A twelve-month short forward contract on a dividend-paying stock is entered into when the stock price is 133 and this stock is expected to pay a dividend of 5 in seven months. In addition, the risk-free rate of interest is 7.13% p.a. with semi-annual compounding for all maturities. Answer the following questions: (1) What are the forward price and the initial value of this forward contract? (2) Nine months later, the price of the stock is $152 and the risk-free rate of interest remains the same. What are the forward price and the value of the forward contract?
Question
A twelve-month short forward contract on a dividend-paying stock is entered into when the stock price is 133 and this stock is expected to pay a dividend of 5 in seven months. In addition, the risk-free rate of interest is 7.13% p.a. with semi-annual compounding for all maturities. Answer the following questions: (1) What are the forward price and the initial value of this forward contract? (2) Nine months later, the price of the stock is $152 and the risk-free rate of interest remains the same. What are the forward price and the value of the forward contract?
Solution
(1) The forward price of a dividend-paying stock can be calculated using the formula:
F = Se^(rT) - Ie^(rT)
where: F is the forward price, S is the spot price, r is the risk-free rate, T is the time to maturity, and I is the present value of dividends.
First, we need to calculate the present value of the dividend. The dividend is paid in 7 months, so the present value is:
I = 5 / (1 + (0.0713/2))^(7/6) = 4.81
Then, we can calculate the forward price:
F = 133e^(0.07131) - 4.81e^(0.07131) = 138.91
The initial value of a forward contract is always zero.
(2) Nine months later, the forward contract has three months left until maturity. The present value of the dividend is now zero, because it has already been paid. The forward price is:
F = 152e^(0.0713*0.25) = 155.26
The value of the forward contract is the difference between the forward price and the spot price, discounted back to the present:
V = (F - S) / (1 + r)^T = (155.26 - 152) / (1 + 0.0713)^0.25 = 3.07
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