# VaR/inverse cdf of transformation of normal variables

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.

• 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. – whuber Jan 8 at 20:47
• There is no typo? Because then $X_2$ is a useless component of the exercise. – Xi'an Jan 8 at 21:02
• No, it is used in other parts of the exercise :) – Chris Jan 9 at 9:39