Day count and accrual

Day count fraction

The quantity that converts a calendar period into a fraction of a year, or of a coupon period, so that a rate can be turned into an amount. It is a term of the instrument. Two otherwise identical bonds with different bases pay different amounts on the same dates.

Field	Value
Formula	Interest = notional * rate * D/B, where D is the convention's day count for the period and B its denominator
Worked	Reference bond, 2026-05-15 to 2026-08-27: on ACT/ACT (ICMA), 100 * 0.045 * (104/184)/2 = 1.271739 per 100 of par
Same period, ACT/360	100 * 0.045 * 104/360 = 1.300000
Same period, 30/360	100 * 0.045 * 102/360 = 1.275000
Spread	0.028261 per 100 of par, or 2,826.09 on 10,000,000 par

• ACT/ACT (ICMA) is the odd one out structurally: it is defined per coupon period, not per year, so it has no fixed denominator. Code that models it as D/B with B = 365 or 366 will be wrong on any period that is not exactly half a year, which for a 15 May / 15 November schedule is every period.
• The four ACT-numerator conventions differ only in the denominator, so on any single period they scale each other exactly. The three 30-numerator conventions differ only in the month-end adjustment, so they agree on most dates and diverge precisely where semiannual schedules land.
• A basis mismatch shows up as a settlement break, not a valuation error, which is why it is usually caught by operations rather than by trading. The cheapest control is to reconcile the accrued figure, not the price, on every new issue.
• Confirm the basis against the offering document or the ISDA confirmation, not against a data vendor's default. Vendors carry a basis field and they carry it wrong often enough to matter.

Source: 2006 ISDA Definitions, Section 4.16 (Day Count Fraction); ICMA Rule 251

Also described at: Wikipedia · Wikidata · 2006 ISDA Definitions

30/360, Bond Basis (30U/360)

Every month is treated as 30 days and every year as 360. The US variant adds three month-end adjustments, two of which concern February, and it is the presence or absence of those adjustments that separates this convention from the two others sharing the 30/360 name.

Field	Value
Formula	D = 360*(Y2-Y1) + 30*(M2-M1) + (D2-D1), applying in order: if D1 is the last day of February and D2 is the last day of February then D2 = 30; if D1 is the last day of February then D1 = 30; if D1 is 31 then D1 = 30; if D2 is 31 and D1 is now 30 then D2 = 30. B = 360
Worked, reference bond	2026-05-15 to 2026-08-27: no adjustment applies, so D = 360*0 + 30*(8-5) + (27-15) = 102, D/B = 102/360 = 0.283333333
Worked, February start	2026-02-28 to 2026-08-31: D1 is the last day of February so D1 = 30; D2 is 31 and D1 is 30 so D2 = 30; D = 30*(8-2) + (30-30) = 180
Worked, February end	2026-08-31 to 2027-02-28: D1 = 31 becomes 30; D2 = 28 is untouched because D1 was not a February month end; D = 360 + 30*(2-8) + (28-30) = 178
The consequence	Those two half-years sum to 358 days, not 360, on a convention whose entire purpose is that a year contains 360

• The asymmetry in the last field is real and is the single most common 30/360 bug. A bond with 28 February and 31 August coupon dates accrues 180/360 in one half and 178/360 in the other, so its annual coupon on a strict day-count basis is 358/360 of the stated rate. Bond documentation resolves this by paying an exact half-coupon regardless; swap documentation does not, and pays the day-count amount.
• The order of the adjustments matters. Applying the D2 = 31 rule before the D1 = 31 rule changes the answer on a 31st-to-31st pair, and both orders appear in production code.
• The February rules are the US addition and are absent from 30E/360. If a system produces 182 where another produces 180 on a February-to-August period, the difference is almost always this, not a bug.
• This is the default basis for US corporate, agency and municipal bonds and for the fixed leg of a USD swap, which means a corporate bond and its asset swap usually share the basis while the bond and a Treasury do not.

Source: 2006 ISDA Definitions, Section 4.16(f) (30/360, 360/360, Bond Basis)

Also described at: Wikipedia · Wikidata

30E/360, Eurobond Basis

The same 30-day-month arithmetic with a single, symmetric adjustment: any 31st becomes a 30th, at either end of the period, unconditionally. February is not special.

Field	Value
Formula	D = 360*(Y2-Y1) + 30*(M2-M1) + (D2-D1), where D1 = min(D1, 30) and D2 = min(D2, 30). B = 360
Worked, reference bond	2026-05-15 to 2026-08-27: D = 102, identical to Bond Basis because neither date is a 31st
Worked, February start	2026-02-28 to 2026-08-31: D1 = 28 stands, D2 = 31 becomes 30, so D = 30*6 + (30-28) = 182 against 180 on Bond Basis
Worked, leap February	2024-02-29 to 2024-08-31: D = 181 against 180 on Bond Basis
Difference in interest	On 10,000,000 at 4.500 percent, a 2-day divergence is 2,500.00

• Because it has no February rule, 30E/360 gives a longer count than Bond Basis for a February-to-month-end period and the same count for the reverse. The bias is directional, not random.
• The name is unhelpful. This is the older ICMA 30/360, and ICMA's own current recommendation for new Eurobond issuance is ACT/ACT (ICMA), so a bond described as Eurobond Basis is usually an older one.
• Of the three 30-day variants this is the only one that never inspects whether a date is a month end, which makes it the cheapest to implement correctly and the easiest to confuse with the other two.

Source: 2006 ISDA Definitions, Section 4.16(g) (30E/360, Eurobond Basis)

Also described at: Wikipedia · Wikidata · ICMA rules and recommendations for the secondary market

30E/360 (ISDA), German basis

The 30-day-month arithmetic with the month-end test generalised: any date that is the last day of its month becomes a 30th, whatever that date happens to be. The single exception is a termination date falling in February, which is left alone.

Field	Value
Formula	D = 360*(Y2-Y1) + 30*(M2-M1) + (D2-D1), where D1 = 30 if D1 is the last day of its month, and D2 = 30 if D2 is the last day of its month unless D2 is the termination date and falls in February. B = 360
Worked, January to February	2026-01-31 to 2026-02-28: both are month ends, so both become 30 and D = 30*(2-1) + 0 = 30, against 28 on Bond Basis and 28 actual days
Worked, the reverse leg	2026-08-31 to 2027-02-28: both month ends become 30, D = 180, against 178 on Bond Basis
The property that buys	180 + 180 = 360, so a month-end year does sum to 360
Termination exception	If the second date is the final maturity and falls in February, it is not rolled to 30, so the last period is shorter than the others

• This is the only 30-day variant under which a full year of month-end periods sums to exactly 360. That is the point of it, and it is why German domestic instruments use it.
• The February termination carve-out exists so that the final period ends on the real maturity date rather than a notional 30 February. It makes the last coupon a different length from every other coupon, which is correct and still surprises people.
• The month-end test needs a calendar, not a comparison against 31. Code that tests D == 31 implements 30E/360 and not this convention, and the two agree on most dates, so the bug survives testing.

Source: 2006 ISDA Definitions, Section 4.16(h) (30E/360 (ISDA))

Also described at: Wikipedia · Wikidata · 2006 ISDA Definitions

ACT/ACT (ICMA), Rule 251

Accrual measured against the coupon period rather than against the year: actual days elapsed in the period, over actual days in the period, times one coupon. It is the only common convention that guarantees the sum of a year's accruals equals the stated annual coupon.

Field	Value
Formula	Accrued = (C/f) * A/E, where A is actual days from the last coupon date to settlement and E is actual days in the current coupon period. Expressed as a year fraction, D/B = A/(E*f)
Worked	Reference bond: A = 104, E = 184, C/f = 2.250, so accrued = 2.250 * 104/184 = 1.271739 per 100 of par
As a year fraction	104/(184 * 2) = 0.282608696
Period lengths on this bond	15 May to 15 November is 184 days; 15 November to 15 May is 181 days in a normal year and 182 across a leap February
Self-check	Each period accrues to exactly 2.250 at its end regardless of length, so the annual total is exactly 4.500

• The denominator changes every period. On this bond the same 30 days of holding accrues 0.366848 in the 184-day period and 0.372928 in the 181-day period. Daily accrual is not a constant, and any system that caches a per-day amount is wrong twice a year.
• This is the accrual convention for US Treasury notes and bonds, gilts, and most sovereign and modern Eurobond issuance. It is also the reason a Treasury and a US corporate bond of identical coupon and dates report different accrued interest.
• For irregular first or last periods the rule constructs notional quasi-coupon periods of the regular length and sums the fractions, which is what makes a long first coupon computable at all. See the entry on irregular periods.
• ICMA's formulation is stated per period, so it does not need to know what a year is. That is a feature: no leap-year handling is required anywhere in the calculation.

Source: ICMA Rule 251 (Accrued Interest Calculation); 2006 ISDA Definitions, Section 4.16(c) (ACT/ACT (ICMA))

Also described at: Wikipedia · Wikidata · ICMA rules and recommendations for the secondary market

ACT/ACT (ISDA)

Actual days split at each 31 December, with days falling in a leap year divided by 366 and days in a non-leap year by 365. It shares its name and numerator with ACT/ACT (ICMA) and is a different function.

Field	Value
Formula	D/B = (days in the period falling in non-leap years)/365 + (days falling in leap years)/366
Worked, reference bond accrual	2026-05-15 to 2026-08-27: 104/365 = 0.284931507
Worked, across a year end	2026-11-15 to 2027-05-15: 47/365 + 134/365 = 0.495890411
Worked, across a leap February	2027-11-15 to 2028-05-15: 47/365 + 135/366 = 0.497619582
Full year check	2026-05-15 to 2027-05-15: 231/365 + 134/365 = 1.000000000 exactly
What it does not do	A regular semiannual period gives 0.495890411 of a year, so 4.500 percent applied to it is 2.231507 and not 2.250000

• The last field is the reason this convention is not used for coupon accrual. It measures a fraction of a calendar year, and half a calendar year is not half a coupon period, so it does not reproduce the coupon. Applying it to a bond produces a coupon that varies period to period.
• It is exact over any whole number of years, which is the property it was designed for and which ACT/365F lacks.
• The split happens at each 31 December inclusive, so a period spanning three calendar years has three terms. Implementations that handle only two are common and only fail on periods longer than about a year.
• When a confirmation says ACT/ACT with no qualifier, the question of which one is a real ambiguity with a real cash consequence, and it should be resolved against the document rather than assumed from the asset class.

Source: 2006 ISDA Definitions, Section 4.16(b) (ACT/ACT, ACT/ACT (ISDA), Actual/365, Act/365)

Also described at: Wikipedia · Wikidata · 2006 ISDA Definitions

ACT/360

Actual days over a 360-day year. Because the calendar year has 365 or 366 days and the denominator has 360, a full year accrues more than the stated rate.

Field	Value
Formula	D/B = actual days / 360. Over a 365-day year the fraction is 365/360 = 1.013889
Worked	Reference bond period: 104/360 = 0.288888889, giving 1.300000 per 100 of par at 4.500 percent
Annual overstatement	A 4.000 percent ACT/360 rate held for a full 365-day year pays 4.000 * 365/360 = 4.055556 percent
Equivalent ACT/365F rate	4.000 * 365/360 = 4.055556 percent on an ACT/365F basis is the same cash
On 10,000,000 for 90 days	10,000,000 * 0.04 * 90/360 = 100,000.00 against 98,630.14 on ACT/365F

• This is the USD and EUR money-market basis and the basis for SOFR and EURIBOR floating legs, repo, and commercial paper. A SOFR coupon quoted at 4.000 percent is an ACT/360 rate and is not directly comparable with a bond yield quoted on ACT/ACT.
• The 365/360 factor of 1.013889 is the single most useful number in cross-basis comparison. A quoted money-market rate must be multiplied by it before being set against a 365-basis rate, and the correction is worth about 5.6 basis points on a 4 percent rate.
• The denominator does not change in a leap year, so a leap year accrues 366/360 of the stated rate. That is intentional and is the definition, not an error to correct.
• Because the numerator is actual days, an ACT/360 period ending on an adjusted business day accrues over the adjusted length. Business-day adjustment therefore changes the interest amount on a floating leg, which it does not do on a bond paying an exact half-coupon.

Also described at: Wikipedia · Wikidata

ACT/365F, Actual/365 Fixed

Actual days over a fixed 365-day denominator, in leap years as well as ordinary ones. The word Fixed distinguishes it from ACT/ACT (ISDA), which is sometimes also written Actual/365.

Field	Value
Formula	D/B = actual days / 365, always
Worked	Reference bond period: 104/365 = 0.284931507, giving 1.282192 per 100 of par
Leap year	2024-01-01 to 2025-01-01 is 366 days, so D/B = 366/365 = 1.002739726 of a year
Against ACT/ACT (ISDA)	The same leap year on ACT/ACT (ISDA) is 366/366 = 1.000000000
Cost of the difference	On 10,000,000 at 4.000 percent for that year, 401,095.89 against 400,000.00

• The Fixed suffix is load-bearing. Actual/365 without it is ambiguous and has been read both ways in production systems.
• This is the money-market and floating-leg basis for GBP, AUD, CAD, JPY and HKD. A SONIA leg and a SOFR leg of the same quoted spread are not economically equal, and the wedge is the 365/360 factor.
• Cross-currency basis quoting conventions inherit this split, so a basis spread quoted against SOFR and one quoted against SONIA are on different denominators before any spread is compared.

Also described at: Wikipedia · Wikidata

NL/365, Actual/365 No Leap

Actual days with every 29 February inside the period deleted, over a fixed 365-day denominator. It exists so that a period of a given calendar length always produces the same interest amount regardless of where it sits.

Field	Value
Formula	D = actual days minus the count of 29 February dates in (start, end]. B = 365
Worked, reference bond	2026-05-15 to 2026-08-27: no 29 February in the period, so D = 104, the same as ACT/365F
Worked, spanning a leap day	2024-01-01 to 2024-06-30: actual 181 days, less one for 29 February, so D = 180
Worked, a leap year	2024-01-01 to 2025-01-01: D = 365, so D/B = 1.000000000 exactly
The property	Any period of n calendar days accrues the same amount whether or not it contains a leap day, so 29 February is an interest-free day

• Appears in some US agency documentation and in mortgage and loan servicing systems, and rarely in traded bonds. If a bond claims this basis, confirm it against the document rather than the vendor field.
• The rule is stated on the half-open interval, so a period starting on 29 February does not lose the day and one ending on it does. Both readings exist in code and they differ by one day of interest once every four years.
• It is the only common convention under which a borrower pays nothing on a specific calendar date, which is worth knowing before explaining a servicing statement to anyone.

Accrued interest

The share of the coming coupon that has economically been earned by the seller as at settlement date. It is compensation for holding period, not part of the negotiated price, and it is why the number a bond trades at is not the number that settles.

Field	Value
Formula	AI = (C/f) * (A/E) on ACT/ACT (ICMA), or AI = C * D/B on a fraction-of-year convention. Settlement date counts as a day of the buyer, not the seller
Worked	Reference bond: 2.250 * 104/184 = 1.271739 per 100 of par
On 10,000,000 par	127,173.91
Daily accrual in this period	2.250/184 = 0.012228 per 100 of par per day, or 1,222.83 on 10,000,000
Daily accrual next period	2.250/181 = 0.012431, a different number for the same instrument

• The convention is that accrual runs from the last coupon date up to but excluding settlement date. A one-day error in that boundary is the most common accrued break and is worth a full day of coupon, which on 10,000,000 of this bond is about 122.
• Accrued interest is not a component of yield. Yield is computed from the full price; accrued is the bridge between the quoted price and the full price. Adding accrued into a price and then quoting it as a price is a category error that produces a yield roughly a basis point wrong here.
• Because accrued is linear in time while the full price compounds, the quoted price of a bond trading exactly at its own coupon yield is slightly below par between coupon dates. On this bond at a yield of exactly 4.500 percent the quoted price is not 100 but roughly 99.9938. That gap is a convention artefact, not a mispricing.
• Defaulted, deferred-coupon and pay-in-kind instruments trade flat, meaning no accrued changes hands. Whether a distressed bond trades flat is a negotiated point and it moves the settlement amount by the whole accrued figure.

Source: ICMA Rule 251; MSRB Rule G-33 (Calculations) for municipal securities

Also described at: Wikipedia · Wikidata · 31 CFR Part 356, Appendix B (formulas and tables)

Clean price, dirty price and invoice amount

Three numbers that are routinely conflated. The clean or quoted price is what is negotiated and what appears on a screen. The dirty or full price is clean plus accrued and is the number every valuation formula uses. The invoice amount is the dirty price scaled to the traded par amount, and is the only one that moves cash.

Field	Value
Formula	P_full = P + AI. Invoice = par * P_full/100. Yield, duration, DV01 and convexity are all functions of P_full, never of P
Worked, per 100	98.750000 + 1.271739 = 100.021739
Worked, 10,000,000 par	principal 9,875,000.00 + accrued 127,173.91 = 10,002,173.91
Why it matters	Discounting the reference bond's cash flows at its yield gives 100.021739, which is the full price. Setting that equal to the quoted price would put the yield out by roughly 18.8 basis points
Per 1,000 par	principal 987.50000, accrued 12.71739, invoice 1000.21739

• The quoted price is quoted because it is the part the market has an opinion about. Accrued is arithmetic, so quoting it would add noise to the negotiation without adding information. That is the whole reason the clean convention exists.
• A clean price is comparable across settlement dates and a dirty price is not, because the dirty price sawtooths upward through each coupon period and drops by the coupon on payment. Any time series of dirty prices contains that sawtooth and it is not a return.
• Rounding is a settlement matter with its own conventions: many markets round the invoice amount to the cent and some round accrued per 1,000 or per 1,000,000 of par before scaling. Which rounding applies is a documented term, and the difference is real money on a large ticket.
• Some markets quote dirty. Certain money-market instruments, most short bills, and a number of emerging-market conventions trade on a full-price or discount basis, so the assumption that a quoted number is clean is itself market-specific.

Also described at: Wikipedia · Wikidata

Ex-dividend dates and negative accrued interest

Where a market fixes the coupon recipient a few days before payment, a trade settling inside that window transfers the bond without the coming coupon. The buyer is compensated by paying negative accrued: the invoice amount falls below the quoted price.

Field	Value
Formula	Inside the ex-dividend period, AI = (C/f) * (A/E) - (C/f) = -(C/f) * (E-A)/E, which is negative and grows toward zero as settlement approaches the coupon date
Convention	Gilts go ex-dividend seven business days before the coupon date; the record date is the close of business on the day before the ex-dividend date
Worked	Reference bond schedule, coupon date 2026-11-15, settlement 2026-11-10, so A = 179 and E = 184. Normal accrued would be 2.250 * 179/184 = 2.188859
Ex-dividend accrued	2.188859 - 2.250 = -0.061141, which equals -2.250 * 5/184
Invoice	At a quoted price of 98.750 the full price is 98.688859, below the quoted price
Five days earlier	Settlement 2026-11-04: accrued = -0.134511, a larger negative because more of the coupon is still to come

• US Treasuries and US corporates have no ex-dividend period; the coupon follows the position on the payment date, so accrued is never negative. The concept is a gilt, JGB and some-European convention and code written only against US instruments will reject a negative accrued as invalid input.
• The sign convention makes the arithmetic continuous: accrued jumps from just under a full coupon to minus a small amount at the ex-date, and the full price therefore drops by exactly one coupon on the ex-date rather than on the payment date. Marking a gilt through its ex-date without this produces a spurious one-day loss equal to the coupon.
• Special ex and special cum trades let the parties override the default, which means the ex-dividend status of a specific trade is a trade term and not purely a function of the settlement date.
• The window is in business days, so a holiday inside it moves the ex-date. Deriving it with a fixed seven calendar days is wrong roughly whenever it matters.

Source: ICMA Rule 251; UK Debt Management Office gilt market conventions

Also described at: Wikipedia · Wikidata

Irregular first and last coupon periods

A bond whose issue date does not fall on its coupon schedule has a first period that is shorter or longer than the rest, and its first coupon is correspondingly short or long. ACT/ACT (ICMA) handles this by constructing notional quasi-coupon periods of the regular length and summing fractions across them.

Field	Value
Formula	Short first period: coupon = (C/f) * A/E, where E is the regular quasi-coupon period containing the issue date. Long first period: coupon = (C/f) * [A1/E1 + A2/E2], summed over each quasi-coupon period the long period touches
Setup	Reference bond schedule with an assumed issue date of 2026-03-02, so the first period runs 2026-03-02 to 2026-05-15 inside the 181-day quasi-coupon period 2025-11-15 to 2026-05-15
Worked, short first coupon	A = 74, E = 181, so the first coupon is 2.250 * 74/181 = 0.919890 rather than 2.250
Worked, long first coupon	If instead the first payment were deferred to 2026-11-15, it would be 2.250 * [74/181 + 184/184] = 3.169890
Why quasi-periods	Without them a long period has no well-defined E, since it spans two schedule intervals of different actual lengths

• The quasi-coupon period is notional: no payment occurs at its start, and it exists only to supply a denominator. It is generated from the schedule anchor, which is normally the maturity date rolled backwards, not the issue date rolled forwards.
• The yield calculation for the first period uses the same w exponent machinery as any other period, with w measured against the quasi-period. A pricing routine that derives w from the actual first period length will misprice a new issue for its entire first period and then be correct forever afterwards, which makes the bug hard to find.
• Schedule generation direction matters independently of the day count. A forward-generated schedule from a 31 January issue date and a backward-generated one from a 30 April maturity produce different intermediate dates, hence different E, hence different accrued.
• A long first coupon is a small financing decision by the issuer and shows up as a first-period yield that differs from the bond's quoted yield. It is not a mispricing and it does not persist.

Source: ICMA Rule 251 (quasi-coupon periods)

Also described at: 2006 ISDA Definitions