Portfolio Optimization

The problem of investing money is common to citizens, families and companies. In this chapter, we introduce the decision framework of the portfolio selection problem in general terms. We describe the basic concepts of financial assets, capital to invest, performance (rate of return) and risk (measure of dispersion) possibly with the use of examples. The portfolio selection process is introduced as a scientific approach to the portfolio optimization problem for buy-and-hold investors. Basic mathematical notation is described for portfolio optimization with the definition of the set of feasible portfolios through a system of linear equation and inequalities. We define the portfolio rate of return as random variable and formally introduce the expected return maximization and risk minimization as portfolio optimization objectives. An important part of the chapter is devoted to Markowitz quadratic mean-variance model with its return-risk frontier as first historical contribution in terms of an optimization model for portfolio selection. We then introduce risk measures as dispersion measures with emphasis on their drawbacks and the overcome of their weaknesses through the safety measures. The concepts of minimum risk portfolio, maximum safety portfolio and mean-risk efficient frontier are also discussed. Finally, we deal with the two approaches to handle the portfolio problem, namely bounding approach and the trade-off analysis.