Alpha calculation inconsistent across methodologies

I'm fairly new to finance, and this does not make sense to me.

Consider benchmark & active monthly returns as shown here:

If I do a line of best fit, I get an intercept of 8.4%

Which is meant as the alpha of the active strategy.

However, if I do a cumulative multiplication of benchmark returns (eg. 1.03 * 1.06 * ... *1.03) vs active (1.06 * 1.02 * ... * 1.01) I get benchmark total returns of 214.6% vs 225.8% - ie. a delta of ~11%.

I can't get my head around the relationship between the 11% and the 8.4% from the linear regression.

EDIT: Fixed some arithmetic mistakes.

• By "Active" do you mean monthly returns coming from a fund that is actively managed? Or do you mean active returns in excess of the benchmark returns? Typically, the latter is what is meant when using the word "active". May 17, 2022 at 18:21
• Also, I'm getting a different cumulative product of the benchmark/active returns (215% and 226%, respectively). May 17, 2022 at 18:23
• Here I was using Active in the first sense of the words (the returns of the managed portfolio). You'd have to subtract the columns to get the active return.
– MYK
May 18, 2022 at 9:33

I think there are a couple points to make here:

1. The frequency of the returns here is important. Because you fitted your model using monthly returns, the model is going to be modelling monthly returns as well. So this 8.39% intercept you found indicates that independent of the benchmark portfolio, your active portfolio has an expected monthly return of 8.39%. The fact that your active portfolio outperforms your benchmark portfolio by a margin different than your intercept term is due to you comparing annual returns. Furthermore, I'd add that this outperformance isn't necessarily going to be the monthly expected return annualized due to the volatility of your active portfolio.

2. A linear regression is used to help explain the relationship between two variables and potentially help create a model from those findings. You may have seen regressions done in the context of CAPM (Capital Asset Pricing Model) which are of the form $$\mu_i = \beta_{M,i}\mu_M + \alpha_i$$ Where $$\mu_i$$ is the expected (excess) return of asset $$i$$, $$\beta_{M,i}$$ is the beta/coefficient asset $$i$$ has to the market portfolio ($$M$$), $$\mu_M$$ is the expected (excess) return of the market portfolio, and $$\alpha_i$$ is the intercept term which resembles the amount of $$\mu_i$$ that is not explained by the market portfolio. These models are most helpful when you have some reason to believe that the two assets (asset $$i$$ and the market portfolio $$M$$ in the CAPM example) have a relationship (in other words, you have reason to believe that the market portfolio can explain part of the return of asset $$i$$). If you have that basis, then you have a bit of a stronger footing behind stating that asset $$i$$ actually exhibits some $$\alpha$$. By looking at the two return streams for your benchmark and active portfolios, they do not look to be very related, so I'd argue your $$\beta$$ term doesn't offer much helpful information, and your $$\alpha$$ term therefore doesn't either.

Hope this helps.

• It does - it's a great answer. The data is made up, but I'm trying to understand the numbers. You made the distinction of monthly and annual returns, but 8.39*12 is nowhere near 15%
– MYK
May 18, 2022 at 9:36
• Also, there was a mistake - the difference in the annual returns should be 11%
– MYK
May 18, 2022 at 9:41
• 8.39% is the amount of the monthly return the active portfolio earns on average independently of the benchmark portfolio, not in addition to the benchmark portfolio. We don't expect the active portfolio to earn 8.39% above the benchmark, we expect it to earn 8.39% independent of the benchmark. This independence means that the benchmark can very well outperform the active strategy since the active strategy won't necessarily do well when the benchmark does well (in fact, your regression shows that it has a negative beta, implying it will move in opposite directions). May 18, 2022 at 13:18
• Thank you; that answers my question.
– MYK
May 18, 2022 at 14:00