Assume that there are only $2$ assets in the market and that short selling is possible.

Consider the two assets A and $B$ for which the prices $p_A(t)$ and $p_B(t),t=0,1,2,$dots$30$ observed over $31$ periods are given in the Excel file 'Data$-$Exam.xIsx' available on Moodle.

Exercise $1$

$1.$a For each asset compute the series of the $30($linear$)$ returns corresponding to the $30$ periods $t=1,2,$dots,$30.$

For the obtained series of returns, compute the characteristic values $E(R_A),{}_A^2($and $($:${}_A\},{}_B^2($and ${}_B),$ as well as the correlation coefficient ${}_{AB}.$

Plot the values of the linear returns over time.

$1.$b Consider the $9$ portfolios obtained as a combination of A and $B$ with a fraction of capital invested in A equal to $0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}4,0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}7,0\mathrm{.}8,0\mathrm{.}9,$ respectively. For each such portfolio P compute $E(R_P)$ and ${}_P.$

$1.$c Plot in the plane $({}_P,E(R_P))$ the points corresponding to assets A and $B,$ and those corresponding to the $9$ portfolios found in Exercise $1.$b to obtain an approximation of the opportunity set in the given market.

Exercise $2$

$2.$a Considering that short selling is possible, assume now that the investor wants to sell short asset $A,$ in order to invest more in asset $B.$ Therefore consider portfolio $P_{SS}$ which corresponds to investing in $B$ a quantity $x_B=1\mathrm{.}20.$ Compute the values $E(R_{P_{SS}})$ and ${}_{P_{SS}}$ also for this portfolio and include the corresponding point in the previous plot.

$2.$b In the plane $({}_P,E(R_P))$ consider the point which represents a portfolio with the following characteristic values: $E(R_P)=0\mathrm{.}02$ and ${}_P=0\mathrm{.}06.$ Add also this point in the previous plot. In this market with only two assets A and $B$ establish the nature of all the portfolios represented by points A$,$ B$,$ P$3,$ P$4,$ P$5,$ P$6,$ P$7,$ P$8,$ P$9,$ Pss and P $($possible$,$ impossible, efficient, dominated, etc.$).$