The time value of money calculations can be used to calculate a bond price.

A business will issue bonds if it wants to obtain funding from long term investors by way of loans. The bond will stipulate the interest rate known as the coupon rate, and the term to be used, known as the maturity date. Throughout the term the investor will receive the interest (coupon) payments, and at the maturity date will receive payment of the principal amount invested.

## Bond Price Example

The future bond cash flow is presented in the diagram below:

To find the current bond price an investor would pay for the bond these cash flows need to be discounted back to today, the start of period 1. The discount rate to use will depend on the risk associated with the cash flows from the bond.

*Note: Do not confuse the interest rate on the bond (in this case 8%) which determines the amount of the future cash flows, with the discount rate (i) which determines the level of risk associated with the cash flows.*

## Discount Rate Higher than the Bond Coupon Rate

Suppose the discount rate was 10%, then the price of the bond is given by the present value of two cash flows:

### Cash Flow 1 – The Principal

The lump sum principal amount of 10,000 is received in 18 months time at the maturity date, and the present value is calculated using the present value of a lump sum formula

### Cash Flow 2 – The Interest Payments

The interest payments of 400 are received every 6 months, and the present value is calculated using the present value of an annuity formula.

As interest is paid every 6 months, a period is defined as 6 months, and there are three, 6 month periods in the 18 month term.

The bond price is calculated as follows:

n = 3 (6 month periods) i = 10% / 2 = 5% (per 6 month period) FV = Principal = 10,000 Pmt = Interest = 400 per 6 month period Bond price = PV Cash flow 1 + PV Cash flow 2 Bond price = FV / (1 + i)^{n}+ Pmt x (1 - 1 / (1 + i)^{n}) / i Bond price = Principal / (1 + i)^{n}+ Interest x (1 - 1 / (1 + i)^{n}) / i Bond price = 10,000 / (1 + 5%)^{3}+ 400 x (1 - 1 / (1 + 5%)^{3}) / 5% Bond price = 9,727.68

The present value of the cash flows from the bond is 9,727.68, this is what the investor should be prepared to pay for this bond if the discount rate is 10%. The price is lower than the par value of the bond because the market rate (10%) is higher than the interest rate on the bond (8%).

## Discount Rate Equal to the Bond Coupon Rate

If the market rate was the same as the interest rate on the bond 8%, then the bond price would be given as follows:

n = 3 (6 month periods) i = 8% / 2 = 4% (per 6 month period) FV = Principal = 10,000 Pmt = Interest = 400 per 6 month period Bond price = FV / (1 + i)^{n}+ Pmt x (1 - 1 / (1 + i)^{n}) / i Bond price = Principal / (1 + i)^{n}+ Interest x (1 - 1 / (1 + i)^{n}) / i Bond price = 10,000 / (1 + 4%)^{3}+ 400 x (1 - 1 / (1 + 4%)^{3}) / 4% Bond price = 10,000

In this special circumstance as the market rate and the bond rate are equal (8%), the price of the bond is the same as the par value of the bond 10,000.

## Discount Rate Lower than the Bond Coupon Rate

If the market rate was lower than the interest rate on the bond, say 6%, then an investor should be prepared to pay more than the par value of the bond as follows:

n = 3 (6 month periods) i = 6% / 2 = 3% (per 6 month period) FV = Principal = 10,000 Pmt = Interest = 400 per 6 month period Bond price = FV / (1 + i)^{n}+ Pmt x (1 - 1 / (1 + i)^{n}) / i Bond price = Principal / (1 + i)^{n}+ Interest x (1 - 1 / (1 + i)^{n}) / i Bond price = 10,000 / (1 + 3%)^{3}+ 400 x (1 - 1 / (1 + 3%)^{3}) / 3% Bond price = 10,282.86

The bond price is the net present value of all the cash flows from the bond. The bond cash flows are determined by the principal, and the coupon rate on the bond, and the net present value of the cash flows is determined by the term of the bond and the discount rate, reflecting the risk to the cash flows. It is important to note that there is an inverse relationship between the market discount rate (i) and the bond price. When the market discount rate increases, the bond price decreases and vice versa.

## About the Author

Chartered accountant Michael Brown is the founder and CEO of Plan Projections. He has worked as an accountant and consultant for more than 25 years and has built financial models for all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a BSc from Loughborough University.