Homework Answsers / Business & Finance
Efficient Diversification
Bodie, Kane, and Marcus
Essentials of Investments Eleventh Edition
6
Chapter
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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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
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6.2 Asset Allocation with Two Risky Assets
Covariance Calculations
Correlation Coefficient
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Spreadsheet 6.1 Capital Market Expectations
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Spreadsheet 6.2 Variance 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
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
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6.3 The Optimal Risky Portfolio with a Risk-Free Asset
Calculating Optimal Risky Portfolio
Two risky assets
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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
Determination of optimal risky portfolio
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
Where:
: component of return due to movements in overall market
: security’s responsiveness to market
: 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 and Graphical 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
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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
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Figure 6.12 Scatter Diagram for Ford
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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
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6.5 A Single-Index Stock Market
Using Security Analysis with Index Model
In
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CAPM and APT
Bodie, Kane, and Marcus
Essentials of Investments Eleventh Edition
7
Chapter
7.1 The Capital Asset Pricing Model
Capital Asset Pricing Model (CAPM)
Security’s required rate of return relates to systematic risk measured by beta
Market Portfolio (M)
Each security held in proportion to market value
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7.1 The Capital Asset Pricing Model: Assumptions
| Market Assumptions | Investor Assumptions |
| All investors are price takers | Investors plan for the same (single-period) horizon |
| All information relevant to security analysis is free and publicly available. | Investors are efficient users of analytical methods investors have homogeneous expectations. |
| All securities are publicly owned and traded. | Investors are rational, mean-variance optimizers. |
| No taxes on investment returns. | |
| No transaction costs. | |
| Lending and borrowing at the same risk-free rate are unlimited. |
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7.1 The Capital Asset Pricing Model
Hypothetical Equilibrium
All investors choose to hold market portfolio
Market portfolio is on efficient frontier, optimal risky portfolio
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7.1 The Capital Asset Pricing Model
Hypothetical Equilibrium
Risk premium on market portfolio is proportional to variance of market portfolio and investor’s risk aversion
Risk premium on individual assets
Proportional to risk premium on market portfolio
Proportional to beta coefficient of security on market portfolio
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Figure 7.1 Efficient Frontier and Capital Market Line
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7.1 The Capital Asset Pricing Model
Passive Strategy is Efficient
Mutual fund theorem: All investors desire same portfolio of risky assets, can be satisfied by single mutual fund composed of that portfolio
If passive strategy is costless and efficient, why follow active strategy?
If no one does security analysis, what brings about efficiency of market portfolio?
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7.1 The Capital Asset Pricing Model
Risk Premium of Market Portfolio
Demand drives prices, lowers expected rate of return/risk premiums
When premiums fall, investors move funds into risk-free asset
Equilibrium risk premium of market portfolio proportional to
Risk of market
Risk aversion of average investor
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7.1 The Capital Asset Pricing Model
Expected Returns on Individual Securities
Expected return-beta relationship
Implication of CAPM that security risk premiums (expected excess returns) will be proportional to beta
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7.1 The Capital Asset Pricing Model
The Security Market Line (SML)
Represents expected return-beta relationship of CAPM
Graphs individual asset risk premiums as function of asset risk
Alpha
Abnormal rate of return on security in excess of that predicted by equilibrium model (CAPM)
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Figure 7.2 The SML and a Positive-Alpha Stock
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7.1 The Capital Asset Pricing Model
Applications of CAPM
Use SML as benchmark for fair return on risky asset
SML provides “hurdle rate” for internal projects
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7.2 CAPM and Index Models
Index Model, Realized Returns, Mean-Beta Equation
: HPR
i: Asset
t: Period
: Intercept of security characteristic line
: Slope of security characteristic line
: Index return
: Firm-specific effects
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7.2 CAPM and Index Models
Estimating Index Model
, excess return
Residual = Actual return Predicted return for Google
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7.2 CAPM and Index Models: SCL
Security Characteristic Line (SCL)
Plot of security’s expected excess return over risk-free rate as function of excess return on market
Required rate = Risk-free rate + β x Expected excess return of index
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7.2 CAPM and Index Models
Predicting Betas
Mean reversion
Betas move towards mean over time
To predict future betas, adjust estimates from historical data to account for regression towards 1.0
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7.3 CAPM and the Real World
CAPM is false based on validity of its assumptions
Useful predictor of expected returns
Untestable as a theory
Principles still valid
Investors should diversify
Systematic risk is the risk that matters
Well-diversified risky portfolio can be suitable for wide range of investors
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7.4 Multifactor Models and CAPM
Multifactor models
Models of security returns that respond to several systematic factors
Two-index portfolio in realized returns
Two-factor SML
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7.4 Multifactor Models and CAPM
Fama-French Three-Factor Model
Estimation results
Three aspects of successful specification
Higher adjusted R-square
Lower residual SD
Smaller value of alpha
