Assignment 4 - Answers

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Victoria University *

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3180

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Finance

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Feb 20, 2024

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Assignment 4 Problem 5.15. The current price of silver is $30 per ounce. The storage costs are $0.48 per ounce per year payable quarterly in advance. Assuming that interest rates are 10% per annum for all maturities, calculate the futures price of silver for delivery in nine months. Answer: $0.48 ÷ 4 = $0.12 storage cost for every quarter The present value of the storage costs for nine months: 0.12 + 0.12 e −0.1×0.25 + 0.12 e −0.1×0.5 = $0.351 The futures price: F 0 = (30 + 0.351) e 0.1×0.75 = $32.72 --------------------------------------------------------------------------------------------------------------------- Problem 5.24. What is the cost of carry for (a) a non-dividend-paying stock, (b) a stock index, (c) a commodity with storage costs, and (d) a foreign currency? Answer: a) the risk-free rate c = r b) the excess of the risk-free rate over the dividend yield c = r – q ( q is the dividend yield) c) the risk-free rate plus the storage cost minus the asset income c = r + g - m ( g is the storage cost, m is the income earned on the asset) d) the excess of the domestic risk-free rate over the foreign risk-free rate c = r - r f --------------------------------------------------------------------------------------------------------------------- Problem 5.25. In early 2012, the spot exchange rate between the Swiss Franc and U.S. dollar was 1.0404 ($ per franc). Interest rates in the U.S. and Switzerland were 0.25% and 0% per annum, respectively, with continuous compounding. The three-month forward exchange rate was1.0300 ($ per franc). What arbitrage strategy was possible? How does your answer change if the forward exchange rate is 1.0500 ($ per franc). Answer: The theoretical forward exchange rate 1.0404 e 0.0025×0.25 = 1.041
If the actual forward exchange rate is 1.03 , then the arbitrage strategy would be: a) borrow Swiss francs b) convert the Swiss francs to 1.0404 dollars and invest the dollars for three months @ 0.25% c) buy Swiss francs at 1.03 in the three-month forward market In three months, the arbitrage would make 1.0404e 0.0025×0.25 = 1.041 A total of 1.3 dollars needed to buy the Swiss francs and the total gain is 0.011 If the actual forward exchange rate is 1.05 , then arbitrage strategy would be: a) borrow dollars b) convert the dollars to 1.0404 Swiss francs and invest for three months @ 0% c) enter into a forward contract to sell 1.0404 Swiss francs in three months In three months, the arbitrage would make 1.0404 Swiss francs The forward contract converts these to (1.05)/1.0404 = $1.0092 A total of e 0.0025×0.25 = 1.0006 is needed to pay off the loan The total profit would be $0.0086 --------------------------------------------------------------------------------------------------------------------- Problem 5.26. An index is 1,200. The three-month risk-free rate is 3% per annum and the dividend yield over the next three months is 1.2% per annum. The six-month risk-free rate is 3.5% per annum and the dividend yield over the next six months is 1% per annum. Estimate the futures price of the index for three-month and six-month contracts. All interest rates and dividend yields are continuously compounded. Answer: The futures price for the 3-month contract: 1200 e (0.03-0.012)×0.25 = $1205.41 The futures price for the 6-month contract: 1200 e (0.035-0.01)×0.5 = $1215.09 ---------------------------------------------------------------------------------------------------------------------
Problem 5.29. A stock is expected to pay a dividend of $1 per share in two months and in five months. The stock price is $50, and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has just taken a short position in a six-month forward contract on the stock. a) What are the forward price and the initial value of the forward contract? b) Three months later, the price of the stock is $48, and the risk-free rate of interest is still 8% per annum. What are the forward price and the value of the short position in the forward contract? Answer: a) The present value of the income is: I = e −0.08×2/12 + e −0.08×5/12 = 1.9540 The forward price is: F 0 = (50 – 1.9540) e 0.08×0.5 = $50.1 The initial value of the forward contract is zero b) The present value of the income in three months is: I = e −0.08×2/12 = 0.9868 The value of the short forward contract is: ƒ = – (48 – 0.9868 – 50.01 e −0.08×3/12 ) = $2.01 The forward price is: F 0 = (48 – 0.9868) e 0.08×3/12 = $47.96
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