Reader question: Why can a bond be quoted at one price but settle for a different cash amount, even when nobody has changed the quoted market price?
The answer is usually accrued interest. A coupon bond earns interest through the coupon period, but coupons are paid only on scheduled dates. If ownership changes between coupon dates, the buyer and seller need a rule for allocating the coupon earned up to settlement. Bond systems therefore distinguish the quoted clean price from the dirty price, which normally includes accrued interest.
In the simplest case:
dirty price = clean price + accrued interest.
The arithmetic is easy. The calendar logic is not.
Page role and boundary
This article teaches fixed-income calculation logic. It is not investment advice, a recommendation to buy or sell bonds, or a substitute for the conventions and legal terms of a specific security. Different bond markets can use different day-count methods, ex-coupon rules, settlement conventions and treatment of irregular coupon periods.
The page is distinct from Bukit Timah Tutor’s yield-curve construction and callable-bond OAS articles. Those pages ask how future cash flows are discounted or option-adjusted. This page asks how a bond transaction allocates already-earned coupon interest between coupon dates.
The core accrued-interest fraction
For a regular coupon bond, accrued interest can be represented as:
AI = coupon payment × day-count fraction elapsed in the coupon period.
If a bond has face value F, annual coupon rate c and m coupon payments per year, the regular coupon payment is:
C = F × c / m.
Then, under a convention that expresses the elapsed fraction as d/D:
AI = C × d/D.
The danger is assuming that d and D always mean literal calendar days. They depend on the security’s day-count convention.
Worked example with a simple regular coupon
Suppose a bond has:
- face value 1,000;
- annual coupon rate 6%;
- semiannual coupons;
- coupon payment = 1,000 × 0.06 / 2 = 30;
- settlement occurs halfway through the coupon period under the applicable convention.
The accrued interest is approximately:
30 × 0.5 = 15.
If the quoted clean price is 101.20 per 100 of face value, the clean cash price for 1,000 face is 1,012. The approximate dirty cash amount is then:
1,012 + 15 = 1,027.
That example is intentionally simple. A production algorithm must derive the coupon schedule, settlement date, previous and next coupon dates, day-count convention, coupon frequency, face amount, ex-coupon status and any irregular-period rules before it can trust the number 15.
Why clean and dirty prices both exist
If market quotations included accrued interest, the displayed bond price would mechanically rise as the next coupon approached and drop when the coupon was paid, even if required yield and credit conditions did not change. Quoting clean prices removes much of that mechanical coupon accrual from the displayed market price.
The dirty price is closer to the actual cash exchanged at settlement. The clean price is often more useful for comparing market-price movement over time. Neither is “more real” in every context; they answer different questions.
Day-count conventions are algorithms, not labels
A day-count convention defines how a system converts dates into a year or coupon-period fraction. Common fixed-income families include Actual/Actual variants and 30/360 variants. They can produce different accrued-interest amounts for the same calendar dates.
For example, under a 30/360-style convention, months are treated according to a rule-based 30-day structure rather than simply counting every calendar day. MSRB Rule G-33 provides explicit day-counting requirements for municipal-securities calculations and illustrates how day counts are part of the prescribed formula, not a free assumption.
A robust algorithm therefore stores the convention as a function:
yearFraction(start date, end date, convention, reference period).
If two systems agree on coupon rate and settlement date but use different day-count functions, they can disagree on accrued interest while both arithmetic engines appear internally consistent.
Coupon schedule generation comes before accrual
The engine first needs to know which coupon period contains the settlement date. For a regular semiannual bond, that may be easy. For real securities, the schedule can include:
- short first coupons;
- long first coupons;
- short or long final coupons;
- end-of-month rules;
- business-day adjustments;
- holidays in the relevant market calendar;
- different payment frequency from a textbook assumption.
MSRB guidance is useful here because it explicitly notes that non-standard security features can create anomalies if a calculator applies a setting that was not designed for the instrument. The general lesson is broader than municipal bonds: do not force irregular cash flows through a regular-period formula without checking the contract and convention.
Settlement date matters more than trade date for accrued interest
Accrued interest is normally determined to the transaction’s settlement date under the applicable market rules. Changing settlement by one business day can change accrued interest even when trade date and clean price are unchanged.
This creates a clean verification test: take the same bond and clean price, shift settlement by one valid day, and confirm that accrued interest moves by exactly the amount implied by the day-count convention unless an ex-coupon or coupon-date boundary changes the rule.
Reopened government securities show why accrued interest is economically necessary
US Treasury securities can be reopened after their original issue. TreasuryDirect explains that a reopened security keeps the original maturity date and coupon rate, and a purchaser may need to pay accrued interest. That accrued amount is then included back with the first interest payment received by the purchaser.
This is a useful counterexample to the idea that accrued interest is merely a secondary-market convention. Even at a reopening auction, settlement can occur after interest has already accrued from the underlying security’s coupon schedule.
The ex-coupon case can reverse the sign
The simple rule “dirty = clean + positive accrued interest” has an important counterexample. Some markets use ex-coupon or ex-dividend periods. The UK Debt Management Office explains that, for gilts settling after the ex-dividend date, the seller remains entitled to the upcoming coupon even though the buyer owns the gilt at settlement. In that period, accrued interest can be negative and is often called rebate interest.
The sign flips because the allocation problem flips. Instead of the buyer compensating the seller for interest already earned, the seller must effectively compensate the buyer for the portion of the coupon period the buyer will own without receiving that upcoming coupon.
This is exactly the kind of boundary case that catches weak bond calculators.
A useful state machine
Accrued-interest calculation can be represented as a small state machine:
- identify security and contractual coupon schedule;
- determine valid settlement date;
- locate previous and next coupon boundaries;
- determine whether settlement is cum-coupon or ex-coupon;
- select the correct day-count convention;
- calculate elapsed or rebate fraction;
- calculate accrued interest;
- combine with clean price using the market’s cash-settlement convention;
- round only at the prescribed stage;
- reconcile the final amount to an independent calculator or market source.
Each transition depends on dates and security terms. The formula at the centre is the easy part.
Inputs and outputs
A generic accrued-interest engine needs:
- security identifier;
- face or par amount;
- coupon rate;
- coupon frequency;
- dated date or accrual start where relevant;
- coupon schedule and maturity;
- trade and settlement dates;
- business-day calendar;
- day-count convention;
- ex-coupon or ex-dividend rules;
- clean price;
- rounding and quotation convention.
Outputs include accrued or rebate interest, dirty price, cash settlement amount and the intermediate dates and fractions needed to explain the result.
Failure modes
Wrong day-count convention: the engine uses Actual/Actual where the security requires a 30/360 rule, or vice versa.
Trade-date accrual: interest is calculated to trade date when the market convention requires settlement date.
Coupon-frequency assumption: the calculator hard-codes semiannual payments for a security with another frequency.
Stub-period failure: a short or long first coupon is treated as a normal period.
Holiday error: settlement is placed on an invalid date or business-day adjustment is applied incorrectly.
Ex-coupon blindness: the engine continues to add positive accrued interest when the market convention requires rebate interest.
Rounding too early: the accrued fraction is rounded before multiplication, creating systematic cash differences across large positions.
Price-unit mismatch: a price quoted per 100 of face value is accidentally treated as the full cash price.
Diagnostics and verification
- Coupon-boundary test: accrued interest should normally reset at a coupon date under the relevant convention.
- One-day delta test: move settlement by one valid day and compare the change with the expected daily accrual.
- Convention cross-check: calculate the same dates under two conventions to make sure the system is actually using the specified one.
- Ex-coupon sign test: move settlement across the ex-coupon boundary and confirm the sign and ownership logic change correctly.
- Reopening test: verify that a reopened security uses the original coupon schedule rather than treating reopening as a new coupon start.
- Stub test: construct short and long first-coupon cases and compare with the documented market formula.
- Independent recomputation: compare a sample with an authoritative market calculator, rulebook formula or independently written script.
- Cash reconciliation: verify clean cash price + accrued or rebate interest = settlement cash amount, subject to any other transaction components.
- Round-trip test: remove accrued interest from the dirty price and recover the clean price within the permitted rounding tolerance.
Evidence polarity: what supports and what weakens a result?
Confidence rises when the coupon schedule matches the security’s official terms, the day-count convention is documented, the settlement date is valid, an independent source reproduces the accrued amount and boundary tests behave as expected.
Confidence falls when two systems disagree on previous or next coupon date, when the ex-coupon flag is undocumented, when the same security is assigned different conventions in different source systems, when one-day settlement shifts produce irregular jumps away from boundaries, or when differences disappear only after unexplained manual overrides.
Alternatives answer different questions
Accrued interest allocates coupon ownership between dates. It does not value the bond’s future cash flows. Discounted-cash-flow valuation, yield-to-maturity calculations, spread models and option-adjusted spread answer different questions.
For a floating-rate instrument, coupon determination can itself depend on a reference rate, observation window and compounding convention. Bukit Timah Tutor’s article on SOFR floating-rate interest algorithms shows why “interest accrued” can become a more complex process when the coupon rate is not fixed in advance.
What would falsify confidence in the accrued-interest engine?
Confidence should fall if coupon-boundary tests do not reset correctly; if the engine cannot reproduce the market rulebook’s examples; if changing settlement date leaves accrual unchanged when it should move; if ex-coupon cases retain the ordinary positive-accrual sign; if reopened securities generate a new schedule incorrectly; or if independent systems disagree because the underlying security terms cannot be traced to an authoritative source.
A good accrued-interest engine should be explainable from dates and rules. If the only defence of a number is “the system produced it,” verification has failed.
Connections across Bukit Timah Tutor
Accrued interest sits between cash-flow schedules and market valuation. Connect it to yield-curve algorithms for present-value mathematics, repo pricing for bond collateral and funding, and independent price verification for the control question: can the final price and its inputs be reproduced from independent evidence?
Update triggers
The mathematics of accrued interest is stable, but market conventions and settlement practices can change. Re-check this article when a relevant market changes standard settlement timing, ex-coupon rules, day-count definitions, coupon-frequency practices or official calculation guidance.
Primary and high-quality references
- Municipal Securities Rulemaking Board, Rule G-33 — Calculations, including accrued-interest, dollar-price, yield and day-count provisions.
- US Treasury, TreasuryDirect, Reopenings, including treatment of accrued interest on reopened securities.
- US Treasury, TreasuryDirect, When Auctions Happen, including examples where accrued interest is added to purchase price.
- UK Debt Management Office, About Gilts, including accrued interest and ex-dividend rebate-interest examples.
- Financial products Markup Language, day-count fraction definitions, for machine-readable convention terminology.
Educational boundary: This article teaches date arithmetic and bond-settlement conventions. It does not recommend any security, price or investment action.