Consider an arbitrary random variable X and let Y $=$ aX $+$ b for constants a $>0$ and b $$ R$.$ Show that VaR$($X $+$ Y $)=$ VaR$($X$)+$ VaR$($Y $)$ for $(0,1).$Exercise $2\mathrm{.}15($VaR and ES for a discrete distribution$)$

The following table contains the net profits on two lines of business A and B of a company

$xYZ$ :

a$)$ Calculate $Va{R}_{0\mathrm{.}95}$ and $E{S}_{0\mathrm{.}95}$ for each of the business lines A and $B.$

$2$ Basic Concepts in Risk Management

b$)$ Calculate $Va{R}_{0\mathrm{.}95}$ and $E{S}_{0\mathrm{.}95}$ for the combined profits of A and $B.$

How do your answers fit in with your knowledge of the coherence of the value$-$at$-$risk and

expected shortfall risk measures?