The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 595 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 595

a 2 + n – 2 P 2

n – 1 P 1 = ---------------------------

1 + r 2

Substituting this equation into equation (20.2), we get

a 1 a 2 + n – 2 P 2

nP 0 = -------------------+ ---------------------------------------

( 1 + r 1 ) ( 1 + r 1 )( 1 + r 2 )

Repeating the same substitution recursively, up to the maturity of the

debt instrument, we find

nP 0 = -------------------+ ---------------------------------------+ ... + ------------------------------------------------------------------ (20.3)

a 1 a 2 an

( 1 + r 1 ) ( 1 + r 1 )( 1 + r 2 ) ( 1 + r 1 )( 1 + r 2 )...( 1 + rn )

In other words, the debt instrument must equal the sum of the present

value of the payments that the debtor is required to make until maturity.

Let’s illustrate the principles to this point. Assume that the length of

a period is one year. Suppose that an investor purchases a 4-year debt

instrument with the following payments promised by the borrower:

Year Interest Payment Principal Repayment Cash Flow

1 $100 $0 $100

2 120 0 120

3 140 0 140

4 150 1,000 1,150

In terms of our notation: a 1 = $100; a 2 = $120; a 3 = $140; a 4 =

$1,150. Assume that the 1-year rates for the next four years are: r 1 =

0.07; r 2 = 0.08; r 3 = 0.09; r 4 = 0.10. The current value or price of this

debt instrument today, denoted 4 P 0 , using equation (20.3) is then

100 120 140

40 P = ----------------+ ---------------------------------+ --------------------------------------------------

(1.07) (1.07)(1.08) (1.07)(1.08)(1.09)

1,150

+ ------------------------------------------------------------------- = $1,138.43

(1.07)(1.08)(1.09)(1.10)