The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 594 Wednesday, February 4, 2004 1:33 PM

594 The Mathematics of Financial Modeling and Investment Management

The estimation of the term structure of interest rates is referred to as

term structure modeling. We will explain how this is done in this chapter.

BASIC PRINCIPLES OF VALUATION OF DEBT INSTRUMENTS

A useful way of understanding the valuation of debt instruments and

how this relates to interest rates is to use the principle that, in perfect

markets, all riskless instruments have the same short-term return which

must coincide with the riskless short-term rate for that period. This con-

dition may be expected to be enforced through arbitrage. The 1-period

rate of return from, say, an instrument with maturity n and a cash flow

denoted by (a 1 , ..., an), consists of the cash payment, a 1 , plus the capital

gain, or the difference between the next-period price and the current

price of the security, expressed as a percentage of initial value.

Let us denote by nPj the price j periods (j < n) from the present of an

instrument maturing n periods later; the capital gain for the current

period is: n–1P 1 – nP 0. Hence the condition that the 1-period return

from holding the instrument must be equal to the short-term rate for the

forthcoming period, denoted by r 1 , can be written as

a 1 + (n – 1 P 1 – nP 0 )

----------------------------------------------- = r 1 (20.1)

nP 0

Solving for nP 0 ,

a 1 + n – 1 P 1

nP 0 = --------------------------- (20.2)

1 + r 1

The reason why the right-hand side of equation (20.2) must be the

equilibrium price of the n-period asset is that, as can be verified, if the

current price, nP 0 , were larger than the right-hand side of equation

(20.2), then the 1-period return of the debt instrument, given by equa-

tion (20.1), would be smaller than the return r 1 obtainable by investing

in the 1-period debt instrument. As a result, no one would want to hold

it, causing its price to drop. Similarly, if nP 0 is smaller than the right-

hand side of equation (20.2), this yield for the debt instrument would be

larger than r 1 , and everyone would want to hold it.

Next we observe that n– 1 P 1 must satisfy an equation like equation

(20.2), or