Value, Publish, Allot: Why the Order Cannot Change

A scheme prices what it holds, works one figure per unit out of that pricing and publishes it, and only then turns money received into units by dividing by the figure it published. The order is forced rather than chosen: units cannot be worked out before a figure exists, and a figure worked out after units are issued rests on money those units were issued for.

Here is the shape of the problem before a single number arrives. On one working day a scheme takes money in from somebody joining, watches the prices of everything it holds move, and hears from somebody who wants out. Three separate jobs have to be done about that, and they have to be done in one particular order. With the order right, nobody ends the day worse off than they started it. With the order wrong, value moves between two sets of people who never agreed to move it, in an amount that can be put to the paise. The joins between the three jobs carry the risk, not any one job on its own.

Everything below runs on a single scheme. Girnar Asset Management Limited, an invented manager, operates the Girnar Large Cap Equity Fund. The fund is open ended and holds equity. Its net assetsThe total the scheme could realise on what it holds, after taking off whatever it still has to settle. come to Rs 4,200 crore. Its units outstandingEvery unit alive in every name at one instant, added up. The count is the divisor in the figure per unit. come to 120.00 crore. One over the other puts a single unit at Rs 35.00, with nothing at all left over, and 3,80,000 folios sit behind it. Kalyani Bhagat looks after the holdings and Sohail Merchant looks after operations.

The three jobs have plain names, and it is worth fixing them before anything else. ValuationThe exercise of putting a price on each thing the scheme holds, one by one. A total is then added up from those prices. puts a price against every holding. The strikeA figure is worked out once, from the inputs of that day. It then stands as the figure of record for that day. and the publicationPutting a figure out where anybody may read it. Publication turns a working number into one that can be transacted on. that follows it turn that priced list into a single figure per unit and put it where it can be used. AllotmentCreating units and writing them against a name. Money the scheme has received becomes somebody holding. then turns money received into units at that published figure. Value, publish, allot. Each of the three consumes exactly what the one before it produced. The sequence is arithmetic rather than habit.

Several things this chain leans on are settled elsewhere. The nature of a scheme, who runs it and who stands between the manager and the holders are settled where schemes are structured. The definition of a unit, how the figure per unit is arrived at and which day a given order is priced on are settled where prices and units are dealt with. The running charge is set against the assets before anything is divided, and a published scheme return is already net of that charge. Both are settled where costs are dealt with. The register that an allotment gets written into is covered separately, as is the method behind the pricing of hard holdings. The join is what remains: what has to be finished before the next stage may start, what each stage hands on, and what it costs when one of them arrives late or wrong.

What are the three stages, and what does each one produce?

Take them one at a time and ask, of each, what it leaves behind on the desk. The first stage prices the holdings. The first stage produces neither a summary nor an opinion, but a list, one line per holding, each line carrying a rupee figure. The second stage adds that list up, sets against it whatever the scheme has to pay out, and divides the result by the units alive at that moment. The second stage produces one number. The third stage takes an amount of money that has arrived and divides it by that one number. The third stage produces a quantity of units against a name.

Each of those three outputs is a different kind of object. A list. A number. A quantity written against somebody. Without the first there is nothing to add up, so the second cannot be reached. A division needs something to divide by, so the third cannot be reached without the second. The order is not a policy anybody chose and could unchoose; it is the only order in which the three arithmetics are defined at all.

Think of a housing society working out this month's maintenance. Somebody first writes down every bill the building has to meet: the lift contract, the guard, the water tanker, the electricity for the common areas. Those written down bills are the priced list. Somebody then adds those bills up and divides by the number of flats. Out comes one figure per flat. Only then can any individual flat be told what it has to pay. Nobody can be billed before the total is known, and the total cannot be known before the bills are written down. Move the steps around and the arithmetic simply has nothing to work on.

Why can the figure per unit not be worked out first?

Because it is not measured, it is computed, and the thing it is computed from is the priced list. There is no instrument anywhere that reads a figure per unit off a scheme. The figure exists only as the result of a division whose numerator is a total built out of the first stage. Take the first stage away and the numerator is missing; there is no fallback source for it.

The distinction between computing and measuring has a consequence which is easy to walk straight past. If the published figure is questioned, the thing being questioned is almost never the division. A division of one total by one count carries no judgement in it: there is exactly one right answer and a calculator will find it. Every judgement in the chain sits upstream, in what price went against which holding and why. A holder who disputes the published figure per unit is disputing the pricing behind it, whether or not they know that is what they are doing.

The consequence changes where a search for a wrong figure should start. Somebody who suspects a figure is wrong and goes hunting through the division is hunting in the one room where nothing could have gone wrong. The room to search is the priced list: the holdings with a clean price available, the holdings without one, and whatever stood in for the missing prices. Pricing a holding that has no clean price is covered separately.

Why must that figure be published before any units are issued?

For a reason so ordinary that it hides in plain sight: a division needs a divisor. Somebody has sent Rs 1,00,000/- and wants units. The number of units is the amount divided by a figure. If the figure has not been struck, there is nothing in the denominator, and no amount of urgency creates one.

An order therefore sits, visibly, between the moment it is received and the moment units appear against a name. The gap is easy to read as slowness, or as somebody not getting round to it. The gap is neither. The wait between a received order and an allotted holding is the mechanism doing its work, not the mechanism failing to. An order that was turned into units the instant it arrived would have been divided by a figure that had not yet been worked out, which means it would have been divided by yesterday's, or by a guess.

How long that gap is, when it opens and when it must close are set by the market regulator, and all three move. The reason the gap exists at all is durable, and does not move.

What does each stage actually hand the one after it?

Three deliveries are the most useful way to hold the handover. The first stage hands over a priced schedule of holdings. The second stage receives that schedule and hands over one published figure per unit. The third stage receives that figure and hands over a unit quantity written against a name in the register.

The receiving stage cannot start on a partial handover at all. A priced schedule with one holding still unpriced does not produce a slightly uncertain total; it produces no total. A party outside the manager has to be able to read the figure, so a figure struck but not yet published cannot be transacted on. And a unit quantity written into the register with no published figure behind it has no basis at all. Each join is a gate rather than a slope: it is either complete and the next stage begins, or it is not and the next stage has nothing.

The pattern is familiar from somewhere ordinary. A wedding caterer cannot start plating until the count of guests is final, and the count of guests is not final until every reply is in. One reply outstanding does not mean a slightly smaller order; it means the order is not placeable. A gate looks like that.

What goes wrong when money is counted before its units exist?

The centre of the argument is arithmetic, so it is worked in full rather than described. One purchase of Rs 1,00,000/- goes through the Girnar Large Cap Equity Fund exactly as it stands, twice, in the two possible orders.

In the wrong order, the money is taken into the scheme assets first and the figure per unit is struck before the units it bought have been issued. Net assets go from Rs 42,00,00,00,000/- to Rs 42,00,01,00,000/-. Units outstanding stay where they were, at 1,20,00,00,000. Divide the new total by the old count: the figure per unit is Rs 35.0000833, with the last digit repeating without end, or exactly Rs 35 plus one twelve thousandth of a rupee.

The rise looks like nothing at all, and that is precisely the trouble. Take the rise on its own. It comes to Rs 0.0000833 a unit and a bit, exactly one twelve thousandth of a rupee. Now multiply it by the 1,20,00,00,000 units that were already standing. One twelve thousandth of a rupee across 1,20,00,00,000 units is Rs 1,00,000/-, exactly, with no remainder in any direction. The whole of the money that came in has been written into the recorded claim of the people who were already there, and the person who paid it holds nothing at all at that instant. Give that its plain name. The movement is a transferValue moving from one set of people to another with nobody paying anybody. A mispriced entry or exit produces exactly that., and nobody decided to make one.

How big is that error, seen from each of its two ends?

The same event has two honest descriptions and they sound like descriptions of different things. Per unit it is one hundred and twentieth of a single paise. There is nothing at that size for a statement to show, so a statement carrying paise would show a change of exactly zero. Across the scheme it is the entire purchase. Both are correct, at the same instant, about the same event.

The same figure can be reached by placing it against the conversion this sequence already carries. One full paise a unit on this scheme is Rs 0.01 across 1,20,00,00,000 units. The total comes to Rs 1,20,00,000/-, or Rs 1.20 crore. The error is Rs 1,00,000/-, and Rs 1,00,000/- is one hundred and twentieth of Rs 1.20 crore. So the per unit size of the error is one hundred and twentieth of a paise. The direct route gave the same answer. Two routes that share no arithmetic land on the same figure. The figure is checked rather than merely plausible.

And now the honest part. The arithmetic forces it, and it is easy to state too loosely. The Rs 1,00,000/- is the size of the misstatement in the published figure, measured across the units it was published against. The misstatement is not automatically the size of what anybody permanently keeps. How much anybody permanently keeps depends on who transacts while the figure is wrong, and the arithmetic below settles that rather than leaving an impression.

Does the value stay where the wrong order put it?

Two checks were just run on that Rs 1,00,000/-, and the first thing to notice is that they were not two checks. One took the rise per unit and multiplied it by the 1,20,00,00,000 units standing. The other took the change in net assets, divided it by the same 1,20,00,00,000 units to get the rise, and multiplied it straight back. Set that out and the unit count cancels. The net assets term goes, so net assets plus the purchase over the units, less net assets over the units, is simply the purchase over the units. Multiply that by the units and the unit count goes too, leaving the purchase. The two checks are one equation rearranged, and the unit count cancels clean out of it. Neither could ever have disagreed with the other, so the pair is a restatement rather than a check.

Now for a check that really can come apart: two ends that share no arithmetic at all, walking up to the same damage. Follow the money past the strike. The person who paid Rs 1,00,000/- is eventually issued units. Issued at the inflated figure, they receive Rs 1,00,000/- divided by Rs 35.0000833 and on, or 2,857.136054 units and a little more, against the 2,857.142857 and on they would have had. The buyer is short by 20,000 divided by 29,40,007 of a unit, about 0.006803 of one. Value that shortfall at the figure it was issued at and it comes to exactly 5 divided by 21 of a rupee, or Rs 0.238095 and on.

Now come at it from the other end. Once those units are written, the assets have not moved but the count has, so the figure settles back to Rs 35.00 and a whisker, exactly Rs 35 plus 1 divided by 5,04,00,24,000 of a rupee. The standing holders finish with their 1,20,00,00,000 units at that figure. The total is Rs 42,00,00,00,000/- plus exactly 50,000 divided by 2,10,001 of a rupee, or Rs 0.238094 and on.

The two do not match, and the gap is stated rather than rounded away. The residue is exactly 5 divided by 44,10,021 of a rupee, roughly one ten lakhth of one, and it leans one way every time rather than wobbling either side of zero. The reason is plain on inspection. The figure itself falls back the moment the missing units land, so valuing the shortfall at the inflated figure always overstates the damage a little. The residue is one sided, it does not cancel, and it is the difference between the figure the shortfall was suffered at and the figure it ends up being measured at.

There is a second one sided residue in this chain, and it belongs to allotment rather than to valuation, so it is named here and covered separately. A register cannot carry an endless run of decimals, and how many places it carries is set by regulation. At three places, purely as an illustration, 2,857.142857 and on becomes 2,857.143. The rounded figure sits exactly 1 divided by 7,000 of a unit above the exact quotient. At Rs 35.00 a unit that excess is Rs 0.005, exactly half a paise, and the rounded holding multiplied back is Rs 1,00,000.005, exactly halfway between two paise. Rounding up is one sided, so those residues do not cancel across many orders either.

So what is the honest size of a sequencing error? Two numbers, and they answer different questions. While the pair is apart, the recorded claim of the standing holders overstates by exactly Rs 1,00,000/-, and anybody transacting on that figure transacts on the overstatement. Once the pair closes and nobody has dealt in between, what actually changes hands is about twenty four paise. A sequencing error has one size while the gap is open and another once it shuts, and which of the two matters depends entirely on whether anybody transacted in the gap.

Why does a correctly ordered issue leave the figure per unit alone?

Run the same purchase the other way round. Price the holdings, strike Rs 35.00, publish it, and only then issue units at it. Rs 1,00,000/- divided by Rs 35.00 is 2,857.142857 units and the sixes and so on without end, exactly twenty thousand sevenths of a unit.

Now recompute. Net assets are Rs 42,00,01,00,000/-. Units outstanding are 1,20,00,02,857.142857 and the rest of the run. Divide one by the other. Out comes Rs 35.00, to the last place, with nothing whatsoever carried over. A correctly ordered issue costs the standing holders precisely nothing, and that follows from the division itself rather than from any courtesy anybody extended.

The reason is the whole engine of the argument, and worth holding on to. Money and units entered in the same ratio the pool already stood in. Rs 1,00,000/- against twenty thousand sevenths of a unit is Rs 35.00 a unit, and the pool was already Rs 35.00 a unit. Adding to a ratio in the ratio it already holds cannot move the ratio, and the figure per unit is nothing more than that ratio.

Does the same mistake run the other way on the way out?

The same mistake does run the other way, and the mirror is more exact than most people expect. RedemptionThe way out of a scheme: units are cancelled and the person leaving is paid what those units came to. is a pair as well, a cancellation of units and a payment of money, and the pair can come apart in the same way.

Cancel the units first and let the money leave later. Net assets have not moved, units outstanding have fallen, so the same pool is now divided by fewer units and the published figure leans up. Pay the money out first and let the cancellation follow, and the pool has shrunk while the count has not, so the figure leans down. Which half of the pair lands first is what decides the direction, and neither direction is the safe one to be careless in.

Put the two transactions together and a pattern falls out. On the way in, money first leans the figure up and units first leans it down. On the way out, units first leans it up and money first leans it down. Four cases, two directions, and in every one of them the published figure is wrong for as long as the pair is apart.

Who runs each stage, and where is the join between them tested?

The stages are not all done by the same party, and that separation is the whole point of the arrangement rather than an accident of who was available. The operations team inside the asset manager prices the holdings and strikes the figure. The registrar and transfer agent writes the allotment into the register of units. The custodian holds the securities the scheme has bought and confirms independently what is there. The trustee company and the auditor test that what was written matches what was struck.

Look at where those responsibilities overlap and one point stands out. The unit count appears in two records kept by two different parties: the count the manager divided by when it struck the figure, and the count the register holds after the allotment was written. Two separately kept records must produce the same quantity at exactly one point, so the join between the struck figure and the register is where this chain is most often tested.

A disagreement therefore shows at that join first rather than on a statement. A statement is only a readout of the register, so it will look entirely orderly even where the register itself disagrees with the count the figure was struck against. The check has to bite where the two records meet, not downstream of one of them.

Where do the outer limits on each stage come from?

Every one of the three stages needs an outer limit, or the day has no closing point. Pricing could always wait for one more quote. A strike could always wait for pricing. An allotment could always wait for a strike. Without a stated limit on each, an order could sit unallotted indefinitely and nobody would have broken any rule. There would be no rule to break.

The limits exist, and all three are set by the market regulator: when a figure per unit has to be out, which day a given order is priced on, and by when an allotment has to be made. All three move. The reason each limit has to exist is durable, and anybody holding that reason can look the current limit up at sebi.gov.in whenever it matters. A limit printed into a reference text does not merely age when the limit moves. The printed limit turns false.

How do the two orders compare, line by line, on one scheme?

Here is the entire thing on one scheme, both orders side by side, with every intermediate figure shown so that nothing has to be taken on trust.

Two lines in that table are worth staring at together. The rise per unit in the wrong order is too small to write in paise, and the same rise across the scheme is the whole purchase. Written per unit the error hides, written across the scheme it is the whole purchase, and neither description is any more correct than the other.

The arithmetic does not scale up to a whole day. The starting record carries no flow of any kind for this scheme: no count of orders in a day, no rupee total received, nothing. So one purchase is worked and the account stops there. Multiplying by a day of orders would mean inventing the day, and an invented total dressed as an aggregate is worse than no aggregate at all.

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