$($Investment Management $)($ Finance $)$

MR Y can invest in two financial securities$,$ security A and security B$.$ The table below gives a

description of the states of the world, their respective probabilities, and the return of each security

in each state.

a$.$ Compute the expected return and the standard deviation of returns of a portfolio which

places equal weights in security A and security $B.$

b$.$

c$.$ If the risk$-$free interest rate is $RF=2\%,$ estimate the optimal risky portfolio for the investor

when short sales are allowed. Calculate the expected return and standard deviation of

the optimal risky portfolio.

d$.$ Suppose that there is a third security, security C$.$ Its return is $-10\%$ when the market is

Bear, and $5\%$ when the market is normal. When the market is Bull, security C has $30\%$

chance to give a return of $20\%$ and $70\%$ to give a return of $80\%.$ Find the optimal risky

portfolio for MR Y$,$ when he can invest in securities $A,B,$ and $C$ and short sales are

allowed. Calculate the expected return and standard deviation of the optimal risky

portfolio.