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I have the following exercise to solve as good preparation for an exam:

NOTE: $VaR_p(X)$ = Value at risk = $F^{-1}_X(p)$

Consider the bivariate normal random vector $(X_1, X_2)$. The marginals are independent and standard normal distributed. The random vector $(Y_1, Y_2)$ follows the same distribution as $(X_1, V X_1)$, where the random variable V , independent of X1, is such that $P[V = 1] = P[V = −1] = 1/2$

Take p > 0.75. Prove that the following relations hold: $$VaR_p[Y_1 + Y_2] = 2\Phi^{−1}(2p-1)$$

Things I was already able to prove and that can be helpful are:

(a) $Y_1$ and $Y_2$ are both standard normal distributed.

(b) $Corr[Y_1, Y_2]$ = 0.

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    $\begingroup$ I find it helpful to draw a diagram of the support of $(Y_1,Y_2).$ From that it should be clear what the distribution of $Y_1+Y_2$ is. Alternatively, if you think better algebraically than visually, write down the simplest possible expression for $Y_1+Y_2 = X_1+VX_1 = X_1(1+V)$ that you can and work from there. $\endgroup$ – whuber Jan 8 at 20:47
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    $\begingroup$ There is no typo? Because then $X_2$ is a useless component of the exercise. $\endgroup$ – Xi'an Jan 8 at 21:02
  • $\begingroup$ No, it is used in other parts of the exercise :) $\endgroup$ – Chris Jan 9 at 9:39

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