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.
Mathematical Description 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 X {\displaystyle X} is ρ ( X ) {\displaystyle \rho (X)} . A risk measure ρ : L → R ∪ { + ∞ } {\displaystyle \rho :{\mathcal {L}}\to \mathbb {R} \cup \{+\infty \}} should have certain properties:
Normalized
ρ ( 0 ) = 0 {\displaystyle \rho (0)=0}
Translative
I f a ∈ R a n d Z ∈ L , t h e n ρ ( Z + a ) = ρ ( Z ) − a {\displaystyle \mathrm {If} \;a\in \mathbb {R} \;\mathrm {and} \;Z\in {\mathcal {L}},\;\mathrm {then} \;\rho (Z+a)=\rho (Z)-a}
Monotone
I f Z 1 , Z 2 ∈ L a n d Z 1 ≤ Z 2 , t h e n ρ ( Z 2 ) ≤ ρ ( Z 1 ) {\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})}
Set-valued In a situation with R d {\displaystyle \mathbb {R} ^{d}} -valued portfolios such that risk can be measured in m ≤ d {\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.
Mathematically A set-valued risk measure is a function R : L d p → F M {\displaystyle R:L_{d}^{p}\rightarrow \mathbb {F} _{M}} , where L d p {\displaystyle L_{d}^{p}} is a d {\displaystyle d} -dimensional Lp space, F M = { D ⊆ M : D = c l ( D + K M ) } {\displaystyle \mathbb {F} _{M}=\{D\subseteq M:D=cl(D+K_{M})\}} , and K M = K ∩ M {\displaystyle K_{M}=K\cap M} where K {\displaystyle K} is a constant solvency cone and M {\displaystyle M} is the set of portfolios of the m {\displaystyle m} reference assets. R {\displaystyle R} must have the following properties:
Normalized
K M ⊆ R ( 0 ) and R ( 0 ) ∩ − int K M = ∅ {\displaystyle K_{M}\subseteq R(0){\text{ and }}R(0)\cap -\operatorname {int} K_{M}=\emptyset }
Translative in M
∀ X ∈ L d p , ∀ u ∈ M : R ( X + u 1 ) = R ( X ) − u {\displaystyle \forall X\in L_{d}^{p},\forall u\in M:R(X+u1)=R(X)-u}
Monotone
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