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Why One Weighted Average Cannot Reveal Multiple Individual Returns

Article Quant Q&A · Author: quant4u
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Summary

The document asks whether individual returns can be recovered from a single weighted average when the component weights are known. Its example gives one overall return formed from three unknown returns and their respective weights. The answer points out the central constraint: one equation with three unknowns does not uniquely determine each return. Many different sets of individual returns can produce the same weighted total.

To identify all unknown returns, one needs at least as many independent equations as unknowns. Even that is a necessary condition rather than a guarantee: the equations must provide independent information. The document offers no additional data or method for estimating the components, so the individual values cannot be solved from the stated average alone. This is a basic identifiability issue in portfolio return analysis, not a limitation of a particular calculation technique.

Key ideas

  • A weighted average provides one constraint on its component returns.
  • One equation with multiple unknown returns cannot uniquely determine every component.
  • Recovering all individual returns requires enough independent equations to identify them.
  • Having as many equations as unknowns is necessary, but the equations must also be independent.

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StatisticsPortfolio construction
Full text
# Am I able to find individual returns from total weighted average of returns?


# Am I able to find individual returns from total weighted average of returns?












As titled states… I am trying to figure out how to solve for individual return given average weighted total return and weights of individual returns?

For example:

2% = (r1 x 0.2) + (r2 x 0.5) + (r3 x 0.3)

What is r1, r2, and r3?

## Answer by Bob Jansen (score 5)

https://quant.stackexchange.com/a/73474

You have one equation and three unknowns, as you found out this can’t work. You need at least as many independent equations as unknowns. I don’t see how you can make this idea work.

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