1. Portfolio Models MGT 4850 Spring 2007 University of Lethbridge

2. Introduction Portfolio basic calculations Two-Asset examples –Correlation and Covariance –Trend line Portfolio Means and Variances Matrix Notation Efficient Portfolios

3. Review of Matrices a matrix (plural matrices) is a rectangular table of numbers, consisting of abstract quantities that can be added and multiplied.numbersabstract quantities that can be added and multiplied

4. Adding and multiplying matrices Sum Scalar multiplication

5. Matrix multiplication Well-defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix (m rows, p columns).

6. Matrix multiplication Note that the number of of columns of the left matrix is the same as the number of rows of the right matrix, e. g. A*B →A(3x4) and B(4x6) then product C(3x6). Row*Column if A(1x8); B(8*1) →scalar Column*Row if A(6x1); B(1x5) →C(6x5)

7. Matrix multiplication properties: (AB)C = A(BC) for all k-by-m matrices A, m-by-n matrices B and n-by-p matrices C ("associativity"). (A + B)C = AC + BC for all m-by-n matrices A and B and n-by-k matrices C ("right distributivity"). C(A + B) = CA + CB for all m-by-n matrices A and B and k-by-m matrices C ("left distributivity").

8. The Mathematics of Diversification Linear combinations Single-index model Multi-index model Stochastic Dominance

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

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

11. portfolio variance For an n-security portfolio, the portfolio variance is:

12. 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

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

14. 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

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

16. 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

17. 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:

18. 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

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

20. Basic Mechanics of Portfolio calculations Two Asset Example p.132 Continuously compounded monthly returns – mean variance std deviation Covariance and variance calculations p.133 Correlation coefficient as the square root of the regression R 2 Portfolio mean and variance p.135

21. Portfolio Mean and Variance Matrix notation; column vector Γ for the weights transpose is a row vector Γ T Expected return on each asset as a column vector or E its transpose E T Expected return on the portfolio is a scalar (row*column) Portfolio variance Γ T S Γ (S var/cov matrix)