A risk measure summarises a chosen aspect of financial uncertainty. Compare expected loss, VaR and expected shortfall on one distribution, with explicit units and tail conventions.
Risk measure
A risk measure is a numerical rule for summarising a chosen aspect of financial uncertainty: the spread of portfolio results, a loss percentile, or the severity of an adverse scenario. It helps compare positions and set constraints, but its meaning depends on the question the rule answers. A small average loss can coexist with a large rare loss.
For measures built from a loss distribution, the notation ρ(L) means “apply the chosen summary rule to the uncertain loss L”. Here, positive L means a loss and negative L means a gain. The distribution describes possible outcomes and their probabilities; the risk measure compresses selected information into a number.
One distribution, three different answers
Suppose a fictional portfolio has the following stipulated one-period loss distribution, in USD thousands. These probabilities are assumptions for the example, rather than estimates from market data.
Loss
Probability
0
90%
10
9%
100
1%
Three summaries of this same distribution answer different questions:
What is the average loss in the worst 5% of probability mass?
28
The expected loss is
E[L]=0.90(0)+0.09(10)+0.01(100)=1.9.
It is an average, not a predicted outcome that must occur. In fact, neither 1.9 nor 28 is a possible realised loss in this model.
For VaR, use the lower inverse-CDF convention:
VaRα(L)=inf{ℓ:P(L≤ℓ)≥α}.
The cumulative probability is 90% at loss zero and 99% at loss 10. Thus 95% VaR is 10. Only 1% of probability lies strictly above that threshold: with discrete outcomes, “95% VaR” does not require an exceedance probability of exactly 5%.
Expected shortfall looks further into the tail. Its quantile-integral definition allocates exactly the worst 5%: all 1% at loss 100 and 4 percentage points of the 9% mass at loss 10. Therefore,
ES0.95(L)=0.050.04(10)+0.01(100)=28.
Averaging only losses strictly above VaR would give 100, but would select just 1% of probability. Including the entire atom at 10 would select 10%. Neither is the required 5% tail. The Acerbi–Tasche paper develops the boundary correction needed for such discrete distributions.
Synthetic one-period losses in USD thousands: zero with probability 90%, 10 with 9%, and 100 with 1%. Expected loss is 1.9, 95% VaR is 10, 95% ES is 28, and the largest stipulated loss is 100. ES uses 4% at loss 10 plus 1% at loss 100.
Original finite-model comparison, in USD thousands. The largest stipulated loss is a bound within this assumed model; the mean, VaR and ES are different summaries of it.
Choose the question before the number
Expected loss describes the average cost across outcomes. It does not show how concentrated that cost is in rare outcomes. VaR locates a quantile but does not describe severity beyond it. ES averages a specified tail, so it responds when the rare largest loss becomes more severe even if VaR stays unchanged.
Other measures retain different information. Standard deviation measures spread around the mean, counting deviations on both sides; it has the same units as the underlying loss. Variance has squared units. Maximum drawdown depends on the order of portfolio values along a path, information that a terminal loss distribution alone does not retain. A worst-scenario loss depends on which stress scenarios were selected and need not come with probabilities.
For the weighted-average and spread calculations behind these distinctions, Expectation, Variance and Covariance develops both from probability weights. The public page is a preview; the full lesson is available with a free signed-in account.
What a reported measure must identify
A usable number states the portfolio scope, valuation date, currency or return basis, loss sign, time horizon, method and assumptions. Quantile and tail measures also need a confidence level and a convention for probability at the cutoff. Comparisons require a consistent basis: a one-period dollar ES and an annual percentage volatility answer different questions.
The risk model supplies the assumed or estimated losses; the measure summarises them; a risk limit sets an authorised threshold for that particular metric. Changing the summary cannot repair omitted exposures, unreliable probabilities or a poor valuation model. Stress testing can expose vulnerabilities outside the fitted distribution, although its own scenario selection also needs scrutiny.
In the example, 100 is the largest loss allowed by the stipulated finite model. It is not established as the portfolio's real-world maximum. VaR of 10 and ES of 28 establish neither a maximum possible loss nor an amount that will be lost next period.
Sources
Alexander J. McNeil, Rüdiger Frey and Paul Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, revised edition (Princeton University Press, 2015), §2.3.1, pp. 61–64, on alternative measurement approaches and model limitations; §2.3.2, Definition 2.8, p. 64, on loss-positive VaR; §2.3.4, pp. 69–70, especially Definition 2.12, on variance and quantile-integral ES. The example and figure here are original.
Carlo Acerbi and Dirk Tasche, On the coherence of Expected Shortfall, revised manuscript dated April 19, 2002, Definitions 2.1 and 2.6 and Proposition 3.2. The paper's lower-tail profit-and-loss convention is translated here into an upper-tail positive-loss convention.
