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I'm working through Options, Futures and Other Derivatives (beginner trying to understand investment banking). I've more or less followed the discussion of interest rates, forward rates and forward rate agreements.

However, I'm struggling to understand why the forward rate ends up where it is. I've read that one way to think about it is to do with arbitrage opportunities, and indeed one of the questions at the end of that chapter sets up the following question:

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

I've figured out that the 9-month forward rate is 8%. But now I'm stuck. What actions could the bank take to arbitrage in this situation?

EDIT: I know the answer is something like borrow money for 9 months at 6%, and lend it out for 6 months, and then invest it and buy an FRA at 7 or something, but I just can't figure out the steps.

EDIT: @Student T - thanks for your answers. I'm trying to figure out the math to see how it would work

Borrow let's say $100 for 9 months at 6% fixed:

Cost to me is FV = PV * e^Rcn = 100*e^(0.06 *.75) = 104.60 ie I pay 4.60

Lend out $100 for 6 months at 5% fixed:

100*e^(0.05 * .5) = 102.53

Then invest that 102.53 at LIBOR floating rate, I think?

If I have sold the FRA at the start too, for a fixed rate on 102.53, then at the end I stand to get 102.53*e^(0.07 * 0.25) = 104.34, so I gain 4.34

But if these figures are right, that means that I paid 4.60 borrowing costs and only made 4.34 on my investments. Did I go wrong somewhere? Or is the correct answer that there are no arbitrage opportunities for the bank? Would it have helped if I bought the FRA rather than sold it?

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  • $\begingroup$ Thanks. I understand how to calculate the forward rate. My question is what actions can the bank take to profit from this arbitrage situation? $\endgroup$ – user384842 Jan 24 '15 at 16:22
  • $\begingroup$ Thanks, but if the payoff is the same, I wouldn't have thought that's an arbitrage opportunity? $\endgroup$ – user384842 Jan 24 '15 at 16:24
  • $\begingroup$ At t=0, borrow $1 to 9m, lend that dollar to 6m. The investment is zero. At t=6, you get the loan back. Invest that loan at the FRA rate to 9m. At 9m, you get the investment back but you also need to pay off you loan from t=0. Any difference is arbitrage. $\endgroup$ – SmallChess Jan 24 '15 at 16:51
  • $\begingroup$ This setup critically depends on the FRA rate from 6m to 9m, which is by definition the forward rate. $\endgroup$ – SmallChess Jan 24 '15 at 16:53
  • $\begingroup$ Thanks for your reply. When I tried it that way round, I couldn't guarantee a profit. But doing it the other way round - lending for 9 months and buying the FRA (I think!!) gets me a guaranteed profit. Not 100% sure, but I don't see a flaw in it. $\endgroup$ – user384842 Jan 24 '15 at 20:56
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Well, I think I have the answer to my own question.

T0: I lend $100 for 9mo, I get the 6% rate, which makes me 100*e^(.06*.75) = 4.60

T0: I borrow $100 for 6 months at 5%, which costs me 100*e^(.05*.5) = (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) = (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.

EDIT: I got it slightly wrong. You don't borrow $100 for 3 months, you borrow $102.53 to pay off the 6 month loan AND interest. Then you're left with $0.26

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  • $\begingroup$ If you borrow $100 at t0 for 6m, you'd have to pay back the loan at 6m. By then you wouldn't have any money for the long position in FRA. If you have a long position in FRA, it only makes sense that you receive money at t=6m so you can immediately reinvest the money into the FRA contract. $\endgroup$ – SmallChess Jan 28 '15 at 0:41

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