Formula, in Risk & Portfolio Construction
Historical VaR Formula
Historical Value-at-Risk is the empirical quantile of a portfolio's past returns at a chosen confidence level. To find the 95% one-day VaR, sort the historical daily returns and read off the loss at the 5th percentile. It makes no distributional assumption, letting the data's own fat tails and skew speak, but it can only show losses that already occurred in the sample.
4 variables. Published May 26, 2026. By AI Fin Hub Research.
VaR_alpha = - Quantile_(1 - alpha)( {R_1, R_2, ..., R_n} )
Index of the quantile observation in sorted ascending returns: rank = (1 - alpha) x n Variables
alpha
Confidence level
The probability the loss will not exceed the VaR, typically 0.95 or 0.99. The VaR sits at the (1 - alpha) quantile of the return distribution: a 95% VaR uses the 5th percentile.
R_i
Historical return
Each past periodic return in the lookback window. Historical VaR uses the actual observed returns directly with no parametric fit.
n
Sample size
Number of historical observations. More observations give a more stable tail estimate; too few and the quantile jumps between individual data points.
Quantile_(1 - alpha)
Empirical quantile
The return value below which a (1 - alpha) fraction of observations fall. VaR is reported as its negative so a loss is a positive VaR number.
Step by step
- 1
Collect the historical periodic returns over the lookback window.
Use the most recent 100 daily returns.
- 2
Sort the returns from most negative to most positive.
The worst 5 of 100 returns occupy the bottom of the sorted list.
- 3
Locate the (1 - alpha) quantile. For 95% over 100 observations, that is the 5th-worst return.
The 5th-worst daily return is -2.6%.
- 4
Report VaR as the absolute value of that quantile loss, scaling to a currency amount by multiplying by portfolio value.
On a 1,000,000 portfolio, 95% one-day VaR is 2.6% x 1,000,000 = 26,000.
Worked example
One-day 95% historical VaR on a 1,000,000 equity book
Lookback observations
100 daily returns
Confidence level
95%
5th-worst daily return
-2.6%
Portfolio value
1,000,000
At 95% confidence over n = 100, the quantile index is (1 - 0.95) x 100 = 5, so the VaR return is the 5th-worst observation, -2.6%. VaR = 0.026 x 1,000,000 = 26,000.
One-day 95% historical VaR of 26,000. There is a 5% chance the book loses more than 26,000 in a single day, based purely on the last 100 days. The method is honest about realized tail shape but blind to any crash bigger than what the window contains, so it should be paired with stress tests for losses outside the sample.
Common variations
Run the numbers next
- Calculators Returns Distribution Analyzer
Paste a returns CSV. Histogram, normal QQ plot, skewness, excess kurtosis, Jarque-Bera test, tail-weight index. See why Sharpe alone misleads.
- Calculators Risk-Adjusted Returns Calculator
Paste a returns CSV. Sharpe, Sortino, Calmar, Omega, alpha, beta, tracking error, information ratio, max drawdown, and tail moments — plus.
Sources
- Value at Risk: The New Benchmark for Managing Financial Risk , Philippe Jorion, McGraw-Hill (2006)
- RiskMetrics Technical Document , J.P. Morgan / Reuters (1996)
Keep the topic connected
- Formula Parametric VaR Formula
The parametric VaR formula: Value-at-Risk from the mean, volatility, and a normal z-score. The variance-covariance method, with a worked example.
- Formula CVaR / Expected Shortfall Formula
The CVaR formula: the average loss in the worst tail beyond VaR. Expected shortfall, a coherent risk measure favored by Basel, with a worked example.
- Learn Value at Risk (VaR)
Value at Risk: the loss threshold you'll exceed with probability α. Why historical VaR is brittle and what it doesn't tell you about the tail.
- Learn Expected Shortfall (CVaR)
Expected shortfall: the average loss given a VaR breach. Why regulators are migrating from VaR and what ES catches that VaR misses.