Net Asset Value and Units: Work the Arithmetic

The calculator below builds a scheme's net assets out of its own asset and liability lines, divides them by units outstanding to get the value per unit, and turns a rupee amount into a unit count at the value applied, so Rs 1,00,000/- at Rs 35.00 becomes 2,857.143 units. The one thing it cannot settle is which day's value will attach to an application.

The panel's default figures are the worked example the whole calculation runs on, set out below as ordinary text. Investments of Rs 4,158.60 crore, cash and bank balances of Rs 52.40 crore and receivables of Rs 18.75 crore give total assets of Rs 4,229.75 crore. Payables on purchases of Rs 21.10 crore, redemptions payable and other current liabilities of Rs 6.45 crore and accrued unpaid expenses of Rs 2.20 crore give total liabilities of Rs 29.75 crore. The second total subtracted from the first leaves net assets of Rs 4,200.00 crore. Divided by 120.00 crore units outstanding, the value per unit is Rs 35.00 exactly, with nothing dropped in the rounding. On the application side, Rs 1,00,000/- at the nil stamp duty the panel opens on is Rs 1,00,000/- that buys units. At Rs 35.00 a unit that amount comes to 2,857.142857 and onward, recorded as 2,857.143 units. The recorded units come back to Rs 1,00,000.005, a residue of Rs 0.005 above what went in.

The arithmetic is easy and the discipline about the inputs is the hard part. Eight of the panel's ten figure fields belong to the scheme and can be checked against a document; two belong to the applicant, the amount and the duty assumed. A calculator that gets the division right while accepting one holder's unit count in the units outstanding field has done nobody any favours at all. Knowing which number goes in which slot matters more here than anything the arithmetic then does with it.

One scheme runs through every example. Girnar Asset Management Limited, an invented asset manager, operates the Girnar Large Cap Equity Fund, an open ended equity scheme whose net assetsEverything the scheme holds less everything it owes, on the day the figure is struck. of Rs 4,200 crore over units outstandingThe count of units in issue across every holder, which is what the pool gets divided by. of 120.00 crore give a value per unitNet assets divided by units outstanding: what one unit is worth on the day. of Rs 35.00 exactly. Kalyani Bhagat manages the portfolio and Sohail Merchant heads operations. The trustee company, the custodian, the registrar and transfer agent and the auditor are referred to by role and never named.

Three things are settled elsewhere and none of them is rebuilt here: what the value per unit means and how a scheme strikes it out of its books; what a unit is as a legal interest, and why unit counts run to three decimal places rather than stopping at whole numbers; and which day's figure attaches to an application. The third of those turns on when the application and the money reach the scheme, and SEBI settles and revises it. The arithmetic below takes all three as given and does one job with them: it computes.

What does this calculator actually hold together?

The calculator adds, subtracts, divides and multiplies the figures handed to it, and that is the whole specification. The calculator looks nothing up, it does not know what day it is, it holds no record of any scheme, and it has no view about what happens next. Given a set of numbers it works on those numbers whether or not they belong together. A calculator is a machine for being fast rather than a machine for being right, and the difference between those two things is the person filling the fields.

A weighing scale in a vegetable market is accurate: a two kilogram weight on one pan reads two kilograms all day. The scale has nothing to say about whether that weight is honest, whether the vegetables are the ones asked for, or whether the rate per kilo is the one being charged. A scale answers the question it was built for and no other, and this calculator has the same limits.

So the honest description is a narrow one. The panel builds net assets out of the scheme's own lines, divides them by units in issue, and turns an amount into a count at a value supplied to it. Everything else a reader might reasonably want from it sits outside it, and a section below names each of those limits and its cause.

How do the three quantities sit against each other?

In one relationship, written three ways. Net assets over units outstanding gives the value per unit; net assets over the value per unit gives units outstanding; the value per unit times units outstanding gives net assets back. The three forms are not three facts to be memorised but one identityA relationship that holds by definition rather than by observation, so it can be rearranged freely and stays true. turned around, in the way a speed, a time and a distance are one relationship rather than three. A reader who can rearrange this once can rearrange it for the rest of their life.

Run all three on the scheme and watch them close. Rs 4,200 crore over 120.00 crore units is Rs 35.00 a unit; Rs 4,200 crore over Rs 35.00 a unit is 120.00 crore units; Rs 35.00 a unit times 120.00 crore units is Rs 4,200 crore. The reason to write the second and the third out rather than assert them is that almost nobody has ever done them. Most people meet this relationship only in its first form, published once a day, and never see it turned around.

The turning around earns its keep. Where the published value per unit and the scheme's stated net assets are in hand but the units in issue are nowhere stated, the second form recovers them in one division. Where the unit count and the value per unit are known and a stated net assets figure needs a sanity check, the third form does that. Neither is a trick, and the only reason they feel unfamiliar is that published disclosure runs in one direction and habit follows disclosure.

What happens when the scheme's own figures move?

The panel at the top takes figures off documents. The panel below does a different job: the scheme's side of the arithmetic can be moved and the effect watched. The three quantities move against fixed scales, the conversion boxes recompute in both directions, the holding is drawn inside the scheme at a magnification the drawing declares out loud, and the curve at the bottom shows where the current reading sits. The sliders open on the same figures as the calculator above.

Two habits are worth carrying into it. The check line under the sliders recomputes the relationship in all three directions every time a control moves, and reports whether the three agree exactly rather than approximately. The residue box is the other one to follow: it will often show a few paise that seem to have appeared from nowhere, and the two sections that follow are about where they come from.

How does a rupee amount become a unit count?

Divide the amount by the value per unit. Rs 1,00,000/- at Rs 35.00 a unit is 2,857.142857142857 and onward, a decimal that repeats and never settles. A repeating decimal is not a defect in the example. Division normally does this, and a whole number would be the surprising outcome. The amount is fixed and the value per unit is fixed, so the count lands wherever the division puts it, and the only question left is where the record stops writing digits.

The Girnar Large Cap Equity Fund states that it records unit counts to three decimal places, so 2,857.142857 and onward is written down as 2,857.143 units. The sentence claims less than it appears to. The scheme states this convention; nothing in the sentence says three decimals is required, or usual, or a limit. A printed rule that has since moved is wrong rather than merely dated, so how many decimals a scheme has to keep is a matter for SEBI at sebi.gov.in.

The household version is the sweet shop. The shopkeeper cannot cut a laddoo into the exact grams a five hundred rupee note buys, so he weighs to the nearest gram and settles the difference in the price. A scheme does the reverse. A unit is only a number in a register and can be cut as finely as the register allows, so the scheme keeps the money exact and cuts the unit. The rounding conventionThe rule a record keeper follows about how many decimal places to keep and what to do with the digits beyond them. is where it stops cutting.

Going out, Rs 1,00,000/- became 2,857.143 units. Coming home, 2,857.143 units at Rs 35.00 comes to Rs 1,00,000.005. Any tool that quietly rounded the return leg back to a tidy Rs 1,00,000/- would have hidden the most instructive thing in the whole calculation.

One more property of this division is worth seeing as a shape. Hold the amount at Rs 1,00,000/- and let the value per unit vary. At Rs 10.00 a unit it buys 10,000 units, at Rs 20.00 it buys 5,000, at Rs 35.00 it buys 2,857.143, at Rs 100.00 it buys 1,000. Dividing a fixed amount by a bigger and bigger number gets closer and closer to nothing without ever arriving, so the count falls in a curve that keeps flattening and never touches the floor.

The falling curve has one practical consequence and one dangerous misreading. The consequence is that a small difference in the value per unit moves the count a lot when the value is low and very little when it is high. The misreading is to treat a scheme with a low value per unit as better value because the same money buys more units. More units of a smaller thing is the same amount of money. The curve is a fact about division and says nothing about whether one scheme is preferable to another.

How does a unit count become rupees again?

Multiply the count by the value per unit and look hard at what comes out. The 2,857.143 units the register now holds, multiplied by Rs 35.00, give Rs 1,00,000.005, and that is exact rather than rounded: 2,857.143 times 35 is 100,000.005 with nothing left over. The trip out and the trip home differ by Rs 0.005, half of one paisa. The half paisa is the three decimal convention becoming visible, and it is the one place on the record where the convention shows itself.

Now the awkward part. Rs 1,00,000.005 sits exactly halfway between Rs 1,00,000.00 and Rs 1,00,000.01, so it cannot be written to the paisa without somebody choosing a rule. Rounded half up it becomes Rs 1,00,000.01; rounded half down, or half to even, it becomes Rs 1,00,000.00. The scheme's record states the three decimal unit convention and is silent on which rule applies to a rupee figure landing on a half paisa. The exact product is Rs 1,00,000.005, and saying so plainly is the honest position rather than a gap in the teaching.

Why not just suppress it? Because a residueThe small leftover that appears when a rounded figure is put back through the arithmetic it came out of. that has been swept away teaches a reader that the arithmetic is exact when it is not, and then the first statement carrying a few odd paise looks like an error in the scheme's records rather than the ordinary consequence of writing a number to three places. Half a paisa is trivial in itself. The habit of noticing where a convention bites is not, and it transfers to every rounded figure a reader will ever meet.

Why is the divisor units outstanding and not the units one holder holds?

Because the relationship is about a whole pool and everything in issue against it, not about any one holder. Net assets are the scheme's assets less its liabilities across every holding in it, so dividing by every unit in issue gives what one unit is worth. Dividing instead by the units in one person's folio sets a number from one side of the scheme against a number from the other, and the answer is not wrong so much as meaningless. Most fields in this arithmetic describe the whole scheme and two describe one applicant, and mixing them produces a figure that corresponds to nothing at all.

Do it once and see how badly it goes. Net assets are Rs 4,200 crore and there are 1,20,00,00,000 units in issue, and that division gives Rs 35.00. Put 2,857.143 in the divisor slot instead and the answer is roughly Rs 1.47 crore. No unit in the scheme is worth anything like that. The divisor was about 4,20,000 times too small, so the answer came out about 4,20,000 times too large, and a reader who did not already know what to expect cannot catch that from the number alone.

A housing society has two hundred flats and a common electricity bill of Rs 2,00,000/- for the month. Divided by two hundred flats, each flat's share is Rs 1,000/-. Divided by the four people living in one flat, the answer is Rs 50,000/-, and that is nobody's share of anything. The arithmetic did not fail; the wrong count went in.

There is a related figure people often want at this point, and it is worth doing properly. What share of the scheme does that holding represent? 2,857.143 units divided by 1,20,00,00,000 units is about 0.000238 per cent of the scheme. The division of one unit count by another is legitimate because both numbers are unit counts. The illegitimate one was putting a unit count where a whole scheme count belonged.

What can this calculator not settle?

Three things, each with a different cause, and each worth taking one at a time.

The calculator does not know which day's value per unit will attach to an application. The panel divided by whatever number sat in that field, and that number could be today's published figure, yesterday's, or a guess. Which figure actually attaches, the applicable valueThe figure that ends up being used for a particular application, decided by an attachment rule rather than by whoever is doing the arithmetic., turns on when the application and the money reach the scheme, on conditions SEBI sets and revises and publishes at sebi.gov.in. The figure has not been struck yet, so nobody at the moment of applying knows the answer either. The wait is a property of the arrangement, not a shortcoming of the tool.

The calculator does not produce a return. A return needs two figures and the period between them. The panel holds one figure and has no clock. Nothing in it grows, compounds, projects or forecasts, and nothing in it should be read as suggesting what any amount becomes later. Returns over a period, and the discipline of saying whether a figure is gross or net, are settled elsewhere.

The calculator applies no charge of its own. The single deduction anywhere in the calculator is the stamp duty typed into that field. The rate is set in law, so the field opens at nil. A scheme's running expenses are accrued against its assets before the value per unit is struck, so the value per unit already carries them, and deducting an expense ratio from the panel's output would charge the same amount twice. No load and no tax is applied either, and what a scheme may charge is SEBI's territory.

How should what comes out of it be read?

As arithmetic on the numbers typed in, and no better than they were. The narrowness sounds like modesty and it is the operating instruction. A value per unit taken from the wrong day makes the count wrong by exactly that much; units outstanding taken from an old disclosure make the share of the scheme wrong in proportion. The accuracy of the output is entirely the accuracy of the inputs, and the tool has no way of signalling that an input is stale.

Which leaves the one discipline worth carrying away. Most of the fields come from the scheme and can be checked against a document; two come from the applicant. There is never a good reason to supply a scheme figure from memory or from an earlier day when the current record is a few clicks away. The amount being put in, or the unit count already held, is the only input that is genuinely the applicant's, and the only one the scheme cannot check.

One last framing, and a reader who takes one thing away should take this one. A unit count from any calculator, this one included, is arithmetic on an assumption rather than a statement about an outcome. The assumption is that the value per unit supplied is the one that will apply. The assumption is reasonable to work with while thinking, it is not a fact, and treating it as one does harm rather than good.

The whole worked instance in one place

Here is every figure the two panels open on, with the operation that produced it, so the arithmetic survives whether or not either of them is ever touched.

Every row above reconciles in both directions, and reconciling in both directions is the check worth doing on any calculator before trusting it. The value per unit multiplied back by units outstanding brings net assets back. The recorded count multiplied back by the value per unit lands on Rs 1,00,000.005 rather than the Rs 1,00,000/- that went in, and the half paisa is the convention rather than a fault.

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.