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Chapter 8
Risk, Return,
and Portfolio
Theory
Chapter 8 Outline
8.1 Measuring
Returns
• Ex post vs. ex
ante returns
• Total return
• Measuring
average
returns
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8.2
Measuring
Risk
• The standard
deviation: ex
post
• The standard
deviation: ex
ante
8.3 Expected
Return and
Risk for
Portfolios
• Calculating a
portfolio’s
return
• Covariance
• Correlation
coefficient
8.4 The
Efficient
Frontier
• Modern
Portfolio
Theory
• Harry
Markowitz
• Efficient
portfolio
8.5
Diversification
• Random
diversification
• Unique risk
• Market risk
• Total risk
8.1 Measuring Returns on
Investment
Definitions:
 Ex post return-past
or historical returns
 Ex ante returns-expected returns
 Income yield-return earned by investors as a
periodic cash flow
 Capital gain-measures the appreciation (or
depreciation) in the price of the asset from
some starting price
3
Where CF1 is the expected cash flows to be received, P0 is the
purchase price today, and P1 = selling price 1 year from today
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Total return
 Total return-
the sum of the income yield and the
capital gain (or loss) yield
 Example: At the end of 2009, IBM had a stock
price of $130.90 and at the end of 2010 IBM had a
stock price $146.76. During 2010, IBM paid four
dividends, totaling $2.50. The return on IBM stock
for 2010 is:
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Total Return
Answer:
$2.50 $146.76−130.90
Total return2010 =
+
$130.90
130.90
=1.191% + 12.12% = 14.03%
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Measuring Average Returns

Arithmetic mean or average mean-sum of all observations
divided by the total number of observations

Geometric mean-average or compound growth rate over
multiple time periods
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Why do the arithmetic mean return
and the geometric mean return differ?
 The
arithmetic mean simply averages the annual
rates of return without taking into account that
the amount invested varies across time.
 The geometric mean is a better average return
estimate when we are interested in the rate of
return performance of an investment over time.
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Estimating Expected Returns on
Investment
The expected return is often estimated
based on historical averages, but the
problem is that there is no guarantee that
the past will repeat itself.
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Estimating Expected Returns on
Investment
 Example:
Suppose you have two possible returns on investment:
The expected return is the weighted average of the possible returns, where the
weights are the probabilities:
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Expected return, E(r) = 0.07
Estimating expected returns
Suppose we have the following estimates for
the return on an investment:
Probability
Return
Best case
30%
25%
Most likely case
60%
10%
Worst case
10%
-5%
What is the expected value return, based on
these estimates?
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Estimating expected returns
Best case
Most likely case
Worst case
Total
Probability Return
30%
25%
60%
10%
10%
-5%
100%
Probability x
Return
7.5%
6.0%
-0.5%
13.0%
Expected
return
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8.2 Measuring Risk


Risk is the probability of incurring harm, and
for financial managers, harm generally
means losing money or earning an
inadequate rate of return.
Risk is the probability that the actual return
from an investment is less than the expected
return.
This means that the more variable the possible
return, the greater the risk.
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Methods of Measuring Risk
Standard Deviation: Ex Post
Uses information that has occurred (ex post)
Ex post standard deviation (σ) =
Where σ is the standard deviation, is the average return, xi is the observation
in year i, and N is the number of observations
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Methods of Measuring Risk
The Standard Deviation: Ex Ante
Formulated based on expectations about the
future cash flows or returns of an asset.
Ex ante standard deviation (σ) =
Where r is one of the possible outcomes, E(r) is the calculated expected value
of the possible outcomes, and pi is the probability of the occurrence
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Ex-ante risk
Probability
Return
Deviation
from
Weighted
Expected expected Squared
squared
value
value
deviation deviation
Best case
30%
25%
7.5%
12%
0.0144 0.00432
Most likely case
60%
10%
6.0%
-3%
0.0009 0.00054
Worst case
10%
-5%
-0.5%
-18%
Expected return =
13.0%
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 =
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2
0.0324
0.00324
Variance = 0.00810
0.00810 = 0.09 𝑜𝑟 9%
Methods of Measuring Risk
Ex Post Standard
Deviation
Ex Ante Standard
Deviation
Perspective
Looking back on what
has occurred.
Looking forward to what
may happen.
Observations
Observations from the
past.
Possible outcomes in the
future.
Calculation of
the variance
Average of the squared
deviations of observations
from the mean of the
observations.
Weighted average of the
squared deviations of
possible outcomes from the
expected outcome, where
the weights are the
probabilities.
Formula for the
standard
deviation (σ)
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8.3 Expected Return and Risk
for Portfolios
A
portfolio is a collection of assets, such as
stocks and bonds, that are combined and
considered a single asset.
 Investors should diversify their investments
so that they are not unnecessarily exposed
to a single negative event
“Don’t put all your eggs in one basket”
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Calculating a Portfolio’s Return
The expected return on a portfolio is the
weighted average of the expected returns on
the individual assets in the portfolio:
where E(rp) represents the expected return on the
portfolio, E(ri) represents the expected return on asset i,
and wi represents the portfolio weight of asset i.
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Calculating a Portfolio’s Return
Good
OK
Bad
Probability
25%
60%
15%
Return on
A
15%
10%
5%
Probability Probability
Return on x Return x Return
B
on A
on B
20%
3.75%
5.00%
12%
6.00%
7.20%
-10%
0.75%
-1.50%
10.50%
10.70%
Expected
return on A
Expected
return on B
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Calculating a Portfolio’s Return
Estimating standard deviation:
where σpis the portfolio standard deviation and
COVxy is the covariance of the returns on X and Y
Covariance-a statistical measure of the
degree to which two or more series move
together
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Estimating a standard
deviations of returns
Good
Weighted
deviation
Return Return Probability x Probability x squared for
Probability on A
on B
Return on A Return on B Return on A
25%
15%
20%
3.75%
5.00%
0.000506
Weighted
deviation
squared for
Return on A
0.002162
OK
60%
10%
12%
6.00%
7.20%
0.000015
0.000101
Bad
15%
5%
-10%
0.75%
-1.50%
0.000454
0.006427
10.50%
10.70%
0.000975
0.008691
Standard deviation of A = 3.1225%
Standard deviation of B = 9.3226%
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Estimating a Covariance
Good
OK
Bad
Probabilit Probabilit Deviation Deviation
y x Return y x Return for Return for Return Product of
Probability
on A
on B
on A
on B
deviations
25%
3.75%
5.00%
4.500%
9.300% 0.0041850
60%
6.00%
7.20%
-0.500%
1.300% -0.0000650
15%
0.75%
-1.50%
-5.500% -20.700% 0.0113850
10.50%
10.70%
Probability
weighted
products
0.0010463
-0.0000390
0.0017078
0.0027150
Covariance
15%-10.5%
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20%-10.7%
Correlation Coefficient
Correlation coefficient- a statistical measure
that identifies how asset returns move in
relation to one another; denoted by p


Ranges from -1 (perfect negative correlation) to +1
(perfect positive correlation)
Related to covariance and individual standard
deviations:
where ρxy is the correlation coefficient, COV is
covariance, subxy are the variables, and σ is the
standard deviation
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Correlation Examples
No correlation:
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Correlation Examples
Perfect positive and perfect negative
correlation:
26
Correlation Examples
Positive and negative correlation:
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Correlation
From our example,
0.0027150
Correlation=
= 0.93268
0.31225 × 0.093226
Possible return
Return on A
30%
20%
10%
0%
-10%
-20%
Good
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Return on B
OK
Scenario
Bad
Word of Caution
The essential problem is that our models are
still too simple to capture the full array of
governing variables that drive global
economic reality.
A model, of necessity, is an abstraction from
the full detail of the real world.
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8.4 The Efficient Frontier
There is a mix of assets that minimizes a
portfolio’s standard deviation
Modern Portfolio Theory
 Modern
portfolio theory is a set of
theories that explain how rational
investors, who are risk averse, can select a
set of investments that maximize the
expected return for a given level of risk
 Harry Markowitz is the “father” of modern
portfolio theory, was awarded the 1990
Nobel Prize in Economics
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Modern Portfolio Theory
 Harry Markowitz showed investors how to
diversify their portfolios based on several
assumptions:



Investors are rational decision makers
Investors are risk averse
Investor preferences are based on a portfolio’s
expected return and risk (as measured by variance
or standard deviation)
 Introduced the notion of an efficient portfolio
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The Efficient Portfolio
The efficient portfolio is a collection of
investments that offers the highest expected
return for a given level of risk, or,
equivalently, offers the lowest risk for a given
expected return
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The Efficient Frontier
34
The Efficient Frontier
Portfolios are either:
 Attainable (lie on the minimum variance
frontier), or
 Dominated (lower level of expected return for a
given level of risk than another portfolio)
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The Efficient Frontier
 Rational, risk-averse investors are interested in
holding only those portfolios, such as Portfolio
II, that offer the highest expected return for
their given level of risk.
 A more aggressive (i.e., less risk averse)
investor might choose Portfolio II, whereas a
more conservative (i.e., more risk averse)
investor might prefer Portfolio V (i.e., the
MVP).
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8.5 Diversification
The reduction of risk by investing funds
across several assets
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Types of Risk
Random diversification or naïve
diversification is the act of randomly buying
assets without regard to relevant investment
characteristics (e.g., “dartboard”)
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Types of Risk
 Unique
risk, nonsystematic risk, or
diversifiable risk-a company-specific part
of total risk that is eliminated by
diversification
 Market risk, systematic risk, beta risk, or
nondiversifiable risk-a systematic part of
total risk that cannot be eliminated by
diversification
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Types of Risk
 Total risk
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= Market risk + Unique risk
Summarizing Risks
Risk that DISAPPEARS as you
diversify
• Diversifiable risk
• Nonsystematic risk
• Unique risk
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Risk that REMAINS even
when you diversify
•
•
•
•
Market risk
Systematic risk
Nondiversifiable risk
Beta risk
Summary
 In financial decision making and analysis, we
use both ex post and ex ante returns on
investments: We use ex post returns when we
look back at what has happened, and we use
ex ante returns when we look forward, into
the future.
 One measure of risk is the standard deviation,
which is a measure of the dispersion of
possible outcomes.
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Summary
 When we invest in more than one investment,
there may be some form of diversification,
which is the reduction in risk from combining
investments whose returns are not perfectly
correlated.
 If we consider all possible investments and
their respective expected return and risk,
there are sets of investments that are better
than others in terms of return and risk. These
sets make up the efficient frontier.
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Summary
 If we consider a company as a portfolio of
investments, diversification plays a role in
financial decision making. Financial managers
need to consider not only what an investment
looks like in terms of its return and risk as a
stand-alone investment, but more important,
how it fits into the company’s portfolio of
investments.
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Problem #1
Scenario Probability Outcome
Good
30%
$40
Normal
50%
$20
Bad
20%
$10
What is the expected value and standard
deviation for this probability distribution?
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Problem #2
The Key Company is evaluating two projects:
 Project 1 has a 40% chance of generating a
return of 20% and a 60% change of generating a
return of -10%.
 Project 2 has a 20% chance of generating a
return of 30% and an 80% chance of return of 5%.
Which project is riskier? Why?
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Problem #3
Suppose the covariance between the returns
on project A and B is -0.0045. And suppose
the standard deviations of A and B are 0.1
and 0.3, respectively. What is the correlation
between A and B’s returns?
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