1. ERASMUS Ponzan Richard Szabó Óbuda University Efficient Portfolios (Lagrange – multiplication process) Part 1. Richard Szabó, Óbuda University

2. ERASMUS Ponzan Richard Szabó Óbuda University Acknowledgement Special Thanks : Prof. Janos SZAZ (Corvinus University, former President of the Budapest Stock Exchange) PhD András MEDVE (Dean, Óbuda University, Economic Department) Erasmus Program (Poznan Wyzsza Szkola Logistykiy, Óbuda University)

3. ERASMUS Ponzan Richard Szabó Óbuda University 1. Efficient Portfolios 2. Theoretical Background: Lagrange multiplication process 3. Example: Selected Hungarian Blue Chips & Selected ISE papers (Practice) 4. Conclusion

4. ERASMUS Ponzan Richard Szabó Óbuda University PART 1: Efficient Portfolios How to make efficient portfolio?

5. ERASMUS Ponzan Richard Szabó Óbuda University How to make efficient portfolio? Portfolio is a basket of selected securities Included 2, 3, 4 … or more securities We don’t know the future, therefore the indicators of the portfolios are variable.

6. ERASMUS Ponzan Richard Szabó Óbuda University Variable can be analyzed by our moments General form of the weighed moment: The unpaired moments can be originated by the first unpaired moment M(  ) The paired moments can be originated by the first paired moment  (Standard Deviation)

7. ERASMUS Ponzan Richard Szabó Óbuda University the first unpaired moment M(  ) (Mean) is the Yield the first paired moment  (Standard Deviation) is the Risk The „general investor” wish : more and more Yield and no (or the lowest) risk

8. ERASMUS Ponzan Richard Szabó Óbuda University Mathematical: maximalisation the yield, zero risk or minimalisation the risk

9. ERASMUS Ponzan Richard Szabó Óbuda University Mathematical :

10. ERASMUS Ponzan Richard Szabó Óbuda University Yield of portfolio Normal formula Weighted average

11. ERASMUS Ponzan Richard Szabó Óbuda University Risk of portfolio Variance (Deviation) second moment Square average of the differences between the yield and the arithmetical average

12. ERASMUS Ponzan Richard Szabó Óbuda University Variance covariance matrix cov11cov12cov13cov1n cov21cov22cov23cov2n covn1covn2covnn w1 w2 wn w1w1 w2w2 wnwn

13. ERASMUS Ponzan Richard Szabó Óbuda University Risk of Portfolio with three elements

14. ERASMUS Ponzan Richard Szabó Óbuda University And

15. ERASMUS Ponzan Richard Szabó Óbuda University

16. WHAT ARE THE INDEPENDENT VARIABLE ?

17. ERASMUS Ponzan Richard Szabó Óbuda University The LAGRANGE function

18. ERASMUS Ponzan Richard Szabó Óbuda University Must be derivated by and 0

19. ERASMUS Ponzan Richard Szabó Óbuda University all derivated values are equal as 0

20. ERASMUS Ponzan Richard Szabó Óbuda University Simplificated

21. ERASMUS Ponzan Richard Szabó Óbuda University or

22. ERASMUS Ponzan Richard Szabó Óbuda University The solution

23. ERASMUS Ponzan Richard Szabó Óbuda University

24. THANK YOU for the ATTENTION!