Solutions to Further Problems Risk Management and Financial Institutions
Second Edition
John C. Hull
Chapter 1: Introduction
1.15. 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 in figure below. w1 w2 ?P ?P 0.0 1.0 12% 20% 0.2 0.8 11.2% 17.05% 0.4 0.6 10.4% 14.69% 0.6 0.4 9.6% 13.22% 0.8 0.2 8.8% 12.97% 1.0 0.0 8.0% 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 of returns corresponding to an expected return of 20% is 39%. 1.17.
(a) The bank can be 99% certain that profit will better than 0.8?2.33×2 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% 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 0.2 is
0.05 + 0.2 × (?0.30 ? 0.05) = ?0.02
or –2%. The actual return of –10% is worse than the expected return. The portfolio manager has achieved an alpha of –8%!
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Chapter 2: Banks
2.15. 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 million 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 behavior 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 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 further 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 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, 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.
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Chapter 3: Insurance Companies and Pension Funds
3.16. (Spreadsheet 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: 0.011858
Year 2: (1?0.011858) × 0.012966 = 0.012812
Year 3: (1?0.011858) × (1?0.012966) × 0.014123 = 0.013775
The expected payouts at times 0.5, 1.5, 2.5 are therefore $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 is Year 0: 1
Year 1: 1?0.011858 = 0.988142 Year 2: 1?0.011858?0.012812 = 0.97533
The present value of the premiums received per dollar of premium is therefore 2.797986. The minimum premium is or $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 diversify away the risk, the minimum cost of reinsurance is
or $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 above 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 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. (Spreadsheet 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 $239,005.31. 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
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