Chapter
6
Efficient Diversification
Bodie, Kane, and Marcus
Essentials of Investments
12th Edition
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
• Ex) a risky stock and a risky bond
𝐸(𝑟𝑝 ) = 𝑊1 𝑟1 + 𝑊2 𝑟2
W1 = Proportion of funds in security 1
W2 = Proportion of funds in security 2
r1 = Expected return on security 1
r 2 = Expected return on security 2
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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
S
Cov(rS , rB ) =  p (i )[rS (i ) − E (rS )][rB (i ) − E (rB )]
i =1
• Correlation Coefficient
ρ SB
Cov(rS , rB )
=
σS  σB
Cov(rS , rB ) = ρ SB σ S σ B
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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
• Rule 1) Weighted average of returns on components, with
investment proportions as weights
• Rule 2) Weighted average of expected returns on components, with
portfolio proportions as weights
• Rule 3) Variance of portfolio:
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6.2 Asset Allocation with Two Risky Assets
• Exercise)
𝑬 𝒓𝑩 = 𝟓%; 𝝈𝑩 = 𝟖%; 𝑬 𝒓𝒔 = 𝟏𝟎%; 𝝈𝒔 = 𝟏𝟗%; 𝝆𝑩𝑺 = 𝟎. 𝟐
𝒓𝑷 = 𝒘𝟏 𝒓𝑩 + 𝒘𝟐 𝒓𝒔 , 𝒘𝟏 = 𝟎. 𝟒; 𝒘𝟐 = 𝟎. 𝟔
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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
SP =
E (rP ) − rf
P
• Optimal Risky Portfolio
• Best combination of risky and safe assets to form portfolio
• Two risky assets by maximizing 𝑆𝑝
wB =
[ E (rB ) − rf ] S2 − [ E (rs ) − rf ] B S  BS
[ E (rB ) − rf ] S2 + [ E (rs ) − rf ] B2 − [ E (rB ) − rf + E (rs ) − rf ] B S  BS
wS = 1 − wB
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Optimal Risky Portfolio
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Optimal Risky Portfolio
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Optimal Risky Portfolio
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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
𝑤𝐵 𝑂 = 0.568
𝑤𝑠 𝑂 = 0.432
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The Complete Portfolio
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Figure 6.7 The Complete Portfolio
Complete portfolio, C
55% of wealth in portfolio O
45% in Treasury bills
•
•
𝐸 𝑟𝐶 = 3% × 0.45 + 7.16% × 0.55 = 5.29%
𝜎𝑐 = 0.55 × 10.15% = 5.58%
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Figure 6.8 Portfolio Composition: Asset Allocation Solution
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Concept check 6.4
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Concept check 6.4
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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
1. Maximize risk premium for any level standard deviation
2. Minimize standard deviation for any level risk premium
3. 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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Markowitz Model
• Generalize the portfolio construction problem into the
case of many risky securities and a risk-free asset
1.
Determine the risk-return opportunities available to the investor
(minimum variance frontier)
• The portion of the frontier that lies above the global minimum-variance
portfolio is called the efficient frontier of risky assets
2.
Find the CAL with the highest Sharpe ratio
3.
The investor chooses the appropriate mix between the optimal
risky portfolio P and T-bills
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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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CALs with various portfolio
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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(regression model)
𝑹𝒊 = 𝜷𝒊 𝑹𝑴 + 𝜶𝒊 + 𝒆𝒊
Where:
• β𝑖 𝑅𝑀 : component of return due to movements in overall market
• β𝑖 : security’s responsiveness to market
• α𝑖 : stock’s expected excess return if market factor is neutral, i.e. market-
index excess return is zero
• 𝑒𝑖 : Component attributable to unexpected events relevant only to this
security (firm-specific)
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6.5 A Single-Index Stock Market
Variance(𝑅𝑖 )
=Variance (𝛼𝑖 + 𝛽𝑖 𝑅𝑀 + 𝑒𝑖 )
=Variance(𝛽𝑖 𝑅𝑀 )+Variance(𝑒𝑖 )
=Systematic risk + Firm-specific risk
• Statistical Representation of Single-Index Model
• Security Characteristic Line (SCL)
• Plot of security’s best predicted excess return from excess return of
market
• Algebraic representation of regression line:
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𝐸(𝑅𝐷 ห𝑅𝑀 ) = 𝛼𝐷 + 𝛣𝐷 𝑅𝑀
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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
Systematic (or explained) Variance
=
Total Variance
2 2
2
𝛽𝐷 𝜎𝑀
𝛽𝐷2 𝜎𝑀
=
= 2 2
𝜎𝐷2
𝛽𝐷 𝜎𝑀 + 𝜎 2 (𝑒𝐷 )
𝑅2
• A higher R-square tells us that systematic factors dominate firm-
specific factors in determining the return on the stock
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Figure 6.12 Scatter Diagram for Disney
𝑅𝐷𝑖𝑠𝑛𝑒𝑦 𝑡 = 0.23 + 1.046 ∗ 𝑅𝑀 𝑡 + 𝑒𝐷𝑖𝑠𝑛𝑒𝑦 (𝑡)
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Figure 6.13 Various Scatter Diagrams
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Basic Statistics
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Linear Regression
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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
n
eP =  wi ei
i =1
• Portfolio nonsystematic variance
n
 =  wi2 e2
2
eP
i =1
i
• Predicting betas – it exhibit a mean-reversion property
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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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Portfolio with Single Index Security
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Portfolio with Single Index Security
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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, 𝑆1 √𝑇, grows with investment horizon
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Table 6.4 Investment Risk for Different Horizons
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