# Dirty Price

Dirty price (also called full price) is the price of a bond inclusive of interest accrued on the bond since the last coupon date. It is the amount that the buyer of a bond must pay to its seller in exchange for the bond. It is also called invoice price, price plus accrued interest, cum-coupon price, all-in-one price and settlement price.

Dirty price of a bond is important in the context of a bond transactions carried out between two coupon dates. In such a situation, most markets require the buyer of the bond to compensate the seller for the amount of interest accrued between the last coupon date and the transaction date. It is because the buyer will receive the full coupon amount on the next coupon date though he will hold the bond only for a fraction of the coupon period.

However, many market participants quote bonds at a price that does not take into account the portion attributable to accrued interest. Such quoted price is called clean price and it equals dirty price minus accrued interest. It is because it is easy to relate a clean price with movements in interest rates, yields, growth rates and other economic phenomena.

## Formula

Dirty price equals the present value of the bond coupon payments and face value at the settlement date.

Since the dirty price is most relevant between coupon dates, we must discount the first coupon payment only for a fraction of the coupon period and this time difference applies to every coupon period. The bond value function can be expressed as follows:

$$\text{Dirty Price}=\frac{\text{C} _ \text{1}}{{(\text{1}+\text{r})}^{\text{1}-\text{u}/\text{T}}}+\frac{\text{C} _ \text{2}}{{(\text{1}+\text{r})}^{\text{2}-\text{u}/\text{T}}}+\text{...}\\+\frac{\text{C} _ \text{n}+\text{F}}{{(\text{1}+\text{r})}^{\text{n}-\text{u}/\text{T}}}$$

Where C1, C2 and Cn are the coupon payments, r is the periodic market discount rate, u is the number of days since the settlement date (bond valuation date) and the next coupon date, T is the total number of days in a coupon period and F is the face value.

Alternatively, we can value the bond at the last coupon date and arrive at the present value on bond at settlement date using hte following formula:

$$\text{Dirty Price}\ ={\rm \text{BV}} _ {\text{LC}}\times{(\text{1}+\frac{\text{YTM}}{\text{m}})}^\frac{\text{t}}{\text{T}}$$

Where BVLC is the bond value at the last coupon date, YTM is the annual yield to maturity, t is the number of days since the last coupon date and T is the total number of days in the coupon period.

If the clean price is given, dirty price equals the clean price plus interest accrued since last coupon date.

## Example

Adobe Systems Inc. (NASDAQ: ADBE) has $600 million worth of bond payable outstanding. The$1,000 par, 3.25% semi-annual coupon bonds are due to mature on 1 February 2015. The coupon dates are 1 February and 1 August. They follow 30/360 day count convention and next coupon is due on 1 August 2013. Yvonne Chien bought 1,000 such bonds from Charles Schwab on 20 July 2013 when their yield was 0.89%. The market requires buyer to compensate seller for the accrued interest. How much Yvonne must pay CS?

Yvonne must pay the dirty price.

We need to work out the bond value as at the last coupon date i.e. 1 February 2013 which equals \$1,046.68. Let's call it BVLC.

To get to a value as at 20 July 2013, must compound the BVLC by the fraction of t/T. Where t is the number of days since last coupon period i.e. 169 and T is the total number of days in a coupon period i.e. 180.

$$\text{Dirty Price}\ =\text{\1,046.68}\times{(\text{1}+\frac{\text{0.89%}}{\text{2}})}^\frac{\text{169}}{\text{180}}=\text{\1,051.05}$$