$3.$ Answer all parts of the question

a$.$ Modern portfolio theory is built on a mean$-$variance framework. What assumptions are necessary for this framework to

be a credible working hypothesis? Are these assumptions valid? $(30$ marks$)$

b$.$ Assume now the existence of a riskless asset and of N risky assets. $(50$ marks$)$

i$.$ Represent on a graph the opportunity set and the minimum variance frontier. Identify the efficient frontier. What

is the two$-$fund separation theorem? Why is the two$-$fund separation theorem important?

ii$.$ Discuss the impact of short selling $($of both risky assets and the risk$-$free asset$)$ on the opportunity set, the

minimum variance frontier and the efficient frontier.

iii. What is the impact on the efficient frontier of having two different rates for borrowing and lending $($with the

lending rate being smaller that the borrowing rata$)?$

c$.$ Suppose that you have a risky asset that provides you with an expected return of $12\%$ per year with a standard deviation

of $20\%.$ Consider a risk$-$free asset that provides you with a $3\%$ risk$-$free return. $(20$ marks$)$

i$.$ If you have $$100,000$ and invest $80\%$ into the risky asset and $20\%$ into the risk$-$free asset, what is the expected

return and risk of your portfolio?

ii$.$ How much will your portfolio be worth if the realised return on the risky asset is $15\%?$

iii. If you cannot borrow money, what is the maximum possible expected return on your portfolio, and what is the

minimum?

iv$.$ If you are allowed to borrow money at the risk$-$free rate, how can you get a portfolio with an $18\%$ expected

return and what is the risk $($or standard deviation$)$ of this portfolio?