Chapter 5: Models and Equations 285
Writing this in terms of yieldsy(t;T)wehave
Z(t;T)=e−y(t;T)(T−t)and so
F(t;T)=y(t;T)+∂y
∂T.This is the relationship between yields and forward
rates when everything is differentiable with respect to
maturity.
In the less-than-perfect real world we must do with only
a discrete set of data points. We continue to assume
that we have zero-coupon bonds but now we will only
have a discrete set of them. We can still find an implied
forward rate curve as follows. (In this I have made the
simplifying assumption that rates are piecewise con-
stant. In practice one uses other functional forms to
achieve smoothness.)
Rank the bonds according to maturity, with the shortest
maturity first. The market prices of the bonds will be
denoted byZiMwhereiis the position of the bond in
the ranking.
Using only the first bond, ask the question ‘What inter-
est rate is implied by the market price of the bond?’
The answer is given byy 1 , the solution of
ZM 1 =e−r^1 (T^1 −t),i.e.
r 1 =−ln(ZM 1 )
T 1 −t.This rate will be the rate that we use for discounting
between the present and the maturity dateT 1 of the
first bond. And it will be applied toallinstruments
whenever we want to discount over this period.