May 26

Interest rate swap as sequence of FRAs - Practice Question (Par 4 difficulty)

by David Harper, CFA, FRM, CIPM

It's tedious question, but it gives practice to a few ideas.

Assume a bank is the floating-rate payer in an interest rate swap with a $100 million notional. The counterparties exchange 5% fixed rate for LIBOR. The swap tenor is six months (i.e., time between swap settlements) and the swap has a remaining life of 15 months.

The LIBOR curve will be unchanged at the following continuously compounded spot rates: [3 months = 3.0%, 6 months = 3.5%, 9 months and = 4.0%, and 15 months = 5%]. That is, a straight line of LIBOR/swap zero rates.

Let's value the swap as a sequence of forward rate agreements (FRAs):

What are the two relevant forward rates, in continuously compounded terms? Specifically, the first forward rate is f(0.25,0.75) or FRA 3 x 9. The second forward rate is f(0.75, 1.25) or FRA 9 x 15.
Convert the forward rates into their semi-annual equivalent rates.
Compute the floating-rate swap payments as if the forward rates are realized (i.e., as if the LIBOR rates used to determine the floating-rate payments are indeed predicted by the forward rates)
Discount the net cash flows to value the swap.

(try before peeking!)

Answer:

The calculations are found in the EditGrid spreadsheet below.

What did we learn:

The 3 x 9 forward rate is inferred by the three and nine month spot rates.
We convert the continuous rate of 4.5% to a semiannual rate with = 2*[EXP(4.5%/2)-1] = 4.55%. Note: FRM candidate must be able to do this calc.
Then is a matter of present valuing the three net cash flows
Note the first floating rate is 3.5% paid in three months: it pays the six month rate, not the three month rate. (We only discount at three months)

EditGrid: