QUESTION $1.$

Consider a portfolio equally invested in two stocks: A and $B.$ Current market value of this portfolio is $100$ million. The daily volatility for stock A$\text{'}$s returns is $0\mathrm{.}09,$ and the daily volatility for stock B$\text{'}$s returns is $0\mathrm{.}05.$ The correlation between returns on these stocks is $0\mathrm{.}5.$

$1\mathrm{.}1)$ Define value at risk $(VaR)$ for a portfolio of stocks. Illustrate your definition with a relevant diagram.

$(15\%)$

$1\mathrm{.}2)$ Briefly explain the variance$-$covariance method to compute VaR. Compute VaR$(1$day, $95\%)$ of the portfolio described above, assuming the standard normal deviate for the $95\%$ confidence level is $1\mathrm{.}645.$

$(15\%)$

$1\mathrm{.}3)$ Compute VaR$(4-$day, $95\%)$ of the above portfolio. Explain your approach. In times of a crisis, would you adjust your calculation upwards or downwards, or would you apply no adjustment? Explain your answer.

$(20\%)$