# Given three stocks what is the fraction of each stock's risk is diversified away

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.

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 $$\sigma^2 - \frac{\sigma^2}{3} = \frac{2\sigma^2}{3},$$