# Value at Risk (VaR): Normal distribution with gamma distributed volatility

If I was to do a 99% VaR calculation on a portfolio with normally distributed returns $$\mathcal{N} (\mu,\sigma)$$, the 99% VaR would be $$\mu - 2.33\sigma$$.

Instead of having a constant volatility, let's say volatility is gamma distributed, i.e. $$\sigma \sim \Gamma(k, \theta)$$.

Explain in as many details as possible (either derive a formula or explain a numerical solution using a computer program) how to compute the VaR of the portfolio when returns have a normal distribution conditional on sigma, and sigma is distributed according to a gamma distribution.

• Just idea: try to generate $\sigma$ from gamma distribution, save it and then generate returns with normal distribution $\mathcal{N}(\mu,\sigma)$. Generate another $\sigma$ and so on. Collect set of returns generated in this way and calculate VaR. – Martin Vesely Apr 5 '20 at 21:55

You might want to have a look at the conjugate priors to the normal distribution (with known mean) Your setup will result in a $$t$$-distributed return with updated dispersion parameter.