CVaR

CVaR, short for Conditional Value at Risk, is a risk measure used in finance and risk management to assess the expected loss in the tail of the loss distribution beyond a specified confidence level α (0 < α < 1). It complements Value at Risk (VaR) by incorporating information about the size of extreme losses rather than only the cutoff point.

For a random loss L, the Value at Risk at level α is VaR_α(L) = inf{l : P(L ≤ l) ≥ α}. The Conditional Value at Risk is CVaR_α(L) = E[L | L ≥ VaR_α(L)]. For continuous loss distributions, this equals the average of losses in the tail beyond VaR_α. An equivalent representation is CVaR_α(L) = (1/(1-α)) ∫_{α}^{1} VaR_u(L) du.

Computationally, CVaR can be computed analytically for simple distributions or estimated from data. A common optimization form is CVaR_α(L) = min_z { z + (1/(1-α)) E[(L - z)^+] }, where (x)^+ = max(x,0). This convex formulation enables optimization over portfolios using linear or convex programming, with L often expressed as a linear function of portfolio weights.

Applications and limitations: CVaR is used in portfolio optimization, risk budgeting, and regulatory capital frameworks (Basel II/III). It is a coherent risk measure, addressing some limitations of VaR by reflecting tail risk. Limitations include estimation error in the tail, dependence on the choice of α, and higher computational demands for large, complex portfolios.