Teoria De Malkowitz

Portfolio optimization models and mean-variance spanning tests

Wei-Peng Chen*
Department of Finance, Hsih-Shin University, Taiwan
c8145666@ms18.hinet.net

Huimin Chung
Graduate Institute of Finance, National Chiao Tung University, Taiwan

Keng-Yu Ho
Department of Finance, National Central University, Taiwan
kengyuho@cc.ncu.edu.tw

Tsui-Ling Hsu
Graduate Institute of Finance,National Chiao Tung University, Taiwan
tracy.shu@msa.hinet.net

Prepared for Handbook of Quantitative Finance and Risk Management

In this chapter we introduce the theory and the application of computer program of modern portfolio theory. The notion of diversification is age-old “don't put your eggs in one basket”, obviously predates economic theory. However a formal model showing how to makethe most of the power of diversification was not devised until 1952, a feat for which Harry Markowitz eventually won Nobel Prize in economics.
Markowitz portfolio shows that as you add assets to an investment portfolio the total risk of that portfolio - as measured by the variance (or standard deviation) of total return - declines continuously, but the expected return of the portfolio is aweighted average of the expected returns of the individual assets. In other words, by investing in portfolios rather than in individual assets, investors could lower the total risk of investing without sacrificing return.
In the second part we introduce the mean-variance spanning test which follows directly from the portfolio optimization problem.

INTRODUCTION OF MARKOWITZ PORTFOLIO-SELECTIONMODEL
Harry Markowitz (1952, 1959) developed his portfolio-selection technique, which came to be called modern portfolio theory (MPT). Prior to Markowitz's work, security-selection models focused primarily on the returns generated by investment opportunities. Standard investment advice was to identify those securities that offered the best opportunities for gain with the least risk and thenconstruct a portfolio from these. Following this advice, an investor might conclude that railroad stocks all offered good risk-reward characteristics and compile a portfolio entirely from these. The Markowitz theory retained the emphasis on return; but it elevated risk to a coequal level of importance, and the concept of portfolio risk was born. Whereas risk has been considered an important factor andvariance an accepted way of measuring risk, Markowitz was the first to clearly and rigorously show how the variance of a portfolio can be reduced through the impact of diversification, he proposed that investors focus on selecting portfolios based on their overall risk-reward characteristics instead of merely compiling portfolios from securities that each individually have attractive risk-rewardcharacteristics.
A Markowitz portfolio model is one where no added diversification can lower the portfolio's risk for a given return expectation (alternately, no additional expected return can be gained without increasing the risk of the portfolio). The Markowitz Efficient Frontier is the set of all portfolios of which expected returns reach the maximum given a certain level of risk.

TheMarkowitz model is based on several assumptions concerning the behavior of investors and financial markets:
1. A probability distribution of possible returns over some holding period can be estimated by investors.
2. Investors have single-period utility functions in which they maximize utility within the framework of diminishing marginal utility of wealth.
3. Variability about thepossible values of return is used by investors to measure risk.
4. Investors care only about the means and variance of the returns of their portfolios over a particular period.
5. Expected return and risk as used by investors are measured by the first two moments of the probability distribution of returns-expected value and variance.
6. Return is desirable; risk is to be...