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In
financial mathematics Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that requir ...
, a risk measure is used to determine the amount of an
asset In financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. Assets represent value of ownership that can ...
or set of assets (traditionally
currency A currency, "in circulation", from la, currens, -entis, literally meaning "running" or "traversing" is a standardization of money in any form, in use or circulation as a medium of exchange, for example banknotes and coins. A more general d ...
) to be kept in reserve. The purpose of this reserve is to make the
risks In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environme ...
taken by
financial institutions Financial institutions, sometimes called banking institutions, are business entities that provide services as intermediaries for different types of financial monetary transactions. Broadly speaking, there are three major types of financial inst ...
, such as banks and insurance companies, acceptable to the regulator. In recent years attention has turned towards 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 \rho: \mathcal \to \mathbb \cup \ should have certain properties: ; Normalized : \rho(0) = 0 ; Translative : \mathrm\; a \in \mathbb \; \mathrm \; Z \in \mathcal ,\;\mathrm\; \rho(Z + a) = \rho(Z) - a ; Monotone : \mathrm\; Z_1,Z_2 \in \mathcal \;\mathrm\; Z_1 \leq Z_2 ,\; \mathrm \; \rho(Z_2) \leq \rho(Z_1)


Set-valued

In a situation with \mathbb^d-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 cost In economics and related disciplines, a transaction cost is a cost in making any economic trade when participating in a market. Oliver E. Williamson defines transaction costs as the costs of running an economic system of companies, and unlike pr ...
s.


Mathematically

A set-valued risk measure is a function R: L_d^p \rightarrow \mathbb_M, where L_d^p is a d-dimensional
Lp space In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbak ...
, \mathbb_M = \, and K_M = K \cap M where K is a constant
solvency cone The solvency cone is a concept used in financial mathematics which models the possible trades in the financial market. This is of particular interest to markets with transaction costs. Specifically, it is the convex cone of portfolios that can be ...
and M is the set of portfolios of the m reference assets. R must have the following properties: ; Normalized : K_M \subseteq R(0) \; \mathrm \; R(0) \cap -\mathrmK_M = \emptyset ; Translative in M : \forall X \in L_d^p, \forall u \in M: R(X + u1) = R(X) - u ; Monotone : \forall X_2 - X_1 \in L_d^p(K) \Rightarrow R(X_2) \supseteq R(X_1)


Examples

*
Value at risk Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by ...
*
Expected shortfall Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the wor ...
* Superposed risk measures *
Entropic value at risk In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The en ...
* Drawdown * Tail conditional expectation *
Entropic risk measure In financial mathematics (concerned with mathematical modeling of financial markets), the entropic risk measure is a risk measure which depends on the risk aversion of the user through the exponential utility function. It is a possible alternative ...
*
Superhedging price The superhedging price is a coherent risk measure. The superhedging price of a portfolio (A) is equivalent to the smallest amount necessary to be paid for an admissible portfolio (B) at the current time so that at some specified future time the va ...
* Expectile


Variance

Variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
(or
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
) is not a risk measure in the above sense. This can be seen since it has neither the translation property nor monotonicity. That is, Var(X + a) = Var(X) \neq Var(X) - a for all a \in \mathbb, and a simple counterexample for monotonicity can be found. The standard deviation is a
deviation risk measure In financial mathematics, a deviation risk measure is a function to quantify financial risk (and not necessarily downside risk) in a different method than a general risk measure. Deviation risk measures generalize the concept of standard deviation. ...
. To avoid any confusion, note that deviation risk measures, such as
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
and
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
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 R_(X) = R(X) and A_ = A.


Risk measure to acceptance set

* If \rho is a (scalar) risk measure then A_ = \ is an acceptance set. * If R is a set-valued risk measure then A_R = \ is an acceptance set.


Acceptance set to risk measure

* If A is an acceptance set (in 1-d) then \rho_A(X) = \inf\ defines a (scalar) risk measure. * If A is an acceptance set then R_A(X) = \ is a set-valued risk measure.


Relation with deviation risk measure

There is a one-to-one relationship between a
deviation risk measure In financial mathematics, a deviation risk measure is a function to quantify financial risk (and not necessarily downside risk) in a different method than a general risk measure. Deviation risk measures generalize the concept of standard deviation. ...
''D'' and an expectation-bounded risk measure \rho where for any X \in \mathcal^2 * D(X) = \rho(X - \mathbb * \rho(X) = D(X) - \mathbb /math>. \rho is called expectation bounded if it satisfies \rho(X) > \mathbb X/math> for any nonconstant ''X'' and \rho(X) = \mathbb X/math> for any constant ''X''.


See also

* Coherent risk measure * Dynamic risk measure * Managerial risk accounting * Risk management * Risk metric - the abstract concept that a risk measure quantifies * RiskMetrics - a model for risk management * Spectral risk measure *
Distortion risk measure In financial mathematics and economics, a distortion risk measure is a type of risk measure which is related to the cumulative distribution function of the return of a financial portfolio. Mathematical definition The function \rho_g: L^p \to \ma ...
*
Value at risk Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by ...
*
Conditional value-at-risk Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the wor ...
*
Entropic value at risk In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The en ...
* Risk return ratio


References


Further reading

* * * * {{Authority control Actuarial science Financial risk modeling