Risk measure

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:

; 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)

Set-valued

In a situation with \mathbb{R}^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 costs.

Mathematically

A set-valued risk measure is a function R: L_d^p \rightarrow \mathbb{F}_M, where L_d^p is a d-dimensional 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:

; Normalized : K_M \subseteq R(0) \text{ and } R(0) \cap -\operatorname{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

• Value at risk
• Expected shortfall
• Superposed risk measures{{cite journal | article-number = 2050012
• Entropic value at risk
• Drawdown
• Tail conditional expectation
• Entropic risk measure
• Superhedging price
• Expectile

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, 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. 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 R_{A_R}(X) = R(X) and A_{R_A} = A.

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 set-valued 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 1-d) 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 set-valued 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 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.

References

References

1. (1999). "Coherent Measures of Risk". Mathematical Finance.
2. (2004). "Vector–valued coherent risk measures". Finance and Stochastics.
3. (2010). "Duality for Set-Valued Measures of Risk". SIAM Journal on Financial Mathematics.
4. Andreas H. Hamel. (2011). "Set-valued risk measures for conical market models". Mathematics and Financial Economics.
5. (22 January 2003). "Deviation Measures in Risk Analysis and Optimization".