# Portfolio Optimization

Portfolio optimization is usually based on the following single period model of investment. At the beginning of a period, an investor allocates the capital among various financial assets (e.g., stocks, bonds and mutual funds), thus assigning a nonnegative weight (share of the capital) to each asset. During the investment period, an asset generates a random rate of return. This results in a change of capital invested (observed at the end of the period) which is measured by the weighted average of the individual rates of return. *Portfolio optimization* is the problem of determining, according to some criterion, the optimal proportions of the capital invested in each asset at the beginning of the period.

Markowitz (1952) provided the fundamental basis for Modern Portfolio Theory modelling the portfolio optimization problem as a return-risk bicriteria optimization problem where the expected return is maximized while the variance as a scalar risk measure is minimized. The trade-off between risk and return is different for each investor, but the preferences of all investors lie upon a curve which is usually called the frontier of efficient portfolios, or simply the *efficient frontier*. The efficient frontier corresponds to the upper portion of the curve shown in the figure below. It consists of all points that minimize the portfolio risk for a specific value of expected portfolio return or, conversely, maximize the portfolio return given a value of risk.

When trading in financial market, some real-life features have to be taken into consideration, especially by private investors. Indeed, since private investors trade assets in small/moderate amounts, when investing they have to consider real-life features such as fixed transaction costs, minimum transaction lots, maximum number of assets composing the portfolio (the so-called cardinality constraint), minimum and maximal capital invested in each asset.

Following Markowitz quadratic model for portfolio optimization, several alternative bicriteria models have been proposed in the literature, where the variance is replaced with another risk measure. For the case of rates of return distributed as discrete random variables, some of the latter models have the relevant advantage to be Linear Programming (LP) solvable. The linear structure of the mathematical programming models makes it possible to include the aforementioned real-life features without afflicting too much the complexity of the problem (e.g., see Mansini *et al.* (2003)), while this is not true for quadratic programming formulations.

**References**

Markowitz, H.. **Portfolio selection**. *The Journal of Finance* (1952) 7: 77-91.

Mansini, R., Ogryczak, W., and Speranza, M.G.. **LP solvable models for portfolio optimization: A classification and computational comparison**. *IMA Journal of Management Mathematics* (2003) 14: 187-220.