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 towards convex and coherent risk measurement.
Contents

Mathematically 1

Setvalued 2

Examples 3

Well known risk measures 3.1

Variance 3.2

Relation to acceptance set 4

Risk measure to acceptance set 4.1

Acceptance set to risk measure 4.2

Relation with deviation risk measure 5

See also 6

References 7

Further reading 8
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{L} \to \mathbb{R} \cup \{+\infty\} should have certain properties:^{[1]}

Normalized

\rho(0) = 0

Translative

\mathrm{If}\; a \in \mathbb{R} \; \mathrm{and} \; Z \in \mathcal{L} ,\;\mathrm{then}\; \rho(Z + a) = \rho(Z)  a

Monotone

\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} \;\mathrm{and}\; Z_1 \leq Z_2 ,\; \mathrm{then} \; \rho(Z_2) \leq \rho(Z_1)
Setvalued
In a situation with \mathbb{R}^dvalued 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. Setvalued risk measures are useful for markets with transaction costs.^{[2]}
Mathematically
A setvalued risk measure is a function R: L_d^p \rightarrow \mathbb{F}_M, where L_d^p is a ddimensional Lp space, \mathbb{F}_M = \{D \subseteq M: D = cl (D + K_M)\}, and K_M = K \cap M 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:^{[3]}

Normalized

K_M \subseteq R(0) \; \mathrm{and} \; R(0) \cap \mathrm{int}K_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
Well known risk measures
Variance
Variance (or standard deviation) is not a risk measure. 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{R}, and a simple counterexample for monotonicity can be found. The standard deviation is a deviation risk measure.
Relation to acceptance set
There is a onetoone correspondence between an acceptance set and a corresponding risk measure. As defined below it can be shown that R_{A_R}(X) = R(X) and A_{R_A} = A.^{[4]}
Risk measure to acceptance set

If \rho is a (scalar) risk measure then A_{\rho} = \{X \in L^p: \rho(X) \leq 0\} is an acceptance set.

If R is a setvalued risk measure then A_R = \{X \in L^p_d: 0 \in R(X)\} is an acceptance set.
Acceptance set to risk measure

If A is an acceptance set (in 1d) then \rho_A(X) = \inf\{u \in \mathbb{R}: X + u1 \in A\} defines a (scalar) risk measure.

If A is an acceptance set then R_A(X) = \{u \in M: X + u1 \in A\} is a setvalued risk measure.
Relation with deviation risk measure
There is a onetoone relationship between a deviation risk measure D and an expectationbounded risk measure \rho where for any X \in \mathcal{L}^2

D(X) = \rho(X  \mathbb{E}[X])

\rho(X) = D(X)  \mathbb{E}[X].
\rho is called expectation bounded if it satisfies \rho(X) > \mathbb{E}[X] for any nonconstant X and \rho(X) = \mathbb{E}[X] for any constant X.^{[5]}
See also
References

^ Artzner, Philippe; Delbaen, Freddy; Eber, JeanMarc; Heath, David (1999). "Coherent Measures of Risk" (pdf). Mathematical Finance 9 (3): 203–228.

^ Jouini, Elyes; Meddeb, Moncef; Touzi, Nizar (2004). "Vector–valued coherent risk measures". Finance and Stochastics 8 (4): 531–552.

^ Hamel, A. H.; Heyde, F. (2010). "Duality for SetValued Measures of Risk" (pdf). SIAM Journal on Financial Mathematics 1 (1): 66–95.

^ Andreas H. Hamel; Frank Heyde; Birgit Rudloff (2011). "Setvalued risk measures for conical market models" (pdf). Mathematics and Financial Economics 5 (1): 1–28.

^ Rockafellar, Tyrrell; Uryasev, Stanislav; Zabarankin, Michael (2002). "Deviation Measures in Risk Analysis and Optimization" (pdf). Retrieved October 13, 2011.
Further reading

Crouhy, Michel; D. Galai; R. Mark (2001). Risk Management.

Kevin, Dowd (2005). Measuring Market Risk (2nd ed.).

Foellmer, Hans; Schied, Alexander (2004). Stochastic Finance. de Gruyter Series in Mathematics 27. Berlin:

Shapiro, Alexander; Dentcheva, Darinka;
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