Question

I currently own a portfolio of stocks worth $10 million that has a beta of 1.2.  It has perfect positive correlation with the S&P500 index.  The risk-free rate is 3%, and the market risk premium for the S&P500 is 8%.  The S&P500 is currently valued at 2500.  The notional value of one contract is $250*S&P value.

1. Calculate the one-year futures price on the S&P500 index.

2. If I want to own a risk-free bond instead of the index, explain how I can do this without selling my portfolio of stocks and purchasing bonds.  Calculate the number of futures contracts I need to achieve this and specify if the contracts are long or short.

3. Suppose that instead of wanting to own a risk-free bond, I instead want to own the Russell 3000 index.  Explain how I can do this without selling my portfolio of stocks that is correlated with the S&P500 and buying the Russell 3000 (HINT: futures are available on both the S&P500 and the Russell 3000).  No calculations are required – simply explain the necessary steps.

Answer 1

1. Futures price of S&P500:

F = S * (1 + r )t

F = futures price

S = spot price

r = risk free rate

t = time

(assuming there is no storage cost and no income yield from the underlying)

F = 2500 * ( 1 + 0.03)1 = 2575 = value od S&P after one year

notinal value of one futures contract = S&P * 250$ = 2575*250$ = $ 643,750

2. to answer this part lets understand some relationships:

As asked in the question we want to own a risk free bond. So we will look at the first relationship:

The payoff by holding the underlying asset ( portfolio of stocks) and selling the futures contract of the same will be exactly same as holding the risk free bond without having to purchase the same.

number of futures contract needed to be sold : beta * notional principal / future contact value

N = 1.2 * 10,000,000 / 643750 = 18.64077 or 19 futures contracts have to be sold

3. here the investor wants to replicate the russel index without selling the stock portfolio and without buying the russel index itself. For this investor should:

step 1: Long the underlying asset ie continue to hold the stock portfolio of 10 million dollars as it is

step 2 : Borrow funds at the risk free rate of 3%

step 3: Using the borrowed money buy the russel futures contract and not the russel index itself