hi there,

I am looking for

-- a numerical method

and

-- a closed form solution

for the expected return formulation of the classical mean variance portfolio optimization problem.

max_{w}(w' \mu)

s.t.

w' \Sigma w = \sigma^{2}_{0}

w' i = 1

Could someone please let me know some references about this?

Good news: This equation exists. They figured out the precise portfolio maximum return with no constraints computation using matrix algebra at NYU and published it in a paper. Check their papers. It will give you negative weights if you use it (the optimal portfolio value includes short positions). There is no practical application because most people don't maximize returns without constraints.

Bad news: I didn't write it down because I use linear programming. I have it saved somewhere in an excel file, but I don't want to look for it that badly. If you really, really need it I can look.

You can easily compute portfolio return using matrix algebra. Corrected - Portfolio variance = W'CW, where W = weight matrix and C = covariance matrix. Your portfolio returns is just weights times position variances. Use linear programming to solve for the weights while minimizing standard deviation given a return. The sum of the weights must be equal to 1. The individual weights may or may not have to be greater than zero, depending up whether or not you have the ability to hold short positions.