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MGT 4850 Spring 2008 University of Lethbridge

Portfolio Models http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html#Benchmarks. MGT 4850 Spring 2008 University of Lethbridge. Introduction. Portfolio basic calculations Two-Asset examples Correlation and Covariance Trend line Portfolio Means and Variances

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MGT 4850 Spring 2008 University of Lethbridge

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  1. Portfolio Modelshttp://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html#Benchmarks MGT 4850 Spring 2008 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.

  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 productAB 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 (n2 – 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 R2 • Portfolio mean and variance p.135

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

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