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1、Solutions to Further ProblemsRisk Management and Financial InstitutionsSecond EditionJohn C. HullChapter 1: Introduction1.15. The impact of investing w1 in the first investment and w2 = 1 w1 in the second investment is shown in the table below. The range of possible risk-return trade-offs is shown i
2、n figure below.w1w2mPsP0.01.012%20%0.20.811.2%17.05%0.40.610.4%14.69%0.60.49.6%13.22%0.80.28.8%12.97%1.00.08.0%14.00%1.16. In this case the efficient frontier is as shown in the figure below. The standard deviation of returns corresponding to an expected return of 10% is 9%. The standard deviation o
3、f returns corresponding to an expected return of 20% is 39%.1.17.(a) The bank can be 99% certain that profit will better than 0.82.332 or 3.85% of assets. It therefore needs equity equal to 3.85% of assets to be 99% certain that it will have a positive equity at the year end.(b) The bank can be 99.9
4、% certain that profit will be greater than 0.8 3.09 2 or 5.38% of assets. It therefore needs equity equal to 5.38% of assets to be 99.9% certain that it will have a positive equity at the year end.1.18. When the expected return on the market is 30% the expected return on a portfolio with a beta of 0
5、.2 is0.05 + 0.2 (0.30 0.05) = 0.02or 2%. The actual return of 10% is worse than the expected return. The portfolio manager has achieved an alpha of 8%!Chapter 2: Banks2.15. There is a 99.9% chance that the profit will not be worse than 0.6 3.090 2.0 = $5.58 million. Regulators will require $0.58 mil
6、lion of additional capital.2.16. Deposit insurance makes depositors less concerned about the financial health of a bank. As a result, banks may be able to take more risk without being in danger of losing deposits. This is an example of moral hazard. (The existence of the insurance changes the behavi
7、or of the parties involved with the result that the expected payout on the insurance contract is higher.) Regulatory requirements that banks keep sufficient capital for the risks they are taking reduce their incentive to take risks. One approach (used in the US) to avoiding the moral hazard problem
8、is to make the premiums that banks have to pay for deposit insurance dependent on an assessment of the risks they are taking. 2.17. When ranked from lowest to highest the bidders are G, D, E and F, A, C, H, and B. Individuals G, D, E, and F bid for 170, 000 shares in total. Individual A bid for a fu
9、rther 60,000 shares. The price paid by the investors is therefore the price bid by A (i.e., $50). Individuals G, D, E, and F get the whole amount of the shares they bid for. Individual A gets 40,000 shares. 2.18. If it succeeds in selling all 10 million shares in a best efforts arrangement, its fee
10、will be $2 million. If it is able to sell the shares for $10.20, this will also be its profit in a firm commitment arrangement. The decision is likely to hinge on a) an estimate of the probability of selling the shares for more than $10.20 and b) the investment banks appetite for risk. For example,
11、if the bank is 95% certain that it will be able to sell the shares for more than $10.20, it is likely to choose a firm commitment. But if assesses the probability of this to be only 50% or 60% it is likely to choose a best efforts arrangement. Chapter 3: Insurance Companies and Pension Funds3.16. (S
12、preadsheet Provided). The unconditional probability of the man dying in years one, two, and three can be calculated from Table 3.1 as follows:Year 1: 0.011858Year 2: (10.011858) 0.012966 = 0.012812Year 3: (10.011858) (10.012966) 0.014123 = 0.013775The expected payouts at times 0.5, 1.5, 2.5 are ther
13、efore $59,290.00, $64,061.25, and $68,872.91. These have a present value of $175,598.60. The survival probability of the man isYear 0: 1Year 1: 10.011858 = 0.988142Year 2: 10.0118580.012812 = 0.97533The present value of the premiums received per dollar of premium is therefore 2.797986. The minimum p
14、remium isor $62,758.92.3.17(a) The losses in millions of dollars are normally distributed with mean 150 and standard deviation 50. The payout from the reinsurance contract is therefore normally distributed with mean 90 and standard deviation 30. Assuming that the reinsurance company feels it can div
15、ersify away the risk, the minimum cost of reinsurance isor $85.71 million. (This assumes that the interest rate is compounded annually.)(b) The probability that losses will be greater than $200 million is the probability that a normally distributed variable is greater than one standard deviation abo
16、ve the mean. This is 0.1587. The expected payoff in millions of dollars is therefore 0.1587 100=15.87 and the value of the contract is or $15.11 million.3.18. The value of a bond increases when interest rates fall. The value of the bond portfolio should therefore increase. However, a lower discount
17、rate will be used in determining the value of the pension fund liabilities. This will increase the value of the liabilities. The net effect on the pension plan is likely to be negative. This is because the interest rate decrease affects 100% of the liabilities and only 40% of the assets. 3.19. (Spre
18、adsheet Provided) The salary of the employee makes no difference to the answer. (This is because it has the effect of scaling all numbers up or down.) If we assume the initial salary is $100,000 and that the real growth rate of 2% is annually compounded, the final salary at the end of 45 years is $2
19、39,005.31. The spreadsheet is used in conjunction with Solver to show that the required contribution rate is 25.02% (employee plus employer). The value of the contribution grows to $2,420,354.51 by the end of the 45 year working life. (This assumes that the real return of 1.5% is annually compounded
20、.) This value reduces to zero over the following 18 years under the assumptions made. This calculation confirms the point made in Section 3.12 that defined benefit plans require higher contribution rates that those that exist in practice.Chapter 4: Mutual Funds and Hedge Funds4.15. The investor pays
21、 tax on dividends of $200 and $300 in year 2021 and 2021, respectively. The investor also has to pay tax on realized capital gains by the fund. This means tax will be paid on capital gains of $500 and $300 in year 2021 and 2021, respectively The result of all this is that the basis for the shares in
22、creases from $50 to $63. The sale at $59 in year 2021 leads to a capital loss of $4 per share or $400 in total.4.16. The investors overall return is 1.08 0.92 1.12 0.88 1 = 0.0207or 2.07% for the four years.4.17. The overall return on the investments is the average of 5%, 1%, 10%, 15%, and 20% or 8.
23、2%. The hedge fund fees are 2%, 2.2%, 4%, 5%, and 6%. These average 3.84%. The returns earned by the fund of funds after hedge fund fees are therefore 7%, 1.2%, 6%, 10%, and 14%. These average 4.36%. The fund of funds fee is 1% + 0.436% or 1.436% leaving 2.924% for the investor. The return earned is
24、 therefore divided as shown in the table below. This example explains why funds of funds have declined in popularity. Return earned by hedge funds8.200%Fees to hedge funds3.840%Fees to fund of funds1.436%Return to investor2.924%4.18. The plot is shown in the chart below. If the hedge fund return is
25、negative, the pension fund return is 2% less than the hedge fund return. If it is positive, the pension fund return is less than the hedge fund return by 2% plus 20% of the return. Chapter 5: Financial Instruments5.30. There is a margin call when more than $1,000 is lost from the margin account. Thi
26、s happens when the futures price of wheat rises by more than 1,000/5,000 = 0.20. There is a margin call when the futures price of wheat rises above 270 cents. An amount, $1,500, can be withdrawn from the margin account when the futures price of wheat falls by 1,500/5,000 = 0.30. The withdrawal can t
27、ake place when the futures price falls to 220 cents. 5.31. The investment in call options entails higher risks but can lead to higher returns. If the stock price stays at $94, an investor who buys call options loses $9,400 whereas an investor who buys shares neither gains nor loses anything. If the
28、stock price rises to $120, the investor who buys call options gains2000 (120 95) 9400 = $40, 600An investor who buys shares gains100 (120 94) = $2, 600The strategies are equally profitable if the stock price rises to a level, S, where100 (S 94) = 2000(S 95) 9400orS = 100The option strategy is theref
29、ore more profitable if the stock price rises above $100.5.32. Suppose ST is the price of oil at the bonds maturity. In addition to $1000 the Standard Oil bond pays:ST ST $2 : 170 (ST 25) ST $40: 2, 550This is the payoff from 170 call options on oil with a strike price of 25 less the payoff from 170
30、call options on oil with a strike price of 40. The bond is therefore equivalent to a regular bond plus a long position in 170 call options on oil with a strike price of $25 plus a short position in 170 call options on oil with a strike price of $40. The investor has what is termed a bull spread on o
31、il.5.33. The arbitrageur could borrow money to buy 100 ounces of gold today and short futures contracts on 100 ounces of gold for delivery in one year. This means that gold is purchased for $500 per ounce and sold for $700 per ounce. The return (40% per annum) is far greater than the 10% cost of the
32、 borrowed funds. This is such a profitable opportunity that the arbitrageur should buy as many ounces of gold as possible and short futures contracts on the same number of ounces. Unfortunately, arbitrage opportunities as profitable as this rarely, if ever, arise in practice.5.34.(a) By entering int
33、o a three-year s it receives 6.21% and pays LIBOR the company earns 5.71% for three years.(b) By entering into a five-year s it receives 6.47% and pays LIBOR the company earns 5.97% for five years.(c) By entering into a s it receives 6.83% and pays LIBOR for ten years the company earns 6.33% for ten
34、 years.5.35. The position is the same as a European call to buy the asset for K on the date.5.36. (a) When the CP rate is 6.5% and Treasury rates are 6% with semiannual compounding, the CMT% is 6% and an Excel spreadsheet can be used to show that the price of a 30-year bond with a 6.25% coupon is ab
35、out 103.46. The spread is zero and the rate paid by P&G is 5.75%. (b) When the CP rate is 7.5% and Treasury rates are 7% with semiannual compounding, the CMT% is 7% and the price of a 30-year bond with a 6.25% coupon is about 90.65. The spread is thereforemax0, (98.5 7/5.78 90.65)/100or 28.64%. The
36、rate paid by P&G is 35.39%.5.37. The trader has to provide 60% of the price of the stock or $2,400. There is a margin call when the margin account balance as a percent of the value of the shares falls below 30%. When the share price is S the margin account balance is 2400 + 200 (S20) and the value o
37、f the position is 200S. There is a margin call when 2400 + 200 (S20) 0.3 200 Sor140 S 1600or S 11.43that is, when the stock price is less than $11.43.Chapter 6: How Traders Manage Their Exposures6.15. With the notation of the text, the increase in the value of the portfolio isThis is0.5 50 32 + 25 4
38、 = 325The result should be an increase in the value of the portfolio of $325.6.16. The price, delta, gamma, vega, theta, and rho of the option are 3.7008, 0.6274, 0.050, 0.1135, 0.00596, and 0.1512. When the stock price increases to 30.1, the option price increases to 3.7638. The change in the optio
39、n price is 3.7638 3.7008 = 0.0630. Delta predicts a change in the option price of 0.6274 0.1 = 0.0627 which is very close. When the stock price increases to 30.1, delta increases to 0.6324. The size of the increase in delta is 0.6324 0.6274 = 0.005. Gamma predicts an increase of 0.050 0.1 = 0.005 wh
40、ich is (to three decimal places) the same. When the volatility increases from 25% to 26%, the option price increases by 0.1136 from 3.7008 to 3.8144. This is consistent with the vega value of 0.1135. When the time to maturity is changed from 1 to 11/365 the option price reduces by 0.006 from 3.7008
41、to 3.6948. This is consistent with a theta of 0.00596. Finally when the interest rate increases from 5% to 6% the value of the option increases by 0.1527 from 3.7008 to 3.8535. This is consistent with a rho of 0.1512.6.17. The delta of the portfolio is1, 000 0.50 500 0.80 2,000 (0.40) 500 0.70 = 450
42、The gamma of the portfolio is1, 000 2.2 500 0.6 2,000 1.3 500 1.8 = 6,000The vega of the portfolio is1, 000 1.8 500 0.2 2,000 0.7 500 1.4 = 4,000(a) A long position in 4,000 traded options will give a gamma-neutral portfolio since the long position has a gamma of 4, 000 1.5 = +6,000. The delta of th
43、e whole portfolio (including traded options) is then:4, 000 0.6 450 = 1, 950Hence, in addition to the 4,000 traded options, a short position in 1,950 is necessary so that the portfolio is both gamma and delta neutral.(b) A long position in 5,000 traded options will give a vega-neutral portfolio sinc
44、e the long position has a vega of 5, 000 0.8 = +4,000. The delta of the whole portfolio (including traded options) is then5, 000 0.6 450 = 2, 550Hence, in addition to the 5,000 traded options, a short position in 2,550 is necessary so that the portfolio is both vega and delta neutral.6.18. Let w1 be
45、 the position in the first traded option and w2 be the position in the second traded option. We require:6, 000 = 1.5w1 + 0.5w24, 000 = 0.8w1 + 0.6w2The solution to these equations can easily be seen to be w1 = 3,200, w2 = 2,400. The whole portfolio then has a delta of450 + 3,200 0.6 + 2,400 0.1 = 1,
46、710Therefore the portfolio can be made delta, gamma and vega neutral by taking a long position in 3,200 of the first traded option, a long position in 2,400 of the second traded option and a short position in 1,710.6.19. (Spreadsheet Provided) Consider the first week. The portfolio consists of a sho
47、rt position in 100,000 options and a long position in 52,200 shares. The value of the option changes from $240,053 at the beginning of the week to $188,760 at the end of the week for a gain of $51,293.The value of the shares change from 52,200 49 = $2,557, 800 to 52,200 48.12 = $2,511,864 for a loss
48、 of $45,936. The net gain is 51,293 45,936 = $5,357. The gamma and theta (per year) of the portfolio are 6,554.4 and 430,533 so that equation (6.2) predicts the gain as430,533 1/52 + 0.5 6,554.4 (48.12 49)2 = 5,742The results for all 20 weeks are shown in the following table.WeekActual Gain ($)Predicted Gain ($)15,3575,74225,6896