Clean price, dirty price and accrued interest · RiskAtoms Academy

Know these first

L1.4.03 Pricing a bond
L1.3.06 Day counts and rate conventions

By the end you will be able to

1Convert between a bond's quoted (clean) price and the settlement (dirty) price.
2Explain why quoted prices leave out accrued interest.
3Calculate accrued interest using a day-count convention.
4Explain the sawtooth pattern of the dirty price over time.

Why it matters

You buy a bond quoted at 98.50. The bill says 99.30. Nobody has made a mistake. Where did the extra 0.80 come from?

Core explanation

Paying the seller for the interest they earned

Coupons are paid on fixed dates, but bonds change hands on any day. Suppose a bond pays its coupon every six months and you buy it halfway through. On the next coupon date you receive the whole coupon, even though the seller held the bond for the first half of the period. That is not fair to the seller.

So at the trade, you pay the seller for the interest they have earned so far. That amount is the accrued interest. It is worked out to the settlement date, the day the money and the bond actually change hands.

Clean price

What the screen shows. The price of the bond with no accrued interest.

+ Accrued interest

The seller's share of the next coupon, for the days since the last one.

= Dirty price

What the buyer pays. Also called the full price or invoice price.

Why quote clean? The dirty price creeps up every day as interest builds, then drops on each coupon date when the coupon is paid out. That sawtooth has nothing to do with the market. If dealers quoted dirty prices, a bond could look as if it had jumped or fallen when only the calendar had moved. The clean price strips the calendar out, so a change in the quote means a change in the market.

Scrub through timeTwo years in the life of a bond

A 5-year 4% bond paying coupons twice a year, with the yield held at 5% throughout.

Months from today9.00

Clean

96.207

Accrued

1.000

Dirty

97.207

The gap between the two lines is the accrued interest. It grows each day and resets to zero when the coupon is paid. The clean price drifts slowly up towards 100 as maturity nears (L1.4.06).

Counting the days. Each market has a day-count convention (L1.3.06). Government bonds in the US, the UK and the euro area use ACT/ACT: actual days since the last coupon, divided by actual days in the coupon period. Many US corporate bonds use 30/360, which treats every month as 30 days and the year as 360, so every half-year period is 180 days.

Ex-dividend periods. Some markets, such as UK gilts, close the register a few days before each coupon date. A buyer in that ex-dividend period does not get the coming coupon, so the accrued interest is negative: the seller pays the buyer for the days still to run.

The formal version

With an annual coupon $C$ paid $m$ times a year, each coupon is $C/m$ . The accrued interest is the share of it that has built up:

\text{AI} = \frac{C}{m} \times \frac{\text{days since the last coupon}}{\text{days in the coupon period}}

Both day counts follow the bond's convention. Then, in either direction:

P_{\text{dirty}} = P_{\text{clean}} + \text{AI} \qquad\qquad P_{\text{clean}} = P_{\text{dirty}} - \text{AI}

The pricing formula from L1.4.03 gives the dirty price on a coupon date, where accrued interest is zero and clean equals dirty. Between coupon dates, the present value of the cash flows is the dirty price; subtract the accrued interest to get the quote.

Try it

Start with the worked example below: quoted at 98.50, settling on 28 Mar 2028. Slide the settlement date to the end of the period, then switch the day count to 30/360 and watch the accrued interest change.

Try itFrom quote to settlement

Open the full bond pricer →

Quoted (clean) price98.50

Coupon rate4.00%

Coupons a year

Day count

Settlement date (last coupon 15 Jan 2028, next 15 Jul 2028)28 Mar 2028

Buyer pays (dirty price)

99.3022

Clean price98.5000

+ Accrued0.8022

Days accrued73 / 182

Coupon per period2.00

Share of the next coupon earned by the seller

Worked example

Buying a US Treasury note between coupons

A 4% US Treasury note pays coupons on 15 January and 15 July, so each coupon is 2 per 100 of face value. Treasuries use ACT/ACT. You buy at a quoted price of 98.50, for settlement on 28 Mar 2028.

From quote to invoice

Step 1 of 6

1
Find the coupon period
Last coupon 15 Jan 2028, next coupon 15 Jul 2028: 182 actual days (2028 is a leap year, so February has 29).

Under 30/360 the days accrued happen to match here (two 30-day months plus 13 days), but the period is 180 days instead of 182, so the accrued interest is a little higher. Always check the convention before you settle a trade.

Regulatory lens · looking ahead

No rules apply yet. In accounts, banks and insurers usually report accrued interest on their bonds separately from the clean value (L4.7). Getting the day count wrong is a classic settlement error that operations teams check for.

Sparks

SparkA price that moves when nothing happens

Track a bond's dirty price for a year with yields frozen and you get a saw blade: up every day, down on each coupon date. Track the clean price and you get an almost flat line. That is the whole case for quoting clean.

SparkWhy 30/360?

Before computers, 30/360 made interest easy to work out by hand: every month the same length, every half-year 180 days. Many markets still use it for corporate bonds and swaps.

Key takeaways

Bonds are quoted clean, without accrued interest. Buyers pay the dirty price: clean plus accrued.
Accrued interest is the coupon per period times days accrued over days in the period, counted by the bond's day-count convention.
ACT/ACT counts actual days and is used for most government bonds; 30/360 treats every month as 30 days.
The dirty price rises daily and drops by the coupon on each coupon date: the sawtooth. The clean price stays smooth.
In an ex-dividend period the buyer misses the coming coupon, so accrued interest is negative.

Reaction

Check what stuck

Question 1 · objective 3

A bond pays a 6% coupon twice a year (3 per period) and uses ACT/ACT. The last coupon was on 31 Jan 2027 and the next is on 31 Jul 2027, a period of 181 days. Settlement is on 22 Mar 2027, 50 days later. What is the accrued interest per 100, to four decimal places?

Question 2 · objective 1

The same bond is quoted at 101.25 clean. What does the buyer pay per 100 of face value, to two decimal places?

Question 3 · objective 2

Why do bond markets quote clean prices rather than dirty prices?

ABecause accrued interest is taxed separately everywhereBSo that price moves reflect changes in the market, not the coupon calendarCBecause the buyer never pays accrued interestDBecause dirty prices are always higher than 100

Question 4 · objective 4

A coupon of 2 is paid today. With yields unchanged, what happens to the dirty price from yesterday to today?

AIt rises by about 2BIt falls by about 2CIt stays the sameDIt falls to the clean price minus 2

Question 5 · objective 3

A US corporate bond pays 5% twice a year and uses 30/360. The last coupon was on 1 Mar 2027; settlement is on 16 May 2027. What is the accrued interest per 100, to four decimal places?

Glossary

Clean price: The quoted price of a bond, without accrued interest.
Dirty (full) price: The price the buyer actually pays: the clean price plus accrued interest. Also called the full or invoice price.
Accrued interest: The part of the next coupon that the seller has earned by holding the bond since the last coupon date.
Settlement date: The date the buyer pays and receives the bond, usually one or two business days after the trade.
Day-count convention: The rule for counting the days in an accrual period and in a year, such as ACT/ACT or 30/360.
Ex-dividend period: A short period before a coupon date in which a buyer does not receive the coming coupon.

Deep Dives & Origins

Deep Dives

UK Debt Management Office, Formulae for Calculating Gilt Prices from Yields. Advanced.

Origins

UK Debt Management Office, Formulae for Calculating Gilt Prices from Yields.
International Capital Market Association, ICMA Secondary Market Rules & Recommendations(day-count conventions).
Frank J. Fabozzi and Francesco A. Fabozzi, Bond Markets, Analysis, and Strategies, 10th edition, Pearson, 2021.

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Previous · L1.4.04

Yield to maturity

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The price-yield relationship