Earnings at Risk vs Economic Value at Risk

They do not disagree. One asks what a rate move does to twelve months of income; the other asks what it does to the present value of everything on both sides. At Vindhya Commercial Bank Limited, invented, its own 200 basis point rise adds Rs 156 crore to income and takes Rs 840 crore off value. Both are right, and a committee shown one has seen half the position.

Everything here rests on one idea that has to be accepted before the comparison can land: a measure is defined by the question it was built to answer. Two measures laid over one balance sheet under one shock can therefore produce answers of opposite sign without either being wrong, or even surprising. The rest is arithmetic, and the arithmetic turns out to be easier than the idea.

Picture a household that has just fixed its home loan rate for five years and keeps its savings in an account that reprices the week rates move. Ask that household two different questions. What will the next twelve months of interest paid and interest received look like if rates rise? Better. The savings start paying more and the loan does not. What is that loan worth to the lender now that rates have risen? Less. A stream of fixed payments is worth less when money costs more. Same house, same loan, same rate rise, two answers pointing opposite ways. Neither is a mistake. The two measures compared here are that pair of questions asked of an entire bank, and every difference between them follows from the question rather than from the numbers.

A comparison that starts contrasting before both sides are built teaches nothing, so the definitions come first and the contrast second. Two full definitions, then six axes of difference, then the arithmetic, then the one thing about this invented bank that nobody in it has reconciled.

What does earnings at risk actually ask?

Earnings at risk asks one question and only one: if rates move by a stated amount, how much more or less net interest income does this bank collect over the next twelve months? Nothing about value, nothing about the far end of the book, nothing about what anything is worth. Just the interest arriving and the interest leaving over four quarters.

The machinery underneath it is a table, not a model. Every asset and every liability is slotted by the date its rate next changes. The date a rate next changes is a different question from the date an instrument matures. Then the slots are added up. At Vindhya Commercial Bank Limited, invented, that table is a repricing ladder of eight buckets numbered RB1 to RB8, and the buckets that fall inside one year are RB1 up to one month, RB2 over one to three months, RB3 over three to six months and RB4 over six months to one year. Assets repricing inside those four buckets come to more than the liabilities repricing inside them, and the running total is what the measure needs. The whole of the earnings answer is one number off that table, the cumulative repricing gapThe running total of assets less liabilities whose rate resets inside a given period, taken here through the one year point of the bank's own ladder. through RB4, multiplied by the rate move and by the average time left in the year.

Three things follow from that and all three matter later. The measure has a horizon of exactly twelve months, so what happens in month thirteen is invisible to it. Its unit is income, so its answer will eventually appear as a real line in a profit and loss account that somebody signs. And a rise in rates helps a bank whose assets reprice sooner than its liabilities. The sign of the earnings answer is therefore positive at this particular invented bank rather than negative.

What does economic value at risk actually ask?

Economic value at risk asks a completely different question: if rates move by that same stated amount, what happens to the present valueWhat a future cash flow is worth today once the delay in receiving it has been allowed for. Taken here as a known object and derived in the quantitative material. of every cash flow the bank will ever receive and every cash flow it will ever pay, and therefore to the difference between the two? The difference between the two present values is the economic value of equity, and the change in it under a shock is what this measure reports.

The machinery is a sensitivity, not a table. Rate sensitive assets of Rs 84,000 crore carry a modified durationA single number saying how much the value of a position moves for a small change in rates. It is used here as a locked input for this invented bank and is worked out in the fixed income material, not here. of 3.00 years and rate sensitive liabilities of Rs 84,000 crore carry one of 2.50 years, and both are the bank's own locked figures. The difference leaves a duration gapThe modified duration of the assets less the modified duration of the liabilities, being 0.50 years at this invented bank and used here as a given rather than derived. of 0.50 years. The duration gap multiplied by the rate move and by the size of the book gives the answer. Because the assets are the longer side, a rise in rates takes more value off the assets than off the liabilities, so this measure answers a rate rise with a loss.

Three things follow here too. The horizon is not twelve months, it is the whole remaining life of every cash flow on both sides, so a payment due in year nine is inside this answer and outside the other one. The unit is value, and a change in it appears in no financial statement anywhere: nobody books it, nobody reports it as profit, and the only place it surfaces is a risk pack. And the whole computation is a present value calculation, so every assumption about how long a balance sticks around goes straight into the answer.

Where exactly do the two measures differ, axis by axis?

Now that both sides are built, the differences can be laid out without any risk of the reader mistaking one for a variant of the other. There are six of them and the first one causes the other five.

Read down the first column and the result is a measure of trading performance in the ordinary sense: money in, money out, over a year that can be planned around. Read down the second and the result is something closer to a valuation. The two are not a short version and a long version of one thing, they are an income statement question and a balance sheet question, and the only element they share is the size of the rate move applied to them.

A committee paper that puts two numbers on one line hides the separation between them, and the separation is worth seeing drawn. The two builds have three inputs each. Five of those six inputs are different objects taken from different tables. Only the shock is common.

What does one 200 basis point rise do to each answer?

Take Vindhya Commercial Bank Limited, invented, at its reporting date, and apply the bank's own internal scenario of a 200 basis point parallel rise. The scenario is the bank's own, chosen by the bank, and it is not a requirement, a standardised shock or a threshold set by anybody.

The income side first. The cumulative repricing gap through RB4 is 18,000 plus 6,000 less 3,600 less 4,800. The four figures come to plus Rs 15,600 crore. Applying the rise to that gap for an average of half a year of the year remaining gives 15,600 times 2.0 per cent times 0.5, being plus Rs 156 crore of net interest income over twelve months, against net interest income of Rs 2,880 crore, so 5.4 per cent.

The value side next. The duration gap of 0.50 years, times the same 2.0 per cent, times the rate sensitive book of Rs 84,000 crore, gives minus Rs 840 crore of economic value of equity. Against tier 1 capital of Rs 6,600 crore that is 12.7 per cent, and against the Rs 990 crore cap of the bank's own limit L8 it is 84.8 per cent utilisation. Note that carefully: the 84.8 per cent here is limit L8 utilisation on the economic value measure, and it is not the bank's other 84.8 per cent. The other 84.8 per cent is model inventory completeness, 28 registered models against 33 found in use. 840 over 990 reduces to 28 over 33 exactly, so the two fractions are literally identical. Two entirely different objects happen to share one number.

Why is the value answer 5.38 times the income answer at every shock size?

Because in the upward direction both answers are straight lines through zero, and the ratio of two straight lines through zero is a constant. Work out what each one is worth per basis point and the whole table falls out. The income answer is 15,600 times one basis point times half a year, being Rs 0.78 crore for each basis point. The value answer is 0.50 times one basis point times 84,000, being Rs 4.2 crore for each basis point. Divide the second by the first. 4.2 over 0.78 is 5.3846, and the rounded figure is 5.38 times. The shock sits in both numerators and cancels, so the ratio does not change when the shock changes.

So at 50 basis points the pair reads plus Rs 39 crore and minus Rs 210 crore. At 100 it reads plus Rs 78 crore and minus Rs 420 crore. At the bank's own 200 it reads plus Rs 156 crore and minus Rs 840 crore. At about 236 basis points it reads plus Rs 184 crore and minus Rs 990 crore, exactly where the Rs 990 crore cap of limit L8 is reached. At 400 it reads plus Rs 312 crore and minus Rs 1,680 crore. Every one of those pairs sits on the same 5.38. The relationship between the two answers is a property of this balance sheet, not of the size of the shock, and that is worth knowing because it means no shock size will ever make the two answers comparable in size.

Is that gap of 5.38 times only an artefact of the units?

It is a fair suspicion. One number is being measured against a year of income and the other against a whole book, so of course the second is bigger. The way to settle it is to stop comparing the two answers with each other and compare each one with the thing it is naturally read against.

The income answer moves net interest income, so net interest income is what it is read against. Rs 156 crore against Rs 2,880 crore is 5.4 per cent. A change in the value of equity eats into tier 1 capital, so tier 1 capital is what the value answer is read against. Rs 840 crore against Rs 6,600 crore is 12.7 per cent. Now divide one percentage by the other: 12.7 over 5.4 is 2.35. Even after both answers have been turned into shares of their own natural denominators, the value answer is still 2.35 times the income answer, so the gap between them is a real feature of this balance sheet and not a trick of the units.

Can a committee net one answer against the other?

No, and this is the reason the comparison is worth working through. The temptation is not stupid, it is arithmetic that presents itself. Put the two numbers on one slide, plus Rs 156 crore a year of income and minus Rs 840 crore of value, once. Divide. Rs 840 crore over Rs 156 crore a year is 5.38 years. On that reading the bank earns back what it lost in a bit over five and a half years. Five and a half years sounds uncomfortable rather than serious, and the paper moves on.

The break even sentence is wrong three separate times, and every one of the three is worth knowing on its own. The first is double counting. The value figure is a change in the present value of every future cash flow on both sides, and every one of those five and a half years of interest income is already inside it. Adding the income back to a present value that already contains it counts the same rupees twice.

The second is the horizon. The Rs 156 crore is the cumulative gap through RB4 applied for an average half year remaining. The figure describes one year and one arrangement of buckets. In year two the buckets have rolled, a different set of balances reprices, and the gap is a different number. The case for this invented bank gives no second year of income at all, so five and a half years of Rs 156 crore is not an estimate with an error around it; it is one figure repeated five and a half times.

The third is the worst and it is the central finding. The two figures do not rest on the same assumption about the bank's own deposits, and the assumed deposit life is worked through in full below. And the number 5.38 deserves attention at this point: it appeared earlier as the ratio between the two answers, and it has just appeared again as a number of years. The number 5.38 is the same arithmetic wearing two costumes, and the second costume is a fiction: dividing a change in value by a year of income produces a figure with the units of years, but nothing about the bank takes 5.38 years to do anything.

Which limit and which appetite clause sit under each measure?

A symmetrical answer would be the natural expectation. Two measures of the same risk, one control apiece, both watched the same way. Symmetry is not what this invented bank has, and the shape of what it does have is a lesson in itself.

On the value side there is a limitAn operational cap set below the board's stated appetite and monitored between meetings. This invented bank has twelve of them, numbered L1 to L12.. Limit L8 caps economic value of equity sensitivity at 15.0 per cent of tier 1 capital, being Rs 990 crore, measured against a 200 basis point parallel move. Current reading Rs 840 crore, utilisation 84.8 per cent, within the cap.

On the income side there is no limit at all. Not a loose one, not a wide one: not one of this bank's twelve limits is measured on earnings at risk. The only control in the frame that touches income is appetite clauseA board statement of how much of something the institution is willing to accept. This invented bank has eight of them, numbered A1 to A8, and they sit above the limits. A2. Clause A2 says net interest income over the next twelve months does not fall by more than Rs 360 crore under a 200 basis point move in either direction. The bank's worst case is a fall of Rs 336 crore, so A2 runs at 93.3 per cent.

Now look at what that does to the cascade. Appetite clause A2 is about earnings and cascades into limit L8, and limit L8 is about value. The only limit sitting beneath this bank's earnings clause is measured on something else entirely, so the clause is watched by a control that cannot move when the thing the clause is about moves. And the two run at different pressures: 93.3 per cent on the clause against 84.8 per cent on the limit beneath it, so the clause is 8.5 percentage points tighter than its own supposed control, and the two figures are not even about the same quantity.

Which assumed deposit life is each answer standing on?

Here is where the comparison stops being tidy. Both answers depend on what the bank assumes about Rs 36,000 crore of current and savings accounts. Current and savings accounts are non-maturity depositsBalances a customer may withdraw whenever they like, so they have no contractual end date and any date used for them in a table is an assumption about behaviour rather than a fact.: money the customer may take out this afternoon, and money that in practice mostly does not move for years. Every table in the bank has to put a date on that balance. There is no date in the contract, so no table can read the date off one.

The reason for the differences matters more than the differences themselves. Without the reason, a second treatment of the same balance looks like an error. The tables differ because they ask different questions, and none of the four is a check on any of the others. When does the cash leave in ordinary conditions? When does the rate on it change? How long does the balance stay, in value terms? How much of it walks in a thirty day stress? Four questions, four answers, one balance.

The thirty day row is worked in the liquidity material and is quoted only to show the spread. A blended 16.2 per cent against the maturity ladder's 5.0 per cent over the same thirty days is 3.23 times as much and Rs 4,020 crore more. Four tables, four treatments, one balance, in one bank, in one month, and nobody has reconciled them because nobody is required to.

Now the two treatments the comparison needs. The income answer comes from the repricing ladder. The repricing ladder slots the whole Rs 36,000 crore into bucket RB5, one to three years. Bucket RB5 puts the balance outside the one year window entirely. The cumulative gap through RB4 is therefore plus Rs 15,600 crore and the income answer is positive. The value answer comes from a computation that gives the same Rs 36,000 crore an average behavioural life of 0.5 years. Half a year is what makes the duration gap 0.50 years and the value answer negative.

The two reported figures therefore rest on two different assumed lives for the same balance, and there is no single assumption at which both of them arise. Plus Rs 156 crore needs a life of one year or more. Minus Rs 840 crore needs exactly half a year. The two conditions cannot both hold, so the pair on the slide was never one picture of one balance sheet. Nothing about that makes either figure wrong. The mismatch makes the sentence underneath them compulsory.

One more thing belongs elsewhere. The model that sets the 0.5 year assumption is V1, and V1 is one of the three models at this invented bank that have never been validated. The trigger an unvalidated model should set off, who validates it and on what cycle belong to the risk reporting, data and model risk material. The sentence that belongs with the comparison is simpler: the assumption deciding the sign of this bank's headline interest rate number comes out of a model nobody has checked.

What happens if one consistent assumption is run through both?

Running one consistent assumption through both computations is the step that usually gets skipped, so it is worth doing slowly. Pick one assumed average life for the Rs 36,000 crore, feed it to both computations, and read the pair that comes out. Then pick another and read it again.

The value side moves smoothly. Extending the assumed life by one year lengthens the liability side and moves the result by 36,000 times 2.0 per cent, being Rs 720 crore. So the value answer is minus 840 plus 720 times the amount by which the assumed life exceeds half a year, and it is a straight line.

The income side does not move smoothly at all, and the reason is a detail of how the ladder was built. The bank puts the whole Rs 36,000 crore in one bucket rather than spreading it, so the income answer is a step and not a slope. Below an assumed life of one year the entire balance falls inside the one year window at once, the cumulative gap goes from plus Rs 15,600 crore to 15,600 less 36,000, being minus Rs 20,400 crore, and the income answer is minus Rs 204 crore. At one year or above the whole balance is outside the window and the answer is plus Rs 156 crore. Nothing in between exists. A ladder that spread the balance across several buckets would give a ramp; this one gives a cliff, and the swing across it is Rs 360 crore.

Now read the pair at three settings. At half a year, the value computation's own assumption run through both, the answers are minus Rs 204 crore and minus Rs 840 crore: both negative. At two years, the midpoint of the bank's own RB5 bucket run through both, the answers are plus Rs 156 crore and plus Rs 240 crore: both positive. The two answers carry opposite signs only when the assumed life sits between 1.00 year and about 1.6667 years, a range eight months wide, and neither of the bank's own two assumptions is inside it.

Two more things fall out of the same line, and both are reproduced from the invented bank's own published working rather than derived again here. Limit L8 is satisfied only across a range 2.75 years wide, with edges at 0.2917 years and 3.0417 years. The bank's 0.5 years sits 0.2083 years from the near edge, or two and a half months of assumed deposit life. And read across the bank's own RB5 bucket, one to three years, the value result runs from minus Rs 480 crore to plus Rs 960 crore: inside limit L8 the whole way, but utilisation travels from 48.5 per cent, through zero, to 97.0 per cent on the opposite sign, and the top of the bank's own bucket sits about two weeks of assumed life from breaching that limit in the other direction.

What can neither measure say if rates fall instead?

Almost nothing, and that is worth stating plainly rather than papering over. Everything so far has been an upward move. Turn the shock around and the comparison stops being available.

On the income side there is exactly one figure. A 200 basis point fall takes Rs 336 crore off net interest income over twelve months, and against appetite clause A2's Rs 360 crore tolerance that runs at 93.3 per cent. The downward figure is asymmetric with the upward answer, and the reason is that deposit rates at this invented bank are assumed not to fall as far as loan rates do. But that asymmetry comes out of a behavioural assumption the case never publishes, so the downward figure is one observed point and not a line: it may not be divided by 200, scaled to any other shock, or continued in either direction. The upward answer is a gap times a rate times a time, and each of those three is known, so the upward answer can be scaled. The downward one cannot.

On the value side there is nothing at all. No economic value figure exists anywhere in this case for a rate fall. Not a small one, not an approximate one. Flipping the sign of the Rs 840 crore would produce one easily, and that is exactly why the temptation is worth naming. The two sided comparison simply cannot be made in the downward direction, and saying so is better than manufacturing the missing half by symmetry.

Who actually reads this pair, and what do they do with it?

Three sorts of reader, and they want different halves of it. The people running the balance sheet want the income answer. The income answer is the one that turns up in the accounts they are measured on and the one they can plan a year around. The people watching solvency want the value answer. A Rs 840 crore change against tier 1 capital of Rs 6,600 crore is a question about how much of the cushion a rate move eats. An outside analyst reading a disclosure wants both, and mostly gets whichever one flatters the position.

The habit of a careful reader is simple to describe and rare to see. Read each answer against its own denominator rather than against the other one. Ask which table each came from and what that table assumed about balances with no maturity date. Then ask whether the two tables assumed the same thing, and if they did not, refuse to combine the answers at all. A statement of the deposit life each figure used is the single most useful sentence a risk pack can carry under the two numbers. Without it the two figures look like one picture and are not.

There is a governance shape underneath this too, and it belongs to the risk governance material. At this invented bank the committee that sets the behavioural assumption and the committee that sets the limit the assumption decides the answer to are two different bodies, and no single paper puts both decisions in front of the same people. The split is a structural gap between two mandates rather than anybody's mistake, and the gap is why the pair on the slide survives from month to month without anyone reconciling it.

Where do these two measures come from, and what binds an Indian bank?

The mechanism above is jurisdiction free. A gap table and a shock produce an income answer; a duration gap and the same shock produce a value answer. The arithmetic would work in any currency and under any rulebook, and none of it is a requirement.

The measures themselves, the standardised interest rate shocks used to run them, and the outlier test that compares the value result to capital, all come from the Basel work on interest rate risk in the banking book published by the Bank for International Settlements at bis.org. The Basel work is the origin of the shape of these two measures. The rules that actually bind a bank in India, meaning which of these must be computed, on what shocks, with what treatment of balances that have no maturity, what must be reported and from what date, come from the Reserve Bank of India at rbi.org.in. The second source governs anything intended to be relied on; the first explains only where the measures came from.