## What Is a Multi-Factor Model?

A multi-factor model is a financial model that employs multiple factors in its calculations to explain market phenomena and/or equilibrium asset prices. A multi-factor model can be used to explain either an individual security or a portfolio of securities. It does so by comparing two or more factors to analyze relationships between variables and the resulting performance.

### Key Takeaways

- A multi-factor model is a financial modeling strategy in which multiple factors are used to analyze and explain asset prices.
- Multi-factor models reveal which factors have the most impact on the price of an asset.
- Multi-factor portfolios can be constructed using various methods: intersectional, combinational, and sequential modeling.
- The beta of a security measures the systematic risk of a security in relation to the overall market.
- The Fama-French three-factor model is a well-known tool that builds upon the capital asset pricing model, which focuses solely on the market risk factor, by incorporating size and value factors.

## Understanding a Multi-Factor Model

Multi-factor models are used to construct portfolios with certain characteristics, such as risk, or to track indexes. When constructing a multi-factor model, it is difficult to decide how many and which factors to include. Also, models are judged on historical numbers, which might not accurately predict future values.

Multi-factor models also help explain the weight of the different factors used in the models, indicating which factor has more of an impact on the price of an asset.

## Multi-Factor Model Formula

Factors are compared using the following formula:

**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 with respect to each factor including the market (m)

**e** is the error term

**a** is the intercept

## Types 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, and interest.

**Fundamental models:** Fundamental models analyze the relationship between a security's return and its underlying financials, such as earnings, market capitalization, 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.

## Construction of Multi-Factor Models

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, 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.

## Measurement of Beta

The beta of a security measures the systematic risk 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.

## Fama-French Three-Factor Model

One widely used multi-factor model is the Fama-French three-factor model. The Fama-French model has three factors: the size of firms, book-to-market 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.

SMB accounts for publicly traded companies with small market caps that generate higher returns, while HML accounts for value stocks with high book-to-market ratios that generate higher returns in comparison to the market.