Question

A fund manager has a portfolio worth $86 million with a beta of 1.20. The manager is concerned about the performance of the market over the next two months and plans to use three-month futures contracts on the S&P 500 to hedge the risk. The current level of the S&P 500 index is 1,190, the risk-free rate is 2% per annum, and the dividend yield on the index is 4% per annum. The current 3-month S&P 500 index futures price is 1,180, and one contract is on 250 times the index. (a) What position should the fund manager take to eliminate all risk exposure to the market over the next two months? (β*=0) How many three-month S&P 500 futures contracts should the fund manager buy or sell now? (Rounding to the nearest whole number.) (b) Calculate the effect of your strategy on the fund manager’s returns if the level of the S&P 500 index in two months is 1,100. Assume that the 3-month S&P 500 index futures price is 1,090 in two months. What would be the expected value of the fund manager’s hedged portfolio (including the spot and futures positions) in two months?

Please give me the process, thank you!

Answer 1

a) The number of contracts the fund manager should short is

= 0.87 * 86,000,000 / (1180 * 250) = 253.63

Rounding to the nearest whole number, 254 contracts should be shorted.

b) The gain on the short futures position is = (1180 - 1090) * 250 * 254 = 5,715,000

The return on the index is = 4 % * 2 /12 = 0.67% in the form of dividend and =1190 - 11000 / 1190 = - 7.56% in the form of capital gains.

The total return on the index is therefore - 6.9 %.

The risk-free rate is 2% per annum i.e.0.33% for two months.

The return is therefore = -6.9% - 0.33% = - 7.23 % in excess of the risk-free rate.

From the capital asset pricing model we expect the return on the portfolio to be = 1.2 * (- 7.23%) = -8.68% in excess of the risk-free rate.

The portfolio return is therefore -8.68% + 0.33% = - 8.34%.

The loss on the portfolio is or $86,000,000 * (- 8.34%) = -7,174,375.

When this is combined with the gain on the futures the total value of portfolio is = 5,715,000 - 7,174,375 + 86,000,000 = 84,540,625 $