Unformatted text preview: reme events with signiﬁcant magnitude corresponding to the tail part of a probability distribution.
• The two most widely adopted risk measures nowadays in risk management, ValueatRisk (VaR) and conditional ValueatRisk (CVaR) are
both rooted in Roy’s safetyﬁrst principle. 8
VaR and CVaR
• The probability that the ﬁnal wealth R(x) =
exceed a threshold α is represented as
Ψ(x, α) n
i=1 Ri xi does not p (R )d R .
R(x)≤α Given a disarster level β (usually less than 0.1) and a ﬁxed x ∈ X ,
the valueatrisk is deﬁned as
VaRβ (x) max{α ∈ R : Ψ(x, α) ≤ β }. The corresponding conditional valueatrisk, denoted by CVaRβ (x),
is deﬁned as the expected value of the ﬁnal wealth that is below
VaRβ (x), that is,
CVaRβ (x) 1
β R(x)≤VaRβ (x) R (x )p (R )d R . 9
Coherent Risk Measure • A risk measure ρ is a coherent risk measure (Artzner et al., 1999,
Mathematical Finance), if it satisﬁes:
1. Subadditivity: for all random returns X and Y , ρ(X + Y ) ≤
ρ(X ) + ρ(Y );
2. Positive homogeneity: for positive constant λ, ρ(λX ) = λρ(X );
3. Monotonicity: if X ≤ Y for each outcome, then ρ(X ) ≥ ρ(Y );
4. Translation invariance: for constant m, ρ(X + m) = ρ(X ) − m.
• CVaR is coherent, while VaR is not. 10 Probability VaR E ( x) E
CVaR E ( x) Random variable R(x)...
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 Fall '11
 DuanLI
 Probability theory, Value at risk, • Roy, Both Markowitz, Ri xi, meanvariance utility approach

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