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?