Why not a new theory?

Again, I am stuck with the fundamental principle of Beta. Beta, according to traditional measure, equals to [covariance between the market returns and a particular stock’s returns] divided by [variance of the market returns over the same period of time].

So, Beta = Cov(RM,RA)/Var(RM).

Now, Cov(RM,RA)= Average of all the products of (RM-Mean of RM)*(RA-Mean 0f RA) at distinct time periods.

Var(RM)= Average of all the products of (RM-Mean of RM)*(RM-Mean of RM) at distinct time periods.

Now when we divide Cov(RM,RA) by Var(RM), it essentially means that on an average, (RM-Mean of RM) terms are crossed out from both the numerator and the denominator. So, on an average, we end up with (RA-Mean of RA)/ (RM-Mean of RM), which refers to, on an average, how much the stock returns is deviating from its mean return for each unit of deviation of the market return from its mean return. Simply speaking, how much can we expect a stock returns to deviate from its mean return for each unit of market deviation from its mean return. The procedure is simple: Divide the terms after computing the average.

Now, let’s come to my question. Why can’t we calculate [the stock return deviation per unit of market return deviation] for each distinct time period and then average all these values? My intuition is simple and straight-forward. Say, at Time1, we calculate the [stock return deviation per unit of market return deviation] by dividing [(RM-Mean of RM)*(RA-Mean of RA)] by [(RM-Mean of RM)*(RM-Mean of RM)]. For each distinct time periods, we will repeat the process and then average all these values. My mechanism is not fundamentally different from the traditional measure. I am calculating individual [stock return deviation per unit of market return deviation] at each time periods and then average, whereas the traditional measure averages the values of [(RM-Mean of RM)*(RA-Mean of RA)] and [(RM-Mean of RM)*(RM-Mean of RM)] first and then determine the [stock return deviation per unit of market return deviation] by dividing.

My question is why can’t my principle work out? Yes, I am well aware of the statistical principles (regression and all that to calculate Beta). But, please, for the sake of this argument, avoid the statistical mechanism.