A loss distribution assigns probabilities to the possible losses of a position, portfolio or business over a stated horizon. It is the probability law of a loss random variable, usually written . It lets a risk manager ask both how often a loss threshold might be crossed and how severe the outcomes beyond that threshold could be. An average loss alone cannot answer either question.
The sign convention matters: here positive means a loss; negative means a gain. If is the period’s net profit and loss, then . For a fixed portfolio with no external cash flows or distributions, that is . Once cash flows, costs or trading enter, define the P&L consistently before reversing its sign.
Three outcomes, more than one risk question
Consider this stipulated one-period model. All amounts are in USD thousands, and the probabilities are assumptions, not estimates from market data.
| Loss | Economic outcome | Probability |
|---|---|---|
| −10 | Gain USD 10,000 | 80% |
| +20 | Lose USD 20,000 | 15% |
| +100 | Lose USD 100,000 | 5% |
This table is the entire probability mass function: its probabilities sum to one, and it assigns zero probability to other outcomes. Zero probability outside this model’s support is an assumption about the model, not a guarantee about real losses.
The chance of a positive loss is . Yet the expected loss is zero:
The frequent small gain offsets the less frequent losses in the probability-weighted average. No individual outcome breaks even. A zero mean does not remove the USD 100,000 loss scenario.
The variance is in squared USD-thousand units:
Variance measures dispersion on both sides of the mean. To ask about an adverse threshold, look at the right-hand side of the loss distribution.
The same law shown as probability masses and cumulative probability. Filled points include each jump; hollow points exclude the lower value. The 5% strictly beyond 20 is all at 100.
Read the boundary before quoting VaR
Using the lower inverse-CDF convention, value at risk at confidence level is
At 95%, the first loss threshold whose cumulative probability reaches 0.95 is 20, so thousand USD. Exactly 5% of the model’s probability lies strictly above it. VaR is a quantile, not a maximum possible loss: the 100 outcome remains possible.
With a point mass, “at least the threshold” differs from “strictly above it.” Here , whereas . A software percentile that interpolates between outcomes can also differ from the stated inverse-CDF convention. Name the convention when discrete scenarios or repeated losses matter.
Expected shortfall averages the worst probability mass. For an integrable loss, use the quantile-integral definition:
The worst 5% here consists entirely of the 100 loss, giving thousand USD. Averaging all outcomes at or above 20 would instead give 40 thousand USD; that uses 20% of the probability and is not 95% ES. At 90% confidence, the worst 10% combines the 5% at 100 with only 5% of the mass at 20, giving ES of 60 thousand USD. Selecting the required tail mass may mean splitting a probability atom. Acerbi and Tasche’s original paper on ES explains why discontinuities make these definitions consequential.
What must be specified before the curve is useful
A loss distribution needs a portfolio scope, horizon, currency, P&L definition and probability interpretation. A one-day marked trading loss and an annual credit loss are different random variables. A forecast conditional on today’s market state can also differ from a distribution pooled across historical regimes.
In portfolio modelling, two ingredients produce the law: a model of risk-factor changes and a rule that converts those changes into portfolio losses. Mapping Risks: The Loss Operator and Risk Factor Representation develops that bridge through stock and option examples. The full lesson requires membership; a public preview is available.
Historical revaluation, analytical assumptions and Monte Carlo simulation are ways to construct or estimate the distribution. More simulated draws can reduce sampling noise under the chosen model; they do not establish that its probabilities, valuation rules or tail behaviour are correct. Likewise, individual positions’ loss distributions do not determine portfolio losses without their joint distribution.
A risk measure compresses this distribution into a number. Keep the underlying scenarios and assumptions available: two summaries of the same law answer different questions, and a precise summary of a misspecified law can still mislead.
Sources
- Alexander J. McNeil, Rüdiger Frey and Paul Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, revised edition, Princeton University Press, 2015, §2.2.3, pp. 58–61 (constructing loss distributions), §2.3.2, pp. 64–66 (loss CDF and VaR), and §2.3.4, pp. 69–70 (variance and ES). The three-outcome model and illustration here are original.
- Carlo Acerbi and Dirk Tasche, “On the Coherence of Expected Shortfall”, Journal of Banking & Finance 26 (2002), pp. 1487–1503; §§2–3 of the author preprint. The paper uses a P&L variable and tail-probability notation; this article uses positive losses and confidence levels.