The Mean Variance (MV) assumption is the most commonly used function in portfolio optimization. The results of mean-variance analysis hold when (a) investors have quadratic utility functions or (b) returns are normally distributed.

The optimization is typically structured in the following manner:

Let,

N is the number of assets in the investment universe.

w = portfolio weight vector.

m = expected return vector.

D = covariance matrix of dimension N 2.

The mean of a portfolio with weights w i , where i=1 to N is:

(where T is the transpose), and the variance is:

The portfolio is mean-variant efficient for a given level of portfolio expected return l m (where l is a constant) if it satisfies the following conditions:

minimize the variance

subject to the constraint

Frequently, the portfolio weights are normalized so that they sum to unity and they are non-negative (no borrowing).

The typical method for optimization is to use “parametric quadratic programming”. This curve is generated by minimizing the following function for various values of l :

or, maximize: