Regression Analysis and CAPM Quiz

Study Notes

The fundamental relationship between a dependent variable (Y) and an independent variable (X) is expressed through the equation: Y = α + βX + ε.
The coefficient β quantifies the change in Y for every unit change in X, calculated as the covariance of X and Y divided by the variance of X.
The R-squared value indicates the proportion of variation in the dependent variable (Y) that is explained by the independent variable (X).

The Capital Asset Pricing Model (CAPM) describes the expected return of a stock as a function of its beta: (r_it - r_f) = α_i + β_i(r_Mt - r_f) + ε_it
This model breaks down the total risk of a stock into two components: systematic (β_i^2 * σ_M^2) and idiosyncratic (σ_ε^2) risk.

The expected return of an asset (E(r_j)) is estimated using CAPM: E(r_j) = r_f + β_j[E(r_M) - r_f].
This calculation relies on the risk-free rate (r_f), the market risk premium (E(r_M) - r_f), and the asset's beta (β_j).

CAPM suggests a linear relationship between expected excess returns and an asset's beta.
The market risk premium serves as the slope of the Security Market Line (SML), depicting the relationship between expected return and risk.
The expected return of an asset depends solely on its beta, implying that the asset's specific risk is not rewarded.

Time-series analysis investigates whether the alpha (α_i) for all assets is zero, as CAPM predicts.
The two-pass approach involves:
- Estimating betas for each stock individually.
- Running a cross-sectional regression: (r_it - r_ft) = γ_0 + γ_1β_i + ε_i
CAPM should hold if the intercept (γ_0) equals zero and the slope (γ_1) is equivalent to the market risk premium (r_Mt - r_ft).

The Fama-MacBeth (1973) method is widely used to test CAPM. This approach involves creating portfolios based on betas, calculating rolling betas, and conducting cross-sectional regressions.
The research findings highlight:
- An approximately linear relationship between returns and betas.
- A flatter risk-return relationship than predicted by CAPM.
- Lower returns for high-beta stocks than anticipated by CAPM.
- Higher returns for low-beta stocks than predicted by CAPM.

Empirically testing CAPM using market proxies might not be a true test of the model.
Only the true market portfolio, encompassing all assets, can offer a legitimate validation of CAPM.
CAPM may be inherently untestable due to the inability to observe the true market portfolio.

Errors in beta estimation can impact the validity of CAPM results.
The use of market proxies rather than the true market portfolio can lead to inaccurate conclusions.
The market representation might be incomplete, as it typically excludes assets like bonds, real estate, foreign investments, and human capital.

Standard CAPM tests have been challenged by adding factors like squared beta and idiosyncratic risk.
Empirical evidence indicates that the Security Market Line (SML) is flatter than predicted by theory.
Introducing firm size and book-to-market ratio as variables into the model leads to rejection of CAPM, as these factors prove significant while beta does not.

Small firms tend to earn higher returns compared to large firms, known as the size effect.
Stocks with a high book-to-market ratio, often referred to as value stocks, generally earn higher returns than stocks with a low book-to-market ratio, known as growth stocks.
These effects have been observed consistently across various countries.

Multifactor models were developed to account for multiple risk factors beyond market risk.
Examples of such factors include GDP, interest rates, and expected inflation.

Proposed by Ross in 1976, APT provides an alternative to CAPM.
It assumes that returns can be explained by multiple factors, and ultimately, no risk-free arbitrage opportunities exist.
APT distinguishes between systematic (factor) risk and diversifiable idiosyncratic risk.

Developed as an extension of CAPM, it includes three factors: market excess return, SMB (Small Minus Big), and HML (High Minus Low book-to-market).
This model has proven more successful than CAPM in explaining the cross-sectional variation in returns.

This model expands upon the three-factor model by incorporating two additional factors: profitability (RMW) and investment (CMA).

There are concerns regarding data-snooping when it comes to empirically motivated factors.
There is ongoing debate surrounding whether these factors act as proxies for unknown risk factors.