tvAR

TVaR, or Tail Value at Risk, is a risk measure used to assess the expected magnitude of losses in the tail of the loss distribution beyond a specified Value at Risk (VaR) threshold. It is also known as Conditional Tail Expectation (CTE) or Conditional VaR (CVaR) in many texts. TVaR is widely used in finance and insurance to quantify tail risk and to inform capital allocation and risk management decisions.

Definition: For a loss random variable L and confidence level α ∈ (0,1), VaR_α is the α-quantile of L: VaR_α = inf{l: P(L ≤ l) ≥ α}. TVaR_α is the expected loss given that the loss exceeds VaR_α: TVaR_α = E[L | L ≥ VaR_α]. If the distribution is continuous, TVaR_α = (1/(1-α)) ∫_α^1 VaR_u du.

Relation to ES: TVaR and Expected Shortfall (ES) are often used interchangeably; CVaR is a term frequently encountered in risk management literature. TVaR is a coherent risk measure, satisfying properties such as monotonicity, subadditivity, translation invariance, and positive homogeneity under standard definitions.

Computation: In practice, TVaR is estimated via simulation or numerical integration. For parametric models, closed-form expressions exist for some distributions (e.g., normal) to compute ES directly; in other cases, Monte Carlo or historical simulation yields the tail average of losses beyond VaR.

Applications: Used in risk budgeting, regulatory capital calculations, and stress testing. Basel III and other frameworks prefer Expected Shortfall as a market-risk measure due to its sensitivity to tail risk.