Source: Bodie, Kane and Marcus, Investments, 12 ed., McGraw-Hill, 2021
Index model
• Drawbacks to Markowitz procedure
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• Requires a huge number of estimates to fill the covariance matrix
• Model does not provide any guidelines for finding useful estimates of these covariances or the risk premiums
• Introduction of index models
• Simplifies estimation of the covariance matrix • Enhances analysis of security risk premiums
• Optimal risky portfolios constructed using the index model
A Single‐Factor Security Market
• Advantages of the single-index model
• Number of estimates required is a small fraction of what would otherwise be needed
• Specialization of effort in security analysis
r E(r)me iiii
• βi = sensitivity coefficient for firm I
• m = market factor that measures unanticipated
developments in the macroeconomy • ei = firm-specific random variable
Single-Index Model 1
• Regression equation
R(t)R (t)e(t) iiiMi
• Expected return-beta relationship
E(R)E(R ) iiiM
Single-Index Model 2
• Total risk = Systematic risk + Firm-specific risk
(e)
• Covariance = Product of betas × Market-index risk
Cov r , r 2 ij ijM
• Correlation = Product of correlations with the market index
Index Model and Diversification
• Variance of the equally-weighted portfolio of firm-specific components:
2(eP)Var
1n n12
2(e) i
• When n gets large, σ2(ep) becomes negligible
• As diversification increases, the total variance of a portfolio approaches the systematic variance
The Variance of an Equally Weighted Portfolio with Risk Coefficient, βp
Excess Monthly Returns on Amazon and the Market Index
Scatter Diagram
• Access the text alternative for slide images.
Security Characteristic Line (SCL)
Expected excess return when the market excess return is zero
Sensitivity of security i’s return to changes in the return of the market
• Expected excess return of
• Excess return of security i R(t)α βR (t)e(t)
i i iS&P500 i
the market
Excel Output: Regression Statistics
Regression Statistic
Multiple R R-Square Adjusted R-Square Standard Error Observations
0.5351 0.2863 0.2742
Intercept 0.0192 Market index 1.5326
0.0093 0.3150
0.0686 60 Coefficients
Standard Error
t-statistic 2.0645
p-value 0.0434
The Industry Version of the Index Model
• Predicting betas
• Betas tend to drift to 1 over time
• Rosenberg and Guy found the following variables to help predict betas:
1. Variance of earnings
2. Variance of cash flow
3. Growth in earnings per share 4. Market capitalization (firm size) 5. Dividend yield
6. Debt-to-asset ratio
Portfolio Construction and the Single-Index Model 1
• Alpha and security analysis
1. Macroeconomic analysis used to estimate the risk premium and risk of the market index
2. Statistical analysis used to estimate beta coefficients and residual variances, σ2(ei), of all securities
3. Establish expected return of each security absent any contribution from security analysis
4. Security-specific expected return forecasts are derived from various security-valuation models
Portfolio Construction and the Single-Index Model 2
• Single-index model input list:
1. RiskpremiumontheS&P500portfolio
2. StandarddeviationoftheS&P500portfolio
3. nsetsofestimatesof: – Beta coefficients
– Stock residual variances – Alpha values
Portfolio Construction and the Single-Index Model 3
• Optimal risky portfolio in the single-index model
• Objective is to select portfolio weights to maximize the Sharpe ratio of the portfolio
n1 n1 E(RP)PE(RM)P wE(R) w
w w(e)
2 n1 1/2 22 2 1/22 22
[(e)]
P PMPMi1i1
Portfolio Construction: The Process
• Optimal risky portfolio in the single-index model is a combination of two component portfolios:
• Active portfolio, denoted by A
• Market-index portfolio (that is, passive portfolio), denoted by M
Summary of Optimization Procedure 1
1. Compute the initial position of each security:
2. Scalethoseinitialpositions:
3. Computethealphaoftheactiveportfolio:
i 2 (e )
A n w
Summary of Optimization Procedure 2
4. Compute the residual variance of A:
5. Compute the initial position in A:
6. Compute the beta of A:
w0 A 2(eA) A E(R )2
A n w
Summary of Optimization Procedure 3 7. AdjusttheinitialpositioninA:
NoteWhenA 1
w* w0 AA
A 1(1 )w0 AA
Summary of Optimization Procedure 4
8. Optimal risky portfolio now has weights:
w* 1w* MA
w* w* w iAi
9. Calculate the risk premium of P (Optimal risky portfolio):
E(R )(w* w* )E(R )w* PMAAMAA
10. Compute the variance of P:
2 (w* w* )22 [w*(e )]2 PMAAMAA
Optimal Risky Portfolio: Information Ratio
• InformationRatio
• The contribution of the active portfolio depends on the ratio of its alpha to its residual standard deviation (Step 5)
• Calculated as the ratio of alpha to the standard deviation of diversifiable risk
• The information ratio measures the extra return we can obtain from security analysis
Optimal Risky Portfolio:
• The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy):
S2S2A 2
P M (e ) A
Efficient Frontier: Index Model and Full-Covariance Matrix
• Figure 8.4 Efficient frontier constructed from the index model and the full covariance matrix
Portfolios from the Index and Full-Covariance Models
A. Weights in Optimal Risky Portfolio
Market index WMT (Walmart) TGT (Target) VZ (Verizon)
F (Amazon)
GM (General Motors)
B. Portfolio Characteristics Risk premium
Standard deviation
Sharpe ratio
0.10 0.19 0.07 0.08 0.01 −0.03
Index Model
Full-Covariance Model
0.13 0.17 −0.07 −0.14 −0.05 −0.18
0.0605 0.0639 0.1172 0.1238 0.5165 0.5163
Is the Index Model Inferior to the Full-Covariance Model?
• Full Markowitz model is better in principle, but
• The full-covariance matrix invokes estimation risk of thousands of terms
• Cumulative errors may result in a portfolio that is actually inferior
• The single-index model is practical and decentralizes macro and security analysis
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