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Consider an equally weighted portfolio of three stocks, each of which is independently distributed of the others but have the same risk. I.e., $cov(r_i, r_j) = 0$; $\forall i \neq j$, and $\sigma_i = \sigma$; $\forall i$. What fraction of each stock's risk is diversifed away by including it in this portfolio?

Attempted solution - I believe that the fraction of asset $i$'s risk that it contributes to a portfolio is given by $corr(r_i,r_p)$, where $r_p$ is the portfolio return. That is $$corr(r_i,r_p) = \frac{cov(r_i,r_p)}{\sigma_i \sigma_p}$$ Although, I am not sure if this is the case or how I would proceed further. Any suggestions are greatly appreciated.

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In general, the variance of a portfolio is just $$\sigma_p^2 = \sum_i \sum_j w_i w_j \sigma_i \sigma_j \rho_{ij},$$ which intuitively makes sense since we are summing over all weighted standard deviations and their correlations. Since $w_i = \frac{1}{3}$ and $\sigma_i = \sigma$ for all $i = \{1,2,3\}$, and $\rho_{ij} = 0$ for all $i \neq j$, it simplifies to $$\sigma_p^2 = \frac{1}{3^2} \sum_i \sigma ^2 = \frac{1}{3^2} 3\sigma^2 = \frac{\sigma^2}{3} $$ where the summation over $j$ is dropped because of the correlation assumption. The fraction that is diversified away is then just $$\sigma^2 - \frac{\sigma^2}{3} = \frac{2\sigma^2}{3},$$

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  • $\begingroup$ Made it clearer. $\endgroup$ – Forgottenscience Jan 30 '17 at 18:42
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Just buy stock 1 and 2 and short stock 3 (sigma_1^2 + sigma_2^2)/sigma_3^2 times.

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