Parametric VaR (Normal)

Parametric VaR (Normal Distribution)

The parametric (variance-covariance) approach assumes returns follow a normal distribution. Given the mean μ and standard deviation σ of daily returns:

Formula

VaR = -(μ + z * σ)

where z = N⁻¹(1 - confidence) is the inverse normal CDF quantile.

Key quantiles:

• 95% confidence → z ≈ −1.6449
• 99% confidence → z ≈ −2.3263

Implementing the Inverse Normal CDF

Use the identity: N⁻¹(p) = √2 · erfinv(2p − 1)

Implement erfinv via bisection on math.erf.

Example

μ = 0.001, σ = 0.02, 95% confidence:
z = N⁻¹(0.05) ≈ −1.6449
VaR = −(0.001 + (−1.6449)(0.02)) = −(0.001 − 0.03290) ≈ 0.0319