How Independent Stocks Diversify Portfolio Variance
Summary
The document considers an equally weighted portfolio of three stocks with equal volatility and zero pairwise correlation. It asks what fraction of an individual stock’s risk is diversified by including it in that portfolio, and presents a variance-based calculation. The portfolio variance formula sums the weighted variances and covariance terms; under the stated independence and equal-risk assumptions, the covariance terms vanish and the portfolio variance is one third of the variance of a single stock.
Comparing the portfolio variance with the standalone variance shows that two thirds of the variance is diversified away in this setup. The calculation concerns variance, not standard deviation, and depends on the strong assumptions of equal weights, equal volatilities, and zero correlations. The document does not establish that an individual asset’s correlation with the portfolio alone measures its contribution to risk. A brief alternate answer proposes a long-short position but does not explain or support that approach.
Key ideas
- Portfolio variance combines weighted asset variances with pairwise covariance terms.
- With three equally weighted, uncorrelated stocks of equal volatility, portfolio variance is one third of standalone variance.
- Under these assumptions, two thirds of the standalone variance is diversified away.
- The result applies to variance and relies on equal weights, equal risk, and zero correlations.
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Full text
# Given three stocks what is the fraction of each stock's risk is diversified away
# 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.
## Answer by Forgottenscience (score 3, accepted)
https://quant.stackexchange.com/a/32169
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 $$\sigma^2 - \frac{\sigma^2}{3} = \frac{2\sigma^2}{3},$$
## Answer by Phun (score -1)
https://quant.stackexchange.com/a/32225
Just buy stock 1 and 2 and short stock 3 (sigma_1^2 + sigma_2^2)/sigma_3^2 times.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.