1. FINANCIAL TRADING AND MARKET MICRO-STRUCTURE MGT 4850 Spring 2011 University of Lethbridge

2. Topics The power of Numbers Quantitative Finance Risk and Return Asset Pricing Risk Management and Hedging Volatility Models Matrix Algebra

3. MATRIX ALGEBRA Definition –Row vector –Column vector

4. Matrix Addition and Scalar Multiplication Definition: Two matrices A = [a ij ] and B = [b ij ] are said to be equal if Equality of these matrices have the same size, and for each index pair (i, j), a ij = b ij, Matrices that is, corresponding entries of A and B are equal.

5. Matrix Addition and Subtraction Let A = [a ij ] and B = [b ij ] be m × n matrices. Then the sum of the matrices, denoted by A + B, is the m × n matrix defined by the formula A + B = [a ij + b ij ]. The negative of the matrix A, denoted by −A, is defined by the formula −A = [−a ij ]. The difference of A and B, denoted by A−B, is defined by the formula A − B = [a ij − b ij ].

6. Scalar Multiplication Let A = [a ij ] be an m × n matrix and c a scalar. Then the product of the scalar c with the matrix A, denoted by cA, is defined by the formula Scalar cA = [ca ij ].

7. Linear Combinations A linear combination of the matrices A 1,A 2,..., A n is an expression of the form c 1 A 1 + c 2 A 2 + ・ ・ ・ + c n A n

8. Laws of Arithmetic Let A,B,C be matrices of the same size m × n, 0 the m × n zero matrix, and c and d scalars. (1) (Closure Law) A + B is an m × n matrix. (2) (Associative Law) (A + B) + C = A + (B + C) (3) (Commutative Law) A + B = B + A (4) (Identity Law) A + 0 = A (5) (Inverse Law) A + (−A) = 0 (6) (Closure Law) cA is an m × n matrix.

9. Laws of Arithmetic (II) (7) (Associative Law) c(dA) = (cd)A (8) (Distributive Law) (c + d)A = cA + dA (9) (Distributive Law) c(A + B) = cA + cB (10) (Monoidal Law) 1A = A

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

11. 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

12. Adding and multiplying matrices Sum Scalar multiplication

13. 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).

14. 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)

15. 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").

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

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

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

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

20. 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

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

22. 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

23. 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

24. 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

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

26. 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)