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I am building an excel model to calculate rolling alpha at a monthly level to analyze different equity mutual funds. Along with the rolling alpha, I am also calculating rolling absolute returns and beta at a monthly level.

While building the above excel model, I thought of a hypothesis (which might be far-fetched) - "if the alpha of a mutual fund keeps increasing or is at least positive for any given time period (let's assume 3 years), then the beta of the fund should also increase with time. That means with time expected returns will keep increasing as beta is increasing and it will become difficult for the mutual fund to keep increasing alpha

The reason for my hypothesis is explained below:

β implies changes in returns of the fund compared with the market. It is explained by the equation: Rp(Mutual fund return) = β(beta) * Rb(Benchmark return)

But if the fund is consistently beating expected return and has an increasing alpha, then that means either - Rp is increasing faster than β * Rb or Rp is decreasing slower than β * Rb

We will now explore both these scenarios. We know that Jenson's alpha (based on CAPM model) = Mutual fund return - Risk free return - Beta * (Benchmark/index return - Risk free return)

To better represent it in equation form: α = Rp - (Rf + β*(Rb - Rf))

Assumption:

  1. We are calculating rolling returns - Rp, Rb and Rf. α is also calculated for every month, hence at a rolling level. Rf = 0% assumed .
  2. Mutual fund is positively correlated with the market throughout the analysis since if beta changes drastically (maybe positive to negative or vice-versa) then this will mean that the underlying fundamentals of a mutual fund has changed.

So, if my rolling alpha (α) is increasing with time, then mathematically either Rp should increase faster than β * Rb or Rp should decrease slower than β * Rb.

Case 1: Rp is increasing faster than β * Rb

In this case, if Rb is increasing at X% rate then Rp will be increasing at X+𝛿% where 𝛿 is positive. Hence, for α to keep increasing β has to increase too.

The only case where β will decrease is if (Rp - α) increases at a slower rate than Rb. To explain it mathematically: β = (Rp - α) / Rb. Hence, if Rb increases at a faster rate than (Rp - α), then only β will decrease. Question 1: Is this scenario possible in real world? I tried this scenario in excel and it looks reasonable as shown below

enter image description here

Case 2: Rp is decreasing slower than β * Rb

In this case, if Rb is decreasing at X% rate then Rp will be decreasing at X-𝛿% where 𝛿 is positive. Hence, for α to keep increasing β too will decrease.

The only case where β will increase is if (Rp - α) decreases at a slower rate than Rb. Again, to explain it mathematically: β = (Rp - α) / Rb. Hence, if Rb decreases at a faster rate than (Rp - α), then only β will increase.

Question 2: Is this scenario possible in the real world? I have tried this in the excel model too and again it looks reasonable as shown below

enter image description here

However, this brings me to the conclusion - Mathematically, all combinations of Rp, alpha, beta and Rb seem possible. But, intuitively, I feel that as alpha increases, it will have an impact on β and since, β is essentially a regression coefficient between Rp and Rb and it will always try to balance the equation - Rp = β * Rb. That means it will be difficult for the fund to keep generating positive alpha for a longer duration. Question 3: Is this conclusion right?

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If you add or subtract a fixed number to the all of the portfolio returns and regress them against the market returns, you will get the same alpha value as before shocked the portfolio returns.

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