Selecting a reliable profit indicator is not trivial (reliability is an important metric in transfer pricing). A basic function in algebra represents a straight line, such as the prescribed profit indicator model of the OECD in which the expected value of enterprise profits is a linear function of sales, costs or assets:

##### (1) Y =*f*(X) = β X

where the coefficient β is the slope of the line of the joint pairs X and Y representing a profit indicator.

This velocity model is attractive because it's diffuse in economics. Lang (1986), Chapter II, §§ 3 (Straight Line) and 4 (Distance Between Two Points) contains an algebraic discussion of this linear model (1).

In transfer pricing, which can be subject to intense controversy between tax administrations and corporate taxpayers, it's unwise to *prespecify* the functional form of the profit function such as (1). In this context, a preliminary look at the scatter plot of X versus Y is useful to posit a reliable regression relationship between profits and the X-factor (which may be trivariate, as we shall see below).

According to the OECD, Y is net [operating] profits and X can be sales, costs or assets. From the prevailing GAAP or IFRS standards, and from experience, we know that accounting costs and assets are ambiguous. *E*.*g*., cost of goods (known also as “direct costs”) includes one-period change in inventory, which is subject to several valuation methods. In economics, inventory change is regarded as inventory profit (or loss) and is not regarded as direct costs. Depreciation is another dilemma and is also subject to various schedules. Sales are less ambiguous, but we must ascertain that sales exclude excise taxes. Assets present well-known defects leading to recompute with the perpetual inventory method and CAPX time-series data from cash flow statements. See Kuh (1963), pp. 10, 22, 29 and Usher (1980), equations (5) and (6). See also: https://blog.royaltystat.com/return-on-assets-correct-measure

Putting aside these accounting problems, we can identify several conceptual problems that can render model (1) unreliable to determine arm’s length profits. Thus, we must test the empirical validity of the OECD model (1) and not accept it at face value. Also, we can't accept quartiles of the univariate profit ratio β = Y / X as standard practice. Quartiles of these univariate profit ratios are fallible because they may suffer from the defects itemized below.

First, the model (1) may include an *intercept* such that the ratio Y / X becomes a reciprocal function indicating that profits decrease with the increase in the selected X-factor (sales, costs or assets) as asymptote.

Second, the model (1) may be a *power function* (which is also diffuse in economics) indicating that the relationship between profits and the selected X-factor (sales, costs or assets) is linear in double logarithms. This would imply that the profit indicator would be equal to β (Y / X), and not β. See: https://blog.royaltystat.com/profit-margin-using-a-power-function

Third, the model (1) may be *misspecified* because it omits a relevant independent variable (such as assets), rendering the single X-factor slope coefficient biased. See Maddala (1977), § 9-5 (Omission of Relevant Variables). For more details, see Kmenta (1986), § 10-4 (Specification Errors).

The OECD (2010) asset adjustment added to model (1) is spurrious because it lacks economic-theoretic support and produces no reliability measure. Without applying well-defined accounting, economic or statistical principles, this OECD adjustment is arbitrary. It's an Augean task translating the OECD adjustment method into an algebraic functional analysis, which is a first step to verifiable science.

Instead of applying model (1) coupled with the contrived OECD asset adjustment method, we can add a second X-factor and test for asset-intensity using a mixed regression model:

##### (2) Y = β_{1} X_{1} + β_{2} X_{2}

where X_{1} can be sales and X_{2} can be assets.

A ratio version of model (2) is easy to estimate dividing through by X_{1} or X_{2}:

##### (3) M = β_{1} + β_{2} Z

where *e*.*g*. the *profit margin* is M = Y / X_{1} and Z = X_{2} / X_{1}. The variable Z represents asset intensity (assets to sales ratio, also called assets turnover).

The profits model can be also a *quadratic function* such that the profit margin depends on company size:

##### (4) Y = β_{1} X_{1} + β_{2} X_{1}^{2}

##### (5) M = β_{1} + β_{2} X_{1}

where the *profit margin* is M = Y / X_{1} and X_{1} are sales.

For simplicity, we set aside the intercept and the random error component of models (1), (2) and (4). In these models, the intercept may not be zero and the function *f*(X) may not be linear. In many applications, we get a better regression fit using *f*(X) as a power function instead of the OECD linear model (1).

The determination of an arm’s length profit indicator requires advanced knowledge of economics and statistics, because (like in the single X-factor case) the relationship between Y versus X_{1} and X_{2} may be a power function. Moreover, the error component may not be well-behaved, and corrections must be tested to increase the reliability of the regression results. For example, trend, multicollinearity or serial correlation may be present in the model utilized.

The benefits of using regression analysis over quartiles of univariate profit ratios are that we can *test the statistical significance* of the coefficients by examining their *t*-statistics (regression coefficients divided by their standard errors), and we can measure the *explanatory power* of the regression results using the adjusted R^{2}. None of these benefits can be obtained by computing quartiles of univariate profit ratios. Also, without regression analysis, asset intensity adjustments are vapid and can be more easily impeached under audit scrutiny.

#### REFERENCES

Jan Kmenta, *Elements of Econometrics* (2nd edition), Macmillan, 1986 [1971].

Edwin Kuh, *Capital Stock Growth* (*A Micro-Econometric Approach*), North-Holland, 1963.

Serge Lang, *A First Course in Calculus* (5^{th} edition), Springer (1986) [1978].

*Econometrics*, McGraw-Hill (1977).

OECD, “Comparability Adjustments” (July 2010). Accessed on August 31, 2019: https://www.oecd.org/tax/transfer-pricing/45765353.pdf

Dan Usher (editor), *The Measurement of Capital *(*NBER Studies in Income and Wealth Vol. 45*), Chicago University Press, 1980

**Ednaldo Silva**(Ph.D.) is founder and managing director of RoyaltyStat. He helped draft the US transfer pricing regulations and developed the comparable profits method called TNNM by the OECD. He can be contacted at: esilva@royaltystat.com

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