Assume there are two portfolios, perhaps one for each of two asset classes, each subject to a random loss $L_1$ and $L_2$, each of the realised losses conforming to the normal distribution over the time period mandated by the regulator. In reality this might be a portfolio of stocks which yields a certain average loss over a given time horizon and a portfolio of commodities which clearly would report an average loss or gain that is quite different over the same time horizon. If the assumption of normality for each asset is reasonable, we could say that the realised loss one time step into the future would be
$$\mathcal{L}_1 = V_1 l_1$$
$$\mathcal{L}_2 = V_2 l_2$$
where $V_1, V_2$ are the initial porfolio values and the loss random variables
$$l_1 \sim N(\mu, \: \sigma_1^2)$$
$$l_2 \sim N(\mu, \: \sigma_2^2)$$
The question now is, what happens when an investor pairs the portfolios and hence their losses? As VaR has unique characteristics under the assumption of normality, it is prudent to ascertain whether when calculated using the normal distribution a pair of protfolio losses $L_1$ and $L_2$ will retain their normally distributed characteristics. The following can be said about the combined risk arising from $L_1$ and $L_2$
$$ L_1 + L_2 \sim N ( \mu, \: \sigma^2)$$
where
\begin{align} \mu &= V_1 \mu_1 + V_2 \mu_2\\ \\
\sigma^2 &= V_1 \sigma_1^2 + V_2 \sigma_2^2 + 2 V_1 V_2 \rho \sigma_1 \sigma_2 \end{align}