CVaR (Conditional Value at Risk)

CVaR — Conditional Value at Risk, also called Expected Shortfall — answers the question that VaR leaves open: when the loss exceeds the VaR threshold, how much do I actually lose on average? Where VaR is a quantile (a cut-off), CVaR is an expectation (an average). Basel III, ESMA and every institutional risk desk use CVaR as the internal standard because it is a coherent risk measure by definition: it satisfies sub-additivity, so diversification can never make CVaR worse.

Formula

For a confidence level α (typically 95% or 99%) and a return distribution with VaR_α as the corresponding quantile, CVaR is:

CVaR_α = E[ L | L ≥ VaR_α ]

In words: the expected loss given that the loss is at least as bad as VaR. On a 1,000-day historical sample with α = 0.95, VaR is the 50th worst return and CVaR is the arithmetic average of those 50 worst returns.

When to use CVaR instead of VaR

• ·When the loss distribution has fat tails — CVaR captures the magnitude of those tails, VaR ignores it.
• ·When you optimize a portfolio: minimising CVaR yields a convex problem with a unique solution, minimising VaR can yield concentrated portfolios.
• ·When you report risk to investors who care about tail outcomes (institutional clients, regulators).

Numerical example

A 60/40 portfolio simulated with a t-distribution (ν = 5, realistic fat tails for equity) over 5 years of daily returns produces:

• ·Daily 95% VaR ≈ −1.62%
• ·Daily 95% CVaR ≈ −2.34%
• ·CVaR − VaR spread = −0.72% → the tail costs 44% more than the threshold.

Ignoring that 44% means underestimating the expected drawdown on a crisis day by ~70 bps on €100k of capital — the difference between "it went badly" and "I lost a month of expected return in a single day".

How MEDGE Capital uses CVaR

MEDGE exposes both Min CVaR 95% and Min CVaR 99% as portfolio optimization objectives, computed historically over the backtest window. Every risk report shows VaR and CVaR side by side at the same α so the reader sees how much the tail exceeds the threshold — disclosing a risk and understanding it are different acts.