Return-generating models, the market model and beta

Return-generating models

Building the market portfolio from scratch is a huge job: for 1,000 assets you would need 1,000 return estimates, 1,000 standard deviations and 499,500 correlations. A simpler route starts from a known portfolio, such as a broad index, and adds assets one at a time based on their effect on the portfolio. That needs only each asset's systematic risk and an expected return from a return-generating model.

The general form is a multi-factor model: expected excess return equals the sum of factor sensitivities (factor loadings or weights) times expected factor premiums. Every such model includes the market return as one factor.

Factors come in three kinds. Macroeconomic factors are economic variables correlated with returns (growth, interest rates, inflation, employment, confidence). Fundamental factors are company characteristics (earnings and growth, cash flow, research spending, advertising, patents). Statistical factors are found by mining past return data; they may have no economic meaning and can be spurious, so analysts prefer macroeconomic and fundamental models.

Two well-known examples: the Fama-French three-factor model adds relative size and relative book-to-market to the market factor; Carhart's four-factor model adds momentum (relative past returns).

The single-index model and splitting variance

The simplest case uses one factor, the market: $E(R_i) - R_f = \beta_i[E(R_m) - R_f]$. It fits neatly with the CML, which for a diversified portfolio says excess return equals $(\sigma_i/\sigma_m)$ times the market's excess return.

With realised returns, the model gets an error term: $R_i - R_f = \beta_i(R_m - R_f) + e_i$. The error captures non-market surprises and is uncorrelated with the market, so the covariance term vanishes and total variance splits cleanly: $\sigma_i^2 = \beta_i^2\sigma_m^2 + \sigma_{e_i}^2$.

For a well-diversified portfolio, $\sigma_{e_i}^2$ is zero, so total risk is just $\beta_i\sigma_m$. Substituting this into the CML gives exactly the single-index model: the CML and the single-index model agree for diversified portfolios.

The market model and abnormal returns

The market model is the most common single-index implementation: $R_i = \alpha_i + \beta_iR_m + e_i$. Regress historical returns of the asset on historical market returns; the intercept is $\alpha_i$ and the slope is $\beta_i$. To match the single-index model, $\alpha_i = R_f(1 - \beta_i)$.

The fitted line gives the return expected from the market's move on a given day. The difference between the actual return and that expected return is the company-specific or abnormal return, $e_i$.

Calculating and interpreting beta

Because the error term is uncorrelated with the market, $\text{Cov}(R_i, R_m) = \beta_i\sigma_m^2$. So beta is the covariance with the market divided by the market's variance, or equivalently correlation times the ratio of standard deviations. It measures how sensitive the asset is to market moves: its systematic risk.

A positive beta means the asset tends to move with the market; a negative beta means it tends to move against it. A beta of zero means no correlation with the market, as for the risk-free asset (zero covariance) or an asset that is volatile but uncorrelated. Total risk alone tells you nothing about beta.

Substituting the market for the asset gives $\beta_m = 1$, so the average beta of all assets in the market is 1. Most stocks in developed markets are highly correlated with the market (often above 0.70), so consistently negative betas are rare.

In practice beta is estimated by regressing the asset's returns on market returns (the market model). Short windows such as 12 months reflect current risk better but are less accurate because special events distort them; three to five years is more accurate but may miss recent changes in the business. Any beta estimate may differ from the asset's true current or future systematic risk.