# Risk Distribution

## Definition

A Risk Distribution is the core computational tool (building block) of Quantitative Risk Management. Mathematically a risk distribution is the Probability Distribution of a Random Variable.

### Risk Distribution Categories

One can classify risk distributions along a number of interesting characteristics and these tend to be very important in determining fitness of use for particular risk applications

• Parametric or non-parametric
• Obtained as an Analytic Models of via Monte-Carlo Simulation
• Discrete, continuous or mixed support. The nature of the realized risk usually dictates which type is meaningful to use
• Finite, infinite one sided or infinite two sided support. Similarly as above
• Finite of infinite moments / moment generating function. Related to the sub-exponential, heavy-tails and fat-tails properties. NB: It may not always be possible to assert whether an actual phenomenon has infinite moments
• Symmetric or asymmetric.

## Usage

• The most direct application of risk distributions in risk management is the explicit modelling of risk as realisations from a distribution
• The concept forms the basis for the constructions of a variety of risk metrics.
• It is rare that a risk phenomenon is described exclusively by a single parametric distribution. Hence many usage examples are cases where a distribution is used as part of an overall modelling framework (which is multi-dimensional and/or involves time-dependent processes)
• Some distributions (e.g, the uniform and the normal) are used extensively as building blocks to create more realistic distributions
• Some distributions appear primarily in the context of statistical testing that is used in risk management applications
• While in the majority of applications distributions are used in conjuction with historical data (models of risk realisations), in financial applications involving the pricing of risk they may also be used as the implied distributions (of potential future risk events)

### Usage by Risk Type

The quantitative risk management of all major risk types involves risk distributions of various types

### Risk Distribution Catalog

The catalog aims to be a comprehensive list of one dimensional (univariate), parametric risk distributions as they are used in risk management applications. More general (non-parametric) approaches are out of scope.

The underlying probability distribution functions are generally well documented in Wikipedia therefore the table below links to individual entries for further information on the functional form and other mathematical properties of each distribution. The main purpose of the catalog is establish an overview of aspects that are interesting in risk context

Some distributions may coincide for certain parameters or be special cases

The references are not meant to be exhaustive nor setting academic priority, they are merely easily available examples of usage.

Name Discreteness Boundedness Finite Moments Symmetry Comments Market Risk Credit Risk Operational / Insurance Risk
Beta Continuous Finite All No (unless a=b) Widely used as a model for a fractional (percent) outcome Loss-given-default model
Bernoulli Discrete Finite All No (unless p=0.5) Widely used as the archetype of coin toss experiment (n=2) Tree based pricing of options Default / No-default model
Binomial Discrete Finite All No (unless p=0.5) Counts number of events in a sequence of n independent experiments Value-at-risk exceptions Simple model of portfolio default count
Cauchy Continuous Infinite None No (for specific parameters) Example of distribution with no defined expectation (therefore difficult to use in risk management)
Chi-squared Continuous Infinite All No Sum of squared Normal variables Used in credit scorecard validation
Dirac Delta Continuous Infinite All Yes A degenate distribution serving as a link between the continuous and discrete categories
Exponential Continuous Infinite All No Archetype of event arrival time (e.g for a Poisson process) As component in market jump processes As credit event arrival time model As operational risk event arrival time
F Distribution Continuous Infinite 3 No Arrises in statistical testing of variance (F-Test) In Model Validation
Gamma Continuous Infinite All No Generalization of exponential, chi-squared etc distributions As operational risk event severity model
Gumbel Continuous Infinite All No The distribution of the extreme realizations when sampling from various distributions (Extreme Value Theory)
Inverse Gaussian Continuous Infinite All No First passage time of Brownian Motion As operational risk event severity model
Log-Normal Continuous Infinite All No The product of many independent random variables, each of which is positive Archetypal model for asset prices Used in threshold (distance-to-default) credit models As operational risk event severity model
Logistic Continuous Infinite All No Its cumulative distribution function is the logistic function Widely used (implicitly) in logistic regression models
Normal (Gaussian) Continuous Infinite All Yes The most commonly used distribution function due to its expectional tractability, limit theorems and natural emergence in physical phenomena The basis of large fraction of stochastic processes Implicit in many statistical analyses
Poisson Discrete Infinite All No The probability of a given number of events occurring in a fixed interval of time As component in market jump processes Credit event count Operational event count
Student's t Continuous Infinite All Yes Arrises in statistical testing (Student's t Test) In Model Validation
Uniform Continuous Finite All Yes Used as building block for generating samples from other distrubutions
Vasicek Distribution Continuous Finite All No Arrises as a limit distribution of the sum of conditionally independent Bernulli variables Used as a simple model of portfolio loss (also in Basel II as the ASRF model)
Weibull Continuous Infinite All No A generalization of the exponential As parametric model of survival curves As operational risk event severity

## Issues and Challenges

• The availability of a very large number of parametric risk distributions means that modelling univariate risks becomes often a mechanical exercise focusing of measures of fit without regard as to whether the distribution has some intrinsic affinity with the process being modelled