Volatility of a multiple-asset portfolio [closed]

Question

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 1

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 2

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.