Cross Rates with Compounding Growth

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In managerial accounting, we learn that sales is a function of price and volume, and that the growth of both sales and volume are not additive, but rather multiplicative (i.e., geometric).  The product of their growth rates is known as a cross rate, and can be derived by the following formula:

Sales = Price × Volume, and

Sales Growth (Geometric) =[ ( 1 + Price Growth Rate) × (1 + Volume Growth Rate) -1]

Sales Growth (Arithmetic) = Price Growth Rate + Volume Growth Rate

In the attached Microsoft Excel © Workbook,  this relationship is applied in cell F5 for the sale of fuel, beginning with 5,000 gallons and growing at 5% annually, starting at a price of \$3.00/Gallon, growing at 4% annually.  The growth rate is thus 9.2%, not 9.0%, due to the cross rate (i.e., the multiplicative nature of Price × Volume).

 Volume (Gallons) 5,000 5% Price (\$/Gallon) \$3.00 4% Arithmetic Sales Growth 9% Geometric Sales Growth 9.2%

Simple enough….

Now let’s not just calculate a static Year 0 and Year 1 growth rate, but role this forward with these growth rates then compounding.  This can be seen in cells C7:H13.  Clearly the cross rate of 9.2% applied to all future years.

 Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 Volume 5,000 5,250 5,513 5,788 6,078 6,381 Price \$3.00 \$3.12 \$3.24 \$3.37 \$3.51 \$3.65 Sales \$15,000 \$16,380 \$17,887 \$19,533 \$21,330 \$23,292 Volume Growth 5.0% 5.0% 5.0% 5.0% 5.0% Price Growth 4.0% 4.0% 4.0% 4.0% 4.0% Sales Growth 9.2% 9.2% 9.2% 9.2% 9.2%

But this simple math soon gets more complex if we extend either the Price, Volume, or both the Price and Volume to multiple factors.

For example, the 5% annual growth rate for Volume may be driven by two independent factors, factor A at 3% annual growth, and factor B at 2% annual growth, for a combined arithmetic annual growth rate of 5%.  The Price remains at a single Factor at its original 3% annual growth rate.  Based on the fact that both the cumulative Volume, and Price are at their original growth rates, that their resulting cross rate will be 9.2% annually as before.  What we find though is that the cross rates differ (i.e., see Cells C16:H44):

What is occurring here is that each Factor must compound individually.  The formula for this is derived in cells K17:T31, and the resulting formula is:

$\mbox{Correct Growth Rate} = [(1+g_{A})^{n} + (1+g_{B})^{n} + \dotso-1]$

Which is quite different from what we would probably expect:

$\mbox{Incorrect Growth Rate} = [(1+g_{A} + g_{B} + \dotso)^{n}-1]$

As the number of Factors increases, and if both Price and Volume are broken into multiple Factors, the complexity grows, because the Correct Growth Rate equation above would need to be placed into the simplified cross rate formula at the start of this paper, for both the Price and the Volume.

We can clearly see that Volume at 5% growth, and Price at 4% growth, yielded 9.2% growth, not the anticipated 9% growth, but where does this increase of 0.2% growth get assigned, and in what amount?  Is it due to the higher growth in Price or Volume, and what percent of the 0.2%?  This even becomes more complex if we add multiple Factors to each cumulative growth rate.  Another Post will be coming soon to show how we can bring this geometric growth rate back to the various Factors to determine what percent of this growth rate is derived by each factor.  This is important because we may want to quote the growth rate of Price and Volume separately, and if Factor exist for each one, report accurate growth rates for each Factor.

A copy of the Excel file used in this post can be downloaded here.