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:
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).
For period 1, T1 = t1, F1 = f1, I1 = t1-f1.
For period 2*,
T2 = T1+(1+T1)*t2
F2 = F1+(1+T1)*f2
I2 = I1+(1+T1)*(t2-f2)
*T2 = (1+T1)*(1+t2) -1 which simplifies to: T1 + (1+T1)*t2
For period 3,
T3 = T2+(1+T2)*t3
F3 = F2+(1+T2)*f3
I3 = I2+(1+T2)*(t3-f3)
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.
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).
Compounding Effect: the impact of compounding on factor attribution between currency vs percentage cumulative performance
When evaluating cumulative performance between currency or percentage space, the factor attribution order may change. This can happen due to the compounding effect when evaluating percentage cumulative performance.
The compounding effect on cumulative percentage performance inherits a timing element, where factors with higher return impact earlier on will compound more. This means that as a portfolio’s exposures to certain factors change over time, this can cause their relative impact to vary between performance formatted in currency terms or percentage.
Also for portfolio's evaluated using NAV, assuming the performance of certain securities grows over time -- i.e. the portfolio’s value grows over time -- means that leverage can change over time. This means returns calculated later in time, when leverage might be higher, can have greater dollar impact.