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 ${\displaystyle X}$  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]

Mathematically edit

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

${\displaystyle K_{M}\subseteq R(0)\;\mathrm {and} \;R(0)\cap -\mathrm {int} K_{M}=\emptyset }$ 

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 edit

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]

Risk measure to acceptance set edit

• 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.

Acceptance set to risk measure edit

• 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.