Question

A stock is expected to pay a dividend of $1 per share in 2 months and in 5 months. the stock price is $50, and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An inverstor has just taken a short position in a 6-month forward contract on the stock.
a) what are the forward price and the initial value of the forward contract?
b) Three months later, the price of the stock is $48 and the risk-free rate interest is still 8% per annum. what are the forward price and the value of the short position in the forward contract?

Answer 1

Answer:

present value = $1.9539

forward price f1 = $50

present value = $0.9867

forward price f2 = $47.96

value of short position = $2

Explanation:

given data

pay a dividend = $1 per share

time = 2 months

time = 5 months

stock price = $50

rate of interest = 8%

solution

we get here present value that is

present value = principal × $e^{rt}$

so

present value =  $1 × $e^{-0.08*2/12}$ + $1 $e^{-0.08*5/12}$  

present value = $1.9539

and

forward price will be

forward price = ( stock price - present value ) × $e^{rt}$

forward price = ( 50 - 1.9539 ) × $e^{0.08*6/12}$

forward price f1 = $50

and

now we get present value of future dividend that is

present value = $1 × $e^{-0.08*2/12}$

present value = $0.9867

and

forward price is now as

forward price = ( $48 - 0.9867 ) × $e^{-0.08*3/12}$

forward price f2 = $47.96

and

value of short position is  = ( f1 - f2)  × $e^{-0.08*3/12}$

value of short position = ( 50 - 47.96 )  × $e^{-0.08*3/12}$

value of short position = $2