In all our calculations so far we have assumed that the next coupon payment is either 6 months (semi-annual) or 1 year (annual) away. In practice investors are likely to purchase bonds between coupon payments so the next coupon is less that 6 months or 1 year away. Under these circumstances, how do we calculate the fair price of the bond ?

To calculate the fair price of a bond bought between coupon payments we must go through the following three steps:

1. Calculate the number of days until the next coupon payment.      
2. Determine the present value of the cashflows over a fractional period.      
3. Calculate by how much the buyer must compensate the seller for the coupon interest he would have earnt during the fractional period that the bond was held.

Number of days until the next coupon payment

The number of days until the next coupon payment is not as straightforward a question as it would appear.

The answer will depend on the market conventions for the type of bond in question.

Here we will look at two conventions: “actual/actual” and “30/360”. The former method is used for US Treasury securities, the latter for Eurobonds.

Consider a US government bond whose last coupon payment was March 1st. US Treasuries pay a semi-annual coupon so the next coupon payment would be six months later on September 1st.

The bond is purchased with a settlement date of May 10. The actual number of days between May 10 and September 1st is calculated as follows:

However, if the security was a eurobond - because each month is assumed to have 30 days - the number of days until the next coupon payment is 111.

Determining the present value of cash flows for fractional periods

Once you have determined the number of days before the next coupon payment, go through the following steps:

1. calculate the following ratio:

Remember: for an instrument using the “actual/actual” convention the number of days in the coupon period will be the actual number of days, whereas for an instrument using the “30/360” convention it will be 180 or 360 depending on whether it pays semi-annually or annually.

2. Using the ratio calculated above, you can now calculate the fair price of the instrument by applying it to the discounted cash flow formula.

The period in the formula for determining the present value is generally expressed as t - 1 + w. For the first cash flow the period is 1 - 1 + w (or simply w). At maturity, the last period is n - 1 + w, where n is the number of coupon periods to maturity.

Suppose an annual pay eurobond with a coupon of 8% maturing March 1st 2003 is purchased with a settlement date of July 17 1998. What would be the bond’s price with a yield of 6%.

The next coupon payment will be made on March 1st 1999. Because the bond is a eurobond the “30/360” convention applies and there are 224 days between the settlement date and the next coupon payment. The number of days in the coupon period is 360 so:

w = 224 / 360 = 0.6222

The number of coupon payments remaining until maturity is 5. The price of the bond is calculated as follows: