In particular lets consider a zero-beta asset $$i$$ (in the CAPM sense). Let

1. $$R_f$$ be the risk free rate

$$R_f$$ be the risk free rate

2. $$R_i$$ the return on the asset $$i$$

$$R_i$$ the return on the asset $$i$$

3. $$R_m$$ the return on the market portfolio

$$R_m$$ the return on the market portfolio

4. $$\beta=\frac{Cov(R_i,R_m)}{Var(R_m)}$$

$$\beta=\frac{Cov(R_i,R_m)}{Var(R_m)}$$

5. $$E_P (E_Q)$$ the expectation under $$P$$ the historical probability ($$Q$$ the risk neutral probability)

$$E_P (E_Q)$$ the expectation under $$P$$ the historical probability ($$Q$$ the risk neutral probability)

By the martingale properties the follwoing identity holds: $$E_Q[R_i]=R_f$$

By the martingale properties the follwoing identity holds: $$E_Q[R_i]=R_f$$

By the CAPM the following holds:

$$E_P[R_i]=R_f+\beta E_P[R_m-R_f]= E_Q[R_i]+\beta E_P[R_m-R_f]$$

If we assume $$\beta=0$$, then $$E_P[R_i]=E_Q[R_i]$$

My question is: does $$E_P[R_i]=E_Q[R_i]$$ imply $$P=Q$$?

# Can the historical probability be the same as the risk neutral probability measure?

In particular lets consider a zero-beta asset $$i$$ (in the CAPM sense). Let

1. $$R_f$$ be the risk free rate
2. $$R_i$$ the return on the asset $$i$$
3. $$R_m$$ the return on the market portfolio
4. $$\beta=\frac{Cov(R_i,R_m)}{Var(R_m)}$$
5. $$E_P (E_Q)$$ the expectation under $$P$$ the historical probability ($$Q$$ the risk neutral probability)

By the martingale properties the follwoing identity holds: $$E_Q[R_i]=R_f$$

By the CAPM the following holds:

$$E_P[R_i]=R_f+\beta E_P[R_m-R_f]= E_Q[R_i]+\beta E_P[R_m-R_f]$$

If we assume $$\beta=0$$, then $$E_P[R_i]=E_Q[R_i]$$

My question is: does $$E_P[R_i]=E_Q[R_i]$$ imply $$P=Q$$?