Modern Portfolio Theory
Purpose: Review the highlights of portfolio
theory, Value Additivity Principle, Tobin’s
separation theorem
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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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-3-
Demonstration of Diversification
• Let’s start with one
stock picked at
random
Expected
Return
s
-4-
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
-6-
Demonstration of Diversification
• Now, pick another stock
at random
Expected
Return
s
-7-
Demonstration of Diversification
• Now, pick another stock
at random and
• Make a new portfolio
with equal proportions
Expected
Return
s
-8-
Demonstration of Diversification
Each repetition has
reduced impact as we
approach
Expected
Return
CML
Mkt
efficient
frontier
– efficient frontier
– market portfolio
– capital market line
s
-9-
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 estimating probabilities,
see problems 1 through 3 in the problem set
Given the best available expert opinion … 95% certain … return will be somewhere
in the range from 0% to 30%. Assuming a symmetric normal probability
distribution, translate into a mean and standard deviation.
Given the best available expert opinion … 95% certain … return will be somewhere
in the range from -5% to +40%. Assuming a symmetric normal probability
distribution, translate into a mean and standard deviation.
An investment has expected return of 10% with standard deviation of 2%.
Assuming the probability distribution is symmetric normal, what is the probability
of return 12% or higher?
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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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Practice
For practice in calculating probabilities
of beating the T-bill rate,
see problems 4 & 5 in the problem set
An investment has expected return of 12% with standard deviation 3%. Return on
U.S. Treasury bills is 6%. What is the approximate probability that the risky
investment will perform better than Treasury Bills?
An investment has expected return of 10% with standard deviation 2%. Return on
U.S. Treasury bills is 8%. What is the approximate probability that the risky
investment will perform better than Treasury Bills?
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Markowitz Formulae
• Expected Return for
a Portfolio
n
R p = Â wi Ri
i=1
• Variance
n
n
s 2p = Â Â wi w j s i s j rij
i=1 j=1
• Standard deviation is
square root of
variance
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Example Calculations
• Expected Return
for a Portfolio
Security 1
Security 1
Security 2
Security 3
n
R p = Â wi Ri
i=1
Security 2
Security 3
Weight x
Mean
Weight x
Mean
Weight x
Mean
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Example Calculations
• Expected Return=
16.5%
Security 1
Security 1
n
R p = Â wi Ri
i=1
Security 2
Security 3
.2 x 10%
Security 2
.3 x 15%
Security 3
.5 x 20%
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Example Calculations
• Expected Return=
14%
Security 1
Security 1
Security 2
Security 3
n
R p = Â wi Ri
i=1
Security 2
Security 3
.5 x 10%
.2 x 15%
.3 x 20%
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Practice
For practice in calculating expected return
of portfolios,
see problems 6 & 7 in the problem set
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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Example Calculations
• Variance
n
n
s 2p = Â Â wi w j s i s j rij
i=1 j=1
Security 1
Security 1
Security 2
Security 3
Security 2
Security 3
w12 s12
w2 w1s 2 s1r 21
w3w1s 3s1r31
w1w2s1s 2r12
w22 s 22
w3w2 s 3s 2r32
w1w3s1s 3r13 w2 w3s 2s 3r23
w32 s 32
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Practice
For practice in calculating standard
deviation for portfolios,
see problems 8 & 9 in the problem set
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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Practice
For more practice in estimating the mean
return and standard deviation for a
portfolio,
see problem 10 in the problem set
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Example: Mean Return
TCS
TCS
ACU
.4 * 15%
ACU
.6 * 20%
Mean = 6% + 12% = 18%
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Example: standard deviation
TCS
TCS
(.4 * .05)2
ACU
2(.4*.6*.05*.1*.2)
ACU
(.6 * .1)2
Variance = 0.0004 + 0.0036 + (2 * 0.00024) = .00448
Standard Deviation = 6.69%
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Practice
For more practice in estimating the mean
return and standard deviation for a
portfolio,
see problem 10 in the problem set
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Under what circumstances would mean and
standard deviation be insufficient?
• Skewed distributions
– i.e., options
• “Fat-tailed” distributions
– i.e., day-to-day returns for
individual stocks
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Practice
See problem 11 in the problem set
Cost: 10,000
Recovery: 9,000
0.6
0.5
0.4
Possible Outcomes:
-1000
0
15000
0.3
Probablity
0.2
0.1
0
-1000
15000
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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 12 & 13 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
• Now, is there a
dominant portfolio?
• Optimal investment
strategy
– the second separation
theorem
Expected
Return
CML
P*
efficient
frontier
Rf
standard
deviation
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Practice
See problem 14 in the problem set
Someone says that a risk-averse investor should select a
portfolio of low-risk stocks such as utilities, plus longmaturity municipal bonds … “Broad Index too risky” …
“Treasury Bills have too little return”
What is wrong with this advice?
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Practice
See problem 15 in the problem set
Jones is risk averse. Smith is more risk tolerant. Can both be
satisfied with portfolios built around an index fund and a
money market fund?
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Practice: See problem 16 in the problem set
Market portfolio, 95%
confidence interval:
-20% to +40%
Expected Return = 10%
Standard Deviation = 15%
Target portfolio, 95%
confidence interval:
-7.5% to +22.5%
Expected Return = 7.5%
Standard Deviation = 7.5%
Safe Rate is 5%
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Problem 16
• A simple solution:
– Half the money in the market portfolio, and half in treasury
bills
E(R)
CML
10%
7.5%
5%
s
7.5%
15%
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Practice: See problem 17 in the problem set
Market portfolio, 95%
confidence interval:
-4% to +28%
Expected Return = 12%
Standard Deviation = 8%
Target portfolio, 95%
confidence interval:
+2% to +10%
Expected Return = 6%
Standard Deviation = 2%
Safe Rate is 4%
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Problem 17
• A simple solution:
– 25% of the money in the market portfolio, and 75% of the
money in treasury bills
E(R)
CML
12%
6%
4%
s
2%
8%
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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 18 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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