In financial mathematics, a risk measure assigns a numerical value to the risk associated with a financial position or portfolio. In financial management and insurance, risk measures are often used to determine capital reserve requirements to mitigate downside risk to make it acceptable to regulators. In recent years attention has turned to convex and coherent risk measurement.

A risk measure is defined as a mapping from a set of random variables to the real numbers. Depending on context, the random variables may represent portfolio returns or insurance losses. In the former case, risk is associated with the left tail of the distribution, while in the latter it is associated with the right tail.

The common notation for a risk measure associated with a random variable 👁 {\displaystyle X}
is 👁 {\displaystyle \rho (X)}
. A risk measure 👁 {\displaystyle \rho :{\mathcal {L}}\to \mathbb {R} \cup \{+\infty \}}
should have certain properties:^[1]

Normalized

👁 {\displaystyle \rho (0)=0}

Translative

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

Monotone

👁 {\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})}

In a situation with 👁 {\displaystyle \mathbb {R} ^{d}}
-valued portfolios such that risk can be measured in 👁 {\displaystyle 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]

A set-valued risk measure is a function 👁 {\displaystyle R:L_{d}^{p}\rightarrow \mathbb {F} _{M}}
, where 👁 {\displaystyle L_{d}^{p}}
is a 👁 {\displaystyle d}
-dimensional Lp space, 👁 {\displaystyle \mathbb {F} _{M}=\{D\subseteq M:D=cl(D+K_{M})\}}
, and 👁 {\displaystyle K_{M}=K\cap M}
where 👁 {\displaystyle K}
is a constant solvency cone and 👁 {\displaystyle M}
is the set of portfolios of the 👁 {\displaystyle m}
reference assets. 👁 {\displaystyle R}
must have the following properties:^[3]

Normalized

👁 {\displaystyle K_{M}\subseteq R(0){\text{ and }}R(0)\cap -\operatorname {int} K_{M}=\emptyset }

Translative in M

👁 {\displaystyle \forall X\in L_{d}^{p},\forall u\in M:R(X+u1)=R(X)-u}

Monotone

👁 {\displaystyle \forall X_{2}-X_{1}\in L_{d}^{p}(K)\Rightarrow R(X_{2})\supseteq R(X_{1})}

• Value at risk
• Expected shortfall
• Superposed risk measures^[4]
• Entropic value at risk
• Drawdown
• Tail conditional expectation
• Entropic risk measure
• Superhedging price
• Expectile

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, 👁 {\displaystyle Var(X+a)=Var(X)\neq Var(X)-a}
for all 👁 {\displaystyle 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.

There is a one-to-one correspondence between an acceptance set and a corresponding risk measure. As defined below it can be shown that 👁 {\displaystyle R_{A_{R}}(X)=R(X)}
and 👁 {\displaystyle A_{R_{A}}=A}
.^[5]

• If 👁 {\displaystyle \rho }
  is a (scalar) risk measure then 👁 {\displaystyle A_{\rho }=\{X\in L^{p}:\rho (X)\leq 0\}}
  is an acceptance set.
• If 👁 {\displaystyle R}
  is a set-valued risk measure then 👁 {\displaystyle A_{R}=\{X\in L_{d}^{p}:0\in R(X)\}}
  is an acceptance set.

• If 👁 {\displaystyle A}
  is an acceptance set (in 1-d) then 👁 {\displaystyle \rho _{A}(X)=\inf\{u\in \mathbb {R} :X+u1\in A\}}
  defines a (scalar) risk measure.
• If 👁 {\displaystyle A}
  is an acceptance set then 👁 {\displaystyle R_{A}(X)=\{u\in M:X+u1\in A\}}
  is a set-valued risk measure.

There is a one-to-one relationship between a deviation risk measure D and an expectation-bounded risk measure 👁 {\displaystyle \rho }
where for any 👁 {\displaystyle X\in {\mathcal {L}}^{2}}

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

👁 {\displaystyle \rho }
is called expectation bounded if it satisfies 👁 {\displaystyle \rho (X)>\mathbb {E} [-X]}
for any nonconstant X and 👁 {\displaystyle \rho (X)=\mathbb {E} [-X]}
for any constant X.^[6]

• 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. Vol. 27. Berlin: Walter de Gruyter. pp. xi+459. ISBN 978-311-0183467. MR 2169807.
• Shapiro, Alexander; Dentcheva, Darinka; Ruszczyński, Andrzej (2009). Lectures on stochastic programming. Modeling and theory. MPS/SIAM Series on Optimization. Vol. 9. Philadelphia: Society for Industrial and Applied Mathematics. pp. xvi+436. ISBN 978-0898716870. MR 2562798.

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