Question
A stock has an expectation to pay a dividend of $1 per share in 2 months and $1.20 in five months. The price of the stock is $80, and the risk-free rate is 8% per year with continuous compounding for all maturities. An investor has taken a short position in an 8 month forward contract on the stock. 3 months later, the price of the stock is $78 and the risk-free rate of interest is still 8% per annum. What are the forward price and the value of the short position in the forward contract?
Answer
-> Forward price at the start of short position
The present value of an income stream with compounding is given by:
P = Ie-rT
Where, r is the risk-free rate and T is time left to receive payment. The risk-free rate, r, is 8%.
Present value of $1 dividend due in 2 months = 1e-0.08(2/12) = 0.9868
Present value of $1.20 dividend due in 5 months = 1.20e-0.08(5/12) = 1.1606
Total present value of two dividends = 0.9868 + 1.1606 = 2.1474
Forward price, F0, is calculated as below:
F0 = (Share price – Present value of dividends)erT
Share price is $80. The forward contract is for 8 months. Substituting values in the above equation:
F0 = (80 – 2.1474)e0.08(8/12) = $82.1175
-> Three months later
Value of the short position
After three months, only one dividend of $1.2 is due in two months.
Present value of $1.2 dividend due in 2 months = 1.2e-0.08(2/12) = 1.1841
Share price at the end of three months is $78.
Five months before the expiry, present value of F0 is calculated below:
Present value of F0 = 82.1175e-0.08(5/12) = $79.4253
Present value of investment to acquire a share = Current share price – Present value of dividend = 78 – 1.1841
Since the investor has a short position, value of the short forward contract, f, is given by the following calculation:
f = Present value of forward F0 – (Current share price – Present value of dividend) = 79.4253 – (78 – 1.1841) = $2.6094
Forward price at the end of three months, F1, with five months to expiry is calculated below:
F1 = (78 – 1.1841)e0.08(5/12) = $79.4196
