Question

Part 1. Suppose you need to hedge the risk in a stock portfolio that your company owns in it's pension plan using the E-mini S&P500 futures contracts (which has a multiplier of 50). The current value of the portfolio is $325 million and the S&P500 is currently at 2637.72 while the futures price (for next month delivery) is at 2643.25. You estimate that the "beta" of this portfolio is 1.1 while the beta of the S&P500 is 1. What position do you take in futures?

Part 2. In the above question suppose you want a tailed hedge. Now what position do you take in futures?

Part 3. In part 1 above suppose it is 1 month (1/12 of a year) until delivery of the futures and the 1-month interest rate is 3.5% per annum. Using the S&P500 index level and the futures price what must the dividend yield on the S&P500 be in order to preclude arbitrage?

Answer 1

Part (1)

The optimal hedge strategy in case of hedging will be to short "H" number of future contracts calculated as follows:

H = Value of the Portfolio / value of the future contracts = $ 325 mn / (Price of 1 index future x Multiplier) = 325,000,000 / (50 x 2,637.72) = 2,464.25, say 2,464 number of future contracts (rounded off to the nearest integer)

Hence, the hedge strategy will be to short 2,464 numbers of index future contracts.

Part (2)

Suppose we want to tail the hedge. In that case, optimal hedge strategy will be to take position in the fewer number of contracts to begin with and then increase the position as we move closer to the date of expiry of the future contracts.

Optimal hedge ratio = Hedge ratio derived in part (1) / Beta of the portfolio = 2,464 / 1.1 = 2,240

Thus, short 2,240 number of future contracts and increase the position to 2,464 as we move closer to the expiry date of futures.

Part (3)

Let's assume D is dividend yield. Hence under no arbitrage principle,

Price of the future at t = Price of Future today x (1 + (I - D) x t)

where I = interest rate per annum, D = Dividend yield per annum and t = time to expiry = 1 month = 1/12 year

Hence, 2,643.25 = 2,637.72 x [1 + (3.5% - D) x 1/12]

hence D = 3.5% - (2,643.25 / 2,637.72 - 1) x 12 = 0.9842%