A multi-factor model is a combination of various elements or factors that are correlated with asset returns. The model uses said factors to explain market equilibrium and asset prices. In multi-factor models, different factors are associated with certain characteristics (such as risk), and it helps determine the weight or importance of that factor when computing asset price or return.

A typical measure of risk is beta, which measures the systemic risk of a security relative to the market. Multi-factor models are commonly used by asset managers to make investment decisions and assess the relevant risk associated with the investments.

Summary

A multi-factor model is a combination of various elements or factors that are correlated with asset returns. The model uses said factors to explain market equilibrium and asset prices.

The three main types of multi-factor models are Macroeconomic Factor Models, Fundamental Factor Models, and Statistical Factor Models.

The Arbitrage Pricing Theory (APT) is a model that is used to describe the expected return of an asset or portfolio as a linear function of the risk of the assets relative to certain factors.

Types of Multi-Factor Models

Multi-factor models can be used in all industries, be it finance, economics, or mathematics. Broadly, three types of multi-factor models can be classified based on the type of factors employed:

1. Macroeconomic Factor Models

In macroeconomic factor models, the factors are associated with surprises in macroeconomic variables that help explain returns of asset classes. The surprise or incremental return can be calculated as the actual value less the forecasted value, and the mean of the return is typically zero.

2. Fundamental Factor Models

In fundamental models, the factors are characteristics of stocks or companies that can be used to explain the changes in stock prices. Examples of such factors are price-to-earnings ratio, market capitalization, and financial leverage. Fundamental factor models use asset returns, and after determining the factor sensitivities, the returns are calculated by running regressions.

3. Statistical Factor Models

In statistical factor models, statistical methods are applied to historical data of returns and are used to explain covariances in data.

Three, Four, and Five-Factor Models

The construction of multi-factor models is an interesting process, and over time, several different types of models have emerged in the field of finance. Below, we will discuss three different types of multi-factor models and their relevant factors used to derive returns:

1. Fama-French Three-Factor Model

Fama-French uses the factors of size and value to derive asset returns. It is a better approach than the Capital Asset Pricing Model (CAPM), as CAPM only explains 70% of a portfolio’s diversified returns, whereas Fama-French explains roughly 90%.

The Fama-French model employs three factors – namely SMB (small minus big), HML (high minus low), and the portfolio return minus the risk-free rate. SMB characterizes publicly-traded companies with small market caps that generate higher returns, and HML uses value stocks with high book-to-market ratios that generate higher returns relative to the market.

The formula for the Fama-French three-factor model is given in the equation below:

Where:

R_{it} = Total return of a stock or portfolio i at time t

R_{ft}= Risk-free rate of return at time tR_{Mt}= Total market portfolio return at time t

R_{it} – R_{ft}= Expected excess return

R_{Mt} – R_{ft}= Excess return on the market portfolio (index)

SMB_{t}= Size premium (Small minus big)

HML_{t }= Value premium (high munus low)

β = Factor coefficients

2. Cahart Four-Factor Model

The Cahart model builds onto the Fama-French three-factor model and introduces a fourth factor called momentum. The concept of the momentum of an asset can be used to predict future asset returns. It is a bit controversial, as it uses risk-based, as well as behavioral-based, explanations to determine returns. The Cahart model is considered a superior one, given its explanatory power of around 95%.

3. Fama-French Five-Factor Model

The Fama-French five-factor model also builds on the three-factor model and introduces two more factors – Profitability (RMW) and Investment (CMA). It uses the return of stocks with high operating profitability minus the return of stocks with low or negative operating profitability.

At times, the factor is replaced by a quality factor. The investment factor recognizes the level of capital investment used to maintain and grow the business. It is typically negatively correlated with the value factor. Given the number of factors, the Fama-French five-factor model is, at times, not practical to be implemented in certain economies.

Arbitrage Pricing Theory (APT)

The arbitrage pricing theory (APT) is a model that is used to describe the expected return of an asset or portfolio as a linear function of the risk of the assets relative to certain factors. It is similar to the CAPM model but with less strict assumptions.

The assumptions are – asset returns are described by a factor model, asset-specific risk can be eliminated, as there are multiple assets, and assets are priced in a way such that no arbitrage opportunities exist. The equation below gives a general framework of the APT model:

Where:

R_{f}= Risk-free rate of return

β= Sensitivity of the asset or portfolio in relation to a specified factor

RP = Risk premium of the specified factor

The factors used in the APT model are systematic risks that cannot be reduced by the diversification of an investment portfolio. Typically, macroeconomic factors are used, as they are reliable as price predictors. Examples of factors include Gross National Product (GNP), unexpected changes in inflation, and shifts in yield curves.

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