Forward contract

Question

A stock is expected to pay a dividend of $1 per share in two months and in five months. The stock price is $80, and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has just taken a short position in an eight-month forward contract on the stock. Three 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

i) Forward price at the start of short position

Forward price is calculated by compounding the net purchase price. The net purchase price is calculated is the difference between stock price and present value (PV) of dividends.

The present value of an income stream with compounding is given by: P = Ie-rT (Brealey et al., 2010)

Where, r is the risk-free rate and T is time left to receive payment. The risk-free rate, r, is 8%.

PV of $1 dividend due in 2 months = 1e-0.08(2/12) = 0.9868 PV of $1 dividend due in 5 months = 1e-0.08(5/12) = 0.9672

Total PV of two dividends = 0.9868 + 0.9672 = 1.9540

Forward price, F0, is calculated as below (Constantinides et al., 2013): 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 – 1.954)e0.08(8/12) = $82.3215

ii) Three months later Value of the short position After three months, only one dividend of $1 is due in two months.

Present value of $1 dividend due in 2 months = 1e-0.08(2/12) = 0.9868

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 = 82.3215e-0.08(5/12) = $79.6226

Present value of investment to acquire a share = Current share price – Present value of dividend = 78 – 0.9868

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) = 79.6226 – (78 – 0.9868) = $2.6094

Forward price at the end of three months, F1, with five months to expiry is calculated below:

F1 = (78 – 0.9868)e0.08(5/12) = $79.6236

References

Brealey, R.A., Myers, S.C., and Allen, F. (2010). Principles of corporate finance. 11th edn, New York: McGraw-Hill Irwin.

Constantinides, G.M. (2013). Financial derivatives: Futures, forwards, swaps, options, corporate securities, and credit default swaps. Singapore: World Scientific Publishing Co.