- Variable Interest Rates 229
10.2 General Term Structure ...................................
Here we shall discuss a model of bond prices without the condition that the
yield should be independent of maturity.
The pricesB(n, N) of zero-coupon unit bonds with various maturities de-
termine a family of yieldsy(n, N)by
B(n, N)=e−(N−n)τy(n,N).Note that the yields have to be positive, sinceB(n, N) has to be less than 1
forn<N. The functiony(n, N) of two variablesn<Nis called theterm
structure of interest rates. The yieldsy(0,N) dictated by the current prices
are called thespot rates.
Theinitial term structurey(0,N) formed by the spot rates is a function
of one variableN. Typically, it is an increasing function, but other graphs
have also been observed in financial markets. In particular, the initial term
structure may beflat, that is, the yields may be independent ofN,whichis
the case considered in the previous section.
Exercise 10.16
IfB(0,6) = 0.96 dollars, findB(0,3) andB(0,9) such that the initial
term structure is flat.The price of a coupon bond as the present value of future payments can be
written using the spot rates in the following way:
P=C 1 e−τn^1 y(0,n^1 )+C 2 e−τn^2 y(0,n^2 )+···+(CN+F)e−τnNy(0,nN) (10.3)for a bond with couponsC 1 ,C 2 ,...,CNdue at times 0<τn 1 <τn 2 <···<
τnNand with face valueF, maturing at timeτnN.
Despite the fact that for a coupon bond we cannot use a single rate for
discounting future payments, such a rate can be introduced just as an artificial
quantity. It is called theyield to maturity, and is defined to be the numbery
solving the equation
P=C 1 e−τn^1 y+C 2 e−τn^2 y+···+(F+CN)e−τnNy.Yield to maturity provides a convenient simple description of coupon bonds and
is quoted in the financial press. Of course, if the interest rates are independent
of maturity, then this formula is the same as (10.1).