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Economic Value of Equity: Rate Sensitivity of the Balance Sheet

Economic value of equity asks what a rate move does to the present value of everything a bank holds and everything it owes. At Vindhya Commercial Bank Limited, invented, assets of Rs 84,000 crore at a modified duration of 3.00 years against liabilities at 2.50 years give a duration gap of 0.50 years, so the bank's own 200 basis point rise takes Rs 840 crore off. Every input is its own.

Two ideas have to sit together before that sentence earns its keep. The first is that a balance sheet has a value that has nothing to do with what the accounts say it is worth. Every loan, every deposit, every bond and every borrowing is a set of future cash flows, and every set of future cash flows has a present valueWhat a stream of future cash flows is worth today once each payment has been discounted back at a rate. Used here as known; where it comes from belongs to the quantitative subject area. that moves when rates move. The second is that nobody reports that value as a level. Banks measure, report and limit the change in that value under a stated rate scenario. The scenario is therefore part of the number, and a figure quoted without it says nothing at all.

Start with a household. The shape is identical. Suppose a saver has lent Rs 10,00,000/- to a cousin who will repay it in one lump three years from now at a fixed rate, and has borrowed Rs 10,00,000/- from a bank, repayable in a lump two years from now, also fixed. On paper the saver is square: an asset of ten lakh, a liability of ten lakh, nothing left over. Now rates in the market rise sharply. A buyer can now get the new higher rate elsewhere, so the three year claim the saver holds is worth less than ten lakh to anyone who might buy it. The two year obligation is also worth less than ten lakh for the same reason. The three year claim has one more year of discounting working on it, so it falls further than the two year obligation does. The saver was square in rupees and was not square in value, and the difference between those two statements is the whole of this measure.

What is economic value of equity actually measuring?

Economic value of equity is measuring the residual. Take the present value of everything on the asset side, subtract the present value of everything on the liability side, and what is left is the value of the equity in economic rather than accounting terms. Then apply a rate scenario and watch that residual move. At Vindhya Commercial Bank Limited the bank's own 200 basis point rise cuts the present value of rate sensitive assets of Rs 84,000 crore by Rs 5,040 crore and the present value of rate sensitive liabilities of Rs 84,000 crore by Rs 4,200 crore. The two sides both fall. The reported figure is what survives the subtraction: Rs 840 crore.

Notice how small the answer is beside the movements that produced it: Rs 840 crore is 16.7 per cent of the Rs 5,040 crore the asset side alone moved. That proportion is worth holding on to, because it shows how the measure behaves. A headline that is a difference between two large numbers is extremely sensitive to anything that changes either of them slightly. A nudge of two per cent on the asset side does not nudge the answer by two per cent; it moves the answer by a great deal more. Everything uncomfortable about this measure follows from that geometry.

THE HEADLINE IS WHAT SURVIVES A SUBTRACTION OF TWO LARGE MOVEMENTS Vindhya Commercial Bank Limited, invented. The 200 basis point rise is the bank's own internal scenario, not a requirement. Present value of rate sensitive assets, Rs 84,000 crore at 3.00 years, falls by Rs 5,040 crore Present value of rate sensitive liabilities, Rs 84,000 crore at 2.50 years, falls by Rs 4,200 crore The difference, which is the whole of the reported change in economic value of equity minus Rs 840 crore, and this is the number that is reported Rs 840 crore is 16.7 per cent of the Rs 5,040 crore the asset side alone moved. A small residual between two large movements is very sensitive to a small change in either of them.
The bank's own 200 basis point rise moves the asset side Rs 5,040 crore and the liability side Rs 4,200 crore, and the reported figure of Rs 840 crore is only the sliver left after the subtraction, which is why the measure reacts sharply to any input that touches either side.

One more thing about that residual, and it surprises most readers the first time. The residual appears nowhere in the bank's financial statements. It is not a provision, it is not a fair value adjustment, it is not a line in the profit and loss account, and no auditor signs it. The residual is computed for managing rate risk and for nothing else, and that is precisely why it is measured and limited separately. A loss of value of Rs 840 crore can be entirely real, entirely relevant to whether the bank can go on doing what it does, and entirely invisible in the accounts for years.

Try it out

Where in this invented bank's financial statements does the Rs 840 crore appear?

Why is it a change rather than a level?

The level could in principle be computed. Discounting every cash flow on the asset side, discounting every cash flow on the liability side and subtracting the second from the first gives the economic value of the equity today. Almost nobody does it, and there is a good reason. The level depends on the entire shape of the rate curve chosen, on how each balance without a contractual maturity was treated, and on judgements about early repayment that are debatable at every point. Two competent teams would produce two different levels for the same bank on the same evening and neither would be wrong. The level is an argument. Both teams start a change from the same place and only the shock differs, so the change under a stated scenario is a measurement.

So the reported figure is always a movement, and the scenario travels with it. Here that scenario is a parallel shockA rate change applied equally at every maturity, so the whole curve shifts up or down without changing shape. A parallel shock is the simplest scenario shape and the one this invented bank uses internally. of 200 basis points, applied equally at every maturity. The 200 basis point figure is Vindhya Commercial Bank Limited's own internal scenario. No authority requires it and no standard sets it. A change in the size of the shock changes the answer in proportion. A change in the shape of the shock, letting short rates move more than long ones, changes the answer in a way that is not proportional at all. A serious rate risk pack therefore carries several shapes rather than one.

A figure of minus Rs 840 crore with no scenario attached is not a measurement of anything. Minus Rs 840 crore under this bank's own 200 basis point parallel rise is.

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How is the number computed, and what goes into it?

Four inputs and one subtraction. The balance on each side, and the modified durationA number expressing how much the present value of a set of cash flows changes for a small change in rates. Treated here as a locked input; where it comes from belongs to the fixed income subject area. of each side. At Vindhya Commercial Bank Limited, invented, rate sensitive assets are Rs 84,000 crore at a modified duration of 3.00 years and rate sensitive liabilities are Rs 84,000 crore at a modified duration of 2.50 years. Both durations are the bank's own locked figures, taken here as inputs. Where a modified duration comes from is taught in the fixed income subject area.

The whole computation is one line: minus the duration gap, times the shock, times the balance. Minus 0.50 years times 2.0 per cent times Rs 84,000 crore is minus Rs 840 crore. Against tier 1 capital of Rs 6,600 crore that is 12.7 per cent. This bank sets limit L8 at 15.0 per cent of its own tier 1 capital, and therefore at Rs 990 crore, so the utilisation is 84.8 per cent. Every one of those figures belongs to the invented bank. None of them is a regulatory number and none may be read as one.

FOUR INPUTS, ONE SUBTRACTION, AND THE WHOLE MEASURE FITS ON A LINE Vindhya Commercial Bank Limited, invented. Every input below is the bank's own locked figure. RATE SENSITIVE ASSETS Rs 84,000 crore the bank's own figure ASSET DURATION 3.00 years modified, never derived SENSITIVE LIABILITIES Rs 84,000 crore the bank's own figure LIABILITY DURATION 2.50 years modified, never derived minus 0.50 years times 2.0 per cent times Rs 84,000 crore = minus Rs 840 crore 12.7 per cent of tier 1 capital of Rs 6,600 crore, which is the bank's own 84.8 per cent of limit L8 whose cap of Rs 990 crore is also its own
The entire measure is four locked inputs and one subtraction, and the resulting minus Rs 840 crore is what the bank carries at 84.8 per cent of limit L8's own Rs 990 crore cap and at 12.7 per cent of its tier 1 capital.
Try it out

What are the two modified durations in this computation, and what is the gap between them?

What is a duration gap, and what does 0.50 years actually mean?

The duration gapThe modified duration of the asset side less that of the liability side. A single number saying which side of the balance sheet is longer in value terms, and by how much. is the modified duration of the asset side less that of the liability side. Here 3.00 less 2.50 gives 0.50 years. Read it as a statement about which side is longer. The bank's assets behave, for the purposes of value, as though they were half a year further out than its liabilities. A positive gap means the asset side is the longer side, so a rise in rates takes more value off the assets than it takes off the liabilities, and the difference lands on the equity. A negative gap would mean the opposite in every respect.

The same shape appears in an ordinary household. Suppose a salary is renegotiated every six months, so the income repriced recently and will reprice again soon. Suppose the rent is fixed by a thirty month agreement signed last year. The obligations are locked far out and the income resets quickly. If the general level of prices and wages jumps, the income catches up in months and the rent does not move at all. The household comes out comfortable. Reverse it, with income locked for thirty months and rent renegotiated every six, and the same jump is painful. Nothing about the size of the salary or the rent changed in either story. Only the relative length of the two sides changed, and that alone decided whether the move helped or hurt. The duration gap is that relative length, written as one number.

Two cautions before the arithmetic. The first is that 3.00 years and 2.50 years are taken as given and neither is derived; how a modified duration is built from a set of cash flows belongs to the fixed income subject area. The second is that a gap of 0.50 years is not small. Since 2.0 per cent of Rs 84,000 crore is Rs 1,680 crore, each full year of duration gap on a balance sheet of this size is worth Rs 1,680 crore of economic value under the bank's own 200 basis point rise. Half a year of gap is therefore Rs 840 crore, and that is the entire headline number restated as a sentence about length rather than about rupees.

THE SIGN OF THE GAP DECIDES THE DIRECTION, AND THE SIZE DECIDES THE AMOUNT Vindhya Commercial Bank Limited, invented. The shock is held at the bank's own 200 basis point rise throughout this drawing. A NEGATIVE GAP A POSITIVE GAP liabilities are the longer side so a rise in rates adds value assets are the longer side so a rise in rates takes value away minus 1.00 minus 0.50 0 plus 0.50 plus 1.00 duration gap, in years, being the asset duration less the liability duration WHAT THE SAME 200 BASIS POINT RISE DOES TO ECONOMIC VALUE plus Rs 1,680 crore plus Rs 840 crore zero minus Rs 840 crore minus Rs 1,680 crore This invented bank sits here, at a gap of 0.50 years Each full year of duration gap is worth Rs 1,680 crore, being 2.0 per cent of Rs 84,000 crore. The map from gap to rupees is a straight line.
A positive duration gap means a rise in rates takes value off the equity and a negative one means it adds value, and on this balance sheet the map from gap to rupees runs at a straight Rs 1,680 crore for every year of gap.

How much room is there before the limit, and can it be read three ways?

Limit L8 at Vindhya Commercial Bank Limited caps the economic value sensitivity at 15.0 per cent of tier 1 capital, being Rs 990 crore. The measured figure is Rs 840 crore, so the headroomThe distance between what is currently measured and the cap that applies to it, before anything is done about either. Here it is Rs 990 crore less Rs 840 crore. is Rs 150 crore and the utilisation against that cap is 84.8 per cent. Rs 150 crore of headroom answers no question anybody actually has, so a committee never leaves it there. A committee wants to know how much more of the lever it controls it can take on. The same Rs 150 crore can be read as room in the duration gap, as room in the shock, or as room in the liability duration, and a good pack shows all three because different people around the table control different levers.

Read it on the duration gap first. The answer is minus Rs 1,680 crore for every year of gap, so the cap is reached when Rs 990 crore divided by Rs 1,680 crore gives a gap of 0.5893 years. The bank has 0.50 years, so the gap may widen by 0.0893 years, a shade over one month of duration. Read it on the shock next. The bank's basis point valueThe change in value for a one basis point move in rates. Here it is Rs 25.2 crore on the asset side less Rs 21.0 crore on the liability side, being Rs 4.2 crore of equity value, and it is built under the sensitivity measures rather than here. on the equity is Rs 4.2 crore, so the cap is reached at Rs 990 crore divided by Rs 4.2 crore, being 235.7 basis points, against the bank's own 200. Read it on the liability side alone last. Hold the asset duration at 3.00 years, and the liability duration may fall from 2.50 years to 2.4107 years before the same cap binds.

Now the check, and it matters more than any single one of the three answers. All three describe one movement: 0.0893 years, Rs 150 crore, seen from three directions. If the three did not agree, the model would be wrong somewhere. But watch the percentages, because they are not the same and they are not supposed to be. On the gap, 0.0893 years against 0.50 years is 17.9 per cent more gap. On the shock, 35.7 basis points against 200 is 17.9 per cent more shock. On the liability duration, 0.0893 years against 2.50 years is 3.6 per cent, and only 3.6 per cent. The base of 2.50 years is a big one and the gap is a small one. The 3.6 per cent figure is the one that matters most. A three and a half per cent shift in one input moves the headline by the entire remaining headroom. The headline is a difference, and a difference has leverage over both of the numbers that made it.

ONE HEADROOM OF Rs 150 CRORE, READ ON THREE DIFFERENT INPUTS Vindhya Commercial Bank Limited, invented. Limit L8 and its Rs 990 crore cap are the bank's own. READ ON THE BANK HAS THE CAP BINDS AT WHICH MEANS ON THE DURATION GAP shock and balances held still 0.50 years 0.5893 years the gap widens by 0.0893 years, about one month of duration, being 17.9 per cent more gap ON THE SHOCK both durations held still 200 basis points 235.7 basis points the shock rises by 35.7 basis points, at Rs 4.2 crore each, being 17.9 per cent more shock ON THE LIABILITY SIDE asset duration held at 3.00 2.50 years 2.4107 years the same 0.0893 years, but now against a base of 2.50 years, being only 3.6 per cent of it All three rows describe one movement: 0.0893 years, and Rs 150 crore of headroom. The percentages differ because the bases differ, and the smallest of them sits on the input nobody watches.
The same Rs 150 crore of headroom reads as 17.9 per cent more duration gap, 17.9 per cent more shock, or a fall of just 3.6 per cent in the liability duration, and it is that last figure that shows how much leverage a small input has over a headline built as a difference.
Try it out

The headroom to limit L8 is Rs 150 crore. How much bigger would the rate shock have to be before the cap binds?

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What is one month of duration gap worth here?

About Rs 140 crore, and that single sentence is what turns this measure from an abstraction into something an analyst can hold in mind while somebody is talking. The arithmetic is trivial once the gap sensitivity is in hand: Rs 1,680 crore for a full year of gap, divided by twelve, is Rs 140 crore a month. Every month of duration gap this bank adds costs it Rs 140 crore of economic value under its own 200 basis point rise, and its whole remaining headroom is a little over one such month.

The Rs 140 crore a month is the sentence to carry into a room. Somebody proposes buying a longer dated portfolio, or lengthening the asset book by writing more fixed rate loans, and the question stops being philosophical. How many months does this add to the gap? Two? Then it costs about Rs 280 crore of economic value under the bank's own scenario, and the bank has Rs 150 crore of room, so the answer is no unless something on the liability side lengthens to match. A measure computable only at month end is a report; a measure that can be priced in months of duration is a control.

ONE YEAR OF DURATION GAP, CUT INTO TWELVE MONTHS OF Rs 140 CRORE Vindhya Commercial Bank Limited, invented, under its own 200 basis point rise. limit L8 binds here Rs 140 Rs 140 Rs 140 Rs 140 Rs 140 Rs 140 Rs 140 Rs 140 Rs 140 Rs 140 Rs 140 Rs 140 the gap the bank has, 0.50 years, being Rs 840 crore all the room that is left: 0.0893 years, being Rs 150 crore Rs 1,680 crore for a full year of gap, divided by twelve, is Rs 140 crore a month. The bank's remaining room is a little over one of these cells.
Cutting the Rs 1,680 crore per year of duration gap into twelve gives Rs 140 crore a month, which prices any proposal that lengthens the asset book and shows that the bank's whole remaining room is a little over one month of duration.
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Which assumption decides the sign of the answer?

Everything so far has treated the two durations as facts. The two durations are not facts. Each is a conclusion, and one input inside them is doing more work than all the rest put together. Vindhya Commercial Bank Limited holds Rs 36,000 crore of current and savings balances. The balances are contractually repayable on demand. The contract says the money can leave tomorrow morning, and in practice most of it never does. Somebody has to decide how long it stays. The length somebody settles on is a behavioural lifeThe average length of time a balance with no contractual maturity is assumed to remain, based on how such balances have actually behaved rather than on what the contract permits., and a behavioural life is a judgement rather than a measurement.

Two answers to the same question look like a mistake at first sight. Nobody made one, and the reason for two answers is worth having before the two answers appear. The bank has several tables in front of it and each asks a different question about the same Rs 36,000 crore. When does the cash actually leave in ordinary conditions? When does the rate paid on the balance next change? How long does the balance stay, in value terms, for discounting? How much of it walks out in a thirty day stress? Four different questions have four different right answers, and none of them is a check on any of the others. The problem, when it comes, is not that the answers differ. The trouble is that two of the answers feed the same headline and nobody has put them side by side.

Two of them matter for the measure at hand. The economic value computation gives the Rs 36,000 crore an average behavioural life of 0.5 years. The bank's own repricing ladder slots the same Rs 36,000 crore in bucket RB5, at one to three years. Since 2.0 per cent of Rs 36,000 crore is Rs 720 crore, one bank, one balance, one month end and two tables carry a difference of judgement worth Rs 720 crore of economic value for every year of assumed life. The first judgement is set by the asset liability management committee, G4 in this bank's structure. The second sits inside the repricing ladder that the same function produces. Neither has ever been reconciled against the other in a paper anybody has read.

ONE BALANCE, TWO TABLES, TWO JUDGEMENTS, ONE MONTH END Vindhya Commercial Bank Limited, invented. Both treatments below are the bank's own and both are current. Rs 36,000 crore of current and savings balances, contractually repayable on demand TABLE ONE, THE VALUE COMPUTATION what it asks: how long does this balance stay, in value terms? 0.5 years of average life set by committee G4, the asset liability body TABLE TWO, THE REPRICING LADDER what it asks: when does the rate on this balance next change? 1 to 3 years, bucket RB5 the same Rs 36,000 crore, not a rupee apart Two different questions, so two different answers, and neither table is wrong on its own terms. The distance between the two judgements is worth Rs 720 crore of economic value for every year of assumed life.
The value computation and the repricing ladder give the same Rs 36,000 crore two different assumed lives because they ask two different questions, and the distance between those judgements prices out at Rs 720 crore of economic value per year of life.
Try it out

Which of this bank's own tables disagrees with the 0.5 year life used in the value computation?

A warning before a third and a fourth answer

The same Rs 36,000 crore is treated again in this bank's liquidity tables, where the questions are when the cash actually leaves in ordinary conditions and how much of it leaves in a stress. The liquidity treatments are different again, they are produced for different purposes, and they are worked through in the liquidity subject area rather than here. Meet them there knowing that four treatments of one balance in one bank in one month is a fact about four different questions and never evidence that three of the tables are wrong. Of the four treatments, two belong to interest rate risk in the banking book and two to the liquidity sequence.

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What happens to the answer when that one judgement is turned?

The next step is the part worth slowing down for. Everything up to here has been arithmetic that anybody would reproduce the same way. From here the arithmetic sits on top of a judgement, and the judgement moves the answer further than the balance sheet does. The size of the lever comes first. The Rs 36,000 crore of current and savings balances is shocked by 2.0 per cent like everything else, and 2.0 per cent of Rs 36,000 crore is Rs 720 crore. So every extra year of assumed behavioural life on those balances lengthens the liability side and moves the reported figure by Rs 720 crore in the favourable direction, and every year taken off moves it Rs 720 crore the other way.

The mechanism is simple once it is said plainly. A balance assumed to stay half a year is, in value terms, a very short liability. Short liabilities barely move when rates change, so they do almost nothing to offset the fall in the value of long assets. Assume the same balance stays two years instead, and it becomes a much longer liability. Now it falls in value alongside the assets, and that fall is a gain to the equity holder. A liability worth less is an obligation that costs less to discharge. Nothing about the money changed. Only the assumption about how long it sits changed, and that assumption is what decides whether the liability side is short enough to leave the bank exposed or long enough to cover it.

Here is the same thing in a smaller frame. A man runs a tea stall outside an office building and is offered money for the business. The stall has one asset, a set of regulars who come every morning, and one obligation, a rent agreement. If he believes his regulars will keep coming for six months, the stall is worth very little. Six months of custom is all he is selling. If he believes they will keep coming for three years, the same stall is worth a great deal more. Not one cup of tea has changed hands differently in the two stories. The only difference is a belief about how long people stay. A behavioural life is exactly that belief, written down as a number and multiplied by a very large balance.

The line follows. The reported figure is minus Rs 840 crore at the bank's own assumed life of 0.5 years, and it changes by Rs 720 crore for each year added. In a column of thirteen assumed lives the shape becomes obvious, and the shape is the whole finding. The relationship is a straight line in the assumed life, it crosses zero inside the bucket this bank's own repricing ladder uses for the same balance, and it breaches limit L8 at both ends of the range.

Assumed behavioural lifeChange in economic value of equityAgainst the Rs 990 crore cap of limit L8Status
0.0000 yearsminus Rs 1,200 crore121.2 per centbreach, on the negative side
0.2500 yearsminus Rs 1,020 crore103.0 per centbreach, on the negative side
0.2917 yearsminus Rs 990 crore100.0 per centthe near edge of the band
0.5000 yearsminus Rs 840 crore84.8 per centthe bank's own assumption
1.0000 yearsminus Rs 480 crore48.5 per centwithin
1.5000 yearsminus Rs 120 crore12.1 per centwithin
1.6667 yearszero0.0 per centthe sign changes here
2.0000 yearsplus Rs 240 crore24.2 per centthe midpoint of bucket RB5
2.5000 yearsplus Rs 600 crore60.6 per centwithin
3.0000 yearsplus Rs 960 crore97.0 per centthe top of bucket RB5
3.0417 yearsplus Rs 990 crore100.0 per centthe far edge of the band
3.5000 yearsplus Rs 1,320 crore133.3 per centbreach, on the positive side
4.0000 yearsplus Rs 1,680 crore169.7 per centbreach, on the positive side

Two things in that column need saying out loud before anybody moves on. The first is that a figure of plus Rs 1,320 crore is a breach of limit L8 just as surely as minus Rs 1,020 crore is. A balance sheet that gains Rs 1,320 crore of value when rates rise is a balance sheet that loses about that much when they fall, so a limit on rate sensitivity caps the size of the movement and not its direction. The bank's own appetite clause A2 is written in both directions. Sensitivity is a two-sided thing, and a reader who treats only the negative end as a breach has understood half of what a limit on a sensitivity is for.

The second is the row at 1.6667 years. The zero row is not a comfortable place to be. A measure sitting at zero is not a measure saying the bank has no interest rate risk; it is a measure saying that at this particular assumed life, on this particular parallel scenario, the two sides happen to offset. Change the shape of the shock and they will not. Zero on a sensitivity measure is a coincidence of assumptions, never a statement that an exposure has gone away.

ONE STRAIGHT LINE, ONE SIGN CHANGE, AND A CAP BREACHED AT BOTH ENDS Vindhya Commercial Bank Limited, invented. The assumed life applies to Rs 36,000 crore of current and savings balances. the Rs 990 crore cap of limit L8, positive side the Rs 990 crore cap of limit L8, negative side minus Rs 1,200 121.2 per cent minus Rs 990 100.0 per cent minus Rs 840 84.8 per cent zero 0.0 per cent plus Rs 240 24.2 per cent plus Rs 990 100.0 per cent plus Rs 1,680 169.7 per cent 0.0000 0.2917 0.5000 1.6667 2.0000 3.0417 4.0000 assumed average behavioural life of the Rs 36,000 crore, in years, and every column is the same balance sheet
Seven assumed lives applied to the same unchanged balance sheet produce answers running from minus Rs 1,200 crore to plus Rs 1,680 crore, breaching the Rs 990 crore cap of limit L8 at both ends and passing through zero at about 1.6667 years.
Try it out

The bank assumes those balances stay half a year on average and reports minus Rs 840 crore. Before the control below is moved: what does the figure become at an assumed life of two years, the middle of the bucket its own repricing ladder uses?

Play with it

Turn the deposit life and watch the headline cross zero

One control, and it is the reader's own. The slider sets the assumed average behavioural life of the Rs 36,000 crore of current and savings balances at Vindhya Commercial Bank Limited. Moving it changes nothing whatever about the balance sheet and everything about the number the bank reports. The upper bar is the reported change in economic value of equity against the Rs 990 crore cap of limit L8 drawn on both sides of zero. The lower panel is the same answer plotted against the assumed life, with the bank's own repricing bucket RB5 of one to three years shaded across it so the two treatments of the same balance are visible together.

assumed behavioural life: 0.50 years
THE REPORTED CHANGE IN ECONOMIC VALUE OF EQUITY, AGAINST ITS OWN CAP The 200 basis point parallel rise, both modified durations and the Rs 990 crore cap are all this invented bank's own. minus Rs 990 crore cap plus Rs 990 crore cap minus Rs 840 crore the bank's own repricing bucket RB5, one to three years 0 1 2 3 4 assumed average behavioural life of the Rs 36,000 crore, in years, with the two dashed lines marking the cap of limit L8
assumed behavioural life
0.50 years
change in economic value
minus Rs 840 crore
utilisation of limit L8
84.8 per cent
from the near band edge
0.2083 years
At an assumed behavioural life of 0.50 years, the bank's own 200 basis point rise changes economic value of equity by minus Rs 840 crore, being 84.8 per cent of its own Rs 990 crore cap under limit L8, and within it.
Educational illustration. Solved points, all of them this invented bank's own: 0.0000 years minus Rs 1,200 crore at 121.2 per cent; 0.2500 minus Rs 1,020 crore at 103.0; 0.2917 minus Rs 990 crore at exactly 100.0; 0.5000 minus Rs 840 crore at 84.8; 1.0000 minus Rs 480 crore at 48.5; 1.5000 minus Rs 120 crore at 12.1; 1.6667 zero; 2.0000 plus Rs 240 crore at 24.2; 2.5000 plus Rs 600 crore at 60.6; 3.0000 plus Rs 960 crore at 97.0; 3.0417 plus Rs 990 crore at exactly 100.0; 3.5000 plus Rs 1,320 crore at 133.3; 4.0000 plus Rs 1,680 crore at 169.7. The band inside which limit L8 is satisfied runs from 0.2917 years to 3.0417 years and is 2.75 years wide. The bank's own repricing ladder slots the same Rs 36,000 crore at one to three years. The model that produces this assumption has never been validated. Turning this control changes no rupee of the balance sheet and moves only the assumed life.

Inside which band of assumed lives is limit L8 actually satisfied?

The control above has a region where the bank stays inside its own cap and two regions where it does not, and the edges of that region are worth computing exactly rather than eyeballing off a chart. The cap is reached on the negative side when the answer is minus Rs 990 crore. Working back from minus Rs 840 crore at 0.5 years, that needs Rs 150 crore of extra loss, and at Rs 720 crore per year of life it takes 0.2083 years of life away. The near edge therefore sits at 0.2917 years. On the positive side the cap is reached at plus Rs 990 crore. Getting there needs Rs 1,830 crore of movement in the favourable direction from minus Rs 840 crore, and at Rs 720 crore per year that is 2.5417 years added. The far edge sits at 3.0417 years, so limit L8 is satisfied only between 0.2917 years and 3.0417 years, a band 2.75 years wide.

Two and three quarter years sounds roomy. The band is not roomy, and the reason is where the bank is standing inside it. The bank's own assumption of 0.5 years sits 0.2083 years from the near edge, or two and a half months. Shorten the assumed life of those balances by two and a half months and this bank breaches limit L8 on the negative side, with no rupee of its balance sheet having moved and no rate in the market having changed. That is a shorter distance than the ordinary revision cycle of a behavioural study, and it is a shorter distance than the difference between two analysts looking at the same deposit history.

Now walk to the other end of the band. The finding turns from uncomfortable into odd there. The bank's own repricing ladder puts the same Rs 36,000 crore in bucket RB5, one to three years. Take the top of its own bucket, three years, as the assumption. The answer there is plus Rs 960 crore, at 97.0 per cent of the cap. The far edge of the band is 3.0417 years. The distance between the top of this bank's own repricing bucket and a breach of its own limit L8 on the positive side is 0.0417 years, about two weeks of assumed deposit life. So one of the bank's own two treatments sits two and a half months from a breach in one direction, and the top of the other sits two weeks from a breach in the other.

Hold both facts at once and the picture is complete. There is no assumption anywhere in the bank's own stated range that is comfortably far from a limit breach, and the two ends of that range breach the same limit with opposite signs. The headroom is not measured in the units a committee imagines, so a committee looking at 84.8 per cent utilisation and thinking of it as sixteen points of headroom has been given a false sense of distance. The real distance to this limit is not Rs 150 crore of value, it is two and a half months of somebody's judgement about how long a savings balance sits.

THE BAND WHERE LIMIT L8 IS SATISFIED, AND WHERE THE BANK STANDS IN IT Vindhya Commercial Bank Limited, invented. Limit L8 and its Rs 990 crore cap are the bank's own. 0.2917 3.0417 LIMIT L8 IS SATISFIED HERE a band 2.75 years wide breach, positive side value rises past the cap 0 1 2 3 4 the bank stands here, at 0.50 years the bank's own repricing bucket RB5, one to three years 0.2083 years, two and a half months, from the bank's own assumption to the near edge 0.0417 years, about two weeks, from the top of its own bucket to the far edge
Limit L8 is satisfied only between an assumed life of 0.2917 years and one of 3.0417 years, and the bank sits two and a half months from the near edge while the top of its own repricing bucket sits about two weeks from the far one.
Try it out

Between which two assumed deposit lives is limit L8 satisfied at this invented bank?

Try it out

The top of this bank's own repricing bucket RB5 is three years. How far is an assumption sitting there from breaching limit L8 on the other side?

Who sets that assumption, and has the model behind it ever been checked?

Follow the decision to a room. At Vindhya Commercial Bank Limited the behavioural assumptions are set by committee G4, the asset liability management committee. G4 has 9 members, meets monthly and is chaired by the chief executive. The cap that the resulting number runs against, limit L8, is set by committee G2, the board risk management committee. G2 has 5 members, meets 6 times a year and is chaired by an independent director. One body decides the assumption that determines the number and a different body decides the cap the number is measured against, and neither meeting has the other one's decision on its agenda. That is not a scandal. The split is an ordinary division of labour, and it is exactly how a number can drift a long way without anyone in either room feeling that anything has been decided.

Behind committee G4's assumption sits a model. In this bank's model inventoryThe register a bank keeps of every model it uses, with what each one does, how material it is and when it was last independently reviewed. A register is a claim about what is running, and only a sweep of the systems tests the claim. that model is V1, the behavioural deposit life model, and it is the model that puts 0.5 years against Rs 36,000 crore. The inventory holds 28 registered models at month 12. Against the bank's own twelve month cycle, 19 are validated and current, 6 are overdue and 3 have never been validated at all. The three never validated are V1, the behavioural deposit life model; V2, the collateral haircut model; and V3, a spreadsheet early warning scorecard kept by one person with no documented specification. The model that decides the sign of this bank's headline interest rate risk number is one of the three in its own register that have never been independently checked even once.

There is a second count worth putting beside the first, and it needs care because of a coincidence in this case. A month 12 sweep found 33 models actually in use against the 28 in the register, so the register was 28 over 33 complete, being 84.8 per cent, and 5 models were running unregistered. The 84.8 per cent completeness of the model inventory is a different statistic altogether from the 84.8 per cent utilisation of limit L8, being Rs 840 crore against a Rs 990 crore cap. Two unrelated quantities in one invented bank happen to round to the same figure, and anybody who lets them sit unlabelled in one paragraph has produced a sentence that means nothing. Which 84.8 is meant has to be named, every single time.

Now the part that has no easy remedy. Somebody reading the last two paragraphs will reach for the obvious fix: never mind the review, just test the model against what happened. Testing against outcomes works beautifully for the traded measure, where a figure is produced every morning and a realised outcome arrives by the evening, so 250 measures can be laid against 250 outcomes and counted. Testing against outcomes does not work here at all. The thing V1 predicts is how long a balance stays, and that is only observable over years. There is no daily outcome to score it against, there is no run of 250 of anything, and by the time enough time has passed to test the assumption the balance sheet it was applied to has been replaced twice over. A model that cannot be tested against outcomes is precisely the model that most needs independent validationAn independent review asking whether a model is fit for the use it is put to, covering its data, its assumptions, its build and its limitations. A validation can find a model unfit even when that model tests perfectly against outcomes., and this one has had neither.

The next step for V1, who ought to do the validating, what an unvalidated model should trigger and how a model inventory is governed all belong to the risk reporting, data and model risk sequence. The narrower point is harder to argue with: the number on the front of the pack depends on V1, V1 has never been checked, and V1 cannot be backtested.

A REGISTER OF MODELS, AND THE ONE THAT DECIDES THE SIGN Vindhya Commercial Bank Limited, invented. Every count below is the bank's own at month 12. 33 MODELS FOUND IN USE AT THE MONTH 12 SWEEP 28 registered, so the inventory is 84.8 per cent complete on that count 5 running unregistered THE 28 REGISTERED MODELS, BY VALIDATION STATUS 19 current 6 overdue against the bank's own twelve month cycle 3 never validated THE THREE THAT HAVE NEVER BEEN VALIDATED V1 the behavioural deposit life model, V2 the collateral haircut model, V3 a spreadsheet early warning scorecard. V1 is the model that puts 0.5 years against Rs 36,000 crore and therefore decides the sign of the headline. And V1 cannot be backtested either, however long anybody waits. How long a deposit stays is observable over years, so there is no daily outcome to score the model against.
Of 33 models found in use only 28 are registered, and of those 28 three have never been validated at all, one of them being the very model that sets the deposit life assumption this bank's headline rate risk number turns on.
Try it out

Why can the behavioural deposit life model not simply be backtested the way this bank's traded measure is?

Who actually uses this number, and what do they do with it?

A measure earns its place when somebody changes a decision because of it, so it is worth walking through the three people who genuinely act on this one at this invented bank. The three are not reading it for the same reason and should not be reading it the same way.

The first is the person proposing a transaction. A treasury or a lending team brings a proposal that lengthens the asset side: a longer dated investment, a tranche of fixed rate loans, a restructuring that pushes repayments further out. Without the measure, the discussion about such a proposal is qualitative and the loudest voice tends to win it. With the measure, the proposal has a price in the units of the measure. Each month of duration gap it adds costs about Rs 140 crore of economic value under the bank's own scenario, and the bank has Rs 150 crore of headroom, so a proposal that adds two months of gap does not fit and no amount of enthusiasm makes it fit. That is what it looks like when a measure becomes a control rather than a report.

The second is a member of committee G2, and their job is not arithmetic. Their job is to notice the input the pack leaves out. A board risk paper that says minus Rs 840 crore, 84.8 per cent of limit L8, within, has given them a number and hidden its most important input. The question worth asking in that room is not whether Rs 840 crore is large. The question is this: what assumed life sits inside the figure, who set it, when was it last looked at, and what does the figure become at the other assumption the bank's own repricing ladder uses? A single sensitivity line showing the answer at 0.5 years, at 1.6667 years and at 2.0 years would change what that committee understands more than any refinement of the arithmetic ever could.

The third is an outsider. Somebody placing a large wholesale deposit with this bank, or lending it money, or writing about it, reads a published sensitivity figure and wants to know what it is worth as information. The honest answer is that it is worth exactly as much as the behavioural assumptions inside it, and those are rarely published. Two banks reporting the same value sensitivity may hold quite different exposures, and two banks reporting different figures may hold the same one, if their deposit life assumptions differ by a year. That is not a reason to ignore a published sensitivity. A published sensitivity is a statement about a bank's judgement as much as a statement about its balance sheet, so ask for the assumption alongside the number.

There is a household version of this too, and it is closer than it looks. A person deciding whether to prepay a long home loan is doing this computation without the vocabulary. Their asset is a stream of future earnings, their liability is a long fixed obligation, and their answer depends entirely on an assumption about how long they will hold the job. Assume a long stable income and the long liability is comfortable. Assume a short one and it is not. Nobody in that household would call the assumption a model, and it is still doing more work in the decision than any of the numbers on the loan statement.

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What is the commonest way this measure goes wrong?

Reading a judgement as though it were a measurement

Look again at how the result appears in a pack. Economic value of equity, 200 basis point parallel rise: minus Rs 840 crore. Limit L8: Rs 990 crore. Utilisation: 84.8 per cent. Status: within. Four lines, all of them true, and every visual convention in them says measurement. The figure is stated to the crore. The figure carries a scenario, sits against a cap and gets a status word. Everything about the presentation invites the reader to treat it as a fact about the balance sheet, in the way that the deposit total of Rs 76,800 crore is a fact about the balance sheet.

The figure is not one fact but two: a balance sheet and a behavioural judgement, and the judgement is doing more work than the balance sheet. Turn the judgement and watch. At 0.5 years, minus Rs 840 crore at 84.8 per cent. At 1.5 years, minus Rs 120 crore at 12.1 per cent. At about 1.6667 years, exactly zero. At 2.0 years, the middle of the bucket this bank's own repricing ladder puts the same balance in, plus Rs 240 crore. Across the full range of assumptions anybody could defend, the answer runs from minus Rs 1,200 crore to plus Rs 1,680 crore, a spread of Rs 2,880 crore, and not one rupee of the balance sheet moves anywhere in that sweep.

Notice the choice the bank has not made. The bank is not choosing between a defensible assumption and an indefensible one. Both ends of this argument are its own. The bank is using two different answers to the same question in two of its own tables, and nobody has put them side by side. The reconciliation would take a single sheet and has never been written, because no meeting has both tables on its agenda at once.

Then three things make it worse, and they compound. First, the band inside which limit L8 is satisfied runs from 0.2917 years to 3.0417 years, and the bank sits 0.2083 years from the near edge, so a shortening of the assumed life by two and a half months breaches the limit on the negative side. Second, the top of its own repricing bucket is three years and the far edge of the band is 3.0417 years, so an assumption taken from the top of its own bucket sits about two weeks of assumed deposit life from breaching the same limit in the opposite direction. Third, the model that sets the assumption is V1, one of the three in this bank's register that have never been validated. V1 cannot be backtested either. How long a deposit stays is only observable over years.

Put together, the single most important input to this bank's headline interest rate risk number has never been independently checked, structurally cannot be tested against outcomes, and is set in a different room from the one that sets the cap it runs against. The number is still worth computing. The number is worth computing alongside the assumption that produced it, and a pack that reports the first without the second has reported a judgement dressed as a measurement.

WHAT THE PACK SAYS, AND WHAT IS ACTUALLY INSIDE IT Vindhya Commercial Bank Limited, invented. Every figure below is the bank's own. AS IT APPEARS IN THE COMMITTEE PACK Economic value of equity, 200 basis point parallel rise: minus Rs 840 crore Limit L8: Rs 990 crore. Utilisation: 84.8 per cent. Status: within. AND HERE ARE THE TWO THINGS THAT PRODUCED IT The balance sheet Rs 84,000 crore on each side, both modified durations locked, audited books. Moves the answer by nothing in this sweep. One behavioural judgement how long Rs 36,000 crore of demand repayable balances stay. Model V1, never validated. Moves the answer across Rs 2,880 crore. the reported minus Rs 840 crore, inside a range the judgement alone can produce minus Rs 1,200 zero plus Rs 1,680 cap cap
The four lines that reach a committee carry no trace of the judgement inside them, and that judgement on its own can move the reported figure across Rs 2,880 crore while the balance sheet stays exactly where it is.

Where does the standard come from, and what does an Indian bank have to do?

The mechanism is jurisdiction free. Two present values, two durations, one shock and one subtraction would work the same way in any country and in any currency. The numbers that make the measure a requirement are not jurisdiction free. Being precise about who publishes what is worth the trouble.

The Bank for International Settlements, at bis.org, is the origin. The Basel Committee's standard on interest rate risk in the banking book holds the economic value measure as a supervisory concept, alongside an earnings measure, a set of standardised interest rate shock scenarios and an outlier testA supervisory device that compares a bank's measured value sensitivity against a stated threshold. A bank beyond the threshold receives closer attention, and the threshold itself is set by the authority. that compares a measured value sensitivity against a threshold. The Basel Committee's standard is the intellectual source of the measure worked here, and naming it is the first half of the job.

The Reserve Bank of India, at rbi.org.in, is what binds. An Indian bank does not comply with a Basel document. An Indian bank complies with what the Reserve Bank of India requires: what must be computed, on which scenarios, how often, in what form, reported to whom, with what treatment of behavioural assumptions, held against with what capital, and from what date each of those applies. Vindhya Commercial Bank Limited is an Indian bank, so the second list is the one that governs it and the first is the one that explains it. The Indian Banks Association, at iba.org.in, carries banking operational convention in India.

Now the accounting. Every single number in this guide belongs to one invented bank: the 200 basis point parallel rise is its own internal scenario, the modified durations of 3.00 and 2.50 years are its own locked inputs, the Rs 990 crore cap of limit L8 is its own limit set at 15.0 per cent of its own tier 1 capital by its own committee G2, and the 0.5 year deposit life is its own assumption produced by its own unvalidated model V1. Every shock size, outlier threshold, behavioural cap and effective date set by an authority is read at the two sites named above. Those figures move, and a teaching text does not.

Jurisdiction

Where the figures set by authority are confirmed

The Bank for International Settlements at bis.org publishes the interest rate risk in the banking book standard, the economic value and earnings measures within it, the standardised interest rate shock scenarios and the outlier test. The Reserve Bank of India at rbi.org.in sets what an Indian bank must actually compute, on what scenarios, how often, in what form, to whom it must be reported, how behavioural assumptions may be treated and from what date each requirement applies. The Indian Banks Association at iba.org.in carries banking operational convention. Every shock size, outlier threshold, behavioural cap and effective date must be confirmed at those sources before it is relied on for anything.

Try it out

Where does the Rs 990 crore cap used throughout come from?

What the reader can now do

The measure can be computed from four inputs and one subtraction, and each of the four can be named. A duration gap of 0.50 years reads as a statement about which side of a balance sheet is longer, and it carries a price: Rs 1,680 crore for a full year of gap on Rs 84,000 crore under a 200 basis point move, and about Rs 140 crore for a month of it. The headroom to a cap can be solved three different ways, their agreement being the check, and the same Rs 150 crore can be explained as 17.9 per cent on the gap and only 3.6 per cent on the liability duration.

The harder thing is also within reach. Minus Rs 840 crore against a Rs 990 crore cap, at 84.8 per cent, status within, invites the question nobody in the room is asking: what assumed life is inside this, who set it, and what does the number become at the other life this bank's own tables already use? The assumption can be named, priced at Rs 720 crore a year, and its band edges located at 0.2917 and 3.0417 years. Those edges place the bank two and a half months from a breach on one side, and the top of its own repricing bucket two weeks from a breach on the other. And it can be said, without any of it being an accusation, that the model producing the input is V1, that it has never been validated, and that it cannot be backtested.

This guide covers economic value of equity as a measure: what it is, why it is reported as a change under a stated scenario rather than as a level, the computation from two balances and two locked durations, the duration gap as a number, the headroom to limit L8 solved three ways, the price of a month of duration gap, the deposit life assumption solved end to end and the band inside which the limit is satisfied. Where modified duration and present value come from belongs to the fixed income and quantitative subject areas and neither is derived here. The repricing ladder RB1 to RB8 is drawn in full elsewhere in this sequence and appears here only as the second treatment of the same Rs 36,000 crore. Earnings at risk is a different measure and the comparison between the two is covered separately; the figure carried here is that the same 200 basis point rise adds Rs 156 crore to net interest income over twelve months. Economic value sensitivity as its own subject, the seven step method for measuring interest rate risk, the hedging policy and the traded measures on the other book are each covered separately. Model validation, the governance of a model inventory and what an unvalidated model such as V1 ought to trigger belong to the risk reporting, data and model risk sequence. The liquidity treatment of the same Rs 36,000 crore, in both the maturity ladder and the thirty day computation, belongs to the liquidity sequence. Committees G2 and G4 as objects, and the limit framework that produced limit L8, belong to the risk governance sequence. A swap, a forward, an option, a bond and a government security are named in this subject area and taught under fixed income and derivatives.
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Sources

SourceDocumentSite
Reserve Bank of IndiaWhat actually binds an Indian bank on interest rate risk in the banking book: what must be computed, on what scenarios, how often, in what form, to whom it is reported, how behavioural assumptions may be treated and from what daterbi.org.in
Bank for International SettlementsThe Basel Committee standard on interest rate risk in the banking book, the economic value and earnings measures within it, the standardised interest rate shock scenarios and the outlier testbis.org
Indian Banks AssociationBanking operational convention in Indiaiba.org.in

Vindhya Commercial Bank Limited is invented.
Educational material. Not advice on any investment, tax, budget or market position.

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