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1

There is no right or wrong, just those 2 conventions are different, each one with its pros/cons. In general what is more important is to be clear about conventions used to avoid miscommunication and mistakes. Now if you calculate returns over an interval where the magnitudes are meant to be small then mathematically speaking the difference between raw ...

4

You can simply start with the definition of gross returns \begin{align*} R_{t+1}&=\frac{D_{t+1}+P_{t+1}}{P_t} \\ &=\frac{1+P_{t+1}/D_{t+1}}{P_t/D_t}\frac{D_{t+1}}{D_t}, \end{align*} where the first fraction contains now your price dividend ratio. Going to log-returns, \begin{align*} r_{t+1} &= \ln\left(1+\frac{P_{t+1}}{D_{t+1}}\right) - \ln\left(\...

0

The second expression is just another representation of the former and has nothing to do with continuous compounding. Instead note that $\log(a)-\log(b)=\log\left(\frac{a}{b}\right)$ from which the result should become immediately clear.

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