Why CVaR should replace VaR in retail

CVaR (Conditional Value at Risk) is the average loss conditional on the VaR threshold being breached at a given confidence level. Where VaR is a quantile — the cut-off above which 5% of losses fall — CVaR is an expectation: the arithmetic mean of those tail losses. Basel III, ESMA and every institutional risk desk use CVaR as the internal standard because it is sub-additive (diversification can never make it worse) and tail-aware (silent on neither the shape nor the magnitude of the worst 5%).

95% VaR remains the most cited risk metric on retail portals, but it leaves the answer half-finished: it tells you the loss threshold not expected to be exceeded 95 days out of 100 and says nothing about how bad the breach is when it happens. CVaR closes that gap.

Definitions in two lines

• ·VaR_α = maximum loss threshold in the worst (1−α)% of cases. Quantile of the return distribution.
• ·CVaR_α = average loss conditional on the VaR threshold being breached. Expectation of the tail.

On a sample of 1,000 days with α = 0.95, VaR is the 50th worst return. CVaR is the average of the 50 worst. By construction, CVaR ≤ VaR (more negative) and captures how "fat" the tail is.

Why VaR fails: subadditivity

A coherent risk measure must satisfy four properties (Artzner et al., 1999): monotonicity, positive homogeneity, translation invariance and — the most important — subadditivity. Subadditivity means diversification cannot increase risk: ρ(A + B) ≤ ρ(A) + ρ(B).

VaR is not subadditive. There are real cases — especially with discrete distributions or highly asymmetric tails — where the VaR of a diversified portfolio is strictly greater than the sum of the VaRs of the individual assets. Consequence: optimizing a portfolio by minimizing VaR can paradoxically push you toward concentration.

Numerical example

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

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

Over a 21-trading-day month, ignoring that extra 44% means underestimating the expected drawdown on a crisis day by roughly 70 bps. On €100k of capital, that is the difference between "it went badly" and "I lost a month of expected return in a single day".

What MEDGE does

In Portfolio → Optimization, both "Min CVaR 95%" and "Min CVaR 99%" are available as optimization objectives. The pipeline computes CVaR historically over the backtest window and uses linear programming with ε-quantile to push to the efficient frontier the combination that minimizes tail expectation, not the quantile.

In the institutional report, every risk metric is shown as a pair (VaR + CVaR) for the same α — so the reader always sees how much the tail exceeds the threshold. That is the difference between disclosing a risk and actually having understood it.

If you are comparing tools that compute these metrics, see how MEDGE Capital stacks up against Portfolio Visualizer — the long-standing US-focused alternative — on CVaR coverage, optimization breadth and pricing: /vs/portfolio-visualizer.