Equal-Weighted Portfolio Volatility with Average Correlation
Summary
The document derives the variance of an equal-weighted portfolio from the covariance terms between its assets. The key correction is that the double sum contains two kinds of terms: each asset’s variance on the diagonal, where the asset is paired with itself, and the covariances between distinct assets. There are n diagonal terms and n(n−1) off-diagonal terms. Weighting each by the squared equal weight gives portfolio variance as the average individual variance contribution plus the average-correlation contribution.
Under the stated simplifications of equal asset volatility and a common average pairwise correlation, the resulting variance is the asset variance multiplied by a weighted combination of one and the average correlation. This resolves why treating every covariance term as an off-diagonal correlation incorrectly omits the diagonal terms. The result relies on equal weights and representative average volatility and correlation; real portfolios may have heterogeneous volatilities and pairwise correlations, in which case the full covariance matrix is needed.
Key ideas
- Portfolio variance is a double sum over all pairs of asset returns.
- The sum includes n self-variance terms and n(n−1) cross-covariance terms.
- Equal weights scale each term by the square of one divided by the number of assets.
- With common volatility and average correlation, portfolio variance combines individual variance and cross-asset covariance contributions.
- Heterogeneous assets require using their full covariance matrix.
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Full text
# Standard deviation of large equal-weighted portfolios
# Standard deviation of large equal-weighted portfolios
Say I've got a portfolio of shares with the following parameters: Let $n$ be the number of shares in the portfolio, let $\bar\sigma$ be the average standard deviation (volatility/risk) for each share, let $\bar\rho$ be the average correlation between each pair of shares.
I would like to determine the portfolio's volatility. For $n=1$, this is simple, $\bar\sigma=\sigma_P$.
For $n=50$, I did the following, which I know is wrong, but I can't spot my mistake:
Let $R_P$ be the portfolio's return, $R_i$ an individual asset's return and let $x_i$ be an asset's weight in the portfolio.
$$ Var(R_P) = Cov(R_P, R_P) = Cov(\Sigma x_iR_i, R_P)=\Sigma x_iCov(R_i, R_P)=\Sigma_i\Sigma_jx_ix_jCov(R_i, R_j)=\Sigma_i\Sigma_jx_ix_j\rho_{ij}\sigma_i\sigma_j. $$
Now, for equal-weighted portfolios, instead of $x_i, x_j$, we can write $1/n$; also, we can write $\sigma_i=\sigma_j=\bar\sigma$ and $\rho_{i,j}=\bar\rho$:
$$ \Sigma_i\Sigma_j\frac{1}{n^2}\bar\rho\bar\sigma^2=\frac{1}{n^2}\bar\rho\bar\sigma^2n^2=\bar\rho\bar\sigma^2 = Var(R_P). $$
Thus, we'd have $\sigma_P=\sqrt{\bar\rho\bar\sigma^2}$. This is true for $n=1$ but it doesn't make sense for multiple shares, does it? I should have forgotten $n$ somewhere.
Why is this wrong?
Please don't be too harsh with me, I'm new to this topic.
## Answer by Hans-Peter Schrei (score 1, accepted)
https://quant.stackexchange.com/a/78435
This is incorrect at the step where you evaluate the double summation:
$Var(R_P)=\sum_{i=1}^n\sum_{j=1}^n\frac{1}{n^2} Cov(R_i,R_j)$
You then have to consider two cases $i=j$ and $i \neq j$. For $i=j$, there are $n$ terms of $\frac{1}{n^2}\overline{\sigma}^2$. For $i\neq j$, there are $n(n-1)$ terms of $\frac{1}{n^2}\overline{\rho}\overline{\sigma}^2$.
This results in $Var(R_P)=\frac{1}{n}\overline{\sigma}^2+\frac{n-1}{n}\overline{\rho}\overline{\sigma}^2$.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.