For performance attribution, we calculate the cumulative total return as the geometric sum over the total daily returns. The Factor and idio cumulative returns are a little different. We perform the calculations in a way that ensures that cumulative factor return + cumulative idio return = cumulative total return. Here is the algorithm we use:
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Assume we have 3 time periods (1-3) and the following types of daily returns total (t), factor (f), idio (i). We also have the following cumulative returns, total (T), factor (F), idio (I).
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For period 1, T1 = t1, F1 = f1, I1 = t1-f1.
For period 2*,
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*T2 = (1+T1)*(1+t2) -1 which simplifies to: T1 + (1+T1)*t2
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For period 3,

and so on... Here's a simple example below that showcases how the calculations work:

This algorithm was designed to support analyzing a portfolio's total return and drilling down into its factor contributors, using total return as the base of analysis. This base encounters a different geometric expansion than the individual factor and idiosyncratic geometric expansions**.

Supporting Math

The geometric sum of A + geometric sum of B does not equal to geometric sum of (A+B), as seen in this table, where 1.5% daily growth on the security will expand more rapidly than its individual market and idio components that grow at 1% and .5%, respectively.
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The cumulative compounded effect of the market on the total securities return is the number represented in the platform.. when looking at a factor's return, we can think of it as holding the factor independently (as seen in the factor profile).