Portfolio Models
MGT 4850
Spring 2009
University of Lethbridge
Introduction
• Portfolio basic calculations
• Two-Asset examples
– Correlation and Covariance
– Trend line
• Portfolio Means and Variances
• Matrix Notation
• Efficient Portfolios
Return Stats 240
Covariance and Correlation 242
Graph 244
Portfolio Mean and STD 245
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:
n
E ( R p )    xi E ( Ri ) 
i 1
where xi  proportion of portfolio
invested in security i and
n
x
i 1
i
1
Two-Security Case
• For a two-security portfolio containing
Stock A and Stock B, the variance is:
  x   x   2 xA xB  AB A B
2
p
2
A
2
A
2
B
2
B
portfolio variance
• For an n-security portfolio, the portfolio
variance is:
n
n
   xi x j ij i j
2
p
i 1 j 1
where xi  proportion of total investment in Security i
ij  correlation coefficient between
Security i and Security 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:
   A B  AB
xA  2
2
 A   B  2 A B  AB
2
B
xB  1  x A
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
Single-Index Model
• Computational advantages
• Portfolio statistics with the single-index
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
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:
i 
COV ( Ri , Rm )
 m2
where Rm  return on the market index
 m2  variance of the market returns
Ri  return on Security i
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:
Ri  ai   im I m   i1 I1   i 2 I 2  ...   in I n
where ai  constant
I m  return on the market index
I j  return on an industry index
 ij  Security i's beta for industry index j
 im  Security i's market beta
Ri  return on Security i
Basic Mechanics of Portfolio
calculations
• Two Asset Example
• Continuously compounded monthly
returns – mean variance std deviation
• Covariance and variance calculations
• Correlation coefficient as the square root
of the regression R2
• Portfolio mean and variance
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)