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# Flat price and full price of bonds

Unlike equity securities in which the price quoted is the same as the price paid, the price paid in case of debt securities may differ from the price quoted.

When a bond is traded between coupon dates, the price paid includes the bond price plus interest accrued since the last coupon date.

## Full price vs flat price

The bond price inclusive of the accrued interest is called full price and the price exclusive of the accrued interest is called flat price. The following equation expresses the relationship between the flat price and the full price

$Full\ Price = Flat\ Price + Accrued\ Interest$

Since the accrued interest is zero just after a coupon date and reaches a maximum as the bond approaches the next coupon date, the full price of a bond see-saws till maturity. Therefore, dealers quote a flat price for debt securities and interest accrued since the last coupon payment is added to determine the amount actually paid by the buyer of the securities on the settlement date.

## Accrued interest

Accrued interest (AI) is calculated by multiplying coupon payment (PMT) with days since the last coupon (t) and divided by the total number of days in a coupon period (T):

$Accrued\ Interest =PMT\times\frac{t}{T}$

Different day count conventions are in bond pricing, 30/360 (mainly for corporate bonds) and actual/actual being the two most popular.

The flat price of the bond is the price determined exactly on a coupon date (as illustrated in bond pricing using yield to maturity and spot rates).

## Calculating full price

There are two ways to determine the full price directly:

• Discounting each bond cash flow for (n – t/T) periods where n being the number of periods of each cash flow.
• Growing the flat bond price on the last coupon date by a factor of (1+r)t/T.
$Full\ Price\ =\ Last\ Flat\ Price\times{(1+r)}^\frac{t}{T}$