Modern Portfolio Theory
Purpose: Review the highlights of portfolio
theory, Value Additivity Principle, Tobin’s
separation theorem
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Learning Objectives Satisfied:
4. Risk and Rates of Return
Objectives: Understand the following concepts:
4Risk of an individual asset and that of a portfolio
4How is risk-return tradeoff affected by
diversification?
4Market risk, diversifiable risk
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Why take a portfolio viewpoint?
• Diversification reduces risk
– Additional securities increase reward,
reduce risk
– Small holdings are inefficient
• Individual securities are fungible
(unless you have inside information)
– Any stock is a perfect substitute for another
in a diversified portfolio
– All portfolios have the same probability of
beating the safe investment
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Demonstration of Diversification
• Let’s start with one
stock picked at
random
Expected
Return
s
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Demonstration of Diversification
• Let’s start with one
stock picked at
random
• Then, pick another
and invest equally in
each
Expected
Return
s
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Demonstration of Diversification
This forms a portfolio
in which:
• E(R) is average of the
two
• sd is less than the
weighted average
Expected
Return
s
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Demonstration of Diversification
• Now, pick another stock
at random
Expected
Return
s
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Demonstration of Diversification
• Now, pick another stock
at random and
• Make a new portfolio
with equal proportions
Expected
Return
s
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Demonstration of Diversification
Each repetition has
reduced impact as we
approach
– efficient frontier
– market portfolio
– capital market line
Expected
Return
CML
Mkt
efficient
frontier
s
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Highlights of Portfolio Theory
• Fundamental
Assumption
– Two dimensions:
Risk & Reward
• Law of One Price
Reward
Market Line
Risk
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Measuring risk and reward for
diversified portfolios
• Why mean and standard deviation of
return were chosen
– Mean = best estimate of future performance
– Standard deviation defines the confidence
interval around this estimate
• Together
– They express the probability of an event
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We Know
• 67% of all events within
1 s.d. above or below the
mean
• 95% of all events within
2 s.d. about the mean
• 99% of all events within
3 s.d. about the mean
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Examples
Assume:
– Expected Return =
10%
– Standard Deviation =
5%
Find:
Probability return will be 5% to
15%
Probability return will be 0% to
20%
Probability of return -5% to +25%
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Examples
Assume:
–
–
Expected Return = 10%
Standard Deviation = 5%
Find:
Probability of return greater than
20%
Probability of losing anything
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Examples
Assume:
– Expected Return = 10%
– Standard Deviation = 5%
Find:
Probability of return 5% or more
Probability of return 0% or more
Probability of return better than
negative 5%
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Practice
For practice in calculating expected return
of portfolios,
see problems 1 & 2 in Problem Set #6
Calculate the expected return for a portfolio made from equal proportions
of investments with expected returns of 10%, 12%, and 14%.
Calculate the expected return for a portfolio with $200 invested in stock A,
$300 in stock B, and $500 in stock C. Expected returns for the individual
stocks are 10% for stock A, 12% for stock B, and 14% for stock C.
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Practice
For practice in calculating standard
deviation for portfolios,
see problems 3 & 4 in Problem Set #6
8.
Suppose a portfolio is has equal proportions of investments with standard
deviation of 10%, 12%, and 14%, respectively. Which of the following
could possibly be the standard deviation of the returns for the portfolio?
A. 14%
B. 15%
C. 9%
D. 20%
9.
Suppose a portfolio includes $200 invested in stock A, $300 in stock B, and
$500 in stock C. Standard deviations for the individual stocks are 10% for
stock A, 12% for stock B, and 14% for stock C. Which of the following
could possibly be the standard deviation of the returns for the portfolio?
A. 14%
B. 13%
C. 8%
D. 15%
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Efficient Frontier
• Is there a dominant
portfolio?
• Why is efficient
frontier concave?
• Could all investors
agree on an “optimal
portfolio”?
Expected
Return
B
A
C
efficient
frontier
standard
deviation
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Practice
For practice in recognizing dominant
portfolios,
see problems 5 & 6 in the problem set
A:
B:
C:
12%
12%
14%
5%
7%
5%
A:
B:
C:
15%
15%
14%
9%
7%
8%
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Tobin's capital market theory
• The capital market
line
Expected
Return
• Now, is there a
dominant portfolio?
CML
P*
• Optimal investment
strategy
efficient
frontier
Rf
standard
deviation
– the second separation
theorem
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Implication:
• A simple investment rule is implied by the linear
relationship between mean return and standard
deviation:
– All portfolios on the CML have the same probability of earning
a higher return than Treasury Bills
E(R)
CML
9%
7%
5%
s
2%
4%
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Diversification reduces risk
• A relatively small
portfolio (12 to 15
securities) does a
very good job.
• Portfolio
performance reverts
to mean
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Practice
See problem 7 in the problem set
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This leads to the Value Additivity Principle:
• Diversification has no market value
– If it did, there would be easy arbitrage opportunities
– Conclusion: the value of the whole just equals the sum of the
values of the parts
• This realization serves as the springboard into Asset
Pricing Theory
– Which computes value based on an asset’s contribution to the
risk and return of a portfolio
• Does completing the market add value?
– Answer: Yes
– Conclusion: value of whole may be less than sum of parts
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Discussion Questions
•
Why pay for an investment manager?
•
Who can pick stocks?
•
Who can time the market?
•
Do people need professional help with asset allocation?
•
Why revise portfolio?
•
Who is best advisor: broker, accountant, or lawyer?
When capital markets are efficient
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