Market Conventions: Day Count, Compounding and Quotation

A market convention is a decision about how a quantity gets expressed: the rule for counting days, the rule for how often a rate is applied inside a period, and the rule for how the result is written. Mathematics implies none of the three. All three change the figure as written while leaving the underlying value untouched.

Everything about market conventions follows from one sentence. A rate is not a number until its conventions are stated. The claim sounds like an overstatement and it is not one. Hand somebody the figure 0.052119 with nothing attached to it and they cannot grow one rupee by it, cannot discount anything with it, and cannot compare it with anything else. What arrives with a bare rate is not a quantity but the raw material of one, and the conventions are what finish it.

Here is the everyday version, and it involves no money at all. Somebody says the water is at twenty-five. Somebody else, standing at the same tap, says it is at seventy-seven. Neither is mistaken and neither has measured anything different. One is reading a Celsius scale and the other a Fahrenheit scale, and the water is exactly as warm as it was before either of them spoke. Subtracting twenty-five from seventy-seven and announcing that the second tap is fifty-two hotter produces a number with a decimal point and no meaning. In finance that same subtraction is much harder to spot. Both rates are written as per cent a year, and neither carries a scale name on its face.

Everything below is worked on one invented quantity, held fixed from the first line to the last: a one year bond price of 0.949216, a price for a single payment of one unit a year from now, produced by the invented rate model this sequence carries. Behind it sits the same standard process this subject area has used throughout, written S with a time subscript, starting at Rs 100/-, drifting at 8 per cent a year with a volatility of 20 per cent a year, watched over one year, with a risk-free rate of 5 per cent a year continuously compounded. None of those five figures moves anywhere below. Only the way the answer gets written down moves.

What is a market convention, and what makes it a decision rather than a fact?

A market conventionA decision about how to express a quantity, agreed between the people using it and not implied by any mathematics. is an agreement about expression. A convention sits at the point where a computed quantity has to be turned into something a person can write on a line, and it settles the questions the computation left open. How are the days counted? How often does the rate get applied inside the period? How many decimal places survive, and in what units? Every one of those is a question the mathematics is silent on, and silence is the whole point: the mathematics finished its job when it produced the price.

Look at what a convention cannot do. The limits are the cleanest way to see what a convention is. A convention cannot make the bond worth more. A convention cannot change how much money arrives, or when. A convention cannot move the risk-free rate, the volatility or the horizon. A convention changes the writing and never the thing written about. A difference produced by a convention therefore carries no information about anything except which convention was used.

People underrate conventions because the word sounds like housekeeping. The word suggests the difference between spelling out a date and abbreviating it. The arithmetic below says otherwise, and it says so by a margin nobody expects: on this one bond price, the choice of compounding convention alone moves the quoted rate further than the most carefully derived correction in the entire rate reading order does. A decision about notation outweighing a derived correction is the reason a convention has to be stated beside every quoted rate.

What does a day count convention decide?

A day count conventionThe rule for counting the days in a period and the days in a year, which together turn a period into a fraction of a year. settles two separate questions and it is worth keeping them apart, because they fail in different ways. The first is how many days there are in the period. The second is how many days count as a year. Together they produce a single fraction, and that fraction is what a rate a year gets multiplied by to give the growth over a shorter stretch.

The first question is less obvious than it looks. Between the fourth of a month and the fourth of the next month, is the period thirty days or thirty-one? Does the count include the day the period opens, the day it closes, or neither? A tenant who asks how many nights they are paying for and a landlord who answers with how many days they were on the premises can differ by one without either of them being wrong, and neither answer is derivable from arithmetic. Someone has to decide, and having decided, has to say so.

The second question is stranger still. Answering it means declaring how long a year is. A year is not a fixed number of days for this purpose. Some counting rules divide by 365, some by 360, and neither divisor is a claim about the calendar. Because 360 splits evenly into twelve months of thirty days, dividing by 360 was originally a convenience for hand arithmetic. The rule survives because the people using it agreed to keep using it.

Now put ninety days at 5 per cent through both divisors and read the two answers. Counting against 365 gives an accrual factor of 0.246575 and a growth of 0.012329. Counting against 360 gives an accrual factor of 0.250000 exactly and a growth of 0.012500. The difference is 0.000171 on a single quarter, and it comes entirely from a choice of denominator that no calculation anywhere required. On a face of Rs 100/- that is Rs 1.232877/- against Rs 1.250000/-, a difference of Rs 0.017123/-.

There is a small exact fact hiding in that pair, and it is worth carrying away because it holds far beyond this example. The ratio of the two accruals does not depend on the rate and does not depend on how many days the period runs. Whatever sits on top, dividing by 360 instead of 365 lifts the answer by exactly one seventy-second.

What does a compounding convention decide?

A compounding conventionThe rule for how often a quoted rate is applied within the period, which fixes how a rate figure turns into a growth factor. settles one question: how often, inside the period, does the rate get applied to a balance that already includes what it earned earlier? Once a year, twice, four times, twelve times, or without interruption. The quoted figure has to be whatever makes the arithmetic land on the price that already exists, so the choice of rhythm alone changes the figure that gets quoted.

The direction of the reasoning matters here, and it runs the reverse of what most people expect. The price is not being computed from the rate. The price is already there, fixed at 0.949216, and the rate is being read out of it. So the question is not what the rate produces. The question is what figure, applied in this particular rhythm, reproduces the price already in hand. Change the rhythm and the figure that solves that equation changes with it.

Work the five readings and set them in a row. The same 0.949216 gives 0.052119 continuously compounded, 0.052232 compounded monthly, 0.052460 quarterly, 0.052804 semi-annually and 0.053501 annually. Every one of the five is correct. Every one of the five describes the identical bond, paying the identical unit on the identical day.

Read the column of rates from the bottom upward and the mechanism is visible without any algebra. The less often a rate is applied, the higher it has to be quoted. The earlier applications are not there to do any of the work. Applying something once a year is the laziest possible rhythm, so it needs the largest figure; applying it without interruption is the busiest, so it needs the smallest. The ordering of the five readings is a feature of the mathematics, but the choice of which of the five gets written on the line is not.

How much is a compounding convention actually worth?

Now the arithmetic that gives all this its point. Subtracting the smallest of the five readings from the largest gives a figure with a name. The convention spreadThe distance between the highest and lowest figures one underlying value produces under different conventions. Here it is 0.001382. on this bond price is 0.001382, running from 0.052119 continuously compounded up to 0.053501 annually. The spread is 2.651814 per cent of the continuous reading, produced by a decision about notation and by nothing else.

Set that beside something the rate reading order worked hard for. There the curve of zero rates was shown never to reach the long-run level of the short rate, falling short of it by 0.000200 forever, and that shortfall was traced to the curvature of the exponential and to the squared volatility of the rate divided by twice the squared speed of reversion. The shortfall is a real property of the model, derived rather than asserted, and it does not shrink at any maturity.

The convention spread of 0.001382 is 6.91 times that shortfall, and unlike the shortfall it is not a feature of anything. One of the two numbers is a consequence of the mathematics. The other is a choice somebody made about notation. The ordering is worth sitting with. An effect that takes an afternoon of study to derive is seven times smaller than an effect a single word in a footnote can produce.

What is a quotation convention, and what does it throw away?

The third convention is the one nobody names, and it is the one that operates on the finished figure. A quotation conventionThe rule for how a computed figure is presented: the units it is written in, the number of decimal places kept, and the sign carried. settles the units, the precision and the presentation. Is the figure written as a decimal or as a per cent? In hundredths of a point? To how many places? Every one of those is a decision, and every one of them can throw information away permanently.

Take the continuous reading of 0.052119 and write it several ways. As a per cent it is 5.2119 per cent. In basis pointsHundredths of a percentage point. A rate of 5.2119 per cent is 521.19 basis points. it is 521.19. The three writings are the same number wearing three costumes, and any of them can be turned back into the others without loss. Then round it. As a per cent to two decimal places it is 5.21 per cent, and the difference between 5.2119 and 5.21 has gone for good. As a decimal to two places it is 0.05, and now something much worse has happened: the figure has collapsed onto the flat rate of 5 per cent exactly, and the whole quantity the rate model was built to produce has vanished into a rounding rule.

Rounding a rate written as a per cent to two decimal places discards up to 0.005 percentage points, a quarter of the entire convexity shortfall. Rounding the same rate written as a decimal to two places discards 0.002119, more than ten times that shortfall and more than the whole convention spread. The rounding rule is doing more damage than either of the effects isolated earlier, and the damage is invisible. A rounded figure looks exactly as authoritative as an unrounded one.

Why is a rate without its conventions not a number?

Here is the claim stated as sharply as it can be. A rate on its own is an instruction that has not been given yet. A rate is wanted for a growth factorThe multiplier that turns one unit today into what it becomes at the end of the period. It is what a rate exists to produce., the multiplier that turns one rupee today into what it becomes at the horizon. The rate figure is only the raw material; the convention is the recipe. Without the recipe there is no dish.

Now break the rule deliberately and watch what happens. Breaking it is the fastest way to feel the size of the effect. Take the figure 0.052119, correct under continuous compounding, and apply it under a different convention, as somebody handed a bare number with no label would have to. Applied annually it gives a growth of 1.052119 and implies a bond price of 0.950463. Applied monthly it gives 1.053382 and implies 0.949323. Only the continuous route returns 0.949216, the price the figure came from.

The spread of implied prices is 0.001247, or Rs 0.124692/- on a face of Rs 100/-. Applying a correct rate under the wrong convention produces a wrong price, and there is nothing about the arithmetic, the figure or the answer that would ever look suspicious to the person doing it. Every step was valid. The single missing fact was one word.

What does a comparison between two differently quoted rates measure?

Everything above converges on one practical hazard, and the hazard belongs to model risk rather than to vocabulary. Two figures land on the desk. One says 0.052119 and one says 0.053501. The difference is 0.001382, a substantial-looking quantity in a world where the convexity shortfall of 0.000200 was worth a long derivation. So the analyst writes down that the second describes a higher value than the first, and carries that conclusion forward.

Nothing has been measured. Both figures describe the identical bond price of 0.949216. The difference between them is the difference between two ways of writing one number, and it says as much about value as the difference between twenty-five and seventy-seven says about how warm the water is.

How does somebody with a rate in front of them actually use this?

Concretely, and it needs no market access, no data and no software. Anybody handed a rate runs three questions before doing anything with it, and none of the three requires knowing anything about the instrument.

1. Ask how the days are countedBoth halves: how the days in the period are counted, and how many days count as a year. If the answer is not in the source the figure came from, it is in the contract, and if it is in neither then the figure is not usable yet. On a quarter at 5 per cent the two common divisors are worth 0.000171 apart, exactly one seventy-second of the accrual either way.
2. Ask how often the rate is appliedContinuously, annually, or somewhere between. The compounding question carries the most weight and gets asked the least. The answer is a single word, and its absence is not conspicuous. On the invented bond price used above, the answer is worth 0.001382, or 6.91 times the derived convexity shortfall.
3. Ask what was rounded away and in what unitsA figure written to two decimal places as a per cent has already lost up to 0.005 percentage points. The same figure written to two decimal places as a decimal has lost 0.002119 and landed on a completely different quantity. Read the precision before reading the number.

Then apply the one rule that follows from all three. Two rates are never subtracted until all three conventions are confirmed to match on both of them; where they do not match, one is converted to the other's conventions first and the subtraction comes afterwards. Converting is easy: the conversion formulas above take any of the five readings back to the price and out again under any other convention, with nothing lost in either direction. The subtraction without the conversion is the unrecoverable move.

The same discipline runs the other way for whoever writes a figure down. The counting rule, the compounding frequency, the units and the precision all belong beside the number rather than in a note somewhere else, and all four of them together. The household version is the recipe card that says two hundred grams rather than one cup. The person who reads it will not have the same cup. A figure that has to be interpreted by somebody who was not there needs its conventions attached to it, and the cost of attaching them is one line.

The one year bond price of 0.949216, the rate model behind it and the standard process it sits on are invented.
Educational material. Not advice on any investment, tax, budget or market position.