Parametric Value at Risk (VaR)
Parametric Value at Risk (VaR)
Value at risk assuming returns are normally distributed with the given mean and volatility: the loss at the (1 − c) tail of that normal curve, z × σ − μ, where z is the standard normal quantile for the confidence level (1.645 at 95%, 2.326 at 99%).
When to use: Use when you have a volatility estimate rather than a long return history, or to compare portfolios on a common assumption. Real returns have fatter tails than the normal curve, so parametric VaR tends to understate rare losses.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| VaR | Value at Risk | Loss at the chosen confidence, as a positive fraction | % |
| MeanReturn | Mean Return | Expected periodic return, as a decimal | % |
| Sigma | Volatility | Standard deviation of periodic returns, as a decimal | % |
| Confidence | Confidence Level | Probability the loss will not be exceeded, as a decimal (0.95 for 95%) | % |
Real-Life Examples
Example 1: Monthly 95% VaR
A portfolio with a 0.8% mean monthly return and 4% monthly volatility, at 95% confidence.
Given
Step-by-Step
One month in twenty, the portfolio is expected to lose 5.8% or more. The positive mean return trims the figure slightly from the pure 1.645σ = 6.58%.
Example 2: Daily 99% VaR
A fund with a 0.1% mean daily return and 2% daily volatility, at 99% confidence.
Given
Step-by-Step
On the worst day in a hundred, expect a loss of at least 4.55% under the normal assumption. Actual markets deliver such days more often than the curve predicts.
Frequently Asked Questions
Under the normal assumption with independent returns, volatility scales with the square root of time, so a 10-day VaR is roughly the 1-day VaR times √10 (ignoring the small mean term).
When the mean return is large relative to z × σ the tail loss is actually a gain. That is a sign the horizon is long or the volatility low; VaR is most meaningful over short horizons.