Investment Science: The price of a bond and it's yield

In the previous post, we saw an introduction about bonds, in this post we will see the formula for the price of a bond, before that we will introduce what is called the yield of a bond.

The Yield to Maturity (YTM) of a bond is the internal rate of return of a bond at it's market price, the price of a bond and it's yield are inversely proportional, if the price goes down, the yield goes up and vice-versa. A par-bond is one where the yield is equal to the interest rate (coupon rate).

Given the above definition, the price of a bond (P) is given by,

where

P = Price of the bond
F = Face value of the bond
C = Coupon payment per year
m = Number of periods in an year the coupon is paid (Typically m = 2 or coupons are paid every 6 months)
n = Number of coupon periods left (with m coupon periods per year)
λ = Yield to maturity

Lets look at an example from Investment Science by Luenberger to calculate the price of a bond (Problem 3.9).

Problem 3.9

An 8% bond with 18 years to maturity has a yield of 9%. What is the price of this bond.

In this problem, it is implied that the face value of the bond F = $100 (unless specified otherwise).

m = 2 (Unless explicitly specified, coupons are paid every 6 months in an year) 
λ = 9%, λ/2 = 4.5%

n = 18, n * m = 36
C = 8% or the coupon payment paid per year = 8% * 100 = $8

Substituting the above values in the Bond price formula, we get

P = $91.164