Unit Allotment: Turning an Amount Into a Number of Units

An allotment is one division. The rupees paid are divided by the published value per unit that a rule attaches to the order, and the quotient is carried to a fixed number of decimals. The carrying leaves a residue with a rupee value. Which day's published figure belongs in the divisor is settled by the market regulator.

The opening setting is the worked example the rest of this guide uses. Rs 1,00,000/- is paid, nothing is deducted, the value applied is Rs 35.00 a unit, and the record carries 2,857.143 units. Moving one box changes it: with the funds becoming available after the cut-off, the applied value jumps to Rs 35.28, and the units to 2,834.467. The arithmetic never varies, so the divisor is the only thing that moves.

The word does quiet work, so start with it. An allotment is the point at which units come into existence and are written against one account: not a promise, not a request, but a figure that now exists. The arithmetic of an allotment is one division, and it is never the difficulty. The difficulty is deciding what goes into the divisor, and what to do with the part of the answer that will not fit.

Girnar Asset Management Limited operates the invented Girnar Large Cap Equity Fund, an open endedA scheme that keeps taking money in and paying money out on a continuing basis, rather than closing its doors after one subscription period. equity scheme whose net assetsThe amount a scheme is left with once everything it owes has come off everything it holds. stand at Rs 4,200 crore against units outstandingThe count of units in existence across every account in the scheme at a given point. of 120.00 crore. The second under the first gives Rs 35.00 exactly for one unit, and that is the divisor, worked out here rather than quoted. The three following days carry values of Rs 35.28, Rs 35.11 and Rs 34.95, and the timing boxes have something to bite on. The scheme is held across 3,80,000 accounts, operations are headed by Sohail Merchant, the equity portfolio is managed by Kalyani Bhagat, and the order is a payment of Rs 1,00,000/-.

The definition of a unit, the striking of its value each day, the contents of an account record and the working of exit terms are settled elsewhere. The arithmetic of the moment is what is left: a division, a rounding, a residue and a check.

How does an amount become a number of units?

Divide. The answer is that short. The rupees paid are divided by the published value of one unit, and the answer is the number of units. No other step, factor, adjustment or table sits inside that computation. A reader who has been assuming there is a hidden second step has just had the most important misconception about allotment removed.

Two things can come off the money before the division, and neither is part of it: a transaction charge where one applies, and stamp duty on a purchase. Both reduce the figure that gets divided rather than changing how the dividing is done, and the calculator shows them as two lines above the net investible. A rate that takes Rs 20/- off Rs 1,00,000/- leaves Rs 99,980.00 over Rs 35.00 to give 2,856.571 units. The rate in force is set by rule and gets revised.

Consider buying sweets for a wedding. The shop prices them by the kilo and the buyer hands over the money set aside; the weight is whatever that money buys, and it lands on some awkward figure the shopkeeper reads off the scale. An allotment works the same way round: the holder chooses the money, the scheme supplies the price, and the units are the consequence rather than the choice.

The divisor is itself the result of a division: net assets over units outstanding, Rs 4,200 crore over 120.00 crore units, giving Rs 35.00 exactly. Both figures are about to move, so being able to divide them rather than read them off is what allows a check, later on, that the movement leaves the answer where it was.

Which published figure does the amount get divided by?

Here the arithmetic stops. The amount is divided by the applicable valueThe published value per unit that a rule attaches to a particular order, as distinct from whatever value happened to be on display when the order was placed., and two things bear on which figure that is: whether a valid application has been received, and whether the money is available for the scheme to use. Receipt and availability are different events, they can happen in either order and on different days, and the market regulator makes a rule that hangs the outcome on the pair of them together.

Rules of this shape get revised, and a reference carrying the old version has not gone stale; it has gone plainly wrong while wearing the same confident face. So the calculator takes both timings as entered facts, read off the transaction confirmation. A calculator that answers a question whose answer moves is worse than one that declines it, so the refusal is a design decision rather than a hole. The Securities and Exchange Board of India (SEBI) makes that rule, and sebi.gov.in carries the version in force.

What happens when the money is credited before the cut-off and realised after it?

A different published value goes into the divisor, and a different number of units comes out. Work the calculator at the top rather than reading about it. Leave the application before the cut-off, and change the funds box to say the money became available for the scheme to use after the cut-off on the same day.

The application condition was satisfied on day D. The funds condition was not. The money only became usable after the cut-off, and that pushes the order to the next business dayA day on which the scheme is open for the work in question, which its own documents list and which is not the same thing as a calendar day.. The applicable value is the one published for the later of the two days, so an order whose paperwork was in early is priced at a day it never touched. The applied value moves to Rs 35.28, the units to 2,834.467, and the holder is 22.676 units short of what they wrote down.

Two things about that shortfall are easy to get backwards. Nothing was lost at the moment of allotment. Rs 1,00,000/- was paid, and units worth almost exactly that were received, priced at the value that applied. RealisationThe point at which money paid in has actually become available for the receiving side to use, as distinct from the point at which it left the payer. changed the price rather than taking money away, and a higher price buys less. And the direction is not fixed. Set the business day box to say the day after day D is not one: the funds wait, Rs 35.11 applies, and 2,848.191 units are allotted, short by 8.952 rather than 22.676. Where the later day's value is the lower, the same delay hands over more units.

One boundary case is worth landing on. People expect trouble there and find none. Set both timing boxes to exactly at the cut-off: both conditions are met on day D, and the answer is the opening 2,857.143 units. A cut-off is a boundary that includes its own edge rather than a gap that swallows what lands on it. What decides the outcome is which side of the line each event fell on, never how close it came.

Why does the division almost never come out even?

Because the two numbers going into it have nothing to do with each other. A person chooses the amount; a portfolio's worth produces the value per unit. Nothing pushes them towards dividing neatly. A fractional unitA holding recorded as part of a unit rather than as whole units only. is the ordinary outcome of an allotment, and a whole number would be the genuine surprise. Most people expect exactly the reverse.

Work this one. Rs 1,00,000/- over Rs 35.00 is 20,000 over 7, and seven goes into 20,000 two thousand eight hundred and fifty seven times with one left over. So the quotientThe answer to a division, as distinct from the remainder that the division leaves over. is 2,857 units plus one seventh, and one seventh as a decimal never finishes: 2,857.142857, then the block 142857 for ever. The recurring decimal is the honest answer, and no record can hold it.

The problem is a familiar one that usually goes unnamed. Three people splitting a Rs 500/- note get Rs 166.66 each and there are two paise standing on the table that nobody can take. The arithmetic of allotment is that everyday problem, arriving at industrial scale, on every order, every working day.

What exactly does the rounding leave behind?

A record cannot store a decimal that never ends, so the quotient is carried to a fixed number of places. Here it is carried to three, and 2,857.143 units is what the record holds. How many decimals a unit figure is recorded to is prescribed rather than chosen, and the prescription can be revised. Three decimals is this worked record's stated convention and nothing more. Where it stands is a SEBI matter, read at sebi.gov.in and in the scheme documents.

Now measure what the carrying did. 143 thousandths less one seventh is one over seven thousand, so the difference between 2,857.143 and 2,857 units and one seventh is one seven thousandth of a unit. The difference is the rounding residueThe exact difference between the answer a division actually gives and the shortened figure that gets recorded in its place., and the first thing to record is its direction. The carried figure is above the exact quotient, so this account has been credited with slightly more of a unit than the money strictly bought. Change the carrying box to cut the rest off and the direction reverses: 2,857.142, three parts in 3,500 below, worth Rs 0.03 the other way.

A fraction of a unit means nothing until it is priced, so give the residue its rupee value. Set Rs 35.00 against one seven thousandth of a unit and the answer is Rs 0.005, half a paisa and nothing over. Thirty five over seven thousand is one two hundredth, and one two hundredth of a rupee is half a paisa exactly.

Then look at what the carried figure is worth in total. Take 2,857.143 units at Rs 35.00 and the total lands on Rs 1,00,000.005, half a paisa above the money paid in and exactly halfway between two paise. There is no way to set that figure down to the paisa unless somebody first settles quietly on a rounding rule, and the whole difficulty lives inside that single sentence.

Where does that half a paisa actually go?

The half a paisa changes hands, rather than evaporating or living only on a printout. After the allotment the scheme holds Rs 1,00,000/- more in money and 2,857.143 more units, and those units carry a claim on the pool worth Rs 1,00,000.005. The extra half a paisa of claim was not paid for by the incoming account, so it came from every account that was already there.

Spread half a paisa across the 1,20,00,02,857.143 units then in existence and the effect on any one unit sits far below the precision at which a value per unit is published. The residue is invisible in the published figure and completely visible in the total, so it has to be tracked as an amount rather than watched for in the price. A household will never see it. An operations record has to carry it.

Why must the allotment register fix a rounding convention in advance?

Because Rs 1,00,000.005 is exactly on the boundary between two paise, and there is more than one defensible thing to do with a figure sitting there. Go up, go down, or go to whichever neighbour is even: each is a coherent rule, and they give different answers on this very order. The answer is not determined by the arithmetic but by the choice.

A convention settled before the run is a rule. The same convention settled after the figures are on the screen is a decision taken by somebody who can already see which way it goes. So the allotment register fixes its convention in advance and writes it down where it can be pointed at. The arithmetic is identical either way. The difference is whether anyone can later show the treatment was not chosen to suit the result.

The second reason is that one of the choices is one sided. Under a nearest convention residues fall on both sides, so across many allotments they largely offset without ever cancelling exactly. If instead the convention is always to round up, every residue points the same way, none of them ever offsets another, and the pile simply grows with the number of allotments processed.

A thousand allotments, each rounded up by half a paisa, is Rs 5/-. Carrying the same one sided treatment across as many allotments as this invented scheme has accounts, 3,80,000 of them, turns the half paisa into Rs 1,900/-. The count of allotments actually processed is not part of the record, so Rs 1,900/- is an illustration rather than a claim about any scheme. A residue that always points one way is a slope, and a slope has to be measured.

Does an allotment move the published value per unit?

No, and one division settles it. Net assets rise by the money received, so Rs 42,00,00,00,000/- becomes Rs 42,00,01,00,000/-, and units outstanding rise by the units created, so 1,20,00,00,000 becomes 1,20,00,02,857 and one seventh. The new pair divided gives Rs 35.00 exactly, with no remainder. An allotment made at the value at which the units were priced costs the standing holders precisely nothing, not approximately nothing, and the division proves it rather than asserting it.

The property has a name worth carrying. The invarianceThe property of a quantity that stays exactly where it was after some operation is performed on the things it is computed from. here is neither coincidence nor approximation: put money and units in beside each other in the proportion that already holds and the proportion is untouched. The invariance is also why the steps of an operating day cannot be shuffled. The property holds only when units are created at the value the pool was measured at, so the value has to be struck and published before any units come into existence.

One honest correction. The result above uses the exact quotient, and the record holds 2,857.143 instead, so the value per unit lands a hair below Rs 35.00 rather than on it. The gap is the half a paisa from the previous part, spread across every unit in existence. The allotment is exactly neutral, and only the rounding moves anything.

Which of these checks can actually fail?

Two checks appeared above and they look like independent confirmations. Check one: 2,857.143 units at Rs 35.00 is Rs 1,00,000.005. Check two: half a paisa on top of Rs 1,00,000/- is Rs 1,00,000.005. The two checks are the same equation written out in a different order, so they cannot possibly disagree, and finding agreement says nothing beyond that the multiplication is consistent with itself. The last two rows of the calculator are that pair, always agreeing, there to make the identity visible rather than to test anything. The invariance check holds by algebra too, so it confirms the arithmetic and never the facts.

A check earns its place only when there is a real way for it to come out wrong. Take the money that actually arrived in the scheme's bank account for a run and set it against the amounts written on the applications. Nothing forces those to agree: a payment can be dishonoured, arrive short, arrive for a different account or not arrive at all. A check that cannot fail is a formality. A check that can fail is the only kind that carries information, so an operations record puts the money it received beside the money it was told to expect.

A second check of the same sort belongs beside it, and the arithmetic above feeds it. The unit figures written to every account in a run are added up and set against the increase in units outstanding. The two totals can differ, and rounding residues are one of the reasons why.

What does the whole allotment look like, worked end to end?

Everything in one place, each line carrying its own arithmetic beside it so that no row has to be taken on faith, and the residue turns up where it is made.

Two lines deserve a second look. The residue line is exact rather than approximate. 143 sevens are 1,001, so 143 over 1,000 minus 1 over 7 is exactly 1 over 7,000. And the last line returns Rs 35.00 with nothing after the decimal point. The invariance holds to the letter rather than to a displayed precision.

What does this tool deliberately not compute?

Four things sit outside what the calculator computes. Which day's published value applies is not decided here, and both timings are taken as entered. Exit terms and any charge on the way out are not applied. Whether anybody should place an order goes unanswered, and no return or projection is produced. Every one of those four would be something other than arithmetic, and this tool is arithmetic and then it stops.

The first of the four is not restraint but computability. Deciding which published value applies needs the moment a valid application was received, the moment the money became available for the scheme to use, and the rule joining the two. The calculator holds the first two only because they were entered, and the third not at all. So its honest output is the units figure and a sentence naming the input it assumed.

Where do the applicable value rule and the rounding convention come from?

From the market regulator, and three routings hide in one sentence. Which day's published value attaches to an order, the cut off arrangement behind it, and how many decimal places a unit figure is recorded to are all set by SEBI, the last of them written into the scheme documents as well. Each can be revised, and the versions in force are read on sebi.gov.in. The Association of Mutual Funds in India (AMFI) puts out operating material for the industry as a whole, at amfiindia.com. For units held in dematerialised form the depositories concerned are the National Securities Depository Limited (NSDL) and Central Depository Services Limited (CDSL).

The durable part survives every revision: the division, the residue it leaves, the direction of that residue and the invariance of the value per unit. Changing the prescribed number of decimals leaves all four statements untouched; only the size of the residue moves. Surviving a changed prescription is the test for whether something is durable arithmetic or belongs with whoever decides it.

Girnar Asset Management Limited, the Girnar Large Cap Equity Fund, Sohail Merchant and Kalyani Bhagat are invented.
Educational material. Not advice on any investment, tax, budget or market position.