Question

A stock is expected to pay a dividend of $1.5 per share in three months and in six months. The stock price is $80, and the risk-free rate of interest is 5% per annum with continuous compounding for all maturities. An investor has just taken a long position in a eight-month forward contract on the stock.

(a) What are the forward price and the initial value of the forward contract?

(b) Four months later, the price of the stock is $82 and the risk-free rate of interest is still 5% per annum. What are the forward price and the value of the long position in the forward contract?

Answer 1

a)

Present value of first dividend = 1.5 / e^{^(0.05 * 3/12)}

Present value of first dividend = 1.5 / e^{^(0.0125)}

Note e = 2.71828

Present value of first dividend = 1.5 / 2.71828^{^0.0125}

Present value of first dividend = 1.5 / 1.012578

Present value of first dividend = 1.481367

Present value of second dividend = 1.5 / e^{^(0.05 * 6/12)}

Present value of second dividend = 1.5 / 1.025315

Present value of second dividend = 1.462965

Total present value of dividends = 1.481367 + 1.462965 = 2.944332

Forward price = (Stock price - dividends)e^{^(r * t)}

Forward price = (80 - 2.944332)e^{^(0.05 * 8/12)}

Forward price = 77.055668e^0.03333

Forward price = $79.68

Initial value of the forward contract at the beginning will always be zero.

b)

Present value of second dividend = 1.5 / e^{^(0.05 * 2/12)}

Present value of second dividend = 1.5 / 1.008368

Present value of second dividend = 1.4876

Forward price = (82 - 1.4876)e^{^(0.05 * 4/12)}

Forward price = 80.5124e^{^0.016667}

Forward price = $81.87

Value of long position = (81.87 - 79.68) / e^{^(0.05 * 4/12)}

Value of long position = (2.19) / 1.016806

Value of long position = $2.15