May 30

Forward rates (FRM building block)

by David Harper, CFA, FRM, CIPM

FRM |

A couple of recent conversation threads in the forum highlight the importance of forward rates as an FRM building block.

Consider my unimaginative, unrealistic spot rate term structure: [2% @ 0.5 years, 2.5% @ 1.0 years, 3% @ 1.5 years, 3.5% at 2 years and 4% at 2.5 years]. Here is something to ponder until it sinks in: this term structure already contains, implicitly, a set of forward rates:

image

The numbers behind this chart:

forward_example

As usual, the compounding frequency matters a bit. Tuckman (who is FRM assigned) and Hull (who is not for this) use the same concept only the compounding frequency is different. Consider the six-month rate starting in six months. We have a few ways to denote this rate:

  • Forward 6 x 12 (borrowing FRA terminology; as in, the rate for a contract that expires in six months and six months later the interest is paid), or
  • F(0.5,1.0) = starting in six months (0.5), the six-month rate (from 0.5 to 1.0). Or, we can be really precise and use a FRA LIBOR notation...
  • F(0,180,180) = the six month rate in six months

Consider the minor procedural difference between Tuckman and Hull:

  • Tuckman (semiannual): F(0.5,1.0) = [(1+2.5%/2)^2/(1+2%/2)-1]*2 = 3.001% (you can't quite see it above)
  • Hull (continuous): F(0.5,1.0) = 2.5%+(2.5%-2%)*(0.5/(1.0-.5)) = 3.0%

Where it comes up

  • In Hull's valuation swaps, both a plain vanilla interest rate swap and a currency swap are valued two ways: as bonds, and a sequence of forward rate agreements (FRAa). It is worth examining these two approaches (I have the XLS uploaded to member page) until you see there is hardly, really a difference.
  • Tuckman has AIM which asks, "Discuss ... the differences maturity has on the return generated by bonds." The key to this AIM is understanding the forward rates above (details here)

Here is a key passage from Tuckman, worth mastering (emphasis mine):

It is no coincidence that when the six-month rate evolves according to the initial forward rate curve investors rolling short-term bonds and investors buying long-term bonds perform equally well. Recall that an investment in a bond is equivalent to a series of forward loans at rates given by the forward rate curve.

EditGrid spreadsheet

This spreadsheet applies my example above; note also the Bond Price is computed (light blue at end) and bond price is the same regardless of whether we use the spots or the forwards, but it differs slightly depending on the compound frequency.

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