Finance: Regression Analysis and CAPM

Regression Analysis

• Regression equation: Y = α + βX + ε
• β (beta) coefficient: Measures the relationship between the independent variable (X) and the dependent variable (Y).
• β = cov(X,Y) / var(X) : Beta is calculated as the covariance between X and Y divided by the variance of X.
• R-squared: Measures the proportion of variation in the dependent variable that is explained by the independent variable.

CAPM and Asset Returns

• Stock return equation: (r_it - r_f) = α_i + β_i(r_Mt - r_f) + ε_it
• Asset return decomposed into systematic and idiosyncratic risk: Systematic risk is measured by beta (β_i) and is the risk that cannot be diversified away. Idiosyncratic risk is specific to the asset and can be diversified away.

Estimating Expected Returns

• Expected return formula: E(r_j) = r_f + β_j[E(r_M) - r_f]
• Inputs for expected return calculation: Risk-free rate, market risk premium, and the stock’s beta.

CAPM Implications

• Expected excess returns are linear in beta: As beta increases, expected excess returns increase linearly.
• The slope of the Security Market Line (SML) is the market risk premium: The market risk premium is the additional return investors expect for holding a market portfolio compared to a risk-free asset.
• Expected returns depend only on beta: According to CAPM, the only factor that determines expected return is beta.

Testing CAPM

• Time-series approach: Tests if the alpha (α_i) for each asset is equal to zero. If alpha is not zero, it suggests that the CAPM does not accurately predict returns.
• Two-pass approach:
  • Step 1: Estimate betas for each stock
  • Step 2: Run a cross-sectional regression: (r_it - r_ft) = γ_0 + γ_1β_i + ε_i
  • If CAPM holds: γ_0 = 0 and γ_1 = (r_Mt - r_ft)

Empirical Evidence

• Fama-MacBeth (1973) method: A popular method to test the CAPM involving portfolio formation, rolling betas, and cross-sectional regressions.
• Findings:
  • There is an approximately linear relationship between returns and betas.
  • The risk-return relationship is flatter than predicted by the CAPM.
  • High-beta stocks: Tend to have lower returns than predicted by the CAPM.
  • Low-beta stocks: Tend to have higher returns than predicted by the CAPM.

Roll's Critique

• CAPM tests using market proxies may not test the CAPM: The market proxy used in testing may not accurately represent the true market portfolio, leading to inaccurate results.
• True market portfolio needed for proper testing: The true market portfolio (including all assets) is needed for accurate testing, but it is unobservable in practice.
• CAPM potentially untestable: It is difficult to measure and test the CAPM due to the difficulty of observing the true market portfolio.

Reasons for CAPM Rejection

• Estimation error in betas: The estimation of betas can introduce errors, leading to inaccuracies in testing the CAPM.
• Use of market proxies instead of true market portfolio: Using a market proxy rather than the true market portfolio can lead to deviations from the CAPM predictions.
• Incomplete market representation: The market should include all assets, such as bonds, real estate, foreign assets, and human capital, to fully test the CAPM.

This quiz covers key concepts in regression analysis, focusing on the regression equation and the beta coefficient. It also explores the Capital Asset Pricing Model (CAPM) and how expected returns are estimated based on systematic and idiosyncratic risks. Perfect for finance students looking to test their understanding of these vital topics.