Accrued Interest and Dirty Prices at Bond Transactions

Why a Bond Transaction Has Two Prices

A bond may be quoted at a price that excludes the interest earned since its last coupon date. At settlement, however, the buyer typically pays for both the bond and that accrued interest. The distinction matters whenever a transaction takes place between coupon dates: multiplying the displayed quote by the face amount alone will not, in general, give the amount the buyer owes.

This tutorial considers a standard fixed-coupon bond traded between coupon dates. The quoted clean price is stated per 100 of face value. The accrued-interest calculation uses the day-count convention specified in the problem, and the dirty price is the clean price plus accrued interest. Assume there are no transaction fees, taxes, ex-coupon rules, or other contractual adjustments unless the problem states them. Such terms can change the settlement amount and must be handled separately.

Definition: The clean price is the bond price quoted excluding accrued interest. Accrued interest is the portion of the current coupon attributed to the period from the previous coupon date through settlement under the specified day-count convention. The dirty price (also called the invoice price) is the amount per 100 of face value including accrued interest: $$ \text{Dirty price}=\text{Clean price}+\text{Accrued interest}. $$

Accrued Interest and Settlement Amount

For a level-coupon bond, calculate accrued interest as a fraction of the coupon for the current coupon period. In the formulas below, $F$ is the bond’s face amount, $c$ is its annual nominal coupon rate, and $m$ is the number of coupon payments per year. The coupon per 100 of face value is $100c/m$. Let $e$ be the number of elapsed days in the coupon period and $D$ the number of days in the full coupon period, both measured according to the applicable convention.

\text{Accrued interest per 100} = \frac{100c}{m}\frac{e}{D}

The fraction $e/D$ applies when the problem specifies a proportional accrual over the coupon period. For Actual/Actual, use the actual elapsed days and the actual number of days in that coupon period. Under a specified 30/360 convention, use the convention’s day counts instead; do not substitute actual calendar days. The precise contractual convention controls.

\begin{aligned} \text{Dirty price per 100} &=\text{Clean price per 100}+\text{Accrued interest per 100},\\ \text{Settlement payment} &=\frac{F}{100}\bigl(\text{Dirty price per 100}\bigr). \end{aligned}

The factor $F/100$ is the number of 100-unit blocks of face value. The price and accrued interest must use the same “per 100” basis before they are added. Equivalently, calculate the clean-price amount on the full face value and add accrued interest on the full face value.

Why Clean Price Plus Accrued Interest Gives the Transaction Price

The clean quote removes the interest earned during the current coupon period from the quoted bond value. But if a bond is sold between coupon dates, the seller has held it during part of that period. Under the standard convention, the buyer compensates the seller for that accrued portion at settlement. The buyer then receives the full scheduled coupon when it becomes payable.

This accounting gives the dirty price directly. If the clean price is $C$ per 100 and the accrued interest is $A$ per 100, the buyer’s price per 100 is the clean component plus the accrued component:

C+A=\text{dirty price per 100}.

For a face amount $F$, there are $F/100$ units of 100 face value. Multiplying each per-100 component by that number gives the total:

\frac{F}{100}(C+A) = \frac{F}{100}C+\frac{F}{100}A.

Thus the total settlement payment has a clean-price part and an accrued-interest part. On a coupon date, elapsed accrual is zero under this setup, so clean and dirty prices agree. This result assumes the stated standard settlement convention; an ex-coupon transaction or a contract with a different settlement rule requires its own treatment.

Identify the coupon and convention.
Find the coupon per 100 for one payment period, and record the day-count basis required by the problem.

Find the elapsed fraction.
Calculate elapsed days divided by days in the full coupon period, using the same stated convention for both quantities.

Add accrued interest to the clean quote.
Keep both amounts on a per-100 basis to obtain the dirty price per 100.

Scale to the face amount and check.
Multiply the dirty price per 100 by $F/100$. Reconcile this total with the clean-price amount plus total accrued interest.

Worked Transactions

Worked Example: Actual/Actual Accrual on a Semiannual Bond

A hypothetical bond has face amount $1,200,000, a 6% annual coupon payable semiannually, and a clean quote of 101.20 per 100. The Actual/Actual coupon period contains 184 days, and settlement is 92 days after the last coupon date. Find the accrued interest per 100, the dirty price, and the settlement payment. Assume no fees or ex-coupon adjustment.

Method. Calculate the semiannual coupon per 100, apply the Actual/Actual elapsed fraction, then add accrued interest to the clean quote. The common mistake to avoid is using the annual coupon per 100 without dividing by two.

The semiannual coupon per 100 is:

\frac{100(0.06)}{2}=\$3.00.

The elapsed fraction is:

\frac{92}{184}=0.5.

Therefore, accrued interest per 100 is:

\$3.00(0.5)=\$1.50.

The dirty price is:

101.20+1.50=102.70\text{ per 100}.

The face amount contains $1{,}200{,}000/100=12{,}000$ units of 100. Thus the settlement payment is:

12{,}000(102.70)=\$1{,}232{,}400.00.

Check. The clean-price amount is $12{,}000(101.20)=\$1{,}214{,}400.00$. Total accrued interest is $12{,}000(1.50)=\$18{,}000.00$. Their sum is $\$1{,}232{,}400.00$, matching the dirty-price calculation.

Worked Example: 30/360 Accrual and a Per-100 Quote

A hypothetical $250,000 bond pays a 4% annual coupon semiannually. Its previous coupon date was November 15, its next coupon date is May 15, and settlement is February 15. The contract specifies 30/360. The clean quote is 102.15 per 100. Find the elapsed fraction, accrued interest per 100, dirty price, and settlement payment. Ignore fees and other adjustments.

Method. Use the 30/360 convention for elapsed days and the full coupon period. Do not use a different February calendar-day count. The semiannual coupon per 100 is:

\frac{100(0.04)}{2}=\$2.00.

Under 30/360, November 15 to February 15 is 90 days, and this coupon period is 180 days. The elapsed fraction is therefore:

\frac{90}{180}=0.5.

Accrued interest per 100 is:

\$2.00(0.5)=\$1.00.

Add this amount to the clean quote:

102.15+1.00=103.15\text{ per 100}.

There are $250{,}000/100=2{,}500$ units of 100 face value. The settlement payment is:

2{,}500(103.15)=\$257{,}875.00.

Check. The clean-price amount is $2{,}500(102.15)=\$255{,}375.00$. Accrued interest on the full face amount is $2{,}500(1.00)=\$2{,}500.00$. The sum is $\$257{,}875.00$, as required.

Worked Example: Actual Days in a Coupon Period of Unequal Length

A hypothetical bond with face amount $480,000 has a 5.2% annual coupon payable semiannually. The current Actual/Actual coupon period contains 181 days, and settlement occurs 43 days after the previous coupon date. Its clean quote is 99.40 per 100. Find the accrued interest per 100, dirty price, and settlement payment. Assume no transaction costs or special settlement provisions.

Method. Apply the given actual-day fraction to the semiannual coupon per 100. Do not replace the 181-day coupon period with a standard 180-day half-year.

The coupon per 100 for the six-month payment period is:

\frac{100(0.052)}{2}=\$2.60.

The elapsed fraction is:

\frac{43}{181}\approx 0.23756906.

Accrued interest per 100 is:

\$2.60\left(\frac{43}{181}\right) =\frac{\$111.80}{181} \approx \$0.61767956.

The dirty price per 100 is:

99.40+0.61767956\approx 100.01767956.

There are $480{,}000/100=4{,}800$ units of 100. Using the unrounded accrued-interest amount, the settlement payment is:

4{,}800\left(99.40+\frac{111.80}{181}\right) \approx \$480{,}084.86.

Check. The clean-price amount is $4{,}800(99.40)=\$477{,}120.00$. Total accrued interest is $4{,}800(111.80/181)\approx \$2{,}964.86$. Adding the two components gives $\$480{,}084.86$, matching the settlement payment rounded to the nearest cent.

Common Mistakes and Practical Checks

Accrued interest is often a small part of a transaction, but a small per-100 error is multiplied by every unit of face value. Keep the units visible throughout the calculation and round only when reporting the final money amount.

Using the annual coupon instead of the coupon for one payment period. For a 4% annual coupon paid semiannually, the coupon per 100 is $100(0.04)/2=\$2.00$, not $4.00.
Mixing day-count conventions. If 30/360 is specified, use its elapsed and period day counts. If Actual/Actual is specified, use the actual days for the stated coupon period.
Adding accrued interest twice. The clean quote excludes accrued interest; the dirty price includes it. Once the dirty price has been calculated, do not add accrued interest again to the settlement amount.
Confusing a price per 100 with a total dollar amount. A dirty price of 103.15 per 100 is not the settlement payment for a $250,000 bond. Multiply by $250{,}000/100$.
Rounding too early. Retain the day-count fraction and accrued-interest amount at full precision until after scaling to the face amount; then round the final dollar payment to cents.

Key takeaway: Under the stated standard convention, find accrued interest per 100 from the coupon and elapsed fraction, add it to the clean quote, and multiply the dirty price by $F/100$. A separate clean-price-plus-accrued-interest check should reproduce the same settlement payment.

Check Your Understanding

For each problem, use only the stated convention and assume there are no fees or special settlement adjustments. Round final dollar payments to the nearest cent.

A hypothetical $600,000 bond pays a 4.8% annual coupon semiannually. Under Actual/Actual, the coupon period is 182 days and 60 days have elapsed. The clean quote is 98.75 per 100. Find accrued interest per 100, dirty price, and settlement payment. (Answer: accrued interest $0.7912 per 100; dirty price 99.5412 per 100; payment $597,247.25.)
A hypothetical $900,000 bond pays a 5% annual coupon semiannually. Under 30/360, 75 of the 180 coupon-period days have elapsed. The clean quote is 101.60 per 100. Find the dirty price and settlement payment. (Answer: accrued interest $1.0417 per 100; dirty price 102.6417 per 100; payment $923,775.00.)
A hypothetical $300,000 bond pays a 6% annual coupon quarterly. The specified Actual/Actual coupon period contains 90 days, of which 27 have elapsed. Its clean quote is 99.85 per 100. Find accrued interest per 100 and the settlement payment. (Answer: accrued interest $0.45 per 100; dirty price 100.30 per 100; payment $300,900.00.)
A hypothetical $1,500,000 bond pays a 3.6% annual coupon semiannually. Under 30/360, 120 of 180 coupon-period days have elapsed. Its clean quote is 100.25 per 100. Find accrued interest per 100 and the buyer’s payment. (Answer: accrued interest $1.20 per 100; dirty price 101.45 per 100; payment $1,521,750.00.)
A hypothetical $400,000 bond pays a 4% annual coupon semiannually and settles exactly on a coupon date. Its clean quote is 102.00 per 100. Under the stated standard convention, find accrued interest and the settlement payment. (Answer: accrued interest $0.00 per 100; dirty price 102.00 per 100; payment $408,000.00.)

What You'll Learn