Dirty Bond Prices

In all major bond markets the convention is to quote price as a clean price. This is the price of the bond as given by the present value of its cash flows, but excluding coupon interest that has accrued on the bond since the last dividend payment. As all bonds accrne interest on a daily basis, even if a bond is held for only one day, interest will have been earned by the bondholder. However, we have referred already to a bond s all-in price, which is the price that is actually paid for the bond in the market. This is also known as the dirty price (or gross price), which is the clean price of a bond plus accrued interest. In other words, the accrued interest must be added to the quoted price to get the total consideration for the bond. 

The accrued interest calculation for a bond is dependent on the day-count basis specified for the bond in question. We have already seen that when bonds are traded in the market the actual consideration that changes hands is made up of the clean price of the bond together with the accrued that has accumulated on the bond since the last coupon payment these two components make up the dirty price of the bond. When calculating the accrued interest, the market will use the appropriate day-count convention for that bond. A particular market will apply one of five different methods to calculate accrued interest these are ... 

It should be noted that the convention in the European bond markets is to quote the clean price and then calculate the accrued interest to obtain the dirty price. However, we want to emphasize that the answer is the same regardless of the order in which clean and dirty prices are calculated. ... 

In this example, one counterparty sells 10 million nominal of the UKT 5.75% 2009 at the spot price of 104.60, this being the market price of the bond at the time. The consideration for this trade is the market value of the stock, which is 10,505,560 as before. Repo interest is calculated on this amount at the rate of 5.75% for one week, and from this the termination proceeds are calculated. The termination proceeds are divided by the nominal amount of stock to obtain the forward dirty price of the bond on the termination date. For various reasons, the main one being that settlement systems deal in clean prices, we require the forward clean price, which is obtained by subtracting from the forward dirty price the accrued interest on the bond on the termination date. At the start of the trade the 5.75% 2009 had 29 days accrued interest, therefore on termination this figure will be 29 + 7 or 36 days. 

The last data point needed is the price of the bond, inclusive of all accrued interest (its dirty price). Since the bond was just issued, it has no accrued interest, and the price observed in the market is 100%. Using the assumptions r = 3.00% and R = 20%, X is found by solving backwards such that the value of the bond s cash flows plus its recovery upon default equal 100% of par. In this example X is found to be approximately 4.09% as illustrated in Exhibit 22.4. 

A bond paying a semiannual coupon has a dirty price of 98.50, an annual coupon of 3 percent, and exactly one year before maturity. The bond therefore has three remaining cash flows two coupon payments of 1.50 each and a redemption payment of 100. Plugging these values into equation (1.20) gives... 

Consider a bond with a dirty price—including the accrued interest the seller is entitled to receive—of 97.89, a coupon of 6 percent, and five years to maturity. FIGURE 1.6 shows the gross redemption yields this... 

Accrued interest compensates sellers for giving up all the next coupon payment even though they will have held their bonds for part of the period since the last coupon payment. A bond s clean price moves with market interest rates. If the market rates are constant during a coupon period, the clean price will be constant as well. In contrast, the dirty price for the same bond will increase steadily as the coupon interest accrues from one coupon payment date until the next ex-dividend date, when it falls by the present value of the amount of the coupon payment. The dirty price at this point is below the clean price, reflecting the fact that accrued interest is now negative. This is because if the bond is traded during the ex-dividend period, the seller, not the buyer, receives the next coupon, and the lower price is the buyer s compensation for this loss. On the coupon date, the accrued interest is zero, so the clean and dirty prices are the same. 

From this it is easy to derive the definition of true yield, which is the discount rate that equates a bond s current market price to the present value of its cash flows. The bond s market price is its dirty price—that is, the price including accrued interest. This is represented in (16.4), where PV is replaced by P, representing the clean price plus AI, representing accrued interest. 

U.S. Treasury price quotes are in ticks, or thirty-seconds of a price point. A half tick is denoted by a plus sign. On May 10, 1994, the 10.25 percent Treasury bond maturing July 21, 1995, was quoted at 104-28+— in other words, an investor would pay 104.28625 for every 100 in face value. It pays coupons on January 21 and July 21. On May 11, 1994, the settlement date, it will have accrued 109 days of interest, for a total of 10.25 X 109/365 x 0.5, or 1.53048 for every 100 of face value. The dirty price of the bond on this date is thus 104-28+ plus 1.53048, or 106.421105. 

This formula calculates the fair price on a coupon payment date, so there is no accrued interest incorporated into the price. Accrued interest is an accounting convention that treats coupon interest as accruing every day a bond is held this accrued amount is added to the discounted present value of the bond (the clean price) to obtain the market value of the bond, known as the dirty price. The price calculation is made as of the bond s settlement date, the date on which it actually changes hands after being traded. For a new bond issue, the settlement date is the day when the investors take delivery of the bond and the issuer receives payment. The settlement date for a bond traded in the secondary market—the market where bonds are bought and sold after they are first issued—is the day the buyer transfers payment to the seller of the bond and the seller transfers the bond to the buyer. 

When investors buy a bond in the market, what they pay is the bond s all-in price, also known as the dirty, or gross price, which is the clean price of a bond plus accrued interest. 

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