Jensen's alpha is the portion of the excess return of a security or a portfolio that is not explained by systematic risk and when I talk about the systematic risk I mean beta. Another way of putting it is that alpha is the difference between the actual return of the security or the portfolio in the return that's predicted by the capital asset pricing model. I've got the capital asset pricing model here and we said that the expected return for a security Ri is equal to the risk-free rate of return (Rf) plus the beta (βi) of the portfolio or the security, multiplied by the market risk premium. Which is just the expected rate of return for the market minus the risk-free rate. So all this right here is gonna tell us the expected return for that security that's share of stock or that portfolio Ri.
Let's just say that we calculate this out, say for this particular stock we're expecting a rate of return of 12% so this is the expected rate of return based on the capital asset pricing model (CAPM). Then in reality it turns out that we have an actual rate of return of 13%. We'd have a difference of 1% here. This stock actually had a higher rate of return than was expected according to the capital asset pricing model (CAPM). So that extra 1% would be the alpha. So we can just add that here in our formula. So if I rearrange the equation to make it easier for you to solve for alpha if you just want to know the formula, so here's an equation to solve for alpha.
Example of Jensen's Alpha
I want to work through an example with some more numbers to make it a little more complicated so make sure you understand. So we've got a risk-free rate of return of 3% then we have an expected rate of return for the market of 11% and then we have a beta let's say we're talking about a particular stock and it has a beta of 1.5. So that's a measure of systemic risk but then the actual rate of return turns out to be 17% and the question is what is alpha? Now if we were talking about portfolio or something here you might look and say "It had a return of 17% and the market had a return of 11%, so you might be thinking it must be 6% that is the Alpha." because clearly this portfolio or this stock outperformed the market, but it doesn't work like that it's not that easy. You have to adjust for risk and so alpha is a risk-adjusted measure of return. So when I say just for the risk I mean adjust for beta, we would expect that when the market goes up this stock or this portfolio would go up by higher than 11% because it has a beta higher than 1 when a stock has a beta higher than 1 that means if the market goes up, it's going to go up even higher than what the market did. So of course it's going to outperform the market index. The question is after accounting for the beta after a risk adjusting is this actual return higher or lower than what we would have expected? That's the same as saying taking the capital asset pricing model in the account, was the expected return is this 17% better or worse than what was predicted.
So we just take our formula from above and we're gonna plug in numbers. We've got an actual return of 17% minus the risk-free rate and then we're gonna subtract the whole thing, we've got the beta times the market risk premium which is 8%. So doing a little math we see that the Alpha is 2%.
Use of Jensen's Alpha
What does that mean let's say we're talking about a portfolio of stocks here, the fund manager who's managing this portfolio is probably gonna point out and say "We actually outperform if you look at the capital asset pricing model we were supposed to have a certain return but we actually had a return that was higher by 2% points." and so they would say that's due to good management on our part and so forth. That would be the argument that's made when you hear about people saying "Oh we're seeking alpha, or we're generating alpha, and so forth." That's what they're talking about.
Calculating Alpha with Regression Analysis
Now if you've ever been familiar with regression analysis I can show the alpha to you in a slightly different way it might make it easier for you to understand. So if we take this equation and over here we've got the excess returns for the market as our independent variable and then as our dependent variable we have the excess return for the security or for the portfolio. So we've got this dependent variable and then we've got our dependent variable, if we were to plot the excess returns of the market and the excess returns the security if we were to actually perform regression analysis and let's say that we had a line
and we did a linear regression and then we've got some different points that are actual points but we have this fitted line here and that fitted line is going to cross the Y-axis at some point. So the point where it crosses the y-axis that's the intercept but we are calling like in terms of statistical analysis you refer to as the interceptor, we can call it in here in finance, we refer to it as alpha. So this is alpha right where the point intersects with the y-axis.
