Problem $1.(15$ points$)$ Suppose an investor wants to invest $V_0=1000$ in $5$ stocks and $1$ risk$-$free asset over an investment horizon of $1$ day.

The return on the risk$-$free asset is $R_0=$exp$(\frac{0\mathrm{.}1}{365})$ and the one$-$day returns $R$ for the stocks have expected values ${}_k$ and standard deviations ${}_k$ given by

${}_1=1\mathrm{.}02,{}_1=0\mathrm{.}05,$

${}_2=1\mathrm{.}20,{}_2=0\mathrm{.}10,$

${}_3=1\mathrm{.}07,{}_3=0\mathrm{.}15,$

${}_4=1\mathrm{.}25,{}_4=0\mathrm{.}10,$

${}_5=0\mathrm{.}75,{}_5=0\mathrm{.}12.$

a$)$ Determine the optimal amounts to be invested today in the stocks and the bond in order to maximize the expected portfolio value tomorrow when the standard deviation of $V_1($the portfolio value tomorrow$)$ is not allowed to exceed ${}_0{V}_0=20.$ The investor supposes that all returns are uncorrelated with each other. Short$-$selling is allowed.

b$)$ Suppose instead that you want to maximize the expected value of $V_1$ but now allowing for a portfolio standard deviation up to ${}_0{V}_0=5.$ Explain how you can compute the new optimal investment amounts from the ones in a$)$ by multiplication with a constant.

c$)$ Under the same conditions as in a$),$ suppose all random returns are normally distributed. What is the distribution of the optimal portfolio value? Plot the density function of the optimal portfolio value.

Use Matlab when necessary