Formula, in Risk & Portfolio Construction
Parametric VaR Formula
Parametric Value-at-Risk assumes returns are normally distributed and computes the loss threshold from the mean, the volatility, and the z-score of the chosen confidence level. Also called the variance-covariance or delta-normal method, it is fast and needs only two moments, but it systematically understates tail risk because real returns have fatter tails than the normal it assumes.
4 variables. Published May 26, 2026. By AI Fin Hub Research.
VaR_alpha = - (mu - z_alpha x sigma) x V
For a zero-mean assumption: VaR_alpha = z_alpha x sigma x V
z_alpha = 1.645 (95%), 2.326 (99%) Variables
mu
Mean return
Expected periodic return of the portfolio. Over short horizons it is often set to zero, since the volatility term dominates the tail at daily frequency.
sigma
Return volatility
Standard deviation of periodic returns. This is the single risk input that drives the VaR magnitude in the parametric method.
z_alpha
Normal z-score
The standard normal quantile for the confidence level: 1.645 for 95%, 2.326 for 99%. It is the number of standard deviations into the left tail at which the VaR cutoff sits.
V
Portfolio value
Current market value of the position or portfolio, used to convert the return-space VaR into a currency loss.
Step by step
- 1
Estimate the periodic mean and volatility of portfolio returns.
Daily mean assumed 0, daily volatility 1.8%.
- 2
Select the z-score for the desired confidence level.
For 99% confidence use z = 2.326.
- 3
Compute the VaR return as z times sigma (subtracting the mean if it is nonzero).
2.326 x 0.018 = 0.04187, a 4.19% loss threshold.
- 4
Multiply by portfolio value for the currency VaR.
On a 2,000,000 portfolio, 0.04187 x 2,000,000 = 83,740.
Worked example
One-day 99% parametric VaR on a 2,000,000 portfolio, zero mean
Daily volatility
1.8%
Mean return
0%
Confidence (z-score)
99% (z = 2.326)
Portfolio value
2,000,000
VaR return = z x sigma = 2.326 x 0.018 = 0.041868. VaR currency = 0.041868 x 2,000,000 = 83,736.
One-day 99% parametric VaR of about 83,700. Under the normal assumption there is a 1% chance of losing more than this in a day. Because true equity returns are leptokurtic, the real probability of breaching this threshold is higher than 1%, so parametric VaR should be backtested (Kupiec, Christoffersen) and supplemented with a fatter-tailed or historical method.
Common variations
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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 Historical VaR Formula
The historical VaR formula: the empirical quantile of past returns at a confidence level. A distribution-free Value-at-Risk method, with an 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.
- Formula Annualized Volatility Formula
The annualized volatility formula scales per-period return standard deviation by the square root of periods per year. Root-time scaling explained.
- 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.