Infinite Series and Its Applications

Bonds consist of fixed coupons and the pricing of the bond involves the use of combinations of infinite series.  A typical example would be the sum of terms that increase at an infinite rate.

This formula can be proved by multiplying both sides of the equation by (1-a) and then cancelling common terms on both sides.

A geometric series with a finite number of terms N can be written as the difference between two infinite series such that all the terms with order N or higher will cancel each other.

The above equation can then be rewritten as follows:

The above formulas can be used to value bonds. The derivation of the formula for consols with fixed coupon rate c, yield y and face value F is given below. A consol is an instrument which has an infinite number of coupon payments at the specified coupon rate –

Let’s consider a bond with finite number of coupons over T periods with principal repaid at maturity. This can be broken down into a portfolio that is –

1)      Long in a consol with coupon rate c

2)      Short a  consol with coupon rate c that starts in T periods

3)      Long a zero coupon bond that pays the face value in T periods

The first two positions ensure that the instrument is composed of the finite number of coupons and the third position ensures that there is a payment of face value at maturity.

The price of the bond is then derived as follows –

The formula can be adjusted for different compounding periods.