A market consists of four stocks. The following information is given:

Stock $1$: r$1=5\%,\backslash$sigma $1=10\%$

Stock$2$: r$2=10\%,\backslash$sigma $2=20\%$

Stock $3$: r$3=15\%,\backslash$sigma $3=30\%$

Stock $4$: r$4=20\%,\backslash$sigma $4=40\%$

$\backslash$rho $12=\backslash$rho $13=\backslash$rho $14=0\mathrm{.}6$

Returns of Stock $2,$ Stock $3$ and Stock $4$ are independent of each other.

Let P be a portfolio of the three stocks with weights $($w$1,$ w$2,$ w$3,$ w$4)$ respectively.

Let $$ denote the risk of such a portfolio.

a$)$ Given the information above, write down the expression for $$

$2$

$/2$ by using the numerical

values of the returns, risks and correlations givenA market consists of four stocks. The following information is given:

Stock $1$: $r_1=5\%,{}_1=10\%$

Stock$2$: $r_2=10\%,{}_2=20\%$

Stock $3$: $r_3=15\%,{}_3=30\%$

Stock $4$: $r_4=20\%,{}_4=40\%$

${}_{12}={}_{13}={}_{14}=-0\mathrm{.}6$

Returns of Stock $2,$ Stock $3$ and Stock $4$ are independent of each other.

Let P be a portfolio of the three stocks with weights $(w_1,w_2,w_3,w_4)$ respectively.

Let ${}_P$ denote the risk of such a portfolio.

a$)$ Given the information above, write down the expression for $\frac{{}_P^2}{2}$ by using the numerical

values of the returns, risks and correlations given.

$(3$ marks$)$

b$)$ Your client wants a portfolio with expected return of $35\%$ and the lowest possible risk.

Write down the Lagrangian needed for the optimization calculation based on the expression

you got in Part a$)$

$(2$ marks$)$