A portfolio manager has an equity portfolio that is valued at $$75$ million. The

portfolio has a current beta of $9$ and a dividend yield of $1\%.$ It is currently August $15$

and the manager is concerned that markets are volatile and the portfolio could lose

value, so they decide to hedge.

a$.$ The manager will use the S&P $500$ index contracts to hedge. The contract is

settled in cash at $$250$ times the contract price. The current S&P index value is

$1484\mathrm{.}43$ and a December S&P $500$ index contract has a price of $1517\mathrm{.}20$

b$.$ Based on these expectations, should they take a short or long futures position

and why?

c$.$ An optimal number of contracts is $N^{**}=($Dollar value of the portfolio$/$dollar

value of one futures contract$)x$ portfolio beta.

d$.$ Based on $($c$)$ above, compute $N^{**}$ and set up the appropriate hedge.

e$.$ On December $15$ the position will be closed. The current S&P $500$ index is

$1410\mathrm{.}20$ and the current contract matures, so convergence takes place. Compute the

percentage loss in the S&P index and the percentage loss in the portfolio, which will

be $(\%$ loss in market $x$ portfolio beta$).$

f$.$ Compute the dollar loss on the portfolio, the dollar change in the futures

position, and any dividends earned on the portfolio $(3$ months$).$ Add these up to get

the total hedged portfolio value.

g$.$ How good was the hedge? Answer this by comparing the change in the market

value of the portfolio to the change in the futures position.