I’ve been trying to answer the following question from the text book Options, Futures and Other Derivatives.

“A bank can borrow or lend at LIBOR. Suppose that the six-month rate is 5% and the nine-month rate is 6%. The rate that can be locked in for the period between six months and nine months using an FRA is 7%. What arbitrage opportunities are open to the bank? All rates are continuously compounded.”

Would someone mind helping me confirm whether my answer is correct, please?

T0: I lend $100 for 9mo, I get the 6% rate, which makes me 100*e^(.06*.75) = 104.60 so profit of 4.60 T0: I borrow $100 for 6 months at 5%, which costs me 100*e^(.05*.5) = 102.53 so loss of 2.53 T0: I buy a 6mo/9mo FRA at 7% T6: Borrow $100 for 3 months at whatever the floating LIBOR rate is (I don’t care what it is because I’m guaranteed to pay 7% on it because of my FRA). Interest I need to pay on the loan is 100*e^(0.07*.25) = 101.76 so loss of 1.76 So I get $4.60 from lending out $100, and I only have costs of 2.53+1.76 = 4.29 Therefore I make a guaranteed $0.31 whatever the market does.

The implied forward rate is 8%. Therefore, you want the short position in the FRA: you want to pay 7% and receive the implied forward rate of 8%. And the way you earn 8% is to be long a 9-month bond at 6% and short a 6-month bond at 5%.

When you say you want to “buy” the FRA it sounds as though you mean you want the long (i.e., receive floating, pay fixed) position; if that’s what you mean by “buy”, then you’re correct in your transactions.

Your calculation of the payoff is incorrect; you should make $0.2547 per $100 (which will be discounted three months at the 3-month LIBOR rate extant when the FRA expires).

By the way, whoever wrote this question doesn’t understand LIBOR: LIBOR isn’t continuously compounded (indeed, LIBOR rates are nominal rates, not effective rates).

Thank you for your full answer - very much appreciated!

Yes, my terminology is probably off on the FRA. I mean that I am the buyer of an FRA, which I think is long, I mean the position where I pay if the reference rate is below the FRA rate. I think that is the equivalent of what you said - that I receive floating and pay fixed.

I’m not sure why I get $0.3059 and you get $0.2547. Is there anything glaringly off with my calculations? I’ve taken the rates to be continuously compounded. I’m not yet looking at day count conventions (if that’s relevant). I double-checked my calculations and I think the arithmetic is right at least.