# Difference Between the Alpha and the Active Return

Here we're going to discuss the difference between the alpha and the active return of a portfolio. Active return is just the difference between the actual return of a portfolio and the actual return of the market index. Let's say that we have a manager of a fund who's managing a portfolio and they earn a return of 13% by actively managing this portfolio and the market index, let's say the S&P500 has a return of 10% during that same time period. We'd say that in this portfolio there was an active return of 3% however this return does not account for the systematic risk of that portfolio and so that's where Alpha comes into place. Alpha is a risk-adjusted measure of return and alpha is the difference between the actual return of the portfolio and the return predicted by the capital asset pricing model. So active return was the actual return and the difference between that and the market indexes return however, alpha is the actual return and then the difference between that and the return predicted by CAPM.

So capital asset pricing model we've got the formula here that's gonna tell us based on the risk-free rate the beta of the portfolio and then we've got our market risk premium. When we put these together we can actually tell okay what is the expected rate of return for this portfolio. Let's just say hypothetically that the expected rate of return for the portfolio was 11% and we say there's a 13% actual return, in that case, this manager would say they had an alpha that they achieved of 2%. You could just add alpha in our CAPM formula. Let me show you a more complex example and if I've rearranged the formula here, if you want to know how to calculate for alpha it's the same thing as above I just rearranged the numbers to make it a little easier for you.

Let's say we have a fund and we've gone a risk-free rate of 3% and then we have an expected return of the market of 11% we have a beta of 1.5 for our portfolio and then turns out we have an actual return of 17%. Let's go about calculating our active return, so the active return is pretty simple we'll assume here that the market return was 11% and the active return heard the actual return was 17% so we just take 17 -11 gives us an active return of 6% but again this does not account for the systematic risk or the beta. Would think about it like this intuitively the market went up by 11% if the beta was 1 then we would also expect the portfolio to go up by 11% but the beta is higher than 1 so when the market goes up by 11% this portfolio should go up by more than 11% because it has a beta higher than 1. It's there's more risk than just a market portfolio, so we would expect the return will be higher. So just knowing active return doesn't necessarily tell us whether this fund manager did a good job or not we need to adjust for the risk that was inherent in the portfolio.

So how do we do that? Well, we take our CAPM formula here and we're gonna solve for the Alpha. So we've got the return of the portfolio was 17% the risk-free rate is 3% then we plug in our beta 1.5 and then we've got the market risk premium is 8%. Doing a little math here tells us that we have an alpha for this portfolio of 2%. So that is basically saying this is the portion of the root excess return that is not explained by systematic risk.