An investment universe includes two assets, A and B$,$ with expected return on asset i of

$r_i$ and variance $v_i$ as set out below :

Asset i Expected return $r_i$ Variance of return $v_i$

A $,r_A=0\mathrm{.}05$

$v_A=0\mathrm{.}16$

B

$r_B=0\mathrm{.}07$

$v_B=0\mathrm{.}25$

The correlation of returns is $c_{AB}=-0\mathrm{.}2.$

In an efficient portfolio, let $a$ be the proportion which is held in asset A$.$

$($a$)(6$ points$)$ Express the portfolio variance $V$ in terms of a quadratic function in a$,$

showing your workings.

$($b$)(4$ points$)$ Let $R$ be the expected return on the portfolio.

Express the portfolio variance $V$ in terms of a quadratic function in $R,$ using your

result from part $($a$)$ and showing your workings.

$($c$)(10$ points$)$ The expression in part $($b$)$ represents the efficient frontier.

An investor uses a utility function that gives rise to an indifference curve

$V=16R-200R^2.$

Determine the two portfolios on the efficient frontier that also lie on the investor's

indifference curve.