ABSTRACT
This chapter introduces a theoretic framework of Markowitz portfolio theory. The set of optimal portfolios include those portfolios with the minimum variance σ2 p for each level of expected return r p . In Markowitz’s original work, short sales are allowed (i.e., negative portfolio weights are permissible). When short sales are allowed, the optimal portfolio weights are derived with Lagrange multipliers and matrix algebra. If short sales are not allowed, quadratic programming needs to be used to derive optimal portfolios (as is demonstrated both with and without the risk-free asset). The chapter demonstrates using the R package to solve this quadratic programming problem and discusses linear programming using the R package since it is useful in portfolio management. Finally, Geometric Brownian motion is assumed for asset returns and solutions are demonstrated based on the approach of finding the portfolio of maximal growth rate using quadratic programming. On the theoretical side, the CAPM is proved as an extension of Markowitz portfolio theory. In addition to sample R codes, sample codes of an analytic solution through matrix algebra in Mathematica are offered so that readers can mimic and practice. The chapter includes a project to construct the Markowitz frontier for a 10-stock portfolio.
