- Variable Interest Rates 233
months from now, at a rate of 10.23%. Does this present an arbitrage op-
portunity? All rates stated in this exercise are continuous compounding
rates.As time passes, the bond prices will change and, consequently, so will the
forward rates. Theforward rate over the interval[M, N] determined at time
n<M <Nis defined by
B(n, N)=B(n, M)e−(N−M)τf(n,M,N),that is,
f(n, M, N)=−lnB(n, N)−lnB(n, M)
(N−M)τ.
Theinstantaneous forward rates f(n, N)=f(n, N, N+ 1) are the forward
rates over a one-step interval. Typically, whenτis one day, the instantaneous
forward rates correspond to overnight deposits or loans. The formula for the
forward rate
f(n, N)=−
lnB(n, N+1)−lnB(n, N)
τ(10.6)
will enable us to reconstruct the bond prices, given the forward rates at a
particular timen.
Example 10.11
Letτ= 121 ,n=0,N=0, 1 , 2 ,3, and suppose that the bond prices are
B(0,1) = 0. 9901 ,
B(0,2) = 0. 9828 ,
B(0,3) = 0. 9726.Then we have the following implied yields
y(0,1)∼= 11 .94%,
y(0,2)∼= 10 .41%,
y(0,3)∼= 11 .11%,and forward rates
f(0,0)∼= 11 .94%,
f(0,1)∼= 8 .88%,
f(0,2)∼= 12 .52%.
Observe that, using the formula for the forward rates, we get
exp(−(0.1194 + 0.0888 + 0.1252)/12)∼= 0 .9726 =B(0,3)which illustrates the next proposition.