# How does one calculate the present value of a bond?

Jul 5, 2015

The present value of a bond is its price.

#### Explanation:

A traditional bond is made up of two kinds of cash flows: a lump sum received at maturity (the face value) and annual (or semi-annual) interest payments. The value of the bond is the present value of these cash flows. In words: the present value of a bond is the present value of the lump sum payment plus the present value of the annual cash flows.

Consider this example.
A 10-year bond with a face (maturity) value of $1,000 and a coupon rate of 5%. Calculation of the present value involves two steps. 1. Present value of the face value. We will use the formula that calculates the present value of a lump sum. $P V = \frac{F V}{1 + r} ^ n$Where, FV = the maturity value r = the discount rate n = the number of years to maturity. We know that FV =$1,000 and n = 10, but what does r = ? First, it is not the coupon rate. It is the rate of return an investor demands when buying the bond - the going market rate. Let us assume that it is 4%.

$P V = \frac{1000}{1.04} ^ 10$

PV = $677 (rounded) 2. Present value of the annual interest payments. For this calculation, we have to use the formula for a present value of an annuity. Where, A = the yearly coupon payment of$50 (.05 * 1,000)

$P V = A \left[\frac{1 - \frac{1}{1 + r} ^ n}{r}\right]$

$P V = 50 \left[\frac{1 - \frac{1}{1.04} ^ 10}{.04}\right]$

PV = $405 When we add the two present values together we get$1,082.
This is the present value of the bond and the price at which it should sell.