Drawdown, VaR and CVaR: measuring the bad days

The idea in plain words

Investors do not mind upside surprises; they mind losing. Maximum drawdown is the deepest fall from a previous high, the pain you would have sat through. Value at Risk (VaR 95%) is the daily loss exceeded on only 5% of days. Conditional VaR (CVaR, also called expected shortfall) is the average of those worst 5% of days, so it also tells you how bad the bad days get.

How it is calculated

Drawdown: track the running peak of the portfolio's value; the drawdown is value ÷ peak − 1; the maximum drawdown is its lowest point. VaR 95%: sort the daily returns, take the 5th percentile, change its sign. CVaR 95%: the average of all daily returns at or below that percentile, sign changed.

DD(t) = V(t) ÷ max_{s≤t} V(s) − 1        MaxDD = min DD(t)
VaR₉₅ = −quantile₀.₀₅(r)
CVaR₉₅ = −mean(r | r ≤ quantile₀.₀₅(r))

How to use it

• In a study's table, compare Max DD and CVaR next to volatility: two models with the same volatility can have very different tails.
• In the live analysis, VaR and CVaR are also shown in dollars for your actual positions.
• Choose CVaR optimisation when the worst days matter more to you than the average swing.

Drawing conclusions

• CVaR is always at least VaR; a big gap means fat tails (rare, large losses).
• A maximum drawdown much larger than three or four times the annual volatility suggests the history contains a crisis, or that the strategy is fragile.

Watch out for

• VaR says nothing about how bad the worst 5% of days are; that is why CVaR exists.
• Two years of daily data have only about 25 days in the worst 5%: the tail estimate is rough.
• Past drawdowns are a floor, not a ceiling, for future ones.

A little history

J.P. Morgan's RiskMetrics (1994) made VaR the industry standard. Artzner, Delbaen, Eber and Heath (1999) showed its flaws and defined "coherent" risk measures, of which CVaR is the best known; bank regulators moved from VaR to expected shortfall in the Basel market-risk rules of 2016–2019.