How to do the Price-Volume-Mix waterfall right
written by Juliane Stephan and Rohit Patel
Introduction
Revenue performance is a complex interplay of factors, often obscuring the true drivers of growth or decline. Even considering a simple case of a single product company is complex because price and volume changes do not add up to revenue changes. While price and volume changes command significant attention, the impact of product mix is harder still to ascertain. The Product-Volume-Mix (PVM) Analysis is a powerful tool to decompose the dynamics at play, but it is a non-trivial analysis that is often done wrong. Making business decisions based on an inaccurate PVM analysis can have a significant negative impact on a business.
How can you spot an inaccurate PVM analysis?
- The answer changes if you flip the time periods around (The attribution to each driver should stay the same, only with inverse signs)
- The answer changes depending on whether the analysis is started with decomposing Price or Volume first (The attribution to each driver should stay the same no matter where you start the analysis)
- The impact of the three drivers needs to add up to the gap between starting period and ending period
This post will walk through a PVM analysis and point out common mistakes that we have come across in the past. If you are interested in a more technical explanation, you can read Rohit’s paper titled “Fair and additive decomposition of metric components” where Example 1 deals with this specific case (PVM Analysis).
The problem statement
Suppose we are trying to break down the change in revenue for a business between “Initial” and “Final” periods into the factors of Price, Volume and Mix. Suppose the business sells m products P1,P2…Pm. Let pi be the price of i-th product and qi be the quantity of the i-th product initially.
To illustrate the issue let’s start by assuming m = 1 such that the business sells only one product. In this case, there are no product mix issues to deal with. Let the initial and final price and quantity of the product be as shown in Table 1
Here, there is 20%($20) change in revenue. How do we attribute this change to the change in price and quantity? The price and quantity changes are 100% and −40% respectively. Let’s start by looking at revenue change due to price change while keeping quantity fixed, that is ($20−$10)×10 = $100. Now let’s look at the change in revenue due to the change in quantity at this new price which comes out to $20 × (6 − 10) = −$80. This adds up nicely to $20 change in revenue and could be our answer.
Except, the choice of calculating the impact of price using initial quantity and impact of quantity using final price appears arbitrary. If we start with looking at the impact of quantity first $10 × (6 − 10) = −$40 and then the impact of price on new quantity ($20 − $10) × 6 = $60 we get very different numbers. These numbers also add up. Table 2 shows the deltas.
Clearly, the two attributions do not match. What’s correct? One important property we want our solution concept to have is consistency. If the flip the initial and final periods (such that revenue change is now −$20) then whatever solution concept we choose should give us the same numbers for impact of price and quantity with a negative sign. Neither of the above solutions have this property. You would get very different numbers in the reverse direction (try it out!). That should not happen.
The answer (in this case) is deceptively simple, average the numbers ($80 being the impact of price and −$60 being the impact of quantity. But is this actually correct? What makes it correct? How does it generalize when you have more than two factors (e.g. when Mix is in the mix)? Will this approach always lead to additive numbers? Let’s look at the case with more than one product and mix changes. Table 3 shows a hypothetical situation with 3 products and a changing mix.
Before we dive in, what does it mean for the mix to change? We consider the mix to remain unchanged when the percentage of each product is constant in the overall sales. This means that the quantity in the final period is a constant multiple of the quantity in the initial period for all products. Naturally, for any real business you would expect the mix to change for any two given periods of comparison.
With this, let’s try to work out the attribution in revenue change to price, volume and mix. In this case, the revenue change is 2320 − 1800 = 520. Now, based on the methodology described above, we need to consider each factor incrementally and in each possible order. First, let’s write down some building blocks in Table 4 (see the accompanying excel worksheet for formulas).
Let’s start with one single permutation order: Price first, then volume, then mix (call it PVM). How do we calculate the contribution of each of the three when using this order (PVM). Keep in mind there are other orders to worry above (e.g. PMV, MPV, MVP etc..) and as we saw in the simple example with only one product, order really matters.
It is easy to calculate the impact of price change alone (initial volume and mix), simply use the price from final period with quantity from initial period to get the revenue, then take difference, this comes out to be $1850−$1800 = $50 as shown in Table 4. Next, to get the impact of volume when price has changed, we look at revenue when both volume and price has changed, and subtract revenue when price has changed (thus giving us incremental change due to volume changes on top of price changes), this comes out to $1951 − $1850 = $101. Finally to get the impact of mix change on top, we can take the final revenue and deduct the revenue when price and volume have changed. This comes out to $2320 − $1951 = $369.
We can now do the same for other combinations. Table 5 shows the impact of each factor when we follow various sequence orders. The accompanying excel worksheet works out all the permutations. Averaging them gives the final answer.
What makes this the right answer? We show in Appendix C of the paper that this solution is always additive and fair. Moreover, it is consistent in the sense that you will get the same answer regardless of which period is tagged “initial” or “final”. The paper also provides the general solution in a more formalized language.
At some point, we did build this solution into QuickAI. You can pop your raw transaction data into the PVM Analysis module and get the answer. The screenshot above can be created by downloading the raw_data sheet of the worksheet and running it on QuickAI. This analysis is also part of the Private Equity Analysis Module.
Appendix
Here are the steps to replicate the analysis above on QuickAI
Raw data from worksheet in CSV format:
