In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator. In recent years attention has turned to convex and coherent risk measurement.

Mathematically

A risk measure is defined as a mapping from a set of random variables to the real numbers. This set of random variables represents portfolio returns. The common notation for a risk measure associated with a random variable X is \rho(X). A risk measure should have certain properties:

Set-valued

In a situation with -valued portfolios such that risk can be measured in m \leq d of the assets, then a set of portfolios is the proper way to depict risk. Set-valued risk measures are useful for markets with transaction costs.

Mathematically

A set-valued risk measure is a function, where L_d^p is a d-dimensional Lp space, , and where K is a constant solvency cone and M is the set of portfolios of the m reference assets. R must have the following properties:

Examples

Variance

Variance (or standard deviation) is not a risk measure in the above sense. This can be seen since it has neither the translation property nor monotonicity. That is, for all, and a simple counterexample for monotonicity can be found. The standard deviation is a deviation risk measure. To avoid any confusion, note that deviation risk measures, such as variance and standard deviation are sometimes called risk measures in different fields.

Relation to acceptance set

There is a one-to-one correspondence between an acceptance set and a corresponding risk measure. As defined below it can be shown that and A_{R_A} = A.

Risk measure to acceptance set

Acceptance set to risk measure

Relation with deviation risk measure

There is a one-to-one relationship between a deviation risk measure D and an expectation-bounded risk measure \rho where for any \rho is called expectation bounded if it satisfies for any nonconstant X and for any constant X.