Accrued interest, flat price and full price

How do you price a bond between coupon dates?

Most trades settle between coupon dates. The buyer then pays the full price : the present value of the remaining cash flows on the trade settlement date , which is the day the buyer pays and the seller delivers. The full price has two parts, the flat price and the accrued interest .

Full price (the invoice price)what the buyer pays on the settlement date

• Flat price (the quoted price) Memorize full price minus accrued interest; moves only when the yield moves
• Accrued interest Memorize the seller’s share of the next coupon; does not depend on the yield

The full price is the bond’s value on the last coupon date, rolled forward at the yield for the fraction of the period that has passed.

$$PV^{Full} = PV \times (1 + r)^{t/T}$$

Here $PV$ is the price on the last coupon date, $r$ the yield per coupon period, $t$ the days from the last coupon to settlement, and $T$ the days in the coupon period. The seller is owed interest for the days the seller held the bond. That is accrued interest.

$$AI = \frac{t}{T} \times PMT$$

$$PV^{Flat} = PV^{Full} - AI$$

Accrued interest does not depend on the yield, so only the flat price responds when yields change.

How are the days counted?

The day count convention states how days are counted and how many days a period has. It sets $t$ and $T$.

Day count conventionhow days are counted, and how many days are in the period

• Actual/actual Use it real days in the period; typical for government bonds
  • Counts real days, leap days included
  • Period length real days between the two coupon dates
• 30/360 Use it typical for many corporate bonds
  • Counts every month 30 days
  • Period length a semiannual period is always 180 days

Under actual/actual , count the real days, leap days included. Under 30/360 , every month has 30 days, and a 31st counts as the 30th.

$$D = 360 (Y_2 - Y_1) + 30 (M_2 - M_1) + (D_2 - D_1)$$

What is the full price of a bond between coupon dates?

Work in two steps: find the value on the last coupon date, then roll it forward.

The flat price (94.481) is only slightly above the coupon-date value (94.352), because most of the growth over 76 days is the accrued interest, which comes back out. Do not report 94.352 or the full price when the question asks for the flat price.

Why do dealers quote the flat price?

If dealers quoted full prices, a bond’s price would rise every day as interest accrues, even with an unchanged yield, and fall sharply on each coupon date. The flat price removes that sawtooth, so a move in the quote signals a move in yields.

Exam tip Accrued interest is the seller’s share of the next coupon. The buyer receives the whole coupon, so the buyer reimburses the seller on the settlement date.

Go deeper Why the exact formula compounds over a fraction of a period

Settlement falls part-way through a period, so the next coupon is only $1 - t/T$ of a period away, the one after that $2 - t/T$, and so on. Discounting each payment at those distances gives the full price directly. Multiplying through by $(1 + r)^{t/T}$ turns it into the coupon-date value times $(1 + r)^{t/T}$, which a calculator solves in one pass. Some markets use a simple-interest fraction instead; the curriculum uses the compound form.

Go deeper Leap years and ex-coupon periods

Under actual/actual, a period that contains 29 February has one more day, so $T$ changes: a 1 January to 1 July period has 181 days in an ordinary year and 182 in a leap year. Some markets also stop accruing to the buyer shortly before a coupon date (the ex-coupon period). That is a market detail and not examinable.