Metrics5 min read

Value at Risk and CVaR

Value at Risk (VaR) and Conditional Value at Risk (CVaR) quantify tail risk by answering two distinct questions: how bad can losses get on a typical bad day, and how bad are losses when that threshold is breached. VaR is a quantile of the loss distribution; CVaR is the expected loss conditional on exceeding that quantile.

For a backtested strategy, these metrics constrain position sizing, capital allocation, and drawdown expectations far more usefully than volatility alone. A returns series with the same standard deviation can have wildly different VaR and CVaR depending on skew and kurtosis.

Computation

VaR at confidence level α (commonly 95% or 99%) is the threshold loss such that the probability of a worse loss equals 1 − α. CVaR is the conditional expectation of the loss given that the loss exceeds VaR.

VaR_α(L) = inf { ℓ ∈ ℝ : P(L > ℓ) ≤ 1 − α }

CVaR_α(L) = E[ L | L ≥ VaR_α(L) ]

In practice, with a vector of historical returns r of length N, the 95% historical VaR is the negative of the 5th percentile of r, and the 95% historical CVaR is the negative of the mean of all returns at or below that 5th percentile. Parametric variants assume a distribution (typically Gaussian or Student-t) and compute the quantile analytically. Monte Carlo variants simulate paths from a fitted process.

VaR_95 ≈ −quantile(r, 0.05)    CVaR_95 ≈ −mean(r[r ≤ quantile(r, 0.05)])

Interpretation

VaR is reported as a positive loss number over a defined horizon. A daily 95% VaR of 1.8% means that on 5% of days — roughly one trading day in twenty — the strategy is expected to lose more than 1.8%. The corresponding CVaR is always larger: it might be 2.7%, meaning the average loss on those bad days is 2.7%.

Typical ranges for daily returns of a moderately leveraged systematic strategy: 95% VaR between 1% and 3%, 99% VaR between 2% and 6%. CVaR/VaR ratios usually fall between 1.3 and 1.8 for near-Gaussian return profiles; ratios above 2.0 indicate heavy tails and warrant scrutiny. Strategies selling optionality (short volatility, mean reversion in trending regimes, carry trades) systematically produce CVaR/VaR ratios above 2.

CVaR is a coherent risk measure; VaR is not. CVaR satisfies subadditivity, meaning the CVaR of a portfolio cannot exceed the sum of the CVaRs of its components. This makes CVaR the preferred metric for portfolio-level risk budgeting and convex optimization.

What These Metrics Do Not Capture

Neither VaR nor CVaR says anything about losses that unfold over multiple periods. A strategy with a benign daily 99% VaR can still drawdown 40% over six months through a sequence of moderate losses none of which individually breach the threshold. Drawdown metrics — maximum drawdown, time underwater, Ulcer Index — address this dimension separately.

Historical VaR is bounded by the worst observation in the sample. If the backtest contains no 2008 or 2020-style event, the historical 99% VaR underestimates the true tail. Parametric Gaussian VaR underestimates it even further because financial returns exhibit excess kurtosis. CVaR partially mitigates this by averaging the tail, but only over the tail that was actually observed.

VaR is not a worst-case loss. A strategy with daily 99% VaR of 3% will, by construction, lose more than 3% on about 2-3 days per year. The 99.9% quantile and beyond — the regime that breaks accounts — is precisely where VaR is least informative.

Both metrics also assume the return-generating process is stationary across the lookback window. Regime shifts (volatility expansion, correlation breakdown, structural changes in liquidity) invalidate the empirical distribution. Rolling estimation and stress-testing against synthetic shocks are necessary complements.

In Kestrel Signal

Kestrel Signal reports historical VaR and CVaR at the 95% and 99% confidence levels on both daily and weekly return series for every backtest. The CVaR/VaR ratio is surfaced explicitly so heavy-tailed strategies are flagged before capital sizing. Rolling 252-day windows are plotted alongside the full-sample estimates so non-stationarity in tail behavior is visible rather than buried in a single aggregate number.