Lecture Notes for Finance 1 (and More).

3.2. ZERO COUPON BONDS AND THE TERM STRUCTURE 15
3.2 Zero coupon bonds and the term structure
Assume throughout this section that the model (π, C) is complete and arbitrage-
free and let d ⊤ = (d1, . . ., dT) be the unique vector of discount factors. Since
there must be at least T securities to have a complete model, C must have
at least T rows. On the other hand if C has exactly T linearly independent
rows, then adding other securities to C will not add any more possibilities of
wealth transfer to the market. Hence we can assume that C is am invertible
T × T matrix.
Definition 6 The payment stream of a zero coupon bond with maturity t is
given by the t ′ th unit vector et of R T .
Next we see why the words discount factors were chosen:
Proposition 4 The price of a zero coupon bond with maturity t is dt.
Proof. Let θt be the portfolio such that C ⊤ θt = et. Then
π ⊤ θt = (Cd) ⊤ θt = d ⊤ C ⊤ θt = d ⊤ et = dt.
Note from the definition of d that we get the value of a stream of payments c
by computing �T t=1 ctdt. In other words, the value of a stream of payments is
obtained by discounting back the individual components. There is nothing
in our definition of d which prevents ds > dt even when s > t, but in the
models we will consider this will not be relevant: It is safe to think of dt as
decreasing in t corresponding to the idea that the longer the maturity of a
zero coupon bond, the smaller is its value at time 0.
From the discount factors we may derive/define various types of interest
rates which are essential in the study of bond markets.:
Definition 7 (Short and forward rates.) The short rate at date 0 is given
by
r0 = 1
− 1.
The (one-period) time t- forward rate at date 0, is equal to
where d0 = 1 by convention.
d1
f(0, t) = dt
dt+1
− 1,
zero coupon bond,
ZCB
discount factors
forward rates
short rate