Skip to content
All library documents

Calculating Portfolio Variance from Weights, Volatility, and Correlations

Article Quant Q&A · Author: idknuttin
Read the original

Summary

The document asks how to calculate the standard deviation of a three-security portfolio when the asset weights, individual expected returns, standard deviations, and pairwise correlations are known. The accepted response gives the core method: combine the weighted individual variances with covariance contributions for each asset pair, then take the square root of the resulting portfolio variance to obtain standard deviation. Individual variances can be obtained by squaring the supplied standard deviations, while covariances use the correlations and the two corresponding standard deviations.

The example includes a short position and weights that sum to one, so the signs of the weights matter in the pairwise terms. However, the displayed source formula is absent from the supplied text, and the response does not work through the numerical calculation. Readers must supply the full covariance-matrix formula themselves; the stated portfolio expected return does not determine portfolio risk.

Key ideas

  • Portfolio variance combines weighted individual variances and pairwise covariance terms.
  • Covariance for an asset pair is determined by its correlation and both assets’ standard deviations.
  • Portfolio standard deviation is the square root of portfolio variance.
  • Short positions affect the signs of the relevant cross terms through their weights.

Tags

StatisticsPortfolio constructionRisk management
Full text
# How do I find the standard deviation of a portfolio?


# How do I find the standard deviation of a portfolio?












Compute the expected return $\mu_V$ and standard deviation $\sigma_V$ of a portfolio consisting of three securities with weights $\omega_1=40\%$, $\omega_2=-20\%$, $\omega_3=80\%$, given that the securities have expected reuturns $\mu_1=8\%$, $\mu_2=10\%$, $\mu_3=6\%$, standard deviations $\sigma_1=0.15$, $\sigma_2=0.05$, $\sigma_3=0.12$, and correlations $\rho_{12}=0.3$, $\rho_{23}=0$, $\rho_{31}=-0.2$.

I know how to compute the expected return of the portfolio, I got $\mu_V=0.06$, but I don't know how to calculate the standard deviation of a portfolio? What is the formula I need to use given the information? Do I need to find the variances given the standard deviations?

## Answer by HyperVol (score 2, accepted)

https://quant.stackexchange.com/a/24442

You can calculate variance of a portfolio/basket by taking direct weighed averages of the components and then adding the relevant correlation terms * weights for each pair.

Can take sqrt of the expression obtained to have Standard deviation.

Exact formula for calculation goes like this :

(source: benetzkorn.com)

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.