Common mistake 1 with Value at Risk & Expected Shortfall

Reporting a 99% VaR from 100 observations - there are only one or two points in the tail, so the estimate is almost meaningless. You need a long sample or a parametric tail model.

Common mistake 2 with Value at Risk & Expected Shortfall

Scaling a one-day VaR to ten days with √10 when returns are autocorrelated or volatility clusters (use a GARCH model instead).

Common mistake 3 with Value at Risk & Expected Shortfall

Treating VaR as a worst case. It is a threshold that is designed to be crossed a known fraction of the time.

Common mistake 4 with Value at Risk & Expected Shortfall

Aggregating VaR across desks by adding. VaR is not sub-additive in general, so the sum is neither an upper nor a lower bound on the whole - which is one of the reasons expected shortfall replaced it in the Basel framework.

EconStatsCraft › Value at Risk & Expected Shortfall

Value at Risk & Expected Shortfall

The loss you should not exceed at a chosen confidence, three ways - historical, Normal, and skew/kurtosis-adjusted - plus the Expected Shortfall beyond it and a Kupiec test of whether the historical figure held up.

What it needs

Return series

The intuition

Value at Risk answers one narrow question: on a bad day, how bad? The 99% one-day VaR is the loss that only the worst 1% of days exceed. It is a quantile of the return distribution and nothing more. Expected shortfall answers the question VaR dodges - GIVEN that you are in that worst 1%, how much do you actually lose on average - and that difference is the whole reason regulators moved to it.

When to use it

Risk reporting and limits, regulatory capital, comparing the tail risk of strategies, or demonstrating why a Normal assumption understates risk for real financial returns.

Where the estimator comes from

For loss L=-R, VaR at level \alpha is the quantile \mathrm{VaR}_\alpha=\inf\{l: P(L\le l)\ge\alpha\}. Expected shortfall is the mean beyond it, \mathrm{ES}_\alpha=E[L\mid L\ge \mathrm{VaR}_\alpha]. Three estimators are in common use. HISTORICAL simply takes the empirical quantile of past returns and assumes no distribution. PARAMETRIC-NORMAL sets \mathrm{VaR}_\alpha=-(\mu+z_\alpha\sigma) with z_{0.99}=-2.326, which is fast and wrong in the tails. The STUDENT-t version replaces z with a t quantile scaled by \sqrt{(\nu-2)/\nu} so the variance still matches \sigma^2; with \nu around 4 to 6, which is typical for daily equity returns, this widens the 99% figure by roughly 15 to 25% against the Normal.

The model

\text{VaR}_{\alpha} = -\,Q_{1-\alpha}(r),\qquad \text{ES}_{\alpha} = -\,\mathbb{E}[\,r \mid r \le Q_{1-\alpha}(r)\,]

What it assumes

Historical VaR assumes the past return sample represents the distribution you face next period - no regime change.
Gaussian VaR assumes returns are Normal, which they are not: real returns are fat-tailed and often left-skewed.
Square-root-of-time scaling to a longer horizon assumes returns are independent across periods with constant variance.
The portfolio composition is fixed over the horizon. VaR computed on a historical return series of a portfolio that has since been rebalanced describes a portfolio nobody holds.

Statistical properties

VaR is not sub-additive: the VaR of a combined portfolio can exceed the sum of the parts. It therefore fails to be a coherent risk measure in the sense of Artzner et al. (1999), and it can penalise diversification. Expected shortfall is coherent.
VaR says nothing about the size of losses beyond it. Two portfolios with identical VaR can have wildly different expected shortfall, and the tail is exactly where the difference matters.
Scaling a one-day figure to ten days by multiplying by the square root of 10 assumes independent, identically distributed returns. Volatility clusters, so this understates risk in turbulent periods and overstates it in calm ones.
The Normal estimate is biased low in the tail for essentially every financial series. Daily equity returns have kurtosis of 5 to 10 against the Normal's 3.
A 99% one-day VaR should be breached about 2.5 times a year. Fewer breaches is not a safer model, it is a mis-calibrated one, and Kupiec (1995) tests exactly this.
Historical VaR cannot produce a loss larger than the worst one already seen, so it is blind to anything unprecedented, and its tail estimate rests on very few observations.

How to read the output

Quote Expected Shortfall, not just VaR, when the tail is fat - ES is the number that reflects HOW bad the bad cases are, and it is what Basel now requires.
If Gaussian VaR is much smaller than historical or Cornish–Fisher VaR, the return distribution has heavy tails and the Normal number is dangerous.
The Kupiec backtest checks whether the historical VaR was breached about as often as it should have been. A failing test means the model is miscalibrated on this data.
VaR says nothing about losses beyond the threshold - two portfolios with the same VaR can have very different worst cases.
Expected shortfall answers the question VaR does not: how bad it is when the limit is breached. Two portfolios with the same VaR can have very different tails, which is why regulators moved towards ES.
Compare the three methods rather than choosing one. When they agree the distributional assumption is not doing much work; when they diverge, the gap IS the model risk and is worth reporting.

A worked reading

99% VaR on 180 monthly returns of a 1,000,000 portfolio: 13.151% historical (131,515), 14.317% Gaussian (143,167) and 14.438% Cornish-Fisher. Expected shortfall is 15.056% historical and 16.510% Gaussian - between 14% and 15% above the corresponding VaR, which is the point of reporting it: on the months the limit is breached, the average loss is well beyond the limit, and VaR alone never says by how much. Now notice what this sample does NOT show. The textbook warning is that the Normal understates the tail, and here it does the opposite - the Gaussian figure is ABOVE the historical one. The reason is in the summary line: excess kurtosis is only 0.239 and skewness 0.049, so these returns are close to Normal and the historical estimate, resting on the handful of worst months actually observed, happens to be the milder of the two. The lesson is the general one rather than the slogan: which method is conservative depends on the data in front of you, so quote all three and look at the kurtosis before deciding which to believe.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Normal and historical VaR differ sharply	Fat tails	Trust the historical or t figure; report the gap
Zero breaches in a long backtest	Over-conservative, not safe	Kupiec test. Capital is being wasted
Breaches arrive in clusters	Volatility clustering unmodelled	Use a GARCH-based conditional VaR
ES barely exceeds VaR	Thin tail, or too few tail observations	Check the sample size beyond the quantile
VaR jumps when one old date drops out	Historical estimator dominated by one event	The window is too short for this confidence level
Portfolio VaR above the sum of the parts	Non-sub-additivity, a known defect	Report expected shortfall as well

Common mistakes

Reporting a 99% VaR from 100 observations - there are only one or two points in the tail, so the estimate is almost meaningless. You need a long sample or a parametric tail model.
Scaling a one-day VaR to ten days with √10 when returns are autocorrelated or volatility clusters (use a GARCH model instead).
Treating VaR as a worst case. It is a threshold that is designed to be crossed a known fraction of the time.
Aggregating VaR across desks by adding. VaR is not sub-additive in general, so the sum is neither an upper nor a lower bound on the whole - which is one of the reasons expected shortfall replaced it in the Basel framework.

Where to read this properly

Artzner, Delbaen, Eber and Heath (1999), "Coherent Measures of Risk", Mathematical Finance 9(3).
Kupiec (1995), "Techniques for Verifying the Accuracy of Risk Measurement Models", Journal of Derivatives 3(2).
McNeil, Frey and Embrechts, Quantitative Risk Management (2015), ch. 2 and 9.
Basel Committee (2019), Minimum Capital Requirements for Market Risk - the move from VaR to ES.

Related Finance & risk

Black–Scholes option pricing · Bond price, duration & convexity · Merton distance to default · EMI / home loan planner · CAPM / beta estimation · Multi-factor model · Portfolio optimization · Performance & drawdown · Time value of money & NPV/IRR · Equity valuation (DDM, FCF, residual income) · Cost of capital (WACC) · Binomial option pricing · Option strategies (payoff & profit) · Forwards, futures & swaps · Credit risk & Basel capital · Extreme value theory (tail risk)

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Value at Risk & Expected Shortfall

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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