- Variable Interest Rates 217
As a consequence of Proposition 10.1, if the yield is independent of maturity
and deterministic (that is,y(n) is known in advance for anyn≥0), then it must
be constant,y(n)=yfor alln. This is the situation in Chapter 2, where all
the bond prices were determined by a single interest rate. The yieldy(n)=y,
independent ofn,is then equal to the constant risk-free interest rate denoted
previously byr.
Historical bond prices show a different picture: The yields implied by the
bond prices recorded in the past clearly vary with time. In an arbitrage-free
model, to admit yields varying with time but independent of maturity we should
allow them to be random, so it is impossible to predict in advance whethery(n)
will be higher or lower thany(0).
We assume, therefore, that at each time instant the yieldy(n)isapositive
random number independent of the maturity of the underlying bond.
Our goal is to analyse the return on a bond investment and the imminent
risk arising from random changes of interest rates. Suppose that we intend to
invest a certain sum of moneyP for a fixed period ofNtime steps. If the
yieldyremains constant, then, as observed in Chapter 2, our terminal wealth
will bePeNτy. This will be our benchmark for designing strategies hedged
against unpredictable interest rate movements.
10.1.1 Investment in Single Bonds ..........................
If we invest in zero-coupon bonds and keep them to maturity, the rate of return
is guaranteed, since the final payment is fixed in advance and is not affected
by any future changes of interest rates. However, if we choose to close out our
investment prior to maturity by selling the bonds, we face the risk that the
interest rates may change in the meantime with an adverse effect on the final
value of the investment.
Example 10.1
Suppose we invest in bonds for a period of six months. Letτ= 121 .We buy a
number of unit bonds that will mature after one year, payingB(0,12) = 0. 9300
for each. This price implies a ratey(0)∼= 7 .26%. Since we are going to sell the
bonds at timen= 6, we are concerned with the priceB(6,12) or, equivalently,
with the corresponding ratey(6). Let us discuss some possible scenarios:
- The rate is stable,y(6) = 7.26%. The bond price isB(6,12)∼= 0 .9644 and
the logarithmic return on the investment is 3.63%, a half of the interest
rate, in line with the additivity of logarithmic returns. - The rate decreases toy(6) = 6.26%, say. (The convention is that 0.01% is