Page 67 - Writing Excel Macros with VBA, 2nd Edition

46 Chapter 2

and r2y. The forward rate that connects these two rates, denoted by f1y,2y, is given by

(1 + r1y)(1 + f1y,2y) = (1 + r2y)2

Effectively, the forward rate ft,u for the period [t, u] is the interest rate for the money
invested between the dates t and u in the future, but agreed upon today.

The preceding is a theoretical construction, and rates obtained in this way are sometimes
called implied forward rates. In practice the actual market forward rates might be
different because of various reasons including transaction costs and only an approximate
knowledge of the term structure of interest rates.

Forward rates are similar to spot rates in their nature, and, therefore, the rules about the
computation of annual equivalents apply. For example, using the six-month annualized spot
rate r6m and the one-year spot rate r1y we get the annualized forward rate f6m,1y between
the sixth month and the year as

(1 + r6m )1/2(1 + f6m,1y )1/2 = 1 + r1y

In general, the forward rate ft,u for the period [t, u] with t and u expressed in years is
determined from this annualized forward rate formula with annual compounding:

(1 + rty)t = (1 + ruy)u(1 + ft,u)u−t (2.10)

For the compounding of m periods per year the formula for the annualized forward rate
fi, j between the ith and j th period is

(1 + r j /m) j = (1 + ri /m)i (1 + fi, j /m) j−i (2.11)

where rk is the annualized spot rate for the interval between today and k periods from today.
The formula for the annualized forward rate with continuous compounding is

eruy ·u = erty ·t · e ft,u ·(u−t) (2.12)

There is a one-to-one relationship between the collection of all spot rates and the collection
of all forward rates. Hence it is sufficient to know either one of these collections in order to
know the term structure of interest rates. As we will see later in the book there are models of
the term structure of interest rates that focus on the specification of the dynamics of forward
rates rather than on the dynamics of spot rates.

Forward rates are not only a theoretical tool. There are forward contracts in the market
trading which one can lock in an interest rate today for a future period. Taf might find such
a trade desirable if he thinks that the future interest rates will be different from the currently
quoted forward rates.

Example 2.8 (Speculating on Forward Rates) The one-year pure discount bond with
nominal $100 trades at $95, and the two-year pure discount bond with nominal $100