# For a mutual fund - if jensen's alpha keeps increasing with time, then how does beta of the mutual fund get impacted with time?

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 ``````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 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?