1. Chapter 5 The Mathematics of Diversification
Portfolio Construction, Management, & Protection, 5e, Robert A. Strong
Copyright ©2009 by South-Western, a division of Thomson Business & Economics. All rights reserved.

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

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

4. 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

5. Linear Combinations
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

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

7. Variance
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

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

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

10. Two Security Case (cont’d)
Example
Assume the following statistics for Stock A and Stock B:
Stock A
Stock B
Expected return
.015
.020
Variance
.050
.060
Standard deviation
.224
.245
Weight
40%
60%
Correlation coefficient
.50

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

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

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

14. 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

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

16. 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?

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

18. Minimum Variance Portfolio (cont’d)
Example (cont’d)
Percentage Invested in Stock A
Portfolio Variance

19. 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

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

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

22. Single-Index Model
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

23. Computational Advantages
A single index drastically reduces the number of computations needed to determine portfolio variance
A security’s beta is an example:

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

25. Portfolio Statistics With the Single-Index Model (cont’d)
Variance of a Portfolio Component:
Covariance of Two Portfolio Components:

26. 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

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