The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 596 Wednesday, February 4, 2004 1:33 PM

596 The Mathematics of Financial Modeling and Investment Management

YIELD-TO-MATURITY MEASURE

Next we must consider how to construct a measure that will permit us

to compare the rate of return of debt instruments having different cash

flows and different maturities. For 1-period debt instruments, the mea-

sure is clear; it is provided by the left-hand side of equation (20.1). But

that approach cannot be generalized readily to long-term debt instru-

ments. For instance, for an instrument with a cash flow (a 1 , a 2 ), the

measure (a 1 + a 2 )/P 0 would not be a useful measure of yield. In the first

place, if we seek a measure that can be used to compare instruments of

different maturities, it must measure return per unit of time. And sec-

ond, the proposed measure ignores the timing of receipts, thus failing to

reflect the time value of money.

The widely accepted solution to this problem is provided by a mea-

sure known as the yield to maturity. It is defined as the interest rate that

makes the present value of the cash flow equal to the market value

(price) of the instrument. Thus for the debt instrument in equation

(20.3), the yield to maturity is the interest rate y that satisfies the fol-

lowing equation:

nP 0 = -----------------+ --------------------+ ... + -------------------- (20.4)

a 1 a 2 an

( 1 + y) ( 1 + y)^2 ( 1 + y)n

In general, the yield to maturity must be found by trial and error or

by using an iterative technique like Newton-Raphson. If the debt instru-

ment is a bond, the cash flow (a 1 ... an) can be written as (C, C, ..., C +

M), where C is the coupon payment and M the maturity value. Equation

(20.4) can be rewritten as

P = C C CM+

-----------------+ --------------------+ ... + -------------------- (20.5)

( 1 + y) ( 1 + y)^2 ( 1 + y)n

After dividing both sides of equation (20.5) by M, to obtain the

price per dollar of maturity value, and factoring C, we obtain

n

P C 1 1

----- = -----∑ -------------------+ -------------------- (20.6)

M Mt = (^1) ( 1 + y)t ( 1 + y)n
Recognizing that the summation on the right-hand side of equation
(20.6) is the sum of a geometric progression, we can rewrite the equa-
tion as