FINANCIAL TRADING AND MARKET MICRO-STRUCTURE

MGT 4850

Spring 2011

University of Lethbridge

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 product AB 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:

1

1

( ) ( )

where proportion of portfolio

invested in security and

1

n

p i ii

i

n

ii

E R x E R

x

i

x

Two-Security Case

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

2 2 2 2 2 2p A A B B A B AB A Bx x x x

portfolio variance

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

2

1 1

where proportion of total investment in Security

correlation coefficient between

Security and Security

n n

p i j ij i ji j

i

ij

x x

x i

i j

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:

2

2 2 2

1

B A B ABA

A B A B AB

B A

x

x x

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

Multi-Index Model (cont’d)

• The general form of a multi-index model:

1 1 2 2 ...

where constant

return on the market index

return on an industry index

Security 's beta for industry index

Security 's market beta

retur

i i im m i i in n

i

m

j

ij

im

i

R a I I I I

a

I

I

i j

i

R

n on Security i

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)