This multiple-period smoothing calculation follows David Carino, Combining Attribution Effects Over Time, Russell Research Commentary (August 1999).

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

Arithmetic Smoothing Calculation

The Carino arithmetic algorithm is similar to the Menchero arithmetic algorithm. The difference lies in the scaling coefficients used for smoothing. These are calculated differently. The single-period Carino coefficients are calculated as:

where if then

The product is then a continuously compounding form of the currency effect. The sum of the continuously compounded currency, cross product, allocation, and selection attribution effects sum to the difference in continuously compounding returns:

To transform back to the difference in returns (instead of the difference in log-returns), the combined period factor is calculated as follows:

where again if then

The formula for the combined currency effect is:

The cross product, allocation, and selection effects are calculated similarly.

Geometric Smoothing Calculation

The Carino geometric algorithm is a variant of the Carino arithmetic algorithm in which the excess return and effects are expressed in geometric form. The approach begins with the arithmetic attribution effects, including scaling them through the factor defined in Carino's arithmetic method. It then transforms the effects into a multiplicative form through an exponential. Through this transformation, the geometric attribution effects are aggregated without a residual to the total period geometric excess return.
More specifically, the smoothed geometric selection effect is produced by the exponential of the Carino smoothed arithmetic allocation effect:
,
and similarly, for the selection and interaction effects:

and
.
Over multiple-periods, these attribution effects are combined multiplicatively. The multiple-period allocation effect is produced by:
.
Similarly, for the selection and interaction effects:

and

The allocation, selection, and interaction effects multiply exactly to the geometric excess return without any residual:

You can calculate a combined period allocation, by compounding:

Combined period selection and interaction are calculated similarly. The combined effects satisfy the multiplicative decomposition: