Estimating Portfolio Risk with Variance and Covariance
Summary
The document explains how to estimate asset and portfolio risk from historical returns. It describes calculating average returns, variance, and standard deviation, with standard deviation expressing return dispersion in the same units as returns. It then introduces sample covariance to describe how two assets’ returns move together and extends the calculation to a variance-covariance matrix for portfolios of any size.
The examples use monthly stock data for Exxon Mobil, American Airlines, and Amazon, and illustrate how portfolio weights and pairwise covariance enter total portfolio variance. The central lesson is that combining assets with low or negative covariance can reduce portfolio risk. The discussion is introductory: the source data, estimation choices, and market conditions affect covariance estimates, and covariance of zero does not generally establish statistical independence. Historical estimates also do not guarantee future portfolio risk.
Key ideas
- Variance measures the average squared deviation of returns from their mean.
- Standard deviation expresses return dispersion in the original units of returns.
- Covariance captures the direction of joint movement between two return series.
- Portfolio variance depends on both asset variances and the covariances between asset pairs.
- A variance-covariance matrix and portfolio weights provide a compact way to estimate multi-asset risk.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.