Return on Equity and the DuPont Decomposition: The Arithmetic

Return on equity is profit after tax divided by equity. For Anjani Stationers that is Rs 30,00,000 over Rs 1,42,00,000, or 21.1 per cent. The DuPont decomposition breaks the same figure into net margin, asset turnover and an equity multiplier. Multiply those three and they land back on 21.1 per cent exactly. Each variant, every step, the reconciliation check and the location of every input in a filing follow in that order.

The panel opens on what Anjani Stationers Private Limited published for the year just closed, and because a figure that lives only inside a script cannot be read by anyone who does not run it, here is that opening setting written out. Revenue from operations Rs 2,70,00,000, operating profit Rs 41,50,000, profit before tax Rs 38,00,000 and profit after tax Rs 30,00,000. Total assets Rs 1,80,00,000 at the year end against Rs 1,33,00,000 a year earlier, and equity Rs 1,42,00,000 against Rs 1,12,00,000. Divide the profit after tax by the closing equity and the return is 21.126761 per cent. Three factors: 0.111111, then 1.500000, then 1.267606. Five factors: 0.789474, then 0.915663, then 0.153704, then 1.500000, then 1.267606. The single division already gave 0.211268, and each chain arrives at that same figure. The two badges at the foot of the panel put the agreement on screen.

Two of the four figures the slider can push cannot move the return at all, and seeing why is worth more than moving it. Raise revenue by a fifth and net margin falls from 0.111111 to 0.092593 while asset turnover rises from 1.500000 to 1.800000, and the product stays at 0.211268. Raise total assets by a fifth and asset turnover falls to 1.250000 while the equity multiplier rises to 1.521127, and the product stays where it was again. Revenue and total assets each appear once above a line and once below one, so any change made to them cancels against itself and the return never hears about it. Profit after tax and equity appear once each and have nothing to cancel against, so pushing either of those moves the return immediately: a fifth more profit takes it to 25.352113 per cent, a fifth more equity takes it down to 17.605634 per cent.

Underneath that single division sit two decisions nobody announces. A closing figure and an average of two dates give different answers on the same profit, so the first decision is which date the denominator is read on. A company that consolidates a subsidiary publishes an equity total and a profit figure in two versions each, so the second decision is which set of accounts the two numbers come from. Every disagreement in this arithmetic traces back to one of those two decisions being made differently on the top and the bottom of the same fraction.

How is return on equity computed?

Return on equityWhat the shareholders earned in a year, set against what they have standing in the business and written as a percentage. The measure reports on their money alone, not on everything the business has to work with. takes the profit that belongs to the shareholders and divides it by what the shareholders have standing in the business. Nothing else enters the calculation. Return on equity is one division, and every complication comes from choosing what goes into the two boxes rather than from the division itself.

Take a household that put Rs 5,00,000 into a small provisions shop and drew Rs 60,000 of profit from it over a year. The return on what the household put in is twelve per cent. The shop's takings, its rent, its stock on the shelf and the credit it gave to regulars are all interesting, and none of them appears anywhere in that sentence. The measure has that shape by design. Return on equity is blind to the whole middle of the business.

Run Anjani Stationers Private Limited through it on the closing balance sheet. Profit after tax for year two is Rs 30,00,000. Equity at the year end is Rs 1,42,00,000. The first divided by the second is 0.211268. As a percentage that is 21.126761, and everybody writes it as 21.1 per cent. The reconciliation check further on compares two numbers at the sixth decimal place, so hold the underlying figure rather than the rounded one. A rounded input fails a test it should pass.

Which equity figure belongs in the denominator, and does averaging change the answer?

Profit is earned across twelve months. Equity on the closing balance sheet is a photograph taken on the last of those days. When equity moved a great deal during the year, the photograph shows a denominator the business did not have for most of the period that produced the profit, and the return comes out understated. Average equityThe mean of the opening and closing equity totals, used as the denominator so that a profit earned across the whole year is divided by a figure that represents the whole year rather than its last day. answers that by taking the mean of the opening and closing totals.

Anjani Stationers opened year two with equity of Rs 1,12,00,000 and closed it at Rs 1,42,00,000, a rise of Rs 30,00,000. The mean of the two is Rs 1,27,00,000. The same Rs 30,00,000 of profit over that denominator is 0.236220, or 23.622047 per cent, against 21.126761 per cent on the closing figure. A spread of just under two and a half percentage points opens up before a single word has been said about the business, purely from which denominator was chosen.

Neither figure is wrong. When equity moved a lot, averaging is the more defensible choice, matching a flow measured over a period against a stock measured across the same period. The closing figure is easier to trace. Finding it needs one sheet of one statement and no comparative column. Publishing the number without saying which basis it is on is never defensible. Write the basisWhich date or dates every input to a ratio was read on, and which set of accounts it came from. A ratio is only comparable with another ratio computed on the same basis. beside the figure every single time. A reader who cannot see it has no way of comparing the number with anybody else's.

Which equity total, and which profit belongs with it?

Anjani Stationers holds seventy per cent of Chitra Binding Works, so it publishes two sets of accounts and three equity totals between them. Standalone equity is Rs 1,42,00,000. Consolidated equity attributable to the owners of the parent is Rs 1,49,00,000. Consolidated equity including the non-controlling interest is Rs 1,59,50,000. Each of those three has exactly one profit figure that belongs beside it, and the rule is that the numerator and the denominator must cover the same set of people.

Standalone equity of Rs 1,42,00,000 pairs with standalone profit after tax of Rs 30,00,000, giving 21.1 per cent. Consolidated owners' equity of Rs 1,49,00,000 pairs with the Rs 37,00,000 of group profit attributable to those same owners, giving 24.8 per cent. Total consolidated equity of Rs 1,59,50,000 pairs with the whole group profit of Rs 40,00,000, giving 25.1 per cent. Three correct answers, all describing something real, all different.

Now cross them. The owners' profit of Rs 37,00,000 over the total equity of Rs 1,59,50,000 is 23.197492 per cent. The whole group profit of Rs 40,00,000 over the owners' equity of Rs 1,49,00,000 is 26.845638 per cent. The two wrong pairings sit 3.65 percentage points apart, they straddle the correct 24.83 per cent on both sides, and neither of them is the return earned by any shareholder anywhere. They are arithmetic on two numbers that were never about the same people.

How is return on equity decomposed in three steps?

The DuPont decomposition, named for the company whose treasury staff set the arrangement out in the 1910s, multiplies the same fraction back together out of three pieces. Net margin sets profit after tax against revenue. Asset turnover sets revenue against total assets. The equity multiplierThe size of the asset base measured in units of the equity underneath it. At 1.27 there are one rupee and twenty seven paise of assets resting on each rupee the shareholders put up, and the extra twenty seven paise came from somewhere else. is total assets over equity. Multiply the three and revenue cancels against revenue and total assets cancels against total assets. What survives is the profit figure sitting over the equity figure, the same division worked at the top.

Think of a courier who earns Rs 40 a parcel, carries eight parcels a trip and makes six trips a day. Multiply the three and the parcels cancel against the parcels and the trips cancel against the trips, leaving Rs 1,920 a day. Nothing has been discovered by multiplying them. The multiplication breaks one number into three that can be pushed on separately. When the daily figure falls, it is clear which of the three to look at. The decomposition is an identity, so it holds by construction and confirms nothing; its entire value is that it shows where to look.

What do the two extra steps in the five-step version isolate?

The five-step version splits net margin into three pieces so that a movement can be traced to the tax line or the interest line without touching the operating business at all. Tax burdenHow much of a rupee of pre-tax profit is still there once the tax charge has been met. At 0.79 the company kept seventy nine paise and the tax authority took the other twenty one. sets profit after tax against profit before tax, Rs 30,00,000 against Rs 38,00,000, or 0.789474. Interest burdenProfit before tax divided by operating profit. Interest burden is the share of operating profit that survives the finance cost, so a figure of 0.92 means eight paise in every rupee of operating profit went to lenders. is profit before tax over earnings before interest and tax (EBIT), Rs 38,00,000 over Rs 41,50,000, or 0.915663. Operating margin is EBIT over revenue, Rs 41,50,000 over Rs 2,70,00,000, or 0.153704.

Multiplying those three returns net margin: 0.789474 times 0.915663 times 0.153704 is 0.111111. Carried on through asset turnover of 1.500000 and the equity multiplier of 1.267606, the product is 0.211268, the same return the three-step version produced and the same one the direct division produced. Both decompositions land on exactly the same number because both are the same fraction written with extra cancelling pairs inserted, and neither adds any information the accounts did not already contain.

One detail explains the failure set out further on. Three of the five drivers are built entirely from flows: tax burden, interest burden and operating margin each divide one income statement figure by another. Flows are measured across the whole year, so those three drivers do not move at all when the basis switches between closing and average. Asset turnover and the equity multiplier move on both. The basis touches exactly the two drivers that have a balance sheet figure in them.

How is a decomposition checked?

Multiply the computed drivers and compare the product against the return computed directly from profit and equity. If the two agree at the sixth decimal place, every input came from where it was meant to come from. A reconciliationA check that two independently computed figures agree. Here it means the product of the drivers is compared with the return computed straight from profit and equity, and the two must match. that closes is worth running on every year of every set of accounts built, and it takes about four seconds.

The check catches an input from the wrong place. The two failures it catches most often are mixing a closing figure with an average one, and mixing a standalone figure with a consolidated one. Both produce a product that misses the stated return by a visible margin. The check never catches a choice that is wrong but applied consistently. Decompose on closing figures throughout when averaging was clearly the right call and the reconciliation closes perfectly. Internal consistency is all the check can see. The reconciliation is a proof of arithmetic and never a proof of judgement, so a set of drivers that reconcile has proved only that they were computed off the same balance sheet.

What happens when a decomposition is forced to reconcile?

Here is what the mixed basis costs once somebody decides to make the numbers agree rather than find out why they do not. The stated return is 23.622047 per cent on average equity. The drivers as computed give 0.211268. The stated return divided by net margin times asset turnover gives the multiplier that would make the line close: 0.236220 divided by 0.166667 is 1.417323. Written in place of 1.267606, the row balances.

Look at what that plugged figure actually is. Total assets of Rs 1,80,00,000, the closing figure, divided by average equity of Rs 1,27,00,000. The forced multiplier takes the assets from the last day of the year and the equity from a twelve month average, so it is a leverage ratio for a balance sheet that existed on no date in the company's history. The plug is not a rounded version of the right answer. It describes a business that never was.

Which of the six combinations can actually be computed?

Two bases crossed with three equity totals gives six cells, and four of them compute from what Anjani Stationers has published. Unrounded, those four are 21.126761 and 23.622047 per cent on standalone equity, 24.832215 per cent on group equity attributable to the owners, and 25.078370 per cent on total group equity. The two that do not are the average basis against either group figure, and the reason is worth stating plainly rather than working around. Averaging needs an opening figure measured on the same basis as the closing one, and the opening group balance sheet is not among the published figures here. A basis that cannot be read on both dates is not a basis that can be used, and the honest entry in that cell is the name of the missing figure rather than a number built from the nearest thing to hand.

The same shortage limits how far a group return can be taken apart. Return on equity on either group figure needs two published numbers and computes cleanly. Decomposing it needs group revenue for the first two drivers, group total assets for the next two, and group profit before tax and group operating profit for the five-step version. Where those are not in the accounts to hand, the return is available and the decomposition is not, and saying so is the correct output.

Where does each input sit in a filing?

Each note gives the position of a line, not what the line means.

Worked down once, the list becomes muscle memory. Profit after tax is the bottom line of the statement of profit and loss. In a set of group accounts the choice is between the total and the share attributable to the owners of the parent, disclosed immediately below it. Revenue is the first line of the same statement. Operating profit is the subtotal directly above finance cost, and profit before tax the subtotal directly above the tax lines. Total assets is the foot of the asset side of the balance sheet. Equity is the balance sheet total, with the non-controlling interest shown separately inside the equity section. Opening equity for averaging is read from the comparative column or from the statement of changes in equity.

Who runs this arithmetic, and what do they do with the result?

A credit analyst at a lender assessing a working capital line runs the decomposition across three years before writing anything, and the first thing they do is fix the basis and never move it again. If year one is on closing figures and year two on average figures, the movement in the multiplier is partly a real change and partly a change of ruler, and the two are impossible to separate afterwards. The reconciliation check is what tells them the ruler stayed the same, and it gets run on every year rather than on the year that looks interesting.

An equity analyst uses it differently. The return is the headline the note opens with, and the drivers are the paragraphs underneath it, so the drivers have to be defensible line by line. A plugged multiplier is so damaging in that setting for exactly that reason: it does not affect the headline at all, and it corrupts every paragraph that follows. A promoter or a controller like Vaidehi Rao runs it a third way, as a housekeeping check. A decomposition that stops reconciling from one month to the next usually means somebody has changed which schedule the balance sheet figures are being pulled from. In all three settings the reconciliation earns its keep not by proving the analysis right but by proving that nothing moved underneath it.

What does this arithmetic leave unsettled?

Return on equity does not say whether 21.1 per cent is good. Answering that needs a comparison, and one business in one year offers none: no peer set, no sourced industry figure. The identity reports positions and never causes, so it does not explain why any driver moved. And a rising equity multiplier lifts return on equity mechanically whether or not a single extra notebook was sold, so the identity cannot tell a company that improved its trading from one that simply borrowed more.

Both decompositions are arithmetic on figures that were already published, and arithmetic on published figures is not evidence about the business. The decomposition holds for any four numbers, which is what makes it safe to compute and useless as a conclusion. The decomposition buys a place to point, and deciding what the pointing means is a different job entirely.

Anjani Stationers Private Limited, Chitra Binding Works and Vaidehi Rao are invented.
Educational material. Not advice on any investment, tax, budget or market position.