# Carhart four-factor model

The Carhart four-factor model is an extension of the Fama–French three-factor model including a momentum factor, also known in the industry as the MOM factor (monthly momentum).[1] Momentum in a stock is described as the tendency for the stock price to continue rising if it is going up and to continue declining if it is going down. The 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 is showing momentum if its prior 12-month average of returns is positive. 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 such that financial analysts incorporate the 52-week price high/low in their Buy/Sell recommendations.[2]

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:

${\displaystyle EXR_{t}=\alpha ^{J}+\beta _{mkt}{\mathit {EXMKT}}_{t}+\epsilon _{t}}$

The intercept in this model is referred to as the “Jensen’s alpha”

2. The Fama-French three-factor model:

${\displaystyle EXR_{t}=\alpha ^{FF}+\beta _{mkt}{\mathit {EXMKT}}_{t}+\beta _{HML}{\mathit {HML}}_{t}+\beta _{SMB}{\mathit {SMB}}_{t}+\epsilon _{t}}$

The intercept in this model is referred to as the “three-factor alpha”

3. The Carhart four-factor model:

${\displaystyle EXR_{t}=\alpha ^{c}+\beta _{mkt}{\mathit {EXMKT}}_{t}+\beta _{HML}{\mathit {HML}}_{t}+\beta _{SMB}{\mathit {SMB}}_{t}+\beta _{UMD}{\mathit {UMD}}_{t}+\epsilon _{t}}$

The intercept in this model is referred to as the “four-factor alpha”

Where EXRt 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 (EXMKT), monthly premium of the book-to-market factor (HML) the monthly premium of the size factor (SMB), and the monthly premium on winners minus losers (UMD) from Fama-French (1993) and Carhart (1997).

A fund has excess returns if it 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.