Investments: Analysis and

Management
Fourteenth Edition
Gerald R. Jensen and Charles P. Jones

Chapter 7
Portfolio Theory
Investment Decisions
• Involve uncertainty
• Focus on expected returns
• Estimates of future returns need to consider and manage risk
• Investors often overly optimistic about expected returns
• Goal is to reduce risk without affecting returns
• Accomplished by building a portfolio
• Diversification is key

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Risk and Return Measures 1

• Ex post Calculations
• Mean (Average) Return ( R ) :


n

()
Rt
R = t =1

• Variance of Return ( 2 ) :

 ( R − R)
n 2

t =1 t
 2
=
n −1
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Risk and Return Measures 2

Ex ante Calculations
• Expected return: E ( R ) =  PS  RS

• Variance of Returns ( 2 ) :

 =  PS  ( RS − E ( R ) )
2 2

• Where, Ps equals probability of state s and Rs equals

return in state s.

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Dealing With Uncertainty
• Risk – the fact that an expected return may not be realized
• Investors must think about return distributions
• Probabilities weight outcomes
• Assigned to each possible outcome to create a distribution
• History provides guide but must be modified for expected
future changes
• Distributions can be discrete or continuous

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Calculating Expected Return
• Expected return for asset “i” E(Ri)
• Weighted average of all possible returns (Ri,s) included in
the probability distribution
• Each outcome weighted by probability of occurrence (Ps)
• Referred to as expected return

E ( Ri ) =  Ps  Ri ,s

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Calculating Risk
• Variance and standard deviation used to quantify and
measure risk
• Measure spread (dispersion) around the mean
• Variance of returns is in percent squared
• Standard deviation of returns (σ) is the square root of
variance and is measured in percent

=  2

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Example

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Modern Portfolio Theory
• Framework for selection of portfolios based on risk and
expected return
• Used, to varying degrees, by financial managers
• Quantifies benefits of diversification
• Security correlations are crucial in determining
portfolio risk
• An asset with high volatility may have low risk

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 Portfolio Expected Return
Portfolio Weights : The percentages of a portfolio’s total
value that are invested in each portfolio asset are referred to
as portfolio weights, which we will denote by w. The
combined portfolio weights are assumed to sum to 100
percent of total investable funds, or 1.0.

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Portfolio Expected Return
• Weighted average of the individual security expected
returns
• Each asset “i” has a weight, w, which represents the
asset’s value as a percent of the portfolio value

n
E ( Rp ) =  wi E ( Ri )
i =1

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 Example
Consider a three-stock portfolio consisting of stocks G, H, and
I with expected returns of 12 percent, 20 percent, and 17
percent, respectively. Assume that 50 percent of investable
funds is invested in security G, 30 percent in H, and 20
percent in I. The expected return on this portfolio is:

12
Portfolio Return 1

13
Portfolio Risk 1

• Portfolio risk is measured by the variance or standard

deviation of portfolio returns
• Portfolio variance is impacted by two characteristics:
1. The variance in returns for the individual assets
included in the portfolio
2. The co-movement of returns for the individual assets
included in the portfolio

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Portfolio Risk 2

• Portfolio risk is “not” the weighted average of individual

security risks
n
 p2   wi i 2
i =1

• The risk of individual securities is “not” the crucial

consideration
• Diversification almost always lowers risk
• An asset with high σ may add little to portfolio risk

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Portfolio Risk 3

Variance of a Portfolio ( p2 ):

 p2 =  wi 2 i 2 +  wiw j ij

σij = covariance of asset i and asset j

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 Calculating Portfolio Risk
THE TWO-SECURITY CASE
The risk of a portfolio, as measured by the standard
deviation of returns, for the case of two securities, 1 and 2, is

lThe variance of each security, as shown by σ11 and σ22 in Equation

l The covariance between securities, as shown by ρ1,2σ1σ2 in Equation
l The portfolio weights for each security, as shown by the wi’s in Equation

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Example
Example
Solution
The Impact of the Correlation Coefficient

The standard deviation of the portfolio is directly affected by

the correlation between the two stocks. Portfolio risk will be
reduced as the correlation coefficient moves from +1.0
downward, everything else constant
Example
The risk of this portfolio clearly depends heavily on the value
of the third term, which in turn depends on the correlation
coefficient between the returns for SEUT and PI. To assess the
potential impact of the correlation, consider the following
cases: a ρ of 11, 10.5, 10.29, 0, 0.5, and 1.0. Calculating
portfolio risk under each of these scenarios produces the
following portfolio risks:

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Securities weights and Portfolio Risk
Risk Reduction in Portfolios 2

• Random (or naïve) diversification

• Diversifying without looking at how security returns are
related to each other
• Marginal risk reduction gets smaller as securities are added
• Random diversification is beneficial but not optimal
• Risk reduction kicks in as securities added
• Research suggests it takes a large number of securities to
eliminate majority of risk

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Risk Reduction in Portfolios 1

• Market risk affects all firms, cannot be diversified away

• It is systematic i.e., part of the system
• The larger the number of securities, the smaller the
exposure to any particular risk
• “Insurance principle”
• Only issue is how many securities to hold

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Security Co-movement
• Correlation (ρij) and covariance (σij) measure the
tendency for security returns to move in the same or
opposite directions

( (
 ij =  Ps  ( Ri , s − E ( Ri ) )  R j , s − E ( R j ) ))
 ij
ij =
 i  j

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Correlation (ρij)
−1  ij  +1

ρij > 0 securities move together

ρij < 0 securities move apart
ρij = 0 no tendency one way or the other
ρij = −1 perfect negative correlation
ρij = +1 perfect positive correlation

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Correlation and Portfolio Risk

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Returns to H-Tech

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Returns to Giffen

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Portfolio: 50% Giffen & 50% H-Tech

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Correlation Coefficient
• When does diversification pay?
• With perfect positive correlation, risk is a weighted average,
therefore, no diversification benefit
• With perfect negative correlation, expected return can be
assured
• With zero correlation, significant risk reduction can be
achieved
• Cannot eliminate risk
• Negative correlation or low positive correlation is ideal, but
unlikely

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Calculating Portfolio Risk 1

• Three inputs to calculate portfolio risk

1. Variance (risk) of each security

2. Covariance between each pair of securities
3. 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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Calculating Portfolio Risk 2

• Generalizations
• The lower the correlation/covariance between securities,
the better
• As the number of securities increases:
• Number of covariances grows quickly

• The importance of covariance relationships increases

• The importance of each individual security’s risk

decreases

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Simplifying Markowitz Calculations
• Markowitz full-covariance model
• Requires a covariance between the returns of all
securities in order to calculate portfolio variance

• n  ( n − 1) 2 set of unique covariances for n securities

• Markowitz suggests using an index to which all
securities are related

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