Consider two stocks, Stock $$ and Stock $,$ and the following information.

$(1)$ The expected rate of return and the standard deviation of stock $$ are $=$

$0\mathrm{.}1$ and $=0\mathrm{.}3.$

$(2)$ The expected rate of return and the standard deviation of stock $$ are $=$

$0\mathrm{.}07$ and $=0\mathrm{.}14.$

$(3)$ The covariance between the rates of return is $,=0\mathrm{.}01.$

$2$

There is also a risk$-$free asset whose rate of return is $=0.$

$2\mathrm{.}1\mathrm{.}1.$ It can be shown that in the $$market portfolio$,$ the weights of securities $$ and

$$ should respectively be equal to $$ and to $1,$ where$1$

$=$

$()$

$2()$

$()$

$2(+2)+()$

$2$

$,$

where $$ denotes a lending or borrowing rate.

Calculate the values of the weights, as well as the expected rate of return of the market

portfolio and its standard deviation.

$2\mathrm{.}1\mathrm{.}2.$ Which positions should a $$mean$-$variance$$ investor take in the market portfolio and in the risk$-$free asset if his or her target rate of return were equal to $($i$)6\%,($ii$)$

$10\%?$ What would be the standard deviation of the rates of return faced by the investor

in the two cases?

$2\mathrm{.}1\mathrm{.}3.$ Suppose that the investors in $2\mathrm{.}1\mathrm{.}2$ can invest funds at the risk$-$free rate of

$0\%,$ but can only borrow funds at the rate of $4\%.$ Would you change your answer to $($i$)$

and$/$or $($ii$)$ in $2\mathrm{.}1\mathrm{.}2?$