Real Return: Why You Divide Rather Than Subtract

A real return is what a nominal return buys once prices have moved. The money grew and the prices grew, and the figure required is the ratio between those two growths. A real return is therefore a division rather than a subtraction. Subtracting is a shortcut that stays close at small rates and drifts steadily as either rate rises.

Two things already built are used here. The price rate itself, what a price index is and what sits inside its basket, is handled where prices are built, and the rate is taken as given. The arithmetic was met earlier from the other end: in the notes on reading a commodity price move, two separate moves had to be combined, and growth factors were found to multiply rather than add. The same finding runs backwards here, with two growths that have to be separated and factors that separate by dividing. Anyone holding the multiplication already holds most of the division.

Where each rate in this guide comes from: the price rate of 6.70 per cent is the one these notes have been carrying since prices were built, and the nominal rates are settings chosen so that the arithmetic can be watched. Both belong to the invented Republic of Sankhya, whose figures were built for teaching and measured nowhere.

What is a real return actually measuring?

The setting is a kitchen table rather than a spreadsheet. A household sets aside Rs 1,00,000/- at the beginning of a year. At the end of the year the account shows Rs 1,08,000/-. Over the same twelve months, the shopping this household does every week has been getting dearer, and the identical list of things that cost Rs 1,00,000/- across the year now costs Rs 1,06,700/-. Somebody asks the obvious question at the table: is the household better off, and by how much?

Two answers are honest here, and they are not the same number. The money answer is Rs 8,000/-, the amount the account gained. The buying answer is smaller. The shopping ran after the money and caught most of it: of the Rs 8,000/- that arrived, Rs 6,700/- was needed simply to stand where the household was already standing. The Rs 1,300/- left over is what a real return is about.

Two things moved rather than one, and the figure required is the relationship between the two movements rather than the size of either of them on its own. Hold on to that one sentence. Every step below is a consequence of it. The nominalMeasured in the money itself, with no adjustment made for what the money will buy. A rate, an amount or a wage described as nominal is being quoted at face value and nothing has been taken out of it. figure describes the money. The price levelA single number standing for what a stated collection of goods and services costs, so that its change over time can be measured. What that collection contains and how the number is built are handled where prices are built. describes what things cost. A real return is neither of them. A real return is the comparison between them.

Why is it a division rather than a subtraction?

Percentages are a way of writing a movement down. Percentages are not the things that moved. Comparing two movements means going back to what each of them did to a quantity, and the tidiest way to hold that is a factor: the number the starting quantity got multiplied by.

A nominal 8.00 per cent means the money at the end is 1.0800 times the money at the start, and a price rate of 6.70 per cent means the same shopping costs 1.0670 times what it did. So write down what the household can buy. At the start, one full list. At the end, 1.0800 times the money against prices 1.0670 times as high. Divide 1.0800 by 1.0670 and the household can buy 1.012184 lists. Take away the one list it could already buy and 0.012184 of a list is left. As a percentage that is 1.22 per cent.

The percentages are not the two quantities being compared; the factors are, and two factors relate to each other by division. The money is a multiplier, the prices are a multiplier, and the only operation that separates one multiplier sitting on another is division. Writing 1.0800 over 1.0670 and taking the quotientThe result of dividing one number by another. The word is worth having because it names the answer without naming either of the two numbers that produced it. is the whole method, and taking one away at the end is only converting a factor back into a rate.

Where does each input actually come from?

Seven things end up on the desk and only the first three go into the division itself, so the useful thing to write down is not what each one means but where each one is sitting when it is needed. Two of those three are somebody else's published number, to be fetched. The third is not a number at all and is the one people forget.

The period is the one input nobody publishes, and being unpublished is exactly what makes it break more often than the rest. Every other row can be fetched and held up against whoever put it out. Nothing anywhere states the period, so a mismatched period cannot be held up against anybody: two numbers taken over different stretches of time divide quite happily and hand back an answer that means nothing. Read the period off both documents and write it beside the answer.

A single rate on a single gain lands in two quite different places depending on the rule, so the last two tax rows are worth one worked pass. Put 25.00 per cent into the panel above and leave the rest at the worked example. Applied to the whole gain measured in money, the tax is 25.00 per cent of the Rs 8,000/-, so Rs 2,000/- goes, the money ends at Rs 1,06,000/-, and against a list now costing Rs 1,06,700/- the real return is minus 0.66 per cent. Applied instead to what is left once the starting cost has been restated for prices, the base is the Rs 1,300/- rather than the Rs 8,000/-, so Rs 325/- goes, the money ends at Rs 1,07,675/-, and the real return is 0.91 per cent. One rate and one gain, and the answer falls on either side of zero purely according to which rule the authority happens to state. Which rule applies where is settled by the authority that states it, and the panel works both without preferring either.

How large is the error from subtracting?

Now the shortcut. Taking 6.70 away from 8.00 gives 1.30. The correct answer was 1.21837, printed as 1.22. The shortcut is out by 0.08163, and since both figures are themselves percentages, the difference between them is measured in pointsThe unit for the distance between two percentages. A rate that slips from 8.00 per cent down to 6.70 per cent has lost 1.30 points, and describing that as a loss of 1.30 per cent would say something quite different. rather than in per cent. The error is 0.08 points, and stated like that it sounds like nothing worth an argument.

So tabulate it instead. In every one of the five settings below the nominal rate is exactly 1.30 points above the price rate, so the shortcut hands back 1.30 every time and never moves. The division does move. The same 1.30 points of gap is being spread over a bigger and bigger price level.

The shortcut's answer does not move by so much as a hundredth down the whole table. The true answer falls from 1.28 to 1.07, so every bit of the widening gap belongs to the shortcut. A reader who saw only the subtraction column would conclude that all five settings deliver the same thing. The last row hands back roughly five sixths of what the first row does, and the subtraction is blind to the whole difference. The gap between the rates is the only quantity subtraction looks at.

Leave the price rate alone and raise the nominal rate instead and the gap opens too. Against a price rate held at 6.70 per cent, a nominal 7.00 gives 0.28 by dividing and 0.30 by subtracting, an error of 0.02 points; a nominal 12.00 gives 4.97 against 5.30, an error of 0.33; a nominal 20.00 gives 12.46 against 13.30, an error of 0.84. Either rate rising opens the gap, in ways the section after this one shows are not the same.

If any of this feels familiar, it should. The notes on reading a commodity price move ran into the identical structure from the opposite direction. There, a commodity dearer by 20.00 per cent where it is quoted, met by a quoting currency that itself gained 5.00 per cent, landed at home as a move of 26.00 rather than the 25.00 that adding hands over. The factors 1.2000 and 1.0500 have a product of 1.2600. Growth factors combine by multiplying and separate by dividing, and the mistake in both directions is treating percentages as though they were the quantities themselves. Learnt once, it holds both ways round.

When does the shortcut stop being harmless?

An error of 0.08 points does sound trivial. But 0.08 points of what? The answer being produced is 1.22 per cent. Dividing the error by the answer gives 0.08163 over 1.21837, or 6.70 per cent.

The shortcut is wrong by 6.70 per cent of the answer it is producing, and 6.70 is not a coincidence. The error as a share of the answer is always exactly the price rate. The proof takes one line. The correct answer is the gap between the rates divided by the price factor, and the shortcut is that same gap with no division done, so the shortcut is the correct answer multiplied by the price factor and overshoots by the correct answer multiplied by the price rate. Divide the overshoot by the answer and the gap between the rates cancels, leaving the price rate standing alone.

Run it back through the table above and the identity holds in every row: at a price rate of 21.70 per cent the error is 21.70 per cent of the answer, roughly a fifth of the figure just handed to somebody. The identity also says something the points column hides. The nominal rate is nowhere in the identity, so raising it makes the error bigger in points and leaves its share of the answer untouched. The absolute error grows with the gap between the rates; the relative error tracks the price rate and nothing else.

So the shortcut stops being harmless in three situations, and they are not one situation wearing three hats. The first is a high price rate, and the price rate is the share of the answer being thrown away. The second is two rates close together. The answer is then small, and a fixed relative error eats a small answer as easily as a large one. The third is compoundingApplying a rate repeatedly, so that each period works on the result the period before it produced rather than on the original amount. How this builds up over several periods is worked through where growth over time is built., and it is the worst of the three because it is the one nobody watches. Carry the correct 1.22 per cent through five periods and it accumulates to 6.24 per cent; carry the shortcut's 1.30 through the same five and it accumulates to 6.67. An error of 0.08 points after one period is 0.43 after five, all of it from a step somebody decided was too small to bother with.

What happens when prices run ahead of the nominal rate?

Nothing special happens. The same division runs and hands back a number below zero, and that number is a perfectly ordinary result rather than a sign that something has gone wrong.

Work one. A nominal 5.00 per cent against the same price rate of 6.70 per cent gives 1.0500 over 1.0670, or 0.984067. The quotient sits below one, and that is the whole message: the household ends able to buy 0.984067 of the list it could buy at the start. Take one away and minus 0.015933 is left, or minus 1.59 per cent. At the kitchen table, the account holds Rs 1,05,000/- against a list now costing Rs 1,06,700/-, so it is Rs 1,700/- short of what it used to cover.

A negative real return is the same division producing a quotient below one, and it is an outcome rather than an error. One habit is worth building around it, and it is about writing rather than arithmetic. Decide once whether the minus sign is carried by the number or by the words around it, and never let both do it. A figure printed as minus 1.59 inside a sentence that also says the return fell is one careless edit away from saying the opposite. Here the number carries the sign, and the one place direction is put into words uses the size alone.

The second habit is about rounding, and it matters here more than anywhere else in this guide. When a figure that can go below zero is rounded, the practice is to round its magnitudeThe size of a number with its sign set aside, so that minus 1.59 and 1.59 have the same magnitude. Useful whenever how big something is has to be discussed separately from which way it points. and then put the sign back on. The common rounding routines move a value exactly on a half towards zero, so a true minus 1.585 prints as minus 1.58 while a true 1.585 prints as 1.59, and one figure has then been rounded two ways depending on which side of zero it fell. Round the magnitude away from zero and both print the same last digit.

What does a real return leave out?

Three things, and the last is the one that stays.

A real return does not say whether the return was worth having. Worth is a judgement about what else could have been done with the money, what had to be given up and how much uncertainty was accepted, and none of those quantities appears anywhere in the division. The arithmetic here takes two rates and hands back a third. The arithmetic has no view.

About the future it says nothing whatever. Every figure above describes a period that has finished. A real return worked for a period that has already happened is a measurement; the same arithmetic pointed at a period still ahead is two assumptions with a division between them, and it should be labelled that way.

And the price index divided by is somebody else's basket, a limitation the figure itself can never correct for. An index measures a stated collection of things in stated proportions, and those proportions are chosen to stand for a population rather than for any one household. If a quarter of a household's spending goes on rent and the index gives housing a tenth of its basket weightThe share of the index that one category of spending accounts for, so that a move in that category counts for that much of the whole. How the weights are set and revised is handled where prices are built., then a period in which rents move sharply and everything else holds still moves that household's cost of living far more than it moves the index. Its true real return that period was worse than the computed figure, and nothing in the division could have shown it. The only fix is to read the weights, and that is why they sit among the inputs earlier even though the division never uses them.

How does somebody doing this for a living handle it?

Somebody doing this for a living divides every time, without pausing over whether this particular case is one where the shortcut would have been close enough. The habit is not fussiness. The habit comes from a property of the error: it is invisible at the moment it is made and becomes visible much later, in somebody else's hands.

Consider what a research analyst is doing when a real return is produced. The figure goes into a note, and the note is read by somebody holding a figure of their own, worked from different underlying numbers by a method they did not describe. The two are set side by side and they differ. Nobody in the room can say what the difference means. A difference arising from method looks exactly like one arising from substance. Half a working day goes into establishing which it was.

The reason to divide is not that the answer is materially better in any one case, it is that the shortcut costs nothing to avoid and its error is undetectable in the only place it ever shows up. The same runs for a lender setting what one loan earns after prices against what another earns, for a household asking whether an arrangement kept pace with its own costs, and for a policy office putting a series of these figures side by side across several periods. In each, the single figure is not the product. The comparison is, and a comparison is worth only as much as the consistency of the method under both sides of it.

Two more habits travel with it. Write the period beside every real return. A figure with no period attached cannot be compared with anything. And write which price index was used. Whose basket the reader is looking through is a permanent limitation rather than a footnote.

The Republic of Sankhya, its currency and the household these notes follow are invented.
Educational material. Not advice on any investment, tax, budget or market position.