Mark hatemets

$$ 1 + R_n = \prod\limits_{i=1}^n (1 + fr_i) $$

$$ f \in (0,1], \ \ r_i \in (-\infty, 1] $$

$$ 1+R_n = \prod\limits_{i=1}^n (1 + fr_i) \Rightarrow \ln(1+R_n) = \ln\left(\prod\limits_{i=1}^n (1 + fr_i)\right) = \sum\limits_{i=1}^n \ln(1+fr_i) \Rightarrow $$

$$ 1+R_n=e^{\displaystyle \sum\limits_{i=1}^n \ln(1+fr_i)} \Rightarrow R_n = e^{\displaystyle \sum\limits_{i=1}^n \ln(1+fr_i)} - 1 $$

$$ G(f) = \max_f \left(\sum\limits_{i=1}^n \ln(1+fr_i)\right) $$

$$ \dfrac{dG(f)}{df} = 0 \Rightarrow \dfrac{d}{df}\ln(1+fr_i) = \dfrac{r_i}{1+fr_i} \Rightarrow G'(f)=\sum\limits_{i=1}^n \dfrac{r_i}{1+fr_i} $$

$$ \sum\limits_{i=1}^n \dfrac{r_i}{1+f^*r_i} = 0 $$

$$ f^* = \underset{f}{\arg\max} \ \dfrac{1}{n}\sum\limits_{i=1}^n\ln(1+fr_i) $$

$$ r_g^*=\dfrac{1}{n}\sum\limits_{i=1}^n \ln(1+f^*r_i) $$

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