This multiple-period smoothing calculation follows Jose G. Menchero, "An Optimized Approach to Linking Attribution Effects Over Time," The Journal of Performance Measurement (Fall 2000).

For more information about the Menchero arithmetic smoothing algorithm, see article #11781, "Multiple Period Attribution and Contribution Smoothing Algorithms," in the Eagle Knowledge Center.

Arithmetic Smoothing Calculation

The Menchero arithmetic algorithm extends a single-period attribution analysis to multiple-periods by noting that the returns of the portfolio geometrically compound over T periods for the portfolio single-time period return,  , as follows:

The benchmark multiple-period return, , is as follows:

For small single-period returns, the relative performance is approximately as follows:

As the single-period returns become larger this breaks down due to the characteristic scaling arising from geometric compounding. The Menchero algorithm compensates for this by introducing the following scaling factor, , in the previous approximation equation:

The Menchero algorithm then deduces the correct scaling law that ensures any residual will be small. It is simply the ratio of the average arithmetic excess return to the average geometric excess return:

Next, the Menchero algorithm introduces corrective terms, , that distribute the small residual among the time periods so that the following holds exactly:

The coefficients are the final optimized linking coefficients:

Finally, the Menchero algorithm derives optimal values for the small values:

Where is a dummy index of summation.
The following sections summarize the required calculations.

Calculate Total Period Statistics

1. Calculate the geometrically compounded portfolio and benchmark returns for the total period and the excess difference:

1. Calculate the multi-period scaling factor:

1. Calculate single-period excess returns for each period:

1. Calculate the sum of the single-period excess returns across periods:

1. Calculate the sum of the squared single-period excess returns across periods:

Calculate Single Period Statistics

1. Calculate the single-period scaling factor adjustments using the above previously calculated components:

1. Calculate the single-period scaling factors:

1. Calculate the single-period fixed income attribution factors across the segments. Superscript A is introduced simply to note an arithmetic style of attribution is being smoothed:

1. Multiply each single-period attribution factor times the single-period scaling factors:

1. Calculate multi-period attribution statistics and add the single-period attribution factors to form the multi-period attribution factors: