We start by looking at the capital asset pricing model and we modeled the expected return of security

**(Ri)**as a function of the risk-free rate of return plus**beta**for that security which is a measure of systemic risk and then we multiply that by the market premium. So beta times the market premium plus the risk-free rate that told us the expected return for security but then we said well we can actually have other factors beyond just looking at systemic risk. Maybe different types of securities respond differently to different macroeconomic factors, maybe there are other things we could be thinking about. Fama and French both came along and said look if you look historically actually small-cap stocks have outperformed large-cap stocks and when I say small-cap I’m talking about the capitalization the market value of the company’s equity.So we said we should control for the size we should include an extra factor, remember that the pricing model is just a single factor model but then Fama and French put together a three-factor model where we still have market risk as a factor but then they included

**size**which I just said so we basically put together a portfolio of small-cap companies and then we go long on that, we buy a bunch of small-cap companies and then we go short sell a bunch of large-cap company. Then that before those returns are based on the size that would be the second factor. They also notice that historically companies that had a higher book-to-market ratio which we call value stocks. The opposite is if you have a low book-to-market ratio you’d be a growth stock. So value stocks tend to do outperform the growth stocks and so they say “Let’s put together a portfolio which we call high minus low.” Where we go long and we purchase the high book-to-market security or book the market of companies and then we sell short ones that have a low book-to-market. So we’re basically going along with value stocks and selling short growth stocks. That’s Fama French’s three-factor model, we’ve got market risk we’ve got sizes and then we’ve got value stocks of looking at.Then Carhart came along and he noticed that typically in the past a company, we looked at their prior one-year return. For example, if we were looking at some company and we say “Oh over the last year they had a

**15% positive**return and then some other company maybe they had a**negative 2%**return.” If you look at the prior one-year return that actually has some predictive value for the future. So what Carhart said is look we should also have a fourth factor called**momentum**. We’ve looked at size with value stocks – that’s what Fama French Model added but Carhart added momentum. If you think about this as a regression this might help you, these all factors are each independent variables that we would have in a regression then the expected return**(Ri)**is our dependent variable.Let’s say we’re to take a portfolio of firms and we were to rank companies, so we had let’s say

**500 companies**in the**S&P 500**or we’re looking at all the companies on the New York Stock Exchange or something like that and we were to rank these firms based on their stock return for the prior year. For the prior 1 year, we rank them and we took the top ones, for example, let’s say the top**30%**of firms based on ranked on their stock return we took the top 30%. So we bought a bunch of all the stocks that were the top performers in the prior year and then we went short on all the ones that were like the bottom 30% in rank. The ones that had the worst returns over the prior year we sold those short. That’s what momentum is. So if we look at the returns, that would be this fourth factor here.What we’re seeing with this Fama French Carhart model is that, yes the capital asset model that’s great and we can look at we can learn a lot about the expected returns of a security by looking at systemic risk but we can also learn more we can explain or of the expected returns of stocks by not just look at market risk but looking at the size and the value stocks outperforming growth stocks. Let’s also consider that the firms that did the best over the prior year there tends to be some momentum that they’re gonna do better than the stocks that did poorly in the prior year.

So we’ll have a more powerful model when we’re trying to explain or trying to predict the returns of different securities.