Why CVaR should replace VaR in retail

95% Value-at-Risk is the most widely used risk measure on retail-investor portals. It means just one thing: there is a 5% probability that the daily loss will exceed the threshold. By how much it is exceeded is not specified. Conditional VaR — also known as Expected Shortfall — answers exactly that second question, and it is the reason Basel, ESMA and any institutional manager has adopted it as an internal standard.

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.