Value-at-Risk and Expected Shortfall

Interactive exploration of VaR, ES, coherent risk measures, time-horizon scaling, and portfolio risk decomposition

Value-at-Risk (VaR) and Expected Shortfall (ES) are summary risk measures that compress the total risk of a portfolio into a single number (see Hull 2023, chap. 11; Christoffersen 2012, chap. 1). VaR was pioneered by JPMorgan in the early 1990s and rapidly became an industry standard. ES addresses key shortcomings of VaR and is now preferred by regulators for market risk capital calculations.

1. VaR and ES from the normal distribution

When portfolio returns are normally distributed with mean \(\mu\) and standard deviation \(\sigma\), VaR and ES have closed-form expressions:

\[ \text{VaR}_\alpha = -\mu - \sigma \, \Phi^{-1}(1-\alpha) \]

\[ \text{ES}_\alpha = -\mu + \sigma \, \frac{\phi(\Phi^{-1}(1-\alpha))}{1 - \alpha} \]

where \(\alpha\) is the confidence level, \(1-\alpha\) is the tail probability, \(\Phi^{-1}\) is the inverse standard normal CDF, and \(\phi\) is the standard normal PDF. Note that \(\Phi^{-1}(1-\alpha)\) is negative for \(\alpha > 0.5\), so both VaR and ES are positive.

Tip

How to experiment

Increase the confidence level to push VaR further into the left tail of the return distribution. Under the normal distribution assumed here, the ES/VaR ratio actually decreases toward 1 at higher confidence levels (with zero mean). This would not necessarily hold for fat-tailed distributions, where the ratio can increase with confidence. Set the mean to zero (the standard assumption for short horizons) to see VaR become proportional to \(\sigma\).

2. VaR’s blind spot — tail risk explorer

VaR tells us the threshold that losses will not exceed with a given probability, but says nothing about what happens beyond that threshold. Two portfolios can have the same VaR but dramatically different tail risk. ES captures this distinction.

Tip

How to experiment

Increase the catastrophic loss multiplier or its probability. Watch how Portfolio B’s ES grows dramatically while its VaR remains identical to Portfolio A’s. This is the fundamental weakness that allows VaR to be “gamed” by traders hiding tail risk.

3. Subadditivity violation

A risk measure is subadditive if \(\rho(A + B) \leq \rho(A) + \rho(B)\) — combining portfolios should not increase measured risk. VaR can violate this, perversely suggesting that diversification increases risk. ES always satisfies subadditivity (see Artzner et al. 1999).

Tip

How to experiment

Adjust the default probability and confidence level to find combinations where VaR(A+B) > VaR(A) + VaR(B). When the individual VaR captures only the “no default” outcome but the combined portfolio’s tail includes single-default events, the violation appears. Compare with ES — it always recognizes diversification benefits.

4. Spectral risk measures

A risk measure can be characterized by the weights it assigns to quantiles of the return distribution. Expressed in terms of return quantiles (where \(q = 0\) is the worst outcome and \(q = 1\) the best), a risk measure is coherent if and only if its weight function is non-increasing (see Artzner et al. 1999).

  • VaR places 100% weight on a single quantile (the \((1-\alpha)\)th percentile) — a spike then drop to zero, hence not coherent.
  • ES gives equal weight to all quantiles below \(1-\alpha\) (the left tail) — non-increasing, hence coherent.
  • Exponential spectral measures assign exponentially decreasing weight from the worst outcomes toward the centre, reflecting higher risk aversion to extreme losses.
Tip

How to experiment

Adjust the confidence level to shift where VaR and ES begin. Change the risk aversion parameter \(\gamma\) to see how the exponential spectral measure concentrates weight on the worst outcomes. Lower \(\gamma\) gives more weight to the extreme tail; higher \(\gamma\) produces a flatter curve.

5. Time horizon scaling and autocorrelation

A common approximation scales risk measures from one day to \(T\) days using the square-root-of-time rule:

\[T\text{-day VaR} = 1\text{-day VaR} \times \sqrt{T}\]

This is exact when daily changes are i.i.d. normal with zero mean. When there is first-order autocorrelation \(\rho\) in daily changes, the \(T\)-day standard deviation becomes:

\[\sigma_T = \sigma_1 \sqrt{T + 2(T-1)\rho + 2(T-2)\rho^2 + \cdots + 2\rho^{T-1}}\]

Positive autocorrelation makes the square-root rule underestimate multi-day risk.

Tip

How to experiment

  1. Start with \(\rho = 0\) to see the pure square-root-of-time scaling.
  2. Increase \(\rho\) to 0.05–0.15 to see how autocorrelation amplifies multi-day risk.
  3. Push the time horizon to 250 days (one year) to see the cumulative effect.

6. Confidence level conversion

Under the normality assumption, VaR and ES at one confidence level can be converted to another without re-estimating the model:

\[\text{VaR}(\alpha^*) = \text{VaR}(\alpha) \times \frac{\Phi^{-1}(\alpha^*)}{\Phi^{-1}(\alpha)}\]

\[\text{ES}(\alpha^*) = \text{ES}(\alpha) \times \frac{(1-\alpha) \, e^{-(Y^{*2} - Y^2)/2}}{1-\alpha^*}\]

where \(Y = \Phi^{-1}(\alpha)\) and \(Y^* = \Phi^{-1}(\alpha^*)\).

Tip

How to experiment

Set a known VaR at a source confidence level (e.g., 95%) and see how it converts to other levels. Notice how VaR grows roughly linearly with the z-score, while ES grows faster because the conditional tail expectation becomes more extreme.

7. ES from discrete distributions

For discrete return distributions, VaR is determined by the \((1-\alpha)\) quantile of the cumulative distribution of returns. ES is the expected loss conditional on being in the left tail below \(-\text{VaR}\). Reading these from a step-function CDF requires careful handling of the probability mass at the VaR boundary.

Tip

How to experiment

Switch between preset scenarios to see how different loss structures affect VaR and ES. Adjust the confidence level to move the VaR threshold through different regions of the distribution. Watch the “Tail decomposition” tab to see exactly how ES is computed from the discrete outcomes.

References

Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, and David Heath. 1999. “Coherent Measures of Risk.” Mathematical Finance 9 (3): 203–28.
Christoffersen, Peter F. 2012. Elements of Financial Risk Management. 2nd ed. Academic Press.
Hull, John. 2023. Risk Management and Financial Institutions. 6th ed. John Wiley & Sons.