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Portfolio Theory
(Jones chapter 7)
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Investment Decisions
Involve uncertainty
 Focus on expected returns
- Estimates of future returns needed to consider

and manage risk

Goal is to reduce risk without affecting
returns
- Accomplished by building a portfolio
- Diversification is key
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Dealing With Uncertainty
Risk that an expected return will not be
realized
 Investors must think about return
distributions, not just a single return
 Probabilities weight outcomes
- Should be assigned to each possible outcome to

-
create a distribution
Can be discrete or continuous
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Calculating Expected Return

Expected value
- The single most likely outcome from a particular probability
distribution
- The weighted average of all possible return outcomes
- Referred to as an ex ante or expected return
m
E(R )   R ipri
i 1
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Example: Given the following probability distribution,
calculate the expected return of security XYZ.

Security XYZ's
Potential return
20%
30%
-40%
50%
10%
Probability
0.3
0.2
0.1
0.1
0.3
Solution:
E(R) = Ripri = (20)(0.3) + (30)(0.2) + (- 40)(0.1) +
(50)(0.1) + (10)(0.3) = 22 percent
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Calculating Risk

Variance and standard deviation used to
quantify and measure risk
- Measures the spread in the probability
-
distribution
Variance of returns: σ² = (Ri - E(R))²pri
Standard deviation of returns:
σ =(σ²)1/2
Ex ante rather than ex post σ relevant
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Portfolio Expected Return

Weighted average of the individual security
expected returns
- Each portfolio asset has a weight, w, which
represents the percent of the total portfolio
value
n
E(R p )   w iE(R i )
i1
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Portfolio Risk



Portfolio risk not simply the sum of individual security risks
Emphasis on the risk of the entire portfolio and not on risk of
individual securities in the portfolio
Measured by the variance or standard deviation of the portfolio’s
return
- Portfolio risk is not a weighted average of the risk of the
individual securities in the portfolio
n
   wi
2
p
i1
8
2
i
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Risk Reduction in Portfolios

Random diversification
- Diversifying without looking at relevant investment
-
characteristics
Marginal risk reduction gets smaller and smaller as
more securities are added
Correlation drives the diversification benefits
 A large number of securities is not required for
significant risk reduction
 International diversification benefits

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Portfolio Risk and Diversification
p %
Portfolio risk
35
20
Market Risk
0
10
20
30
40
......
Number of securities in portfolio
10
100+
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The benefits of diversification

Come from the correlation between asset returns

The smaller the correlation, the greater the risk reduction
potential  greater the benefit of diversification

If r = +1.0, no risk reduction is possible
 Adding extra securities with lower corr/cov with the existing
ones decreases the total risk of the portfolio
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Markowitz Diversification

Non-random diversification
- Active measurement and management of
-
portfolio risk
Investigate relationships between portfolio
securities before making a decision to invest
Takes advantage of expected return and risk for
individual securities and how security returns
move together
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Measuring Portfolio Risk

Needed to calculate risk of a portfolio:
- Weighted individual security risks
Calculated by a weighted variance using the
proportion of funds in each security
For security i: (wi × i)2
- Weighted comovements between returns
Return covariances are weighted using the
proportion of funds in each security
For securities i, j: 2wiwj × ij
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Portfolio Risk and Return

Expected Portfolio Return
N
ER P    Wi ER i 
i 1

Standard Deviation of Portfolio Returns
SDRP  
N
W
i 1
i
 VARRi   Wi  W j  COV Ri R j 
N
2
N
i 1 j 1
j i
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Calculating Portfolio Risk
Encompasses three factors
- Variance (risk) of each security
- Covariance between each pair of securities
- Portfolio weights for each security
 Goal: select weights to determine the
minimum variance combination for a given
level of expected return

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% in stocks
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50.00%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.08%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.00%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Return
Example: NOT on the EXAM
Portfolo Risk and Return Combinations
12.0%
11.0%
100%
stocks
10.0%
9.0%
8.0%
7.0%
6.0%
100%
bonds
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
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Learning objectives
Know the concept of uncertainty
Know how to calculate expected return (probabilities)
Know how to calculate portfolio expected return (weights)
Concept of risk, portfolio risk
Firm and market specific risks; correlation; diversification
Know the concepts of correlation and diversification
NOT on the EXAM p. 189 to 196:
- covariance section p. 189-190
-calculations with standard deviations formula 7-12 section
“the n-security case”
End of chapter 7.1 to 7.5; 7.26; problems 7.1, 7-2
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