Question

A fund manager has a portfolio worth $200 million with a beta against the S&P 500 of 1.2. The manager is concerned about the performance of the market over the next 2 months and plans to use 3-month futures contracts on the S&P 500 to hedge the risk. The current 3-month futures price is 2500 and one contract is written on 250 times the index. The risk free rate is 4% per annum and the dividend yield on the index is 2% per annum. The spot S&P 500 index is 2480. Given the information

Calculate the effect of the hedging strategy on the fund manager’s returns if the index spot price in 2 months is 2000, 2350 and 2700. Assume that the then 1-month S&P 500 futures price is equal to 1.005 times the corresponding index spot price at this time. Assume that the expected rate of return on the unhedged portfolio is given by the Capital Asset Pricing Model.

Answer 1

As a first step we need to find the hedging strategy. Since, fund manager has long position in the portfolio, he must short the futures contract to get an offsetting position.

The number of contracts the fund manager should short = (Portfolio value x Beta of the portfolio) / (Current future price x multiplier) = (200,000,000 x 1.2) / (2,500 x 250) = 320
     
Dividend yield in 2 months time = 2% per annum prorated for 2 months = 2% x 2 / 12 = 0.3333%

Please see the table below. Please be guided by the second column titled “Linkage” to understand the mathematics. The cells highlighted in yellow contain your answer. Figures in parenthesis, if any, mean negative values. All financials are in $.

Situation	Linkage	1	2	3
Current index level	A	2,480.00	2,480.00	2,480.00
Expected index level 2 months after	B	2,000.00	2,350.00	2,700.00
Return on index in 2 months	C = B/A - 1	-19.35%	-5.24%	8.87%
[+] Dividend yield in 2 months	D = 2% x 2 / 12	0.33%	0.33%	0.33%
Total return on index	E = C + D	-19.02%	-4.91%	9.20%
1-month S&P 500 futures price	F = 1.005 x B	2,010.00	2,361.75	2,713.50
The gain on the short futures position	G=(2,500-F) x 320 x 250	39,200,000.00	11,060,000.00	-17,080,000.00
CAPM return	H = 4% + 1.2 x (E - 4%)	-23.63%	-6.69%	10.25%
Gain / (Loss) on the portfolio	I = 200,000,000 x H	(47,251,612.90)	(13,380,645.16)	20,490,322.58
Net gain	J = G + I	(8,051,612.90)	(2,320,645.16)	3,410,322.58