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Tools · Value at risk

Two methods, one series, different answers.

Value at risk has two standard calculations. One reads the loss straight off the observations you have; the other assumes a normal distribution and computes it. On the same series they disagree, and the size and direction of the disagreement is more informative than either figure — it tells you how far the returns depart from the distribution the second method assumes. Both are computed here, with expected shortfall alongside, because value at risk does not say how bad the bad case is.

Calculator

This page is showing an example series. Over 36 months at 95% confidence, historical value at risk is 6.65% and the normal calculation gives 6.66%; expected shortfall is 8.09% historically and 8.53% on the normal assumption. Enable JavaScript to paste your own; the tables, chart and method on this page do not need it.

Monthly returns, as per cent or as decimals. The horizon of the figures is the period of the returns you paste: monthly data gives a monthly value at risk.

Higher confidence means a rarer loss, which on a short series means fewer observations behind the historical figure.

VaR, historical
6.65%
read off the observations
VaR, normal
6.66%
mean and standard deviation
ES, historical
8.09%
mean of the tail
ES, normal
8.53%
from the normal density
Months in the tail
2
of 36 pasted
Skewness
0.24
zero if normal
Excess kurtosis
−0.19
zero if normal
Disagreement
—
historical minus normal, VaR
Every month, worst to best95% threshold marked
The 36 example months sorted worst to best. The baseline is the historical 95% value at risk, 6.65%, which 2 months fell at or below.-10%0%10%20%061218243035Monthly returnMonths, worst to best
The 36 example months sorted worst to best. The baseline is the historical 95% value at risk, 6.65%, which 2 months fell at or below.-10%0%10%20%01835Monthly returnMonths, worst to best

The baseline is the historical value-at-risk threshold. The shape of the left-hand end is the question: a normal distribution approaches its tail in a particular way, and a real return series usually does not. Where the observed tail drops away faster than the line suggests, the normal calculation is understating the loss.

Historical and parametric value at risk and expected shortfall for the example series, by confidence level
ConfidenceVaR, historicalVaR, normalES, historicalES, normalMonths in the tail
90%3.53%5.03%6.33%7.15%4 of 36
95%6.65%6.66%8.09%8.53%2 of 36
99%9.53%9.71%9.53%11.23%1 of 36

Method

How each figure is calculated.

All four are reported as positive losses. Every one is a statement about the period of the returns pasted, not about a year.

Historical value at risk

VaR = −(the worst ⌈(1 − c)n⌉ observations) → last

Sort the returns, take the worst ceiling of (1 − c) × n of them, and report the least bad of those as the threshold. The quantile is therefore always an observation that actually happened rather than an interpolation between two, which is why the count is reported alongside it. At 95% on 36 months that count is 2. At 99% it is 1.

Parametric value at risk

VaR = −μ + zσ

The sample mean and the sample standard deviation, with z = 1.2816 at 90%, 1.6449 at 95% and 2.3263 at 99%. This uses every observation rather than only the tail, so it is far more stable — and it is stable around whatever a normal distribution would imply, which may not be what the series does.

Historical expected shortfall

ES = −mean(the tail)

The average of the observations in the tail, rather than its boundary. It answers the question value at risk leaves open: given that the threshold was breached, how bad was it. On a short series it is an average of very few numbers, and at high confidence it can be an average of one.

Parametric expected shortfall

ES = −μ + σ φ(z) ÷ (1 − c)

Where φ is the standard normal density. Under a normal distribution expected shortfall sits a fixed proportion beyond value at risk — about 1.25 times at 95%. If the historical pair are further apart than that, the tail is heavier than normal; if closer, thinner.

Skewness and excess kurtosis

Reported because they explain the disagreement rather than merely recording it. Negative skewness means the large moves are more often losses; positive excess kurtosis means more extreme observations than a normal distribution produces. Either pushes the true tail beyond what the parametric figure computes. The example series reads 0.24 and −0.19.

Scaling to a longer horizon

There is no scaling here, deliberately. The common practice of multiplying a one-period figure by the square root of the number of periods assumes independent, identically distributed, symmetric returns — the same assumption the parametric figure makes, applied a second time. Where that assumption fails it fails worse over longer horizons, so a scaled figure is less reliable than the one it came from, not more.

Limits

What value at risk does not tell you.

  1. It is a threshold, not a worst case. A 95% figure is expected to be breached one period in twenty. It says nothing about the size of the breach, which is what expected shortfall is for, and nothing at all about the worst possible outcome.
  2. The historical figure at high confidence rests on almost nothing. At 99% on 36 months the tail contains 1 observation. A single month decides the figure, and a different sample period would give a different answer with no change in the underlying risk.
  3. The parametric figure is precise about the wrong distribution. It will always produce a smooth, stable number, including for a series with a visibly fat tail. Stability is not accuracy, and the two moments reported above are the check on it.
  4. Value at risk is not additive across positions. The value at risk of a portfolio is not the sum of the value at risk of its holdings, and it can exceed that sum. Expected shortfall does not have this defect, which is one reason regulators moved towards it.
  5. It is estimated from the past. Both methods describe a sample. Neither observes a risk that has not yet appeared in the data, which is precisely the risk that matters most.
  6. The horizon is the period you pasted. Monthly returns give a monthly figure. Nothing here converts it, for the reason given above.
General information

This calculator is provided for general information only and is directed to wholesale and professional investors. It is not personal advice: it does not take into account the objectives, financial situation or needs of any person, and it is not an offer, invitation or recommendation to acquire any financial product. Investing involves risk, including the possible loss of capital. Past performance is not a reliable indicator of future performance.

Definitions

Terms on this page.

Each links to the glossary entry, which states the convention the term assumes as well as what it means.

  • Value at risk — a loss not expected to be exceeded, with a stated probability.
  • Expected shortfall — the average loss when the threshold is breached.
  • Skewness — asymmetry of the return distribution.
  • Excess kurtosis — how much of the variance comes from rare large moves.
  • Volatility — the standard deviation the parametric figure is built on.

Questions

Common questions.

What is the difference between value at risk and expected shortfall?

Value at risk is the boundary of the tail: a loss that should be exceeded only one period in twenty at 95% confidence. Expected shortfall is the average loss in those periods where the boundary is crossed. Value at risk tells you where the bad cases start; expected shortfall tells you how bad they are once you are in them. Expected shortfall is also additive across positions in a way value at risk is not.

Which method should I use, historical or parametric?

Neither alone. The historical figure uses the actual shape of the returns but rests on very few observations in the tail. The parametric figure uses every observation but imposes a shape the returns may not have. Reading them together, with the skewness and excess kurtosis, tells you more than choosing one: a large gap between them is itself the finding.

Why do the two figures disagree on my series?

Because the returns are not normally distributed, which almost no return series is. Negative skewness or positive excess kurtosis puts more weight in the left tail than a normal distribution allows, so the historical figure comes out worse than the parametric one. If the parametric figure is the worse of the two, the sample tail is thinner than normal — which on a short record usually means the sample simply has not contained a bad month yet.

How many observations do I need?

For the historical figure, enough that the tail is not one or two months. At 95% confidence the tail is one twentieth of the sample, so sixty months gives three observations and a hundred and twenty gives six. At 99% it is one hundredth, which needs several hundred observations before the figure means much. The count is reported above so you can see what your own figure rests on.

Can I get an annual figure from monthly returns?

Not reliably, and this calculator does not try. Scaling by the square root of twelve assumes independent, symmetric, identically distributed returns, which is the assumption the parametric method already makes and the one the disagreement above is measuring. A scaled figure inherits that assumption twice over. If you need an annual figure, compute it from annual observations.

Is anything I paste sent to KyperX Capital?

No. The series is parsed and every figure computed in your browser. Nothing is transmitted, recorded or stored, and this website receives none of it.