Question

A stock is expected to pay a dividend of$1 per share in 3-month and in 9-month. The stock price is$80. An investor has just taken a long position in a 12-month futures contract on the stock. The zero ratesfor 3-, 6-, 9-, 12-month are 5%, 5.5%, 6% and 6.5% per annum with continuous compounding, respectively.6-month later, the stock price is$82. The zero rates happen to be the implied forward rates 6-month ago.What is the value of the long position in the futures contract?

Answer 1

Given Information

Current Stock Price (S₀) =$80

Maturity of futures contract = 12 months

Dividend of $1 per share in 3 and 9 months

Continuously compounded zero rates are

r₃ = 5%            r₆ = 5.5%         r₉= 6%             r₁₂=6.5%

Now, Futures Contract price for long position at time=0 is given by

F₀ = (S₀ – PVD) e^r12

PVD (Present Value of Dividends) = 1* e ^(-r3*3/12) + 1 * e ^(-r9*9/12) =1* e ^(-0.05*3/12) + 1 * e ^(-0.06*9/12)

So, PVD = $ 1.944

So F₀ = (80 – 1.944) * e^0.065 = $ 83.30

Now, after 6 months

S₆ = $82

Zero rates for next 3 months & 6 months happen to be implied forward rates six months ago.

Calculation of implied forward rate for months 6 to 9 (f_6to9)

e^(r9*9/12) = e^(r6*6/12) * e^(f6to9*3/12)

=> f_6to9 = ln [e^(r9*9/12) / e^(r6*6/12) ] * (12/3) = ln [e^(0.06*9/12) / e^(0.055*6/12) ] * (12/3)

=> f_6to9 = 7.00%

Similarly, Calculation of implied forward rate for months 9 to 12 (f_9to12)

e^(r12*12/12) = e^(r9*9/12) * e^{(f9to12*3/12)}

=> f_9to12 = ln [e^(r12*12/12) / e^(r9*9/12) ] * (12/3) = ln [e^{(0.065*12/12)} / e^(0.06*9/12) ] * (12/3)

=> f_9to12 = 8.00%

Calculation of futures price after 6 months

F₆ = (S₆ – PVD) * e^(f6to9*3/12) * e^{(f9to12*3/12)}

After 6 months, there is only one dividend left of $1 to be paid after six months

So, PVD = 1 * e^(f6to9*3/12) = 1 * e^(0.07*3/12) = $0.983

Hence, F₆ = (S₆ – PVD) * e^(f6to9*3/12) * e^{(f9to12*3/12)} = (82 – 0.983) * e^(0.07*3/12) * e^(0.08*3/12)

=> F₆ = $84.11

Value of Futures contract to long position = Current Futures price – Last Futures price = F₆ – F₀

=> Value of Futures contract to long position = 84.11 – 83.30 = $0.81 (Answer)