First of all, according to the capital asset pricing model, there is such a formula:

R 1=RF+β 1F(RM-RF), where R 1 is a single asset, RF is the risk-free rate of return, RM is the expected rate of return in the market, and β 1F is the correlation coefficient between assets and the market, which is equal to CoV (r 1.

Then according to the topic, Ra can be understood as the random rate of return of an asset called a, corresponding to the above R 1.

Then look, we can see that the title tells you that RP is the yield of any effective portfolio in the market, so RP corresponds to RM above.

Then look, the title tells you that ROP is the return rate of zero beta portfolio, which is the return rate of risk-free portfolio, corresponding to RF in the formula.

So let's prove:

First, we input a generation of RA=ROP+βAP(RP-ROP) at a time, and then open the brackets to get it.

RA=ROP+βAP*RP-βAP*ROP, and then we merge similar items.

RA=( 1-βAP)*ROP+βAP*RP

Well, after getting the above polynomial, we need to see whether βAP meets the definition of β coefficient, and then we can find the topic and tell you that βap=COV(RA, RP)/VAR(RP) meets the definition.

It is not difficult to find that β0=0 (constant term is 0) β 1=( 1-βAP) and so on, so we proved its completion in theory.

As for the symbol of the last term in the regression equation, I think it is because the random term in the regression equation represents the influence of random events on the rate of return. If it is not written in the definition formula, it will not affect your proof. Can be ignored.