Capital Asset Pricing and Arbitrage Pricing Theory
Capital Asset Pricing and Arbitrage Pricing Theory Notes
7.1 The Capital Asset Pricing Model
Assumptions:
Markets are competitive and equally profitable.
No investor is wealthy enough to individually affect prices.
All information is publicly available; all securities are public.
There are no taxes on returns and no transaction costs.
Unlimited borrowing/lending is available at the risk-free rate.
Investors are alike except for differences in initial wealth and risk aversion.
Investors plan for a single-period horizon; they are rational and mean-variance optimizers.
All investors use the same inputs and consider identical portfolio opportunity sets.
Equilibrium:
All investors choose to hold the market portfolio.
The market portfolio contains all securities, with the proportion of each security corresponding to its market value as a percentage of total market value.
The market portfolio lies on the efficient frontier, representing the optimal risky portfolio.
Expected Return on Individual Securities:
The risk premium on individual securities is a function of the security's contribution to the risk of the market portfolio.
The individual security’s risk premium is determined by the covariance of its returns with the assets making up the market portfolio.
Example (Dell):
Rearranging provides the CAPM's expected return-beta relationship.
The Security Market Line (SML):
Represents the expected return-beta relationship as described by the CAPM.
Graphs individual asset risk premiums against asset risk.
Alpha:
Defined as the abnormal rate of return on a security that exceeds the rate predicted by the equilibrium model (CAPM).
Applications of CAPM:
The SML serves as a benchmark for fair returns on risky assets.
The SML provides a “hurdle rate” for evaluating internal projects.
Portfolio Beta:
The beta of a portfolio is the weighted average of the betas of the individual stocks within the portfolio.
Example Calculation:
If the portfolio consists of stocks with weights:
40% with a beta of 1.2
60% with a beta of 1.5,
Then Portfolio Beta = 0.4 imes 1.2 + 0.6 imes 1.5 = 1.38.
7.2 CAPM and Index Models
Estimation of the Index Model:
Utilizes historical data from T-bills, S&P 500, and individual securities.
Involves regressing risk premiums for individual stocks against those for the S&P 500.
The slope of the regression (beta) represents the beta for the individual stock.
Index Model Mean-Beta Equation:
r{it} - r{ft} = \alphai + \betai (r{Mt} - r{ft}) + e_{it} where:
r_{it}: Holding period return for asset $i$ at time $t$.
r_{ft}: Risk-free return.
\alpha_i: Intercept of the security characteristic line.
\beta_i: Slope of the security characteristic line.
r_M: Index return.
e_{it}: Firm-specific effects.
Expected Return Equation:
E(r{it}) - r{ft} = \alphai + \betai [E(r{Mt}) - r{ft}].
Figure 7.4 Scatter Diagram for Google vs. S&P 500 (01/06-12/10):
Displays excess rate of return on Google arranged against the excess rate of return on S&P 500.
Table of Regression Statistics for Google (S&P 500), 01/06-12/10:
R = 0.5914
R-squared = 0.3497
Adjusted R-squared = 0.3385
Standard Error of regression = 8.4585
Total observations = 60
Regression equation:
Google (excess return) = 0.8751 + 1.2031 × S&P 500 (excess return).
ANOVA results:
Regression degrees of freedom = 1, Sum of Squares = 2231.50, Mean Square = 2231.50, F-statistic = 31.19, p-level = 0.0000.
Residual degrees of freedom = 58, Sum of Squares = 4149.65, Mean Square = 71.55, Total = 59, Sum of Squares = 6381.15.
Coefficients:
Intercept: 0.8751 ext{ (SE: 1.0920, t: 0.8013, p: 0.4262)}
S&P 500: 1.2031 ext{ (SE: 0.2154, t: 5.5848, p: 0.0000)}
7.3 CAPM and the Real World
The CAPM was introduced by Sharpe in 1964.
Numerous tests followed the theory, including:
Roll’s critique in 1977.
Fama and French study in 1992.
Validations and Critiques:
CAPM is considered false based on the validity of its original assumptions but remains a useful predictor of expected returns.
CAPM is regarded as untestable as a theory; nonetheless, its principles are still relevant:
Investors should diversify their portfolios.
Systematic risk remains the risk that matters most.
A well-diversified risky portfolio can serve a wide range of investors effectively.
7.4 Multifactor Models and CAPM
Limitations of CAPM:
The market portfolio is not directly observable in actual markets.
Research indicates other factors can influence returns beyond those accounted for in CAPM.
Fama-French Three-Factor Model:
Suggests returns are tied to factors other than merely market returns:
Size.
Book value relative to market value.
This three-factor model offers a better explanation of return variations compared to CAPM.
7.5 Arbitrage Pricing Theory (APT)
Arbitrage:
Defined as the practice of capitalizing on relative mispricing in order to generate riskless profits.
Arbitrage Pricing Theory (APT):
Describes risk-return relationships originating from no-arbitrage conditions in expansive capital markets.
APT asserts that for well-diversified portfolios (with no firm-specific risk), the no-arbitrage condition yields the same risk-return relationship as CAPM.
Comparisons Between APT and CAPM:
APT applies to well-diversified portfolios but not necessarily to individual stocks.
Some individual stocks can be mispriced and not align with the SML within the APT framework.
APT is more versatile, deriving expected return and beta relationships without relying on the existence of a market portfolio.
APT can be adapted to include multifactor models, extending its applicability to various risk factors.