# Value-at-Risk of two dependent lognormal random variables

Hello, (I searched the wohole literature and internet and I have no idea myself for this important question)

Let $(X_1,X_2)$ be a multivariate normal random vector ($X_1$ and $X_2$ need not be independent). Is it possible to calculate $$VaR_{\alpha}\bigl(e^{X_1}+e^{X_2}\bigr)$$ analyticaly? Or is it even possible to calulate it in terms of $$VaR_{\alpha}(e^{X_1}) \ \text{ and} \ VaR_{\alpha}(e^{X_2}),$$ i.e. is there a representation (a function $g(\cdot,\cdot)$) of the form $$VaR_{\alpha}(e^{X_1}+e^{X_2})=g\Bigl(VaR_{\alpha}\bigl(e^{X_1}\bigr),VaR_{\alpha}\bigl(e^{X_2}\bigr)\Bigr)$$.

I would be very thakful for hints.

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Can you solve the problem if $X_1$ and $X_2$ are independent? – Shai Covo Mar 17 '11 at 16:54
If $X_1$ and $X_2$ were independent then the distribution function of the sum $e^{X_1}+e^{X_2}$ is a convolution. But this way of trying it has the disadvantage that you dont see if it is analyticaly tractable. – bobbey Mar 17 '11 at 17:58
You may want to ask this question on a sister site devoted to quantitative finance: quant.stackexchange.com – Andrey Rekalo Mar 17 '11 at 18:03
thank you for the suggestion, I was not aware of this page. – bobbey Mar 17 '11 at 18:17
Some useful references are given in mathoverflow.net/questions/30619/… (see the accepted answer; you are not going to find something better than approximations). – Shai Covo Mar 17 '11 at 20:27