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.

  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.