Difference between Delta-Gamma and Delta-normal method for VaR

Question

I was reading the differences between Delta-Gamma and Delta-normal method for VaR. One of the difference I found is mentioned below, but I can't understand it's importance. Can anybody please explain this?

The improved accuracy by the Delta-Gamma method comes at the cost of at least some reduced tractability relative to the Delta-Normal model. In using Delta- Gamma approach we might lose normality in our portfolio return even if changes in the underlying risk factors are normally distributed

Thanks

Answer 1

Delta-Normal VaR means the portfolio is approximated as a linear function $a + \Delta^T X$ of gaussian risk factors $X$ with variance covariance $V$, so the portfolio distribution is also Gaussian, with variance $\Delta^T V \Delta$, and its VaR is computed explicitly as a percentile on the Gaussian probability distribution function.

Delta-Gamma VaR means the portfolio is approximated as a quadratic function $a + \Delta^T X + \frac{1}{2} X^T \Gamma X$ of gaussian risk factors $X$. The portfolio distribution is no longer Gaussian (because of the quadratic term) and its probability distribution function must be computed trough a semi-explicit numerical method such as a fast Fourier Transform applied to its moment generating function.

See Stefan R. Jaschke risk management for financial institutions http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.543.8751&rep=rep1&type=pdf for a good overview of this topic.