Transcript ch06.ppt

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:
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
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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:
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
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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
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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
Variance
Standard deviation
.015
.050
.224
.020
.060
.245
Weight
Correlation coefficient
40%
60%
.50
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Two Security Case (cont’d)
Example (cont’d)
What is the expected return and variance of this twosecurity portfolio?
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Two Security Case (cont’d)
Example (cont’d)
Solution: The expected return of this two-security
portfolio is:
n
E ( R p )    xi E ( Ri ) 
i 1
  x A E ( RA )    xB E ( RB ) 
  0.4(0.015)    0.6(0.020) 
 0.018  1.80%
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Two Security Case (cont’d)
Example (cont’d)
Solution (cont’d): The variance of this two-security
portfolio is:
 2p  xA2 A2  xB2 B2  2 xA xB  AB A B
 (.4) (.05)  (.6) (.06)  2(.4)(.6)(.5)(.224)(.245)
2
2
 .0080  .0216  .0132
 .0428
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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:
   A B  AB
xA  2
2
 A   B  2 A B  AB
2
B
xB  1  x A
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Minimum Variance
Portfolio (cont’d)
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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Minimum Variance
Portfolio (cont’d)
Example (cont’d)
Solution: The weights of the minimum variance portfolios
in this case are:
 B2   A B  AB
.06  (.224)(.245)(.5)
xA  2

 59.07%
2
 A   B  2 A B  AB .05  .06  2(.224)(.245)(.5)
xB  1  xA  1  .5907  40.93%
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Minimum Variance
Portfolio (cont’d)
Example (cont’d)
1.2
Weight A
1
0.8
0.6
0.4
0.2
0
0
0.01
0.02
0.03
0.04
Portfolio Variance
0.05
0.06
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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:
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
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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:
i 
COV ( Ri , Rm )
 m2
where Rm  return on the market index
 m2  variance of the market returns
Ri  return on Security i
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Portfolio Statistics With the
Single-Index Model
 Beta
of a portfolio:
n
 p   xi  i
i 1
 Variance
of a portfolio:
 2p   p2 m2   ep2
  p2 m2
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Portfolio Statistics With the
Single-Index Model (cont’d)
 Variance
of a portfolio component:
    
2
i
 Covariance
2
i
2
m
2
ei
of two portfolio components:
 AB   A  B m2
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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:
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
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