Accrued Interest & Full (Dirty) Price

Bonds pay coupons on fixed dates, but they trade every day in between. If you buy a bond three-quarters of the way through a coupon period, the seller has been "earning" that coupon the whole time — you owe them their share of it, even though YOU will collect the whole coupon from the issuer at the next payment date.

That share is accrued interest: the coupon, prorated by how much of the period has already elapsed. The price quoted in the market — the flat (clean) price — deliberately EXCLUDES this, because otherwise the quoted price would jump upward every single day as interest accrues and then crash back down at each coupon date, making price charts useless for spotting real value changes. The full (dirty) price — flat price plus accrued interest — is what the buyer actually pays in cash at settlement.

A coupon of size $C$ is earned smoothly over the whole coupon period of $T$ days. If $t$ days have passed since the last coupon, the seller has earned the fraction of the coupon proportional to that elapsed time:

The market's quoted price already strips this out to stay comparable day to day — so the actual cash changing hands adds the accrued piece back on top:

At the instant just after a coupon is paid, $t=0$ and full price equals flat price; just before the next coupon, $t$ approaches $T$ and the buyer is paying for nearly the whole upcoming coupon in accrued interest alone.

A bond trades between coupon dates and you need either the cash settlement amount (full price) or the interest owed to the seller (accrued interest) separately from the quoted price.

The straight-line accrual assumption (interest earned evenly, day by day) is a market convention, not a law of finance — real day-count rules differ by market (actual/actual, 30/360, actual/360) and produce slightly different $t$ and $T$ counts for the exact same trade date. The formula also assumes the bond pays on schedule; if a coupon is missed or the bond defaults, accrued interest becomes a legal claim rather than a simple prorated number.

Accrued interest accounting became a standardized bond-market convention as government and corporate debt markets matured in the 19th and 20th centuries — without it, coupon-paying bonds would show artificial "sawtooth" price patterns that made comparing bonds, or even the same bond over time, needlessly confusing. Different markets settled on different day-count conventions (actual/actual for US Treasuries, 30/360 for many corporates) to compute $t$ and $T$, which is why the exact accrued-interest figure on the same bond can differ slightly depending which market convention applies.