MR 1 – Estimating Market Risk Measures – An Introduction and Overview

Returns express the same gains and losses relative to the amount invested, and they come in two forms. The arithmetic (or simple) return divides the period’s P/L by the opening value.

Built into this definition is the assumption that the interim payment D_t sits idle and earns nothing during the period. Over a short period that does little harm. Over a long horizon, however, income is normally reinvested and earns its own return, so the arithmetic return understates what the holder really gets and should not be relied upon for long horizons.

Geometric returns

The geometric (or compound) return takes the natural log of the gross price relative, which amounts to assuming that interim income is reinvested continuously.

Two features make the geometric return the more natural choice for risk work. First, it never implies a negative asset price: however large and negative R_t becomes, the implied end value is the opening value times e^R, which stays above zero. With arithmetic returns, a return below -100% would mean a negative price, which seldom has any economic meaning (in practice this is more a matter of principle, because any fitted distribution is only an approximation in the far tails). Second, geometric returns are convenient to combine. Returns on a foreign currency position come out the same whichever currency is treated as the reference, and the geometric return over several periods is just the sum of the single-period geometric returns. Arithmetic returns have neither property.

How far apart are the two returns?

Writing the geometric return as ln(1 + r_t) and expanding the logarithm as a Taylor series shows exactly how the two measures relate.

When r_t is small its square and higher powers are negligible, so R_t is approximately equal to r_t. As returns grow, the squared term starts to bite and the geometric return falls increasingly below the arithmetic one. Short holding periods produce small returns, which is why the choice hardly matters for daily VaR, while long horizons produce larger returns where the gap becomes material and the geometric form, which allows for reinvested income, is the one to use.

Historical simulation (HS) is the most direct route to a VaR: sort the observed losses and read the answer off the sorted list, with no distribution assumed at all. Take 1000 loss observations and a 95% confidence level. The confidence level leaves a 5% tail, and 5% of 1000 is 50, so 50 observations belong to the tail. The VaR is then taken as the next observation in, the 51st highest loss, which marks the boundary between the tail and the rest of the sample.

With finite samples there is some arbitrariness here. The true VaR is a quantile that marks the boundary between tail and non-tail outcomes, and one could argue for the 50th observation, the 51st, or some blend of the two. The difference is a matter of approximation only, and the 51st is a reasonable convention. Stated generally, with n observations and confidence level α, the HS VaR is the following order statistic.

Doing it in software

On a spreadsheet the job is a single function call. In Excel, LARGE(Loss_data, 51) returns the 51st largest entry of an array named Loss_data, and the general case is LARGE(Loss_data, (1 – alpha)*n + 1). In MATLAB the losses are first put in order with Loss_data = sort(Loss_data), after which the ordered element is picked by its position. Built-in percentile functions such as Excel’s PERCENTILE or MATLAB’s prctile also produce HS quantiles, but they interpolate in ways that are not visible to the reader, and the Excel version can be unreliable, so the explicit ordered-observation approach is more transparent.

Reading HS VaR from a histogram or a cumulative frequency plot

For 1000 simulated standard normal L/P observations, a histogram of the losses with a vertical marker at the 95% VaR shows the marker sitting at 1.704, with the top 5% of losses to its right and the other 95% to its left. The same answer can be read from the empirical cumulative frequency function, which plots each ordered loss against its cumulative frequency: with n observations the ith smallest has frequency i/n. To read the VaR, start at the confidence level on the vertical axis, move across until the line meets the curve, then drop straight down to the loss axis.

A parametric approach replaces the raw sorted data with an explicit statistical distribution. The analyst names the distribution the observations are assumed to come from, fits its parameters, and reads the VaR off the fitted curve rather than from individual data points. Two choices therefore have to be made together: which distribution to use, and which kind of data (P/L, L/P, arithmetic returns or geometric returns) it describes.

Normal P/L

If P/L is normal with mean μ_P/L and standard deviation σ_P/L, the VaR at confidence level α is the negative of the P/L value that leaves a fraction 1 – α of the distribution below it.

Here z_α is the standard normal variate with a probability α to its left and 1 – α to its right. At a 95% confidence level z_0.95 = 1.645, and at 99% z_0.99 = 2.326. In practice the true mean and standard deviation are unknown, so their sample estimates m_P/L and s_P/L are plugged in, and the result is an estimate of the VaR rather than the VaR itself.

When P/L is standard normal, the cut-off between the lowest 5% and the upper 95% of outcomes is -1.645. The VaR is the negative of that cut-off, so it equals 1.645.

Normal L/P

Working in loss/profit form simply flips the sign of the mean while leaving the standard deviation unchanged: μ_L/P = -μ_P/L and σ_L/P = σ_P/L. The VaR formula therefore loses its minus sign.

This version is easier to picture because the horizontal axis now measures money lost. The VaR is the loss with 5% of the density above it and 95% below, or equivalently the loss at which the cumulative distribution function reaches 0.95. For standard normal losses it is again 1.645, exactly as it should be.

Assumptions can be placed on returns instead of on money amounts. Suppose arithmetic returns r_t are normal with mean μ_r and standard deviation σ_r. The starting point is a critical return r* that the actual return beats with probability α, which means r* sits z_α standard deviations below the mean.

The next step converts that return into a money loss. With no interim payment, the arithmetic return is the change in value divided by the opening value, and a change in value is just a loss with its sign reversed.

At the critical point the end value is P*_t and the loss equals the VaR, so the critical return is the negative of VaR divided by P_t-1 (Equation 1.13). Putting this together with Equation 1.11 and solving for the VaR gives the result below.

This formula is consistent with the earlier ones. With α = 0.95, μ_r = 0, σ_r = 1 and an opening value of 1, it returns a VaR of 1.645, the same as the standard normal P/L and L/P cases. That is no coincidence: with a portfolio worth 1, a return is the same number as a P/L, so the three sets of assumptions describe the same distribution.

The weakness of this approach is the same as for normal P/L: a normal distribution puts some probability on returns below -100%, and therefore on a negative asset value. With realistic parameters over short horizons that probability is tiny, but it is never zero, which motivates the lognormal approach in the next section.

Every model so far allows the asset value to turn negative with some positive probability. Switching the normality assumption from arithmetic to geometric returns removes that flaw. If geometric returns are normal with mean μ_R and standard deviation σ_R, and any interim payment is zero or is continually reinvested in the asset (as with a mutual fund that ploughs its income back), then the log of P_t is normal, which is the same as saying P_t is lognormally distributed. A lognormal price is never negative and its density has a long right-hand tail.

Since the loss is the difference between P_t-1, which is known, and P_t, which is random, the loss inherits the shape of the price distribution in reverse. Normal geometric returns therefore imply a lognormally distributed VaR, and the L/P density is a mirror image of the price density, with a hard upper limit at the full value of the position.

The lognormal distribution in brief

A random variable X is lognormal when ln X is normal. Its parameters are the mean μ and standard deviation σ of ln X, and an equivalent description uses m and σ, where m = exp(μ) is the median of X. The density exists only for positive x.

Writing w = exp(σ²) keeps the moments compact.

The skewness is always positive, which confirms the long right tail, and the kurtosis starts just above 3 for a very small σ and climbs steeply as σ grows. With σ = 1, for example, w = e, the skewness is about 6.18 and the kurtosis about 113.9.

Deriving the lognormal VaR

The critical geometric return R* is exceeded with probability α, so R* = μ_R – σ_Rz_α (Equation 1.15). Unwinding the definition of the geometric return turns that critical return into a critical price P*, and the VaR is the drop from P_t-1 to P*.

Because the exponential is always positive, P* is always positive, and the VaR can never exceed the starting value P_t-1. Under the deliberately unrealistic settings μ_R = 0, σ_R = 1 and P_t-1 = 1, the 95% lognormal VaR is 1 – exp(-1.645) = 0.807, well below the 1.645 that a normal model would give with the same parameters.

Expected shortfall is the probability-weighted average of the losses in the tail beyond the VaR, and among the coherent risk measures it is the easiest to estimate. For standard normal losses the 95% VaR is 1.645 and the matching 95% ES is 2.063, a figure that answers the question VaR leaves open: how bad is the typical loss once the VaR has been breached?

Because ES averages over the tail, it can be built from quantiles. Closed-form ES formulas exist for only a few parametric families, such as the elliptical distributions (which include the normal) and the generalised Pareto distribution. The average-tail-VaR method, by contrast, works for any well-behaved distribution, parametric or not. The recipe has three steps.

First, cut the tail beyond the VaR into n slices that each carry the same probability. Second, compute the VaR at each of the n – 1 internal cut points, which are VaRs at confidence levels above α; these are the tail VaRs. Third, take the simple average of those tail VaRs as the ES estimate.

A ten-slice illustration

Take standard normal losses (mean 0, standard deviation 1), a 95% ES and a deliberately small n = 10. The 5% tail is split into ten slices of 0.5% each, giving nine tail VaRs at confidence levels of 95.5%, 96.0% and so on up to 99.5%.

The 99.5% entry is shown here as it is commonly printed, 2.5738. The 99.5% standard normal quantile is in fact 2.5758, and it is 2.5758 that reproduces the 2.0250 average.

The tail VaRs rise from 1.6954 to 2.5758, and their average, 2.0250, is the ES estimate. It sits a little below the true 2.063 because ten slices cannot reach far enough into the tail: the most extreme 0.5% slice contributes only its lower boundary.

How accuracy improves with more slices

Repeating the calculation with more slices shows the estimate creeping upward toward the true value. The true value is commonly printed as 2.0630, but the exact 95% normal ES is 2.0627 to four decimals, so the gap column below is measured from 2.0627.

Even 50 slices bring the estimate within about 0.011 of the truth, and a few thousand slices make the error negligible. Any modern computer can run thousands of slices in a fraction of a second, so ES estimates of this kind are practical in real time.

Any coherent risk measure can be expressed as the quantiles q_p of the loss distribution averaged with weights from a user-chosen weighting function φ(p), also called the risk-aversion function.

ES is the special case in which every quantile in the tail gets the same weight and every quantile outside it gets nothing.

A more general spectral measure lets the weights grow as losses become more extreme, which expresses a risk-averse view of large losses. The method for estimating it is the same average-VaR routine, with the equal weights swapped for the φ(p) weights of the chosen measure.

The exponential weighting function

A popular choice is the exponential spectrum, in which a single parameter γ governs how sharply weight piles up on the worst outcomes.

The denominator makes the weights integrate to 1, and a smaller γ means stronger aversion to large losses. Suppose risk aversion is represented by γ = 0.05, losses are standard normal, and the whole loss distribution (not just a tail) is cut into n = 10 equal-probability slices. That gives n – 1 = 9 quantiles at confidence levels from 10% to 90%, ranging from -1.2816 to 1.2816. Each quantile is multiplied by its weight, and the measure is the average of those products.

The nine products sum to 3.8049, and 3.8049 / 9 = 0.4228. Published versions of this calculation report the result as 0.4228, 0.4226 or 0.4227; recomputing from unrounded quantiles and weights gives 0.42275.

The weights make the point vividly: the 90% quantile alone carries a weight of 2.7067 and contributes 3.4689 of the 3.8049 total, while everything below the median contributes almost nothing. With only ten slices the estimate of 0.4228 is far from the true value, because the exponential spectrum puts its heaviest weight on the extreme tail, exactly the region a coarse grid misses.

The estimates climb steadily and settle in the region of about 1.854. Compared with ES, a spectral measure with strong risk aversion needs a far finer grid before the estimate stabilises, yet the computing burden remains modest for any reasonable machine.

Since the estimate converges as n grows, a practical stopping rule is to begin with a modest n, keep doubling it, and watch how much the estimate moves. Each doubling halves the width of the slices, and the change it produces is the halving error.

The doubling stops once ε_n drops below a tolerance that reflects the accuracy the user needs. For the exponential spectral measure with γ = 0.05 and standard normal losses, starting from an arbitrary 100 slices gives the sequence below.

The halving error at 100 slices compares against the 50-slice estimate of 1.3739, which is why it is as large as 0.2114. From there it roughly halves with each doubling. At 6400 slices the estimate is 1.8477 with a halving error of 0.0055; doubling to 12,800 lifts the estimate to 1.8506 and cuts the halving error to 0.0029. A user with a tolerance of 0.005 would stop at 12,800 slices.

Crude averaging versus numerical integration

Because Equation 1.17 is an integral, one might expect formal quadrature rules such as the trapezoidal rule or Simpson’s rule to beat a simple weighted average of quantiles. For exponential spectral measures the crude method holds up surprisingly well. The reason is that the spectral weights are largest at the most extreme loss, whereas standard quadrature rules deliberately cut the weights on the two end points, which damages accuracy exactly where the spectral measure cares most. Refined versions of the quadrature methods can work around this and should in principle do better, but the simple method is not a poor choice.

Why this is good news for a VaR engine

The common thread is that every coherent risk measure is a weighted average of quantiles. A firm that already estimates VaR has the data, pricing routines and quantile machinery it needs, and moving to ES or a spectral measure often means changing only the last few lines of code that combine the quantiles. The switching cost from VaR to a coherent measure is therefore low.

Every risk figure produced from data is an estimate, and the true value is never observed. An estimate that is very imprecise is close to useless, because the truth could lie almost anywhere; an estimate known to be tight carries real information. Good practice is therefore to report every risk estimate alongside a measure of its precision, either a standard error or, generally more informative, a confidence interval. This section works with the standard error of a quantile, which is the standard error of a VaR.

Deriving the quantile standard error

The argument follows Kendall and Stuart. Let F(x) be a distribution function, either parametric or an empirical cumulative histogram, with density f(x). Take a sample of n and a bin width h. The probability dF that k – 1 observations fall below q – h/2, one falls inside q ± h/2, and the remaining n – k lie above q + h/2 is proportional to the following.

Call the proportion k/n by the name p. For large n the expression behaves like p^np(1 – p)^n(1-p). Perturbing p by a small amount ε, taking logs and expanding to second order shows that dF is approximately proportional to a normal kernel in ε.

So ε is asymptotically normal with variance p(1 – p)/n. Since a small change in probability maps to a change in the quantile through dF(q) = f(q)dq, dividing by the squared density converts this into the variance of the quantile itself.

The formula reveals three drivers. A larger sample size n shrinks the standard error. It rises as p moves further into the tail, because f(q) gets smaller there, so the more extreme the quantile the less precise its estimate. And it depends on the assumed density f(.) and on the bin width h, whose choice is largely arbitrary.

From standard error to confidence interval

The usual textbook construction then applies. A 90% interval for a quantile is the estimate plus or minus 1.645 standard errors.

Weaknesses of the quantile standard error approach

The method is simple and reasonably credible when samples are large, but it has clear limits. It rests on asymptotic theory and so needs large samples. It can deliver imprecise estimates and wide intervals, as the example showed, and its answer depends on the arbitrary bin width. Finally, it produces symmetric intervals, which mislead for extreme quantiles: observations thin out as one moves into the tail, so the true intervals there are asymmetric. Order-statistics and bootstrap intervals tend to be considerably tighter than the ones produced here.

A natural worry is that ES, by averaging over the tail, is harder to pin down than VaR. Yamai and Yoshiba (2001b), one of the first studies on the question, compared VaR and ES estimators under Lévy distributions with different stability parameters. For near-normal Lévy distributions the two estimators had similar standard errors, but for very heavy-tailed distributions the ES estimators had much larger standard errors. Their explanation was that heavier tails make ES more exposed to large but rare losses, which pointed to the gloomy conclusion that fat tails might generally make ES estimators less accurate than VaR estimators.

Later work suggests that conclusion is too pessimistic. Inui and Kijima (2003) showed that a Richardson extrapolation method can deliver ES estimators that are unbiased and have standard errors comparable to those of VaR estimators. Acerbi (2004) confirmed that heavier tails push up ES standard errors more than VaR standard errors, but found the relative accuracy of the two to be roughly comparable over empirically realistic ranges. Because the answer varies from case to case, the sound habit is to estimate the standard error every time a risk measure is estimated.

An asymptotic formula for coherent risk measures

One route starts from the observation that coherent risk measures belong to the class of L-estimators (weighted averages of order statistics), a class whose members become normally distributed in large samples. Taking N discrete points in the density, as N grows the variance of the estimator of Equation 1.17 approaches the double integral below.

The integral is evaluated with a suitable numerical integration routine. For ES the expression simplifies to a double integral over the tail region up to the quantile F^-1(α).

Combined with a good integration method this gives reliable results even when N is fairly small. Because the estimators are asymptotically normal, confidence intervals then follow from the textbook form of Equation 1.27, centred on a good point estimate of the risk measure and using the estimated standard error.

The bootstrap alternative

A second route avoids the algebra altogether. Draw a large number of resamples from the fitted or historical distribution, compute the risk measure on each, and take the standard deviation of those bootstrapped estimates as the standard error. Better still, the same collection of bootstrapped estimates can be used directly to read off a confidence interval, which need not be symmetric.

The whole subject of market risk measurement can be organised around three decisions taken in a natural order, and keeping them in mind gives a frame of reference for every technique that follows.

Which risk measure?

The leading choice, and the most important, is what to estimate: VaR, ES or some other coherent measure. It has to come first, because there is no point deciding how to estimate something before deciding what that something is.

Which level of analysis?

The second choice is whether to measure risk for the portfolio as a whole or to build it up from the individual positions. Working at portfolio level treats the portfolio as a fixed unit with a given composition, which leads to a univariate stochastic problem and keeps things simple. Working at position level allows the composition to change inside the analysis, which is more flexible, but it requires a multivariate stochastic framework with variance-covariance matrices, copulas and related machinery. The trade-off is that the portfolio level is easier but more limiting, while the position level is more flexible but considerably more work.

Which estimation method?

With the measure and the level settled, the remaining choice is the estimation method, usually framed as the VaR trinity:

Each family raises its own set of difficult issues, and each can feed the quantile-based ES and spectral calculations described earlier.

A fresh data set should never go straight into an estimation routine. Even highly reputable vendors deliver data with errors in them, and nobody can be sure that a misplaced decimal point or a mistyped value has not crept in somewhere along the chain. Real data should always be treated with some suspicion, and a short preliminary analysis both reveals problems and builds a feel for what the numbers look like. At a minimum it covers the first two of the following steps, and ideally all three.

The first and most important step is simply to look at the data and ask whether anything appears wrong. Does the pattern of observations make sense? Do any values stand out as suspicious? This informal look, jokingly called the interocular trauma test, is the easiest check to run and the one most worth running on every new data set.

The second step is to draw a histogram and compute summary statistics: mean, standard deviation, skewness, kurtosis and the extremes. For risk work the interest is in departures from normality, such as skewness, excess kurtosis and outliers, and any such evidence should shape whether, and which, parametric distribution is fitted.

The third step is to ask which distribution might fit, using diagnostic tools such as quantile-quantile plots, covered in the next section.

Reading a histogram and its summary statistics

Consider 100 random observations whose histogram shows one dominant central peak, suggesting a unimodal distribution, together with a hint of negative skew and a few large outliers on the far left that might signal a fat left tail. In fact the data come from a Student-t with five degrees of freedom, a distribution that is unimodal, symmetric and heavy tailed, but with real data the true distribution is unknown, and that is precisely why the preliminary analysis matters.

The gap between the mean of -0.099 and the mode of -0.030 is small next to the standard deviation of 1.363. The skewness of -0.503 is noticeably negative and the kurtosis of 3.985 is somewhat above the normal value of 3. The extremes are lopsided too: the minimum of -4.660 is about 4.56 below the mean, while the maximum of 3.010 is only about 3.11 above it. Faced with these numbers from real data, an analyst would worry about skewness and fat tails. Here the apparent skew is just sampling noise, since the parent distribution is symmetric, which is a useful reminder that small samples can mislead.

Depending on the context, formal tests may also be worthwhile, for example testing whether the sample mean, standard deviation and other moments are consistent with a null hypothesis such as normality. The governing principle is that only sample estimates are ever available, so the analyst should extract as much information from them as possible.

A quantile-quantile (QQ) plot puts the quantiles of the sample on one axis and the matching quantiles of a chosen reference distribution on the other. If the sample comes from the reference distribution, the points line up along a straight line; where they bend away, the shape of the bend says how the two distributions differ.

Two contrasting cases

Plot 500 draws from a standard normal against a standard normal reference and the result is close to a straight line: the central observations fit tightly and the extreme tail points stray only a little. That is the classic picture of a sample that matches its reference population. Repeat the exercise with 500 draws from a Student-t having five degrees of freedom, again against a normal reference, and the picture changes. The middle of the plot is still roughly linear, but both tails turn steeper, a sign that the sample has heavier tails than the reference.

Four uses of a QQ plot

First, it screens candidate distributions. A sample drawn from the reference distribution gives a linear QQ plot, and so does a sample from the same distribution with a different location and scale, because a linear transformation leaves the plot straight. Trying several candidate references and discarding any that produce a curved plot leaves a shortlist of plausible distributions.

Second, it gives rough location and scale estimates. A linear transformation of one distribution only shifts the intercept and changes the slope of the QQ plot, so the intercept and slope of a straight QQ plot indicate the location and scale of the data. In the normal-against-normal case the intercept is about 0 and the slope about 1, which points to a standard normal rather than just any normal. Such rough values are a useful yardstick against which to judge more sophisticated estimates, and good starting values for iterative fitting algorithms.

Third, it diagnoses tail weight. If the tails of the sample are heavier than those of the reference, the plot steepens at its ends even when the middle is close to linear; tails with slopes that differ from the central section signal tails that are fatter or slimmer than those of the reference.

Fourth, it exposes outliers. Observations contaminated by large errors stand out clearly from the rest of the plot, even when the other observations agree well with the reference distribution.

A tail diagnostic to use alongside the QQ plot

Another tool for the tails is the mean excess function (MEF). For a threshold u, it measures how far X is expected to overshoot u, conditional on X being above u. Each distribution has its own characteristic MEF, so comparing the empirical MEF with the theoretical MEF of a candidate distribution shows whether that candidate fits the tails. The results need care, because observations become scarce as X grows, so the right-hand end of an empirical MEF rests on very few data points.