# Volatility of a multiple-asset portfolio [closed]

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.

• Just a note: $\sigma^{port}$ as defined in your formula is the portfolio variance. May 16, 2017 at 13:44
• Whats the difference in the terminology?
– WJA
May 16, 2017 at 14:08
• Typically $\sigma$ denotes volatility. Variance is vol squared $\sigma^2$. May 16, 2017 at 14:36

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:

$$\mu^{port} = \sum_{i=1}^N \omega_i \mu_i$$

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.

• The response by Manish is good. I just wanted to make a couple of clarifications because I think they might be helpful. -The "s" for volatility in the formula is the standard deviation of each asset. -The result of the formula is for portfolio variance. If standard deviation is what you mean by "volatility" in your question then take the square root of the result. May 16, 2017 at 19:01