$(46$ POINTS$)$ A pension fund manager is co

nsidering investing in $2$ mutual funds: a stock fund

$(S)$ and a long$-$term corporate bond fund $(B).$ The following table presents the expected return and

standard deviation of the stock fund and of the bond fund:

The correlation between the stock fund and the bond fund is $=0\mathrm{.}3.$ The covariance $(r_E,r_D)$ is:

Cov$(r_E,r_D)={}_E{}_D=\mathrm{.}3(\mathrm{.}26)(\mathrm{.}12)=0\mathrm{.}0094$

$($a$)$ Find the mininimum variance portfolio. That is$,$

$($i$)$ Find the proportion invested in the stock fund, $w_E^{MVP}$ and the proportion invested in the bond

fund, $w_D^{MVP}=1-w_E^{MVP}.$

$($ii$)$ Find the expected return of the minimum variance portfolio, $E(r_{MVP})$ and its standard devia$-$

tion, ${}_{MVP}.$

$($b$)($i$)$ Find the portfolio frontier of the $2$ risky funds. Vary $w_E$ and $w_D=1-w_E$ in increments of

$\mathrm{.}10.$ For each of these pair of values calculate $E{r}_P,{}_P^2$ and ${}_{P.}$ With these data complete the followingtable: $($Use excel to perform these calculations and provide your excel sheet.$)$

and $($ii$)$ draw with precision the portfolio frontier on a graph with $E(r_P)$ on the vertical axis and ${}_P$

on the horizontal axis. On that graph also identify the minimum variance portfolio and the efficient

portfolio frontier.

$($c$)$ A risk averse investor with the following utility function

$=E{r}_P-A\frac{1}{2}{}_P^2$

wants to choose how much to invest in each of the corporate bond fund and equity fund to maximize

his$/$her utility. Let $A=5.$ Find the optimal choice of each fund.

$($d$)$ A Treasury Bill money market fund is now available with a $3\%$ safe return $(r_f=\mathrm{.}03).$ Draw the

capital market line $($CML$).$ Identify the intercept on the vertical axis and the tangency point with the

efficient portfolio frontier for risky assets. What is the efficient set in this case?$($ii$)$ Find the expected return of the market portfolio, $E(r_M)$ and its standard deviation, ${}_M.$

$($iii$)$ Find the Sharpe ratio.

$(f)$ A risk averse investor with the following utility function

$=E{r}_P-A\frac{1}{2}{}_P^2$

wants to choose how much to invest in the money market fund yielding a safe rate of return $r_f$ as well

as in the market portfolio, $M$ to maximize his$/$her utility. Let $A=5,$ find the opitmal choice of each

fund.

$($g$)$ Another investor requires a target expected return of $12\%,$ that is $E(r_{\text{T}\text{A}\text{R}\text{G}\text{E}\text{T}})=0\mathrm{.}12.$ Find the

standard deviation of her portfolio, ${}_{\text{T}\text{A}\text{R}\text{G}\text{E}\text{T}}.$ What proportion will she invest in the money market

fund, $w_{\text{M}\text{O}\text{N}\text{E}\text{Y}}?$ What proportion will she invest in the equity fund, $w_S?$ What proportion will she

invest in the bond fund, $w_B?$

$($e$)$ Find the market portfolio.

$($i$)$ That is find the proportion invested in the stock fund, $w_E^M$ and the proportion invested in the

bond fund, $w_D^M=1-w_S^M$ of the market portfolio