In 3schwartz/SpecialeScrAndFun: Script and functions used for master thesis.

knitr::opts_chunk$set(echo = TRUE)

Using that the Taylor Series is given as $$ f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 $$

for $f(x)$ one get for $e^x$ around $0$ that $$ e^x\approx e^0 + x + \frac{x^2}{2} + \frac{x^3}{3} ? 1 + x + O(x^2) $$ Let $r_i$ be the log return for asset $i$. Since this is log returns they can assume to be small such that $$ \sum_i w_i e^{r_i} \approx \sum_i w_i (1+r_i) = 1 + \sum_i w_i r_i $$

assuming $\sum_{i=1} w_i = 1$. In our case one have equl weights between assets. Hence each weight has a sice of $w_i=w=\frac{1}{S_t}$, where $S_t$ is the total number of assets in the portfolio.
Now let $R_i$ be the simple return of asset $i$ such that $\sum_i w_i R_i$ is the simple cumulated return. This can be written as

$$ \sum_i w_i R_i = \sum_i w_i e^{\ln R_i} = \sum_i w_i e^{r_i} $$ Now taking the log on above I get the cumulated log return.
Using a taylor expanssion of $ln(1+x)$ around $0$ we get $$ \ln(1+x) \approx \ln(1) + x - \frac{x^2}{2} + \frac{x^3}{3} \ = x + O(x^2) $$ again for $x$ small.

Hence the cumulated log return can be calculated as $$ \ln\left(\sum_i w_i e^{r_i} \right) = \ln(1 + \sum_i w_i r_i) \ = \sum_i w_i r_i $$

hence, the cumulated log return can be calculated as the sum of the log return for each asset times its weights.

3schwartz/SpecialeScrAndFun documentation built on May 4, 2019, 6:29 a.m.