- Variable Interest Rates 235
We can see that the yields also increase. (See Exercise 10.20 below for a gener-
alisation of this.)
However, the forward rates do not have to increase with maturity even if
the yields do:
f(0,0) = 9.20%,
f(0,1) = 9.80%,
f(0,2) = 9.56%,
y(0,1) = 9.20%,
y(0,2) = 9.50%,
y(0,3)∼= 9 .52%.Exercise 10.19
Can a forward rate be negative?Exercise 10.20
Prove that iff(n, N) increases withN, then the same is true fory(n, N).10.2.2 Money Market Account .............................
Theshort rateis defined byr(n)=f(n, n).An alternative expression isr(n)=
y(n, n+1),so this is a rate valid for one step starting at timen.The short
rates are unknown in advance, except for the current one,r(0).It is important
to distinguish betweenr(n)andf(0,n).Both rates apply to a single step from
timenton+ 1, but the former is random, whereas the latter is known at the
present moment and determined by the initial term structure.
The money market account denoted byA(n),n≥ 1 ,is defined by
A(n)=exp{τ(r(0) +r(1) +···+r(n−1))}withA(0) = 1,and represents the value at timenof one dollar invested in
an account attracting interest given by the short rate under continuous com-
pounding. For example, ifτ= 3651 ,then the interest is given by the overnight
rate.
The money market account defined in Chapter 2 was a deterministic
sequence independent of the particular way the initial dollar is invested.
HereA(n) is random and, as will be seen below, in general different from
exp{τny(0,n)}, the latter being deterministic and constructed by using zero-
coupon bonds maturing at timen.
Example 10.13
In the setting introduced in Example 10.11, suppose that the bond prices change