Day Count Conventions: Why the Same Rate Pays Differently

A day count convention is the rule inside a swap agreement for turning the two dates of a period into a fraction of a year. The rule comes in two halves: how the days between those dates get counted, and how many days a year is taken to have. The fraction it produces multiplies the rate, so one rate on one notional pays differently under different conventions.

Everything else about a period can be settled and the amount still will not be. The notional can be known. The rate on each side can be known. The day the money moves can be known. What is left is a question nobody can answer by looking outdoors: how much of a year is this stretch of calendar? A rate is quoted annualisedExpressed as though it ran for a whole year, which is how rates are almost always quoted even when nothing actually runs a whole year., and periods are almost never a year. The convention is the agreement's own written answer to that question, and it is a term of the contract in exactly the way the rate is.

Two invented sides sit under one agreement. Chitrakoot Cements Limited has undertaken to pay at 7.20 per cent a year, and in return it collects whatever reading the floating benchmark puts up for that period. On the other side of the same document is Saranga Capital Limited, whose obligation runs the other way: it pays against the floating benchmark and collects the 7.20 per cent. A notional of Rs 1,000 crore sits underneath both undertakings and is never handed over by anybody. The notional is there so that a rate has something to be a rate of.

One thing is worth settling before the arithmetic starts. Valuing this agreement today would take a dated run of the readings the floating benchmark might go on to post, and no such run was ever assembled. Only the opening period carries a reading. Anything attached to a later period below will be a fraction, and a fraction is a piece of the agreement rather than a piece of the market.

What is a day count convention, and what does it actually produce?

Most readers meet day count conventions as a list of names. Somebody writes four or five of them on a slide, everybody copies them down, and nobody says why the list exists. So start at the other end, with the output rather than the catalogue. A day count convention is a machine that takes two dates and returns exactly one number, a decimal somewhere between nil and a little over one, and that decimal is the whole of its job. The decimal is called the day count fraction. Once it is in hand, the period's arithmetic is finished: the annual rate multiplied by the fraction, multiplied in turn by the notional, gives the leg.

Seeing it this way changes what has to be remembered. Every convention that has ever been written into an agreement is a different route to the same kind of answer. Anybody who has grasped why that decimal is wanted can work out what any named convention is doing, without ever having memorised its name. Grasping the purpose beats knowing three names. The fourth agreement a reader picks up will use a fourth name, and a name on its own computes no period.

Here is the familiar version. A flat is let at a monthly rent and the tenant moves in on the eighteenth. What is owed for that first stub of a month? No answer sits in nature waiting to be looked up. The landlord may count the days actually lived in and divide by the days that month happens to have. Or both sides may have agreed in advance, in writing, that any part month counts as a half. Neither is more honest than the other. The bill is different under each, so what matters is that the two of them settled it before the tenant moved in. An argument on the doorstep is worse than either.

The rent on that stub month is a day count convention, doing the job it does everywhere. The swap version is the same idea with more zeroes attached and a document that has to survive scrutiny years later, so the choice is written down rather than assumed. A line in an agreement stating what the day count is, is the tenant and the landlord settling the eighteenth of the month, done properly.

What are the two halves of the rule, and who chooses each?

Almost every misunderstanding here starts in the same place: treating the convention as one choice when it is two. A day count convention is a pair of rules, and the pair is chosen together and written together. Take them apart and each becomes obvious.

The half above the line decides how the days between the opening date and the closing date get counted. There might seem to be one way to count days between two dates, and there is not. The calendar is uneven, some months run to thirty days and some to thirty one and one runs shorter still, and some conventions smooth that unevenness rather than reporting it. Smoothing it is a legitimate choice. Reporting it is a legitimate choice. Neither is a discovery about time.

The half below the line decides how long a year is taken to be for the purpose of this arithmetic. Notice the phrasing. Not how long a year is, which the calendar settles, but how long a year is taken to be, which the parties settle. A leap yearA year the calendar stretches to 366 days so that the count stays in step with the seasons, which is one reason a year has no single length to divide by. exists, so real years are not all the same length, and a rule that had to change its denominator every fourth year would be a nuisance in a document meant to run for a decade. Some conventions therefore fix the denominator and accept a small imprecision in exchange for never having to argue about it.

The two-part shape gives a reading skill worth more than any list. In a convention written as a single expression with a dividing line or a slash in it, the part above or before the line is the numerator rule and the part after is the denominator rule, and knowing that is most of what the name means. Decode it that way and an unfamiliar name stops being unfamiliar.

Why does a rate a year need converting at all?

The conversion step gets skipped so often that it is worth slowing right down on. A rate is quoted for a year. Annual quotation is a habit of the market, not a statement about when anything gets paid. A period is a stretch between two dates and it is almost never a year: it might be three months, it might be six, it might be an awkward opening stub that runs from the day the agreement started to the first scheduled date and matches nothing.

So the arithmetic reaches a question it cannot avoid. What proportion of the annual rate does this particular stretch earn? To prorateTo cut an amount down to the slice of time it actually covers, so that a charge quoted for a whole year turns into a charge for the stretch that ran. anything requires a proportion, and a proportion needs a rule, and here is the uncomfortable part: no answer is waiting in nature to be discovered, because months are of different lengths and years are of different lengths, so every possible answer has to pick a rule and live with a different rule producing a slightly different number.

So the fraction is not measured. The fraction is agreed. A day count fraction is a term of the contract rather than a fact about time, which is why two agreements written over the identical pair of dates can honestly produce two different amounts and neither one is a mistake. Nothing has gone wrong. The two documents said different things about how to count, and the arithmetic obeyed.

Part periods are already settled this way everywhere else without notice. A telephone bill for a plan joined mid month, a school fee for a term started halfway through, a maintenance charge on a flat bought in October: in every case somebody had to decide what a part period counts as, and in every case the decision was written into an agreement rather than deduced from first principles. The unusual feature of a swap is only that the stakes make the decision worth reading carefully.

One period, worked at three different fractions

The best way to see how much the fraction carries is to change nothing else. The notional holds. Both rates hold. The fraction moves on its own, and the sum that actually travels moves with it.

Start with the settings this material has been using throughout. Apply 7.20 per cent to a full year and the fixed side lands on Rs 72.00 crore gross. Do the same on the floating side, where the opening reading posted 6.00 per cent, and Rs 60.00 crore gross drops out. Set the second figure against the first and Rs 12.00 crore is what remains to travel. That Rs 12.00 crore is the net difference for the period, and Chitrakoot Cements is the side that pays it across.

The shape of that product matters more than the numbers in it. Three things multiply together. Three equal partners produce the answer: the notional, the distance between the two rates, and the day count fraction, and only the last of them is missing from the front sheet of the agreement. That asymmetry is why it gets left at its previous value, and why leaving it there is a full sized error rather than a rounding matter.

Now move the fraction and hold everything else. At a fraction of 1.0000, one full year, the period settles at Rs 12.00 crore of net difference. At a declared teaching fraction of 0.5000, a half length period, it settles at Rs 6.00 crore of net difference. At a declared teaching fraction of 0.2500, a quarter length period, it settles at Rs 3.00 crore of net difference. Neither 0.5000 nor 0.2500 is any market's convention, and both are declared here purely so the proportion becomes visible.

Halving the fraction halves the payment, with the notional and both rates untouched. Quartering it quarters the payment. There is nothing subtle in that relationship, and its plainness is the point: a factor sitting in a product behaves like a factor, and the amount tracks it exactly.

One more reading closes a loop. The Rs 12.00 crore that changes hands, set against the Rs 1,000 crore the agreement is written over, gives twelve parts in a thousand. Twelve in a thousand is 1.2 per cent, and 1.2 per cent is the very gap the arithmetic started from. The gap goes out and comes home, but only because the fraction happened to be one. Any other fraction and the two stop matching, which is a quick way to notice that the fraction was there all along even when nobody wrote it down.

A curve would have told a different story. A curve would mean the fraction had some threshold in it, or some diminishing effect, and that a small change near one end mattered more than the same change near the other. There is no such structure here. The line is straight and it passes through the origin, so a given proportional change in the fraction produces the identical proportional change in the settlement wherever on the range the setting happens to sit. A reader who has internalised the straightness will never again describe the fraction as a refinement.

Can the two legs use different conventions, and what does that do to netting?

Nothing in the structure of a swap requires both legs to count a period the same way. In practice they often do not. Each leg's convention tends to follow the practice for the kind of rate it references, and the two references are different animals. So one document can carry two conventions, and both are correct, and each applies only to its own side.

When that happens the two legs produce different fractions for the same stretch of calendar, which is a sentence worth reading twice. Not different dates. Not different periods. The same opening date, the same closing date, the same stretch of days, and two different decimals describing it, because two different rules were applied to it.

The consequence lands on something already established. There is a quick route to the settlement that works beautifully most of the time: the notional multiplied by the gap between the two rates gives the net without either leg ever being computed. The shortcut saves a step and is exactly right whenever both legs share one fraction. Where the fractions differ, that shortcut stops working, and each leg has to be built in full before anything is subtracted.

The loss of the shortcut is easy to feel cheated by, so it is worth being precise about. The shortcut was never a property of a swap. The shortcut was a consequence of a condition that usually holds. Algebraically, the notional times the fixed rate times a fraction, less the notional times the floating rate times the same fraction, factors into the notional times the gap times that fraction only because the fraction is common to both terms. Give the two terms two different fractions and there is nothing left to factor out. The condition failed, so the consequence disappeared, and the underlying instrument did not change at all.

The everyday version: two flatmates split a bill and one of them proposes simply taking the difference in the two shares. Taking the difference works while both are charged over the same period. The moment one is charged for a full month and the other for a part month, the difference in their shares stops meaning anything, and each has to work out what they actually owe before anybody compares.

Why is no convention name worth carrying away?

Two reasons sit behind the absence of any convention name, and the second matters more than the first.

The first reason is routing. Which counting method an agreement of a given kind carries in this market is settled by an authority rather than by argument, and the row below names that authority.

The second reason is about the reader rather than about the source. A reader who memorises a convention name will carry it into the next agreement they open, and the next agreement will say whatever it says. Memorising is the mechanism by which the error in the failure block below gets made. Memorising feels like knowledge and behaves like a habit, and habits are precisely what a document is supposed to override.

So the rule that stands instead is short. The pair is read out of the document at hand, every single time, including the times when it seems already known. Everything above supports doing that with any pair: the fraction is the output, the numerator half is distinguishable from the denominator half, and both are choices somebody wrote down. Anybody holding those three facts can decode a name never seen before. Decoding is a durable skill and a list of names is a perishable one.

Why is only the opening period worked here?

Every worked figure above belongs to the opening period, and there is a reason that runs deeper than convenience. The opening period is the only one where every input exists. The counting rule is written down, the two dates are written down, the fixed rate is written down, and the floating benchmark has been read once. Reach for the second period and one of those four goes missing.

The limit here is unusual in one way: it cuts in an unexpected direction. The counting rule for every future period is knowable this morning, since it sits in the agreement, and the dates are knowable too, which means every future fraction could be computed right now. Nothing about a later period's fraction depends on anything anybody is waiting for. The one missing input is the rate the fraction will eventually be applied to.

The split between what is fixed and what is open is a sharper way of seeing a swap than the usual one. People describe it loosely as an arrangement whose future is unknown, which is too broad to be useful. Most of the future is completely known: the calendar, the counting, the fixed side, and therefore every fraction in every row. Exactly one thing is open, and it is the reading on the floating side. Everything unresolved about the amounts traces back to that single gap, and nothing else.

The fractions for later periods are computable in principle. The amounts are not computable at all: the readings were never gathered. Two different kinds of blank, sitting in adjacent columns, and it is worth being able to tell them apart.

Who actually has to fill this decimal in?

Forms are more useful here than job titles. A day count fraction shows up as a specific empty box on several different sheets of paper, and each box is filled by somebody with a different reason to care.

On a settlement calculation sheet there is a field between the rate and the amount, and whoever prepares that sheet has to put a number in it before the amount can be produced. The preparer is not exercising judgement, only transcribing a rule from a document and applying it to two dates. The job therefore looks clerical, and the error in it is persistent for that reason: nobody expects a transcription step to be where the money goes.

On a cash forecast there is a row for each expected settlement date, and the box beside it is an amount. The person filling that in has to work the fraction for a period that has not happened yet, which they can do, and then guess at a benchmark reading, which they cannot. Their honest output is a range with the fraction pinned and the rate open, and a forecast that pretends otherwise is presenting an assumption as an observation.

On a confirmation being checked against a system, the box is a comparison rather than a value: does the counting rule the system holds match the counting rule the document names? The comparison takes a minute on the day the agreement is set up and is the single cheapest control in this whole subject. On the day of setup nothing has gone wrong yet, and the comparison is the control most often skipped.

And on a household's own paperwork there is no box for it at all. The absence is itself the lesson. Somebody upstream already applied a day count fraction and handed down a finished figure, so a borrower repaying a loan never fills one in. The number arriving in the letter is the output of exactly this arithmetic performed by somebody else, and knowing that a rule sat behind it is what lets a person ask a sensible question about a bill instead of accepting it as arithmetic that fell out of the sky.

What goes wrong with a fraction, and why does it survive review?

Three things go wrong, and they share a property that makes all three of them dangerous.

The first is a fraction carried forward from the previous period because this period looked about the same length. The second is a numerator taken from one convention and a denominator from another, producing a decimal that belongs to neither rule and cannot be traced back to any document. The third is the two sides counting one stretch differently. The confirmation says one thing and somebody's static dataThe settings a system holds for an arrangement between transactions, typed in once at setup and rarely looked at again. says another.

All three share one property: the answer always comes out the right order of magnitude, and that is the whole difficulty. Nothing looks absurd. No amount arrives with an extra zero. Controls are built to catch things that look wrong, and none of these do, so no control fires. The wrong answer is a plausible amount sitting in a plausible field on a plausible sheet, differing from the right answer by a percentage point or two.

One comparison would have caught it and the other would not. Comparing the two amounts shows that they differ and says nothing about why, which is exactly the information that supports writing it off. Comparing the two fractions shows immediately that one of them came from somewhere other than the document, and a difference with a named cause is never written off. The cheap control is not a tighter threshold on the amount. The cheap control is looking one column to the left.

A breakA difference between two records of the same event that will not go away by itself, and that somebody has to explain rather than absorb. that repeats has a signature, and the signature is regularity. A genuine rounding artefact wanders: sometimes up, sometimes down, sometimes absent. A counting disagreement does not wander at all. A counting disagreement arrives the same size and the same direction every period, and the moment somebody notices that pattern the diagnosis takes about ten minutes. The reason it goes unnoticed for years is that each period is closed by a different person on a different day, and nobody is looking at the sequence.

What does the counting method not settle?

Reading a pair of rules off a document and turning them into a decimal is document work plus arithmetic. Whether an arrangement of this shape belongs anywhere near a particular balance sheet is a completely different question, covered separately. Somebody would have to start from what that party already owes and the basis its cost sits on, then take in the stretch of time involved, then look at a period where the difference travelled the other way. None of that was collected here, so the question stays open rather than being answered cautiously.

A fraction is also not evidence of anything beyond itself. Whichever pair is written into the document was written in with the rate beside it, and the two were quoted together. A convention that produces a larger fraction is not therefore a better convention, and one that produces a smaller fraction is not a worse one. Nothing about the counting rule bears on whether the arrangement is a good idea, who is ahead, or what happens next. The counting rule shows how to turn two dates into one decimal, correctly, every time, a smaller claim and a far more useful one.

Being the side that hands money across in a given period is not a failure and is not a defeat. A period settles where the two rates put it, and the fraction only decides how much of a year that settlement was measured over.

Chitrakoot Cements Limited and Saranga Capital Limited are invented.
Educational material. Not advice on any investment, tax, budget or market position.