A probability distribution assigns likelihoods to financial outcomes, helping distinguish expected loss, loss thresholds, and uncertainty in risk models.
A probability distribution describes the possible values of an uncertain quantity and how likely they are. In finance, that quantity might be a portfolio’s one-day loss, a project’s annual cash flow, or the number of borrowers who default within a year.
A distribution answers more than “What is the expected result?” It can also describe the probability of losing money or exceeding a loss threshold. Those answers depend on the data and assumptions used to construct the distribution.
| Representation | What it measures | Financial interpretation |
|---|---|---|
| Probability mass function, or PMF | Probability assigned to each discrete outcome | The chance of exactly three defaults |
| Probability density function, or PDF | Density whose area over an interval gives its probability | The chance of a modeled return falling between two values |
| Cumulative distribution function, or CDF | Probability that the outcome is at or below a threshold | The chance that loss does not exceed $10,000 |
For a discrete distribution, all outcome probabilities must be nonnegative and sum to one. For a continuous model described by a density, the density is nonnegative and its total area equals one. Density at a point is not the probability of that exact outcome. NIST: What Is a Probability Distribution.
For example, suppose a toy payoff is uniformly distributed between $0 and $0.50. Its density is 2 per dollar, but the probability of a payoff between $0.10 and $0.20 is only 20%: density 2 multiplied by interval width $0.10. A density can exceed one without implying a probability above 100%. In this continuous model, any one exact payoff has probability zero; a recorded amount rounded to cents represents an interval instead.
Let L denote a loss, with larger positive values meaning larger losses. Its cumulative distribution and probability of exceeding a threshold x are:
The CDF works for discrete and continuous distributions. The second expression is also called the survival function or complementary CDF. NIST: Related Distribution Functions.
Suppose an analyst uses this simplified distribution for next year’s loss on a credit exposure. The probabilities are illustrative assumptions, not estimates for an actual borrower.
| Annual loss | Assumed probability | Contribution to expected loss |
|---|---|---|
| $0 | 70% | $0 |
| $10,000 | 25% | $2,500 |
| $50,000 | 5% | $2,500 |
| Total | 100% | $5,000 |
The expected loss is the sum of each loss multiplied by its probability:
These numbers answer different questions:
The last two answers differ because the distribution assigns 25% probability to exactly $10,000. Threshold wording matters for discrete outcomes.
Here, the CDF at $10,000 is 95%. Under the convention that selects the smallest loss with cumulative probability of at least 95%, the model’s one-year 95% value at risk is $10,000. It is not the maximum loss: the model still includes $50,000 outcomes.
Start with what the variable can represent. A default count cannot be negative or fractional. A model of a strictly positive price has different constraints from a model of changes in that price.
A binomial distribution can model a default count among a fixed number of borrowers when default indicators are independent and all have the same default probability. Shared economic exposures or different borrower probabilities break that simple specification. NIST: Binomial Distribution, NIST: Statistical Glossary.
A normal distribution is symmetric; a lognormal distribution takes strictly positive values in its usual unshifted form. Neither label establishes that it fits a particular financial series. A price model also does not automatically describe returns, losses, or a derivative’s payoff. NIST: Normal Distribution, NIST: Lognormal Distribution.
An empirical distribution assigns weights to observed outcomes rather than imposing a named curve. It preserves the selected sample’s information, but an event absent from the sample receives no weight unless the modeling process explicitly adds it. Monte Carlo simulation samples a chosen model; simulation does not determine which distribution is appropriate.
Ignoring scope and horizon. A one-day return distribution cannot simply be relabeled as an annual loss distribution. Currency, position size, fees, and the treatment of gains must also be clear.
Treating averages as complete risk measures. Means and standard deviations do not generally identify a distribution. Some heavy-tailed distributions do not even have finite means or variances.
Ignoring dependence. Separate distributions for two borrowers do not determine how often they default together. Portfolio loss requires joint assumptions, not just individual probabilities.
Equating a good fit with reliable forecasts. Estimates can change with the sample period, unusual observations, or changing economic conditions. Compare alternative assumptions and disclose uncertainty rather than presenting probabilities as settled facts.
The examples illustrate financial modeling. They are not forecasts, capital requirements, or personalized investment advice.