4.7. APPENDIX: THE FORWARD FORWARD AND THE FORWARD RATE AGREEMENT 163
replicate a currency-X spot deposit or loan by swapping a currency-Y spot deposit
or loan into currency X. Well, the same holds for forward deposits and loans. For
instance, in the few years whenusdorgbphadFRAmarkets but minor European
currencies had not (yet), pros replicated the missingFRA’s by swappingusdorgbp
FRA’s into, say,nlgvia a forward-forward currency swap, in or out. Such swaps are
described in Chapter 5, and consist of a currency forward in one direction combined
with a second currency forward, in the other direction. In short, when the starting
date of a deposit or loan is not spot butndays forward, we just replace the spot
leg of the swap by the appropriate forward leg.
4.7.4 Forward Interest Rates as the Core of the Term Structure(s)
Remember that forward exchange rates, being the risk-adjusted expectations, are
central in any theory of exchange rates. In the same way, forward interest rates can
be viewed as the core of every theory of interest rates. The standardexpectations the-
oryhypothesizes that forward interest rates are equal to expected future spot rates,
and Hicks added arisk premium, arguing—to use a post-Hicksian terminology—
that the beta risk of a bond is higher the longer its time to maturity. Modern
versions would rather state everything in terms ofpnprices rather than interest
rates, but would agree with the basic intuition of the old theories: forward rates
reflect expectations corrected for risks.
Various theories or models differ as to how expectations evolve and risk premia
are set, but once the forward rates are set, the entire term structure follows. We
illustrate this with a numerical example, and meanwhile initiate you to the various
interest-rate concepts: spot rates, yields at par for bullet bonds, and other yields at
par.
We start from the first row in Table 4.3, which shows a set of forward rates.
For simplicity of notation, current timetis taken to be zero, so that a one-period
forward rate looks liker 0 f,n− 1 ,nrather than the more laboriousrt,tf+n− 1 ,t+n. For some
reason—mainly expectations, one would presume—there is a strong “hump” in the
forward rates: they peak at the 3-to-4 year horizon. (A period is of unspecified
length, in the theories; but let’s agree they are years).^10 The initial spot rate and
the forward rate with starting date 0 are, of course, the same. Below we show you
the formulas to be used in a spreadsheet to generate all possible term structures
(TS).
The TS of spot ratesis obtained in two steps. First we cumulate the forward(^10) For this reason the only non-arbitrary theory is one that works with continuous time, where a
period lasts dtyears. But for intro courses this has obvious drawbacks.