Chapter 6 The Mathematics of Diversification

1. Chapter 6The Mathematics of Diversification

2. O! This learning, what a thing it is! - William Shakespeare

3. Outline • Introduction • Linear combinations • Single-index model • Multi-index model

4. Introduction • The reason for portfolio theory mathematics: • To show why diversification is a good idea • To show why diversification makes sense logically

5. Introduction (cont’d) • Harry Markowitz’s efficient portfolios: • Those portfolios providing the maximum return for their level of risk • Those portfolios providing the minimum risk for a certain level of return

6. Linear Combinations • Introduction • Return • Variance

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

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

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

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

13. Two Security Case (cont’d) Example Assume the following statistics for Stock A and Stock B:

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

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

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

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

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

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

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

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

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 (n2 – n)/2 • Any portfolio construction technique using the full covariance matrix is called a Markowitz model

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

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

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

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