Question

A fund manager has a portfolio worth $50 million with a beta of 0.80. 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 1250, the risk-free rate is 6% per annum, and the dividend yield on the index is 3% per annum. The current 3-month S&P 500 index futures price is 1259, 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 market index in two months is 1,000. Assume that the one-month futures price is 0.25% higher than the market index level 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?

Answer 1

Solution to Problem 2(a)

The Portfolio Hedging Program is a managed hedge program which uses stock index futures and/or options to transfer price risk associated with long equity positions. Instead of selling individuals stocks investors can create a substitute sale through a short position in index futures and/or options. To participate in such a program, determinations would need to be made as to how an investor‟s stock portfolio correlates to a particular index. Once the “beta”, correlation of an individual stock or group of stocks to an index, is calculated, the proper hedge ratio can be determined as under:

Number of contracts to hedge = [Weighted beta of portfolio(β) x $ Value of portfolio(P)] divided by $ Value of index

No. of contracts = 0.80 x $50,000,000 / $1,250 x 250 = 128

The investor should short 128 contracts to eliminate exposure to the market.

Solution to Problem 2(b)

If the index in two month is 1,000 the 1-month future price which is 0.25% higher than the index level at this time can be calculated as under:

Futures price = 1,000 * 1.0025 = 1,002.50

Thus the gain on the short futures position = (1,259 − 1,002.50) × 250 × 128 = $8,208,000

The return on the index for the index value of 1000 can be calculated as follows:

Return in the form of Dividend (3/6)      0.50%

Return in the form of Capital Gains (1000-1250)/1250    -20.00%

Total Returns -19.50%

Risk free rate for two months (6%*2/12) 1.00%

Returns in excess of risk free rate -20.50%

Expected Return on portfolio in excess of risk free rate from CAPM

(0.87*-20.5%) -16.4%

Portfolio Return -15.4%

The loss on the portfolio is 0.154 ×$50,000,000 or $7,700,000. When this is combined with the gain on the futures the total gain is $508,000. The calculations for the return on index at other index levels are the same.

( $8,208,000 - $7,700,000 = $508,000 )