Regression Analysis for Financial Relationships › Factor Exposures and the CAPM Beta

The single most important application of regression in finance is estimating how an asset responds to systematic risk factors. This relationship, called factor exposure or factor loading, tells us how much of an asset's return can be explained by common factors versus idiosyncratic risk. Understanding factor exposures is essential for portfolio construction, risk management, and performance evaluation. You cannot diversify away systematic risk, so these exposures determine how much of that risk a portfolio carries.

The simplest factor model relates individual stock returns to market returns. This approach, developed in the early days of modern finance, recognizes that stocks tend to move together because they are all affected by common economic forces. Broad market movements, driven by changes in interest rates, economic growth expectations, and investor sentiment, influence virtually all stocks to varying degrees.

$$r_{i,t} = \alpha_i + \beta_i r_{m,t} + \varepsilon_{i,t}$$

where:

• $r_{i,t}$: return on stock $i$ at time $t$
• $r_{m,t}$: return on the market index
• $\beta_i$: beta coefficient, measuring sensitivity to market movements
• $\alpha_i$: alpha coefficient, measuring return unexplained by the market
• $\varepsilon_{i,t}$: idiosyncratic error term

This is called the market model or single-index model. The coefficient $\beta_i$ measures the stock's systematic risk, representing how much the stock moves when the market moves. A stock with $\beta = 1.2$ rises 1.2% on average when the market rises 1%, and falls 1.2% when the market falls 1%. This amplification makes high-beta stocks more volatile than the market and more risky in a portfolio context. Conversely, a stock with $\beta = 0.7$ is defensive: it dampens market movements rather than amplifying them.

The idiosyncratic error term $\varepsilon_{i,t}$ captures firm-specific events: earnings surprises, management changes, product launches, and other news that affects only this particular company. Unlike systematic risk, idiosyncratic risk can be diversified away by holding many stocks, because positive surprises at some firms offset negative surprises at others.

The Capital Asset Pricing Model, which we'll derive formally in Part IV, states that the expected excess return on any asset should be proportional to its beta. This fundamental result connects risk and return: assets with higher systematic risk must offer higher expected returns to compensate investors for bearing that risk. The CAPM gives the theoretical foundation for using beta as a measure of risk.

$$E[r_i - r_f] = \beta_i \cdot E[r_m - r_f]$$

where:

• $E[r_i - r_f]$: expected excess return on asset $i$
• $E[r_m - r_f]$: expected market risk premium
• $r_f$: risk-free interest rate
• $\beta_i$: asset's beta

The left side represents the expected return above the risk-free rate, which is the premium investors demand for taking risk. The right side says this premium should equal beta times the market risk premium. An asset with beta of 1.5 should earn 1.5 times the market risk premium, because it carries 1.5 times the market's systematic risk.

To test this empirically, we regress excess returns:

$$r_{i,t} - r_{f,t} = \alpha_i + \beta_i (r_{m,t} - r_{f,t}) + \varepsilon_{i,t}$$

where:

• $r_{i,t} - r_{f,t}$: excess return on asset $i$
• $r_{m,t} - r_{f,t}$: excess return on the market portfolio
• $\alpha_i$: Jensen's alpha measuring abnormal performance
• $\beta_i$: systematic risk exposure
• $\varepsilon_{i,t}$: idiosyncratic error term

If the CAPM holds, the intercept $\alpha_i$ should be zero. This is because all expected return should be explained by beta exposure to the market. A significantly positive alpha suggests the stock earns more than its risk-adjusted required return: it generates excess risk-adjusted performance. A negative alpha means the stock underperforms relative to its risk. This interpretation makes alpha regression one of the most scrutinized statistical tests in investment management. Every active manager claims to generate alpha, and every client wants evidence to support that claim.

Let's estimate the market beta for a stock using historical data. We'll use synthetic data that mimics realistic return characteristics.