Question

A hedge fund manager has a portfolio worth $50 million with a beta of 1.25. The manager is concerned about the performance of the market over the next two months He plans to uses three-month stock market index futures to hedge his market exposure. The current stock market index level is 2,500 and one contract is on 250 times the futures price. The continuously compounded interest rate is 3% and the dividend yield on the stock market index is 2%.

a) What is the fair futures (forward) price today?

b) What position should the fund manager take to hedge his market exposure?

c) Calculate the effect of the hedging strategy on the fund manager’s return if the index in two months is 2,250, 2,500, or 2,750. (Hint: For each scenario, compute the fair forward price with one month maturity left and same interest rate and dividend yield. Then, you can compute the total profits or losses he incurs on his hedging position.)

Answer 1

a]

Futures price today = index price * e^rt,

where r = continuously compounded interest rate - dividend yield

t = time to expiration of contract in years

Futures price today = 2500 * e^{(0.03-0.02)*(3/12)} = 2506.26

b]

The manager should take a short position as they wish to hedge against a fall in the stock market

Number of futures to sell = (portfolio value * portfolio beta) / (futures price * contract size)

Number of futures to sell = ($50 million * 1.25) / ($2506.26 * 250) = 99.75. As fractional futures contracts cannot be sold, we round this off to 99

c]

If the index in 2 months is 2250

Futures price = 2250 * e^{(0.03-0.02)*(1/12)} = 2251.88

Profit on futures = (beginning futures price - ending futures price) * number of contracts sold * contract size

Profit on futures = (2506.26 - 2251.88) * 99 * 250 = $6,295,955

Loss on portfolio = ((2500 - 2250) / 2500) * beta = 10% * 1.25 = 12.50%

Loss on portfolio (in $) = $50 million * 12.50% = $6.25 million

Net loss = Loss on portfolio - Profit on futures = $6.25 million - $6,295,955 = $45,955

If the index in 2 months is 2500

Futures price = 2500 * e^{(0.03-0.02)*(1/12)} = 2502.08

Profit on futures = (beginning futures price - ending futures price) * number of contracts sold * contract size

Profit on futures = (2506.26 - 2502.08) * 99 * 250 = $103,351

Loss on portfolio = zero (since the index has remained unchanged at 2500)

Net profit = Profit on futures = $103,351

If the index in 2 months is 2750

Futures price = 2750 * e^{(0.03-0.02)*(1/12)} = 2752.29

Loss on futures = (ending futures price - beginning futures price) * number of contracts sold * contract size

Loss on futures = (2752.29 - 2506.26) * 99 * 250 = $6,089,307.39

Profit on portfolio = ((2750 - 2500) / 2500) * beta = 10% * 1.25 = 12.50%

Profit on portfolio (in $) = $50 million * 12.50% = $6.25 million

Net profit = Profit on portfolio - loss on futures = $6.25 million - $6,089,307 = $160,693