Calculating Portfolio Volatility from Weights, Correlations, and Standard Deviations

Summary

The note explains how to calculate portfolio volatility when asset weights, return standard deviations, and a correlation matrix are available. It first converts correlations into covariances by placing the asset standard deviations on a diagonal matrix and multiplying that matrix on both sides of the correlation matrix. The portfolio variance is then obtained by applying the weight vector to the covariance matrix and its transpose, and portfolio volatility is the square root of that variance.

This is a standard covariance-based calculation and provides a direct answer to the question about using portfolio weights with a covariance matrix. The document gives no numerical example, data, or discussion of estimation choices. Its result depends on the inputs being aligned in the same asset order and on the correlations and standard deviations being appropriate for the return horizon under study.

Key ideas

• Convert correlations to covariances by scaling with asset standard deviations on both sides.
• Portfolio variance is the weights-covariance-weights quadratic form.
• Take the square root of portfolio variance to obtain portfolio volatility.
• Use consistent asset ordering and return horizons for all input matrices and vectors.

Tags

# calculating portfolio volatility


# calculating portfolio volatility












Given:

vector of portfolio weights $W = [w_1 w_1 ]$

correlation matrix $C = \left( \begin{array}{ccc} a & b \\ d & e \end{array} \right) $

standard deviation of the asset returns $S = [s_1 s_2]$

How can I calculate the portfolio volatility?

If I had the covariance I'd just say $W* Cov * W^{T}$, correct?

## Answer by Physcs Envy (score 2)

https://quant.stackexchange.com/a/18997

You are correct in your basic approach. Given the correlation matrix $\textbf{C}$ and standard deviation matrix $\textbf{S}$ where standard deviations occupy the diagonal and zeros the rest (i.e. $s_{i,j} = \sigma_i | i = j$ and $s_{i,j} = 0 | i \neq j$), the covariance matrix can be found as $\textbf{R} = \textbf{SCS}$. Then your portfolio standard deviation is $\sqrt{\textbf{w}^T\textbf{Rw}}$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.