Solutions to Further Problems Risk Management and Financial Institutions
Second Edition
John C. Hull
Chapter 1: Introduction
. 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 12% 20% % % % % % % % % % % . 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%. .
(a) The bank can be 99% certain that profit will better than ×2 or –% of assets. It therefore
needs equity equal to % of assets to be 99% certain that it will have a positive equity at the year end.
(b) The bank can be % certain that profit will be greater than × 2 or –% of assets. It therefore needs equity equal to % of assets to be % certain that it will have a positive equity at the year end.
. When the expected return on the market is 30% the expected return on a portfolio with a beta of is
+ × ( =
or –2%. The actual return of –10% is worse than the expected return. The portfolio manager has achieved an alpha of –8%!
Chapter 2: Banks
. There is a % chance that the profit will not be worse than × = $ million. Regulators will require $ million of additional capital.
. 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.
. 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 ., $50). Individuals G, D, E, and F get the whole amount of the shares they bid for. Individual A gets 40,000 shares. . 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 $, 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 $ 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 $, 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 Funds
. (Spreadsheet Provided). The unconditional probability of the man dying in years one, two, and three can be calculated from Table as follows: Year 1: Year 2: (1 × = Year 3: (1 × (1 × =
The expected payouts at times , , are therefore $59,, $64,, and $68,. These have a present value of $175,. The survival probability of the man is Year 0: 1 Year 1: 1 = Year 2: 1 =
The present value of the premiums received per dollar of premium is therefore . The minimum premium is or $62,.
(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 $ 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 . The expected payoff in millions of dollars is therefore × 100= and the value of the contract is or $ million.
. 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.
. (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,. The spreadsheet is used in conjunction with Solver to show that the required contribution rate is % (employee plus employer). The value of the contribution grows to $2,420, by the end of the 45 year working life. (This assumes that the real return of % is annually compounded.) This value reduces to zero over the following 18 years under the assumptions made. This calculation confirms the point made in Section that defined benefit plans require higher contribution rates that those that exist in practice.
Chapter 4: Mutual Funds and Hedge Funds
. The investor pays tax on dividends of $200 and $300 in year 2009 and 2010, 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 2009 and 2010, respectively The result of all this is that the basis for the shares increases from $50 to $63. The sale at $59 in year 2011 leads to a capital loss of $4 per share or $400 in total. . The investors overall return is
× × × – 1 =
or % for the four years.
. The overall return on the investments is the average of 5%, 1%, 10%, 15%, and 20% or %.
The hedge fund fees are 2%, %, 4%, 5%, and 6%. These average %. The returns earned by the fund of funds after hedge fund fees are therefore 7%, %, 6%, 10%, and 14%. These average %. The fund of funds fee is 1% + % or % leaving % for the investor. The return earned is
therefore divided as shown in the table below. This example explains why funds of funds have declined in popularity.
Return earned by hedge funds Fees to hedge funds Fees to fund of funds Return to investor % % % % . The plot is shown in the chart below. If the hedge fund return is 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.
风险管理与金融机构约翰第二版答案
