The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 598 Wednesday, February 4, 2004 1:33 PM

598 The Mathematics of Financial Modeling and Investment Management

Premium Par Yield

In general, equation (20.7) and equation (20.8) cannot be solved explic-

itly for y (for n > 2); these equations must be solved by trial and error or

by using an iterative technique—with one important exception. It is

apparent from equation (20.7) that the par value, P/M, increases as the

coupon rate, C/M, increases. Now consider a bond whose coupon rate is

such that the corresponding value of P/M is one—that is, the bond sells

at par. Then equation (20.7) becomes:

C 1 – ( 1 + y) –n 1

1 = ----- -------------------------------- + -------------------- (20.9)

M y ( 1 + y)n

Equation (20.9) can be solved explicitly for y; the solution is y = C/

M. In other words, if a bond sells at par, its yield to maturity is the same

as its coupon rate; for example, if a 7.75%, 20-year bond sells at par, its

yield to maturity is 7.75%. This means that, for a bond to be issued at

par, the coupon rate offered must be the same as the market-required

yield for that maturity. The coupon rate of an n-period bond selling at

par may be labeled the n-period par yield.

It can also be verified from equation (20.9) that if the coupon rate

on a bond is less than the required yield to maturity, or par yield, the

bond will sell at a discount; the converse is true for a bond with a cou-

pon above par yield. The explanation for this relation is self-evident: if

the cash payment per period—namely, the coupon is below the required

yield per period, the difference must be made up by an increase in price,

or capital gain, over the life of the bond. This requires that the price of

the bond be lower than its maturity value. In the United States, bonds

(other than zero-coupon bonds) customarily are issued with a yield to

maturity as to insure that the issue sells at close to par.

Reinvestment of Cash Flow and Yield

The yield to maturity takes into account the coupon income and any

capital gain or loss that the investor will realize by holding the bond to

maturity. The measure has its shortcomings, however. We might think

that if we acquire for P a bond of maturity n and yield y, then at matu-

rity we can count on obtaining a terminal value equal to P(1 + y)n. This

inference is not justified. By multiplying both sides of equation (20.5) by

(1 + y)n, we obtain

P(1 + y)n = C(1 + y)n–1 + C(1 + y)n–2 + C + M