Absolute Return: Why a Long Period Flatters the Number

An absolute return states the whole change over a holding period without saying how long that period was. Over one year it equals the annual figure. Over twenty it does not, and converting between them is a root rather than a division, so dividing an absolute figure by the number of years overstates the annual rate every time.

Here is what sits underneath that. A return is a comparison of two values, one at the start and one at the end, and the stretch of time between them is part of the measure rather than a footnote to it. An absolute returnThe whole change from the start of a holding to the end of it, stated as a proportion of the starting value, with no rate a year attached. does the comparison honestly and then leaves the stretch of time out of the number it prints. Nothing is falsified when that happens. But what each figure was measuring is no longer visible, so a figure that has dropped its period can no longer be set beside any other.

Girnar Asset Management Limited, an invented asset manager, runs an open ended equity scheme called the Girnar Large Cap Equity Fund. For one stated year that scheme returned 13.4 per cent net. The same scheme is offered at an expense ratioThe charge running against a scheme's assets each year, quoted as a percentage of those assets. of 0.85 per cent and at one of 1.65 per cent, a difference of 0.80 percentage pointsThe unit for the difference between two percentages. Moving from 0.85 per cent to 1.65 per cent is a rise of 0.80 percentage points, not of 0.80 per cent. a year, and the holdings behind the two are identical. The portfolio is managed by Kalyani Bhagat, and operations are headed by Sohail Merchant.

A scheme, a unit, how the value of a unit is struck, and what the charge does to that value each day are all covered separately. So is the structure behind those two charges, which appears here only as arithmetic. The ground ahead is narrow: the difference between a total and a rate, and the one calculation that converts between them correctly.

What is an absolute return, and what does it not carry?

An absolute return is the whole change from the start of a holding periodThe stretch of time between the starting value and the ending value. to the end of it, written as a proportion of the starting value. A value at the start, a value at the end, the difference between them, divided by the start. Those four steps are the entire construction, and there is nothing wrong with any of them.

Now look at what came out. The figure carries a change and a direction. The figure does not carry the length of the stretch it covered. The period is not hidden inside an absolute return waiting to be recovered; it was never put in, so the number alone cannot say whether the change took eleven months or eleven years. Every difficulty that follows comes from that single missing piece, and so does every fix.

A shop on a street corner makes the point. The owner says takings have grown by seventy per cent since the shop opened. Nothing can be done with that sentence until somebody asks when it opened, and the answer changes the meaning completely: seventy per cent since last Diwali is a different shop from seventy per cent since 2009. The sentence was true either way. The claim was just not usable until the second sentence arrived.

Over exactly one year, do the two measures differ at all?

No, and that is where the confusion is born. An annual returnThe rate a year that, applied over the whole period, produces the total change actually recorded. is the rate a year that would produce the recorded change over the recorded stretch. When the stretch is exactly one year, the rate a year that produces the change is simply the change. There is nothing to convert, no root to take, no division to do. Over exactly one year an absolute return and an annual return are the same figure, arrived at by the same arithmetic, and a reader who has only ever seen one year figures has never had the chance to notice that the two measures are different objects.

Then the period stops being one year and the two come apart, and they come apart in one direction only. For any period longer than a year, and for any change in the upward direction, the absolute figure is the larger of the two, and it gets larger relative to the annual rate the longer the stretch runs. The asymmetry is not a rhetorical trick anybody applied but a consequence of the arithmetic, worked through next.

Why does a longer period make the same annual rate look large?

Because of compoundingEach year's growth applying to a base that already includes the growth of the years before it, rather than to the original base., covered separately as arithmetic and applied here only. Each year's growth does not land on the amount the holding began with. The growth lands on the amount carried at the start of that year, and that amount already includes everything the earlier years added. So the total change outruns the annual rate multiplied by the count of years, and the gap between those two widens as the count grows.

Run it on the invented cost gap. The gap is a stated annual figure rather than an observed outcome, so compounding it is safe. The two versions of the scheme differ by 0.80 percentage points a year and hold identical portfolios, so each year the version carrying the lower charge keeps a factor of 1.008 more of whatever the holdings did. After five years the factor is 1.008 multiplied by itself five times, or about 1.040645, so the terminal valuesThe amount each version is worth at the end of the period, once every year of the difference has been applied. differ by about 4.065 per cent. After ten years the factor is about 1.082942, a difference of about 8.294 per cent. After twenty it is about 1.172764, a difference of about 17.276 per cent.

Nothing dishonest has happened anywhere in that sequence. The absolute figure of about 17.276 per cent is exactly right, and it becomes misleading only at the moment somebody quotes it without the twenty years standing next to it. A sweet shop that grew takings a little every year for two decades can advertise a large total change and be telling the plain truth, while the total says almost nothing about what any single year in the shop was like.

The second panel is doing real work, and it has not been stretched by cutting the bottom off an axis, the usual way a small difference is made to look big. Both panels start at zero. The right hand one simply plots a different quantity, the difference itself, and gives each point of that difference about 13.8 times the height. A difference of 0.065 points cannot be drawn at a scale that also has to hold 17.276 points, so the honest move is to draw it twice rather than to draw it once and exaggerate.

What does the whole calculation look like written out?

Here is the build in one place, on the invented cost gap, with each figure rounded only at the end and the direction of every rounding stated. The unrounded values are carried right through the chain, so the row that reports about 17.276 per cent is not built on the row that reports about 8.294 per cent.

The last pair of rows is the strongest check available. The identity closes in both directions: adding twenty separate yearly amounts and raising one factor to the twentieth power land on the same 17.2764043480, and taking the twentieth root of the total hands back the 0.80 per cent a year the arithmetic started from. A single figure that only reconciles one way can be a coincidence. One that reconciles both ways is checked.

Two roundings deserve to be named rather than buried. The five year figure of 4.0645140513 rounds up to 4.065 and the twenty year figure of 17.2764043480 rounds down to 17.276, so the pair of them do not drift in the same direction and neither should be treated as exact. And where 17.3 per cent appears below, that is the twenty year figure rounded to one place. The whole size of the error depends a great deal on which version of the number is picked up.

Why is turning an absolute figure back into an annual rate a root and not a division?

Because the growth compounded on the way out, so it has to be uncompounded on the way back. Going forward, the annual rate was applied twenty times over, each time to a base that already carried the earlier applications. The operation that undoes twenty applications of a factor is the twentieth rootThe number which, multiplied by itself the stated number of times, gives back the value it started from. The twentieth root of a factor is what that factor grew by each year., not a division by twenty.

Dividing by the number of years is not a sloppy version of the right answer. Division is the correct answer to a different question, namely what the rate would have been if the growth had been added in equal slices to the original base and had never compounded at all. Equal slices are real arithmetic and simply not what happened to the money. Division always overstates the annual rate, it never understates it, and the overstatement grows with both the length of the period and the size of the total.

On the numbers: the twentieth root of 1.1727640435 is 1.008, so the rate a year is 0.800 per cent, and the check row in the table above already showed that closing. Division instead gives 17.2764043480 over twenty, or 0.8638202174 per cent, about 0.864 per cent. The overstatement is about 0.0638 of a percentage point, a shade under eight per cent of the true 0.800 per cent rate. The rounded 17.3 per cent taken into the same division lands on 0.865 per cent instead, and the overstatement reads as 0.065 of a point. Both of those wrong answers are in circulation, and the difference between them is only which version of the total the person picked up first.

Some of the numbers in this arithmetic sit close enough together to be mistaken for each other. The lower of the two invented charges is 0.85 per cent of assets a year. The readings that division produces run from 0.800 up to about 0.864 per cent a year of growth, and they pass straight through 0.85 on the way. The charge and the converted rates are unrelated quantities that happen to land near one another, one a charge on a pool of assets and the others badly converted rates of change, and a reader who lets them touch has manufactured a connection that the arithmetic never made.

How large is the error, and where does it grow fastest?

The error starts at nothing and widens the whole way. Dividing by one and taking a first root are both the identity, so at one year the division reading and the root reading are the same number. At five years the division reading is about 0.813 per cent against the true 0.800, an overstatement of about 0.013 of a point. At ten years it is about 0.829, an overstatement of about 0.029. At twenty it is about 0.864, an overstatement of about 0.0638.

The shape of that error is what makes the mistake dangerous rather than merely wrong: it is smallest over short periods, where a reader could check it in their head, and largest over long ones, where nobody does. A person who divides a three year figure and is out by a hundredth of a point has made an error that will never change a single decision. The same habit applied to a twenty year figure produces a rate that is out by about a twelfth of itself, and it will look completely ordinary sitting in a table.

The second marker does something worth catching as the years are dragged. At sixteen years the division reading lands on 0.850 per cent a year, exactly the number the lower of the two invented charges carries. The two have nothing to do with one another. One is a charge on assets and the other is a mis-converted rate of change, and they collide at that one setting purely because the arithmetic put them there. The collision is a small reminder that a number matching another number is not evidence of any relationship between them.

What does an absolute figure hide from the reader?

Three things, and they are worth naming one at a time. The figure hides the length of the period, and that omission is what everything above has turned on. Hidden too is every individual year inside that period, including the ones that went backwards. And it hides the shape of the change, meaning whether it arrived steadily across the whole stretch or in one short burst with a long flat stretch on either side.

The shape of the change gets underrated most. Two holdings can produce an identical total change over an identical period, one by climbing a little every year and the other by doing nothing for years and then moving in one stretch, and no absolute figure can tell them apart. The practical consequence is blunt: two absolute figures over different periods are not comparable at all, and a reader holding two such figures with no periods attached is holding nothing they can act on.

When is an absolute figure the honest number to quote?

When the question is about one specific amount of money over one specific stretch, a real question and a common one. For what happened between the day money went in and the day it came out, the absolute figure is the most concrete number available, and it is what the bank balance did. An annualised rate is a construction and the total change is an observation, so no annualised rate can be pointed at with the same directness.

The condition attached is the whole of it: quoted with its period, an absolute figure is the most usable number here, and quoted without one it is the least usable number here, and nothing about the figure itself changes between those two states. That is why the measure is better described as incomplete than as dishonest. Two different periods cannot be compared until both are put on a rate a year, so the moment two records are set beside each other the absolute figure stops being the right tool, no matter how carefully its period is stated.

What happens when this arithmetic is tried on the one year this record holds?

Nothing, and refusing is the correct outcome rather than a shortcoming. For one stated year, the Girnar Large Cap Equity Fund recorded a net return of 13.4 per cent. Because that period is exactly one year, its absolute and annual figures are the same number, so there is no conversion available to perform and none needed. The one year case is the boundary case set out above, sitting in the record itself.

Now the harder half. The record is a single year attached to a single scheme, and nothing sits behind it: no earlier year, no series running month to month, and none of the range's other schemes. The arithmetic worked above stops at the edge of one observed year. One observed year will not annualise into anything, will not stretch forwards or backwards, and will not sit beside any figure whose basis has gone unstated. The compounding worked above was run on a stated annual charge gap, which is a known constant that does not change because a year turned out one way or another. Running the same arithmetic on one observed year would produce a number that looks like a calculation and behaves like a forecast, and the tidiness of the output is exactly what would make it convincing.

So the honest sentence about that figure is short and it stays short. A net 13.4 per cent, from the Girnar Large Cap Equity Fund, across one stated year: one scheme, one period. The arithmetic on it is exact and its reach is very small, and neither of those cancels the other.

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.