Value at Risk vs Expected Shortfall: Measure the Loss Beyond the Threshold

Value at Risk vs Expected Shortfall: Measure the Loss Beyond the Threshold

Value at Risk gives a portfolio a loss threshold for a chosen time horizon and confidence level. Expected Shortfall asks the next question: if that threshold is breached, how large is the average loss in the tail? Used together, they can organise risk discussions—but neither predicts the worst possible outcome.

বাংলায় পড়ুন: Value at Risk বনাম Expected Shortfall: Threshold-এর পরের Loss মাপুন

Learning objective: By the end of this tutorial, you should be able to interpret VaR and Expected Shortfall, calculate a simple historical example, compare common estimation methods and recognise when stress testing is still required.

1. Start with the question each measure answers

The Basel Framework defines Value at Risk (VaR) as a measure of the worst expected loss from market movements over a given horizon and at a stated confidence level. It defines Expected Shortfall (ES) as the average of potential losses beyond VaR at that confidence level.

In plain language:

• VaR is a threshold. A one-day 95% VaR of £2,600 says the model estimates that losses should not exceed £2,600 on about 95% of days under the assumptions and data used.
• Expected Shortfall is a tail average. A one-day 95% ES of £4,100 says the average loss within the model's worst 5% of days is £4,100.

The remaining 5% is not capped at £4,100. Individual losses inside that tail can be much larger.

2. Always state three inputs

A VaR or ES number is incomplete without:

1. Portfolio value and currency — for example, a £100,000 portfolio.
2. Horizon — one trading day, ten days or another period.
3. Confidence level — commonly 95%, 97.5% or 99%, depending on the purpose.

Changing any one of these can materially change the result. A one-day 95% VaR cannot be compared directly with a ten-day 99% VaR.

Regulatory practice also differs from a trader's dashboard. The Basel internal-model framework uses a 97.5% one-tailed Expected Shortfall with liquidity-horizon adjustments for market-risk capital. That does not make 97.5% a universal setting for every portfolio or decision.

3. A worked historical example

Assume a hypothetical portfolio is worth £100,000. You collect 100 comparable daily profit-and-loss observations, rank them from the largest loss to the largest gain and focus on the worst five losses:

Rank among 100 days	Loss	Portfolio return
Worst	£6,200	−6.20%
2nd worst	£4,800	−4.80%
3rd worst	£3,800	−3.80%
4th worst	£3,100	−3.10%
5th worst	£2,600	−2.60%

Historical 95% VaR

With this simple convention, the 95% VaR is the boundary of the worst 5% of observations:

95% one-day VaR = £2,600, or 2.60% of the portfolio.

Quantile conventions differ across software and small samples, so another implementation may interpolate between observations. Record the convention rather than pretending the estimate is exact.

Historical 95% Expected Shortfall

Average the five losses in the worst 5%:

ES = (£6,200 + £4,800 + £3,800 + £3,100 + £2,600) ÷ 5 = £4,100.

Therefore:

95% one-day Expected Shortfall = £4,100, or 4.10%.

VaR identifies the threshold. ES shows that the average bad-tail day was £1,500 worse than that threshold in this sample.

4. What VaR can hide

Consider two portfolios with the same £2,600 VaR:

• Portfolio A has tail losses clustered around £2,700–£3,200.
• Portfolio B has several similar losses but also one £12,000 loss.

Their VaR can be equal even though Portfolio B has more severe tail damage. Expected Shortfall is designed to respond to that difference because it averages tail outcomes.

This is one reason the Basel market-risk framework moved its internal-model capital measure from VaR towards Expected Shortfall under stress. The BIS executive summary also notes the framework's treatment of market illiquidity.

5. Three common estimation methods

Historical simulation

Apply the portfolio to observed historical market moves, then rank the resulting profit and loss.

Strengths: transparent, captures some non-normal features and is easy to explain.

Weaknesses: assumes the chosen history is representative; rare events absent from the sample remain invisible; old regimes may dominate or recent shocks may be underrepresented.

Parametric estimation

Estimate risk from a statistical distribution using volatility and correlations. A basic variance-covariance model often assumes returns are normally distributed and positions behave approximately linearly.

Strengths: fast and compact for simple portfolios.

Weaknesses: normal distributions can understate fat tails; options and other nonlinear positions require more careful treatment; correlations may change during stress.

Monte Carlo simulation

Simulate many possible risk-factor paths using an assumed model, revalue the portfolio and estimate the loss distribution.

Strengths: flexible and useful for nonlinear portfolios.

Weaknesses: the output is only as credible as the model, parameters, scenarios and valuation engine. More simulations reduce sampling noise but do not fix a wrong model.

6. Why the horizon is difficult

Risk is sometimes scaled from one day to several days using the square-root-of-time rule. That approximation assumes independent, similarly distributed returns and stable volatility. Markets often violate those assumptions during stress.

It also ignores execution. A portfolio may be theoretically sellable at the screen price but difficult to exit without moving the market. Liquidity horizons, gaps and trading halts can make realised losses larger than a simple multi-day scaling suggests.

For concentrated or illiquid positions, revalue under explicit multi-day and liquidity scenarios rather than relying only on a scaling shortcut.

7. Backtest the threshold

A VaR model should be compared with realised outcomes. If a one-day 95% VaR is calibrated reasonably, losses should exceed the threshold on roughly 5% of comparable days over a sufficiently large sample.

Too many breaches may mean volatility is understated, correlations have changed, nonlinear risk is missing or the data window no longer represents the market. Too few breaches do not automatically prove excellence; the model may be excessively conservative or the sample may simply be calm.

Expected Shortfall needs a separate review: examine whether the size of tail losses is being captured, not only the number of VaR breaches.

8. Add stress tests and scenarios

VaR and ES are statistical summaries of a modelled distribution. They cannot tell you everything about a specific event. Add scenarios such as:

• a 10% overnight equity gap;
• a simultaneous equity fall and volatility spike;
• a sharp currency move against unhedged holdings;
• oil or gold moving while market liquidity deteriorates;
• correlations moving towards one during a sell-off;
• an exchange halt or inability to close a leveraged position.

An ECB risk-management speech described statistical risk models and comprehensive stress testing as complementary parts of the toolbox. That remains a useful principle for an individual trader: the statistic and the scenario answer different questions.

9. Common mistakes

Reading VaR as a maximum loss

It is not. VaR marks a quantile. Losses beyond it can be much larger.

Quoting a number without horizon or confidence

“VaR is £2,600” is incomplete. State “one-day 95% VaR of £2,600 on a £100,000 portfolio”.

Using too little data

A 99% historical estimate from only 100 observations is effectively driven by one or two days. Tail estimates need enough relevant observations, and even large samples may omit the next type of crisis.

Assuming correlation is stable

Diversification observed in calm periods may weaken during stress. Recalculate risk under higher-correlation scenarios.

Ignoring changing positions

Yesterday's VaR may describe yesterday's portfolio. New size, leverage, options exposure or currency risk can make it stale.

Treating model precision as certainty

£2,603.47 can look scientific while the assumptions are fragile. Use sensible rounding and report limitations.

10. Practical checklist

• Define portfolio value, base currency, horizon and confidence level.
• Choose historical, parametric or Monte Carlo estimation and document the assumptions.
• Include fees, financing and nonlinear revaluation where material.
• Calculate both VaR and Expected Shortfall from the same loss distribution.
• Check how results change under different data windows and confidence levels.
• Backtest VaR breaches and inspect the severity of tail losses.
• Add explicit gap, volatility, correlation and liquidity stress tests.
• Recalculate after material changes in position size or portfolio composition.
• Use the result to inform risk limits, not to promise a safe maximum loss.

The core lesson

VaR tells you where the modelled tail begins. Expected Shortfall tells you the average loss after you enter that tail. The combination is more informative than either number alone, but the result still depends on history, distribution assumptions, liquidity and portfolio structure.

Use VaR and ES as dashboard instruments. Use stress tests to ask what the dashboard has never seen.

Educational risk disclosure

This tutorial was prepared with AI-assisted research and writing under The Money Man editorial workflow. The portfolio and losses are hypothetical and are used only to explain the calculations. This is educational information, not personalised investment advice, a regulatory capital calculation or a recommendation to trade. Statistical risk estimates can fail during gaps, regime changes, illiquidity and extreme events; leveraged losses can exceed the amount initially deposited.