Consider a $1m$ portfolio consisting of $60\%$ invested in asset A and $40\%$ invested in asset B$.$ Daily volatility of asset A returns is $0\mathrm{.}09,$ while daily volatility of asset $B$ returns is $0\mathrm{.}05.$

$1\mathrm{.}1)$ What is Value at Risk $($VaR$)?$ Under which assumptions can one compute VaR from the above data? If the standard normal deviate factor is $1\mathrm{.}645$ at the $95\%$ level, what are individual $1-$day VaRs at $95\%$ confidence on each asset in this portfolio? Would you expect the portfolio's VaR to be greater or smaller than the sum of these two assets' VaRs? Explain your answer.

$(15\%)$

$1\mathrm{.}2)$ Comment on the following statement: "VaR is not a subadditive risk measure, therefore it is not appropriate to use it in risk measurement".

$(15\%)$

$1\mathrm{.}3)$ Define conditional value at risk $($CVaR$).$ Can assets A and $B$ above have identical CVaRs? Illustrate your answer with a relevant diagram.

$(20\%)$