$($Portfolio VaR$)$ Suppose there are two investments A and B$.$ Either investment

A or B has a $8\%$ chance of a loss of $$8$ million, a $12\%$ chance of a loss of $$1$

million, a $30\%$ chance of a profit of $$1$ million and a $50\%$ chance of a profit of $$4$

million. The outcomes of these two investments are independent of each other.

$($a$)$ What is the $90\%$VaR of the loss of investment A$?$ How about investment

B$?$

$($b$)$ What is the $90\%$ for a portfolio consisting of both investments A and B$?$

$($Hint: write out the probabilities of all possible portfolio outcomes.$)$

$($c$)$ Is the summation of the $90\%$VaRs of the individual investments greater

or smaller than the $90\%$VaR of the portfolio? If we measure the risk of

an investment or portfolio using VaR, does this suggest that diversification

must decrease risk? $($Intuitively$,$ putting A and B in a portfolio is a form

of diversification.$)$