Projecting Asset Returns into Market and Residual Components

Summary

The document asks how to interpret the component of an asset’s return that is uncorrelated with the market. It contrasts subtracting a beta-scaled market return, which is the usual linear projection residual, with subtracting a beta-scaled asset return, and questions how this connects to the decomposition of total risk into systematic and nonsystematic risk.

The central concept is regression: the market-related component of the asset return is its beta times the market return, while the residual is the asset return minus that fitted component. With centered returns and beta defined by covariance over market variance, the residual is uncorrelated with the market. The two components are orthogonal, so their variances add by the Pythagorean identity; the returns themselves do not generally add as lengths or vectors in ordinary space. The document raises the conceptual question but gives no worked resolution, data, or discussion of assumptions such as linearity and the chosen market benchmark.

Key ideas

• The market-related component of an asset return is represented by beta times the market return.
• The regression residual is the asset return minus its fitted market component.
• When beta is defined by covariance divided by market variance, the residual is uncorrelated with the market return.
• Orthogonality explains why systematic and residual variances can be added.
• The decomposition depends on the selected market benchmark and linear model.

Tags

# Part of return on asset uncorrelated with market


# Part of return on asset uncorrelated with market












I've been told (and have done problems) involving the part of the return of an asset $X$ that is uncorrelated with returns on the market $M$, which can be written as $X - \beta X$. This sort of makes sense, but if someone asked me to explain what "the part of $X$ that is uncorrelated with $M$" is, I would've guessed $X - \beta M$ (per the Pythagorean theorem).

Is there an intuitive explanation for this? I guess $X - \beta M$ is not actually parallel to $X$ in general, so it's not "part of $X$" in that sense, but still.

I suppose this is very similar to "total risk = systematic risk + nonsystematic risk," which makes sense financially, but is also strange to me because I would think we would need to appeal to Pythagoras to get an additive identity.

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