Question

You enter into a five-to-eight-month forward rate agreement with a firm. You agree to lend the firm a 3-month loan of $5 million starting 5 months from now, with a quarterly compounded forward interest rate of 2.5% per annum. Currently, the continuously compounded 5-month and 8-month interest rates are 3% per annum and 3.5% per annum, respectively.

1) What is the implied forward rate for the 3-month period starting 5 months from now?

2) What is the present value of this forward rate agreement to you now?

Answer 1

e^{8 month interest rate * (8 months / 12 months)} = e^{5 month interest rate * (5 months / 12 months)} * e^{Forward rate * (3 months / 12 months)}

e^{3.5% * (8 / 12)} = e^{3% * (5 / 12)} * e^{Forward rate * (3 / 12)}

e^{Forward rate * (3 / 12)} = e^{3.5% * (8 / 12)} / e^{3% * (5 / 12)}

e^{Forward rate * (3 / 12)} = 1.01089

Forward rate * (3 / 12) = ln(1.01089)

Forward rate * (3 / 12) = 0.01083

Forward rate = 0.01083 * (12 / 3)

Forward rate = 4.33%

Forward rate with quarterly compounding = Compounding frequency * (e^{(Forward rate / Compounding frequency)} - 1)

Forward rate with quarterly compounding = 4 * (e^{(4.33% / 4)} - 1)

Forward rate with quarterly compounding = 4.35%

Value of Forward Contract = Principal * (Lending rate - Forward rate) * ((8 months / 12 months) - (5 months / 12 months) * e ^{- 8 month interest rate * (8 months / 12 months)}

Value of Forward Contract = $5,000,000 * (2.5% - 4.35%) * ((8/12) - (5/12)) * e ^{- 3.5% * (8 / 12)}

Value of Forward Contract = - $22,591.66