Market Risk, Liquidity Risk and What Value at Risk Sees

Market risk is the loss that comes from prices moving. Liquidity risk is the loss that comes from not being able to trade at the quoted price. Value at Risk states a loss threshold that a stated confidence level is not expected to breach over a stated horizon. The measure says nothing about how far past that threshold a loss can travel. Conditional Value at Risk answers that second question.

A portfolio in the ordinary case is covered earlier in this sequence. An investment committee eventually asks the question out loud: what does a bad outcome look like, in rupees, and what does the number handed to that committee actually promise?

The running example is the Anantara Multi-Asset Portfolio, an invented discretionary mandate of Rs 500 crore run by Faiz Ahmad Ansari for an invented charitable endowment whose investment committee is chaired by Rukmini Deshpande. Its policy weights are equity 60.0 per cent at Rs 300 crore, fixed income 30.0 per cent at Rs 150 crore and cash 10.0 per cent at Rs 50 crore, summing to Rs 500 crore exactly. Every figure here belongs to one invented, stated twelve month period and to no real market.

What is market risk when the thing at risk is a whole portfolio?

Market riskThe chance of losing money because prices move against the holder. It applies to a whole holding at once and does not depend on anything being bought or sold. for a single instrument is easy to picture: the price moves against the holder and the value falls. For a portfolio what matters is not the twenty eight separate stories underneath the equity sleeve but the shared exposure they sit on top of, so the size of a portfolio's market risk is governed by what its holdings have in common and not by how many of them there are. A hundred names resting on one condition is one position wearing a hundred labels, so a portfolio can hold a hundred names and still carry one large market risk.

The household version. Three savings decisions taken carefully over ten years: a deposit at the bank near the office, an equity plan started when the salary first allowed it, and a flat bought in the same town. Each was defended on its own, and all three depend on one employer continuing to employ one person in one place. The exposure was never visible in any single decision, so nobody chose it and everybody has it.

Volatility is the input everything downstream is built from, and it is symmetric: deviations are squared before they are averaged, so a rise and a fall of the same size count the same. A symmetric measure is about to be used to say something about one side of the distribution, and that translation is where the assumptions enter.

Using the mandate's own stated assumptions of 18.0 per cent volatility for equity, 5.0 per cent for fixed income, 0.5 per cent for cash and a correlation of 0.20 between equity and fixed income with cash taken as uncorrelated, the portfolio variance works out at 125.3725, whose square root is 11.196986 per cent, carried in the record as 11.20 per cent. Split that variance by where it comes from and the equity sleeve accounts for 119.88 of the 125.3725, fixed income for 5.49 and cash for 0.0025. Those three shares are 95.62 per cent, 4.38 per cent and 0.002 per cent of the variance. Sixty per cent of the money carries 95.62 per cent of the variance, so a conversation about market risk that spends equal time on each sleeve is spending its time in the wrong place.

The same assumptions make diversification concrete. If the three sleeves moved perfectly together the result would be the weighted average of their volatilities. The average works out at 0.6 times 18.0 plus 0.3 times 5.0 plus 0.1 times 0.5, or 12.35 per cent. The portfolio volatility computed with the correlation of 0.20 is 11.20 per cent. The 1.15 percentage points between those two volatilities is the diversification, and diversification of that size exists only because the correlation is 0.20 rather than 1.00. The gap is what market risk management is buying, and it is smaller than most people expect.

What is Liquidity Risk, and why is it a different problem?

Liquidity riskThe chance of losing money because a trade cannot be done at the price on the screen, either because the size involved is large relative to what normally changes hands or because the moment is a bad one. happens for two ordinary reasons. The first is size: the amount that has to be moved is large relative to what normally changes hands, so the sale itself moves the price. The second is the moment: everybody wants the same side of the trade on the same afternoon, and the quoted price is a price for a smaller seller.

The street version. A vegetable seller can clear one crate at the posted price any morning of the week. When the same market is asked to absorb forty crates before noon, the posted price stops being the price. Nothing about the vegetables changed; what changed is the relationship between the amount that had to be sold and what the market was ready to take.

The distinction that matters is this: market risk shows up in the value of what is held whether or not anything is traded, and liquidity risk shows up only when something has to be sold. Liquidity risk is therefore routinely absent from a risk report and present in a crisis. A monthly report has a column for volatility because volatility can be computed from prices that already exist. A monthly report rarely has a column for what selling would cost. The cost of selling is unknown until somebody sells, and by then it is not a forecast, it is a bill.

One part of the Anantara portfolio escapes the question entirely. Cash is Rs 50 crore at the policy weight, 10.0 per cent of the whole, and it needs no buyer. If the endowment asked for Rs 100 crore, half of that request could be met without any security changing hands, and the remaining Rs 50 crore would have to find a buyer at whatever the market offered that week. Cash is not a return decision here, it is the part of the portfolio that can be paid out without asking anybody's permission or accepting anybody's price.

Why do the two risks arrive together rather than separately?

The event that forces a sale and the event that ruins the price are usually one and the same. A shared driver moves, prices fall, and on that same day somebody needs money: a payment falls due, an allocation band is breached and must be brought back inside, a redemption is requested, a call arrives. The sale then meets the fewest willing buyers of the year. A forced sale in a falling market converts a paper loss into a realised one at the worst available price, and the pressure to sell is at its highest exactly when the price on offer is at its worst.

A liquidity assessment made in a calm month is therefore worth less than it looks. Ask any manager in an ordinary week whether the sleeve can be sold and the answer is yes. In an ordinary week it can be. The harder question is whether it can be sold on the worst five days of the period, at a price close to the one in the report. A portfolio can be liquid in every ordinary month and illiquid in the only month it mattered, and an average across months hides exactly the month in question.

What does Value at Risk actually state?

Value at RiskA stated loss level that, over a stated period and at a stated confidence, the loss is not expected to exceed. It marks a point, and says nothing about what lies past that point. is a loss threshold, for a stated horizonThe length of time a risk figure covers. A one day figure and a one year figure describe different things and cannot be compared without converting one to the other., at a stated confidence levelHow often the figure expects not to be exceeded. At 95 per cent the threshold is expected to be crossed in about one period in twenty., under a stated distributional assumption. All four are part of the number. A Value at Risk quoted as a single figure is not a conservative estimate and it is not a rounded one, it is an incomplete sentence, and the missing words are the ones that make it mean anything.

Read in full, the sentence goes like this. Over one year, at 95 per cent confidence, on the assumption that returns are normally distributed around a mean of zero, the loss on this portfolio is not expected to exceed a stated rupee amount. The four rows below say what each clause is doing and what is lost if it goes missing.

The horizon row is the one most often left off and the one that changes the figure most. Hold the same portfolio, the same 95 per cent confidence and the same normal assumption, and scale volatility by the square root of time across an assumed 250 independent periods in the year. The one year threshold of 19.4093 per cent, or Rs 97.05 crore, becomes 5.6030 per cent, or Rs 28.01 crore, over one month of twelve, and 1.2276 per cent, or Rs 6.14 crore, over a single one of those 250 periods. Three different rupee figures, from one portfolio at one confidence level, separated by nothing but the length of the window.

What do the numbers look like on Rs 500 crore?

The worked instance, with every assumption stated rather than buried. The portfolio value is Rs 500 crore. The volatility for the stated twelve month period is 11.8 per cent. The horizon is one year. Returns are assumed normally distributed around a mean of zero. Real return distributions are under no obligation to obey that assumption, and the assumption is by a wide margin the largest thing being taken on trust.

Under that assumption a threshold at a given confidence is simply the standard normal quantile at that confidence multiplied by the volatility. At 95 per cent the quantile is 1.644854, usually quoted as 1.645. Multiplying, 1.644854 times 11.8 gives 19.4093 per cent, and on Rs 500 crore that is Rs 97,04,63,640/-. The record carries the same threshold as 19.41 per cent and Rs 97.05 crore. At 99 per cent the quantile is 2.326348 and 2.326348 times 11.8 gives 27.4509 per cent, carried as 27.45 per cent and worth Rs 137.25 crore. Every rupee amount is computed from the unrounded quantile, so the crore figures reconcile exactly with the percentages beside them.

The shaded region is 5 per cent of the assumed outcomes, so the threshold is designed to be crossed, about one year in twenty. The curve was not measured from the portfolio's own history: this record locks a volatility and a worst fall for one stated year and locks no distribution of returns at all. The shape is constructed and belongs to no portfolio, and treating a curve of this kind as though it were observed is exactly the error at issue.

The choice of volatility matters as much as the choice of confidence. Feed the same arithmetic the policy volatility of 11.196986 per cent instead of the realised 11.8 per cent and the 95 per cent one-year threshold moves from Rs 97.05 crore to Rs 92.09 crore, a difference of Rs 4.96 crore produced by nothing except which volatility went into the box. Neither figure is wrong. The two figures answer different questions: one asks what the mandate was designed to carry and the other asks what the stated year actually delivered.

What does Value at Risk leave out?

Value at Risk states where the thresholdThe dividing line a risk figure names. Outcomes on one side of it are the ordinary case and outcomes on the other side are the ones the figure has grouped together and stopped describing. is and nothing about what lies on the far side: the threshold marks where the tailThe set of outcomes worse than a stated threshold. It is the part of the picture a threshold groups into one bucket and then declines to describe. begins, not where it ends, and everything inside that tail is collapsed into one statement: it is the worst 5 per cent. How much worse is a question the number was never built to answer.

Picture twenty ordinary years, one square each, with one square being a year in which the threshold is crossed. The threshold says nothing whatsoever about what is written inside that square, and two portfolios with identical thresholds can have wildly different things written there.

Value at Risk vs Conditional Value at Risk: which question does each answer?

Conditional Value at RiskThe average of all the losses that are worse than a stated threshold. It answers how bad the bad case is on average, rather than how often it happens. is the average loss given that the threshold was breached. Same distribution, same horizon, same confidence, different question: the threshold asks where the worst 5 per cent begins, and the conditional figure asks what the average outcome inside it looks like. The conditional figure is sometimes called expected shortfall, and the two names describe the same calculation.

On the Anantara figures, at 95 per cent over one year, the conditional figure is 24.34 per cent, or Rs 121.70 crore, against a threshold of Rs 97.05 crore. The conditional figure is the average of a set of numbers every one of which is worse than the threshold, so the conditional figure is always larger than the threshold it belongs to. Reporting both is the minimum for either to be useful.

The movement between the two levels is the part people predict wrongly. Push the confidence from 95 to 99 per cent over the same one year and the threshold moves out from Rs 97.05 crore to Rs 137.25 crore. The conditional figure moves out too, from Rs 121.70 crore to Rs 157.25 crore. The distance between them narrows from Rs 24.65 crore to Rs 19.99 crore, and at 99.5 per cent it is still Rs 18.65 crore. There is always a region beyond wherever the line is drawn, and the average of that region is always worse than the line, so pushing the confidence level higher moves the threshold outward and never catches the tail.

Two portfolios can report the same threshold and carry completely different tails. Take two constructed distributions, both with an identical 95 per cent one-year threshold of Rs 97.05 crore. The first is the normal shape used above, whose average loss beyond that line is Rs 121.70 crore. The second is constructed so that beyond the same line, four percentage points of probability sit at a loss of Rs 97.05 crore and one percentage point sits at a loss of Rs 300 crore. Its average loss beyond the line is four times Rs 97.05 crore plus one times Rs 300 crore, all divided by five, or Rs 137.64 crore. Same threshold, a tail Rs 15.94 crore heavier.

The Anantara mandate gives a small clue in the same direction. The record states a worst peak to trough fall in the stated year of 9.7 per cent, or Rs 48.5 crore, against 8.1 per cent for the composite benchmark. On the same Rs 500 crore base that benchmark fall would be Rs 40.5 crore. The comparison is like for like arithmetic on one base, not a claim that the benchmark holds Rs 500 crore. A single realised fall is one observation and a tail is a distribution, so the comparison shows which fell further in that one window and nothing about the shape of either tail.

When the confidence level moves, almost nothing else does: the portfolio value stays at Rs 500 crore, the volatility at 11.8 per cent for the stated year, and the realised worst fall at 9.7 per cent because it already happened. The only input that moves is the standard normal quantile, and it runs from 1.281552 at 90 per cent to 2.575829 at 99.5 per cent. Everything the control below moves is driven by that single multiplier, and that is worth knowing before any control of this kind is trusted.

What does a year with no breach prove about the model?

Very little, and this is the section most risk reviews skip. The stated year's worst fall of 9.7 per cent, or Rs 48.5 crore, sat at 49.98 per cent of the 95 per cent one-year threshold of 19.41 per cent, or Rs 97.05 crore. The fall was almost exactly half the threshold. A threshold designed to be crossed about one year in twenty simply cannot be tested by one year of anything, so half is neither reassuring nor alarming.

At a one-year horizon and 95 per cent confidence, one year of history contains exactly one observation, and the expected number of breaches in it is 0.05. If the model is right, the chance of seeing no breach in one year is 95 per cent, and it falls only slowly as the record lengthens: fourteen years are needed before the chance of having seen at least one breach passes one half. A quiet year is exactly what the model predicts whether or not the model is any good, so a quiet year distinguishes nothing.

Worse, a quiet year barely separates a good model from a bad one. Suppose the true chance of a breach were double what the model says, at 10 per cent a year rather than 5. In one year the two stories predict a quiet outcome 95 and 90 per cent of the time respectively. Five percentage points apart is a gap no single year can distinguish. BacktestingChecking a risk model against what actually happened, by counting how often the stated threshold was crossed over a long record and comparing that count with how often it should have been. is a counting exercise over many periods, and it is not something one annual report can do.

What will this record not support, and why stop there?

Measuring liquidity risk properly needs traded volume: how much of a holding normally changes hands in a day, and what share of that a seller could take without moving the price. The Anantara record carries none of that. The record carries a portfolio value, sleeve weights, a volatility, a beta, a tracking error, a turnover figure and a concentration position, every one of them about prices or about weights. Not one is about volume.

Something can still be said, and there is a clear point where it stops. The largest holding is 4.6 per cent of the portfolio, or Rs 23 crore, and measured against the Rs 300 crore equity sleeve the same holding is 7.7 per cent. Both figures are right and answer different questions, so the base is named every time. The top ten holdings together, at Rs 155 crore, read the same two ways. Turnover over the year was 34 per cent, and that trading carried a cost no return figure in this guide shows. How many days it would take to sell any of those amounts cannot be said. The calculation needs a traded volume this record does not contain, and a plausible looking figure produced without it would be worth nothing.

How does a committee actually use this on a Tuesday?

Not by reading a single number. The useful output is four lines, each carrying its own labels so that nobody has to remember them: the threshold with its horizon and its confidence, the average loss beyond it on the same assumption, what actually happened in the period, and what could not be computed and why.

A report that says what it could not compute is telling a committee where to send its next question. A report that quietly fills the gap with a plausible figure is telling the committee nothing while sounding more complete. The habit worth building is that every risk figure arrives with its labels attached and every missing figure arrives named as missing.

The same four lines work at household scale with no arithmetic at all. The first is the ordinary bad month for this household in rupees. The second is a genuinely bad month, a different and larger question. The third is the worst month that actually happened last year. Then the fourth: what would the flat fetch if it had to be sold inside a week rather than over six months? The last question is the liquidity question, and it is the one a household almost never asks until the week it matters.

Which measure answers which question?

Each of the four measures set out here answers a narrow question well and the other three badly, as the map below sets out. The measure that needs a word beyond its row is drawdownThe fall from a high point to a following low point, measured inside a stated window. A different window gives a different figure, which is why the window is always quoted.. A different window gives a different figure, so the window is quoted every time. No one of the four is the risk figure, and a report that carries only one of them has answered one question and left the others open.