Intuition behind log return of portfolio = weighted sum of log returns

Question

Suppose we have $n$ assets, each of which has weight $w_i$ in the portfolio. The log return of asset $i$ is denoted by $r_i$.

What's the intuition why this holds approximately:

$$ ln \left( \sum_i w_i e^{r_{i,t}}\right) \approx \sum_i w_i r_{i,t} $$

Answer 1

The above relation really only approximately. If you consider arithmetic retunrs then it is exact.

For the approximation you just need to look at the Taylor series of the exponential: $$ e^x = 1 + x + \text{ terms of higher order}. $$ These terms of higher order ($x^2$ and $x^3$) become small if $x$ is much smaller than one - which holds true for returns. Thus $$ \sum_i w_i e^{r_i} \approx \sum_i w_i ( 1+ r_i) = 1 + \sum_i w_i r_i, $$ if $\sum_i w_i = 1$.

Then conside that $\ln(1+x) \approx x$ (you can check here) and you are done:

$$ \ln \left(1 + \sum_i w_i r_i \right) \approx \sum_i w_i r_i. $$