Expected Shortfall and Tail Risk: ES, EVT and Stressed VaR

VaR identifies a loss threshold. Expected Shortfall (ES) asks what losses look like after that threshold has been crossed.

That distinction matters in FX because devaluations, interventions, funding stress and abrupt correlation shifts can create losses far beyond a normal-market VaR estimate. A practical tail-risk framework therefore combines threshold measures such as VaR with tail measures such as ES, dedicated stress calibration and, where appropriate, Extreme-Value Theory (EVT).

No tail method should be treated as a guarantee of the worst possible outcome. Its value lies in making severe-loss assumptions explicit and forcing governance to confront the part of the loss distribution that ordinary VaR does not describe.

Expected Shortfall is the expected loss conditional on losses reaching or exceeding the VaR threshold.

Where VaR asks “what threshold is exceeded with a stated probability?”, ES asks “when that threshold is exceeded, how severe is the loss on average?”

Under a normal-loss assumption, ES can be expressed using the normal density function:

where $z_\alpha=\Phi^{-1}(\alpha)$.

The formula is useful, but the same caution that applies to parametric VaR applies here: a normal-based ES remains a normal-model estimate.

The most useful difference between VaR and ES is what happens beyond the boundary.

Two portfolios can have the same 99% VaR but very different loss distributions in the worst 1% of outcomes. ES makes that difference visible.

ES is also treated as a coherent risk measure because it respects sub-additivity. That makes it attractive for portfolio risk and capital allocation where diversification effects matter.

Illustrative Example / Hypothetical Scenario

Illustrative Example / Hypothetical Scenario.

Assume daily portfolio volatility of 1.2% and a normal loss distribution.

At 99% confidence:

Using the normal-density multiplier for 99% ES:

The example illustrates the conceptual point: the expected loss in the tail is greater than the threshold defining entry into the tail.

Extreme-Value Theory focuses explicitly on the tail rather than modelling the entire return distribution in the same way.

A common approach fits a Generalised Pareto Distribution to observations beyond a selected threshold:

where:

• $\xi$ is the shape parameter;
• $\beta$ is the scale parameter;
• the model is fitted only to sufficiently extreme observations.

A larger positive shape parameter corresponds to a heavier tail.

FX markets can experience discontinuous moves associated with devaluations, capital controls, interventions and sudden changes in liquidity. Those observations may sit far outside the assumptions implied by a normal model.

EVT can help answer questions such as:

• how quickly does loss severity grow beyond the ordinary VaR threshold?
• does the portfolio contain exposures whose tail behaviour is not visible in normal-market calibration?
• do emerging-market positions require a separate tail overlay?
• are model results stable when the threshold used for fitting the tail is changed?

EVT is therefore best treated as a tail lens, not as a reason to abandon all other risk measures.

Stressed VaR applies VaR methodology using parameters or observations drawn from a historical stress period rather than the current regime.

Conceptually:

Its value is the contrast between a normal-regime estimate and a stress-calibrated estimate.

Illustrative Example / Hypothetical Scenario

Illustrative Example / Hypothetical Scenario.

If daily volatility rises from 1.2% to 2.8%, a 99% parametric estimate moves from approximately 2.8% of notional to approximately 6.5% before any additional effect from changing correlations or non-linear positions is considered.

That contrast can be more informative for governance than the absolute number because it shows how dependent the operating measure is on the market regime used for calibration.

Stressed VaR has important historical regulatory significance and remains useful as a stress-monitoring concept. For regulatory capital, the applicable treatment must be determined from the current rules in the relevant jurisdiction.

FRTB uses Expected Shortfall under the Internal Models Approach, while supervisory backtesting and internal risk monitoring can continue to use VaR-type measures. The applicable treatment should be checked against the current rules in the relevant jurisdiction because implementation details, liquidity horizons and transitional provisions can differ.

A useful tail-risk stack can include:

1. operating VaR — frequent threshold monitoring;
2. Expected Shortfall — severity beyond the threshold;
3. historical stress calibration — behaviour under known stress periods;
4. EVT or another deep-tail technique — sensitivity to rare extreme observations;
5. scenario analysis — specific market narratives not adequately represented in statistical history;
6. reverse stress testing — identification of conditions that would breach appetite or threaten viability.

The measures do not need to agree. Material divergence is itself information.

If normal VaR remains stable while ES, EVT or stress measures rise sharply, investigate:

• changing volatility;
• changing correlation structure;
• concentration in emerging-market or illiquid currencies;
• non-linear option exposures;
• short or unrepresentative calibration windows;
• data treatment that removes genuine extreme observations;
• model assumptions that force a smooth distribution onto discontinuous markets.

For tail-risk measures, retain:

• threshold selection;
• calibration window;
• stress-period rationale;
• EVT threshold and fitted parameters where used;
• comparison with ordinary VaR;
• material model divergences;
• governance decisions arising from those divergences;
• model limitations;
• current regulatory source records where the measure is used for regulatory purposes.