Expected Shortfall: The Average Loss Beyond the VaR Cut-Off

Expected shortfall is the average of the losses beyond a stated cut-off, so the measure answers how bad the bad days are rather than where the bad days begin. At Vindhya Commercial Bank Limited, an invented bank, the one day figure at 97.5 per cent is Rs 21.9 crore, against a value at risk of Rs 15.6 crore at 99 per cent. Both figures belong to that bank alone.

Two habits have to be in place before that paragraph does any work. The first is knowing what value at risk claims. Value at risk is covered separately, and one sentence carries it here: a loss figure a stated book is not expected to exceed over a stated period on a stated share of days. The second is a question the reader may never have thought to ask of a risk number, and it is the entire reason expected shortfallThe average of the losses beyond a stated cut-off, on the same book over the same horizon, which is a statement about the size of bad days rather than about where they begin. exists. The question to ask of any risk figure is what happens on the days that figure is wrong. A reader who has not asked that will read Rs 21.9 crore as a bigger, sterner version of Rs 15.6 crore and learn nothing from either figure.

What does expected shortfall actually claim?

Expected shortfall makes a claim about a region rather than about a point. Fix a book, fix a horizon, fix a cut-offThe point in the range of outcomes past which a tail measure averages, stated as a confidence level., then look only at the outcomes that lie past that cut-off and take their arithmetic mean. At Vindhya Commercial Bank Limited, invented, the book is the Rs 3,600 crore held for trading portfolio, the horizon is one day, the cut-off is 97.5 per cent, and the mean of the losses past it is Rs 21.9 crore. A tail average with no book, no horizon and no cut-off attached is not a measurement of anything, so every one of those four things has to travel with the figure.

The same idea sits in an ordinary household life. Suppose a household looks back at the months when the budget went wrong, and there were eight of them in the last few years. One measure says the overrun in a bad month starts at about Rs 12,000/-. A different measure adds up the eight actual overruns and divides by eight, and says a bad month costs Rs 31,000/- on average. The two measures are not two versions of one statement. The first tells the household where a bad month begins. The second tells the household what a bad month costs. No household would plan a year on the first figure alone, and a bank should not read its trading book on the first figure alone either.

Notice what has been assumed and what has not. Expected shortfall does not assume the losses past the cut-off are the same size, and it does not assume they follow any particular shape. The measure takes whatever is there and averages it. At this invented bank the figure is produced on a historical basisA measure built by reordering what actually happened over a stated window rather than by assuming a shape, which is how both figures in this guide are produced., meaning the bank reorders what actually happened over a stated window rather than fitting a curve to it. Where the number comes from is a separate question from what the number claims, and the claim has to be settled first.

Why does it matter so much that one measure is an average and the other is a threshold?

Only one of the two measures can move when the bad days get worse. A threshold is a location. A threshold answers where, and once it has answered where, it has finished. An average is a summary of contents. An average answers how much, and it changes whenever the contents change. Value at risk locates a cut-off and expected shortfall describes what lies past one, and no amount of turning the confidence dial converts the first job into the second. The difference between locating and describing is the whole difference between Rs 15.6 crore and Rs 21.9 crore at Vindhya Commercial Bank Limited, invented, and every later contrast follows from it.

Follow the consequence through two imagined books. Both are measured at 99 per cent over one day and both produce a value at risk of Rs 15.6 crore. On the first book the days past the line cost about Rs 16.0 crore, so the line is nearly the worst of it. On the second book the positions behind those days move violently once they move at all, so the days past the line cost about Rs 160 crore. The threshold measure returns the identical figure for both books, and a measure that averages what lies beyond returns figures ten times apart. Anybody choosing between those two books on the threshold alone has been handed no information about the losses that would actually hurt.

Why do the two measures at this bank sit at different confidence levels?

Because a measure that averages a region and a measure that marks a boundary do not need the same confidence levelThe share of outcomes a measure is stated against, and the reason two measures on one book can sit at 97.5 per cent and 99 per cent at once. to be talking about a comparable slice of the year. A threshold at 99 per cent points at one place: the boundary of the worst 1 per cent of days. A tail average at 97.5 per cent reaches across a wider region, the worst 2.5 per cent of days, and reports the mean of all of it. Averaging a wider region can land in a similar place to marking a narrower boundary, and for that reason the two measures at Vindhya Commercial Bank Limited, invented, sit at 97.5 per cent and 99 per cent and are still read side by side.

Put the two slices in day counts and the point becomes concrete. Over the 250 observation days this invented bank uses, the worst 1 per cent is 2.5 days and the worst 2.5 per cent is 6.25 days. The threshold measure is pointing at the boundary of a two and a half day region. The tail measure is averaging a six and a quarter day region. Neither confidence level is a requirement and neither is a rule of arithmetic: 97.5 per cent and 99 per cent are choices this invented bank made and wrote down. Any bank could pair them differently, and the only thing that would then be wrong is a reader who compared the two figures without asking what each was stated against.

What is coherence, and who decided what a risk measure ought to satisfy?

Artzner, Delbaen, Eber and Heath set out the answer in Coherent Measures of Risk, published in 1999, and the attribution is worth carrying because coherenceA set of properties a risk measure can be tested against, set out by Artzner, Delbaen, Eber and Heath in 1999, and the reason the two measures in this guide behave differently when books are added together. is a named set of properties rather than a loose compliment. The paper asked a question nobody had written down cleanly before: what should any sensible risk measure do, whatever formula sits inside it. The paper then listed properties and tested candidate measures against them. The property that separates the two measures in this guide is the one about putting books together.

State the property in plain words. Take two books, measure each on its own, and add the two figures. Now put the same positions into one combined book and measure that. A measure that satisfies the property will never report a combined figure larger than the two separate figures added up. Two books that do not always lose money on the same day partly offset each other, so combining should reduce measured risk or leave it alone. The measure that averages the tail satisfies that property, and the measure that marks a threshold does not satisfy it in general.

Why is that property a governance problem and not just a mathematical curiosity?

Because a measure that can punish aggregation hands a desk a respectable argument for staying separate. Picture a bank with two trading desks whose combined figure comes out above the two desk figures added together. The head of each desk can now say, truthfully, that the bank looks riskier on paper when the two books are read as one, and that reading them separately is the more accurate view. Nothing dishonest has been said, and yet the effect is that nobody ever sees one number for the trading operation. A measure that behaves the other way removes that argument entirely.

The everyday version is a household with two earners keeping two separate accounts. If the way the household measured its own strain made the joint position look worse than the two separate positions added up, the sensible response would be to stop looking at the joint position. Nobody would call that a mathematical objection. Anybody watching would call it a reason nobody has a picture of the whole household. A risk measure that discourages the consolidated view has damaged the reporting line before it has damaged anybody's arithmetic.

What does the ratio of 1.40 times actually describe?

At Vindhya Commercial Bank Limited, invented, Rs 21.9 crore divided by Rs 15.6 crore is 1.40 times, and that single figure is the most misread number in this guide. The ratio describes one book, on one evening, with the tail measure at 97.5 per cent and the threshold measure at 99 per cent, and with the threshold measure produced by one named method out of three the bank could have used. The 1.40 is a fact about a portfolio and a pair of choices, and it is not a property of tails in general. Nobody may carry it to another bank, another book or another evening.

Watch what happens when only the denominator changes. Vindhya Commercial Bank Limited computes its threshold measure three ways on the same portfolio on the same evening, and it reports the historical simulation figure. Put each of the three under the same Rs 21.9 crore and the ratio moves without a rupee of position changing.

Read the spread in that table before moving on. A number people talk about as though it described the shape of a tail moves from 1.35 to 1.66 on nothing but the choice of what sits underneath it. Historical simulation is the only method that bank reports on, so only 1.40 belongs to Vindhya Commercial Bank Limited, invented. The three method figures themselves are covered separately and are used here purely as denominators. Any sentence printing this ratio names the denominator in the same breath, or it has printed a number that means three different things at once.

What happened when this bank tested both measures against its own year?

The year gave a clean answer, and it went the way most readers do not expect. Over 250 observation days Vindhya Commercial Bank Limited, invented, recorded seven exceptionA day on which the realised loss went past the measure taken that morning, and in this invented bank there were seven in 250 days. days, meaning seven days on which the realised lossWhat a book actually lost on a day, as against what any measure said it might. went past the value at risk measured that morning. At 99 per cent the bank expected about 2.5 such days in 250. Seven arrived, and seven is 2.8 times the expectation, an error of 180 per cent on the count. Counting is exactly what the threshold measure invites, and on the count the year was a rout.

Now do the other test on the same seven days. The seven realised losses were Rs 19.4 crore, Rs 24.6 crore, Rs 21.0 crore, Rs 17.8 crore, Rs 28.2 crore, Rs 20.4 crore and Rs 18.6 crore, in the order the year produced them. The seven losses sum to Rs 150.0 crore and average Rs 21.43 crore. The bank's expected shortfall estimate for that same book was Rs 21.9 crore. The estimate was Rs 0.47 crore high, or 2.1 per cent of the estimate itself, against a 180 per cent error on the count. Same book, same year, same seven days, and one of the two measures was nearly right.

Two of those figures collide with other figures in this bank and both are named here so nobody merges them. The Rs 15.6 crore in the X5 row is the measure taken on the morning of month 9 day 2, and it happens to carry the same digits as the bank's reported value at risk. The Rs 18.6 crore in the X7 row is a realised loss, and the same digits appear elsewhere in this bank as a measured figure on a different day entirely. The digits alone do not say which object is meant, so the object has to be named every time one of these numbers is printed.

Why does the fair version of that comparison use 6.25 days rather than seven?

Because the two measures do not sit at the same cut-off, and comparing them without saying so would be a small piece of cheating. Expected shortfall at Vindhya Commercial Bank Limited, invented, sits at 97.5 per cent, so it averages the worst 2.5 per cent of days. Over 250 observation days that is 6.25 days. The seven exceptions are something else: they are the days that went past a measure stated at 99 per cent. Seven and 6.25 are close enough to make the rough comparison worth making. Seven and 6.25 are not the same thing, so the rough comparison is not the end of it.

So run it properly with a fractional cut-offThe arithmetic of averaging a non-whole number of days, used here because 2.5 per cent of 250 days is 6.25 days rather than a whole one.. Order the seven recorded losses from worst to least bad. Take the worst six in full: they sum to Rs 132.2 crore. Then take a quarter of the seventh: 0.25 times Rs 17.8 crore is Rs 4.45 crore. The two parts give Rs 136.65 crore spread over 6.25 days, and Rs 136.65 crore divided by 6.25 is Rs 21.86 crore. Against an estimate of Rs 21.9 crore that is a difference of Rs 0.04 crore, being 0.2 per cent, so the honest comparison lands closer than the rough one did.

Are the seven recorded losses the seven largest losses of the year?

Not proven, and this caveat is not optional. Vindhya Commercial Bank Limited, invented, records a realised loss for seven days out of 250. For the other 243 days the case carries no realised loss at all. Every calculation on these figures is exact on the seven recorded losses and is an estimate of the year's tail only under an assumption nobody has actually made. The caveat is not a technicality dragged in for tidiness. The caveat follows directly from the definition of an exception.

An exception is a loss that went past the measure taken on that particular morning, and the morning measure moved through the year. The morning measure ranged from Rs 14.8 crore to Rs 16.2 crore across the seven exception days alone. So a large loss on a morning when the measure happened to stand high would simply not be recorded as an exception. The bank's own record contains the demonstration. On month 3 day 21 the measured figure stood at Rs 19.2 crore and on month 3 day 22 it stood at Rs 18.6 crore, both well above most of the recorded exception losses, and no realised loss is on record for either day. A loss of Rs 18 crore on either of those two mornings would have left no trace in the exception log at all.

Keep the two Rs 18.6 crore figures apart while reading that paragraph. One is the measured value at risk on month 3 day 22, a figure produced in the morning. The other is exception X7's realised loss on month 11 day 22, a figure produced by the market. The two figures carry the same digits and fall on days numbered 22, and they are different kinds of object in different months. Printed without naming what it is, either one leaves a reader no way of telling which was meant.

Where does the bank's own estimate actually sit among the recorded days?

One more calculation on those seven figures turns a worked example into an instrument a reader can hold. Two averages have appeared so far: the average of all seven recorded losses, Rs 21.43 crore, and the average built on the fair 6.25 day basis, Rs 21.86 crore. Neither of them is Rs 21.9 crore. The bank's own estimate does not sit on top of any average that can be built from a whole number of its recorded days, and that is not a defect in the estimate. A figure produced on a 500 day window and then tested against 250 days of what actually happened should be expected to land off every one of those averages.

Consider how the averages behave as the set widens. With the single worst day alone, the average is that day, Rs 28.20 crore. Adding the second worst drops it to Rs 26.40 crore. Every day added is smaller than the ones already in, so every step down the list pulls the average down again, and it never turns back up. The steady fall is a property of averaging a list in size order and not a claim about this bank. The whole sequence, exact on the invented bank's own recorded figures, runs Rs 28.20, Rs 26.40, Rs 24.60, Rs 23.55, Rs 22.72, Rs 22.03 and finally Rs 21.43 crore.

Now read the estimate against that ladder. Rs 21.9 crore falls between the sixth rung at Rs 22.03 crore and the seventh at Rs 21.43 crore. The estimate is below one rung and above the other, so no whole number of worst days reproduces it. The closest the recorded days come is the fractional point at 6.25 days, landing at Rs 21.86 crore, and even that is Rs 0.04 crore away. An estimate that lands inside a gap between two observable averages rather than on either of them is behaving exactly as an estimate should. Nothing in the way the estimate is built would make it match one of them to the paisa, so a figure that did match would be the suspicious result.

The control does something particular to the seventh bar past 6.25 days. The size of the loss on month 6 day 3 is a fact about that day, so the bar's height never changes. The share of that loss entering the average is what changes. A fractional weight is not a fudge, it is the honest way to average a share of days that does not land on a whole day. Two and a half per cent of 250 observation days is 6.25 days, and there is no arrangement of the calendar that turns that into a whole number. Anybody who computes a tail average on a finite window meets this, and the two usual responses are to round to the nearest whole day or to weight the boundary day fractionally. Rounding would quietly change the answer and then not say so, so the worked figures take the second.

What does expected shortfall still not say?

The quickest way to misuse a good measure is to believe it has closed the question. Expected shortfall has not closed it. Expected shortfall is silent about three things, and each silence matters in a different way.

The first silence is the worst day. An average is a single figure standing in for a set of figures, and it can stand in for very different sets. Rs 21.9 crore is what this invented bank expects the bad days to cost on average once they have begun. Rs 21.9 crore is not what the worst of them costs. The seven recorded days show the distance plainly: they average Rs 21.43 crore and the worst of them, exception X5 on month 9 day 2, lost Rs 28.2 crore, or 31.6 per cent more than the average of the set it belongs to. The average of a set never gives the largest member of the set, and no amount of averaging will make it do so. A reader who takes Rs 21.9 crore as a picture of the worst plausible day has made the same category error as the reader who took Rs 15.6 crore as one, just one step further along.

The second silence is the shape of what lies beyond. Two books can produce the identical tail average and be spread quite differently around it. One might lose an amount close to Rs 21.9 crore on almost every bad day. The other might sit near Rs 18 crore on most of its bad days and then, rarely, lose a great deal more, with the two effects cancelling out into the same average. The two books present the same figure to a committee and are not the same object at all. An average locates a centre and says nothing about the spread around that centre, and for that reason a tail figure is a better answer than a threshold figure rather than a complete one. The bank's own look-back windowThe stretch of past days a historical measure is computed over. Here it is 500 days for the measure and 250 days for the count of realised losses against it, and neither can say anything about a day outside itself. holds the days it holds, and the arithmetic reports their centre.

The third silence is the largest one, and it cannot be repaired by any adjustment to the arithmetic. Both figures in this guide are built by reordering days that actually occurred inside a window. The measure runs on 500 days and the count of realised losses runs on 250. A day whose kind has never occurred inside that window is not in the data being reordered, so it contributes nothing to either figure. Lowenstein, When Genius Failed, 2000, is the standing account of what a modelled tail does when the world produces a day outside its own history, and the lesson survives every improvement to the measure. Averaging the days that have been seen produces a better answer about those days and no answer at all about a day that has not. The seventh bar in the control above can be given any weight at all and it will never conjure an eighth.

What can this case not compute about expected shortfall?

Here is the honest inventory, and skipping it would teach a habit worse than any single wrong figure. Vindhya Commercial Bank Limited, invented, provides two numbers about traded risk on the Rs 3,600 crore held for trading book inside its Rs 96,000 crore balance sheet: a threshold figure of Rs 15.6 crore at 99 per cent and a tail average of Rs 21.9 crore at 97.5 per cent. The record adds seven realised losses and the morning figures those seven were measured against. The record holds nothing else. Everything this guide computes is built from those figures and nothing in it may be built from anything else.

So the following cannot be produced here, and the reason is the same in each case. There is no expected shortfall at 99 per cent for this book. There is no threshold figure at 97.5 per cent either. There is no return series, no volatility and no distribution of daily outcomes, so nothing at all joins the two published points to each other. Producing a figure at a cut-off the case does not carry would mean assuming a shape for the outcomes between them, and the moment a shape is assumed a number has been invented that other work on the same invented bank will then contradict.

The mistake would be an easy one to make. A ratio of 1.40 times is already in hand. Reaching for it, multiplying Rs 15.6 crore by something and announcing a tail figure at 99 per cent is tempting. The ratio of 1.40 times describes two particular figures on one particular book on one particular day. The ratio is not a conversion factor and not a property of tails in general. The relationship between a threshold at one cut-off and an average beyond another depends entirely on the shape of the outcomes, and this case records no shape. The 1.40 is an observation about two numbers and never a rule for making a third.

How a committee actually uses the second number

A measure that changes no decision is an ornament. The claim is settled, and the use somebody makes of the claim on a Tuesday is not. Take the invented bank's own reporting route. Committee G7 receives the traded position, the figure measured each morning and the count of days on which a realised loss went past it. Devendra Achar, head of treasury at that invented bank, sits inside that route. Limit L5 caps the threshold measure at Rs 18.0 crore and the current reading is Rs 15.6 crore, running at 86.7 per cent of the cap. Every one of those figures belongs to that bank alone.

Now watch what each measure lets that committee do. The threshold figure and its cap answer a question about permission: is the desk inside the box it was given? At 86.7 per cent the answer is yes, with a little room. Permission is a genuine and necessary question, and it is the only question a cap can answer. The cap cannot tell the committee what a day outside the box would cost, and that is precisely the question a committee asks the moment a cap is approached. The tail average is the figure that answers it, so a committee reading both is reading a permission and a consequence at the same time, rather than a permission twice.

Brought down to a household, the pairing becomes obvious. A household says: in a normal month the overspend does not go past Rs 4,000. The Rs 4,000 figure is a threshold, and it is useful, and it is exactly the shape of a cap. Then comes the second question. In the months when the household does overspend, what does it actually cost? If the honest answer is Rs 6,000, the household has a mild seasonal wobble. If the honest answer is Rs 40,000, and the months that go wrong are the months when a medical bill or a wedding arrives, the household has something completely different on its hands, and the Rs 4,000 threshold said nothing about which of the two it was living with. The threshold gives how often things go wrong and the tail average gives what going wrong is like, and every household that has ever been surprised by a bad month has been surprised by the second one.

A street vendor outside a single office building runs the same pair without writing either down. Most days takings are steady, and the vendor knows the level below which they rarely fall. The number that governs whether the vendor keeps a cash cushion is the other one: on the days takings do collapse, when the building is shut or the rain does not stop, how far do they collapse? A vendor who has only ever thought about the ordinary bad day and never about the size of the extraordinary one is carrying a cushion sized against the wrong figure. The bank is doing the same arithmetic at Rs 3,600 crore and with a committee attached.

There is one more practical use, and it is the reason the coherence argument earlier is not decoration. When a committee aggregates several desks into one number, it needs the aggregate to behave. A measure that can report more risk for the combined book than for the two desks measured separately hands every desk head a reason to argue that their book should be looked at on its own. The separate-book argument fragments the one view the committee exists to hold, so it is a governance failure long before it is a mathematical one. The measure that averages the tail behaves under aggregation and the measure that locates a threshold does not in general, so the choice of measure quietly decides how hard it is to hold one view of the institution. Whether either measure is required for any regulatory purpose in India is a separate question, and the sources named below hold the answer.