The two most widely adopted risk measures nowadays in

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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, Value-at-Risk (VaR) and conditional Value-at-Risk (CVaR) are both rooted in Roy’s safety-first 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 value-at-risk is defined as VaRβ (x) max{α ∈ R : Ψ(x, α) ≤ β }. The corresponding conditional value-at-risk, 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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