Multiple linear regression serves as the foundational tool in our quantitative framework. Its strategic importance lies in its ability to systematically model and understand the relationships between a dependent variable, such as an asset’s return, and a set of explanatory variables. This powerful model allows us to decompose performance, identify the key drivers of returns, and understand a portfolio’s exposure to various market factors.

The general form of the multiple linear regression model for a given population is expressed as:

y = β0 + β1×1 + β2×2 + . . . + βkxk + ε

Where:

• y is the dependent variable (e.g., portfolio return).
• β0 is the constant intercept, representing the value of y when all explanatory variables are zero.
• Each βj (beta) is a sensitivity factor, quantifying how much the portfolio’s return (y) is expected to change for every one-unit change in a given market driver (xk).
• xk represents the kth explanatory or independent variable (e.g., a macroeconomic factor or style index).
• ε is the residual, or “alpha,” which captures the portion of the return unexplained by the model’s factors—a key indicator of manager skill.

The validity of the model rests on three core assumptions regarding the error term, ε:

1. The errors are normally distributed with a mean of zero.
2. The variance of the errors is constant across all observations (a condition known as homoscedasticity).
3. The error terms are independent of each other over time.

To determine the model’s parameters (the β coefficients), we employ the Ordinary Least Squares (OLS) estimation method. OLS works by identifying the coefficients that minimize the sum of the squared residuals. This method is statistically robust, providing the best linear unbiased estimate (BLUE) of the model’s parameters.

2.1 Application: Style and Performance Analysis

A primary application of multiple regression is in style analysis, which allows us to understand a portfolio manager’s underlying investment strategy. Following the benchmark methodology developed by William F. Sharpe, we can regress a portfolio’s returns against a set of style-specific market indexes. This reveals the manager’s implicit bets on different market segments, such as value versus growth or large-cap versus small-cap stocks.

For example, a style benchmark can be constructed using a constrained regression against relevant style indexes:

Sharpe benchmark = 0.43 × (FRC Price-Driven Index) + 0.13 × (FRC Earnings-Growth Index) + 0.44 × (FRC 2000 Index)

The performance of the manager can then be evaluated by calculating the “Added value residual,” defined as:

Added value residual = Actual return − Sharpe benchmark return

This residual represents the portion of a manager’s return that cannot be explained by their exposure to the defined investment styles. A consistently positive residual provides compelling evidence that a manager delivers true alpha through security selection or market timing, rather than simply providing generic exposure to a market style that could be replicated passively and at lower cost.

While linear regression is a powerful tool for analyzing average relationships, effective portfolio risk management requires a more dynamic approach to forecasting the component of risk that changes over time: volatility.