Question

How does one calculate the present value of a bond?

Answer 1

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.
`PV = (FV)/(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%.

`PV = (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)

`PV = A[(1-(1)/(1 + r)^n)/r]`

`PV = 50[(1-(1)/(1.04)^10)/.04]`

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.