Taylor Expansion for the Return averaged over k periods?

Question

this is my first question here. I need help to understand the Taylor Expansion which gives the (2.2.5) equation (see the pictures). Thanks (pictures from: Schmidt - Quantitative Finance for Physicists. An Introduction. 2005)

definition simple return

taylor expansion

Answer 1

What the book refers to in fact is the Taylor expansion of the $\log(1+x)$ function about $x_0=0$, that is $\log(1+x) = x + o(x)$, where $o(x)$ is the error term.

The equation (2.2.4) can be rearranged into $$ 1+ \tilde R(t,k) = \left[\Pi_{i=0}^{k-1} \big(1+R(t-i)\big)\right]^{1/k}. $$ Taking logarithm, $$ \log \big(1+\tilde R(t,k) \big) = \frac 1 k \sum_{i=0}^{k-1} \log \big(1+R(t-i)\big), $$ $$ \tilde R(t,k) + o(\tilde R(t,k)) = \frac 1 k \sum_{i=0}^{k-1} \big(R(t-i) + o(R(t-i))\big). $$ When the interest rates are small, the error terms can be neglected and the approximate formule (2.2.5) is obtained: $$ \tilde R(t,k) \approx \frac 1 k \sum_{i=0}^{k-1} R(t-i). $$