Economic Value Sensitivity: The Duration Gap Under a Shock

The calculator below turns the three inputs behind one number. At Vindhya Commercial Bank Limited, invented, a duration gap of 0.50 years under its own 200 basis point rise gives minus Rs 840 crore, filling 84.8 per cent of the Rs 990 crore cap that bank sets itself. Move the assumed deposit life to two years and the same shock gives plus Rs 240 crore.

Between those two figures no transaction took place. Nothing was bought. Nothing was sold. No deposit left and no loan was written. One judgement was turned from one plausible value to another plausible value, and a reported figure travelled from a loss of Rs 840 crore to a gain of Rs 240 crore. A figure that moves that far while the balance sheet stands still is the whole subject of this measure.

What does this tool compute, and from what inputs?

The tool computes one thing and refuses everything else. Given a duration gapThe modified duration of the assets less the modified duration of the liabilities, in years. Here it arrives as a fixed number, and deriving it is covered separately., a rate move applied equally across every maturity, and the size of the rate sensitive balance sheet, it returns the change in the value of the equity. The change in the value of equity is the whole computation. A single multiplication with a minus sign in front of it does all of it, and a calculator built around one multiplication is not built for the arithmetic. Two of the three things being multiplied are decisions somebody made, and the tool is the only place those decisions can be seen priced.

Here is the sentence in the order the machine performs it. Take the duration gap. Multiply it by the rate move expressed as a decimal. Multiply that by the money on the rate sensitive part of the balance sheet. Put a minus sign in front. A rise in rates costs a bank whose assets run longer than its funding. At Vindhya Commercial Bank Limited, invented, that reads minus 0.50 times 2.0 per cent times Rs 84,000 crore. The product is minus Rs 840 crore. Every number this tool will ever print is that one line with one of its three factors changed.

Six figures go into it and every one of them is a locked inputA figure this invented case fixes. The tool allows it to be moved only to see what would follow, never to change the case itself. belonging to this invented bank at its month 12 reporting date. Rate sensitive assets of Rs 84,000 crore at a modified duration of 3.00 years. Rate sensitive liabilities of the same Rs 84,000 crore at a modified duration of 2.50 years. Demand repayable current and savings balances of Rs 36,000 crore sitting inside those liabilities. A parallel shockA rate change applied equally at every maturity, so the whole curve moves up or down together rather than twisting. The 200 basis points used here is this invented bank's own internal scenario. of 200 basis points, being the bank's own internal scenario rather than a requirement issued by anybody. And limit L8, the bank's own cap on this measure, set at 15.0 per cent of tier 1 capital of Rs 6,600 crore, being Rs 990 crore.

Notice what is not on that list. There is no yield curve, no cash flow schedule, no discount factor and no security. The tool never opens a bond, never asks what a swap pays and never rebuilds a present value. The tool takes the modified durations as given numbers, the way a load calculation takes the strength of a steel section as a given number rather than re-deriving metallurgy at the site. A tool that recomputed its own inputs would be a different tool, and the reason this one can be turned quickly is precisely that it does not.

One more property decides how every number the tool prints should be read. The output is a change, not a level. The tool never states what the equity of this invented bank is worth. The tool states how much that worth moves when rates move by a stated amount. A change of minus Rs 840 crore is not a valuation and cannot be compared with the Rs 7,680 crore of equity on the balance sheet as though the second were being reduced to the first. The change can be compared with the cap the bank set for itself, and the tool prints that comparison beside every reading.

The comparison has a name on the committee papers of this invented bank. Limit utilisationThe measured figure expressed as a percentage of the cap it runs against. Above one hundred per cent the cap has been passed and the position is a breach. is the measured figure written as a percentage of the cap it runs against. At minus Rs 840 crore against a Rs 990 crore cap, this reads 84.8 per cent, and 84.8 per cent here means limit L8 utilisation and nothing else. The label is worth pinning down once. Elsewhere in this invented bank, 84.8 per cent turns up attached to a completely different object, and a reader who carries the number without the label will eventually attach it to the wrong one.

How much does one year of assumed deposit life move the answer?

Rs 720 crore, and the reason is a single multiplication that can be done standing up. The behavioural deposit lifeThe average length of time a balance that is repayable on demand is assumed to actually stay. It is estimated, not contracted, and it is the control that moves this answer furthest. applies to Rs 36,000 crore of current and savings balances at this invented bank. The current and savings balances are legally repayable the moment somebody asks. Nobody prices them that way. In practice a very large share of them sits still for years, so the bank estimates an average life and treats them as funding of that length. Lengthening the estimate by one year lengthens one year of duration on Rs 36,000 crore. At a 2.0 per cent move, Rs 36,000 crore times 2.0 per cent is Rs 720 crore.

The everyday version is a shop that takes advances from regulars. Legally every regular could walk in tomorrow and ask for the money back. In practice the shopkeeper knows that the float never drops below a certain level, so he treats it as money he can rely on for a while and commits it to stock. If he assumes the float stays six months he behaves one way. If he assumes it stays two years he behaves quite differently. The money is the same money, and the belief about how long it stays is the only thing that moved. The assumed life is not a fact about the deposits, it is a belief about the depositors, and it enters the arithmetic exactly where a fact would.

Because Rs 36,000 crore and 2.0 per cent are both held still, the relationship is a straight line with no curve in it anywhere. Start from the bank's own reading of minus Rs 840 crore at an assumed life of 0.50 years. Add Rs 720 crore for every extra year and subtract Rs 720 crore for every year removed. The line gives minus Rs 1,200 crore at zero, minus Rs 480 crore at one year, minus Rs 120 crore at a year and a half, plus Rs 240 crore at two years, plus Rs 960 crore at three, and plus Rs 1,680 crore at four. Every one of those is the same line read at a different point.

Read the shaded column on that picture rather than the line for a moment. The shading does the harder work, and it marks the one to three year repricing bucketThe slot in a bank's own repricing ladder into which a balance is placed by when its rate next changes. This invented bank puts the same Rs 36,000 crore at one to three years there. into which this bank's own repricing ladder places exactly the same Rs 36,000 crore. Inside that column the tool reads anything from minus Rs 480 crore at one year to plus Rs 960 crore at three. The bank's own two tables, side by side, imply answers that sit on opposite sides of zero, and the picture is the only place they have ever appeared together.

A ladder of readings is worth more here than a picture of a slope. A committee paper would carry the ladder, and the two places the line touches the cap are easier to see in a row than on a diagonal. Here are eleven readings of the same line, all at the bank's own duration gap and the bank's own shock, with only the assumed life moved.

One caution about that ladder, and it is the kind of thing a tool must say about itself. Read the two edge rows as printed figures rather than as exact ones. At an assumed life of 0.2917 years the underlying answer is minus Rs 989.98 crore, and at 3.0417 years it is plus Rs 990.02 crore, so both print as Rs 990 crore and both print as 100.0 per cent while sitting a whisker on either side of the cap. The exact edges are 0.291667 and 3.041667 years. A figure that prints as exactly the cap is not the same thing as a figure that is exactly the cap, and near an edge that distinction decides whether a report says within or says breach.

Where does the answer cross zero, and what is that point worth knowing?

At about 1.6667 years, and the arithmetic is short enough to check on the back of the agenda. The bank starts at minus Rs 840 crore. Each extra year of assumed life is worth Rs 720 crore. So the extra life needed to wipe out Rs 840 crore is 840 divided by 720, being 1.1667 years. Adding that to the starting assumption of 0.50 years gives 1.6667 years. Above that the reported figure is a gain and below it a loss.

Now the part that matters. A zero crossingThe setting of an input at which a computed answer is exactly nought, so that either side of it the same measure reports the opposite sign. would be a curiosity if it landed somewhere nobody would ever choose. A crossing at nine years, or at three weeks, would show only that a straight line eventually gets to zero. The crossing lands at one year and eight months, and one year and eight months sits inside the one to three year bucket that this same bank uses for these same balances in the table next door. The crossing is not in a region the bank would find absurd, it is in a region the bank has already committed to in writing somewhere else.

Think about what that does to a reader of the committee paper. The paper says the change in the value of equity is minus Rs 840 crore, being 84.8 per cent of the cap of limit L8, status within. The reader has no way of knowing that a different but equally defensible assumption, drawn from the bank's own repricing ladder, would have printed a positive number. Neither figure would look odd. Neither would be flagged. The paper is silent about the one input that decides which of them is printed, and silence about an input is the most expensive kind of silence a report can carry.

Two readings of that tool are worth taking away, and neither depends on the slider having been touched. Turning the assumed deposit life at the bank's own gap and shock gives minus Rs 1,200 crore at zero years, minus Rs 1,020 crore at a quarter of a year, minus Rs 990 crore at 0.2917, minus Rs 840 crore at half a year, minus Rs 480 crore at one year, minus Rs 120 crore at a year and a half, zero at about 1.6667, plus Rs 240 crore at two years, plus Rs 600 crore at two and a half, plus Rs 960 crore at three, plus Rs 990 crore at 3.0417, plus Rs 1,320 crore at three and a half and plus Rs 1,680 crore at four. Thirteen readings of one balance sheet, six of them outside a cap that the bank reports itself comfortably inside.

Turning the duration gap at the bank's own shock gives plus Rs 840 crore at a gap of minus 0.50 years, zero at a gap of zero, minus Rs 420 crore at 0.25, minus Rs 840 crore at the bank's own 0.50, minus Rs 990 crore at about 0.5893, minus Rs 1,260 crore at 0.75 and minus Rs 1,680 crore at 1.00. Turning the shock at the bank's own gap gives minus Rs 210 crore at 50 basis points, minus Rs 420 crore at 100, minus Rs 840 crore at the bank's own 200, minus Rs 1,260 crore at 300 and minus Rs 1,680 crore at 400, with the cap reached between the 235 and the 236 basis point step.

Between which two assumed lives does limit L8 actually hold?

Between 0.2917 years and 3.0417 years, and that pair of numbers is the most useful thing in this guide. Below the first the answer passes the Rs 990 crore cap on the negative side. Too short an assumed life makes the funding look shorter than the assets and the loss too large. Above the second the answer passes the same cap on the positive side. Too long an assumed life makes the funding look longer than the assets and the gain too large. Both are breaches of limit L8. A cap on the size of a move is breached by a large gain exactly as it is by a large loss, and a reader who has only ever thought about the loss side will be surprised by that the first time it happens.

Risk governance publishes the same pair of edges, solving the same line. The arithmetic behind it is the arithmetic already set out above. Rs 990 crore of cap less Rs 840 crore of current reading leaves Rs 150 crore of room on the loss side, and at Rs 720 crore per year that is 0.2083 years, so the near edge sits 0.2083 years below the bank's own 0.50. On the gain side the distance is Rs 990 crore plus Rs 840 crore, being Rs 1,830 crore. At Rs 720 crore per year that is 2.5417 years above 0.50.

The bandThe stretch of assumed deposit lives inside which the reported figure stays within the cap. Outside it at either end the same cap is passed. between those edges is 2.75 years wide. The width sounds generous until the bank's position inside it is examined. The bank does not sit in the middle. The bank sits 0.2083 years from the near edge, or two and a half months of assumed deposit life, and 2.5417 years from the far one. Put plainly, this invented bank is twelve times closer to breaching on the loss side than on the gain side, and no market event is needed to push it over. Somebody deciding that these balances are a little less sticky than last year's study suggested would be enough.

The magnified strip at the foot of that picture is there because the honest version of the far edge is invisible at full scale. The top of the bank's own repricing bucket is 3.00 years. The far edge of the band is 3.0417 years. The two are 0.0417 years apart, or about half a month of assumed deposit life, and at full scale that is a gap of eight pixels nobody would notice. An assumption taken from the very top of this bank's own ladder would sit about two weeks of assumed life away from breaching the same limit with the opposite sign, and the picture had to be magnified twenty times before that could be seen at all.

How much duration gap is one month worth in rupees?

About Rs 140 crore, and this is the single sentence that makes the tool usable when the tool is not to hand. On a Rs 84,000 crore rate sensitive balance sheet at a 2.0 per cent move, Rs 84,000 crore times 2.0 per cent is Rs 1,680 crore, so the answer moves Rs 1,680 crore for every full year of duration gap. Divide by twelve and a month of gap is Rs 140 crore. With Rs 140 crore a month committed to memory, any proposal that lengthens or shortens this bank's asset book can be priced against its own cap in the meeting where the proposal is made.

Watch how that works in practice. Somebody proposes buying a block of longer dated government securities that would add a month to the weighted duration of the asset side. The tool is not open. Nobody has a model. But the gap goes from 0.50 to about 0.5833 years, that is one month, so the answer moves from minus Rs 840 crore to about minus Rs 980 crore, and the utilisation of limit L8 goes from 84.8 per cent to about 99 per cent. The proposal has just consumed almost the entire remaining headroom on a limit set by committee G2, and everybody in the room now knows it before anybody has left the room.

The cap arrives at a gap of about 0.5893 years, a little over seven months. Read that as a staircase and it is very concrete. Six months of gap is Rs 840 crore, and that is where the bank stands. The seventh month takes it to Rs 980 crore, still inside. The cap is passed a very short way into the eighth. There is not quite one and a tenth months of duration gap left on this balance sheet before a cap the bank set itself is reached, and a bank that never states the figure in months will never notice how little that is.

What happens when the shock is turned instead of the assumption?

The same answer arrives by a different road, and comparing the roads is how they are proved to be one road. With the assumed life and the duration gap held at the bank's own settings and only the size of the rate move turning, 0.50 years times one basis point times Rs 84,000 crore is Rs 4.2 crore, so the answer is Rs 4.2 crore for every basis point. Fifty basis points gives minus Rs 210 crore. A hundred gives minus Rs 420 crore. The bank's own two hundred gives minus Rs 840 crore. Four hundred gives minus Rs 1,680 crore.

The cap is reached at 235.714 basis points. Since the tool moves in whole basis points, there is no step that lands on it: the 235 step reads minus Rs 987 crore at 99.7 per cent and the 236 step reads minus Rs 991 crore at 100.1 per cent, so the cap is crossed between them. The distinction is worth stating carefully. A report that records the crossing as 236 basis points giving exactly Rs 990 crore at exactly 100.0 per cent has rounded the input up and printed the output as though the input had been exact. Rounding an input and then reporting the output as an exact figure is one of the quieter ways a table stops being true about itself.

Now put the three roads beside each other. The answer in rupees may rise from Rs 840 crore to Rs 990 crore, a rise of 17.9 per cent. The duration gap may widen from 0.5000 to 0.5893 years, a rise of 17.9 per cent. The shock may rise from 200 to 235.7 basis points, a rise of 17.9 per cent. Three different questions, one answer, and the reason is that the result is a single product of a gap, a shock and a balance sheet size, so stretching any one of those factors by a given proportion stretches the product by exactly the same proportion.

The assumed deposit life is the exception, and the exception is instructive rather than annoying. The assumed life is not one of the three factors. The assumption sits inside the duration gap, changing the gap by Rs 36,000 crore of duration for every year it moves, so its route to the cap is not a proportional stretch at all. The near edge requires the assumed life to fall from 0.5000 to 0.2917 years, a fall of 41.7 per cent rather than 17.9. Anybody who reads the three agreeing numbers and assumes the fourth control agrees too will be out by more than a factor of two.

Why do the bank's own two tables give one balance two different lives?

Because the two tables are asking different questions of the same money, and neither question is wrong. The repricing ladder at this invented bank asks when the rate on a balance next changes. The bank does not in practice move those rates often or quickly, so on that question the ladder puts the Rs 36,000 crore of current and savings balances at one to three years. The value computation asks how long the money stays, and on that question a separate study produced an average of half a year. Different questions, different answers, and a bank is entitled to both.

A bank is not entitled to silence about the difference. One balance of Rs 36,000 crore is being given a life of half a year in one committee paper and one to three years in another, and the two papers never appear on the same agenda. Shading the repricing bucket across the same axis as the assumed life is the whole point of the picture: the disagreement stops being something a reader has to be told about and becomes something they can see, along with the fact that every setting inside the shaded region produces a number different from the one the bank actually reports.

There is a household version of this and it is uncomfortably close. A couple planning around a monthly transfer from a working son can treat it as money that will keep arriving for six months, and plan cautiously, or as money that will keep arriving for three years, and commit to a longer obligation. Both are honest estimates. Dishonesty would begin with one of them planning on six months, the other planning on three years, and neither of them ever saying the number out loud. The money is the same money. The plans are incompatible. Nobody has lied.

What can this tool not show, and why does it refuse?

Three things, and the refusals are as much a part of the instrument as the readings. The first and most important is the falling rate direction. Everything drawn in this guide is a rate rise. The line is straight, it runs through the origin and each of its three factors is a locked figure, so any point on it can be computed honestly. A rate fall is a different matter entirely at this invented bank. The case carries exactly one point for it: minus Rs 336 crore of net interest income over twelve months under a 200 basis point fall.

The single point carries two reasons the tool cannot use it. The figure is an income figure and not a value figure, so it is not even measuring the same thing this tool measures. And it was produced by a behavioural assumption about how far deposit rates follow loan rates downward, an assumption the case does not publish. Dividing minus Rs 336 crore by two hundred and drawing a straight line through it would be inventing a shape, and a tool that draws a shape it cannot compute has stopped being a tool and become a picture.

The second refusal is any move that is not parallel. A real interest rate change almost never lifts every maturity by the same amount, and the shapes that matter, a steepening, a flattening, a twist about some point in the middle, produce answers that a single duration gap cannot generate. Vindhya Commercial Bank Limited runs one internal parallel scenario. The tool runs the same one. The tool has no basis for anything else and says so rather than offering a plausible looking control that would be quietly meaningless.

The third is the effect on income. The same 200 basis point rise that takes Rs 840 crore off the value of this bank's equity adds Rs 156 crore to its net interest income over twelve months, and those two figures point in opposite directions on purpose because they answer different questions. Working that pair is a separate exercise with its own arithmetic, and this tool computes only the value side. The value effect and the income effect are two answers to one rate rise, and a measure that reports only one of them is incomplete rather than wrong.

Has the model behind this assumption ever been checked?

No. In this invented bank's own model register, the behavioural deposit life model is recorded as never validatedA status in an institution's own model register meaning that no independent review of whether the model is fit for the use it is put to has ever been performed., and it is one of only three carrying that status out of twenty eight registered. The status matters because it changes how every number the tool prints should be read. Who should validate a model, on what cycle, to what standard, and what an institution ought to do about one that has never been reviewed, is covered separately under risk reporting, data and model risk.

One property of this particular model belongs with the arithmetic above rather than with the governance. The model cannot be tested against outcomes either. A trading measure can be counted against realised daily losses because the outcome shows up the next morning. The outcome this model predicts is how long a demand repayable balance stays, and that is only observable over years. A model that cannot be checked against outcomes and has never been reviewed for fitness is producing the single input that decides whether this bank's headline rate risk number is a loss or a gain.

The finding is not an accusation of incompetence. Nobody at this invented bank did anything wrong. A committee set a behavioural assumption, as committees must. A different committee set a cap, as committees must. The register recorded the model honestly, including the fact that it had never been reviewed. Every part worked. The gap is between the parts, and the calculator above is the only place in the whole institution where the assumption and the cap it decides appear on one screen.

What is the commonest way this number gets misread?

Who actually uses a tool like this, and what do they do with it?

Four readers, and each of them is reaching for a different thing. The one closest to the position comes first. Devendra Achar, head of treasury at this invented bank, does not open a tool like this to admire a number. He opens it to find out what he is still allowed to do this quarter. The reading he takes away is Rs 140 crore a month of duration gap and Rs 4.2 crore a basis point. Those two figures let him price any proposal that crosses his desk against the Rs 150 crore of headroom he has under limit L8 before anybody builds anything. A limit reader is never asking how large the risk is, only how much more of it this morning's decision is permitted to add.

Second, somebody sitting on the committee that sets the behavioural assumption. Their use of the tool is the opposite of the treasurer's: they are not looking at the answer at all, they are looking at how much the answer depends on them. The band tells them that their assumption has to stay between 0.2917 and 3.0417 years for the bank to remain inside a cap they do not set and may not even receive a paper about. The fact is uncomfortable and useful at once. Most committees setting a behavioural input have no idea which caps elsewhere in the institution depend on their answer.

Third, somebody lending wholesale money to this bank, or placing a large deposit with it, and reading whatever it discloses. Their question is not the level of the number, it is the width of the band around it. Reading minus Rs 840 crore at 84.8 per cent and stopping there tells them the position at one setting of one judgement. Asking what the assumed behavioural life is, and what the answer would be at the top and the bottom of the bank's own repricing bucket, tells them the range the bank is actually operating inside. The first question gets a number. The second gets an understanding of how much the number is worth.

Fourth, and this is the reader nobody hands the tool to, a household with a fixed rate home loan and a term deposit at the same bank. The household will never see limit L8 and never hear the phrase duration gap, and stands nonetheless inside the same arithmetic from the other side: the household is the demand repayable balance whose assumed life is being estimated, and the borrower whose loan runs longer than the funding behind it. The exposure does not require anybody to be watching it, and an institution builds a cap and a committee and a monthly paper around a number nobody outside it will ever read for exactly that reason.

One last practical note that belongs to all four. Every figure this tool prints belongs to Vindhya Commercial Bank Limited, invented, at its month 12 reporting date. The 200 basis point scenario is that bank's own choice of shock. The Rs 990 crore cap is that bank's own choice of limit, being 15.0 per cent of its own tier 1 capital of Rs 6,600 crore. The 0.50 year assumed life is that bank's own estimate. None of the three is a requirement issued by anybody, and copying any of them into a real institution because a teaching example used them is the error that turns somebody else's judgement into a rule.

Where do the inputs come from, and what binds an Indian bank?

What this establishes

A reported economic value sensitivity can now be met with the one question that decides what it is worth: what was the behavioural life assumption behind it? A change in that assumption carries a rupee figure without a model. One year is Rs 720 crore on Rs 36,000 crore at a 2.0 per cent move. A change in the duration gap carries one too. One month is about Rs 140 crore on Rs 84,000 crore. And a change in the shock carries one. One basis point is Rs 4.2 crore.

The band can be stated. Between 0.2917 and 3.0417 years of assumed deposit life, limit L8 holds at this invented bank; outside it at either end the same Rs 990 crore cap is passed, once with a loss and once with a gain. The bank sits 0.2083 years, being two and a half months, from the near edge and 2.5417 years from the far one. The two distances show which side the pressure is on. And the answer crosses zero at about 1.6667 years, inside the one to three year bucket this same bank's own repricing ladder already uses for the same money.

The refusals can be stated too. The tool draws the rising direction and not the falling one. The case carries a single point for a fall, and that point is an income figure produced by an assumption nobody published. The tool draws a parallel move and not a twist. The tool computes a value effect and not an income effect. Knowing which of those three a tool is not computing is worth as much as knowing what it computes. Every one of the three is a question somebody will eventually ask of a number that cannot answer it.