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CONJUGATE VARIANCE

Variance calculation inevitable results in a "common" area that has to be attributed in some way.

ALLOCATION ON TWO DIMENSIONS

When variance is calculated on two dimensions - price and volume - the common area that must be attributed can be depicted as a rectangle. The area, represented in yellow below, represents the intersection of the volume and of the price variance effect.

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One method is simply to attribute all this common area either to volume or to price variance. This reflects a widespread but somewhat arbitrary practice.

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'Mparanza avoids this issue by allocating the common area equally between price and volume variance.

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The formula we use is the following

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ON THREE DIMENSIONS

Things get more complicated when variance is calculated on three dimensions -for instance price, volume and margin rate"- to identify the factors behind margin change.

The common rectangle area becomes a tri-dimensional box, or rather a collection of boxes in the tridimensional space.

We can visualize the initial situation as a blue Duplo parallelepiped.

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The simple, one-dimensional, variances are represented as the green, yellow and red blocks. For simplicity, we are showing positive variances :-).

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We can visualize the pairwise common areas as the orange, grey and white blocks. These blocks need each to be divided in two, into triangular-based prisms. The volume of each prism must be attributed to the corresponding variance dimension.

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We can visualize the threesome common area as the pink block. This volume needs to be divided in three equal slices, and each slice needs to be attributed to a variance dimension.

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It can be difficult to visualize a cube divided by three, so here you go.

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We use this approach to calculate three-dimensional variance in the options 2,3,6 above. Option 4, being four-dimensional, would be too complicated to calculate with attributions, so we use a simpler, somewhat volume-variance-biased, formula.

A huge thanks to Ludovico Ruggeri Laderchi for the explanation, Duplo pictures and formulas.