46 Chapter 5The Mathematics of Diversification

◆ A portfolio’s performance is the result of the performance of its components • The return realized on a portfolio is a linear combination of the returns on the individual investments

◆ The reason for portfolio theory mathematics:

• To show why diversification is a good idea

• To show why diversification makes sense

logically

Introduction (cont’d)

◆ Harry Markowitz’s efficient portfolios:

• Those portfolios providing the maximum return

for their level of risk

• Those portfolios providing the minimum risk

for a certain level of return

◆ A portfolio’s performance is the result of the

performance of its components

• The return realized on a portfolio is a linear

combination of the returns on the individual

investments

• The variance of the portfolio is not a linear

combination of component variances

where proportion of portfolio

invested in security and 1

n

i i

n

x

i x

◆ Introduction

◆ Two-security case

◆ Minimum variance portfolio

◆ Correlation and risk reduction

◆ The n-security case

◆ Understanding portfolio variance is the essence

of understanding the mathematics of

diversification

• The variance of a linear combination of random

variables is not a weighted average of the

component variances

where proportion of total investment in Security

correlation coefficient between Security and Security

n n

p i j ij i j

i j i

Two Security Case (cont’d)

Two Security Case (cont’d)

Two Security Case (cont’d)

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:

Minimum Variance Portfolio (cont’d)

Example (cont’d)

Solution: The weights of the minimum variance portfolios

in the previous case are:

Correlation and Risk Reduction

◆ Portfolio risk decreases as the correlation

coefficient in the returns of two securities

decreases

◆ Risk reduction is greatest when the securities are perfectly negatively correlated

◆ If the securities are perfectly positively

correlated, there is no risk reduction

The n-Security Case

◆ For an n-security portfolio, the 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 j

i j i

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

Example of Variance-Covariance Matrix Computation in Excel

Portfolio Mathematics (Matrix Form)

◆ Define w as the (vertical) vector of weights on the different assets.

◆ Define the (vertical) vector of expected returns

◆ Let V be their variance-covariance matrix

◆ The variance of the portfolio is thus:

Portfolio optimization consists of minimizing this variance subject to the constraint of achieving a given expected return.

p w Vw

µ

Portfolio Variance in the 2-asset case

Covariance Between Two Portfolios

(Matrix Form)

◆ Define w 1 as the (vertical) vector of weights on

the different assets in portfolio P 1 .

◆ Define w 2 as the (vertical) vector of weights on

the different assets in portfolio P 2 .

◆ Define the (vertical) vector of expected returns

◆ Let V be their variance-covariance matrix

◆ The covariance between the two portfolios is:

1 , 2 1 ' 2 2 ' 1 (by symmetry)

µ

The Optimization Problem

=   M = M 

µ µ µ

Plugging (1) into (2) yields:

[ ] ( ) ( )

( ) ( )

{ { ( ){

1 1

1 1

1

( 1) ( )

( 2) ( 2)

,

1 1'

E R V

( )

( 2) (2 )

(2 1) (2 2)

E R V

◆ The last equation solves the mean-variance

portfolio problem The equation gives us the

optimal weights achieving the lowest portfolio variance given a desired expected portfolio

Global Minimum Variance Portfolio

◆ In a similar fashion, we can solve for the global minimum variance portfolio:

The global minimum variance portfolio is the efficient frontier portfolio that displays the

absolute minimum variance.

Another Way to Derive the Variance Efficient Portfolio Frontier

Mean-◆ Make use of the following property: if two

portfolios lie on the efficient frontier, any linear combination of these portfolios will also lie on the frontier Therefore, just find two mean-

variance efficient portfolios, and compute/plot the mean and standard deviation of various

linear combinations of these portfolios.

Some Excel Tips

◆ To give a name to an array (i.e., to name a

matrix or a vector):

• Highlight the array (the numbers defining the

matrix)

• Click on ‘Insert’, then ‘Name’, and finally

‘Define’ and type in the desired name

Excel Tips (Cont’d)

◆ To compute the inverse of a matrix previously named (as an example) “V”:

• Type the following formula: ‘=minverse(V)’

and click ENTER

• Re-select the cell where you just entered the

formula, and highlight a larger area/array of the size that you predict the inverse matrix will

take

• Press F2, then CTRL + SHIFT + ENTER

Excel Tips (end)

◆ To multiply two matrices named “V” and “W”:

• Type the following formula: ‘=mmult(V,W)’

and click ENTER

• Re-select the cell where you just entered the

formula, and highlight a larger area/array of the size that you predict the product matrix will

take

• Press F2, then CTRL + SHIFT + ENTER

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:

2

2

( , )

where return on the market index

variance of the market returns return on Security

i m i

m m

m

COV R R R

β

σ σ

Portfolio Statistics With the

Portfolio Statistics With the Single-Index Model (cont’d)

◆ Variance of a portfolio component:

◆ Covariance of two portfolio components:

( , ) ( , ) ( , ) ( , ) ( , )

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

a I I

i R

β β