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 $X$ 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

$\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} \;\mathrm{and}\; Z_1 \leq Z_2 ,\; \mathrm{then} \; \rho(Z_2) \leq \rho(Z_1)$

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.^[2]

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:^[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

• Value at risk
• Expected shortfall
• Tail conditional expectation
• Entropic risk measure
• Superhedging price
• Expectile

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 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 $A_{R_A} = A$.^[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 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.^[5]

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
• Value at risk
• Conditional value-at-risk
• Entropic value at risk
• Risk return ratio

References

1. Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (1999). "Coherent Measures of Risk" (pdf). Mathematical Finance 9 (3): 203–228. doi:10.1111/1467-9965.00068. Retrieved February 3, 2011.
2. Jouini, Elyes; Meddeb, Moncef; Touzi, Nizar (2004). "Vector–valued coherent risk measures". Finance and Stochastics 8 (4): 531–552. doi:10.1007/s00780-004-0127-6.
3. Hamel, A. H.; Heyde, F. (2010). "Duality for Set-Valued Measures of Risk" (pdf). SIAM Journal on Financial Mathematics 1 (1): 66–95. doi:10.1137/080743494. Retrieved August 17, 2012.
4. Andreas H. Hamel; Frank Heyde; Birgit Rudloff (2011). "Set-valued risk measures for conical market models" (pdf). Mathematics and Financial Economics 5 (1): 1–28. doi:10.1007/s11579-011-0047-0. Retrieved April 20, 2012.
5. 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. McGraw-Hill. pp. 752 pages. ISBN 0-07-135731-9.
• Kevin, Dowd (2005). Measuring Market Risk (2nd ed.). John Wiley & Sons. pp. 410 pages. ISBN 0-470-01303-6.

• Foellmer, Hans; Schied, Alexander (2004). Stochastic Finance. de Gruyter Series in Mathematics 27. Berlin: Walter de Gruyter. pp. xi+459. ISBN 311-0183463. MR 2169807.
• 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. MR 2562798.