# 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?

• Hi, and welcome. Can you be more precise what you want to achieve, firming up your first sentence: "Are there any known & generally accepted methods of scaling returns to match a specific target in performance attribution?" Ask it like we were 5 year olds, and you're more likely to get a precise, and well-argued, answer ;-) – demully Nov 7 '19 at 3:13
• @demully hi! thanks for the advice! I just made an edit, please let me know if this still not clear enough – user134788 Nov 7 '19 at 3:18
• Ah OK, I think I get it (I think). Ln() your prices to give you log-prices, that can be subtracted to give you log-returns, that can be added and subtracted, no problems. Don't worry - this is actually the default assumption in financial mathematics ;-) You just need to E(sum(log-returns)) when you exit to give you an actual price/value. – demully Nov 7 '19 at 3:41