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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.

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  • $\begingroup$ 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. $\endgroup$
    – Martin Vesely
    Apr 5, 2020 at 21:55

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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.

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