Exam $1\mathrm{.}1($Mean variance and CAPM$)$ Suppose the investment universe consists of only three assets, with rates of return $11,12,$ and $13,$ respectively. The covariance matrix, V$,$ and vector of expected rates of return E$[$r$]$ are:

V$=$

$=$

$[0\mathrm{.}4$

$0$

$0$

$00\mathrm{.}5-0\mathrm{.}3$ E$[$r$]$

$0-0\mathrm{.}30\mathrm{.}5$

$[0\mathrm{.}25]=0\mathrm{.}07$

$0\mathrm{.}07$

$=$

$(0\mathrm{.}5,0\mathrm{.}25,0\mathrm{.}25).$ Show

$1.$ Consider portfolio A with portfolio weights $($x$1,$ x$2,$ x$3)$ that the expected return and return variance of portfolio A are E$[$ra$]$ Var$[$rA$]=0\mathrm{.}125$ respectively. $(4$ points$)$

$=0\mathrm{.}16$ and

$2.$ Find the portfolio weights of the minimum$-$variance portfolio. $(6$ points$)$

$3.$ Suppose that a risk$-$free asset with rate of return r$$ exists, and that portfolio A is in fact the market portfolio, i$.$e$.$ the efficient portfolio of risky assets. Show that in this case it must be that rf $=0\mathrm{.}01.(6$ points$)$

$4.$ Assume that all the assumptions underlying the CAPM hold, and again that A is the market portfolio. What is the beta of asset $1?(3$ points$)$

$5.$ Suppose asset B has a beta equal to $0\mathrm{.}5.$ What is the covariance of this asset with the market portfolio A$?(3$ points$)$