Question

A stock is expected to pay a dividend of $0.50 per share in two month, in five months and in eight
months. The stock price is $20, and the risk-free rate of interest is 5% per annum with continuous
compounding for all maturities. You have just taken a short position in a nine-month forward contract
on the stock. Seven months later, the price of the stock has become $23 and the risk-free rate of
interest is still 5% per annum. What is the value your position seven months later?

Answer 1

Computation of the 9 months forward price:

We know that F = ( Spot price - Dividend ) e^rt

First Compute the present value of the dividend.

We know that PV of Dividend = $ 0.5/(e^r*2/12) + $ 0.5/( e^r*5/12) + $ 0.5/(e^r*8/12)

= $ 0.5/(e^0.05*2/12) + $ 0.5/(e^0.05*5/12) + $ 0.5/(e^0.05*8/12) [ Refer calculation below]

= $ 0.5/1.00836884 + $ 0.5/1.0210544 + $ 0.5/1.033899

= 0.49585+0.48968+0.48360

= $1.46913

So the present value of the dividend is $ 1.46913.

Calculations

e ^ (0.05*2/12) = 2.7183^(0.008333) = 1.00836884

e^(0.05*5/12)= 2.7183^(0.0208333) = 1.0210544

e^(0.05*8/12) = .7183^(0.033333) = 1.033899

Forward price = ( Spot price - Dividend) e^0.05*9/12

= ( $ 20-$ 1.46913) e^0.0375

= $18.5308 e^0.0375

= $ 18.5308*1.038216

= $19.2389

Hence the forward price after 9 months is $ 19.2389

Given the stock price rises to $ 23 after seven months.

Forward price after 2 months from the 7th month.

Present value of the Dividend after 7 months = $ 0.5/ e^0.05*1/12

= $ 0.5/( e^0.00416)

= $ 0.5/1.004169

= $0.4979

Future price = ( $ 23-$ 0.4979)* e ^0.05*2/12

= $ 22.5021* e^0.0083333

= $ 22.5021*1.008368

= $ 22.6903

Note :When we take the short position, the price should decrease. But here the share price is increased.So we will be incurring loss on forward contract.

Value of the position after seven months later = ( intial forward price - $ 22.6903)/ e^0.05*2/12

= ( $19.2389-$ 22.6903) / e^0.05*2/12

=   -3.4514/1.008368

= -$ 3.42276.

Hence the value of the forward after seven months later is -$3.42276.

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