## The concepts behind bond pricing

##### Published: Feb 2003

There is no calculation this month because we want to explain a concept we will be using over the next few months. The concept is how to produce a bond price.

A bond is issued by a bank or corporation for a specific period or term. At the end of the bond’s term, the current bond holders are repaid the initial investment plus a final interest payment (known as the coupon). Many bonds also pay the current bond holder interim interest payments (or coupons) on a regular basis (for example, annually, semi-annually or quarterly). The exception is a zero-coupon bond.

The potential bond investor will know the cash value of the bond and its associated coupons in advance. The investor will discount those future cash flows to a present value. The net present value of all the future cash flows is the price that the potential investor is prepared to pay.

The price of the bond is therefore dependent not only upon the coupon rate, but also the internal rate of return of the investment. The internal rate of return is the constant rate at which the future cash flows are discounted to a present value which is equal to the price of the bond. This internal rate of return then becomes the yield of the bond. So by calculating the value of the future cash flows and taking the bond price today, the yield is calculated.

This means that:

• The amount of each future cash flow is dependent on the coupon rate.
• The net present value of these future cash flows is dependent on the internal rate of return – the rate at which the future cash flow is discounted back to a present value

The bond price will also have to incorporate a value for the ‘accrued coupon’ if the bond is sold part way through an interest period. Because the holder of the bond receives the coupon on the coupon date, the seller of the bond will want to be compensated for that period between the coupon dates when they held the bond.

These relationships can be represented:

$$Price = \Sigma \:^t \frac{CF_t}{1 + \frac{i}{n}d_t \: \times \: n \: / \:year}$$

Where i = yield, n is the number of interest payments a year, d t is the number of days until cash flow t ( CFt),the year is the number of days convention considers as part of the year and t is the number of outstanding cash flows.

– this says that the price of the bond is equal to the sum of the future cash flows each of which is discounted to a present value.

But the markets use a more convenient method to calculate bond prices which we will look at next month.