Estimate VaR using a parametric approach for both normal and lognormal

In contrast to the historical simulation method, the parametric approach (e.g., the delta- normal approach) explicitly assumes a distribution for the underlying observations.

We will analyze two cases:

(1) VaR for returns that follow a normal distribution, and

(2) VaR for returns that follow a lognormal distribution.

Intuitively, the VaR for a given confidence level denotes the point that separates the tail losses from the remaining distribution. The VaR cutoff will be in the left tail of the returns distribution. Usually, the calculated value at risk is negative, but is typically reported as a positive value since the negative amount is implied (i.e., it is the value that is at risk).

Normal VaR

The normal distribution is usually used to simulate the stock return. In this case, the VaR at significance level $$\alpha$$ is:

$$VaR(\alpha\%) = (\mu_R + \sigma_Rz_\alpha) P_{t-1}$$

In practice, the population parameters are not likely known, in which case the researcher will use the sample mean and standard deviation.

Lognormal VaR

Lognormal is used to simulate stock price as it’s always larger than 0. To calculate VaR in lognoram form. We need to transform the  non-logarithmized price mean and variance, denoted m and v, into a loarithmized mean and variance.

$$\mu = ln(\frac{m_{p/l}}{\sqrt{1+\frac{v_{p/l}}{m_{p/l}^2}}})$$.

$$\sigma^2 = ln(1+\frac{v_{p/l}}{m_{p/l}^2})$$ .

$$VaR(\alpha\%) = e^{\mu_{p/l} + \sigma_{p/l} z_\alpha}$$