Understanding Revenue Changes

Managers focus highly on revenue.  Increases are good and decreases are bad.  But what drives those increases and decreases?

Most executives will blandly state revenue is largely the result of volume and sales volume and revenue are often conflated.  However, revenue is the product of both volume and price.  As such, there are two key approaches to improving revenue: increase volume or increase prices.

But which impacts revenue the most?  How do you prove it?

Revenue variance analysis, part of the larger profit analysis discussed in recent academic papers, is under renewed scrutiny. There are four different approaches to a revenue variance analysis.  While each sums up to the same change in revenue, they disagree on what drove the change.

To demonstrate, we will examine a simple case of a single product in a single currency under the full knowledge that more complicated equations can be used to examine multiple products in multiple currencies and under various levels of scrutiny.  Fortunately, such added complexity is unnecessary to demonstrate our claim.

Unresolved Revenue Change

The change in revenue is simply the revenue in the current period less that in a reference period, preferably of equal duration.  When no attempt is made to resolve this into the impacts of changes in prices and volume, we have simply a metric of the change in revenue.

Using the subscript 2 for the current period and 1 for the reference period and denoting prices and volume with P and Q respectively, we find the change in revenue ΔRev.

ΔRev = P₂Q₂ – P₁Q₁

When both prices and volumes are changing, four different situations arise which would change the revenue.  (1) Price and volume both increase denoted as P↑V↑.  (2) Price increases while volumes decrease denoted as P↑V↓.  (3) Price decreases while volumes increase denoted as P↓V↑.  (4) Price and volumes both decrease denoted as P↓V↓.

Graphically, the change in revenue is the difference between the size of the two boxes shown above.  The blue box is the current period revenue, and the orange box is the reference period revenue.  In two cases, the blue and orange boxes completely overlap.  In the other two, a rectangular area is not contained in either box.

Unresolved Joint Variance

In a simple approach, this change in revenue is attributed to changes in price and volume as well as the joint variance in both.  Unfortunately, the joint variance has no clear managerial meaning and, worse, doesn’t send a consistent message.

Defining the change in price and quantity as ΔP = P₂ – P₁ and ΔQ = Q₂ – Q₁, respectively, we write

ΔRev = P₁ΔQ + ΔPQ₁ + ΔPΔQ

In this approach, the first term would be attributed to the impact of volume changes, the second to the impact of price changes, and the third cross term would be the joint variance.

Examining our four different cases, the challenge of the joint variance becomes clear.

In the above graphic, the purple box illustrates the impact of volume changes, the green box illustrates that for price changes, and the grey box illustrates that for the joint variance.

Algebraically and as shown graphically, the joint variance is positive in two instances: P↑V↑ and P↓V↓.  While having a positive joint variance when both price and volumes increase is expected, reporting that same positive joint variance when both price and volumes are down is problematic.  What should managers think when they see a positive joint variance and nothing else?  That revenues are up or down?

While the joint variance can be algebraically defined, it has little to no meaning for managerial decision-making.  For this reason, most people have dropped this approach.

Unbalanced Impact Attribution

As a historical standard approach, the joint variance is removed by measuring the impact of price and volume changes and alternating the period of reference used to define the impacts.  We are left with simple attribution of the change in revenue to impacts due to changes in volumes and prices, but the lack of balance results in excess attributions in some cases and under attributions in others.

ΔRev = P₁ΔQ + ΔPQ₂

In the unbalanced approach, the first term is attributed to the impact of volumes and the second to the impact of changes in price. Graphically, our four different cases are shown below.

For P↑V↑, the previously unresolved joint variance is now completely attributed to the impact of price changes.  But, for P↓V↓, that same previously unresolved joint variance is being completely attributed to the impact of changes in volume.  Why should pricing get extra credit when things are good and few demerits when things are bad?  Or why should sales get little credit when things are good and all extra demerits when things are bad?  This is not only inconsistent, but it can also lead to bad decision-making as well.

Let us also consider the intermediate cases.  When P↑V↓ the impact previously attributed to the joint variance is completely removed, which seems appropriate.  But when P↓V↑, that same impact previously attributed to the joint variance now becomes attributed to both the impact of changes in price and volume.  Why should we accept zero accounting of this joint impact in some cases and double accounting in others?

The over and under attribution of impacts may lead executives to bad decision-making. It would be nice if the attribution was at least done consistently.  Fortunately, it can by relying on mirror symmetry.

Mirror Symmetry Impact Attribution

As demonstrated in a recent research article, a balanced approach to making attributions is delivered through mirror symmetry.

Defining the average price and volumes across the two periods as P̿ = (P₂ – P₁) / 2 and Q̿ = (Q₂ – Q₁) / 2, respectively, we write

ΔRev = P̿ΔQ + ΔPQ̿

As before, the first term is attributed to the impact of volumes and the second to the impact of changes in price. Graphically, our four different cases are shown below.

As one can see, all cases lead to the same approach of making attributions under this mirror symmetry approach.

Management deserves clarity from their revenue variance analysis, especially if the variance analysis is used in making compensation, budgetary, and personnel decisions.  The unbalanced approach, though currently common and historically practiced, clearly has some significant shortcomings, oddities, and biases.  The mirror-symmetric approach is a much overdue update delivering consistency, clarity, and beauty.  Now that is understandable.