1. 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.”

2. 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.

3. 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.

4. Theoretical framework

5. Theoretical framework

6. Theoretical framework

7. The Markowitz’s Mean-Variance model

8. The Markowitz’s Mean-Variance model

9. The Markowitz’s Mean-Variance model

10. Efficient Portfolios without Short Sales

11. Efficient Portfolios without Short Sales

12. Efficient Portfolios without Short Sales

13. Efficient Portfolios without Short Sales

14. Efficient Portfolios without Short Sales

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

16. 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).

17. 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:

18. Systematic and Non-Systematic Risk

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

20. 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!