# RoyaltyStat Blog

## Ednaldo Silva

Ph.D. Economics from U.C. Berkeley. Founder & Director of RoyaltyStat. Developer of the TNMM = CPM.

### Recent Posts

It’s useful to study the mean and variance of the first-order autoregressive model (AR(1)), which is postulated as univariate:

We can determine an arm's length profit margin (expressed as operating profit divided by net sales) of a controlled taxpayer (“tested party”) by using a first-order autoregressive model, which we can show (like any stable first-order difference equation) to be equivalent to a range of comparable “routine” profit margins plus a weighted random error time series.

The operating profit margin (measured after depreciation and amortization (OMAD)) of 23,151 companies listed in many countries, reflecting fiscal year-end 2015 accounting results, departs from the usually presumed normal distribution. In this sample, OMAD gets a better fit using the Gamma distribution.

Naive analysts use the normal (Gauss-Laplace) distribution to compute the mean and standard deviation, or quartiles, without verifying if the data are abiding. Double fault is committed when they propose a “broad and unconvincing” range of data, such as the interquartile range (IQR), which may become useless to determine arm’s length (or reasonable) royalty rates. In practice, the interquartile range of royalty rates is used to help settle licensing disputes in tax or intellectual property valuation.

Computation of an arm's length operating profit margin is not as easy as it seems. As an example, we consider the directly proportional profit model postulated by the OECD Transfer Pricing Guidelines and the more likely power function that we may encounter when we analyze actual comparable data including small and large enterprises. In cases when the selected comparables include small, medium and large uncontrolled enterprises, an arm's length operating profit margin is more likely to be reliably measured by a double log or power function. The operating profit margin is defined as operating profits (before or after depreciation and amortization) divided by sales revenue.

In transfer pricing, we may encounter a situation in which the statistical residuals among the selected comparables do not have a common variance. This phenomenon is called heteroskedascity. To correct this problem, we can transform or deflate the relevant variables and measure them as ratios. E.g., suppose that we have comparable company coordinated pairs of data on sales (S) and “net” (operating) profits (P), and their bivariate scatter diagram suggests a linear relationship:

If an enterprise makes or buys goods to sell, the cost of goods sold (COGS) can be deducted from net receipts. However, to determine COGS, the inventory at the beginning and end of each tax year must be valued. Consider the symbols:

Zero Intercept Linear Profit Function

The typical OECD TNMM (CPM in the U.S.) prescribes a linear statistical function to test the arm's length character of “net” profits (Y) in terms of the net sales (X):

(1)     Yi = α Xi considering i = 1, 2, …, N comparables

where α is the estimated “net” profit margin. For simplification, we set aside a random error term that is added to equation (1). The controlled taxpayer ("tested party") is the case N + 1.

Non-Linear Profit Function

Instead of equation (1), "net" profits and sales may be represented by a power function:

(2)     Yi = α Xiβ

Power functions are pervasive in economic estimates. Equation (2) states that Yi is proportional to Xiβ . In this case, the profit margin is the slope coefficient of equation (2), which below we show is different from α. A power function is appropriate e.g. when the selection of comparables to the tested party includes small and large companies or when the residual variance is not constant.

Consider the profit margin of selected comparables in the general case when the industry includes two types of uncontrolled (or consolidated) enterprises: (i) innovators that can earn a temporary or persistent excess profit margin, and (ii) imitators that are attracted by the excess profit but whose entry in the industry have the effect of eroding the excess profit margin. As a result, the industry of the controlled taxpayer may exhibit over an audit cycle both a common (or equilibrium) and disequilibrium profit margins earned by innovators and their predators. Here is a schema of this competitive technological and marketing treadmill:

The U.S. Internal Revenue Service (IRS) publishes training materials developed by its International Practice Units (IPUs). On March 7, 2016, the IRS released a new publication about the residual profit split method (RPSM). This IPU provides guidance about how the IRS examiners determine if the RPSM is the “best method” under Section 482, and how to apply such method between a U.S. parent and its controlled foreign affiliates when intangible property is transferred.