Reading the risk measures

This page shows, in three numbers, how large a single day's loss on one listed issue has been. The values come from that issue's own past trading record and say nothing about what happens next. The same issue gives different values over a one-year, three-year or five-year window, because each window covers a different stretch of market conditions.

This is not a tool for choosing issues. A larger value does not rank one issue above another, and the interval shown beside each number is too wide to tell two issues apart.

VaR (Value-at-Risk) is the size that a single day's loss stays below with the stated probability. At the 95% level it is the size exceeded on about one trading day in twenty. It is shown as a share of the amount invested.

cVaR (conditional Value-at-Risk) takes only the days on which the loss ran past VaR and averages the losses on those days. VaR alone says nothing about how far the loss runs past it, which is why the two are published together. By construction this value is always the larger of the two at the same level.

Annualized volatility is the standard deviation of daily log returns over the whole window, restated in yearly units. The first two values are sizes as of the latest settled session while this one compresses the entire window into a single number, so there is no reason for the two kinds of number to agree.

The interval shown beside each value is the range the value would plausibly have taken had this window's observations fallen a little differently. It does not say how far the value will move from here.

(1) Returns. Returns are read straight from the previous-day change and the base price the exchange publishes for that session, and every quantity below is computed on the log return $r$. On a corporate-action day the exchange resets the base price itself, so no further adjustment is applied. The base price is sometimes set from that day's opening price, so listing days, relisting days, the first session after a halt, and any session whose base price moved while the change was zero are dropped from the sample. Sessions in which the price moved beyond the daily limit are dropped as well.

(2) Today's volatility. Volatility is updated every session by an exponentially weighted moving average.

The decay constant $\lambda$ is fixed at 0.94 and never varied by issue. The value is carried over from work that estimated it by minimizing volatility error on currencies and equity indices rather than on individual issues, and letting it vary issue by issue would leave room to choose, after the fact, whichever value passes.

(3) Standardization. Each day's return is divided by that day's volatility.

This lets quiet and violent stretches sit in one sample.

(4) Quantile. The empirical quantile of the standardized values is taken and scaled back by today's volatility.

Here $\mathcal{Z}_t(W)$ is the standardized returns of the $W$ trading days up to yesterday and $\mathcal{S}$ is the subset that sits below the quantile. No normal distribution is assumed, since share returns have heavier tails than a normal one. The rule for computing the quantile is held fixed. The approach follows the line of work that joins volatility updating to historical simulation (Hull and White 1998).

(5) Scale as published. The quantities above are on a log scale, so they are shown on screen as simple-return losses.

A quantile passes through a monotone transform unchanged, so VaR is converted at the point estimate and at both ends of the interval alike. cVaR is a mean rather than a quantile, and converting it afterwards would overstate the loss, so each resample has its tail converted before averaging. Annualized volatility is left on the log scale as a standard deviation restated by $\sqrt{252}$. That restatement changes the units of a standard deviation, whereas the square-root scaling, which is set aside below, stretches a quantile to a longer holding period, and the two are not the same operation.

(6) Estimation windows. Three windows of 252, 756 and 1260 trading days are used. The first 100 returns of each issue serve only to start the calculation and are dropped from the sample. If real observations cover less than 90% of a window, no value is produced.

(7) Interval. The standardized returns are resampled 2000 times in circular blocks of 20 trading days, a percentile interval is read off, and the estimation error of the volatility itself is folded into it. Blocks are used because adjacent days resemble one another and drawing observations one at a time would make the interval narrower than it truly is.

The holding period is fixed at one day. Scaling a one-day value to a longer holding period by the square root of the horizon is not used. That practice is known to understate risk systematically in markets with jumps, and the more so the longer the horizon.

The measure describes how wide an issue's loss distribution is today. Annualized volatility is the average width across the whole window, while VaR and cVaR are the width as of the latest settled session, so in violent stretches the latest width runs far above the window average. A change of regime shows up in the ratio of the latest width to the window average.

Comparisons between issues call for care. The width of the interval shown beside each number is close to the spread of values across issues, so two issues whose numbers differ a little cannot be told apart on that basis.

Sectors and the market as a whole move in step, and that shows up here unchanged. When many issues rise and fall together on the same day, that is a market regime, not a property of any one issue.

VaR and cVaR give the size of a one-day loss as a share of the amount invested. Putting the same amount into different issues can mean very different daily swings, and these numbers make that difference explicit.

They say nothing about what to do with a position. A small value does not mean an issue is safe, and a large one does not mean the risk has to be avoided. The only question these numbers answer is how large a one-day loss has been in the past record.

Breaches also arrive in clusters, something the level by itself does not show. The measure is right about how often the loss exceeds the stated level, but not about when those breaches cluster, so during unsettled stretches the realized rate runs above the stated level.

Every issue was scored before publication against a pre-registered standard. Scoring builds each day's value from data available up to that day and then compares it with the following day's realized return.

Whether exceedances arrive at the stated rate is tested issue by issue (Kupiec 1995), and the standard applies to the share of issues rejected. On the main board that share was 7.2362% over one year, 1.5267% over three years and 1.79% over five years, against a threshold of 10%. At the 99% level over five years it was 2.506%. On the growth board, in the same order, the figures are 6.257%, 1.3308% and 2.5473%.

The gap between the median exceedance rate and the stated level, measured as a share of that level rather than in percentage points, is read alongside it. On the main board it deviated by 9.6036% over one year, 3.0184% over three years and 1.6748% over five years, against a threshold of 20%.

cVaR is checked against a pass criterion of its own. On the main board the share of issues rejected was 1.206% over one year, 0.8724% over three years and 1.5513% over five years, against a threshold of 10%.

Conditional coverage does not pass. Testing whether exceedances arrive back to back (Christoffersen 1998) rejects between 10.6205% and 51.6583% of issues on the main board. That result is disclosed here. The measure gets the frequency of exceedances right but not their timing, and because exceedances cluster in time, the realized rate runs above the stated level through stressed stretches. Between 16.8258% and 19.4975% of issues fail a test that reads the whole distribution (Berkowitz 2001).

Measured against a plain historical simulation that does not rescale for volatility, the median loss ratios are 0.94915534, 0.92948177 and 0.92313356, all below one, and the measure scores better on between 93.5678% and 97.136% of issues.

The 99% level over five years on the growth board has a median of 1898 scorable dates against the 2500 the standard requires, so it is recorded as untested and is not published. Untested means the question could not be settled, not that the answer was no.

Two limits are left uncorrected, namely that estimation error can move a test's true size away from its nominal one (Escanciano and Olmo 2010) and that the tests used here lose power on short samples (Campbell 2005; Kupiec 1995). The regression-form test (Engle and Manganelli 2004) is reported as a figure.