Lecture Presentation Software
to accompany
Investment Analysis and
Portfolio Management
Eighth Edition
by
Frank K. Reilly & Keith C. Brown
Chapter 7
Chapter 7 - An Introduction to
Portfolio Management
Questions to be answered:
• What do we mean by risk aversion and what evidence indicates that
investors are generally risk averse?
• What are the basic assumptions behind the Markowitz portfolio
theory?
• What is meant by risk and what are some of the alternative
measures of risk used in investments?
• How do you compute the expected rate of return for an individual
risky asset or a portfolio of assets?
• How do you compute the standard deviation of rates of return for an
individual risky asset?
• What is meant by the covariance between rates of return and how
do you compute covariance?
Chapter 7 - An Introduction to
Portfolio Management
• What is the relationship between covariance and correlation?
• What is the formula for the standard deviation for a portfolio of
risky assets and how does it differ from the standard deviation of an
individual risky asset?
• Given the formula for the standard deviation of a portfolio, why and
how do you diversify a portfolio?
• What happens to the standard deviation of a portfolio when you
change the correlation between the assets in the portfolio?
• What is the risk-return efficient frontier?
• Is it reasonable for alternative investors to select different portfolios
from the portfolios on the efficient frontier?
• What determines which portfolio on the efficient frontier is selected
by an individual investor?
Background Assumptions
• As an investor, you want to maximize return
for a given level of risk.
• Your portfolio includes all of your assets
and liabilities, not just your traded
securities.
• The relationship between the returns of the
assets in the portfolio is important.
• A good portfolio is not simply a collection
of individually good investments.
Risk Aversion
Given a choice between two assets
with equal rates of return, most
investors will select the asset with the
lower level of risk.
Evidence That
Investors are Risk Averse
• Many investors purchase insurance:
–
–
–
–
Life
Automobile
Health
Disability
Insurance is one of the few
things we buy which we know
has a negative NPV
• The insured trades a known cost (the premium) for
an unknown risk of loss
• The required yield on bonds increases with risk
classifications from AAA to AA to A….
But Not Totally Risk Averse . . .
• Risk preferences may have to do with the
amount of money involved – we are willing
to risk small amounts, but we insure against
large losses
– People buy lottery tickets (negative expected
value but the potential loss is small)
– But also buy insurance (negative expected
value but the potential loss is large)
Which Definition of Risk?
•
Uncertainty of future outcomes
–
–
•
Risk involves both positive & negative outcomes
What we measure with standard deviation
Probability of an adverse outcome
–
–
Ignore outcomes that are better than expected
Investors only really care about negative surprises.
They like positive surprises.
Rates of Return 1900-2003
80%
Stock Market Index Returns
Percentage Return
60%
40%
20%
0%
-20%
1900
1920
1940
-40%
-60%
Year
1960
1980
2000
Source: Ibbotson Associates
Actual market returns exhibit significant fluctuation
around the mean return. Measure the size of the
fluctuations with variance & standard deviation
Measuring Risk
Histogram of Annual Stock Market Returns
# of Years
24
24
19
20
15
16
10
12
3
2
50 to 60
30 to 40
20 to 30
10 to 20
0 to 10
-10 to 0
-20 to -10
Return %
-30 to -20
1
-40 to -30
0
1
-50 to -40
4
4
40 to 50
8
13
12
Each bar shows the number of years that the annual return
was within that range out a total of 102 years of data.
Measuring Risk: 1900 - 2003
Portfolio
Treasury
bills
Standard
Deviation Variance
2.8%
7.9
Government 8.2%
bonds
68.0
Common
stocks
402.6
20.1%
Period
Std. Dev. Of US
Stock Market
1931 – 1940
37.8%
1941 – 1950
14.0%
1951 – 1960
12.1%
1961 – 1970
13.0%
1971 – 1980
15.8%
1981 – 1900
16.5%
1991 - 2003
14.8%
Risk of Individual Common Stocks
Stock
Standard
Deviation
Stock
Standard
Deviation
Amazon
72.9%
GE
28.2%
Dell
53.0%
Coca-Cola
27.3%
Reebok
52.3%
Pfizer
24.3%
Microsoft
47.5%
Heinz
23.7%
Ford
43.8%
ExxonMobil
18.2%
Alcan
30.2%
Nokia
54.0%
Standard deviation over the period January 1999 – December, 2003
Markowitz Portfolio Theory
• Derives the expected rate of return and expected
risk for a portfolio of assets
• Shows that the variance & standard deviation of
the rate of return is a meaningful measure of
portfolio risk
• Derives the formula for computing the variance
& standard deviation of a portfolio, showing
how to effectively diversify a portfolio
Harry Markowitz
• Nobel Laureate (1990)
• In 1952, while still a graduate
student at Chicago, Markowitz
took just one afternoon to convert
the notions of risk & return into a
set of written rules involving the
use of diversification &
optimization. These became the
building blocks for all future
advances in investment theory.
Assumptions of
Markowitz Portfolio Theory
1.
2.
3.
4.
5.
Investors consider each investment alternative as defined
by a probability distribution of expected returns over a
holding period.
Investors maximize one-period expected utility, and
their utility curves demonstrate diminishing marginal
utility of wealth.
Investors estimate the risk of the portfolio on the basis of
the variability of expected returns (assumes that returns
are normally distributed).
Investors base decisions solely on expected return and
risk, so their utility curves are a function of expected
return and the expected variance (or standard deviation)
of returns only.
For a given level of risk, investors prefer higher returns
to lower returns. Similarly, for a given level of expected
returns, investors prefer less risk to more risk.
Markowitz Portfolio Theory
Using these five assumptions, a single asset
or portfolio of assets is considered to be
efficient if no other asset or portfolio of
assets offers a higher expected return with
the same (or lower) risk, or lower risk with
the same (or higher) expected return.
Concept of Dominance
Return
A dominates B & C
D
B dominates C
D does not dominate
A
B
C
Standard Deviation
Expected Rates of Return: Single
Asset
• For an individual asset - sum of the possible
returns multiplied by the corresponding
probability of the return occurring
Expected Rate of Return:
Single Risky Asset
Probability
Possible Rate of
Return
Expected Return
35%
30%
20%
8%
10%
12%
2.8%
3.0%
2.4%
15%
14%
E(R)
2.1%
10.30%
N
E ( RSecurity )   PR
i i
i 1
 .35  8   .30 10   .20 12   .15 14 
 10.30%
Variance (Standard Deviation) of
Returns for an Individual Asset
• Variance is a measure of the dispersion of
returns around the mean
• If returns are tightly clustered around the
mean, variance is low
• If returns are widely dispersed around the
mean, variance is high
• Standard deviation is the square root of the
variance
Variance (Standard Deviation)
of Returns for an Individual
Investment
n
Variance ( )   Pi R i - E(R i ) 
2
2
i 1
Where:
Pi is the probability of Ri occurring
Ri is the ith rate of return
Standard deviation ( ) 
n
 Pi  R i - E(R i ) 
i 1
2
Variance & Standard Deviation:
Example
Calculate the variance
& standard deviation
for an asset with the
following returns &
associated
probabilities.
Probability Return
35%
8%
30%
10%
20%
12%
15%
14%
Variance & Standard Deviation:
Example
n
Variance ( )   Pi  R i -E(R i ) 
2
2
i 1
 0.35  8  10.3 +0.30 10  10.3  0.20 12  10.3  0.15 14  10.3
2
2
2
2
 4.51
Standard deviation ( ) 
n
 P  R -E(R ) 
i 1
i
i
2
i
 0.35  8  10.3 +0.30 10  10.3  0.20 12  10.3  0.15 14  10.3
2
 4.51
 2.12%
2
2
2
Variance & Standard Deviation of
Historical Rates of Return
• The variance & standard deviation we just calculated assumes that we
have a distribution of expected returns
• When we are calculating the variance and standard deviation of
historical returns, the probability for each return occurring is the same.
 R - R
n
2
Variance (  Sample
)=
i 1
i
n-1
 R - R
n
Std Dev (  Sample )=
2
i 1
i
n -1
2
Checking for Normality
• Many of our standard models in finance assume normality
• To check a distribution for normality, check for both skewness and
kurtosis
• A normal distribution is completely described by two parameters:
– Mean
– Variance
• A normal distribution is perfectly symetrical
– 68.26% of the observations lie within one standard deviation of the mean
– 95.44% of the observations lie within two standard deviations of the mean
– 99.74% of the observations lie within three standard deviations of the
mean
Skewness
• Skewness is present when the distribution is not
symmetrical
• A distribution with positive skewness has a large number
of small negative values with a few large positive values –
the distribution has a long right tail (why we buy lottery
tickets)
• A distribution with negative skewness has a large number
of small positive values with a few large negative values –
the distribution has a long left tail (why we buy insurance)
Skewness
• Positive skewness
– Mode < Median < Mean
– Investors like positive skewness because the
mean is greater than the median
• Negative skewness
– Mean < Median < Mode
– to help remember the correct order, this is the
same order as they would be listed in the
dictionary
Skewness
R  R


1
n
Skewness   
n
i 1
3
i
3
 Sample
A symmetrical distribution has a skewness of zero.
The sign indicates whether the skewness is positive or
negative (when we cube the deviation, the sign of the
deviation is retained).
For a sample size of 100 taken from a normal distribution, a
skewness of +/- 0.5 would be considered unusually large.
See the exact formula on the next page.
Exact Formula for Skewness
n

Ri  R



n
i 1
Skewness  

3
n

1
n

2




Sample



3
For small sample sizes (n < 100), should use the exact formula.
For large
samples (over 100 observations) the
approximation is appropriate.
Example: Calculating Skewness
T. Rowe Price Equity Fund, 1993 - 2002
Year
Return
1993
14.8%
1994
4.5%
1995
33.3%
1996
20.3%
1997
28.8%
1998
9.2%
1999
3.8%
2000
13.1%
2001
1.6%
2002
-13.0%
Points to note:
1. The mean is 11.64%
2. 5 observations are greater than
the mean; 5 observations are
smaller than the mean
3. It would appear that the
negative deviations
approximately offset the
positive deviations
Solution
n

Ri  R



n
i 1
Skewness  

3
n

1
n

2




Sample



3

 933.064
10
 

3
10

1
10

2
13.65





 0.05
Interpretation: A skewness of zero
indicates a perfectly symmetrical
distribution. Given a value of -0.05 we
can assume the distribution is almost
symmetrical.
Skewness: S&P 500 Annual &
Monthly Returns, 1926 - 2002
Series
# of
Arithmetic
Periods
Mean
Annual
77
Monthly
924
Std Dev
Skewness
Kurtosis
12.20% 20.49% -0.2943 -0.2207
0.97%
5.65% +0.3964
9.4645
Note the difference in skewness between the
monthly & annual returns. This is typical of many
return series – the nature of the statistic depends
heavily on the time period used.
Kurtosis
• Kurtosis is a statistical measure that indicates
whether a distribution is more or less peaked than
a normal distribution.
• For example, observations may be either more
tightly clustered around the mean (a more peaked
distribution) or more of them may lie in the tails of
the distribution
– Leptokurtic – distribution is more peaked (lepto is from
the word Greek word for slender) & has fatter tails
– Platykurtic – the distribution is less peaked than a
normal distribution (platy is the Greek word for broad)
– Mesokurtic – a “normal” distribution (meso is from the
Greek word for middle)
• A normal distribution has a kurtosis of 3
Calculating Kurtosis
n

Ri  R


n  n  1
i 1
Excess Kurtosis  
4
  n  1 n  2  n  3  Sample


4


1   Ri  R 

3
4


n   Sample 





4

2

3  n  1

  n  2  n  3



Where n is the sample size
Kurtosis
• The kurtosis of a normal distribution is 3
• Excel and many other software packages calculate
kurtosis and then subtract 3, giving a measure
called excess kurtosis (as calculated on the
previous page).
• An excess kurtosis greater than zero indicates a
leptokurtic distribution; an excess kurtosis less
than zero indicates a platykurtic distribution.
Moving From One Risky Asset to
Several Risky Assets
• The return on the risky asset portfolio is calculated
as a weighted average of the return of the assets in
the portfolio
– Weights are the market values of each asset divided by
the total market value of the portfolio
N
E ( RPortfolio )   Wi Ri
i 1
Expected Rate of Return:
Portfolio of Risky Assets
Weight
Expected
Expected
(% of Portfolio) Return (Asset i) Portfolio Return
20%
30%
30%
10%
11%
12%
2.0%
3.3%
3.6%
20%
13%
E(R)
2.6%
11.50%
N
E ( RPortfolio )   Wi Ri
i 1
 .20 10%   .30 11%   .30 12%   .20 13% 
 11.50%
Calculating Risk: Two Risky Assets
• The risk of a single risky asset is calculated
as its standard deviation
• When there are two or more risky assets in a
portfolio, we must also incorporate how the
individual assets move in relation to each
other
• Thus we need to understand covariance &
correlation
Covariance of Returns
• A measure of the degree to which two variables “move
together” relative to their individual mean values over time
– If both returns are typically above their respective means at the
same time, the covariance will be positive
– If one return is typically above its mean when the other return is
below its mean, covariance will be negative
– For two assets, i and j, the covariance of their returns is defined as:
Covij = E{[R i - E(R i )] [R j - E(R j )]}
Covariance of Returns: Example
Date
Wilshire 5000
Lehman T Bond Index
January 2004
2.23%
1.77%
February 2004
1.46%
2.00%
March 2004
-1.07%
1.50%
April 2004
-2.13%
-5.59%
May 2004
1.38%
-0.54%
June 2004
2.08%
0.95%
July 2004
-3.82%
1.73%
August 2004
0.33%
3.74%
September 2004
1.78%
0.84%
October 2004
1.71%
1.51%
November 2004
4.68%
-2.19%
December 2004
3.63%
2.31%
Mean Monthly Return 1.0217%
0.6692%
Covariance of Returns: Example
Covij = E{[R i - E(R i )] [R j - E(R j )]}
 {[2.23 - 1.02] [1.77 - 0.67]  [1.46 - 1.02] [2.00 - 0.67]  [1.07 - 1.02] [1.50 - 0.67] 
[2.13 - 1.02] [5.59 - 0.67]  [1.38 - 1.02] [0.54 - 0.67]  [2.08 - 1.02] [0.95 - 0.67] 
[3.82 - 1.02] [1.73 - 0.67]  [0.33 - 1.02] [3.74 - 0.67]  [1.78 - 1.02] [0.84 - 0.67] 
[1.71 - 1.02] [1.51 - 0.67]  [4.68 - 1.02] [2.19 - 0.67]  [3.63 - 1.02] [2.31 - 0.67]}/11
7.00
11
 0.637

Note that we divided by N -1 rather than N, since we are
dealing with a sample of the data rather than a
population.
Covariance and Correlation
• The correlation coefficient is obtained by dividing the
covariance by the product of the individual standard
deviations
ij 
Covij
 i j
where:
ij  the correlation coefficient (small Greek letter rho)
 i  the standard deviation of R it
 j  the standard deviation of R jt
ij 

Cov ij
 i j
0.637
 2.38  2.46 
 0.109
The correlation between the
Wilshire 5000 and the Lehman
Treasury Bond Index is 0.109
Correlation Coefficient
• Can vary only in the range +1 to -1.
• A value of +1 would indicate perfect
positive correlation.
– This means that returns for the two assets move
together in a completely linear manner.
• A value of –1 would indicate perfect
negative correlation.
– This means that the returns for two assets have
the same percentage movement, but in opposite
directions
Measuring Portfolio Return & Risk:
2 Risky Assets
RPortfolio = x A RA + xB RB
Where : xi = proportion in the i th asset
Ri = return on the i th asset
2
 Portfolio
 x A2 A2  xB2 B2  2 x A xB  AB A B
Where : xi = proportion of the i th asset
 i2 = variance of the i th asset
 i = standard deviation of the i th asset
 AB = correlation coefficient
Variance-Covariance Matrix
The variance of a two stock portfolio is the sum of these
four boxes
Stock A
X A X B AB 
Stock A
Stock B
xAσA
2
2
X A X B  AB  A B
Stock B
X A X B  AB  A B
x Bσ B
2
2
Example
• You are holding the following portfolio of two
risky assets:
Asset A
Return
Asset B
14%
8%
Standard
Deviation
22%
14%
Proportion of
portfolio
40%
60%
Correlation
0.20
Calculate:
1. Return on the
portfolio
2. Risk of the
portfolio
Example: Solution
RPortfolio = x1 R1 + x2 R2
  0.40 14%    0.60 8% 
 10.0%
2
 Portfolio
 x A2 A2  xB2 B2  2 x A xB  AB A B
  0.4 
2
 484    0.6  196   2  0.4  0.6  0.20  22 14 
 177.6
 Portfolio  Variance
 177.6
 13.3%
2
Example: Solution
Stock 1
Stock 1
Stock 2
Stock 2
Many Risky Assets Portfolio
• Return on the portfolio is simply a weighted
average of the returns of the assets within
the portfolio
RPortfolio  X 1R1  X 2 R2  ...  X N RN
Xi = Proportion in asset i
Ri = Return on asset i
Risk: Many Risky Assets
• To calculate the variance of the portfolio, use a variancecovariance matrix
Asset 1
Asset 2
Asset 3
Asset 4
Asset 1
Variance of
Asset 1
Covariance
of Asset 1 &
2
Covariance
of Asset 1 &
3
Covariance
of Asset 1 &
4
Asset 2
Covariance
of Asset 1 &
2
Variance of
Asset 2
Covariance
of Asset 2 &
3
Covariance
of Asset 2 &
4
Asset 3
Covariance
of Asset 1 &
3
Covariance
of Asset 2 &
3
Variance of
Asset 3
Covariance
of Asset 3 &
4
Asset 4
Covariance
of Asset 1 &
4
Covariance
of Asset 2 &
4
Covariance
of Asset 3 &
4
Variance of
Asset 4
Variance-Covariance Matrix
• The variance-covariance matrix shows that
the influence of individual asset risk quickly
diminishes as the size of the portfolio
grows, whereas the influence of covariance
grows quickly.
• For a portfolio of N assets, there are N
variance terms and N2 – N covariance terms
Contribution to Portfolio Risk
2
 Portfolio
=
1
1
 Average variance  +  1 -   Average Covariance 
N
N

As N, the number of securities in the portfolio, increases, portfolio
variance approaches the average covariance
Thus the risk of a well-diversified portfolio depends on the market
risk of the securities in the portfolio.
Market risk is measured by Beta.
Portfolio standard deviation
Measuring Risk
Portfolio risk falls rapidly as the
number of securities in the portfolio
rises. A 1970 study by Fama & Lorie
found that 80% of the unique risk is
diversified away with 8 stocks; 95%
with 32 stocks & 99% with 128
stocks
Unique
risk
Market risk
0
5
10
15
Number of Securities
Canadian studies have found that substantially more stocks
are required in Canada to achieve good diversification, due to
the heavy concentration of resource stocks on the TSX.
Estimation Issues
• Results of portfolio allocation depend on accurate
statistical inputs
• Estimates of
– Expected returns
– Standard deviation
– Correlation coefficient
• Among entire set of assets
• With 100 assets, 4,950 correlation estimates
• Estimation risk refers to potential errors
Estimation Issues
• With assumption that stock returns can be described by a
single market model, the number of correlations required
reduces to the number of assets
• Single index market model:
R i  a i  bi R m   i
bi = the slope coefficient that relates the returns for security i
to the returns for the aggregate stock market
Rm = the returns for the aggregate stock market
έi = error term (lower case Greek letter epsilon)
Estimation Issues
If all the securities are similarly related to the market
and a bi derived for each one, it can be shown that the
correlation coefficient between two securities i and j is
given as:
 m2
ij  bibj
 i j
Where :
ij  the correlatio n between asset i and asset j
 m2  the variance of returns for the aggregate stock market
b i  the slope coefficien t that relates the returns
for security i to the returns for the aggregate stock market
The Efficient Frontier
• The efficient frontier represents that set of portfolios with
the maximum rate of return for every given level of risk, or
the minimum risk for every level of return
• Frontier will be portfolios of investments rather than
individual securities
– An exception is the asset with the highest return
Efficient Frontier
for Alternative Portfolios
E(R)
Efficient
Frontier
A
B
C
Standard Deviation of Return
The Efficient Frontier
and Investor Utility
• An individual investor’s utility curve specifies the tradeoffs he is willing to make between expected return and risk
– the more risk averse the individual, the steeper the slope of his/her
utility curve
• The slope of the efficient frontier decreases steadily as you
move upward
• These two interactions will determine the particular
portfolio selected by an individual investor
• The optimal portfolio has the highest utility for a given
investor
• It lies at the point of tangency between the efficient frontier
and the utility curve with the highest possible utility
Selecting an Optimal Risky
Portfolio
E(R port )
U3’
U2’
More Risk
Averse
Investor
Less Risk
Averse
U1’ Investor
Y
U3
X
U2
U1
E( port )
The Internet
Investments Online
http://www.pionlie.com
http://www.investmentnews.com
http://www.ibbotson.com
http://www.styleadvisor.com
http://www.wagner.com
http://www.effisols.com
http://www.efficientfrontier.com