1 chapter 6 the mathematics of diversification. 2 o! this learning, what a thing it is! - william...

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Chapter 6

The Mathematics of Diversification

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O! This learning, what a thing it is!

- William Shakespeare

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Outline Introduction Linear combinations Single-index model Multi-index model

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Introduction The reason for portfolio theory

mathematics:• To show why diversification is a good idea

• To show why diversification makes sense logically

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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

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Linear Combinations Introduction Return Variance

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Introduction 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

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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

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Variance Introduction Two-security case Minimum variance portfolio Correlation and risk reduction The n-security case

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Introduction 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

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Introduction (cont’d) 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

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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

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Two Security Case (cont’d)Example

Assume the following statistics for Stock A and Stock B:

Stock A Stock B

Expected return .015 .020

Variance .050 .060

Standard deviation .224 .245

Weight 40% 60%

Correlation coefficient .50

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Two Security Case (cont’d)Example (cont’d)

What is the expected return and variance of this two-security portfolio?

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Solution: The expected return of this two-security portfolio is:

1

( ) ( )

( ) ( )

0.4(0.015) 0.6(0.020)

0.018 1.80%

n

p i ii

A A B B

E R x E R

x E R x E R

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Solution (cont’d): The variance of this two-security portfolio is:

2 2 2 2 2

2 2

2

(.4) (.05) (.6) (.06) 2(.4)(.6)(.5)(.224)(.245)

.0080 .0216 .0132

.0428

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

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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

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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

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Example (cont’d)

Assume the same statistics for Stocks A and B as in the previous example. What are the weights of the minimum variance portfolio in this case?

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Example (cont’d)

Solution: The weights of the minimum variance portfolios in this case are:

2

2 2

.06 (.224)(.245)(.5)59.07%

2 .05 .06 2(.224)(.245)(.5)

1 1 .5907 40.93%

B A B ABA

A B A B AB

B A

x

x x

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Example (cont’d)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.01 0.02 0.03 0.04 0.05 0.06

Wei

ght A

Portfolio Variance

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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

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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 ji j

i

ij

x x

x i

i j

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The n-Security Case (cont’d) The equation includes the correlation

coefficient (or covariance) between all pairs of securities in the portfolio

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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

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Single-Index Model Computational advantages Portfolio statistics with the single-index

model

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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

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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 mi

m

m

m

i

COV R R

R

R i

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Portfolio Statistics With the Single-Index Model

Beta of a portfolio:

Variance of a portfolio:1

n

p i ii

x

2 2 2 2

2 2

p p m ep

p m

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

Variance of a portfolio component:

Covariance of two portfolio components:

2 2 2 2i i m ei

2AB A B m

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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

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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

1 chapter 6 the mathematics of diversification. 2 o! this learning, what a thing it is! - william...

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