In portfolio management, the Carhart four-factor model is an extra factor addition in the

Fama–French three-factor model
In asset pricing and portfolio management the Fama–French three-factor model is a statistical model designed in 1992 by Eugene Fama and Kenneth French to describe stock returns. Fama and French were colleagues at the University of Chicago Booth ...

, proposed by Mark Carhart
Mark Carhart is a finance researcher and a science man and quantitative investment manager known for extending the Fama–French three-factor model with a momentum
In Newtonian mechanics, momentum (more specifically linear momentum or t ...

. The Fama-French model, developed in the 1990, argued most stock market
A stock market, equity market, or share market is the aggregation of buyers and sellers of stocks (also called shares), which represent ownership claims on businesses; these may include ''securities'' listed on a public stock exchange, a ...

returns are explained by three factors: risk, price ( value stocks tending to outperform) and company size (smaller company stocks tending to outperform). Carhart added a momentum factor for asset pricing
In financial economics, asset pricing refers to a formal treatment and development of two main pricing principles, outlined below, together with the resultant models.
There have been many models developed for different situations, but correspon ...

of stock
In finance, stock (also capital stock) consists of all the Share (finance), shares by which ownership of a corporation or company is divided.Longman Business English Dictionary: "stock - ''especially AmE'' one of the shares into which owners ...

s. The Four Factor Model is also known in the industry as the Monthly Momentum Factor (MOM). Momentum is the speed or velocity of price changes in a stock, security, or tradable instrument.
Development

The Monthly Momentum Factor(MOM) can be calculated by subtracting the equal weighted average of the lowest performing firms from the equal weighed average of the highest performing firms, lagged one month (Carhart, 1997). A stock would be considered to show momentum if its prior 12-month average of returns is positive, or greater. Similar to the three factor model, momentum factor is defined by self-financing portfolio of (long positive momentum)+(short negative momentum). Momentum strategies continue to be popular in financial markets. Financial analysts often incorporate the 52-week price high/low in their Buy/Sell recommendations. The four factor model is commonly used as an active management and mutual fund evaluation model. Three commonly used methods to adjust a mutual fund's returns for risk are: 1. The market model: :$EXR\_t=\backslash alpha^J+\; \backslash beta\_\backslash mathit\_t+\backslash epsilon\_t$ The intercept in this model is referred to as the "Jensen's alpha". 2. TheFama–French three-factor model
In asset pricing and portfolio management the Fama–French three-factor model is a statistical model designed in 1992 by Eugene Fama and Kenneth French to describe stock returns. Fama and French were colleagues at the University of Chicago Booth ...

:
:$EXR\_t=\backslash alpha^+\; \backslash beta\_\backslash mathit\_t+\; \backslash beta\_\backslash mathit\_t+\; \backslash beta\_\backslash mathit\_t+\backslash epsilon\_t$
The intercept in this model is referred to as the "three-factor alpha".
3. The Carhart four-factor model:
:$EXR\_t=\backslash alpha^c+\; \backslash beta\_\backslash mathit\_t+\; \backslash beta\_\backslash mathit\_t+\; \backslash beta\_\backslash mathit\_t+\; \backslash beta\_\backslash mathit\_t+\backslash epsilon\_t$
The intercept in this model is referred to as the "four-factor alpha".
$EXR\_t$ is the monthly return to the asset of concern in excess of the monthly t-bill rate. We typically use these three models to adjust for risk. In each case, we regress the excess returns of the asset on an intercept (the alpha) and some factors on the right hand side of the equation that attempt to control for market-wide risk factors. The right hand side risk factors are: the monthly return of the CRSP value-weighted index less the risk free rate ($\backslash mathit\_t$), monthly premium of the book-to-market factor ($\backslash mathit\_t$) the monthly premium of the size factor ($\backslash mathit\_t$), and the monthly premium on winners minus losers ($\backslash mathit\_t$) from Fama-French (1993) and Carhart (1997).
A fund manager shows forecasting ability when his fund has a positive and statistically significant alpha.
SMB is a zero-investment portfolio that is long on small capitalization (cap) stocks and short on big cap stocks. Similarly, HML is a zero-investment portfolio that is long on high book-to-market (B/M) stocks and short on low B/M stocks, and UMD is a zero-cost portfolio that is long previous 12-month return winners and short previous 12-month loser stocks.
See also

* Capital asset pricing model (CAPM) * Size premium *Fama–French three-factor model
In asset pricing and portfolio management the Fama–French three-factor model is a statistical model designed in 1992 by Eugene Fama and Kenneth French to describe stock returns. Fama and French were colleagues at the University of Chicago Booth ...

* Momentum factor
*Returns-based style analysis Returns-based style analysis is a statistical technique used in finance to deconstruct the returns of investment strategies using a variety of explanatory variables. The model results in a strategy's exposures to asset classes or other factors, inte ...

References

Financial markets Behavioral finance Technical analysis Financial models {{finance-stub