Return Measures for a Fund: Compute Them Side by Side

A return calculator for a scheme computes the absolute change, the annual equivalent and the gap against a yardstick, and the three answers differ. A fourth, the return the holder's own money earned, is weighted by when each rupee arrived, so it differs again. Every one is only as honest as two labels: the exact period, and whether each figure is gross or net. Without both, the number compares with nothing.

A measure is not a fact about a scheme. A measure is arithmetic applied to a record. The arithmetic answers whatever question was asked of it, and the question asked is not always the question intended. Each measure asks something else, so three measures run on one identical record return three different numbers and every one of them is correct.

One record supplies every figure below. Girnar Asset Management Limited, an invented fund house, operates the Girnar Large Cap Equity Fund, an open ended equity scheme with net assets of Rs 4,200 crore and 120.00 crore units outstanding. Those assets divide to Rs 35.00 a unit. Its expense ratio is 1.65 per cent. For the one stated year it returned 13.4 per cent measured value to value, and its stated benchmark returned 12.1 per cent. Kalyani Bhagat manages the portfolio and Sohail Merchant heads operations.

Four things are covered separately and are used here rather than rebuilt: measuring from one date to another; measuring one window length from many start dates; why a scheme figure and an index figure are not comparable as published; and taking a rootThe number which, multiplied by itself a stated number of times, gives back the starting number. rather than dividing when a change is turned into a yearly rate. The cost material settles the point that a published scheme return already carries the charge. The machinery below is the new part: the two labels made compulsory rather than advisory, and the two places where the calculator stops and returns a word instead of a number.

What does a return calculator compute, and what can it never produce?

Three outputs come out of one set of inputs. The first is the absolute changeThe whole movement from the value at the start of a period to the value at the end, as a percentage of the starting value. over the period entered. The second is the annual equivalentThe steady yearly rate which, compounded across the same number of years, reaches the same total movement. of that change. The third is the gap between the scheme figure and whatever yardstickAny second figure a result is measured against, such as an index return or another scheme's return. figure supplied. One record in, three readings out.

The three are not competing. Ask how far a train went, how fast it went, and how it did against the timetable, and nobody hears the three answers as contradictions. All three return measures are printed as a percentage, so they look like rival estimates of one quantity. Printing is the only reason the three answers are harder to hear apart.

A fourth output is missing on purpose. The tool gives no verdict, no ranking, and no view on whether any figure it prints is one to be pleased about. A number and a judgement are different objects, and this tool makes only the first. A verdict would import a standard, and that standard would be invisible to whoever read the output.

A gap of 1.3 points looks like a verdict. It is not. A gap is a subtraction, and a subtraction inherits every property of the two things subtracted, including whether they were measured the same way. The labels are the only reason the subtraction is worth doing.

Which two labels does every figure it produces have to carry?

The exact period, and the basisThe statement of what a figure has already had taken out of it: before charges, after charges, or neither. of each figure entered. The tool returns no usable output until both labels are supplied. A calculator that fills in a missing label silently produces a confident number that nobody can check, so the refusal is a design decision rather than an inconvenience. The number still appears, with three decimal places, and the assumption behind it has left no trace on the screen.

The period decides which measure even applies, so take the period first. Over a year and a bit, an annual figure is a real conversion. Over eleven months it is a projection wearing arithmetic clothes. Over twenty years the two can stand more than twenty times apart, a spread worked out in full below. None of that can be read off the number itself.

Now the basis, the harder of the two labels. A figure is grossBefore the charge for running the thing has been taken out. when the cost of running the thing has not yet come out of it, and netAfter the charge for running the thing has already come out. when it already has. Readers forget a third state: a figure carrying no cost at all. The thing measured was never something anybody could have held, so nothing was ever taken out of it. An index sits there. Setting a net figure against a costless one and calling the difference performance is the most common error in this whole area, and it is an error of labelling rather than of arithmetic.

Here is the household version. One person says they cleared Rs 62,000/- last month and means what landed in the bank after tax was deducted at source. The other says Rs 65,000/- and means what the offer letter says. The Rs 3,000/- between them describes nothing. Nobody made an arithmetic mistake. Somebody skipped a label.

Why do the measures disagree when they describe one identical record?

Because the annual equivalent is a root of the absolute change rather than a share of it, and a root and a share coincide in one place only. Over exactly one year they are the same figure; over every other period length they are not. The arithmetic behind the root is covered separately.

Run it on the stated year. The scheme returned 13.4 per cent net over one year, so the absolute change is 13.400 per cent. The annual equivalent is the first root of that same change, and the first root of anything is the thing itself, so it is also 13.400 per cent. Over exactly one year the two readings are forced to agree. A single stated year is therefore the one period on which nobody argues about which measure to quote.

The period can also be stretched. An absolute change of 17.300 per cent over twenty years, a figure from cost arithmetic worked out separately rather than any year a scheme recorded, is clean and carries a long period. The annual equivalent is the twentieth root: 1.173 raised to the power of one twentieth, less one, and the answer is 0.801 per cent a year. Against 17.300 that is a ratio of 21.6 to one.

Neither output is more correct. The absolute change answers what happened in total; the annual equivalent answers at what steady rate. The disagreement between them is information about the length of the period rather than noise in the measurement. A wide gap says a long stretch; no gap at all says exactly one year.

The two readings part company a second way, and this is the one that costs money. Hold a steady 0.801 per cent a year and let the years run. The root undoes compounding exactly, so taking the root of whatever total has built up returns 0.801 per cent at every period length. Compounding has made the total more than the years times the rate, and division has no way of knowing that. Dividing the total by the number of years therefore returns 0.801 at one year, 0.831 at ten and 0.865 at twenty, climbing steadily.

Dividing an absolute change by the number of years is not annualising it, and the error grows quietly with the period rather than announcing itself. Here is the check that settles it. Run the divided reading forward: 0.865 per cent a year compounded for twenty years arrives at 18.798 per cent, not the 17.300 it was meant to describe, an overshoot of 1.498 points. Run the root reading forward and 0.801 per cent a year arrives at 17.2997 per cent, three ten-thousandths of a point short, and short only because 0.801 is itself a rounded printout of 0.801015 per cent.

What happens to the two annual readings as the period stretches?

Stretch it yourself. One absolute change is held still at 17.300 per cent while the period runs from three months to twenty years. Two panels are drawn because one is not enough: the left starts at zero and cannot show the difference, the right declares an origin part way up the scale and can. Neither panel is lying and neither is sufficient. Every performance chart is in that situation.

Why does the tool refuse to annualise a period shorter than a year?

Because the arithmetic that would do it is not a conversion, it is a forecast. Turning three months into a year means multiplying the observed stretch by the part that was not observed. The multiplication assumes those nine months behave like the three on record. An assumption about the future dressed as arithmetic is still an assumption, and a calculator that makes it quietly has made a forecast on the reader's behalf.

A stub periodA stretch shorter than the full unit being reported, such as four months inside a year. is not a small year, it is a different object. A vegetable seller who takes Rs 4,000/- on a Tuesday in a wedding season has not established a rate of Rs 14,60,000/- a year, and everybody sees why the moment it is put in rupees on a street. In percentages the arithmetic is tidy, and tidiness reads as authority, so the same projection starts to look like a conversion.

So the words not annualised come back in the field where a number would have appeared, at the same size and in the same place. A refusal in small grey type below an output is a footnote, and nobody moving quickly reads footnotes.

What does the tool do when a basis is unknown?

The tool returns the gap as unavailable rather than a number. A subtraction across two unknown bases produces a number with no subject, and no amount of decimal places rescues it. The benchmark material argues this rule in prose, where it is advice, and advice is followed by people who were going to be careful anyway. Here it becomes machinery. A field that will not accept a blank removes the option of being careless without noticing, and that is the strongest form a rule of method can take.

A third refusal falls out of the same rule. Suppose both bases are stated but do not match and neither can be restated into the other. The charge is known, so a costless figure against a net one can be restated. A net figure against one whose charges are known only to somebody else cannot be restated, and the tool prints both bases and stops.

How does the gross and net restatement change a benchmark gap?

The restatement roughly doubles the gap, and the tool shows both readings on one screen rather than offering a choice. The scheme returned 13.4 per cent net over the stated year, computed from values that already carry the 1.65 per cent charge. The benchmark returned 12.1 per cent carrying no cost at all. Subtracting the second from the first sets a net result against a costless one, so the 1.300 points it produces is not a like for likeA comparison in which both figures have had the same things taken out of them. reading.

Putting both sides on one footing means restating the scheme figure upwards, by either of two routes. The first adds the charge straight back: 13.4 plus 1.65 is about 15.05 per cent, and 15.050 less 12.100 is a gap of 2.950 points. The second route starts from a charge that accrues daily against an asset base moving every day. A daily accrual is multiplicative rather than additive, so the second route divides instead of adding: 1.134 divided by 0.9835 is about 1.153025, which is about 15.302 per cent, and 15.302 less 12.100 is a gap of 3.202 points. Both reconcile backwards on their own route: 15.050 less 1.650 returns exactly 13.400, and 1.153025 times 0.9835 returns exactly 1.134.

The two routes disagree by 0.252 points, and that residue is printed as a row rather than smoothed away. Even the multiplicative route treats a daily accrual as one flat annual deduction, so neither restatement is exact. An approximationA figure close enough to be useful and known not to be exact. is written with the word about attached and its residue shown, rather than as an equality a later reader will quote as one.

Read the last row first. Two careful routes on identical inputs land a quarter of a point apart, and that residue is nobody's mistake: it is what happens when a daily process is described by an annual number. The honest gap sits near three points, more than twice the 1.3 the headline shows. Both rows are printed because a reader shown only one will use whichever one they were shown.

Why is the scheme's stated return not the return a holder got?

Because the scheme's figure is struck from one unit value to another and takes no notice of how much money was riding on it in between. The stated figure is the return on a single rupee left alone from the first day to the last. A holder who put money in twice held one number of units through the early part of the year and a larger number through the later part, so their own answer is weighted by how many rupees were in and for how long. The scheme's figure and the holder's figure are two different sums on one identical record, and neither is a corrected version of the other.

Run it on the Girnar record. A unit is Rs 35.00 on the first day, Rs 31.50 at the half year, and Rs 39.69 at the close. The move from Rs 35.00 to Rs 39.69 is the 13.4 per cent the scheme states. A holder puts Rs 1,00,000/- in on the first day, buying 2857.143 units, and another Rs 1,00,000/- in month 6, buying 3174.603 units because units are cheaper then. The holder ends with 6031.746 units at the close, worth Rs 2,39,400.00/-. The one steady yearly rate that turns Rs 1,00,000/- over twelve months plus Rs 1,00,000/- over six months into Rs 2,39,400.00/- is 26.796 per cent. The scheme's own figure for that same year is 13.400 per cent. Same scheme, same year, same holder, and the two answers stand 13.396 points apart.

The disagreement runs the other way too. Keep the close at Rs 39.69 but let the unit value rise to Rs 42.00 by the half year instead of falling to Rs 31.50, and the same holder adding in month 6 buys dear rather than cheap: their own year comes back at 5.289 per cent against the scheme's unmoved 13.400. Two settings hand back agreement and they are the only two, the first day and the last.

The second error, and who pays for it

Somebody reads the scheme's 13.400 per cent and tells a holder that is what their money did. On the falling path that holder is being told 13.396 points less than their own year produced, and on the rising path 8.111 points more. A scheme return describes the scheme. Turning a scheme return into a sentence about a person needs the dates the rupees arrived on, and the published figure does not carry those dates.

The missing dates are why the instrument asks for the account statement and not only the figures on the disclosure. The two questions look like one because both answers are printed as a percentage over the same twelve months.

What does the tool return when the record is run through it?

Three settings are worth running, and all three are printed here so a reader who never touches the machinery still leaves with every reading. The first is the stated year. Enter 13.4 with the basis set to NET, a period of twelve months, a charge of 1.65 per cent, and a yardstick of 12.1 with its basis set to NO COST AT ALL. Back come an absolute change of 13.400 per cent, an annual equivalent of 13.400 per cent identical to it because the period is exactly one year, a published gap of 1.300 points, a like for like gap of about 2.950 points on the add-back route and about 3.202 points on the division route, and a residue of 0.252 points. On an amount of Rs 1,00,000/- the absolute change is Rs 13,400.00/-.

The second setting is the conversion case and uses no observed year. Enter an absolute change of 17.300 per cent with a period of two hundred and forty months, and the annual equivalent comes back as 0.801 per cent with the division reading of 0.865 per cent beside it, marked as the wrong route. A reader who has never seen the two side by side cannot recognise the mistake when somebody else makes it, so the division reading is shown rather than suppressed.

The third setting is the pair of refusals. Set the period to three months and the annual field returns not annualised; set the yardstick basis to not stated and the gap fields return unavailable rather than zero. A calculator whose only output lives inside the instrument teaches nothing to a reader who never touches it, so every reading named above is stated in words as well.

What can the output of this tool not be used for?

The output cannot establish that a scheme is good, that a manager has skill, or that a result will repeat. An arithmetic answer is exactly as strong as the record it was computed from, and one year for one scheme is one observation. A gap of 3.202 points from a single year has three decimal places and one data point, and the second of those decides what the number can carry.

The record is small, so say its size out loud. There is one year and one scheme, and the month by month path in part two of the instrument is assumed from two entered values rather than observed. A single year cannot be extended forward or backward, cannot be set against any real fund or index, and is not evidence about what selecting holdings delivers.

A person who reads an output and scrolls has already decided the number is the point and the rest is disclaimer, so the assumptions sit beside the answer rather than beneath it.

Try it out

The tool returns a clean like for like gap of about 3.202 points. How strong is that as evidence?

Exactly as strong as one year on one schemeStrong, because the arithmetic behind it is exactStrong enough to compare two ways of running money

Who reaches for this arithmetic on a working day, and what do they do with it?

Three people, and none is doing it out of interest. Sohail Merchant, who heads operations at Girnar Asset Management, runs the restatement backwards when a query arrives: a holder writes in saying the year was 13.4 per cent but the charge is 1.65 per cent, so where is their 11.75 per cent. The answer is that 11.75 per cent takes the same money out twice. Most queries of this kind are label queries, and they resolve in one screen when the label is shown rather than described.

An analyst preparing a note uses it to stop themselves. The temptation on a scheme figure and an index figure over the same year is to subtract and move on, and the field that refuses a blank basis is what interrupts that. The analyst then writes two rows rather than one, with the word about in front of both. The slowness is the product.

What secures a loan is a rupee figure and not a rate, so a lender assessing somebody who holds units cares about none of the gap rows and only about the value of the units held. A household deciding whether they are on track wants part two instead. None of the three can learn from this arithmetic whether the result was any good. A verdict needs a standard the tool does not hold and the surrounding material does not supply.

The error that gets made, and what it costs

A reader enters two figures, reads the gap that comes back, and quotes it as what the manager achieved, having quietly accepted whichever basis the tool happened to default to. Nothing about the output looked wrong. The gap was formatted, it had decimals, and it came out of a box. A calculator confers authority, so a number that came out of a machine is trusted more than the same number written out by a person, and the assumptions behind it are the first thing a user stops looking at.

The second failure is the writer's, and it comes first in time. A tool that fills in an unstated basis, annualises a stub period, or shows a single gap row has made three decisions on the reader's behalf and displayed none of them. The reader cannot be blamed for missing what was never printed. Every reader of that tool inherits all three decisions and none of them ever knows it happened. A mistake that never surfaces is worse than an ordinary one.

The fix is structural rather than a warning label, and its three parts are the whole design of the calculator. The tool refuses to return an output where a label is missing. The tool shows both gap rows together rather than letting anybody choose which one the reader sees. And it prints what it assumed beside the answer rather than beneath it, so a reader who never reads a word of the surrounding text still cannot get an unlabelled number out of it.

India

Who decides which measures a scheme must show, and where is that written down?

The Securities and Exchange Board of India (SEBI) does. SEBI fixes the periods over which a scheme return must be shown, the form the disclosure takes, the yardstick a scheme is measured against and the valuation duties behind the values a measure is computed from, and revises all of them from time to time. The current position is published at sebi.gov.in and is worth reading on the day it is needed, and material at amfiindia.com is description rather than the rule.

Tax is the other rule-bound limb and is covered separately. How a gain on units is classified, what holding period applies to that classification and what rate follows are matters for the tax authority at incometaxindia.gov.in.

Try it out

Which single field, left blank, makes every output on this tool unusable rather than merely incomplete?

The periodThe basis of each figureThe amount in rupees

How each measure works on its own is covered separately, as are the cash a scheme holds and the credit quality of what a debt scheme holds. How a gain is classified and taxed is a matter for the tax authority at incometaxindia.gov.in. The Girnar Broad Market Index Fund and the discretionary mandate covered separately are both measured against different sticks on different bases, and setting two such records side by side without restating both is the exact error the compulsory labels exist to stop. Which periods a scheme must report, and in what form, are SEBI's to set and to change, at sebi.gov.in, with industry level material at amfiindia.com.

References

Source	Document	Where
Securities and Exchange Board of India	The rules fixing the periods over which a scheme return must be shown, the form that disclosure takes, the yardstick a scheme is measured against and the valuation duties behind the values a measure is computed from	sebi.gov.in
Association of Mutual Funds in India	Industry level material on how scheme performance is published and described, which describes practice rather than making any rule	amfiindia.com
The tax authority	The classification of a gain on units and the treatment that follows from it, which is settled by the tax authority rather than by any measure of return	incometaxindia.gov.in

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