Question

3.         Question 3                                                                                                      [Total: 20 marks]

Assume a futures contract with 8 months to maturity is employed to hedge a well-diversified portfolio over the next 6 months.  The contract size of one futures contract is $250 times the index (i.e. the futures on the index). The value of the S&P500 index is currently 1200. The current futures price is $1220. The value of the portfolio that you are hedging is $ 40 million. The Beta of the portfolio is 1.25. The risk-free rate of interest is 1.5 % per year and the dividend yield is 0.5 % per year.  In 6 months, the index drops to 900 with the futures price becoming $ 907.

1. What is the number of futures contracts that you need to short?                        

[5 marks]

2. Calculate the gain or the loss on the futures position after 6 months.                 

[5 marks]

3. What is the expected value of the portfolio at the end of 6 months?                   

[5 marks]

4. What is the final value of the portfolio after the hedge has been applied?          

[5 marks]

Answer 1

Value of Portfolio VF = $40 million = $40,000,000

Current S&P Index = 1200

Current Futures Price F0= $1220

Index Multiplier = $250

= 1.25

Risk free rate = 1.5% per year

Dividend yield =0.5% per year

a) HEdge ratio = * VF / VA

VA = Current Futures Price * Index Multiplier = 305,000

Hedge ratio = 1.25*40,000,000 / 305,000 = 163.93

163.93 futures needs to be short

b) S&P index after 6 month = 900

Futures after 6 month = $907

Gain from Short Future position = H*(F0-F1)*Multiplier where F1 is futures price after 6 month

Gain from Short Future position = 163.93*(1220-907)*250 = $12,827,868.85

Loss on index = (900-1200)/1200 = -0.25 = - 25% in 6 month period

c) Dividend Yield per year = 0.5%

Dividend Yield = 0.5%/2 = 0.25% per six month

Return on index in 6 month = -25% + 0.25% = - 24.75%

Return from portfolio in 6 month = Rf + *(Rm - Rf) = 0.75% + 1.25*(-24.75% +0.75%) = - 31.125%

Expected value of portfolio = Portfolio value(1+Return in 6 month) = 40,000,000*(1-31.125%) = $27,550,000

Loss on Portfolio = initial portfolio value - portfolio value after 6 month = 40,000,000 - 27,550,000 = $12,450,000

d) Final Value = Expected value of portfolio in 6 month + gain from futures = 27,550,000 + $12,827,868.85 = $40,377,868.85 = $40.38 million