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
175,598.60?62,758.92
2.797986or $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
90?85.71 1.05or $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
6 / 40
15.87?15.11 1.05or $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 end of the 45 year working life. (This assumes that the real return of 1.5% 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 3.12 that defined benefit plans require higher contribution rates that those that exist in practice.
7 / 40
Chapter 4: Mutual Funds and Hedge Funds
4.15. 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. 4.16. The investors overall return is
1.08 × 0.92 × 1.12 × 0.88 – 1 = ? 0.0207
or ? 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
8.2%. 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 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
8.200% 3.840% 1.436% 2.924% 8 / 40
4.18. 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.
9 / 40
Chapter 5: Financial Instruments
5.30. There is a margin call when more than $1,000 is lost from the margin account. This 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 take 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 stock price rises to $120, the investor who buys call options gains
2000 × (120 ? 95) ? 9400 = $40, 600 An investor who buys shares gains
100 × (120 ? 94) = $2, 600
The strategies are equally profitable if the stock price rises to a level, S, where
100 × (S ? 94) = 2000(S ? 95) ? 9400 or
S = 100
The option strategy is therefore more profitable if the stock price rises above $100.
5.32. Suppose ST is the price of oil at the bond’s maturity. In addition to $1000 the Standard Oil bond pays:
ST < $25 : 0
$40 > ST > $2 : 170 (ST ? 25) ST > $40: 2, 550
This is the payoff from 170 call options on oil with a strike price of 25 less the payoff from 170 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 oil.
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 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 into 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 years.
10 / 40
风险管理与金融机构-约翰-第二版-答案
