1 Answers to Selected Problems Problem 1.11. The farmer can short 3 contracts that have 3 months to maturity. If the price of cattle falls, the gain on the futures contract will offset the loss on the sale of the cattle. If the price of cattle rises, the gain on the sale of the cattle will be offset by the loss on the futures contract. Using futures contracts to hedge has the advantage that it can at no cost reduce risk to almost zero. Its disadvantage is that the farmer no longer gains from favorable movements in cattle prices. Problem 1.12. The mining company can estimate its production on a month by month basis. It can then short futures contracts to lock in the price received for the gold. For example, if a total of 3,000 ounces are expected to be produced in September 2014 and October 2014, the price received for this production can be hedged by shorting a total of 30 October 2014 contracts. Problem 1.13. The holder of the option will gain if the price of the stock is above $52.50 in March. (This ignores the time value of money.) The option will be exercised if the price of the stock is above $50.00 in March. The profit as a function of the stock price is shown below. Problem 1.14. The seller of the option will lose if the price of the stock is below $56.00 in June. (This ignores the time value of money.) The option will be exercised if the price of the stock is below $60.00 in June. The profit as a function of the stock price is shown below. -50510152020 30 40 50 60 70ProfitStock Price-1001020304050600 20 40 60 80 100 120ProfitStock Price2 Problem 1.20. a) The trader sells 100 million yen for $0.0080 per yen when the exchange rate is $0.0074 per yen. The gain is 100 0 0006 millions of dollars or $60,000. b) The trader sells 100 million yen for $0.0080 per yen when the exchange rate is $0.0091 per yen. The loss is 100 0 0011 millions of dollars or $110,000. Problem 1.21. a) The trader sells for 50 cents per pound something that is worth 48.20 cents per pound. Gain = ($0.5000 − $0.4820) × 50,000 = $900. b) The trader sells for 50 cents per pound something that is worth 51.30 cents per pound. Loss = ($0.5130 − $0.5000) × 50,000 = $650. Problem 2.11. There is a margin call if more than $1,500 is lost on one contract. This happens if the futures price of frozen orange juice falls by more than 10 cents to below 150 cents per lb. $2,000 can be withdrawn from the margin account if there is a gain on one contract of $1,000. This will happen if the futures price rises by 6.67 cents to 166.67 cents per lb. Problem 2.15. The clearing house member is required to provide 20×$2,000 = $40,000 as initial margin for the new contracts. There is a gain of (50,200  50,000) 100  $20,000 on the existing contracts. There is also a loss of (51,000 – 50,200) × 20 = $16,000 on the new contracts. The member must therefore add 40,000 – 20,000 + 16,000 = $36,000 to the margin account. Problem 2.16. Suppose 1F and 2F are the forward exchange rates for the contracts entered into July 1, 2013 and September 1, 2013, respectively. Suppose further that S is the spot rate on January 1, 2014. (All exchange rates are measured as yen per dollar). The payoff from the first contract is 1()SF million yen and the payoff from the second contract is 2()FS million yen. The total payoff is therefore 1 2 2 1( ) ( ) ( )S F F S F F     million yen. Problem 2.23. The total profit is 40,000 × (0.9120 – 0.8830) = $1,160 If you are a hedger this is all taxed in 2014. If you are a speculator 40,000 × (0.9120 – 0.8880) = $960 is taxed in 2013 and 40,000 × (0.8880 – 0.8830) = $200 is taxed in 2014.3 Problem 3.12. Suppose that in Example 3.4 the company decides to use a hedge ratio of 0.8. How does the decision affect the way in which the hedge is implemented and the result? If the hedge ratio is 0.8, the company takes a long position in 16 December oil futures contracts on June 8 when the futures price is $8. It closes out its position on November 10. The spot price and futures price at this time are $95 and $92. The gain on the futures position is (92 − 88)×16,000 = $64,000 The effective cost of the oil is therefore 20,000×95 − 64,000 = $1,836,000 or $91.80 per barrel. (This compares with $91.00 per barrel when the company is fully hedged.) Problem 3.16. The optimal hedge ratio is 120 7 0 614    The beef producer requires a long position in 200000 0 6 120 000    lbs of cattle. The beef producer should therefore take a long position in 3 December contracts closing out the position on November 15. Problem 3.18. A short position in 50 000 301 3 2650 1 500   contracts is required. It will be profitable if the stock outperforms the market in the sense that its return is greater than that predicted by the capital asset pricing model. Problem 4.10. The equivalent rate of interest with quarterly compounding isRwhere 40 1214Re or 0 034( 1) 0 1218Re    The amount of interest paid each quarter is therefore: 0 121810 000 304 554    or $304.55.4 Problem 4.11. The bond pays $2 in 6, 12, 18, and 24 months, and $102 in 30 months. The cash price is 0 04 0 5 0 042 1 0 0 044 1 5 0 046 2 0 048 2 52 2 2 2 102 98 04e e e e e                        Problem 4.14. The forward rates with continuous compounding are as follows: to Year 2: 4.0% Year 3: 5.1% Year 4: 5.7% Year 5: 5.7% Problem 5.9. A one-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding. a) What are the forward price and the initial value of the forward contract? b) Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. What are the forward price and the value of the forward contract? a) The forward price,0F, is given by equation (5.1) as: 0 1 1040 44 21Fe   or $44.21. The initial value of the forward contract is zero. b) The delivery price K in the contract is $44.21. The value of the contract, f, after six months is given by equation (5.5) as: 0 1 0 545 44 21fe      2 95 i.e., it is $2.95. The forward price is: …