The two most widely adopted risk measures nowadays in

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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)...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online