# Dirty Price

Dirty price (also called full price) is the amount that the buyer of a bond must pay to its seller in exchange of the bond. It equals the present value of the bond's future cash flows determined at the transaction date.

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 that has been 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.

The value determined by discounting the future payments is the price already takes into account this accrued interest earned by the seller.

Many markets 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.

## Formula

Dirty price is simply the present value of a bond's future coupon payments and its face value calculated using the following formula:

Dirty Price = PV of coupons and face value calculated as | Cash Flow |

(1 + i)^{t-1+w} |

Where,*t* is the time period of the cash flow*w* is the ratio of the number of days between the transaction date and the next coupon date to the total number of days in a coupon period

## 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?

__Solution__

Yvonne must pay the dirty price as calculated below:

Days between the transaction date and next coupon date = 11 = 10 days of July plus 1 day of August

Days in the coupon period = 180 (since 30/360 day count convention is used)

w = | days between transaction date and next coupon date | = | 11 | = 0.061 |

days in a coupon period | 180 |

Dirty price is equal to the sum of the present value of all the coupon payments and face value which equals $1,051.05. Please download the following spreadsheet and take a look at the formulas used:

Price quoted on Morningstar for the bond as at 20 July 2013 is $1036, which is the clean price. The difference between the Morningstar quote and our calculation is attributable to accrued interest which represents interest earned on the bond between 1 February 2013 and 20 July 2013.

The sum of clean price and accrued interest equals dirty price as demonstrated below:

Clean price ($1036.1) + Accrued Interest = | $1,000 × 3.25% | × | 180-11 | ≈ Dirty Price ($1,051.05) |

2 | 180 |

Written by Obaidullah Jan