Question

The S&P 500 Index is currently at 1,200. You manage a \(6 million indexed equityportfolio. The S&P 500 futures contract has a multiplier of \)250.

a. If you are temporarily bearish on the stock market, how many contracts should yousell to fully eliminate your exposure over the next six months?

b. If T-bills pay 2% per six months and the semi-annual dividend yield is 1%, what is theparity value of the futures price? Show that if the contract is fairly priced, the totalrisk-free proceeds on the hedged strategy in part (a) provide a return equal to theT-bill rate.

c. How would your hedging strategy change if, instead of holding an indexed portfolio,you hold a portfolio of only one stock with a beta of .6? How many contracts wouldyou now choose to sell? Would your hedged position be riskless? What would be thebeta of the hedged position?

Short answer

a. 20 contracts;

b. Yes; .1

c. .6; 12; 0

Step-by-step solution

Given information

Portfolio value = $6,000,000

Each contract = $250 times the index valued

Index value = $1200

Calculation of sale of contracts‘a’

Since each contracts is for $250 times the index valued currently at $1200,

This implies each contract has market exposure of $1200 x $250 = $300,000

To hedge 6 million, no. of contracts required = 6,000,000 / 300000 = 20 contracts

Calculation of parity value of the future price ‘b’

F₀ = S₀(1 + r_f – d)^T

= 1,200 (1 + .02 - .01)²

= 1,212

Action	Initial cash flow	Cash flow at time T
Short 20 futures contract	0	20 x $250 x ($1,212 - ST)
Buy 5000 shares of index @$1200 per share	-6 million	(6 million x .01) + (5,000 x ST)
Total	-6 million	6.06 million (riskless)

Hence this equals to the hedged strategy equaling T-bill rate of 2% = (6.06/6 -1) =.1

Explanation on hedging strategy, contract sale hedged position ‘c’

The stock now swings only .6 as much as market index.

Hence no. of contracts needed = .6 x 20 = 12. These will not be riskless as these are exposed to the unsystematic risks.

Beta of the hedged position = 0 as all systematic risks are hedged