## Bond Pricing – part II

##### Published: Jan 2007

Last month, we looked at how a basic coupon paying bond structure is priced. This month, we explain how to price a bond that has no coupon.

Instead of paying regular interest, a zero-coupon bond yields a return by selling at a discount to its face value which is paid out at maturity. In order to calculate its price, we have to determine the present value of its par value at maturity. This present value – or the difference between the cost now and the principal value at maturity – is the interest earned. If we assume this is compounded semiannually at a required yield we can calculate the price of the bond.

The formula to use is simple:

$$\mathrm{Zero\: bond\: price} = \frac{Value\: at\: maturity}{1\: +\:required\: yield^{years \:until\: maturity}}$$

or

$$\mathrm{P}=\frac{M}{1\:+\:r^n}$$

Where:

• $$\mathrm{P}=\mathrm{ Price}$$
• $$\mathrm{M}= \mathrm{Value\: at\: maturity}$$
• $$\mathrm{r}=\mathrm{ Required\: yield \:per\: annum\: ÷ \:2}$$
• $$\mathrm{n}= \mathrm{Number \:of \:years \:until\: maturity\: ×\: 2}$$
##### Example

An investor wants to purchase a zero-coupon bond with a face value of €1,000. The bond matures in five years and the investor seeks a required yield of 8% per year.

Typically the required yield of zero-coupon bonds is based on two annual interest payments in the same way interest would be paid on an interest bearing bond. This means that the required yield of 8% per year has to be divided by 2, resulting in 4% for every six months. At the same time, the number of interest periods has to be doubled from five years to ten half-year periods.

Using the formula above, the investor would be willing to pay not more than:

$$\frac{€\:1,000}{1\:+\:0.04^{10}}=\mathrm{€675.56}$$

Zero-coupon bonds tend to be long-term investments with a maturity of 10 years or longer. The longer the maturity, the higher the discount on the face value of the zero-coupon bond will be.

Due to the lack of coupon payments, zero-coupon bonds yield no current income, which may deter some investors. However, long-term financial needs in particular can be met with this type of bond. The large discount means that compared to coupon paying bonds, less capital is needed upfront from the investor.

Like any other bond, the price of a zero-coupon bond rises when interest rates fall and falls when interest rates rise. The value of a zero-coupon bond relies solely on the difference between purchasing price and face value. As a result, the price of a zero-coupon bond reacts with much more volatility to interest rate changes than the price of a regular coupon paying bond. In other words, duration is shorter for interest bearing bonds than zero-coupon bonds which have a duration that is equal to maturity. Zero-coupon bonds also effectively lock-in the reinvestment rate and are therefore particularly attractive to investors when interest rates fall.

One issue that US investors may have with zero-coupon bonds is that unless the bond is tax exempt or held in a tax-deferred account, tax on the imputed interest that accrues each year has to be paid annually, even though the interest gain is only realised at maturity.

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