Question

A fund manager has a portfolio worth $100 million with a beta of 1.5. 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 index is 2250, one contract is on 250 times the index, the risk free rate is 2%, and the dividend yield on the index is 1.7% per year. (Assume all the rates are continuously compounded.)

a) What is the theoretical futures price for the three-month futures contract?

b) What position should the fund manger takes to eliminate all exposure to the market over the next two months?

c) Calculate the effect of your strategy on the fund manager’s returns if the level of the market in two months is 2,000, 2,200, 2,500, 2,800, and 3,000.

Answer 1

(a) Theoretical futures price for 3-month contract =

r is the risk-free rate = 0.02

q is the dividend yield = 0.017

T = 3/12 = 0.25

S(0) = 2250

Theoretical futures price for 3-month contract = 2250*e^((0.02-0.017)*0.25) =$2251.688

(b) No. of contracts to hedge = (Weighted Beta of portfolio) * Value of portfolio / Value of Index

No. of contracts to hedge = 1.5*100,000,000/(2250*250) = 266.67

Rounding off to 267

The investor should short 267 contracts to eliminate exposure to the market.

(c) The index price after 2 months is given

The futures price is the price = index price after 2 months*e^(0.02*1/12) = 1.001668*Index price after 2 months

Return in the form of dividend in 2 months = 1.7% *2/12 = 0.2833%

Total returns = Capital gains + Dividend gains

Risk-free rate for 2 months = 2%*2/12 = 0.3333%

Returns in excess of risk-free rate = Total returns - Risk-free rate for 2 months

Expected Return on the portfolio in excess of risk-free rate from CAPM = 1.5*Returns in excess of risk-free rate

Portfolio return = Expected Return on the portfolio in excess of risk-free rate from CAPM + Risk-free rate for 2 months

Total gain = Portfolio gain + Gain on futures

We use the above formulas in excel and calculate the returns for each level of Index levels in 2 months

Index in 2 months	2000	2200	2500	2800	3000
Futures price	2003.336	2203.6696	2504.17	2804.6704	3005.004
Gain on futures	16577496	3205228.2	-16853173.5	-36911575.2	-50283843
Index return	-216.5562222	-42.65629	284.8602778	692.3768444	1008.499
Excess Index return	-222.2222222	-48.88889	277.7777778	684.4444444	1000
Excess portfolio return	-333.3333333	-73.33333	416.6666667	1026.666667	1500
Portfolio return	-327.6673333	-67.10073	423.7491667	1034.599067	1508.499
Portfolio gain	-16383366.67	-3050033	16949966.67	36949966.67	50283300
Total gain	194129.3333	155194.87	96793.16667	38391.46667	-543