2. RETURN MARKET, BETA, DAN MATHEMATIKA DIVERSIFIKASI Pertemuan 12 dan 13 Matakuliah: F0892 - Analisis Kuantitatif Tahun: 2009

3. RETURN MARKET Return market : ialah return dari seluruh usaha yang ada di suatu wilayah tertentu. Karena sukar menghitung return seluruh usaha dalam wilayah tertentu maka bisa diwakilkan dengan menghitung return dari seluruh saham yang tercatat di bursa. (Di Indonesia ialah Bursa Efek Indonesia). Yang digunakan ialah indeks  dapat IHSG, LQ 45, atau Kompas 100.

4. Return market diperoleh dengan menghitung perubahan indeks per hari. Bina Nusantara University 4 IHSG t+1 - IHSG 1

5. MATHEMATIKA DIVERSIFIKASI Bina Nusantara University 5

6. 6 Linear Combinations Introduction Return Variance

7. 7 Introduction A portfolio’s performance is the result of the performance of its components –The return realized on a portfolio is a linear combination of the returns on the individual investments –The variance of the portfolio is not a linear combination of component variances

8. 8 Return The expected return of a portfolio is a weighted average of the expected returns of the components:

9. 9 Variance Introduction Two-security case Minimum variance portfolio Correlation and risk reduction The n-security case

10. 10 Introduction Understanding portfolio variance is the essence of understanding the mathematics of diversification –The variance of a linear combination of random variables is not a weighted average of the component variances

11. 11 Introduction (cont’d) For an n-security portfolio, the portfolio variance is:

12. 12 Two-Security Case For a two-security portfolio containing Stock A and Stock B, the variance is:

13. 13 Two Security Case (cont’d) Example Assume the following statistics for Stock A and Stock B: Stock AStock B Expected return.015.020 Variance.050.060 Standard deviation.224.245 Weight40%60% Correlation coefficient.50

14. 14 Two Security Case (cont’d) Example (cont’d) What is the expected return and variance of this two- security portfolio?

15. 15 Two Security Case (cont’d) Example (cont’d) Solution: The expected return of this two-security portfolio is:

16. 16 Two Security Case (cont’d) Example (cont’d) Solution (cont’d): The variance of this two-security portfolio is:

17. 17 Minimum Variance Portfolio The minimum variance portfolio is the particular combination of securities that will result in the least possible variance Solving for the minimum variance portfolio requires basic calculus

18. 18 Minimum Variance Portfolio (cont’d) For a two-security minimum variance portfolio, the proportions invested in stocks A and B are:

19. 19 Minimum Variance Portfolio (cont’d) Example (cont’d) Assume the same statistics for Stocks A and B as in the previous example. What are the weights of the minimum variance portfolio in this case?

20. 20 Minimum Variance Portfolio (cont’d) Example (cont’d) Solution: The weights of the minimum variance portfolios in this case are:

21. 21 Minimum Variance Portfolio (cont’d) Example (cont’d) Weight A Portfolio Variance

22. 22 Correlation and Risk Reduction Portfolio risk decreases as the correlation coefficient in the returns of two securities decreases Risk reduction is greatest when the securities are perfectly negatively correlated If the securities are perfectly positively correlated, there is no risk reduction

23. 23 The n-Security Case For an n-security portfolio, the variance is:

24. 24 The n-Security Case (cont’d) The equation includes the correlation coefficient (or covariance) between all pairs of securities in the portfolio

25. 25 The n-Security Case (cont’d) A covariance matrix is a tabular presentation of the pairwise combinations of all portfolio components –The required number of covariances to compute a portfolio variance is (n 2 – n)/2 –Any portfolio construction technique using the full covariance matrix is called a Markowitz model

26. 26 Single-Index Model Computational advantages Portfolio statistics with the single-index model

27. 27 Computational Advantages The single-index model compares all securities to a single benchmark –An alternative to comparing a security to each of the others –By observing how two independent securities behave relative to a third value, we learn something about how the securities are likely to behave relative to each other

28. 28 Computational Advantages (cont’d) A single index drastically reduces the number of computations needed to determine portfolio variance –A security’s beta is an example:

29. 29 Portfolio Statistics With the Single-Index Model Beta of a portfolio: Variance of a portfolio:

30. 30 Portfolio Statistics With the Single-Index Model (cont’d) Variance of a portfolio component: Covariance of two portfolio components:

31. 31 Multi-Index Model A multi-index model considers independent variables other than the performance of an overall market index –Of particular interest are industry effects Factors associated with a particular line of business E.g., the performance of grocery stores vs. steel companies in a recession

32. 32 Multi-Index Model (cont’d) The general form of a multi-index model: