Conditional VaR (CVaR/Expected Shortfall)

While VaR answers the question 'What is the maximum loss at 95% confidence?', CVaR answers the deeper question 'If we are in the worst 5% of outcomes, what is the average loss?' This distinction is crucial: two portfolios can have identical VaRs but dramatically different CVaRs if one has a heavier tail. CVaR is therefore a more risk-sensitive and coherent risk measure, satisfying mathematical properties (monotonicity, translation invariance, homogeneity, and sub-additivity) that VaR violates.

Sub-additivity is particularly important for risk aggregation. If CVaR(A) = 10 and CVaR(B) = 8, then CVaR(A+B) ≤ 18. This means CVaR correctly reflects diversification benefits when aggregating risk across a portfolio. VaR can actually increase when combining positions, violating intuitive diversification logic.

For a continuous return distribution, CVaR at confidence level α is the conditional expectation of losses exceeding VaR(α): CVaR_α = E[Loss | Loss > VaR_α]. In historical simulation, it is simply the average of the worst (1−α)% of historical daily losses in the sample.

In the Indian regulatory context, the Basel Committee on Banking Supervision (BCBS) transitioned banks from VaR to ES (Expected Shortfall at 97.5% confidence) under FRTB (Fundamental Review of the Trading Book) for market risk capital. RBI's adoption of FRTB norms for Indian banks was under active consideration in the early 2020s, signalling the shift from VaR to CVaR as the standard regulatory metric.

For Indian fund managers and PMS providers, CVaR reporting provides richer risk disclosure. A PMS with 95% one-month VaR of 8% and CVaR of 15% tells investors that while large losses occur rarely, when they do occur they are very large — a profile very different from a fund with the same VaR but CVaR of 10%. SEBI's push for enhanced risk disclosures in PMS and AIF offering documents made CVaR reporting increasingly common in institutional client reporting.