Dirty Price

The dirty price is the total price a buyer pays for a bond, including the interest that has built up since the last coupon was paid. It is also called the full price, gross price or invoice price. It differs from the clean price, which is the figure quoted in the market and leaves accrued interest out. The dirty price is the clean price plus accrued interest, and it is the amount that changes hands at settlement.

The dirty price is the true economic value of a bond at a given moment. It equals the present value of all the payments the holder will receive. The clean price is a convention that makes quotes easier to compare. Anyone settling a trade, valuing a portfolio, calculating a yield or lending against bonds works from the dirty price.

The relationship is:

where, for a bond with regular coupons:

• $C$ is the coupon for one coupon period.
• $d$ is the number of days from the last coupon date to the settlement date.
• $D$ is the number of days in the coupon period.

The day counts follow the bond's day count convention. Consider a bond that pays a 4 percent coupon once a year, quoted at a clean price of 101.20 per 100 of face value. Settlement falls 73 days after the last coupon date in a 365-day period. Accrued interest is 4 × 73 ÷ 365 = 0.80. The dirty price is 101.20 + 0.80 = 102.00.

An investor buying $5 million of face value pays 5,000,000 × 102.00 ÷ 100 = $5,100,000. Of this, $5,060,000 is principal at the clean price and $40,000 is accrued interest passed to the seller.

The dirty price is what a valuation model produces. For a bond valued on a coupon date:

Here $P$ is the price, $C$ is the coupon per period, $F$ is the face value repaid at maturity, $y$ is the yield per period and $n$ is the number of periods remaining.

Between coupon dates the exponents become fractional, and the result of the calculation is the dirty price directly. The yield to maturity is defined as the rate $y$ that makes this present value equal to the dirty price. To quote a bond, a trader computes the dirty price from a yield and subtracts accrued interest. The steps of bond pricing therefore run from yield to dirty price to clean price.

If yields did not change, the dirty price would still move every day. It rises steadily as interest accrues, reaches a peak just before the coupon date, and falls by the amount of the coupon when the coupon payment is made. Plotted over time it looks like the teeth of a saw. The clean price over the same period would be nearly flat, drifting slowly toward face value as maturity approaches.

The drop on the coupon date is not a loss. The holder receives cash equal to the fall in price, in the same way that a share price falls when it goes ex-dividend. This predictable movement is the reason markets quote the clean price: it removes a daily change that carries no information.

• Settlement. The cash paid on a bond trade is the dirty price times the face amount. Confirmations list the clean price, accrued interest and total separately.
• Repo and collateral. In a repurchase agreement, the cash lent is based on the dirty price of the bonds given as collateral, less any haircut, because that is what the collateral could be sold for.
• Valuation. Funds and trading books are valued through mark-to-market at the dirty price, with accrued interest sometimes shown as a separate line.
• Risk measures. Duration and related sensitivities are calculated on the dirty price, since they describe the percentage change in the amount actually invested.
• Futures delivery. The invoice amount for a bond delivered into a bond futures contract includes accrued interest.

For a zero-coupon bond there is no coupon to accrue, and the clean and dirty prices are the same. A bond in default normally trades flat: the buyer pays no accrued interest because the next coupon is not expected, so the quoted price is the full price. Some markets quote certain instruments dirty as a matter of custom.

In markets that have an ex-dividend period, such as UK government bonds, a buyer who settles during the last few days before a coupon date does not receive that coupon. Accrued interest for those days is negative, and the dirty price is below the clean price. For floating-rate notes the coupon for the current period is known, so accrued interest is computed in the usual way from the rate set at the start of the period. For inflation-linked bonds both the clean price and the accrued interest are scaled by an index ratio in many markets, which adds a further step between the quote and the cash paid.