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FINANCIAL TRADING AND MARKET MICRO-STRUCTUREPowerPoint Presentation

FINANCIAL TRADING AND MARKET MICRO-STRUCTURE

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Topics

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

MATRIX ALGEBRA

- Definition
- Row vector
- Column vector

Matrix Addition and Scalar Multiplication

- Definition: Two matrices A = [aij] and B = [bij ] are said to be equal if Equality of
these matrices have the same size, and for each index pair (i, j), aij = bij , Matrices

that is, corresponding entries of A and B are equal.

Matrix Addition and Subtraction

- Let A = [aij] and B = [bij] 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 = [aij + bij ] .
- The negative of the matrix A, denoted by −A, is defined by the formula −A = [−aij ] .
- The difference of A and B, denoted by A−B, is defined by the formula A − B = [aij − bij ] .

Scalar Multiplication

- Let A = [aij] 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 = [caij ] .

Linear Combinations

- A linear combination of the matrices A1,A2, . . . , An is an expression of the form c1A1 + c2A2 + ・ ・ ・ + cnAn

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.

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

Portfolio Models

- Portfolio basic calculations
- Two-Asset examples
- Correlation and Covariance
- Trend line

- Portfolio Means and Variances
- Matrix Notation
- Efficient Portfolios

Review of Matrices

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

Adding and multiplying matrices

- Sum
- Scalar multiplication

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).

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)

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").

The Mathematics of Diversification

- Linear combinations
- Single-index model
- Multi-index model
- Stochastic Dominance

Return

- The expected return of a portfolio is a weighted average of the expected returns of the components:

Two-Security Case

- For a two-security portfolio containing Stock A and Stock B, the variance is:

portfolio variance

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

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

Minimum Variance Portfolio (cont’d)

- For a two-security minimum variance portfolio, the proportions invested in stocks A and B are:

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

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

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

- Of particular interest are industry effects

Multi-Index Model (cont’d)

- The general form of a multi-index model:

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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