1. S&P 500 index has a dividend yield of 3% today. The current index is trading

Question

1. S&P 500 index has a dividend yield of 3% today. The current index is trading at $1800 and the risk-free interest rate is 5% per annum with continuous compounding. An investor would like to enter into a short one-year forward contract on the index.

A. What would be the fair forward price for this contract?

B. Suppose that six months later, S&P 500 index trades at $2000, the risk-free interest rate is 6% per annum and the annualized dividend yield increased to 4%. What would be the current value of the investor’s short position?

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Explanation / Answer

Forward price = (S0 )e^(r-y)T, where S0 is spot price, r is interest rate and T is time, hence e^rT is continous compounding growth rate. y is known income yield like dividend yield.

Hence Forwad price = ($1800)* e^(5%-3%)*1 = 1800*e^2% = $1836.3624 is the answer

B) Now first we calculate price after 6 months where Spot is $2000, Dividend yield is 4% and risk free rate is 6%. Time is 0.5 years

Forward price at 6 month = 2000*e^(6%-4%)*0.5 = 2020.1003

This price is for after 6 months, we need to have present value of the six month after price. We will use the current risk free rate to discount the price

= 2020/((1.05)^0.5) = 1970.2239

Hence value of short position today = $1836.3624 - 1970.2239 = -133.8615 is the second answer

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