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.
Formula
Copy the exact expression or work through it step by step below.
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
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Sources & References
- Value at Risk: The New Benchmark for Managing Financial Risk — Philippe Jorion, McGraw-Hill (2006)
- RiskMetrics Technical Document — J.P. Morgan / Reuters (1996)
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