Scaling Returns to Match a Target in Performance Attribution
Summary
The document asks how to adjust individual asset returns so that their compounded cumulative return matches a chosen target, and whether there is a formal name for distributing the discrepancy. It contrasts multiplying each return by a common factor—an approach that can make the arithmetic sum hit a target—with compounding the adjusted returns, which may produce a different result. A portfolio example illustrates the mismatch between a target sum and the resulting product of gross returns.
The question also raises how adjustments might behave across multiple periods and when returns are allocated among assets or attribution categories. No answer or accepted algorithm is supplied, so the document establishes the problem rather than recommending a method. Any solution would need to specify the return aggregation convention, what components must remain consistent, and how the adjustment should be allocated. The example does not provide enough information to determine a unique allocation, and it should not be read as evidence that proportional scaling is appropriate for attribution.
Key ideas
- Proportional scaling can make the arithmetic sum of returns match a target without making compounded returns match it.
- Compounded cumulative return is calculated from the product of gross returns across periods.
- The question concerns allocating a return discrepancy across assets or attribution components.
- Extending an adjustment across periods requires a specified aggregation and allocation rule.
- The document poses the problem but provides no accepted method or tested solution.
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Full text
# Error distribution algorithms in performance attribution # Error distribution algorithms in performance attribution Are there any known & generally accepted methods of scaling each return in a period such that the total cumulative return equals a more desired amount? For example, 0.5 and 0.3 have a total cumulative return of 0.95. How should I scale (remove from, in this case) each of returns such that the total cumulative return is then 0.9%? I suppose this could be answered in the generalized scope of error distribution amongst returns. Is there a formal name for this type of algorithm? I would also be interested in how these methods interact when aggregating multiple periods and what might be involved there. Given a portfolio with 3 assets and return for an arbitrary single time period: ``` Asset Return Weight Stock Selection ... A 0.05 ... ... B -0.19 ... ... C 0.23 ... ... Cumulative Return: 0.046115 ``` Let's say we want to "scale" this attribution to a different cumulative return and distribute the error. I'm aware that we can do the following ``` T = Scaling Target = 0.05 R = Sum of Existing Returns = 0.09 Asset Return Weight Stock Selection ... A 0.05 * (T/R) ... ... B -0.19 * (T/R) ... ... C 0.23 * (T/R) ... ... ``` With a scaling target of T = 0.05, this would give us: ``` Asset Return Weight Stock Selection ... A 0.0278 ... ... B -0.106 ... ... C 0.128 ... ... Sum of Returns : 0.05 ``` However, this seems odd that I now have to sum the returns to get my desired 0.05 return, rather than obtaining the cumulative return by taking `(1 + Ri) * (1 + Ri+1) ... (1 + Rn)` as the return calculated in that manner would be 0.0368. It seems like this approach just fudges the numbers and forces us to use summation to obtain the desired return. Is there a more "return-friendly" way of performing this type of scaling such that `(1 + Ri) * (1 + Ri+1) ... (1 + Rn)` would equal my desired return?
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