Risk measure

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.

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 [math]\displaystyle{ X }[/math] is [math]\displaystyle{ \rho(X) }[/math]. A risk measure [math]\displaystyle{ \rho: \mathcal{L} \to \mathbb{R} \cup \{+\infty\} }[/math] should have certain properties:

Normalized

[math]\displaystyle{ \rho(0) = 0 }[/math]

Translative

[math]\displaystyle{ \mathrm{If}\; a \in \mathbb{R} \; \mathrm{and} \; Z \in \mathcal{L} ,\;\mathrm{then}\; \rho(Z + a) = \rho(Z) - a }[/math]

Monotone

[math]\displaystyle{ \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) }[/math]

Set-valued

In a situation with [math]\displaystyle{ \mathbb{R}^d }[/math]-valued portfolios such that risk can be measured in [math]\displaystyle{ m \leq d }[/math] 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 [math]\displaystyle{ R: L_d^p \rightarrow \mathbb{F}_M }[/math], where [math]\displaystyle{ L_d^p }[/math] is a [math]\displaystyle{ d }[/math]-dimensional Lp space, [math]\displaystyle{ \mathbb{F}_M = \{D \subseteq M: D = cl (D + K_M)\} }[/math], and [math]\displaystyle{ K_M = K \cap M }[/math] where [math]\displaystyle{ K }[/math] is a constant solvency cone and [math]\displaystyle{ M }[/math] is the set of portfolios of the [math]\displaystyle{ m }[/math] reference assets. [math]\displaystyle{ R }[/math] must have the following properties:

Normalized

[math]\displaystyle{ K_M \subseteq R(0) \; \mathrm{and} \; R(0) \cap -\mathrm{int}K_M = \emptyset }[/math]

Translative in M

[math]\displaystyle{ \forall X \in L_d^p, \forall u \in M: R(X + u1) = R(X) - u }[/math]

Monotone

[math]\displaystyle{ \forall X_2 - X_1 \in L_d^p(K) \Rightarrow R(X_2) \supseteq R(X_1) }[/math]

Examples

• Value at risk
• Expected shortfall
• Superposed risk measures
• 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, [math]\displaystyle{ Var(X + a) = Var(X) \neq Var(X) - a }[/math] for all [math]\displaystyle{ a \in \mathbb{R} }[/math], 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 [math]\displaystyle{ R_{A_R}(X) = R(X) }[/math] and [math]\displaystyle{ A_{R_A} = A }[/math].

Risk measure to acceptance set

• If [math]\displaystyle{ \rho }[/math] is a (scalar) risk measure then [math]\displaystyle{ A_{\rho} = \{X \in L^p: \rho(X) \leq 0\} }[/math] is an acceptance set.
• If [math]\displaystyle{ R }[/math] is a set-valued risk measure then [math]\displaystyle{ A_R = \{X \in L^p_d: 0 \in R(X)\} }[/math] is an acceptance set.

Acceptance set to risk measure

• If [math]\displaystyle{ A }[/math] is an acceptance set (in 1-d) then [math]\displaystyle{ \rho_A(X) = \inf\{u \in \mathbb{R}: X + u1 \in A\} }[/math] defines a (scalar) risk measure.
• If [math]\displaystyle{ A }[/math] is an acceptance set then [math]\displaystyle{ R_A(X) = \{u \in M: X + u1 \in A\} }[/math] 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 [math]\displaystyle{ \rho }[/math] where for any [math]\displaystyle{ X \in \mathcal{L}^2 }[/math]

• [math]\displaystyle{ D(X) = \rho(X - \mathbb{E}[X]) }[/math]
• [math]\displaystyle{ \rho(X) = D(X) - \mathbb{E}[X] }[/math].

[math]\displaystyle{ \rho }[/math] is called expectation bounded if it satisfies [math]\displaystyle{ \rho(X) \gt \mathbb{E}[-X] }[/math] for any nonconstant X and [math]\displaystyle{ \rho(X) = \mathbb{E}[-X] }[/math] for any constant X.

Further reading

• Crouhy, Michel; D. Galai; R. Mark (2001). Risk Management. McGraw-Hill. pp. 752 pages. ISBN 978-0-07-135731-9. 
• Kevin, Dowd (2005). Measuring Market Risk (2nd ed.). John Wiley & Sons. pp. 410 pages. ISBN 978-0-470-01303-8. 

• Foellmer, Hans; Schied, Alexander (2004). Stochastic Finance. de Gruyter Series in Mathematics. 27. Berlin: Walter de Gruyter. pp. xi+459. ISBN 978-311-0183467. 
• Shapiro, Alexander; Dentcheva, Darinka; Ruszczyński, Andrzej (2009). Lectures on stochastic programming. Modeling and theory. MPS/SIAM Series on Optimization. 9. Philadelphia: Society for Industrial and Applied Mathematics. pp. xvi+436. ISBN 978-0898716870.