Tell apart the full price, the flat price and accrued interest
The idea
The full price, also called the dirty or invoice price, is what a buyer actually pays and equals the present value of the remaining cash flows on the settlement date. Accrued interest is the portion of the next coupon earned by the seller for the days they held the bond, computed on an actual or thirty-day convention depending on the market. The flat or clean price is the full price less accrued interest, and it is what is quoted, precisely because it does not jump downward each time a coupon is paid and so is comparable across dates.
Formulas
Accrued interest
accrued interest = coupon × (t ÷ T)
- coupon
- the full coupon for the period
- t
- days from the last coupon date to settlement
- T
- days in the coupon period
The day count decides t and T, not the calendar: actual/actual for government bonds, 30/360 for most corporates. The same settlement date gives two different numbers under the two conventions.
Full price from the flat price
full price = flat price + accrued interest
- flat price
- the quoted price, which excludes accrued interest
- accrued interest
- the coupon earned by the seller but not yet paid
The full price is what the buyer pays. Bonds are quoted flat so the printed price does not saw up through the coupon period and drop on every coupon date.
A practice question
Which price does a bond buyer actually pay?
- The full price, including accrued interest — correct
- The flat priceWhy not: That is what is quoted, and it deliberately excludes accrued interest. The invoice adds the accrual back.
- The par valueWhy not: Par is repaid at maturity. What changes hands at settlement is a market price plus accrued interest.
The quoted flat price excludes accrued interest, which is added at settlement to compensate the seller for the days they held the bond.
A harder one
Why would a quoted full price appear to drop sharply on a coupon date?
- Because the bond's yield jumpsWhy not: Nothing about a scheduled payment moves the yield. The drop is arithmetic, not a repricing.
- Because the issuer's credit deteriorates on paymentWhy not: Paying a coupon on time is evidence of the opposite.
- Because the accrued interest resets to zero once the coupon is paid — correct
The saw-tooth is an artefact of the accrual rather than a price movement, which is exactly why the flat price is quoted instead.
The hardest tier
A bond is purchased midway between coupon dates. How is accrued interest treated by the buyer?
- It is a cost that reduces the yield earnedWhy not: It is advanced and recovered in full at the next coupon, so it nets to nothing. Treating it as a cost would double-count.
- It is added to the purchase price and recovered when the full coupon is received — correct
- It is refunded by the issuer at settlementWhy not: The issuer is not party to the trade. The transfer is between buyer and seller.
The buyer advances the seller their share and then collects the entire coupon. The transaction is a transfer between the two parties and does not affect the bond's yield.
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