VaR vs CVaR — How Much Do You Actually Lose in the Worst 5%?

4-section structure: Concept / How We Compute / How to Read / Caveats.

1. Concept

"8% 10-year return" sounds great, but what investors really worry about is "how bad is the worst day?" — that's tail risk.

Two complementary metrics:

Metric	Answers
VaR	What's the most I'll lose in a day with 95% confidence?
CVaR	If I do hit the worst 5%, what's the average loss?

Why CVaR Matters More Than VaR

VaR gives you a threshold; it doesn't tell you how bad things are beyond the threshold.

Stock A: VaR 95% = −3%, tail-day average = −3.5%
Stock B: VaR 95% = −3%, tail-day average = −10%

Same VaR, but B's black swans are dramatically worse. CVaR distinguishes them; VaR doesn't. This is why Basel III and modern academia prefer CVaR (Expected Shortfall, ES).

2. How We Compute

Data

Individual stock: past 252 trading days of simple daily returns
Portfolio: daily returns derived from portfolio NAV

Method: Historical Simulation

VaR 95% = 5th percentile of past-252-day returns CVaR 95% = mean of the worst 5% (~12–13 days)

Why Historical, Not Parametric Normal

The common alternative — assuming normal distribution and using mean ± 1.645σ — understates tail risk because stock returns are:

Left-skewed (crashes happen more often than rallies of equal magnitude)
Fat-tailed (extreme events more frequent than normal predicts)

Historical simulation takes empirical tails as-is.

Insufficient Sample

When < 60 days, returns available: false to avoid unstable estimates.

3. How to Read

UI Display (Stock Risk tab / Portfolio VIP block)

VaR 95%   -2.19%     CVaR 95%   -2.73%

Reading:

"95% confidence this stock won't lose more than 2.19% in a day"
"If we do hit the worst 5%, average loss is 2.73%"

Daily VaR 95% Magnitude Reference

VaR 95%	Volatility Profile
> −1.5%	Very low (bonds, utilities)
−1.5% to −2.5%	Typical large-caps
−2.5% to −4%	Growth / mid-caps
−4% to −6%	Small-caps, volatile speculatives
< −6%	Extreme speculation (biotech, small alt-coins, leveraged ETFs)

Portfolio vs Individual VaR

Portfolio VaR < any component's VaR → good diversification (negative correlations offsetting)
Portfolio VaR ≈ max-vol component's VaR → diversification failed (verify via P0.2 correlation matrix)

VaR / CVaR Relationship

Typically CVaR < VaR < 0.

CVaR / VaR ratio ≈ 1 → thin tail (good)
CVaR / VaR ≫ 1 (e.g., 1.5x) → fat tail (bad: extreme events much worse than threshold)

4. Caveats

⚠️ Past Doesn't Guarantee Future

Historical simulation assumes the past distribution repeats. If the past 252 days had no 2008-scale crash, VaR underestimates true risk. Market regime shifts (rate cycles, black swans) distort historical VaR significantly.

Mitigate: combine with stress tests (Phase 2.3), GARCH volatility (Phase 4.4), historical extreme-event flagging (P1C.5).

⚠️ 252 Days Is Not Enough for All Regimes

252 days ≈ 1 trading year. One year may contain no crises (too optimistic) or be entirely inside a crisis (too pessimistic). Academic practice uses 500+ days. We use 252 for consistency with Sharpe; configurable window is planned.

⚠️ VaR / CVaR Are Daily

To convert horizons (normal approximation):

Weekly VaR ≈ daily × √5
Monthly VaR ≈ daily × √21

But this approximation fails for fat tails — extreme risk scales worse than √T.

⚠️ CVaR Is Unstable in Small Samples

CVaR averages only 12–13 tail points in 252-day windows. A single black-swan day dominates the estimate. Smaller windows → noisier CVaR.

⚠️ Not a Buy/Sell Signal

VaR/CVaR measure risk magnitude, not desirability.

High VaR + high expected return → maybe worth it (use Sharpe)
Low VaR + low return → suits conservative investors
High VaR + low return → avoid

Always pair with return analysis.

Try It

Open Stock Analysis → Risk — VaR/CVaR cards sit below Beta/Corr/R²/Std
Open Portfolio — below Rolling Sharpe, portfolio VaR/CVaR cards
Click 📐 to see formula, method, sample size, and tail day count