QUESTION FIVE$(15$ marks$)$

a$)$ Consider an investment whose returns $x$ follow a continuous uniform distribution over

the range $5\%$ to $18\%$ pa$.$

i$)$ Calculate the variance and downside semi$-$variance of returns $(3$ marks$)$

ii$)$ Calculate the shortfall probability and the expected shortfall based on a benchmark

level of $6\%(4$ marks$)$

b$)$ Let $x$ and $Y$ be two identical corporate bonds each with a default probability of $5\%,$ a

notional of $100$ and assume no recovery if default occurs and that defaults are independent.

Compute the $95\%$VaR$(x+Y)$ and thus show that value$-$at$-$risk is generally not sub$-$

additive. $(4$ marks$)$

c$)$ Show that any coherent risk measure $(x)$ is a convex function. $(4$ marks$)$