Accrued Interest: The Part of the Price That Is Not Price

Accrued interest is the slice of the next coupon a seller has already earned by the time the money moves. Work it out from the coupon for one period, the days gone by, the days in the period and a stated day count basis. The accrual is added to the quoted price at settlement. Which basis applies to a given instrument is a rule set elsewhere rather than a result of this arithmetic.

The panel opens on a worked example, and it is set out here in words as well so it can be read without touching anything. A holding of Rs 1,000.00/- of face amount carries 8.50 per cent a year and pays once a year, so one whole coupon period pays Rs 85.00/-. The last coupon fell on 31 December 2025 and the money moves on 30 April 2026, 120 days later on a calendar. On an actual over 365 basis the accrual is Rs 85.00/- times 120 over 365, or Rs 27.9452/-, and against a clean quote of Rs 1,000.00/- the amount payable is Rs 1,027.9452/-.

A coupon is earned smoothly and paid in lumps. Interest builds up every day a bond is held, but the instrument hands the whole payment to whoever is holding it on the payment date, however recently they bought it. A seller who has held the bond for a third of the coupon periodThe stretch from one coupon payment date to the next. Every accrual is worked inside one of those stretches. and then sells has earned a third of that payment and will receive nothing. Accrued interest is the arithmetic that closes that gap. The buyer pays the seller for the part the seller earned, and the whole payment landing later is then a reimbursement rather than a windfall.

What does this calculation work out, and what has to be handed to it?

The calculation works out one number in rupees: the interest earned so far in the current coupon period and not yet paid to anybody. To produce it, five things have to be handed over, and the fifth is the one people forget.

The first is the coupon rateThe yearly rate written into the bond's own terms, which applied to the face amount gives the rupees promised each year., expressed as a rate for a year. The second is the face amount. A rate with nothing underneath it cannot be turned into money. The third is the number of payments in a year, and that number decides how much of a year one coupon period covers. The fourth is a pair of counts: the days gone by since the last coupon date, and the days in the whole period. The fifth is the day count basis, the rule deciding how both of those counts get made.

The day count basis is an input, sitting alongside the face amount and the coupon rate, and not a setting somebody chose once and left alone. The same bond, the same two dates and the same coupon produce different rupee answers under different bases, and none of those answers is wrong. Each is right under the basis it was worked out on, and each is meaningless without that basis printed next to it.

Where does each of those five inputs come from?

Three of them are read off the instrument's own terms, one is read off a calendar that somebody else controls, and one cannot be read off the instrument at all. The table below gives where to look and in what form the number has to arrive. The meaning of each input is covered separately.

The fourth row is the one that catches people. The count does not run to the day the deal was agreed. The count runs to the settlement dateThe day the security and the money actually change hands, a fixed number of working days after the two sides agree the deal., the day the security and the money actually move, and that date is fixed by a cycle rather than by the two people dealing. So the trade dateThe day the two sides agree the terms of a deal, set apart from the money date by a cycle neither of them chooses. and the date the count runs to are ordinarily different days, and using the wrong one shifts the accrual by a day or two of interest every time.

The fifth row is the one that has to stay empty. Which basis attaches to a given instrument, and which settlement cycle it trades on, are written down by the Reserve Bank of India at rbi.org.in for government securitiesInstruments through which a government borrows, and in most rupee markets the reference other borrowing is measured against. and the money marketWhere borrowing and lending is done for very short stretches, often well under a year., and by the Securities and Exchange Board of India (SEBI) at sebi.gov.in where the borrowing is corporate debtBorrowing raised by a company rather than a government, through instruments a buyer can later sell on.. Each of those texts is revised from time to time, and each office's own wording is the current one. A convention that can be rewritten is why the calculation takes the basis as an input instead of choosing one quietly, and a calculator that chooses one without saying so is behind most of the disagreements between two people working out the same accrual.

How is the accrual worked out once those inputs are in?

One multiplication and one division, in that order. Work out what one whole coupon period pays: the coupon rate applied to the face amount and then shared over the payments in a year. Multiply that by the days gone by. Divide by the days in the period. The result is the accrual in rupees.

Two things about that second fraction repay a slow reading. The fraction is a share of one coupon period and not a share of a year, and the two part company the moment a bond stops paying once a year. On the ten year bullet bond the period and the year happen to be the same stretch, so the distinction costs nothing and stays invisible until a schedule that pays more often turns up.

The second thing is quieter and does more damage. Both the number on top of that fraction and the number underneath it are produced by the day count basis. The basis reaches into the sum twice, before there is anything to round, and that is why two people can agree on the bond, agree on the dates, agree on the coupon and still produce two different rupee figures.

The worked run, on the ten year bullet bond

The bond carries Rs 1,000.00/- of face amount and 8.50 per cent a year written into its terms, so one whole coupon period pays Rs 85.00/-. Every figure below is worked on an actual over 365 basis, an assumption stated rather than a rule taken from anywhere.

The right hand column holds the quoted price of Rs 1,000.00/- steady at par. Par holds only if the yield has not moved off 8.50 per cent. To that steady price the column adds the accrual. The only thing changing between the rows is the count of days, so reading down the accrual column gives a straight climb. The last row is the instant before the coupon is paid rather than the instant after: the whole Rs 85.00/- has been earned, the payment falls due, and the accrual drops to nothing at once.

Why is that fraction a share of one period rather than a share of a year?

A bond paying once a year never raises the question at all, and that silence is precisely why it catches people the first time they meet a bond that pays more often. Take the same Rs 85.00/- a year and suppose it arrived instead as two payments of Rs 42.50/-, an arrangement declared here to make the arithmetic visible and attached to no instrument. Suppose 100 days have gone by inside a period of 182 days.

A route is only right or wrong against a basis, so name the basis before anything else: say it is actual days over the actual days in the period. The sum that follows from that takes one period's coupon, Rs 42.50/-, and keeps 100 of its 182 days: Rs 42.50/- times 100 over 182 comes to Rs 23.3516/-. The sum that feels right instead takes the yearly Rs 85.00/- and keeps 100 of the year's 365 days, coming to Rs 23.2877/-. The second route has put a year underneath the fraction where the named basis puts the coupon period. The wrong route lands within seven paise of the right one. A figure that looks about right is never checked, so the wrong route survives.

Worked properly the gap is Rs 0.0640/-. Subtracting the two printed figures instead gives Rs 0.0639/-, one paisa adrift. Printed figures are already rounded, and rounded figures make poor inputs. On Rs 1,000.00/- of face amount that gap is small change, and on a settlement covering a few crore it is not. The gap never announces itself either: both routes produce a number that looks entirely reasonable sitting on its own.

The panel above shows one more thing about that second figure. Set the schedule to two payments a year and the basis to actual over 365, and Rs 23.2877/- comes back as the right answer. A fixed 365 basis puts 182.5 days under the fraction by definition. The shape of the sum is not right or wrong on its own, only against a basis somebody has named.

Why does a screen quote a price with the accrual taken out of it?

Because a price with the accrual inside it is almost unreadable. The accrual climbs steadily for a whole coupon period, reaches the full Rs 85.00/-, and then falls to nothing in a single day when the payment is made. Add that to a price and the price inherits the same jagged pattern, rising for a year and dropping like a stone on one date, over and over, for the whole life of the instrument.

None of that movement carries any information. The sawtooth is a calendar, not the market changing its mind about the borrower and not the yield moving. Taking the accrual out leaves a figure that only moves when something worth watching has moved, and a figure like that is the one everybody looks at. A price quoted with the accrual stripped out is called a clean price, and the amount actually handed over, with the accrual added back, is called a dirty price. The names are ugly and the distinction is not.

Take one moment out of that picture and look at it closely. At 120 days into the coupon period, with the yield still at 8.50 per cent, the screen reads Rs 1,000.00/- and the money that moves is Rs 1,027.9452/- for every Rs 1,000.00/- of face amount. Both figures describe the same bond on the same morning, neither is a mistake, and the question each one answers is not written on the number.

What does the note at settlement actually set out?

Three separate lines, and the habit worth building is reading them as three. The quoted price is one line. The accrual is a second line, worked out from days and a coupon and nothing else. The amount payable is the third, and it is simply the first two added together.

Why does the day count basis change the answer at all?

Because it decides both counts in the fraction, and it can move them in different directions. Hold everything else still and run the same stretch, 31 December to 30 April, twice. Counted as the days fall on a calendar that is 120 days, and an actual over 365 basis puts 365 underneath, so the accrual is Rs 85.00/- times 120 over 365, or Rs 27.9452/-. Counted as thirty day months it is four whole months, also 120 days, but the basis puts 360 underneath, so the accrual is Rs 85.00/- times 120 over 360, or Rs 28.3333/-.

Both of those figures are right under the basis printed beside them, and neither of them is right without one. An accrual without a named basis is an incomplete statement in the same way that a length without a unit is. The basis is therefore printed next to every accrual produced here. Which basis applies to any particular instrument is a matter of rule rather than a matter of arithmetic, and is covered separately.

The gap is also less well behaved than a single worked example suggests. Measure the same bond at twelve month ends through a year, with a coupon date on 31 December, and the gap between the two bases wanders. The gap is widest at the end of April, it narrows to under four paise at the end of October, it points the other way at the end of January and again at the end of February, and it closes to exactly nothing at the end of December.

A wandering gap kills a comfortable assumption. The gap between two bases is not a fixed spread that can be learned once and carried around; it depends on where in the calendar the count falls. A reader who checks the difference in June and finds it under thirty five paise has learned nothing about what it will be in January, where it runs the other way. The December bar is drawn as an empty outline because there is nothing to fill it with: given a whole year, both bases have counted the same year.

Why does the accrual have to come out before any sensitivity is worked?

A day count belongs inside any account of rate risk for one reason. Accrued interest does not respond to a move in the yield at all. It is a count of days multiplied by a coupon that was fixed the day the instrument was written. Whether the yield rose 200 basis points this morning, fell 200, or did not move, the accrual at 120 days on an actual over 365 basis is Rs 27.9452/-, the same figure in all three worlds. No rate is one of its five inputs.

Now look at what that does to a percentage. A percentage price move is a response divided by a base. The record for this bond gives a fall of 12.030 per cent where the yield rises by 200 basis points, worked on a price of Rs 1,000.00/- struck at a coupon date, where the MODIFIED duration reads 6.5613. In rupees the bond gives up Rs 120.30/- and the clean price comes to Rs 879.70/-. The Rs 120.30/- is the response, and it is a real thing that happened to the instrument.

Divide it by a dirty price and the answer shrinks, not because the bond moved less but because the divisor grew. At 120 days into the period the settlement figure is Rs 1,027.9452/-, so the same Rs 120.30/- reads as 11.7030 per cent. The reported sensitivity has fallen by 0.3270 percentage points and nothing whatever has happened to the bond. A response has been divided by a base that contains something which did not respond.

The reading drifts through the year, for calendar reasons alone

The damage is worse than a single understatement. The size of the understatement depends on where in the coupon period the measurement was taken.

Read the third column first: the third column never changes. The bond gave up Rs 120.30/- in every single row. The fourth column moved, from 12.0300 per cent down to 11.0899 per cent, a drift of 0.9401 percentage points across one coupon period. Nobody traded, no yield moved twice, and no assumption changed. Only the date changed.

One more turn, and it is the reason the two ideas here belong together. The base in the denominator is itself a function of the day count basis chosen. At 120 days on a thirty day basis the settlement figure is Rs 1,028.3333/-, so the same Rs 120.30/- reads as 11.6985 per cent rather than 11.7030 per cent. A convention chosen for counting days has quietly walked into a measurement of interest rate risk, where it has no business being at all.

The repair is one line long and it never changes. Strip the accrual out, do every yield and sensitivity calculation on the clean figure, and put the accrual back at settlement where it belongs. Striking a duration or repricing figure on the price at a coupon date does the same work in advance: with the accrual at nothing, there is nothing to strip.

Who actually reaches for this, and what do they do with it?

Start with a version that has nothing to do with bonds. A tenant pays rent on the first of every month and moves out on the twentieth. Nobody thinks the landlord keeps the last ten days, or that the tenant keeps the first twenty free. The month gets split by days, and the split is settled when the keys change hands. The bond version differs in one respect: the payment runs the other way, from the borrower to whoever holds the instrument on the payment date, so the two holders settle it in advance rather than refund it afterwards.

Somebody running a settlement desk uses this every working day, and their use of it is the least glamorous and the most important. The desk receives a note with a price, a day count and a total, and reproduces the total from the first two before releasing anything. When the two figures disagree, the first suspect is never the price. The count is, and behind the count, the basis.

An analyst uses it in a more dangerous place. Computing what a move in rates did to a holding means dividing one number by another, and choosing the divisor is where the work goes wrong. Any percentage that will be compared with another percentage has to be struck on the same kind of base, and clean is the only base that stays put while the calendar moves. Settlement figures taken out of a system, worked into percentage moves and compared across dates, give a series that wanders for reasons nobody in the room can find.

Someone buying a bond for their own account uses it once, briefly, and at the worst possible moment: when comparing two quotes. Comparing two quotes is where the money is lost, and the failure is worth a block of its own.

What can this calculation not work out?

A tool is worth as much for the questions it refuses as for the one it answers, and this one refuses five. None of the refusals is caution. In each case the question is real and its answer simply lives somewhere that arithmetic cannot reach.

One of those deserves a further word. The fourth is worth sitting with, and the reason an accrual can be computed at all without ever mentioning a borrower is simple: the sum has no slot for one. A regulated institution carries the same bond at a different figure again on its own books, settled by a valuation normThe written rule an institution follows in deciding what figure a holding sits at on its own books. A presentation rule, not a market price. rather than by any arithmetic above.

How does the panel round, and what will it refuse to do?

Two habits make a computed figure worth trusting, and both are better stated than left inside the code. The first is that every amount is carried as a whole number of ten thousandths of a rupee, a hundredth of a paisa, and rounded once, half away from zero, at the moment it is printed. No rounded figure is ever fed into a later step. Rounding once at the end is why the quoted price, the accrual and the amount payable add up exactly at every setting rather than nearly. Money itself settles in whole paise, so the panel prints that figure as well and says which of the two it is.

The second habit is refusal. Type a settlement date that sits past the next coupon and the panel prints no accrual at all, only the date the period ends on and the instruction to move the last coupon date forward. Guessing instead would be easy, and the guess would look exactly like a worked answer. An instrument that fills in a missing input on the reader's behalf hands back a number indistinguishable from one it actually worked out. The same refusal covers the basis, the settlement cycle, whether the coupon will be paid and what it is worth after tax.

Driving the calculator into the failure named here shows both habits earning their keep. With the settlement date on the coupon date itself the accrual is nothing, so the quoted price and the amount payable are one figure. Moved to the day before the next coupon, the accrual is nearly a whole period's coupon, so the two stand almost Rs 85.00/- apart on a bond whose quoted price never moved. Anybody comparing the first screen against the second is comparing one bond with itself.