When you combine multiple, single-period arithmetic attributions across time, the attribution segment level effects do not sum to the total effects of the portfolio over the benchmark for the total time period. This is due to compounding over the periods. This problem is dealt with by introducing smoothing coefficients that adjust each of the values slightly to make the effects sum or multiply properly for the full period being analyzed. There are numerous algorithms that exist to achieve this goal. For Brinson-Fachler attribution, you can use the following methods to link attribution effects over time without introducing unexplained residuals:
- Menchero or Carino arithmetic algorithms
- Carino geometric algorithm
- Geometric Method; Multiple-Period Smoothing
- Carino Multiple-Period Smoothing
- Geometric Attribution Method - an industry accepted geometric attribution approach that is independent of a smoothing method
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