let us explore each component. Suppose there are three stocks and the allocated proportion

is $(x,y,z).$ Since they are proportions, the sum of them is equal to $1.$ The required mean

return of the portfolio is equal to $4.$ From the estimation results of the historical data, the

mean return vector of each share of stock is $(1,10,6)$ and their variance are $(1,4,2)($we can

see that the stock with higher mean is also more volatile$).$ The correlation between the first

and second stock is $0\mathrm{.}125.$ Then the variance of the portfolio with allocation $(x,y,z)$ can be

written as

$x^2+0\mathrm{.}5xy+4y^2+2z^2.$

a$.$ We want to minimize the variance of the portfolio to obtain our optimal portfolio. In

the course, we have mainly focused on maximization problem. How should we modify

the objective function such that it becomes an maximization problem? $($Hint: min $f(x)$

is the same as max $-f(x).)$

b$.$ Please write down the equality constraints.

c$.$ Obtain the optimal allocation by applying what we have learned in constrained opti$-$

mization problem.

d$.$ Translate the problem into an univariate unconstrained optimization and solve it$.$