# Understanding Multi-Factor Models: Key Concepts and Formula Explained

![Multi-Factor Model](https://www.investopedia.com/thmb/SL47lMVdPRXjh2SpFEFEldamdhI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Multi-FactorModel_Final_4192607-893ee4d3244c4b7c8506e8dc0f4e81b2.png)

*— Investopedia / Paige McLaughlin*

## What Is a Multi-Factor Model?

A multi-factor model employs multiple factors to analyze and explain market activities and [equilibrium](https://www.investopedia.com/terms/e/equilibrium.asp) asset prices. Typically used to assess individual securities or [portfolios](https://www.investopedia.com/terms/p/portfolio.asp), these models compare factors, offering insights into variable relationships and performance outcomes.

### Key Takeaways

*   Multi-factor models use multiple factors to analyze and explain asset prices and market phenomena.  
    
*   There are several types of multi-factor models, including macroeconomic, fundamental, and statistical models.  
    
*   Multi-factor models can be constructed using combination, sequential, and intersectional methods.  
    
*   The Fama-French three-factor model adds size and value factors to the traditional market risk factor of the capital asset pricing model.  
    
*   Understanding beta is crucial when using multi-factor models to assess the systematic risk of securities.

## How Multi-Factor Models Work

Multi-factor models help build portfolios with specific traits, like risk, or to track [indexes](https://www.investopedia.com/terms/i/index.asp). Choosing how many and which factors to include in a multi-factor model can be challenging. Moreover, models rely on historical data, which may not predict future values accurately.

Multi-factor models also show the weight of each factor, indicating which impacts an asset's price more.

## Formula for Multi-Factor Model Analysis

Use the following formula to compare factors:

**Ri = ai + \_i(m) \* Rm + \_i(1) \* F1 + \_i(2) \* F2 +...+\_i(N) \* FN + ei**

Where:

**Ri** is the return of security

**Rm** is the market return

**F**(1, 2, 3 ... N) is each of the factors used

**\_** is the [beta](https://www.investopedia.com/terms/b/beta.asp) with respect to each factor including the market (m)

**e** is the error term

**a** is the intercept

## Different Categories of Multi-Factor Models

Multi-factor models can be divided into three categories: macroeconomic models, fundamental models, and statistical models.

**Macroeconomic models:** Macroeconomic models compare a security's return to such factors as employment, [inflation](https://www.investopedia.com/terms/i/inflation.asp), and interest.

**Fundamental models:** Fundamental models analyze the relationship between a security's return and its underlying financials, such as [earnings](https://www.investopedia.com/terms/e/earnings.asp), [market capitalization](https://www.investopedia.com/terms/m/marketcapitalization.asp), and debt levels.

**Statistical models:** Statistical models are used to compare the returns of different securities based on the statistical performance of each security in and of itself. Many times, historical data is used in this type of modeling.

## Building Multi-Factor Models: Key Approaches

The three most commonly used models to construct a multi-factor model are a combination model, a sequential model, and an intersectional model.

**Combination model:** In a combination model, multiple single-factor models, which utilize a single factor to distinguish stocks, are combined to create a multi-factor model. For example, stocks may be sorted based on momentum alone in the first pass. Subsequent passes will use other factors, such as [volatility](https://www.investopedia.com/terms/v/volatility.asp), to classify them.

**Sequential model:** A sequential model sorts stocks based on a single factor in a sequential manner to create a multi-factor model. For example, stocks for a specific market capitalization may be sequentially analyzed for various factors, such as value and momentum, sequentially.

**Intersectional model:** In the intersectional model, stocks are sorted based on their intersections for factors. For example, stocks may be sorted and classified based on intersections in value and [momentum](https://www.investopedia.com/terms/m/momentum.asp).

## Understanding Beta in Multi-Factor Models

The beta of a security measures the [systematic risk](https://www.investopedia.com/terms/s/systemic-risk.asp) of a security in relation to the overall market. A beta of 1 indicates that the security theoretically experiences the same degree of volatility as the market and moves in tandem with the market.

A beta greater than 1 indicates the security is theoretically more volatile than the market. Conversely, a beta less than 1 indicates the security is theoretically less volatile than the market.

When multi-factor models are used by investment managers to assess the risk of investments, beta is an important factor that they can use.

## Exploring the Fama-French Three-Factor Model

The Fama-French three-factor model is a widely used multi-factor model. The Fama-French model has three factors: the size of firms, [book-to-market](https://www.investopedia.com/terms/b/booktomarketratio.asp) values, and excess returns on the market. In other words, the three factors used are SMB (small minus big), HML (high minus low), and the portfolio's return less the [risk-free rate of return](https://www.investopedia.com/terms/r/risk-freerate.asp).

SMB accounts for publicly traded companies with small market caps that generate higher returns, while HML accounts for [value stocks](https://www.investopedia.com/terms/v/valuestock.asp) with high book-to-market ratios that generate higher returns in comparison to the market.

## The Bottom Line

Multi-factor models are essential tools in financial modeling, offering insights into asset prices by evaluating multiple factors. By capturing the influence of these factors, such as market size and risk, investors can better understand asset volatility and performance.

Using models like the Fama-French three-factor model, which considers company size, book-to-market values, and market excess returns, investors can construct robust portfolios tailored to specific characteristics like risk. While these models offer valuable predictions of market behavior, it's important to remember they rely on historical data, which may not always predict future performance. Understanding and applying these models can significantly aid investors in making informed financial decisions.