# Clean Price

Clean price (also know as flat price) of a bond is the price that does not take into account the accrued interest on the bond since its last payment date. It equals the present value of the bond's future cash flows minus the interested earned on the bond between the last payment date and the transaction date.

Most financial markets (particularly in the US) quote bonds at their clean prices. Accrued interest is added back to the quoted price to determine the settlement price i.e. the consideration that buyer pays to the seller in exchange of the bond.

## Formula

If we have dirty price and accrued interest values for a bond, we can find the clean price using the following formula:

Clean Price = Dirty Price − Accrued Interest

Dirty price is the present value of future coupon payments and maturity value of the bond determined using the following formula:

Dirty Price = c × F × | 1 − (1 + r)^{-n} | + | F |

r | (1 + r)^{n} |

Where *c* is the periodic coupon rate, *F* is the face value, *r* is the periodic yield to maturity and *n* is the total number of coupon payments.

Accrued interest can be calculated using the following formula:

Accrued Interest = F × | C | × | D |

m | T |

Where *F* is the face value, *C* is the total annual coupon rate, m is the coupon payments per year, *D* is the days since last payment date and *T* is the total number of days between coupon payments.

## Example

Please consider the facts given in the example in article on dirty price.

The dirty price was calculated as $1,051.05 being the present value of its future cash flows. In order to find clean price, we must subtract accrued interest from dirty price.

Accrued Interest = $1,000 × | 3.25% | × | 180 − 11 | = $15.26 |

2 | 180 |

Clean price = $1,051.05 − $15.26 = $1.035.8

Clean price as per our calculation i.e. $1,035.8 ≈ price quoted by Morningstar i.e. $1,036.1

by Obaidullah Jan, ACA, CFA and last modified on

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