Question

Suppose the S&P 500 currently has a level of $1,000. You wish to hedge a

$5,000.000 portfolio that has a beta of 1.5 with the S&P 500. In order to hedge the

portfolio, a 4-month futures contract is used over the next 3 months. The current

price of the future contract is $1,010. The index also pays a dividend yield of 1%

per annum, while the continuously compounded return on a 1-year T-bill is 4%.

(a) How many S&P 500 futures contracts should you short to hedge your

portfolio? Assume each futures contract is for delivery of $250 times the

index.

(b) Suppose now that in 3 months the value of S&P 500 index drops to $900,

while the futures contracts price drops to $902.

(i) What is the gain from shorting the futures contracts?

(ii) What is the expected value of the portfolio at the end of the

3 months?

(iii) What is the expected value of your overall position including the

gain from the hedge? Briefly, discuss the actual return generated

from the overall hedged position.

Answer 1

We have to Hedge a Portfolio Value of  $5,000, 000

Now beta of the Portfolio = 1.5

So Equivalent Index Value of the portfolio =  Portfolio Value * beta = $5,000, 000 * 1.5 = 7500,000

S&p each futures contract is for delivery of $250 times the

So No of Futures contract Required = Equivalent Index Value of the portfolio / ( S&P Index price * ContractSize)

= 7500,000 / ( 250 * 100) = 30

Ans a : 300 S&P 500 futures contracts should you short to hedge the portfolio.

(b) Suppose now that in 3 months the value of S&P 500 index drops to $900,

while the futures contracts price drops to $902.

Gain from Shorting the Future contract = ( Sell Price - Buy Price ) * No of contracts * Contracts Size   

= ( 1010 - 902 ) * 250 * 300 = 810,000

B (I) Gain from Shorting the Future contract  $ 810,000

Price drop in Index = 1000 - 900 = 100

Percentage Drop in Index price = 100 / 1000 = 10%

ExpectedPercentage Drop in Portfolio = Portfolio Beta * Index Price Change = 1.5 * 10% = 15%

expected value of the portfolio at the end of the 3 months = (1 - ExpectedPercentage Drop in Portfolio) * Portfolio Value

= ( 1 - 15% ) *  5,000, 000 = 4250,000

B (II) expected value of the portfolio at the end of the 3 months 4250,000

(iii)

the expected value of your overall position, There will be loss in Portfolio But will be gain in Shorting the future contract.

Return Generated from the Overall Hedge = Profit from the Hedge - Loss from the Portfolio

= 810,000 - Expected loss percentage in Portfolio * Portfolio Value

= 810,000 - 15% * 5000,000 = 60,00