Return on Assets and Return on Equity, Taken Apart

Return on assets divides an institution's profit after tax by everything it has working. Return on equity divides that identical profit by the money its owners put in. The first multiplied by leverage, meaning assets over net worth, gives the second, so what stands between them is nothing but the quantity of borrowed money in the picture. One figure alone cannot tell two lenders apart.

Two lenders carry the same return on equity below. Their figures land on one identical value for that ratio and disagree on everything else. Every ratio here is divided out again at the spot where it is printed. Four of the questions raised below have answers that the Reserve Bank of India settles and then revises. Those four are drawn as rows with the authority named and the value left out. A number written into such a row would look right for a while and then quietly stop being right. Each lender carries one stated year, no second year and no downturn.

One profit figure sits on top of both measures. Only the thing underneath changes. A change confined to the denominator makes the relationship between the two measures something other than a comparison. The relationship is a multiplication, and the thing doing the multiplying is the single piece of information the second measure destroys on its way to producing one tidy number.

The street version comes before any arithmetic arrives, and the figures in it are the very same figures rather than a loose likeness. The picture is of two shopkeepers. Each has Rs 1,00,000/- of personal money tied up in the business and each finishes the year Rs 9,375/- better off. One is sitting on Rs 10,00,000/- of stock, Rs 9,00,000/- of it bought with money somebody lent. The other keeps Rs 5,00,000/- of stock, Rs 4,00,000/- of it borrowed. Identical result on the money each of them personally has in the place. Anyone walking into both would never describe them as one kind of shop.

What does return on assets measure, and on what base?

Profit after taxWhat is left of a year's earnings once every cost, every provision and the tax on the result have all been taken out. Profit after tax is the last line of the income statement. for the period, divided by assets. The definition is the whole of it. A return quoted without its base is unusable, so the base is named in the same breath as the number.

Take Suvarna Commercial Bank Limited, a bank carrying one stated year. Profit after tax of Rs 2,250 crore divided by total assets of Rs 2,40,000 crore is 0.9375 per cent, reported as 0.94 per cent. Now take Rukmini Finance Limited, one stated year again. Profit after tax of Rs 337.50 crore divided by assets under managementHow much a finance company has out on loan and under its control. Here it does the same job that total assets does at a bank, and it is the base that lender's own return is struck on. of Rs 18,000 crore is 1.875 per cent, reported as 1.88 per cent.

Return on assets asks how much each rupee the institution has working produced, and the question contains no reference at all to who supplied that rupee. Look at what is missing from the definition. There is no deposit in it, no borrowing in it, no share capital in it. A rupee lent out is a rupee lent out whether the money behind it came from a depositor, a bond investor or a shareholder, and return on assets is the measure that refuses to care.

What does return on equity measure, and on what base?

The same profit after tax, divided by net worthThe money the owners have in the institution: what was put in at the start plus everything earned and kept since, less anything paid out. Net worth is what would be left if every asset turned into cash and every debt were settled.. At Suvarna Commercial Bank Limited, Rs 2,250 crore divided by net worth of Rs 24,000 crore is 9.375 per cent, reported as 9.38 per cent. At Rukmini Finance Limited, Rs 337.50 crore divided by net worth of Rs 3,600 crore is also 9.375 per cent, reported as 9.38 per cent.

Notice what changed and what did not. The numerator is the identical figure in both measures, unaltered, not adjusted, not restated. Only the denominatorThe number underneath in a fraction, the thing being divided by. Changing it asks a different question of an unchanged number on top. moved, and the entire difference between the two measures is that movement.

A return on equity never travels without both its parts, so here they are even while the measure is only being defined. Suvarna Commercial Bank Limited: 0.9375 per cent earned on each rupee of assets, levered 10.0 times. Rukmini Finance Limited: 1.875 per cent on each rupee, levered 5.0 times. The two products are equal. The equality was designed rather than stumbled into, and it is taken apart properly below.

How is one measure exactly the other multiplied by leverage?

Work it rather than take it. Profit over assets, multiplied by assets over net worth, gives profit over net worth. The asset term appears once on top and once underneath, so it cancels and leaves nothing behind. The derivation is complete in that one line.

So return on equity equals return on assets multiplied by leverage, where leverage here means assets divided by net worth, in times. There is nothing approximate in that statement, no assumption propping it up and no condition under which it stops holding. The statement is a fraction cancelling, so it holds for any institution with any set of books in any year.

The consequence worth sitting with is this: a single reported return on equity is the product of two numbers, and a product never reveals what was multiplied. Twelve is six twos and it is also two sixes and it is also one twelve. Anyone handed a twelve and asked to compare it with another twelve has not been given enough to answer, and no amount of care with the twelve will fix that. Two roads reach one number, and the number cannot say which road it came by.

How can two lenders post the same return on equity and be nothing alike?

Suvarna Commercial Bank Limited and Rukmini Finance Limited both earn 9.38 per cent on equity for their stated year. The two figures are equal by construction rather than by coincidence. A matching figure with no explanation beside it reads like a copying error, and the equality here is the opposite of one.

Here is the route each of them took. At Suvarna Commercial Bank Limited the 9.375 per cent comes out of a return on assets of 0.9375 per cent, shortened to 0.94 in reporting, standing at 10.0 times assets to net worth. At Rukmini Finance Limited the identical 9.375 per cent comes out of a return on assets of 1.875 per cent, shortened to 1.88, standing at 5.0 times. One lender makes half as much on every rupee it has working, the other carries half as many rupees of assets on each rupee of its own money, and those two halvings meet exactly in the middle.

Rank institutions on return on equity alone and these two cannot be told apart. There is not one respect in which the pair are actually alike. One takes deposits and one does not. One pays a great deal less for its money and earns a great deal less on what it lends. One has ten rupees of assets standing on each rupee of its own money and the other has five. The ranking cancelled a doubling in one direction against a doubling in the other and reported a tie.

One part of this looks like a mistake and is not one. Taking the two shortened returns on assets and multiplying each by its leverage figure gives 9.40 per cent both times: 0.94 against 10.0 gives 9.40, and 1.88 against 5.0 gives 9.40. The reported answer at each lender is 9.38. Anybody who does that multiplication and is told nothing about it will fairly decide the arithmetic does not add up.

The reported returns on assets are roundings, and the product ties only before the rounding happens. The exact figures are 0.9375 per cent at Suvarna Commercial Bank Limited and 1.875 per cent at Rukmini Finance Limited. 0.9375 multiplied by 10.0 is 9.375. 1.875 multiplied by 5.0 is 9.375. And 9.375 per cent is what rounds to the reported 9.38 per cent. Nothing is wrong anywhere in that sequence. Printing the shortened figures, printing the answer and leaving the two irreconcilable is what would be wrong. Every unroundedThe figure as it comes out of the division, before it is shortened for reporting. Rounding is a presentation step, and a chain of arithmetic ties at the unrounded values rather than at the printed ones. limb here is printed beside its reported one.

Go back to the two shopkeepers for a moment. The arithmetic there is the same arithmetic rather than a loose analogy. Rs 1,00,000/- of own money became Rs 9,375/- at both shops, or 9.375 per cent on the money each of them put in. The first got there on 0.9375 per cent of Rs 10,00,000/- of stock. The second got there on 1.875 per cent of Rs 5,00,000/-. One shop turns over twice as much and makes half as much on each rupee of it. Same answer at the end of the year, and no sensible person would call those the same shop.

What do the two income ladders look like per rupee of assets?

Restate each income statement as percentages of that institution's own assets and the disagreement becomes complete. Restating turns two sets of books of very different sizes into two things that can be held side by side, and it costs nothing but a division.

At Suvarna Commercial Bank Limited, on assets of Rs 2,40,000 crore: net interest incomeThe gap between what a lender collects in interest and what it hands over in interest. Most of a lending business's earnings sit on this one line. 3.10, other income 1.00, total income 4.10, operating expensesThe cost of running the place for the year: staff, premises, systems and everything else that is not interest and not a loss on lending. 2.10, operating profitThe figure a lender reaches once the running costs are out, and before any charge for lending that has gone wrong, and before tax. 2.00, provisionsAmounts set aside against loans that are not expected to be repaid in full. Provisions reduce profit in the year they are made, whether or not any cash has moved. 0.75, profit before tax 1.25, profit after tax 0.9375. All in per cent of its own assets, for its stated year.

At Rukmini Finance Limited, whose assets under management come to Rs 18,000 crore: net interest income 7.70, no other income at all, operating expenses 3.00, operating profit 4.70, credit costsThe charge a lender takes for lending that has gone wrong, covering both what is set aside and what is written off. Credit costs are the lending equivalent of breakage in a shop. 2.20, profit before tax 2.50, profit after tax 1.875. Same rule, different institution.

Read down those two lists and the pattern nobody expects arrives at the bottom. Profit before tax is exactly double, 1.25 against 2.50, and because both lenders carry tax at 25.0 per cent for the year the doubling survives all the way to 0.9375 against 1.875. Every line above it disagrees as well, and not one of them disagrees by two. Net interest income is 3.10 against 7.70, nearer two and a half times. Operating expenses are 2.10 against 3.00. Credit charges are 0.75 against 2.20, nearly three times. The neat factor of two exists only at the very bottom, and it exists because several untidy differences happen to cancel on the way down.

What moves one measure and leaves the other where it was?

Separating the two limbs is the diagnostic the decomposition buys, and it is the reason a practitioner keeps both beside each other rather than one.

A wider spread moves return on assets. So do lower operating costs, and so do lower credit losses. Anything that changes what a rupee of the book earns or what it costs to run and to lose lands in the first limb, and return on equity follows it upward or downward because return on equity contains it. A spread, a running cost and a credit loss are all statements about the lending.

Raising capital, or returning capital to the owners, moves leverage and nothing else. Put more of the owners' money behind an unchanged book and assets to net worth falls, so return on equity falls with it. Return on assets sits exactly where it was. Its definition carries no funding term anywhere. So if a lender's return on equity has moved and its return on assets has not, nothing about the lending has to have changed at all, and what moved was the amount of its own money standing behind the book.

Take the household version. A person who has been running a small business on Rs 1,00,000/- of their own money and Rs 9,00,000/- borrowed decides to put in another Rs 1,00,000/- and repay some of the loan. The shop sells the same goods to the same customers at the same margin. Every rupee of stock earns exactly what it earned last year. The return on what they personally have tied up in the place, however, is now smaller, and nothing about the shop has changed.

There is one more way to look at the same statement, and it is the way a funding side reads it. At Suvarna Commercial Bank Limited, Rs 24,000 crore set against Rs 2,40,000 crore of assets is 10.0 per cent, so the owners' money is one tenth of everything on the books. At Rukmini Finance Limited, Rs 3,600 crore set against Rs 18,000 crore is 20.0 per cent, or one fifth. One divided by 0.10 is 10.0 and one divided by 0.20 is 5.0, so the share of assets carried by the owners' money and the leverage multiple are the same statement said two ways.

Why does one lender run twice the leverage of the other?

Suvarna Commercial Bank Limited runs 10.0 times assets to net worth and Rukmini Finance Limited runs 5.0 times. Neither multiple is a preference expressed on a Tuesday morning. A leverage ceiling is the kind of number that gets settled from outside.

How far a bank may lever, how far a non-banking finance company may lever, the capital minimum that sits behind both of those limits, and the risk weighted assetsA lender's assets restated so that each one counts according to how risky it is judged to be, rather than at its face amount. The weights are set by the regulator, not by the lender. that make the two requirements differ are every one of them set by the Reserve Bank of India, and they every one of them move.

The question is a good one, and the answer has an address that changes on its own schedule. So the row gets drawn, named, and handed over with the authority sitting inside it and the value left out. A row filled from the source never goes out of date. A printed number looks authoritative only for as long as it takes somebody to revise it.

How does somebody reading a set of accounts actually use the two measures together?

What does the whole comparison look like in one place?

Both institutions, both decompositions and both ladders now sit together, laid out the way somebody working through the accounts would assemble them. Every figure is worked from the two sets of books and every ratio names its base.

Read the table from the bottom two rows upward and the leverage figures stop being arbitrary. Ten per cent of assets carried by the owners is the same thing as ten times. Twenty per cent is the same thing as five times. And read it from the middle and the equality at the bold row is the only agreement in the whole table.

Two settings of the control above carry the teaching. At 10.0 times, the bank's actual leverage and where the control opens, the bank's bar reads 9.375 per cent and stands exactly level with the finance company's. At 5.0 times, the finance company's own leverage, the bank's bar falls to 4.6875 per cent, reported as 4.69, or half the other bar. Level the two lenders' leverage and the difference the equal return on equity was hiding appears immediately, and it is exactly the factor by which their returns on assets differ.

What does an identical return on equity not settle?

An identical return on equity does not say the two institutions are equally good, equally safe, equally durable or equally anything at all. The equality says the product of two numbers came out the same at both of them, and that the two numbers differ by a factor of two in opposite directions.

No verdict is reached here on which of these two ways of lending is preferable. Running more leverage is not the same thing as being reckless. Earning a wider spread is not the same thing as being skilful. Both statements are tempting and neither follows from the arithmetic. Each lender is one institution in one stated year, with no downturn, no second year and no failure of any kind. Evidence for what an approach to lending delivers has to be built out of several years and at least one bad one. Neither set of books here holds that material.

The arithmetic itself holds, together with a precise statement of what the arithmetic cannot settle. Stopping there is not a hedge. The stop marks the difference between a measure and a verdict, and the measure is the part that is actually available.