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

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:

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

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:
$$\ln \left(1 + \sum_i w_i r_i \right) \approx \sum_i w_i r_i.$$
• Just to add that the approximation works well for x within +-10% - see e.g. wolframalpha.com/input/… – rbm Apr 28 '17 at 9:35