Arbitrage opportunities on an FRA?

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.

Many thanks!

I cover the arbitrage transactions in the article I wrote on pricing FRAs:

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%.

Thank you for the resource.

Does my answer look correct?

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.

Thanks again for your help!

The implied forward rate is 8%.

8% compounded continuously for 3 months on $100 is $2.0201.

7% compounded continuously for 3 months on $100 is $1.7654.

The difference is $0.2547.

The problem with your calculation is that the $4.60, $2.53, and $1.76 don’t all occur at the same time.

Thank you! I will take a look and try to see where I am going wrong. Thanks again for the time you have given to this. Very much appreciated…

My pleasure.