Investment Theory and Risk Management, + Website

Markowitz Portfolio Selection Criteria

Efficiency is about minimizing risk. The efficient frontier is the set of risk-minimizing portfolios for any given set of returns. The mean-variance efficient portfolio is the one with minimum risk. Markowitz's contribution was to give us the calculus to solve portfolio problems like this. The objective is to find the weightings (w_i) that minimize the risk for a given portfolio return. Achieving this would generate a weighting of the assets in a portfolio, which would occupy a point on the efficient frontier. How can one achieve this objective? Consider again the risk we solved earlier and reprint here:

We desire to choose the weightings (the w_i and w_j) that minimize this quantity for a given portfolio return. There are several ways to set up this problem and I outline a few here that serve as a platform for our empirical work in Excel. I begin with a very intuitively appealing representation of the problem as a risk-adjusted return maximization in matrix format that solves for the vector w that maximizes the quadratic form given by:

Here w is a k-dimensional row vector of weights , where k stands for the number of assets in the portfolio, r