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I have only been able to find the calculation for computing volatility for two assets, and I don't believe one can solve for covariance of $n$ assets where $n>2$ . Is there a method for solving for a portfolio's volatility through it's components purely mathematically? I can always solve via simulation but mathematically would be more precise.

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    $\begingroup$ Let $X_1,\dotsc,X_n$ be random variables. Then $$\text{Var}\left(\sum_{i=1}^n X_i\right)=\sum_{i=1}^n \text{Var}(X_i)+\sum_{i=1}^n \sum_{j=1}^n \text{Cov}(X_i,X_j)$$. This works for $n=2,3,\dotsc$, so not only for $n=2$. Is that what you are asking? $\endgroup$ Sep 2, 2016 at 19:17
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    $\begingroup$ Also the elements of the covariance matrix are $\rho_{ij} \sigma_i \sigma_j$ so you can compute them if (if you don't have them) from the correlation matrix and the volatilities. Then you can use Richard Hardy's formula. $\endgroup$
    – nbbo2
    Sep 2, 2016 at 19:50
  • $\begingroup$ yes it is @richard hardy $\endgroup$
    – FX_NINJA
    Sep 3, 2016 at 0:46
  • $\begingroup$ It's also really easy with matrix math. $w'\Sigma w$ $\endgroup$
    – John
    Sep 19, 2016 at 15:29

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Is there a method for solving for a portfolio's volatility through it's components purely mathematically?

Take variance as a measure of volatility. Let $X_1,\dotsc,X_n$ be a set of random variables (e.g. asset returns). Let $a_1,\dotsc,a_n$ be a corresponding set of weights (e.g. portfolio weights). Then the variance of the linear combination of the random variables (the volatility of the portfolio) is

$$ \begin{aligned} \text{Var}\left(\sum_{i=1}^n a_i X_i\right) &= \sum_{i=1}^n \text{Var}(a_i X_i)+\sum_{i=1}^n \sum_{j=1}^n \text{Cov}(a_i X_i,a_j X_j) \\ &= \sum_{i=1}^n a_i^2 \text{Var}(X_i)+\sum_{i=1}^n \sum_{j=1}^n a_i a_j \text{Cov}(X_i,X_j). \\ \end{aligned} $$

This works for $n=2,3,\dotsc$, so not only for $n=2$. (Also, it is a very general result. It does not require any distributional assumptions.)

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