Portfolio construction

Drawdown, VaR and CVaR: measuring the bad days

Volatility measures all swings; these three measure the losses: the worst fall from a peak, the loss you exceed on 1 day in 20, and the average loss on those days.

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ایده به زبان ساده

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.

روش محاسبه

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))

نحوه استفاده

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.

نتیجه‌گیری

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.

مثال

Running example (used in every portfolio guide): US500, GER40, gold (XAUUSD) and EURUSD, two years of daily data. Expected returns 8%, 7%, 7% and 1% a year; volatilities 18%, 20%, 15% and 7%. US500 and GER40 move together (correlation about 0.7); gold hardly moves with the indices (about 0.04). Equal weights: VaR 95% of 1.04% a day, CVaR 1.31%, maximum drawdown −14.9% over the two years. All in US500: VaR 1.85%, CVaR 2.17%, drawdown −17.4%. The minimum-variance mix: VaR 0.73%, CVaR 0.90%, drawdown −8.6%.

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مراقب این‌ها باشید

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.

کمی تاریخچه

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.

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