The time value of money formulas can be used to calculate the price of a zero coupon bond.

A business will issue zero coupon bonds when it wants to obtain funding from long term investors by way of debt finance. The bond will stipulate the term to be used, known as the maturity date, and the face value, which is the amount the bondholder will receive back at maturity.

As the name implies, a zero coupon bond does not have a coupon rate and does not make periodic interest payments. In order for the bondholder to get a return on their investment when buying zero coupon bonds, the bond is issued at a discount to its face value (hence the reason why a zero coupon bond is sometimes referred to as a discount bond or deep discount bond).

The zero coupon bond price or value is the present value of all future cash flows expected from the bond. As the bond has no interest payments, the only cash flow is the face value of the bond received at the maturity date.

## Zero Coupon Bond Pricing Example

Suppose for example, the business issued 3 year, zero coupon bonds with a face value of 1,000.

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

Period | 1 | 2 | 3 |
---|---|---|---|

Face value | 1,000 ↑ |

To find the current price an investor would pay for the bond this cash flow needs 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 zero coupon bond.

Suppose the discount rate was 7%, the face value of the bond of 1,000 is received in 3 years time at the maturity date, and the present value is calculated using the zero coupon bond formula which is the same as the present value of a lump sum formula.

The zero coupon bond price is calculated as follows:

n = 3 i = 7% FV = Face value of the bond = 1,000 Zero coupon bond price = FV / (1 + i)^{n}Zero coupon bond price = 1,000 / (1 + 7%)^{3}Zero coupon bond price = 816.30 (rounded to 816)

The present value of the cash flow from the bond is 816, this is what the investor should be prepared to pay for this bond if the discount rate is 7%. The investor pays 816 today and receives the face value of the bond (1,000) at the maturity date, as shown in the cash flow diagram below.

Period | 1 | 2 | 3 |
---|---|---|---|

Price | ↓ 816 | ||

Face value | 1,000 ↑ |

## Zero Coupon Bond Rates

The value of a zero coupon bond will change if the market discount rate changes. Suppose in the above example, the market discount rate increases to 10%, then the bond price would be given as follows:

n = 3 i = 10% FV = Face value of the bond = 1,000 Zero coupon bond price = FV / (1 + i)^{n}Zero coupon bond price = 1,000 / (1 + 10%)^{3}Zero coupon bond price = 751.31 (rounded to 751)

As the face value paid at the maturity date remains the same (1,000), the price investors are willing to pay to buy the zero coupon bonds must fall from 816 to 751, in order from the return to increase from 7% to 10%.

## Zero Coupon Bond Prices and Term to Maturity

The longer the term the zero coupon bond is issued for the lower the bond price will be. Using the example above, if the issue was a 10 year zero coupon bond, then the price at issue would be given as follows:

n = 10 i = 7% FV = Face value of the bond = 1,000 Zero coupon bond price = FV / (1 + i)^{n}Zero coupon bond price = 1,000 / (1 + 10%)^{10}Zero coupon bond price = 508.35 (rounded to 508)

In this example the bondholder has to wait 10 years before they receive the face value of the bond. Assuming the bond discount rate remains the same (7%), then the price investors are willing to pay must fall from 816 to 508 in order to compensate for the term increasing from 3 to 10 years.