The following table gives data on monthly changes in the spot price and the futures price for a certain commodity. Use the data to calculate the optimal hedge ratio.

Spot Price Change

+0.50

+0.61

-0.22

-0.35

+0.79

Futures Price Change

+0.56

+0.63

-0.12

-0.44

+0.60

Spot Price Change

+0.04

+0.15

+0.70

-0.51

-0.41

Futures Price Change

-0.06

+0.01

+0.80

-0.56

-0.46

Hints: 1) for a sample of N observations, x1, x2, ..., xN, the formula for the sample standard deviation is:

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2) for two samples of N observations, x1, x2, ..., xN; y1, y2, ..., yN, the formula for the correlation coefficient is:

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FINA 471 Liang Ma Problem Set 1 This problem set covers materials in Chapters 1-4 of Hull 2017 Due on September 21, 2017 at the beginning of the class Each group should submit one hard copy of the solution. The names (both the first and last names) of each student in the group should be written clearly on the first page of the hard copy. Clarity will be rewarded and disorganization will be penalized. Show your detailed work! 1. On Aug 26, you sold 10 gold futures contracts at a price of $1298/oz. Each contract represents gold 100 oz. The initial margin is USD 5,000 per contract, and the maintenance margin is USD 4,000 per contract. You deposited the initial margin on Aug 26. The subsequent settlement prices are shown in the table below: Aug 26 1297 Aug 27 1284 Aug 28 1283 Aug 29 1304 Aug 30 1307 Aug 31 1315 1) Compute the daily loss/gain, and cumulative loss/gain for each date. 2) Suppose you did not withdraw any money from your account during this period, when would you receive a margin call and how much would you have to deposit to meet the margin call? 2. You manage a $13.5 million portfolio, currently all invested in equities, and has a beta of 1.2. You believe that the market is on the verge of a big but short-lived downturn; you would move your portfolio temporarily into T-bills, but you do not want to incur the transaction costs of liquidating and reestablishing your equity position. Instead, you decide to temporarily hedge your equity holding with S&P 500 index futures contracts. 1) Should you be long or short the contracts? Why? 2) How many contracts should you enter into? The S&P 500 index futures price is now at 1286 and the contract multiplier is $250. 3) Suppose instead of reducing your portfolio beta all the way down to zero, you decide to reduce it to 0.5, how many index futures contracts should you enter into? 3. The following table gives data on monthly changes in the spot price and the futures price for a certain commodity. Use the data to calculate the optimal hedge ratio. FINA 471 Liang Ma Spot Price Change Futures Price Change 0 50 0 56 0 61 0 63 0 22 0 12 0 35 0 44 0 79 0 60 Spot Price Change Futures Price Change 0 04 0 06 0 15 0 01 0 70 0 80 0 51 0 56 0 41 0 46 Hints: 1) for a sample of N observations, x1, x2, ..., xN, the formula for the sample standard deviation is: 1 1 ( ) 2) for two samples of N observations, x1, x2, ..., xN; y1, y2, ..., yN, the formula for the correlation coefficient is: ( )( ) [ ( ) ][ ( ) ] 4. You enter into a five-to-eight-month forward rate agreement with a firm. You agree to lend the firm a 3-month loan of $5 million starting 5 months from now, with a quarterly compounded forward interest rate of 2.5% per annum. Currently, the continuously compounded 5-month and 8-month interest rates are 3% per annum and 3.5% per annum, respectively. 1) What is the implied forward rate for the 3-month period starting 5 months from now? 2) What is the present value of this forward rate agreement to you now