1. Topic 5: Asset Pricing Models Larry Schrenk, Instructor
FIN 377: Investments
Topic 5: Asset Pricing Models
Larry Schrenk, Instructor

2. Overview 7.1 The Capital Asset Pricing Model
7.2 Empirical Tests of the CAPM
7.3 The Market Portfolio: Theory versus Practice
7.4 Arbitrage Pricing Theory
7.5 Multifactor Models and Risk Estimation
Appendix: Multiple Regression in Excel

3. Learning Objectives @

4. Readings
Reilley, et al., Investment Analysis and Portfolio Management, Chap. 7

5. 7.1 The Capital Asset Pricing Model

6. 7.1 The Capital Asset Pricing Model
The capital asset pricing model (CAPM) extends capital market theory in a way that allows investors to evaluate the risk–return trade-off for both diversified portfolios and individual securities
The CAPM:
Redefines the relevant measure of risk from total volatility to just the nondiversifiable portion of that total volatility (systematic risk)
The risk measure is called the beta coefficient and calculates the level of a security’s systematic risk compared to that of the market portfolio

7. 7.1.1 A Conceptual Development of the CAPM
The existence of a risk-free asset resulted in deriving a capital market line (CML) that became the relevant frontier
However, CML cannot be used to measure the expected return on an individual asset
For individual asset (or any portfolio), the relevant risk measure is the asset’s covariance with the market portfolio
That is, for an individual asset i, the relevant risk is not σi, but rather σi riM, where riM is the correlation coefficient between the asset and the market

8. 7.1.1 A Conceptual Development of the CAPM
Inserting this product into the CML and adapting the notation for the ith individual asset:
Let βi=(σi riM) / σM be the asset beta measuring the relative risk with the market, the systematic risk
The CAPM indicates what should be the expected or required rates of return on risky assets
This helps to value an asset by providing an appropriate discount rate to use in dividend valuation models

9. 7.1.2 The Security Market Line
The SML
Is a graphical form of the CAPM
Shows the trade-off between risk and expected return as a straight line intersecting the vertical axis (zero-risk point) at the risk-free rate
Considers only the systematic component of an investment’s volatility
Can be applied to any individual asset or collection of assets

10. 7.1.2 The Security Market Line

11. 7.1.2 The Security Market Line
Determining the Expected Rate of Return for a Risky Asset
Example:

12. 7.1.2 The Security Market Line
Risk-free rate is 5% and the market return is 9%
This implies a market risk premium of 4%

13. 7.1.2 The Security Market Line
Identifying Undervalued and Overvalued Assets
In equilibrium, all assets and all portfolios of assets should plot on the SML
Any security with an estimated return that plots above the SML is underpriced
Any security with an estimated return that plots below the SML is overpriced
A superior investor must derive value estimates for assets that are consistently superior to the consensus market evaluation to earn better risk-adjusted rates of return than the average investor

14. 7.1.2 The Security Market Line
Example:
Compare the required rate of return to the estimated rate of return for a specific risky asset using the SML over a specific investment horizon to determine if it is an appropriate investment

15. 7.1.2 The Security Market Line

16. 7.1.2 The Security Market Line

17. 7.1.2 The Security Market Line

18. 7.1.2 The Security Market Line
Calculating Systematic Risk
A beta coefficient for Security i can be calculated directly from the following formula:
Security betas can also be estimated as the slope coefficient in a regression equation between the returns to the security (Rit) over time and the returns (RMt) to the market portfolio (the security’s characteristic line):
Rit = ai + bi(RMt) + eit

19. 7.1.2 The Security Market Line
The Impact of the Time Interval
The number of observations and time interval used in the calculation of beta vary widely, causing beta to vary
There is no “correct” interval for analysis
Morningstar uses monthly returns over five years
Reuters Analytics uses daily returns over two years
Bloomberg uses weekly returns over two years although the system allows users to change the time intervals

20. 7.1.2 The Security Market Line
The Effect of the Market Proxy
The Standard & Poor’s 500 Composite Index is often used as the proxy because:
It contains large proportion of the total market value of U.S. stocks
It is a value weighted index
Theoretically, the market portfolio should include all U.S. and non-U.S. stocks and bonds, real estate, coins, stamps, art, antiques, and any other marketable risky asset from around the world

21. 7.1.2 The Security Market Line
Computing a Characteristic Line: An Example
The example shows how to estimate a characteristic line for Microsoft Corp (MSFT) using monthly return data from January 2016 to December 2016
Betas for MSFT are calculated using:
The S&P 500 (SPX)
The MSCI World Equity (MXWO) index

22. 7.1.2 The Security Market Line

23. 7.1.2 The Security Market Line

24. 7.1.2 The Security Market Line

25. 7.1.2 The Security Market Line
Industry Characteristic Lines
The characteristic line used to estimate beta value can be computed for sector indexes

26. 7.1.2 The Security Market Line

27. 7.2 Empirical Tests of the CAPM

28. 7.2 Empirical Tests of the CAPM
When testing the CAPM, there are two major questions
How stable is the measure of systematic risk (beta)?
Is there a positive linear relationship as hypothesized between beta and the rate of return on risky assets?

29. 7.2.1 Stability of Beta
Numerous studies have examined the stability of beta and generally concluded that the risk measure was not stable for individual stocks but was stable for portfolios of stocks
The larger the portfolio and the longer the period, the more stable the beta estimate
The betas tended to regress toward the mean
High-beta portfolios tended to decline over time toward 1.00, whereas low beta portfolios tended to increase over time toward unity

30. 7.2.2 Relationship Between Systematic Risk and Return
The ultimate question regarding the CAPM is whether it is useful in explaining the return on risky assets
Specifically, is there a positive linear relationship between the systematic risk and the rates of return on these risky assets?
Study (Jensen) shows that:
Most of the measured SMLs had a positive slope
The slopes change between periods
The intercepts are not zero
The intercepts change between periods

31. 7.2.2 Relationship Between Systematic Risk and Return

32. 7.2.2 Relationship Between Systematic Risk and Return
Effect of a Zero-Beta Portfolio
The characteristic line using a zero-beta portfolio instead of RFR should have a higher intercept and a lower slope coefficient
Several studies have tested this model with its higher intercept and flatter slope and found conflicting results

33. 7.2.2 Relationship Between Systematic Risk and Return
Effect of Size, P/E, and Leverage
Size and P/E are additional risk factors that need to be considered along with beta
Expected returns are a positive function of beta, but investors also require higher returns from relatively small firms and for stocks with relatively low P/E ratios
Bhandari (1988) found that financial leverage also helps explain the cross section of average returns after both beta and size are considered

34. 7.2.2 Relationship Between Systematic Risk and Return
Effect of Book-to-Market Value
Fama and French (1992) concluded that size and book-to-market equity capture the cross-sectional variation in average stock returns associated with size, E/P, book-to-market equity, and leverage
Two variables, BE/ME, appear to subsume E/P and leverage

35. 7.2.3 Additional Issues Effect of Transaction Costs Effect of Taxes
With transactions costs, the SML will be a band of securities, rather than a straight line
Effect of Taxes
Differential tax rates could cause major differences in the CML and SML among investors

36. 7.2.4 Summary of Empirical Results for the CAPM
Early evidence supported the CAPM; there was evidence that the intercepts were generally higher than implied by the RFR that prevailed, which is either consistent with a zero-beta model or the existence of higher borrowing rates
To explain unusual returns, size, the P/E ratio, financial leverage, and the book-to-market value ratio are found to have explanatory power regarding returns beyond beta
Further studies;
Kothari, Shanken, and Sloan (1995) measured beta with annual returns and found substantial compensation for beta risk, which suggested that the results obtained by Fama and French may have been time-period specific
Jagannathan and Wang (1996) employed a conditional CAPM that allows for changes in betas and in the market risk premium and found that this model performed well in explaining the cross section of returns
Reilly and Wright (2004) examined the performance of 31 different asset classes with betas computed using a broad market portfolio proxy; the risk–return relationship was significant and as expected by theory

37. 7.3 The Market Portfolio: Theory versus Practice

38. 7.3 The Market Portfolio: Theory versus Practice
The true market portfolio should
Included all the risky assets in the world
In equilibrium, the assets would be included in the portfolio in proportion to their market value
Using U.S. Index as a market proxy
Most studies use an U.S. index
The U.S. stocks constitutes less than 15% of a truly global risky asset portfolio

39. 7.3 The Market Portfolio: Theory versus Practice
The beta intercept of the SML will differ if
There is an error in selecting the risk-free asset
There is an error in selecting the market portfolio
Using the incorrect SML may lead to incorrect evaluation of a portfolio performance

40. 7.3 The Market Portfolio: Theory versus Practice

41. 7.3 The Market Portfolio: Theory versus Practice

42. 7.4 Arbitrage Pricing Theory

43. 7.4 Arbitrage Pricing Theory
CAPM is criticized because of
The many unrealistic assumptions
The difficulties in selecting a proxy for the market portfolio as a benchmark
An alternative pricing theory with fewer assumptions was developed: Arbitrage Pricing Theory (APT)

44. 7.4 Arbitrage Pricing Theory
Three major assumptions:
Capital markets are perfectly competitive
Investors always prefer more wealth to less wealth with certainty
The stochastic process generating asset returns can be expressed as a linear function of a set of K factors or indexes
In contrast to CAPM, APT does not assume:
Normally distributed security returns
Quadratic utility function
A mean-variance efficient market portfolio 15

45. 7.4 Arbitrage Pricing Theory
Theory assumes that the return-generating process can be represented as a K factor model of the form:
where: 15

46. 7.4 Arbitrage Pricing Theory
The APT requires that in equilibrium the return on a zero-investment, zero-systematic-risk portfolio is zero when the unique effects are fully diversified
This assumption implies that the expected return on any Asset i can be expressed as:
where: 15

47. 7.4 Arbitrage Pricing Theory15

48. 7.4.1 Using the APT Two-stock and a two-factor model example:
Assume that there are two common factors: one related to unexpected changes in the level of inflation and another related to unanticipated changes in the real level of GDP
Risk factor definitions and sensitivities: 15

49. 7.4.1 Using the APT
Assume also that there are two assets (x and y) that have the following sensitivities to these common risk factors: 15

50. 7.4.1 Using the APT 15

51. 7.4.2 Security Valuation with the APT: An Example
Suppose that three stocks (A, B, and C) and two common systematic risk factors (1 and 2) have the following relationship (for simplicity, it is assumed that the zero-beta return [λ0] equals zero):
If λ1 = 4 percent and λ2 = 5 percent, then the returns expected by the market over the next year can be expressed as: 15

52. 7.4.2 Security Valuation with the APT: An Example
Assuming that all three stocks are currently priced at $35 and do not pay a dividend, the following are the expected prices a year from now: 15

53. 7.4.2 Security Valuation with the APT: An Example
If everyone else in the market today begins to believe the future price levels of A, B, and C—but they do not revise their forecasts about the expected factor returns or factor betas for the individual stocks—then the current prices for the three stocks will be adjusted by arbitrage trading to: 15

54. 7.4.3 Empirical Tests of the APT
Roll-Ross Study (1980)
Methodology followed a two-step procedure:
Estimate the expected returns and the factor coefficients from time-series data on individual asset returns
Use these estimates to test the basic cross-sectional pricing conclusion implied by the APT
The authors concluded that the evidence generally supported the APT but acknowledged that their tests were not conclusive

55. 7.4.3 Empirical Tests of the APT
Extensions of the Roll–Ross Tests
Cho, Elton, and Gruber (1984) examined the number of factors in the return-generating process that were priced
Dhrymes, Friend, and Gultekin (1984) reexamined techniques and their limitations and found the number of factors varies with the size of the portfolio
Roll and Ross (1984) pointed out that the number of factors is a secondary issue compared to how well the model can explain the expected return

56. 7.4.3 Empirical Tests of the APT
Connor and Korajczyk (1993) developed a test that identifies the number of factors in a model that allows the unsystematic components of risk to be correlated across assets
Harding (2008) also showed the connection between systematic and unsystematic risk factors

57. 7.4.3 Empirical Tests of the APT
The APT and Stock Market Anomalies
An alternative set of tests of the APT considers how well the theory explains pricing anomalies: the small-firm effect and the January effect
APT Tests of the Small-Firm Effect
Reinganum: Results inconsistent with the APT
Chen: Supported the APT model over CAPM
APT Tests of the January Effect
Gultekin and Gultekin: APT not better than CAPM
Burmeister and McElroy: Effect not captured by model but still rejected CAPM in favor of APT

58. 7.4.3 Empirical Tests of the APT
Is the APT Even Testable?
Shanken (1982)
APT has no advantage because the factors need not be observable, so equivalent sets may conform to different factor structures
Empirical formulation of the APT may yield different implications regarding the expected returns for a given set of securities
Thus, the theory cannot explain differential returns between securities because it cannot identify the relevant factor structure that explains the differential returns returns
A number of subsequent papers, such as Brown and Weinstein (1983), Geweke and Zhou (1996), and Zhang (2009), have proposed new methodologies for testing the APT

59. 7.5 Multifactor Models and Risk Estimation

60. 7.5 Multifactor Models and Risk Estimation
In a multifactor model, the investor chooses the exact number and identity of risk factors, while the APT model does not specify either of them
where:
Fit= Period t return to the jth designated risk factor
Rit = Security i’s return that can be measured as either a nominal or excess return to Security i

61. 7.5.1 Multifactor Models in Practice
A wide variety of empirical factor specifications have been employed in practice
Alternative models attempt to identify a set of economic influences
Two approaches:
Risk factors can be viewed as macroeconomic in nature
Risk factors can also be viewed at a microeconomic level

62. 7.5.1 Multifactor Models in Practice
Macroeconomic-Based Risk Factor Models
Chen, Roll, and Ross (1986):
Where:

63. 7.5.1 Multifactor Models in Practice

64. 7.5.1 Multifactor Models in Practice
Burmeister, Roll, and Ross (1994) analyzed the predictive ability of a model based on the following set of macroeconomic factors:
Confidence risk
Time horizon risk
Inflation risk
Business cycle risk
Market timing risk

65. 7.5.1 Multifactor Models in Practice

66. 7.5.1 Multifactor Models in Practice
Fama and French (1993) developed a multifactor model specifying the risk factors in microeconomic terms using the characteristics of the underlying securities
SMB (i.e. small minus big) = return to a portfolio of small capitalization stocks less the return to a portfolio of large capitalization stocks
HML (i.e. high minus low) = return to a portfolio of stocks with high ratios of book-to-market values less the return to a portfolio of low book-to-market value stocks

67. 7.5.1 Multifactor Models in Practice

68. 7.5.1 Multifactor Models in Practice
Carhart (1997), based on the Fama-French three-factor model, developed a four-factor model by including a risk factor that accounts for the tendency for firms with positive past return to produce positive future return
Where
MOMt = the momentum factor

69. 7.5.1 Multifactor Models in Practice
Fama and French (2015) developed their own extension of the original three-factor model by adding two additional terms to account for company quality: a corporate profitability risk exposure and a corporate investment risk exposure
Where

70. 7.5.1 Multifactor Models in Practice
Extensions of Characteristic-Based Risk Factor Models
One type of security characteristic-based method for defining systematic risk exposures involves the use of index portfolios (e.g. S&P 500, Wilshire 5000) as common risk factors, such as the one by Elton, Gruber, and Blake (1996), who rely on four indexes:
The S&P 500
The Barclays Capital aggregate bond index
The Prudential Bache index of the difference between large- and small-cap stocks
The Prudential Bache index of the difference between value and growth stocks

71. 7.5.1 Multifactor Models in Practice

72. 7.5.1 Multifactor Models in Practice

73. 7.5.2 Estimating Risk in a Multifactor Setting: Examples
Estimating Expected Returns for Individual Stocks
One direct way to employ a multifactor risk model is to use it to estimate the expected return for an individual stock position
In order to do this, the following steps must be taken:
A specific set of K common risk factors (or their proxies) must be identified
The risk premia (Fj) for the factors must be estimated
The sensitivities (bij) of the ith stock to each of those K factors must be estimated
The expected returns can be calculated by combining the results of the previous steps

74. 7.5.2 Estimating Risk in a Multifactor Setting: Examples

75. 7.5.2 Estimating Risk in a Multifactor Setting: Examples
Whichever specific factor risk estimates are used, the expected return for any stock in excess of the risk-free rate (the stock’s expected risk premium) can be calculated with either the three-factor or four-factor version of the formula:

76. 7.5.2 Estimating Risk in a Multifactor Setting: Examples
Comparing Mutual Fund Risk Exposures
To get a better sense of how risk factor sensitivity is estimated at the portfolio level, consider the returns produced by two popular mutual funds: Fidelity’s Contrafund (FCNTX) and T. Rowe Price’s Mid-Cap Value Fund (TRMCX)

77. 7.5.2 Estimating Risk in a Multifactor Setting: Examples

78. 7.5.2 Estimating Risk in a Multifactor Setting: Examples

79. Aappendix: Multiple Regression in Excel

80. Data Analysis ToolPak Installed?

81. Installing Data Analysis ToolPak

82. Installing Data Analysis ToolPak

83. Installing Data Analysis ToolPak

84. Installing Data Analysis ToolPak

85. Multiple Regression in Excel

86. Multiple Regression in Excel

87. Multiple Regression in Excel

88. Multiple Regression in Excel

89. Multiple Regression in Excel