Value at Risk & CVaR
Trading Intelligence
10 min read
var95cvar95
Understand VaR and Conditional Value at Risk (Expected Shortfall) for measuring tail risk in your trade distribution.
Loading
10 min read
Understand VaR and Conditional Value at Risk (Expected Shortfall) for measuring tail risk in your trade distribution.
VaR tells you the door to the danger zone. CVaR tells you what happens once you walk through it.
Value at Risk is a quantile of the loss distribution. The 1-day 95% VaR is the loss level you exceed on only 5% of days under historical conditions — not the maximum loss, not the average loss, just the threshold. It answers the question: "At what loss level do my worst 5% of days begin?"
VaR_alpha(L) = inf{ x : P(L ≤ x) ≥ alpha }
For example, a daily VaR of $500 at 95% confidence means:
VaR is defined by the percentile of the loss distribution. At 95% confidence, VaR is the 5th percentile of your return distribution. At 99% confidence, it is the 1st percentile.
The concept originated in institutional finance -- banks and hedge funds use it to set capital reserves. But for discretionary and systematic traders, VaR provides a concrete, intuitive risk boundary that connects directly to position sizing, risk of ruin, and the shape of your return distribution.
There are three primary approaches:
The simplest method. Take your actual trade returns (or daily PnL), sort them from worst to best, and find the value at the desired percentile.
For 95% VaR with 100 trades:
Advantages: No assumptions about distribution shape. Uses your actual data. Disadvantages: Requires sufficient historical data. Past distribution may not represent future conditions.
Assumes returns follow a normal distribution and calculates VaR using the mean and standard deviation:
VaR = mu − Z × sigma
For 95% confidence, the Z-score is 1.645. For 99%, it is 2.326.
Advantages: Simple formula, easy to compute. Disadvantages: Assumes normality. Crypto daily returns are roughly Student-t with df ~ 3–4. A normal 99% VaR would call a -8 sigma BTC day a once-in-the-universe event; in 2020 and 2022 those days arrived in the same year. Parametric VaR can underestimate the true 99% tail by 3–10x — fat tails and skewness mean the Gaussian sigma simply does not measure what you need it to.
Simulates thousands of possible return paths based on your strategy's statistical properties, then extracts the VaR percentile from the simulated distribution.
Advantages: Can model complex, non-normal distributions. Flexible. Disadvantages: Computationally intensive. Quality depends on the accuracy of input assumptions.
For most active traders, Historical VaR is the most practical and reliable method. It requires no distributional assumptions and directly reflects your actual trading outcomes.
VaR has a critical limitation: it tells you nothing about the severity of losses beyond the threshold.
A 95% VaR of $500 says losses will exceed $500 on 5% of days. But will those bad days produce losses of $510 or $5,000? VaR is silent on this question.
This is known as the "VaR break" problem. Two strategies can have identical VaR but completely different tail risk profiles:
Same 95% VaR (-$500), wildly different tails. CVaR captures the gap; VaR does not.
Both report the same VaR. Strategy B is the one that ends careers. If you size on VaR alone, you cannot tell A from B — and the market will eventually pick B for you. This is not a theoretical defect; it is the reason 2008 happened on top of risk dashboards that were green.
Conditional Value at Risk (CVaR), also called Expected Shortfall (ES), addresses VaR's blind spot. It measures the conditional expectation of loss given that loss has already exceeded VaR. By construction CVaR is always at least as large as VaR — the average of the tail can never be smaller than the boundary that defines the tail.
CVaR_alpha = E[L | L ≥ VaR_alpha]
For 95% CVaR:
Using the example above:
Now the difference is stark. Strategy B's CVaR is four times worse than Strategy A's, revealing the hidden tail risk that VaR missed entirely.
CVaR is a strictly more informative metric than VaR for several reasons:
It looks inside the tail. VaR draws a line in the sand. CVaR explores what lies beyond that line. For traders who care about survival, what happens in the worst 5% of outcomes matters more than where the boundary sits.
It is coherent (Artzner, Delbaen, Eber, Heath 1999). CVaR satisfies subadditivity, which means diversifying a portfolio always reduces or maintains CVaR; VaR does not. After 2008, Basel III's FRTB (BCBS 2019) explicitly replaced VaR with Expected Shortfall at 97.5% for trading-book market-risk capital — the regulators who invented VaR abandoned it as the headline metric.
It penalizes fat tails. Trading returns are fat-tailed. Extreme losses occur more frequently than a normal distribution predicts. CVaR naturally captures this because it averages the actual extreme outcomes, however severe they may be.
It drives better decisions. When you optimize a strategy to minimize CVaR rather than VaR, you are directly reducing the expected damage of worst-case scenarios, not just the probability of crossing a threshold.
Using your last 200 trades at 95% confidence:
Suppose 200 BTC-perp scalps show:
Worst 5% threshold across 200 BTC-perp scalps. On the bad days, your losses begin here.
Average loss inside the worst 5% — twice the VaR. This is what tail days actually cost you.
The 2x gap means your stops are leaking — likely funding flips, liquidation cascades on CME open, or stop-runs through illiquid books. CVaR sizes you for that reality; VaR pretends those days don't exist.
In words:
1.2R or less2.4RIf your CVaR is significantly larger than your VaR, you have fat-tailed downside risk that feeds directly into your risk of ruin estimate. This is common in strategies that occasionally experience gap moves, slippage beyond stops, or correlated drawdowns.
You can use CVaR to set maximum position sizes:
Max position size = Acceptable loss / CVaR per unit
If your account can tolerate a maximum single-trade loss of $1,000 and your 95% CVaR is $200 per contract, you should trade no more than 5 contracts. This ensures that even in worst-case scenarios, your loss stays within acceptable bounds.
Sample size matters. With fewer than 50 trades, the 5th percentile is based on 2-3 data points. The estimate will be noisy and unreliable. Aim for at least 100 trades, ideally 200+.
Stationarity assumption. A historical 99% CVaR fitted to BTC's calm Jan–Feb 2020 said the worst-day loss should average around -8%. On 12 March 2020 BTC lost ~50% intraday. Regime breaks (LUNA, FTX, March 2020) make historical CVaR a lower bound, not a forecast — which is exactly when you need stress tests, not point estimates.
Confidence level choice. 95% is standard but somewhat arbitrary. For capital preservation, 99% may be more appropriate. The tradeoff: higher confidence requires more tail data and produces noisier estimates.
Not a worst case. CVaR is the average of tail losses, not the absolute worst outcome. Your actual worst loss will exceed CVaR by definition.
Hard to backtest. Unlike VaR, CVaR is not "elicitable" (Gneiting 2011) — you cannot score one CVaR forecast against another with a simple loss function. Practitioners backtest VaR exceedances and use CVaR as the sizing metric.
Noisier at the tail. A 99% CVaR on 200 trades averages just two data points. Estimation error is large; bootstrap your confidence interval before trusting the number.
| Dimension | VaR | CVaR |
|---|---|---|
| Definition | Loss at a specific percentile | Average loss beyond that percentile |
| Tail sensitivity | None (ignores losses beyond threshold) | High (averages all tail losses) |
| Coherent risk measure | No | Yes |
| Ease of calculation | Simple | Simple (requires one extra step) |
| Conservative | Less | More |
| Fat-tail awareness | Poor | Strong |
No. VaR is the loss level you exceed on (1 - alpha)% of days — the threshold of the tail, not the floor under it. The maximum loss is unbounded; CVaR estimates the average loss inside the tail.
After 2008 it was clear VaR ignored the magnitude of tail blowups — two strategies with identical VaR can have wildly different worst-case losses. The FRTB framework (BCBS 2019) replaced VaR with Expected Shortfall at 97.5% for trading-book market-risk capital.
95% is fine for routine sizing on 200+ trades. Use 99% only with 1000+ trades or you are estimating from 10 data points — the noise will dominate the signal.
Take your last 200 trade returns, sort them from worst to best, and read the 10th worst (5% of 200) as your 95% VaR. The average of those 10 worst returns is your 95% CVaR.
Adjust the skewness to see how the distribution shape changes. With negative skew, the left tail (losses) extends further — this is where VaR and CVaR become critical risk measures.
The Distribution of Trade Returns gives you the shape; Risk of Ruin gives you the survival probability; Position Sizing gives you the per-trade bet; Optimal Withdrawal & Growth Strategy gives you the long-run drift; Trade Expectancy Trees give you the conditional EV. VaR and CVaR are the tail-risk lens that ties them together — the shape parameter for everything sized off this module.