18,179/1. 02, or $17,822. 6. (Chapter 4, slides, Q2) What is the size of the assets in mutual funds industry in 2008? Solutions: Mutual funds have grown very fast in since the Second World War.
These estimates of assets managed by mutual funds were over $10 trillion by 2008. About 50% of US households own mutual funds. Page 4 of 8 7 (Chapter 4, slides, Q18) Suppose that a hedge fund manager is presented with an opportunity where there is a 0. 4 probability of a 60% profit and a 0. 6 probability of a 60% loss, with the fees earned by the hedge fund manager being 2 plus 20%. (a) What is the expected return of the project? (b) What is the expected profit of the manager if he decide to make this investment? (c) What is the lesson of this example?
Solutions: (a) The expected return of the investment is 0. 4 x 60 + 0. 6 x (-60) or -12%. (b) If the investment produces a 60% profit. the hedge fund ‘s fee is 2 + 0. 2 x 60. or 14%. If the investment produces a 60% loss, the hedge fund’s fee 2% . The expected fee to the hedge fund is therefore 0. 4 x 14 + 0. 6 x 2 = 6. 8 or 6. 8% or the funds under administration . The expected management fee is 2% and the expected incentive fee is 4%. (c) It shows that the fee structure of a hedge fund gives its managers an incentive to take high risks even when expected returns are negative. (Chapter 4, Practice Questions, 9) Give four examples of the trading restrictions that apply to mutual funds trading, but do not apply to hedge funds trading. Solutions: 9. (Chapter 5, slides, Q4) What is the size of the derivative markets in June 2008 (over-the-counter market plus the exchange-traded market)? Page 5 of 8 Solutions: A derivative is a financial instrument – or more simply, an agreement between two people or two parties – that has a value determined by the price of something else (called the underlying).
It is a financial contract with a value linked to the expected future price movements of the asset it is linked to – such as a share or a currency. Using these measures, by June 2008, the over-the-counter market had grown to $683. 7 trillion and the exchange-traded market had grown to $84. 3 trillion. 10. (Chapter 5, Practice Questions, 17) Suppose that USD-sterling spot and forward exchange rates are as follows: What opportunities are open to an arbitrageur in the following situations: (a) a 180-day European call option to buy ? 1 for $1. 57 costs 2 cents and (b) a 90day European put option to sell ? 1 for $1. 64 costs 2 cents? Solutions:
Page 6 of 8 11. (Chapter 6, slides, Q23) Supposethe stock price is $49, the strike price is $50, the risk- free rate is 5%, the stock price volatility is 20% and the time to exercise is 20 weeks or 20/52 years. Table 6. 4 Greek letters calculated using DerivaGem: S=49, K=50, r=5%, ? =20%, and T=20 weeks. Short position Single Option In 100,000 options Value ($) 2. 40 -240,000 Delta (per $) 0. 522 -52,200 Gamma (per $ per $) 0. 066 -6,600 Vega (per %) 0. 121 -12,100 Theta (per day) -0. 012 1,200 Rho (per %) 0. 089 -8,900 (a). When there is an increase of $0. 1 in the stock price with no other changes, the option price increases by ( 0. 522 ). (b). When there is an increase $0. 1 in the stock price with no other changes, the delta of the option increases by ( 0. 0066 ). (c). When there is an increase in volatility of 0. 5 % with no other changes, the option price increases by about ( $0. 0605 ). (d). When one day goes by with no changes to the stock price or its volatility, the option price decreases by ( $0. 012 ). (e). When interest rates increase by 1 % (or 100 basis points) with no other changes, the option price increases by ( $0. 089 ). 12. (ch7, slides, Q9 and Q13) Consider a three-year 10% coupon bond with a face value of $100.
Coupon payments of $5 are made every six months. Suppose that the yield on the bond is 12% per annum with continuous compounding. (a). Calculate the duration. (b). Suppose that the bond yield increases by 10 basis points to 12. 1%. What will be the new bond price? Solutions: Page 7 of 8 (a) (b) We have deltaB= – 94. 2 13 x 2. 653*deltay That is, deltaB = -249. 95*deltay When the yield on the bond increases by 10 basis points (= 0. 1 %), deltay = + 0. 00 I. The duration relationship predicts that deltaB = -249. 95 x 0. 001 = -0. 250. Therefore, the bond price goes down to 94. 213 – 0. 250 = 93. 963. Page 8 of 8