QUESTION ONE$(30$ marks$)$

a$)$ i$)$ Define index tracking $(1$ mark$)$

ii$)$ Outline the two$-$step approach for sparse index tracking $(4$ marks$)$

b$)$ Let $xN(0,{}_x^2)$ and $YN(0,{}_y^2)$ be two loss random variables with their $(\{$:$1-p)$ VaR

being $Z_{1-p}{}_x$ and $Z_{1-p}{}_y$ respectively. Show that VaR$(x+Y)$VaR$(x)+$VaR$(Y)(3$

marks$)$

c$)$ Let $x$ have an Exponential distribution with parameter $.$

i$)$ Given a random sample of size $n,$ by choosing suitable normalizing constants deter$-$

mine the cumulative distribution $F^n$ of the normalized block maxima.$(4$ marks$)$

ii$)$ Compute $\underset{n}{lim}{F}^n(3$ marks$)$