1. Index Models The Capital Asset Pricing Model
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2. Chapter Overview Advantages of a single-factor model
Risk decomposition
Systematic vs. firm-specific
Single-index model and its estimation
Optimal risky portfolio in the index model
Index model vs. Markowitz procedure

3. A Single-Factor Market
Advantages
Reduces the number of inputs for diversification
Easier for security analysts to specialize
Model
βi = response of an individual security’s return to the common factor, m; measure of systematic risk
m = a common macroeconomic factor
ei = firm-specific surprises

4. Single-Index Model Regression equation:
Expected return-beta relationship:

5. Single-Index Model Variance = Systematic risk + Firm-specific risk:
Covariance = Product of betas × Market index risk:

6. Single-Index Model
Correlation = Product of correlations with the market index

7. Index Model and Diversification
Variance of the equally-weighted portfolio of firm-specific components:
When n gets large, σ2(ep) becomes negligible and firm specific risk is diversified away

8. Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk Coefficient βp

9. Figure 8.2 Excess Returns on HP and S&P 500

10. Figure 8.3 Scatter Diagram of HP, the S&P 500, and HP’s SCL

11. Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard

12. Table 8.1 Interpreting the Output
Correlation of HP with the S&P 500 is
The model explains about 52% of the variation in HP
HP’s alpha is 0.86% per month (10.32% annually) but it is not statistically significant
HP’s beta is , but the 95% confidence interval is 1.43 to 2.53

13. Figure 8.4 Excess Returns on Portfolio Assets

14. Portfolio Construction and the Single-Index Model
Alpha and Security Analysis
Use macroeconomic analysis to estimate the risk premium and risk of the market index.
Use statistical analysis to estimate the beta coefficients of all securities and their residual variances, σ2(ei).
Establish the expected return of each security absent any contribution from security analysis.
Use security analysis to develop private forecasts of the expected returns for each security.

15. Portfolio Construction and the Single-Index Model
Single-Index Model Input List
Risk premium on the S&P 500 portfolio
Estimate of the SD of the S&P 500 portfolio
n sets of estimates of
Beta coefficient
Stock residual variances
Alpha values

16. Portfolio Construction and the Single-Index Model
Optimal risky portfolio in the single-index model
Expected return, SD, and Sharpe ratio:

17. Portfolio Construction and the Single-Index Model
Optimal risky portfolio in the single-index model is a combination of
Active portfolio, denoted by A
Market-index portfolio, the passive portfolio, denoted by M

18. Portfolio Construction and the Single-Index Model
Optimal risky portfolio in the single-index model
Modification of active portfolio position:
when

19. Portfolio Construction and the Single-Index Model
The Information Ratio
The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy):

20. Portfolio Construction and the Single-Index Model
The Information Ratio
The contribution of the active portfolio depends on the ratio of its alpha to its residual standard deviation
The information ratio measures the extra return we can obtain from security analysis

21. Figure 8.5 Efficient Frontiers with the Index Model and Full-Covariance Matrix

22. Table 8.2 Portfolios from the Single-Index and Full-Covariance Models

23. Is the Index Model Inferior to the Full-Covariance Model?
Full Markowitz model may be better in principle, but
Using the full-covariance matrix invokes estimation risk of thousands of terms
Cumulative errors may result in a portfolio that is actually inferior to that derived from the single-index model
The single-index model is practical and decentralizes macro and security analysis

24. Industry Version of the Index Model
Use 60 most recent months of price data
Use S&P 500 as proxy for M
Compute total returns that ignore dividends
Estimate index model without excess returns:

25. Industry Version of the Index Model
Adjust beta because
The average beta over all securities is 1; thus, the best forecast of the beta would be that it is 1
Firms may become more “typical” as they age, causing their betas to approach 1

26. Table 8.4 Industry Betas and Adjustment Factors

27. The Capital Asset Pricing Model

28. Capital Asset Pricing Model (CAPM)
It is the equilibrium model that underlies all modern financial theory
Derived using principles of diversification with simplified assumptions
Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development 28

29. Assumptions Investors optimize portfolios a la Markowitz
Investors use identical input list for efficient frontier
Same risk-free rate, tangent CAL and risky portfolio
Market portfolio is aggregation of all risky portfolios and has same weights

30. Resulting Equilibrium Conditions
All investors will hold the same portfolio for risky assets – market portfolio
Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value

31. Figure 9.1 The Efficient Frontier and the Capital Market Line

32. Market Risk Premium
The risk premium on the market portfolio will be proportional to its risk and the degree of risk aversion of the investor:
E(RM) = Ᾱσ2M
Where σ2M is the variance of the market portfolio and Ᾱ is the average degree of risk aversion across investors

33. Return and Risk For Individual Securities
The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio.
An individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio.

34. GE Example Covariance of GE return with the market portfolio:
Therefore, the reward-to-risk ratio for investments in GE would be:

35. GE Example Reward-to-risk ratio for investment in market portfolio:
Reward-to-risk ratios of GE and the market portfolio should be equal:

36. GE Example
The risk premium for GE:
Restating, we obtain:

37. Expected Return-Beta Relationship
CAPM holds for the overall portfolio because:
This also holds for the market portfolio:

38. Figure 9.2 The Security Market Line

39. Figure 9.3 The SML and a Positive-Alpha Stock

40. Single-Index Model and Realized Returns
To move from expected to realized returns, use the index model in excess return form:
The index model beta coefficient is the same as the beta of the CAPM expected return-beta relationship.

41. Assumptions of the CAPM
Individuals
Mean-variance optimizers
Homogeneous expectations
All assets are publicly traded
Markets
All assets are publicly held
All information is available
No taxes
No transaction costs

42. Extensions of the CAPM Zero-Beta Model
Helps to explain positive alphas on low beta stocks and negative alphas on high beta stocks
Consideration of labor income and non-traded assets

43. Extensions of the CAPM Merton’s Multiperiod Model and hedge portfolios
Incorporation of the effects of changes in the real rate of interest and inflation
Consumption-based CAPM
Rubinstein, Lucas, and Breeden
Investors allocate wealth between consumption today and investment for the future

44. Liquidity and the CAPM
Liquidity: The ease and speed with which an asset can be sold at fair market value
Illiquidity Premium: Discount from fair market value the seller must accept to obtain a quick sale.
Measured partly by bid-asked spread
As trading costs are higher, the illiquidity discount will be greater.

45. Figure 9.5 The Relationship Between Illiquidity and Average Returns

46. Liquidity Risk
In a financial crisis, liquidity can unexpectedly dry up.
When liquidity in one stock decreases, it tends to decrease in other stocks at the same time.
Investors demand compensation for liquidity risk
Liquidity betas

47. CAPM and World Academic world Investment Industry
Cannot observe all tradable assets
Impossible to pin down market portfolio
Attempts to validate using regression analysis
Investment Industry
Relies on the single-index CAPM model
Most investors don’t beat the index portfolio