Value at Risk (VaR) estimates a loss threshold at a chosen probability level, while Expected Shortfall (ES) describes the average loss in the tail beyond that threshold under a stated definition. The two measures answer related but different questions about downside risk.
Start with a loss convention
Let L represent loss as a positive number. At confidence level α, VaRα is an α-quantile of the loss distribution. In a continuous distribution, ESα can be described as the expected loss conditional on being at or beyond the VaR threshold.
Sign conventions differ across systems. Some report negative P&L rather than positive loss. State the convention before comparing numbers.
VaR is a threshold, not the maximum possible loss
A one-day 95% VaR of 100,000 does not mean the portfolio cannot lose more than 100,000. It means the model places the 95th percentile of the specified one-day loss distribution at that threshold.
VaR says little by itself about how severe losses can become after the threshold is crossed.
Expected Shortfall looks farther into the tail
ES asks about the average loss in the tail selected by the confidence level. Two portfolios can have similar VaR but different ES if one has much more severe outcomes beyond the quantile.
This is why ES can provide information about tail severity that a quantile threshold alone does not show.
A simple discrete example
Suppose a simplified set of 100 equally likely one-period losses has 95 outcomes at or below 10, and the worst five losses are 12, 14, 18, 26 and 30. Under an empirical convention that takes those worst five observations as the 5% tail, their average is 20.
The precise empirical VaR value can depend on the sample-quantile convention used at a boundary. The example is intended to show the conceptual distinction: VaR identifies a tail threshold; ES summarizes tail severity.
Horizon and confidence level are part of the metric
A daily 95% VaR is not directly comparable with a ten-day 99% VaR. Changing horizon or confidence changes the question, and scaling risk across horizons can require assumptions about dependence and volatility.
The distribution method matters
Historical simulation uses observed or resampled changes; parametric approaches impose a fitted distribution; Monte Carlo approaches simulate from an assumed process. Each can produce different tail estimates from the same portfolio because the assumptions differ.
Monte Carlo Simulation covers simulation design and model risk.
Expected Shortfall has aggregation advantages under standard definitions
In risk-measure theory, standard Expected Shortfall satisfies coherence properties such as subadditivity under its usual mathematical definition, while VaR is not subadditive for every possible loss distribution. That does not make ES assumption-free or automatically accurate.
An ES estimate can still be unstable when tail observations are scarce or the fitted distribution understates extreme losses.
Backtesting tail measures is difficult
VaR can be checked partly by counting how often realized losses exceed the predicted quantile, but calibration also depends on independence and model specification. ES focuses on rarer tail observations, so direct evaluation can require more data and specialized tests.
Neither metric replaces stress testing
VaR and ES summarize a modeled distribution. Scenario & Stress Testing can examine deliberately adverse states that are poorly represented in the historical sample or difficult to assign reliable probabilities.
A robust risk process uses distributional measures and scenario analysis for different purposes.
Use the measure with its assumptions attached
Record the horizon, confidence level, loss sign, data window, valuation method, distribution or simulation method and treatment of changing volatility/correlation.
VaR answers “where is the selected loss threshold?” while Expected Shortfall asks “how severe is the selected tail on average?” Neither number is a guarantee about the largest future loss.