How do i solve below optimization problem analytically :

Given n securities with Expected return vector and Covariance matrix , consider the Portfolio with Expected return _p() = ^T and variance ² _p() = ^T, where Rⁿ is the vector of weights. We denote by e the column vector of Rⁿ with all entries equal to 1.

1 ) Using a Lagrangian approach, provide the analytical solution of

maximize _p() subject to:

e^T = 1

² _p() = ² T

2) Now we take n = 2 and denote the optimal solution by (_T) and the Lagrange multiplier for the risk constraint by ₂(_T). Assume that

= 5%

10%

and = 1% 1%

1% 4%

and consider a grid of _T in the range [2%,30%] by steps of 0.5%.

Plot the ecient frontier, namely the graph of the mapping _{T p}((_T)) (volatility on x-axis and expected return on the y-axis )

Add to that gure the graph of the mapping _{T 2}(_T). How do you interpret ₂?

3) Using a Lagrangian approach, provide the analytical solution of :

minimize ²_p(w) subject to :

e^T = 1

_p() = _T and plot the ecient frontier.