Accrued interest

Accrued interest is the money earned on bonds during the days after its last coupon payment. It can be viewed as a temporarily capital gain on a bond between 2 sequential coupon payments. Investors trying to buy a bond with an annual coupon payment only 1 month before the coupon payment will not succeed in having the full benefit of earning a full coupon in just 1 month time.

Clean and dirty price

The dirty price is just the price investors pay in order to obtain the bond. A part of the money paid is thus compensation for the accrued interest since the last coupon payment. Obviously, the moment after the coupon is distributed, the clean and dirty price are equal. The clean price equals the dirty price adjusted for any accrued interest. The clean prices has the advantage of showing changes in market risk, default risk or interest rate changes better than dirty price. The latter is also impacted by time which affects the bond price.

P_{clean} = P_{dirty} - I_{accrued}

Accrued interest calculation

The accrued interest can be calculated by the annual coupon rate c, multiplied by the nominal N divided by the amount of coupon payments per year CP. Accordingly adjusted for time passed since the last coupon payment which depends on the applied day count convention for that specific bond.

I_{accrued} = \dfrac{c\cdot N}{CP} \cdot \dfrac{\Delta t}{T}

Day count convention

The applied day count convention varies from bond to bond but is largely determined by the type of bond – corporate or government – and continent. The most common day count conventions are: 30/360, Actual/365, Actual/Actual. The former defines 1 month as 30 days and 1 year as 360 days. The latter determines the real amount of days passed since the last coupon payment and the real amount of days per year.

Summary

Accrued interest equals the interest earned since a bond’s last coupon payment. Its calculation largely depends on the day count convention applied on the bond and  can be used determine the bond’s clean price.