Chapter
Efficient Diversification
Bodie, Kane, and Marcus
Essentials of Investments
12th Edition
Adopted for ECON5009 Financial Markets Securities
and Derivatives , University of Glasgow, 2022/2023
6
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6.1 Diversification and Portfolio Risk
• Market, Systematic, & Nondiversifiable Risk
• Risk factors common to whole economy
• Unique, Firm-Specific, Nonsystematic &
Diversifiable Risk
• Risk that can be eliminated by diversification
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Figure 6.1 Risk as Function of Number of Stocks in Portfolio
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Figure 6.2 Risk versus Diversification
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Spreadsheet 6.1 Capital Market Expectations
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6.2 Asset Allocation with Two Risky Assets
• Covariance and Correlation
• Portfolio risk depends on covariance between
returns of assets
• Expected return on two-security portfolio
2211)( rWrWrE p +=
2security on return Expected
1security on return Expected
2security in funds of Proportion
1security in funds of Proportion
2
1
2
1
=
=
=
=
r
r
W
W
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Spreadsheet 6.2 Variance & Standard Deviations of
Returns
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Spreadsheet 6.3 Portfolio Performance
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Spreadsheet 6.4 Return Covariance
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6.2 Asset Allocation with Two Risky Assets
• Covariance Calculations
• Correlation Coefficient
])()()][()()[(),Cov(
1
BS 
=
−−=
S
i
BBSS rEirrEiriprr
BS
BS
SB
rr
σσ
),Cov(
ρ

=
BSSBBS rr σσρ),Cov( =
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6.2 Asset Allocation with Two Risky Assets
• Using Historical Data
• Variability/covariability change slowly over time
• Use realized returns to estimate
• Cannot estimate averages precisely
• Focus for risk on deviations of returns from
average value
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6.2 Asset Allocation with Two Risky Assets
• RoR: Weighted average of returns on components, with
investment proportions as weights
• ERR: Weighted average of expected returns on
components, with portfolio proportions as weights
• Variance of RoR:
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6.2 Asset Allocation with Two Risky Assets
• Risk-Return Trade-Off
• Investment opportunity set
• Available portfolio risk-return combinations
• Mean-Variance Criterion
• If E(rA) ≥ E(rB) and σA ≤ σB
• Portfolio A dominates portfolio B
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Spreadsheet 6.5 Investment Opportunity Set
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Figure 6.3 Investment Opportunity Set
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Figure 6.4 Opportunity Sets: Various Correlation Coefficients
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Spreadsheet 6.6 Opportunity Set -Various Correlation Coefficients
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6.3 The Optimal Risky Portfolio with a Risk-Free Asset
• Slope of CAL is Sharpe Ratio of Risky
Portfolio
• Optimal Risky Portfolio
• Best combination of risky and safe assets to
form portfolio
( )P f
P
P
E r r
S

−
=
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6.3 The Optimal Risky Portfolio with a Risk-Free Asset
• Calculating Optimal Risky Portfolio
• Two risky assets
BSSBfsfBBfsSfB
BSSBfsSfB
B
rrErrErrErrE
rrErrE
w


])()([])([])([
])([])([
22
2
−+−−−+−
−−−
=
BS ww −=1
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Figure 6.5 Two Capital Allocation Lines
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Figure 6.6 Bond, Stock and T-Bill Optimal Allocation
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Figure 6.7 The Complete Portfolio
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Figure 6.8 Portfolio Composition: Asset Allocation Solution
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6.4 Efficient Diversification with Many Risky Assets
• Efficient Frontier of Risky Assets
• Graph representing set of portfolios that
maximizes expected return at each level of
portfolio risk
• Three methods
• Maximize risk premium for any level standard deviation
• Minimize standard deviation for any level risk premium
• Maximize Sharpe ratio for any standard deviation or risk
premium
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Figure 6.9 Portfolios Constructed with Three Stocks
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Figure 6.10 Efficient Frontier: Risky and Individual Assets
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6.4 Efficient Diversification with Many Risky Assets
• Choosing Optimal Risky Portfolio
• Optimal portfolio CAL tangent to efficient frontier
• Separation Property implies portfolio choice,
separated into two tasks
1. Determination of optimal risky portfolio
2. Personal choice of best mix of risky portfolio and risk-
free asset
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6.4 Efficient Diversification with Many Risky Assets
• Optimal Risky Portfolio: Illustration
• Efficiently diversified global portfolio using stock
market indices of six countries
• Standard deviation and correlation estimated
from historical data
• Risk premium forecast generated from
fundamental analysis
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Figure 6.11 Efficient Frontiers & CAL: Table 6.1
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6.5 A Single-Index Stock Market
• Index model: Relates stock returns to returns on
broad market index & firm-specific factors
• Excess return: RoR in excess of risk-free rate
• Beta: Sensitivity of security’s returns to market
factor
• Firm-specific or residual risk: Component of
return variance independent of market factor
• Alpha: Stock’s expected return beyond that induced
by market index
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6.5 A Single-Index Stock Market
• Excess Return
• = β + α +
Where:
• β: component of return due to movements in overall
market
• β: security’s responsiveness to market
• α: stock’s expected excess return if market factor is neutral,
i.e. market-index excess return is zero
• : Component attributable to unexpected events relevant
only to this security (firm-specific)
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6.5 A Single-Index Stock Market
• Statistical Representation of Single-Index
Model
• Security Characteristic Line (SCL)
• Plot of security’s predicted excess return from excess
return of market
• Algebraic representation of regression line
( )D M D D ME R R R= +
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6.5 A Single-Index Stock Market
• Statistical and Graphical Representation of
Single-Index Model
• Ratio of systematic variance to total variance
2
2 2 2 2
2 2 2 2
Systematic Variance
Total Variance
( )
D M D M
D D M De

   
   
=
= =
+
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Figure 6.12 Scatter Diagram for Disney
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Figure 6.13 Various Scatter Diagrams
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6.5 A Single-Index Stock Market
• Diversification in Single-Index Security Market
• In portfolio of n securities with weights
• In securities with nonsystematic risk
• Nonsystematic portion of portfolio return
•
• Portfolio nonsystematic variance
2 2 2
1
P i
n
e i e
i
w 
=
=
1
n
P i i
i
e w e
=
=
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6.5 A Single-Index Stock Market
• Using Security Analysis with Index Model
• Information ratio
• Ratio of alpha to standard deviation of residual
• Active portfolio
• Portfolio formed by optimally combining analyzed stocks
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6.6 Risk of Long-Term Investments
• Why the Unending Confusion?
• Vast majority of financial advisers believe stocks
are less risky if held for long run
• Risk premium grows at rate of horizon, T
• Standard deviation grows at √T
• Sharpe ratio, 1√, grows with investment horizon
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Table 6.4 Investment Risk for Different Horizons