Portfolio Variance and Covariance Matrices for Risk Measurement
Summary
The guide explains how to estimate return variance for a single asset, covariance between asset returns, and total variance for a portfolio. It begins with historical returns, using their average as an expected-return estimate and variance or standard deviation as measures of dispersion. It then explains how the sign of covariance describes whether two return series tend to move together or in opposite directions. For two assets, portfolio variance combines each asset’s weighted variance with a covariance term; for multiple assets, the calculation is expressed as the weight vector multiplied around the covariance matrix.
Examples use monthly stock data, including an oil producer and an airline, to illustrate return calculations and negative sample covariance. The guide argues that combining assets with negative covariance can lower portfolio risk for a given expected return, and shows how pairwise estimates feed into a broader portfolio risk calculation. These examples are educational rather than a reliable forecast: historical covariance can change, and covariance near zero does not by itself establish statistical independence. The article assumes familiarity with statistics and portfolio concepts and does not assess estimation error or out-of-sample performance in depth.
Key ideas
- Asset variance measures dispersion in returns, while standard deviation expresses that dispersion in the original return units.
- Covariance indicates whether two return series tend to move in the same or opposite directions.
- Portfolio variance includes both weighted asset variances and covariances between asset pairs.
- The matrix expression for multi-asset portfolio variance combines portfolio weights with the covariance matrix.
- Negative covariance can reduce portfolio variance, though historical estimates may not persist.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.