The Markowitz’s Mean-Variance model 1952. H. M. Markowitz, Nobel prize winner, set the base of the Modern Portfolio Theory (MPT) 1959. the firts edition of the book Portfolio Selection: Efficient Diversification of Investments How to set a well balanced set of stocks, in the sense of return and risk – portfolio. B. G. Malkiel: ”You should not put all your eggs in one basket.”

The Markowitz’s Mean-Variance model The balance between the return (measured by mean) and risk (measured by variance): MV model (mean-variance model) How to form a portfolio which gives the highest return for a given level of risk or, for a given return has the lowest risk – efficient portfolio.

Theoretical framework – review of basic notation N risky assets with known expected return and variance; Vector of means E(R), variance – covariance matrix S, portfolio investor's vector π, expected portfolio return, portfolio’s variance.

Theoretical framework

Theoretical framework

Theoretical framework

The Markowitz’s Mean-Variance model

The Markowitz’s Mean-Variance model

The Markowitz’s Mean-Variance model

Efficient Portfolios without Short Sales

Efficient Portfolios without Short Sales

Efficient Portfolios without Short Sales

Efficient Portfolios without Short Sales

Efficient Portfolios without Short Sales

Comparison of Short Sales and No Short Sales Efficient Frontiers Standard deviation Mean return No short sale Short sales allowed

The Markowitz’s Mean-Variance model Probably, the most important 20th century’s inovation in the field of investing and portfolio management; Markowitz changed the paradigm of investment management; Simple and empirically tested as reliable at normal market conditions; Individual investors knew that they should “diversify” and not “all their eggs in one basket”. But Markowitz and his followers gave statistical and implementional meaning to these cliches – Modern Portfolio Theory (MPT).

Notice about diversification effects Diversification effects can be quantified. Assuming equal standard deviations and proportions of all securities and their correlations being zero, the portfolio’s standard deviation is given by:

Systematic and Non-Systematic Risk

Some critics of the model Normal distribution assumption Statics of the model (At the time of model’s emergence: numerous and complicated calculations)

Some critics of the model The variance (standard deviation) is acceptable risk measure in the case of normal distribution of returns. Other cases – Alternative Risk Measures!