Quartiles of the profit margin are much abused in transfer pricing. Typically, the profit margin (expressed as profits divided by sales) is computed without information about its logical underpinnings. A source of the problem is equivocal regulatory guidance. Another source is the prevailing use of “best practice” *sans* cogito. In the OECD *Transfer Pricing Guidelines*, ¶ 2.90, “A net profit indicator of net profit divided by sales, or net profit margin*, *is frequently used to determine the arm’s length price of purchases from an associated enterprise for resale to independent customers.” A similar postulate is found in the U.S. Treas. Reg. § 1.482-5(b)(4)(ii)(A)(*Profit level indicators*) in which the profit margin is defined as the “ratio of operating profit to sales”.

It's time to examine the logical foundations of the selected PLI (profit level indicator) because the profit margin defined as a simple ratio of (gross, operating or net) profits to sales may be valid only under special circumstances. Although this same malady applies to other profit indicators that are defined as a fixed proportion to costs or assets, here we concentrate of the profit margin under the OECD TNMM, which is equivalent to the U.S. CPM (comparable profits method).

When we compute the profit margin as profits/sales, we assume that (a) profits are proportional to sales, (b) this assumed relationship has no other constant term (zero intercept), and (c) the relationship has no curvature. This means that the OECD and other country specific transfer pricing regulations assume that *a direct proportional relationship exists* between a dependent variable (Y) and an explanatory variable (X) such that each coordinated pair of (X* _{i}*, Y

*) represents the position of the*

_{i}*i*-th comparable company. In this case, Y is profits and X is net sales (revenue, turnover). In symbols, a linear relationship is prescribed by the OECD and the U.S. transfer pricing regulations:

(1) Y* _{i}* = m X

_{i}where “m” is the unknown profit margin to be determined from multi-year profits and sales data obtained from *i* = 1, 2, ..., *N* comparables.

If we are glued to equation (1), as ill-instructed by the OECD and the U.S. regulations, we may face three problems. First, the intercept in equation (1) may be nonzero, because a certain amount of profits may be unrelated to sales. Second, the relationship between profits and sales may be nonlinear. Third, another explanatory factor, such as a combination of specific assets (A* _{i}*), which may include inventory, property, plant & equipment, can be another determinant of profits. Relevant explanatory variables can't be excluded from the estimating equation because the reliability of the results become compromised. We shall consider these problems seriatim.

#### Non-Zero Intercept

The relationship between profits and sales may be linear, as postulated by the OECD and the U.S. regulations; however, it may contain a nonzero intercept, α ≠ 0. Thus, instead of equation (1), we may have a proportional relationship *plus* a nonzero intercept:

##### (2) Y_{i} = m X_{i} + α,

_{i}

_{i}

which would imply that the profit margin is not a “ratio of operating profit to sales” because of the intercept (alpha) displacement. Equation (2) implies that the profit margin is a reciprocal function in which profits are proportional to the inverse of sales:

##### (3) (Y_{i} /X_{i}) = m + α/ X_{i}

_{i}

_{i}

_{i}

If equation (3) is a good fit based on comparable company data, then “m” is not a good measure of the profit margin as defined by the OECD and the U.S. regulations, and quartiles based on the estimated coefficient “m” are unreliable because the added inverse function (α/ X* _{i}*) is not considered.

#### Non-Linear Relationship

In economics, measuring power relationships is pervasive. Therefore, we may expect that instead of the simple prescriptive equation (1), profits and sales may be represented by a power function:

##### (4) Y_{i} = α X_{i}^{m}

_{i}

_{i}

where the profit margin is the slope coefficient (m (Y* _{i}* /X

*)) of equation (4). This*

_{i}*power function*states that Y

*is proportional to X*

_{i}

_{i}^{m}. Likewise, the profit margin is m (Y

*/X*

_{i}*), and not simply “m”.*

_{i}#### Multiple Factors

Profits may be linear but may also depend on both sales and assets in which case we would have a weighted sum of two explanatory factors:

##### (5) Y_{i} = m X_{i} + n A_{i}

_{i}

_{i}

_{i}

such that the profit margin is not directly proportional to sales; it's proportional to asset intensity (A* _{i}*/X

*) plus an intercept:*

_{i}##### (6) (Y_{i} /X_{i}) = m + n (A_{i}/X_{i})

_{i}

_{i}

_{i}

_{i}

In the case of equation (6), defining the profit margin by “m” is unreliable because the proportionality factor n (A* _{i}*/X

*) is not considered.*

_{i}#### Selection of the PLI

According to the OECD, ¶ 2.76, “In applying the transactional net margin method [TNMM], the selection of the most appropriate net profit indicator should follow the guidance at paragraphs 2.2 and 2.8 in relation to the selection of the most appropriate method to the circumstances of the case.” However, equation (1) can't be assumed to be true à priori. In selecting the empirical functional form of the most appropriate PLI, such as the most appropriate method to measure the net or operating profit margin, it’s important to respect the U.S. Treas. Reg. § 1.482-5(c)(3)(i)(*Data and assumptions*) provision that* “* The reliability of the results derived from the comparable profits method [CPM ≈ TNMM] is affected by the quality of the data *and assumptions used to apply this method.”* Emphases added.

In sum, the direct proportionality assumption underlying equation (1) must be tested. This OECD proportionality assumption may apply in special circumstances (zero intercept, nonlinearity), making the routine calculation of quartiles of the profit margin defined by the simple ratio ((Y/X) = m) unreliable. Thus, the reliance on dubious assumptions and “best practice” claims can't be trusted, and are unlikely to survive audit scrutiny. In RoyaltyStat® we can test these hidden assumptions based on the facts and circumstances of the transfer pricing case under audit. In fact, RoyaltyStat provides (Y* _{i}*, X

*, A*

_{i}*) contemporaneous data for*

_{i}*i*= 1 to 35,000 publicly-listed companies located in many countries, and build-in intelligent tools to test the applicability of equations (1) to (6).

**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

and

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