Calculating Portfolio Volatility from Asset Covariances
Summary
The document explains how to combine separately estimated asset volatilities into a portfolio volatility estimate when the assets have nonzero correlations. Its central method is to form the covariance matrix and calculate the portfolio variance as the weighted quadratic form of that matrix. Portfolio volatility is the square root of the resulting variance. For two assets, the general expression reduces to the familiar weighted sum of individual variances and covariance terms.
An expanded example for five assets illustrates the diagonal contributions from individual variances and the pairwise contributions from covariances. The answer also gives the weighted sum of asset returns as the portfolio return. This approach lets a researcher use individual volatility models, as the question requests, instead of estimating risk only from the historical standard deviation of portfolio returns. Its output depends on having appropriate estimates for every pairwise covariance; the discussion does not specify how to estimate those inputs or account for changing correlations.
Key ideas
- Portfolio variance is the weighted quadratic form of the asset covariance matrix.
- Portfolio volatility is obtained by taking the square root of portfolio variance.
- Individual asset variances contribute according to squared portfolio weights.
- Each pair of assets contributes through its covariance and both assets’ weights.
- The portfolio return is the weighted sum of the constituent asset returns.
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Full text
# Volatility of a multiple-asset portfolio
# Volatility of a multiple-asset portfolio
I have N assets with their individual volatilities $\sigma_{i,t}$. I construct a portfolio using the weights $w_{i,t}$ that I obtained in a matter that is irrelevant.
Now I want to determine the portfolio volatility $\sigma_{port, t}$ by combining the individual volatilities, using the weights and correlations.
I know that for two assets you can do:
$\sigma^2_{port} = w^{2}_1 \sigma^{2}_1 + w^{2}_2 \sigma^{2}_2 + w_1 w_1 \text{Cov}_{1,2}$
But what do you do when you have N assets?
Important: I know that you can calculate the portfolio volatility using the portfolio returns and then simply taking the historical standard deviation. This is not what I am after since the individual volatilites are estimated using their individual model.
## Answer by JejeBelfort (score 6, accepted)
https://quant.stackexchange.com/a/34233
You can generalize the formula from a portfolio composed of 2 assets to a portfolio composed of $N$ assets as follows :
$$ \sigma^2_{port} = \sum_{i=1}^N \sum_{j=1}^N \omega_i \text{cov} (i,j)\omega_j = \sum_{i=1}^N \sum_{j=1}^N \omega_i \sigma_{i,j}\omega_j $$ where $\sigma_{port}$ represents the standard deviation of your portfolio.
Taking $N = 2$ yields to the formula you wrote above.
Besides, denoting by $\mu_i$ the return of asset $i$, the return of your portfolio can be written as:
$$ \mu^{port} = \sum_{i=1}^N \omega_i \mu_i $$
## Answer by user18663 (score 1)
https://quant.stackexchange.com/a/34234
You can continue with the same formula as mentioned above in your question for N assets also. To elaborate the above given answer it should be (taking sample as 5 asset portfolio):-
$$(w_1^2)(s_1^2) + (w_2^2)(s_2^2) + (w_3^2)(s_3^2) + (w_4^2)(s_4^2) + (w_5^2)(s_5^2) + 2(w_1)(w_2)Cov_{1,2} + 2(w_1)(w_3)Cov_{1,3} + 2(w_1)(w_4)Cov_{1,4} + 2(w_1)(w_5)Cov_{1,5} + 2(w_2)(w_3)Cov_{2,3} + 2(w_2)(w_4)Cov_{2,4} + 2(w_2)(w_5)Cov{2,5} + 2(w_3)(w_4)Cov_{3,4} + 2(w_3)(w_5)Cov_{3,5} + 2(w_4)(w_5)Cov_{4,5}$$
where W stands for Weight of the asset and S stands for volatility.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.