Q$.2($Portfolio optimization with no$-$shortselling, $35pts).$ In the lectures, we consider

portfolio optimization problems in which shortselling is allowed. This idealization

allows us to solve the problems analytically using Lagrange duality. The purpose of

this exercise to push us to think about the case where shortselling is not allowed.

To that end, consider a market with three risky assets with respective risk levels ${}_1,{}_2,$

${}_3,$ all strictly positive. The returns of the first two assets are correlated with correlation

coefficient ${}_{12}.$ Assume that The return of the third asset is uncorrelated

with the returns of the first two assets. A risk$-$averse investor with positive intial wealth

wants to choose a portfolio for themself.

First, consider the situation where shortselling is allowed.

a$.(5)$ Show that there is a unique minimum variance portfolio $($MVP$).$ Calculate the

weights of the MVP$.$ Show that the MVP has shortselling.

Now, suppose that shortselling is no longer allowed.

b$.(5)$ In a generic portfolio, denote the weight of the first asset by a variable $s,$ denote

the weight of the second asset by a variable $t.$ Using only $s,t$ as decision vari$-$

ables, formulate the problem of finding an MVP with no$-$shortselling. Express the

objective function as explicitly as possible.

c$.(5)$ Among all portfolios with no$-$shortselling and no investment in the first asset,

find the one with minimum variance of return.

d$.(5)$ Among all portfolios with no$-$shortselling and no investment in the second asset,

find the one with minimum variance of return.

e$.(5)$ Compare the risks of the two portfolios that you find in c$.$ and d$.$ Which one has

smaller risk?

f$.(10)$ Show that the portfolio that wins in e$.$ is indeed the unique optimal solution of

the problem in $b.$

Hint: Draw pictures of the contour curves of the quadratic objective function.