Finance

Rate of Return

Given a portfolio's initial value $P_i$, its final value $P_f$, and the elapsed number of periods of time $t$ between, the portfolio's continuous rate of return $r$ can be found using the following equation

$$ r = \frac{\ln(P_f) - \ln(P_i)}{t} $$

If $P_i$, $r$, and $t$ are known quantities, then $P_f$ can be found using the following equation

$$ P_f = {P_i} * e^{rt} $$

To convert the annual rate of return $r_a$ into the continuous rate of return $r_c$, the following equation can be used

$$ r_a = e^{r_c} - 1 $$

To convert the continuous rate of return into the annual rate of return, the following equation can be used

$$ r_c = \ln(r_a + 1) $$

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