Value at Risk and Expected Shortfall: One Book, Two Measures

This tool puts both traded measures on one invented book. Value at risk reads Rs 15.6 crore, Rs 13.2 crore or Rs 16.2 crore depending on method, expected shortfall reads Rs 21.9 crore, and the ratio between them moves from 1.35 to 1.66 with nothing but the denominator changing. Every figure belongs to Vindhya Commercial Bank Limited, invented.

Both of these numbers have been met already, one at a time. Reading them side by side is the part that has not been done. A risk officer at this invented bank does exactly that at about half past seven every morning. Two figures land on one screen, about one book, produced from one night's positions, and neither of them is wrong. The whole difficulty of reading them is that they answer questions which are near enough alike to be confused and far enough apart to disagree. The calculator below is that screen. Turning the choices shows which numbers move.

What does this tool actually show?

Four locked results, one cap, and four choices that can be turned. The apparatus is that short. The word locked is worth being blunt about before anything else happens, and it is what separates this tool from a calculator. A locked resultA figure the worked case fixes in advance, which is displayed rather than derived, because the inputs that would produce it were never recorded. is a figure this invented case fixes. The tool displays it. The tool does not produce it and could not produce it. The ingredients that would produce it were never written down anywhere in the case.

A weighing machine at the entrance of a gym works the same way. The machine shows a number and the number is trusted, but few of the people who read it could explain how the strain gauge inside converted a deflection into kilograms, and none of them needs to in order to read the dial correctly. The reader does need to know what the dial measures, on whom, and at what moment. Reading a measure honestly is mostly a matter of knowing what sits behind the number, not of being able to rebuild it. Everything on this screen is a dial with its object and its moment written next to it.

The book is Rs 3,600 crore of held for trading positions at Vindhya Commercial Bank Limited, invented, at month 12. The held for trading book is the only book any figure here is measured on. The rest of that invented bank's Rs 26,400 crore investment holding, being Rs 8,400 crore available for sale and Rs 14,400 crore held to maturity, sits in the banking book and is controlled by an entirely different set of measures that this screen never shows.

On that one book the case fixes four results and one cap. Value at riskA loss figure a book is not expected to exceed over a stated horizon on a stated share of outcomes. at one day and 99 per cent reads Rs 15.6 crore on historical simulation, Rs 13.2 crore on variance covariance and Rs 16.2 crore on a Monte Carlo run. Expected shortfallThe average of the losses beyond a stated cut-off, on the same book over the same horizon. at one day and 97.5 per cent reads Rs 21.9 crore. Limit L5 caps the first of those at Rs 18.0 crore. Every one of those five figures is that invented bank's own, and none of them is anybody's requirement.

The four choices are these: the method result on display, the measure the cap is read against, the cap itself, and how bad the worst recorded day of the year was. Nothing else moves. In particular, and this is the sentence the rest of the discussion keeps returning to, the positions in the book never move at all, so anything that changes on this screen changed because of a reporting choice and not because of a risk taken.

Why does one book produce three of one measure and only one of the other?

Because three methods were run for the first measure and one was run for the second, and that is a fact about this invented bank's own practice rather than anything deeper. The asymmetry is still worth pausing on, and it is the first thing a new reader misreads. Seeing three figures under one heading and one figure under the other, most people conclude that the first measure is unstable and the second is solid. The conclusion is exactly backwards. The count of figures records how many times somebody ran something, not how reliable the answers are.

The household version runs like this. Three people at a wedding, asked how many guests turned up, give three numbers. One counted plates, one counted chairs and one counted the entry register. One person asked gives one number. The single number is not more accurate. The single number is less examined. Three answers on one book is evidence that three methods were tried, and it says nothing at all about which of the three is closest to the truth.

The three readings, on one book on one day, are Rs 13.2 crore, Rs 15.6 crore and Rs 16.2 crore. The lowest and the highest sit Rs 3.0 crore apart, and that gap is 22.7 per cent of the lowest. The mean of the three is exactly Rs 15.0 crore and the middle one is the Rs 15.6 crore this invented bank actually publishes, so its chosen method sits above the average of its own three and below its own highest. None of that is a defect. Each method buys its answer by trusting a different thing, and what each one trusts is taught separately from this screen.

The screen adds the consequence, and the consequence lives in the cap. Against limit L5 at Rs 18.0 crore, the three readings are 73.3 per cent, 86.7 per cent and 90.0 per cent of the cap. The spread is 16.7 percentage points on one book on one day, with not one rupee of position changing hands. Headroom to the cap is Rs 4.8 crore, Rs 2.4 crore and Rs 1.8 crore respectively, so on the lowest reading the measured risk would have to rise 36.4 per cent before it touched the cap, and on the highest it would have to rise 11.1 per cent. The distance between a comfortable desk and a watched one, on this book on this evening, was settled by whoever chose the method.

Why do the two measures sit at different cut-offs here?

Because they are different kinds of statement about the same ordered list of outcomes, and this invented bank chose a point on the scale for each of them separately. The first is stated at 99 per cent. The second is stated at 97.5 per cent. Both levels are that bank's own choices, and neither is quoted here as anybody's requirement.

The cut-offThe point in the range of outcomes a measure is stated at, being 99 per cent for one of these two figures and 97.5 per cent for the other in this invented bank. is doing different work in each case, and that is the part worth slowing down for. In the first, the cut-off marks a line: the outcomes are ordered from best to worst, and the loss standing 99 per cent of the way along is the figure. Everything past the line is outside what the figure describes. In the second, the cut-off marks the beginning of a region: the walk goes 97.5 per cent of the way along, and every outcome from there to the end is averaged.

So the second measure starts nearer the middle and covers more ground. The first starts further out and covers none. The combination of a line and a region is why the two cannot be lined up as one dial at two settings, and why the natural instinct, that 99 must be stricter than 97.5, is not a safe guide to which figure comes out larger. One of these numbers is a place on the scale and the other is an average over part of the scale, so moving the cut-off does not do the same thing to both.

Put it in a way anybody can feel. Suppose a school reports its examination results two ways. The first way names the mark that the ninety-ninth pupil out of a hundred beat, a single mark that says nothing about the pupil at the very bottom. The second way takes the worst two and a half pupils out of a hundred and averages their marks, saying a great deal about the bottom and nothing at all about where any single line sits. Both are honest reports of one set of results. Handing over only one of them, and letting the reader assume it is the other, is where the damage begins.

What is the ratio between the two numbers actually describing?

The ratio is the part of the screen that people quote in meetings, so it deserves the most care. The tail figure divided by the threshold figure gives a number that feels like it describes the shape of the losses out in the tail: how much further the average of the bad days sits beyond the line that marks where the bad days start. On this invented bank's own method the arithmetic is Rs 21.9 crore over Rs 15.6 crore, giving 1.40 times, and that figure is locked in the case.

With nothing changed except which method result goes underneath, Rs 21.9 crore over Rs 13.2 crore is 1.66 times. Rs 21.9 crore over Rs 16.2 crore is 1.35 times. The numerator has not moved by a single paisa. The book has not moved. The day has not moved. The number that is supposed to describe the shape of the tail moves from 1.35 to 1.66, nearly a quarter, on the strength of a methodology choice made in a different document.

A bare tail ratioThe tail measure divided by the threshold measure, which describes the shape of the losses only when the figure in the denominator is named alongside it. is close to meaningless. The ratio looks like a property of the book and it is a property of the pairing. Only 1.40 belongs to this invented bank. Historical simulation is what its own policy records as the published method, and any paper reproducing the ratio has to say so in the same sentence rather than later in an appendix.

The everyday form of the same trap is this. A household says its rent is 1.4 times its electricity bill. The claim sounds like a fact about the household until somebody asks which month's electricity bill. The same rent against a summer bill and against a winter bill gives two different multiples, and neither multiple describes the rent. A ratio is only ever as stable as the thing underneath it, and a reader who is not told what is underneath has been handed a number with no object attached.

What happens when the same cap is put under the other measure?

The next reading is the one the whole screen exists for, and it takes ten seconds to produce and rather longer to absorb. Limit L5 is a cap of Rs 18.0 crore. Read against this invented bank's published value at risk of Rs 15.6 crore, limit utilisationA measured figure expressed as a percentage of its cap, which on this book depends on both the method and the measure chosen before any arithmetic starts. is 86.7 per cent and the row on the limit report says the limit is within. Read against the same bank's expected shortfall of Rs 21.9 crore, the same Rs 18.0 crore cap gives 121.7 per cent, and 121.7 per cent is not a warning. On this bank's own convention anything above 100 per cent is a live breach. The row would go to the market risk committee as an open item with a named owner and a date.

Look at what did not happen in between those two sentences. No position was bought or sold. No price moved. No model was re-estimated. Nobody made a mistake. The bank went from comfortably within a cap to in breach of it by changing which of its own two measures the cap was read against, and by changing nothing else whatsoever.

The reason this is possible at all is that a limit is not one decision. A limit is two decisions, welded into a sentence that only ever gets quoted by half. The first decision is the number, Rs 18.0 crore. The second is the measurement basisWhat a cap is measured on, being the second half of every limit and the half that almost never gets quoted when the limit is discussed., and here that basis is one day, 99 per cent, historical simulation, value at risk, on the held for trading book. Everything after the rupee figure in that sentence is doing as much work as the rupee figure itself. Quoting the cap alone quotes the half that cannot move without a board paper and leaves out the half that can move because somebody changed a report template.

A household lives with the same structure without noticing. A rule that the monthly grocery bill will not exceed Rs 12,000/- is a cap. The basis is everything the rule leaves unsaid: whether it counts the vegetable vendor's cash purchases, whether it counts the month a cousin stayed for three weeks, whether it is the calendar month or the salary month. Changing any one of those quietly stops the cap binding without it ever being raised. A cap with no basis stated is not a control. The measure underneath it can be redefined faster than the cap can be changed.

What would the cap have to be, and is that a translation?

The obvious repair suggests itself immediately. If switching to the tail measure puts the bank in breach at 121.7 per cent, then raise the cap so that utilisation reads what it read before. The arithmetic is one line. To hold utilisation at 86.7 per cent on a tail figure of Rs 21.9 crore, the cap has to be Rs 18.0 crore times 1.4038, or Rs 25.27 crore. Check it back the other way and Rs 21.9 crore over Rs 25.27 crore is 86.7 per cent, exactly where the reported figure started.

The repair works, and it is a trap. The arithmetic takes a ratio observed once, on one book, on one evening, on one of three available methods, and promotes it to a conversion factor. The ratio is not a conversion factor. Run the identical arithmetic on the same bank's other two method results and the multiplier is 1.35 or 1.66, producing a cap of Rs 24.33 crore or Rs 29.86 crore instead. Three defensible caps, spread over Rs 5.53 crore, from one repair applied three ways.

And the deeper problem is not the spread across methods, it is time. The 1.40 holds on the shape this book's losses happened to have in the window the measure was computed over. Change the composition of the book, or let the window roll forward, and the shape changes, so the multiplier changes with it. A cap rescaled by a ratio observed once is a limit set by yesterday's tail, and it will keep being set by yesterday's tail long after yesterday has stopped resembling today.

If the institution genuinely wants a cap on the tail measure, the honest route is to decide what loss it is willing to average on its worst days and set a number for that, in the same way the Rs 18.0 crore was decided in the first place. Setting that number is a board conversation about appetite. Multiplying the old cap by 1.4038 is not that conversation. The multiplication is a way of arriving at a new number without ever having the conversation, and it produces a limit whose level nobody has ever agreed to.

What refuses to move when all four choices are turned at once?

Everything above this line was one choice at a time. The screen below turns all four at once, and that is the only way to see that they interact. Turn the method and the ratio moves. Turn the measure and the verdict against the cap moves. Turn the cap and the verdict moves again without either figure changing. Raise the worst recorded day of the year and one reading climbs while another sits completely still.

Two warnings sit on the screen itself and both are load-bearing. The first is that the exposure does not move when the display does: every position in the Rs 3,600 crore book stays exactly where it is at every setting. The second is that the method selector genuinely does nothing to the tail figure. The case records that figure on one method at one cut-off and nowhere else. A control that appears to move a number it cannot move is the fastest way to teach a reader something false, so this one says so on its face.

What happens to the two readings when the worst recorded day gets worse?

Now use the last control on its own. Turning it alone isolates the single sharpest difference between these two ways of looking at a book. Vindhya Commercial Bank Limited recorded seven days in its 250 observation days when the realised loss went past the figure measured that morning. The seven days are numbered X1 to X7. Their losses were Rs 19.4, Rs 24.6, Rs 21.0, Rs 17.8, Rs 28.2, Rs 20.4 and Rs 18.6 crore, summing to Rs 150.0 crore and averaging Rs 21.43 crore.

Take the worst of them, X5 at Rs 28.2 crore, and make it worse. At Rs 50 crore the seven average Rs 24.54 crore. At Rs 100 crore they average Rs 31.69 crore. At Rs 200 crore, a genuinely ruinous single day for a Rs 3,600 crore book, they average Rs 45.97 crore. And the exception countThe number of days on which a realised loss went past the figure measured that morning, which is completely insensitive to how far past it the loss went. reads seven at every single one of those settings. Seven at Rs 28.2 crore. Seven at Rs 200 crore. Seven at any number at all.

A day that goes past the measure is one day however far past it goes, so a count of such days has no way of showing whether the year was uncomfortable or catastrophic. The blindness is not a flaw somebody failed to fix. Counting simply works that way. The moment the decision is made to record whether something happened rather than how much of it happened, the size has been thrown away, and with it come all the robustness and all the blindness of the trade.

The household version is a parking fine. Whether the car overstayed by four minutes or by four hours, the register records one violation. The register is convenient, it is easy to audit, and it is completely silent about the difference between a small lapse and an abandoned vehicle. Knowing what the year actually cost means adding up the amounts, and adding up amounts is a different exercise from counting events.

One caution belongs here before this gets pushed too far, and the case is precise about it. Moving the worst recorded day is a reader's control and not this bank's history: the recorded year stands as recorded. And neither of the two locked figures on the screen moves when that control is turned. Both were measured that morning from the positions rather than from what the day went on to do. The realised average of the seven moves, and the count of them refuses to.

Who decided what a risk measure ought to satisfy?

Four people, in a paper, and the attribution matters because the properties are theirs rather than folklore. Philippe Artzner, Freddy Delbaen, Jean-Marc Eber and David Heath set out in Coherent Measures of Risk, published in 1999, the conditions a measure of risk should meet if it is going to be used the way institutions actually use one. CoherenceA named set of properties a risk measure can be tested against, set out by Artzner, Delbaen, Eber and Heath in 1999. is the label for that set, and the reason it turns up on a screen like this one rather than staying in a journal is the fourth condition.

The four are worth naming plainly. A measure should not call a book that is worse in every outcome less risky than one that is better. Double every position and the measure should double. Add cash to the book and the measure should fall by that cash. And, fourth, combining two books should never produce a measured risk larger than the two books measured separately. The fourth condition is subadditivityThe property that combining two books never makes the measured risk larger than the two measured apart, which is the one that separates these two measures., and it is the one that separates the two figures on this screen.

The averaging measure satisfies it. The threshold measure need not, and the fact that it need not is a structural feature rather than a bug in anybody's code. A measure that can report more risk for two desks combined than for the two desks apart hands every desk in the institution an argument for never being aggregated with anyone. None of that is a mathematical curiosity. The failure is a governance problem. A limit framework runs desk by desk and then adds up, and the adding up is precisely where the property bites.

Feel it in a smaller setting. Two shopkeepers who share a wall each insure against fire separately, and their insurer prices each policy on its own worst case. If a combined policy over both shops came out costing more than the two separate ones, every shopkeeper in the street would refuse to be combined, and the insurer would end up holding a set of separately priced risks that it can never look at as one exposure. An institution whose aggregation measure can misbehave is in that position: the incentive runs the wrong way, and it runs the wrong way quietly.

Two boundaries sit around that property. Whether one measure is therefore the better choice is a separate comparison covered elsewhere, and it turns on more than one property. A numerical instance of the failure is not available either. The case carries no two-desk decomposition of its Rs 3,600 crore book, and inventing one to make the point would put a figure into a record that has to tie across everything else this bank appears in.

What can this tool not compute, and why does it refuse to guess?

Four things, and the refusal is part of the tool rather than an apology for it. There is no expected shortfall at 99 per cent in this invented case. There is no value at risk at 97.5 per cent. There is nothing that joins the two cut-offs, and there is no way to reproduce any of the three method results from anything printed here.

The reason is the same in all four instances. Producing any of them would need a shape: a return series to reorder, a volatility to scale, a correlation matrix to combine positions, or an assumed distribution to interpolate between two points on it. The case records none of those inputs. The case records outcomes. So the honest state of the grid is two filled cells and two empty ones, and the empty ones stay empty.

Filling them is tempting, and the temptation is worth naming because it is so reasonable-sounding. Somebody will observe that the ratio between the two figures is 1.40, and propose that dividing the tail figure by 1.40 gives a value at risk at 97.5 per cent, or that some adjustment gives a tail figure at 99. Every one of those routes assumes the shape of the losses between the two cut-offs, and assuming the shape is exactly the step this case gives nobody the material to take. A tool that assumes a shape has invented a number and then displayed it as though it were a record.

One more silence is worth stating out loud, and it constrains what anybody may say about the seven recorded days. The case records no realised loss for the other 243 observation days. So the seven that went past the measure cannot be shown to be the seven largest losses of the year, and any statement about the shape of the year's losses that depends on that is a statement the record does not support.

Which measure would a cap sit on, and what kind of decision is that?

Having watched the consequences, the question answers itself in shape if not in substance. A cap on the threshold measure controls how often this book is expected to have a bad day. A cap on the tail measure controls how bad those days are on average when they come. The two caps control two different things, and an institution can legitimately want either.

The tool has shown that the choice is not a technical one. The choice looks technical, and it is expressed in the language of confidence levels and estimation methods and usually made by the people who run the models. But look at what actually changes when it is made. The reported utilisation of a limit moves. The point at which a desk gets a phone call moves. The evidence that reaches a committee moves. Choosing what a cap is measured on is a decision about what the institution wants controlled, and that makes it a governance decision wearing technical clothes.

In this invented bank the two halves of that decision sit in two places. The board risk management committee sets every one of the twelve limits and accepts or refuses every breach. The market risk committee receives the value at risk position, the backtest and the currency open position every month. So the number is set in one room and read in another, and the sentence naming its measurement basis travels between them inside a policy document that neither committee spends its meeting time on. The gap between the two rooms is exactly where a measurement basis can move without anybody experiencing the change as a decision.

The argument between the two measures, over which of them a cap should sit on, is made elsewhere. The smaller and harder claim holds regardless. Whichever measure the cap sits on, the sentence naming it belongs beside the cap everywhere the cap is quoted, including in the one line summary that goes to the board.

Who actually reads these two numbers, and what does each one do with it?

Three readers, and they want three different things. One figure cannot serve all of them. Start with the desk itself. A dealing room head reads the threshold figure the way a driver reads a speedometer: it arrives every morning, it is comparable with yesterday, and its job is to say whether today looks like the recent past. Rs 15.6 crore against Rs 15.4 crore yesterday is a quiet morning. Rs 15.6 crore against Rs 11.0 crore last week is a question to answer before lunch. The desk wants a stable series, and for that purpose the threshold figure is genuinely better. The threshold figure is produced the same way every day and it moves for reasons the desk can trace back to positions.

Second, the committee that receives the monthly position. The committee is not looking for a series, it is looking for whether the control worked. The committee needs both figures and the cap sentence. Only one thing tells it whether 86.7 per cent is comfortable, and that is knowing the same book reads 121.7 per cent against the same cap on the other measure. A committee handed the first number alone has been told the desk is within its limit. A committee handed both has been told the desk is within a limit that happens to be drawn on the narrower of the institution's two views of the same book.

Third, the reader furthest from the desk, whether that is a board member, a rating analyst, or somebody outside the institution reading a disclosure. The distant reader cannot audit anything and should not pretend to. Checking for the accompanying sentence is open to them. A traded risk figure that arrives without its book, its horizon, its cut-off, its method and its measure has arrived without the information needed to compare it with anything, including with the same institution's own figure from the previous period. The single most useful skill this screen can leave behind is the reflex of asking what the figure was measured on before asking whether it is large.

A household version of all three is worth carrying because it survives when the arithmetic fades. A relative who says they spent Rs 40,000/- last month has said nothing usable until it is clear whether that is one household or two, whether it counts the school fee paid annually, and whether last month had a wedding in it. Nobody accepts the figure alone from a relative. The figure from a trading book arrives with decimal places and a committee stamp. Accepting it alone is the same error dressed better.

What does this screen actually leave behind?

A reader who arrived able to define both measures separately leaves able to read them together, a different skill and the one an institution actually needs. Four locked results and one cap sit behind the screen: Rs 15.6 crore, Rs 13.2 crore and Rs 16.2 crore for the threshold measure at 99 per cent on three methods, Rs 21.9 crore for the tail measure at 97.5 per cent, and limit L5 at Rs 18.0 crore. Every one of the five belongs to Vindhya Commercial Bank Limited, invented, on its Rs 3,600 crore held for trading book at month 12.

The three utilisations stand at 73.3, 86.7 and 90.0 per cent of one unchanged cap, with a 16.7 percentage point spread between the outer two. The crossing point is where the same cap read against the tail measure gives 121.7 per cent and a breach. The equivalent cap of Rs 25.27 crore is a snapshot rather than a conversion, with Rs 24.33 crore and Rs 29.86 crore standing beside it as equally defensible products of the same arithmetic. The ratio stands at three denominators, 1.66, 1.40 and 1.35, under the rule that it is never printed without the figure underneath it named in the same sentence.

One arithmetic separates counting from averaging: raising this bank's worst recorded day from Rs 28.2 crore to Rs 200 crore takes the average of its seven recorded days from Rs 21.43 crore to Rs 45.97 crore while the count of them stays at seven at every point in between. Beside it stand the four conditions of Artzner, Delbaen, Eber and Heath from 1999, with the fourth of them, about combining books, named as the one that turns a mathematical property into an argument at an aggregation meeting.

Most of all there is the habit. A traded risk figure means nothing until the book, the horizon, the cut-off, the method and the measure are known, and a cap means nothing until the same five things are attached to it too. None of that is a sophisticated insight, and it does not need a model to apply. The habit needs somebody in the room to ask what the figure was measured on, before anybody asks whether it is large.