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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.

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    $\begingroup$ Is it behause $\rho_{ij}$ for $i=j$ is one which I forgot? $\endgroup$ Feb 21 at 23:30

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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 $Var(R_P)=\frac{1}{n}\overline{\sigma}^2+\frac{n-1}{n}\overline{\rho}\overline{\sigma}^2$.

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