Use of the logarithm

Recall the compounding formula:

\[A = P(1+r)^T\]

When compounded semiannually:

\[A = P(1+\frac{r}{2})^{2T}\]

The formula generalizes:

\[A = P(1+\frac{r}{n})^{nT}~~~~ (E_1)\]

Continuous returns is when the time between two compounding periods tends to zero. In other words, the returns are reinvested at every point of time. Thus, we look at the limit:

\[lim_{n \to +\infty}(1+\frac{r}{n})^{nT}\]

Recall that:

\[lim_{x \to +\infty}(1+\frac{a}{x})^{x} = e^a\]

Thus:

\[lim_{n \to +\infty}(1+\frac{r}{n})^{nT} = e^{rT}\]

Using again equation $(E_1)$:

\[A = Pe^{rT}\]

When $T=1$:

\[P_2 = P_1 e^{r}\] \[\ln (\frac{P_2}{P_1}) = r\]

Additive property

\[\ln(\frac{P_2}{P_0}) = \ln(\frac{P_2}{P_1}\frac{P_1}{P_0}) = \ln(\frac{P_2}{P_1}) + \ln(\frac{P_1}{P_0}) = r_2 + r_1\]