Question

An investor has just taken a short position in a one-year forward contract on a dividend paying stock. The stock is expected to pay a dividend of $2 per share in five months and in eleven months. The stock price is currently selling for $100 and the risk-free rate of interest is 8.50% per year with continuous compounding for all maturities.

a. What are the forward price and the initial value of the forward contract? The forward price is (sample answer: $75.50)

and the initial value is (sample answer: $75.50)

b. Six months later, the price of the stock is $105 and the risk-free rate stays the same. What are the forward price and the value of the position in the forward contract? Now the forward price is (sample answer: $75.50)

and the initial value is (sample answer: +$5.50; or -$5.50)

Answer 1

A) Spot Price = $100 so after one year its price will be

Forward price = Spot * (eRF)^T

eRF is continuous compunding risk free rate

T is time

=$100 * (1+ (e0.085)^12/12)

=$100 * 1.0887

=$108.87

Considering effect of Dividends in forward price

Dividend which is expected to receive in 5 months

Dividend = $2 *  (1+ (e0.085)^7/12)

=$2 * (1.08871)^(7/12)

=$2 * 1.0508

= $2.10

Dividend which is expected to receive in 11 months

Dividend = $2 *  (1+ (e0.085)^1/12)

=$2 * (1.08871)^(1/12)

=$2 * 1.0071

= $2.01

so final forward price after considering dividend

= $108.87 - $2.10 - $2.01

=$104.76

Initial Forward value for investor who has taken short position will be $104.76

B) Six months later Spot Price has increased to $105 so lets calculate its forward price at end of year

Spot Price = $105 so after 6 months its price will be

Forward price = Spot * (eRF)^T

eRF is continuous compunding risk free rate

T is time

=$105 * (1+ (e0.085)^6/12)

=$100 * 1.0434

=$109.55

Considering effect of Dividends in forward price

Dividend which is expected to receive in 11th months

Dividend = $2 *  (1+ (e0.085)^1/12)

=$2 * (1.08871)^(1/12)

=$2 * 1.0071

= $2.01

so final forward price after considering dividend

= $109.55 - $2.01

=$107.54

Calculation of value of forward position

= $107.54 - $104.76

= $2.78

Calculation of Present value of this $2.78

= $2.78 / ((eRF)^(T)

= $2.78 / (e0.085)^(6/12)

=$2.78 / (1.0887)^0.5

=$2.78 / 1.0434

=$2.66

As price of forward conract has increased so investor who has taken short position will have loss of $2.66