This may be the most stupid question ever asked here, so sorry in advance for asking it.
Suppose we have a single period security which gives dividend $D_{t+1}$ and has current price $P_t$. By definition we have: $$R_{t+1}\equiv\frac{D_{t+1}}{P_t}\ \ (1)$$ Here is the first trouble:I believe I’m making no assumption by saying that by definition it’s still true that: $$P_{t}\equiv\frac{D_{t+1}}{R_{t+1}} \ \ (2)$$ So if we take expectations of (1) and (2) we have: $$E[R_{t+1}]=E\left[\frac{D_{t+1}}{P_t}\right]= \frac{E[D_{t+1}]}{P_t}\ \ (1^E)$$ $$E[P_{t}]=P_t=E\left[\frac{D_{t+1}}{R_{t+1}}\right] \ \ (2^E)$$ So if we solve $(1^E)$ for price we have the following equivalence : $$P_t=E\left[\frac{D_{t+1}}{R_{t+1}}\right]=\frac{E[D_{t+1}]}{E[R_{t+1}]} \ \ (3)$$ The reason why I’m puzzled is that, in general, $E[\frac{X}{Y}]\neq\frac{E[X]}{E[Y]}$. I can’t see my mistake given that I believe I’ve just worked with definitions (which should hold both ex-post and ex-ante) and taken expectations of them.