# Carhart Four-Factor Model

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21 Aug, 2024

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Table Of Contents

## What Is Carhart Four-Factor Model?

The Carhart Four-Factor Model is a system used to price securities and make investment decisions. It was developed by Mark Carhart in 1997 as an extension of the Fama and French Three-Factor Model. Momentum is the additional factor included in the four-factor model.

The cross-sectional momentum component of the concept improved the explanations surrounding the multi-factor model and mutual fund performances. Although it is expected to give better insights into asset performance than the three-factor model, the factors of uncertainty in stock markets cannot be ignored. This is because investments inherently carry risks associated with market uncertainty.

##### Table of contents

- The Carhart Four-Factor Model, developed by Mark Carhart (1997), is an extension of the Fama and French Three-Factor Model, focusing on another factor or parameter called Momentum.
- It identifies four risk factors, namely, Market, Size, Value, and Momentum, to study and interpret how stock returns vary as these parameters change.
- Though it is used extensively for analysis, the model may not always explain why fluctuations in returns are seen for portfolios arranged by size and momentum.
- Despite its usefulness, it is wise to consider other parameters and investment decision-making frameworks before investing in specific stocks.

### Carhart Four-Factor Model Explained

The Carhart Four-Factor Model is an investment assessment model published in 1997 as an extension of the Fama-French Three-Factor Model. By adding the Momentum Factor, also called Up-Minus-Down (UMD), to the original three factors of beta (market), size, and value, the Carhart model can increase the explanatory power by a significant margin. This means the four-factor model can explain more variations visible in stock returns than the three-factor model. The comprehensive analysis of Carhart’s model is a result of including the additional factor, which shows price trends in a given direction (upward or downward).

The Fama-French Three-Factor Model, first proposed in 1993, and the Carhart Four-Factor Model, introduced in 1997, provides a framework for interpreting and computing the expected stock returns. These models identify four risk factors, namely, Market, Size, Value, and Momentum that help explain the variation in stock returns across different companies.

The market factor is typically measured by the excess return on a stock market index, which reflects the overall level of risk in the market. The size factor captures the tendency of small companies to get more stock returns compared to large companies. In contrast, the value factor reflects the tendency of stocks with high book-to-market ratio to outperform stocks with low book-to-market companies. Finally, the momentum factor represents the tendency of stocks that have performed well in the past to continue to perform well in the future relative to stocks that have performed poorly.

The model offers significant input about expected stock returns, but it is advisable to pair it with other models or frameworks for a reliable analysis to improve investment planning and decision-making. It must be noted that this model is not a substitute for fundamental analysis, as fundamental analysis remains valid in varied situations. In addition, adequate risk management measures must be taken to avoid loss-making situations.

### Formula

The formula takes into account the four factors and the expected stock returns based on these parameters. Here is the formula:

*E(R _{s}) = R_{f }+ βs^{Mkt}* + β_{sMB}*E(R_{sMB}) + β_{HML}*E(R_{HML}) + β_{WML}*E(R_{WML})+ ε*

**:** This represents the market risk premium, which is the excess return that investors expect to earn over a risk-free rate of return.

**R _{f}:** The risk-free rate is a rate of return that an investor could earn with no risk.

**β _{sMkt}:** It refers to the market beta of an asset, which measures the sensitivity of its returns to changes in the overall market.

**β _{SMB}: **This beta factor represents the excess return of small portfolios over big portfolios.

**β _{HML}:** This beta factor represents the excess return of high P/E ratio portfolios over low book-to-market portfolios.

**β _{WML}:** This beta factor represents the excess return of winner portfolios over loser portfolios.

**E(R _{s}):** This is the expected rate of return on the stock or portfolio being analyzed.

**E(R _{Mkt}):** This is the expected return on the overall market.

**E(R _{SMB}):** This is the average expected return on small minus big portfolios.

**E(R _{HML}):** This is the average expected return on high minus low portfolios.

**E(R _{WML}):** This return premium is calculated from the difference between winner and loser portfolios.

**ε:** This is an error term that represents the degree to which this model cannot explain the actual returns of a stock or portfolio.

### Limitations

Despite being widely used, the Carhart Four-Factor Model’s empirical performance is not always effective. In some instances, the model fails to adequately explain the returns or fluctuation in returns for portfolios arranged by size and momentum. This implies that the model might be missing additional market anomalies or risk factors. This means that all the parameters that affect stock performance are likely not being considered.

Several studies have found that the momentum effect on stock returns is less pronounced in portfolios of large-cap stocks, while it is more prominent in portfolios of small-growth stocks. Analysis revealed that the momentum effect was more prominent in portfolios of small-growth stocks, while it was less evident in large-cap value stocks. However, the relationship between momentum and other risk factors may be complex and might vary depending on the market conditions and investment strategies being considered.

The model relies on historical data. Hence, the accuracy of such data is vital to the analysis of future stock performance.

### Carhart Four-Factor Model vs Fama-French Three-Factor Model

The Carhart Four-Factor Model is known to improve upon the Fama-French Three-Factor Model and enhance the overall results by considering an additional parameter about the direction a stock takes. However, the two concepts are different. In the table below, we have highlighted the differences.

Fama-French Three-Factor Model | Carhart Four-Factor Model |
---|---|

The Fama-French Three-Factor Model Was first introduced in 1993. | The Four-Factor Model was introduced in 1997 as an extension of the Fama-French model. |

The initial three estimation parameters were market risk, size, and value. | It adds the momentum component to the initial three factors: market risk, size, and value. |

The Fama-French Three-Factor Model is widely acknowledged as an improvement over the Capital Asset Pricing Model (CAPM) due to its increased ability to explain the cross-section of stock returns. | Carhart’s Four-Factor Model is recognized as an improvement over both the Fama-French Three-Factor Model and the CAPM due to its additional factor, which captures the momentum effect on stock returns. |

### Frequently Asked Questions (FAQs)

**What does MOM refer to in the Carhart Four-Factor Model?**The Monthly Momentum Factor (MOM) is an alternative name used in the financial industry to refer to the Carhart Four-Factor Model. The model includes momentum as an additional factor, which captures the tendency of stocks that have exhibited strong price performance in the recent past.

**Can the Carhart Four-Factor Model be adopted for other asset classes, such as bonds or real estate?**Carhart’s Four-Factor Model may be applied to other asset classes outside stocks, such as bonds or real estate. The fundamental goal of this model is to pinpoint the sources of risk and return within a specific asset class. However, unique risks and limitations related to specific assets (real estate, bonds, etc.) must be considered in addition to the results of this model.

**What is the argument in favor of analysts adding the fourth factor defined by the Carhart Four-Factor Model while computing stock returns?**The noteworthy argument in favor of this model is that it offers a relatively comprehensive view of how stocks might perform in a given period and what returns investors can expect when compared with the three-factor model. It is based on a stock’s tendency to move in a specific direction. Also, it helps investors construct portfolios that focus on stock momentum for investment decisions.

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