*R**eturns on assets* (ROA) are ill-defined and selection of this profit indicator in transfer pricing can lead to intractable controversy between the tax administration and corporate taxpayers. Assets (which combine liabilities and equity) are an accounting quagmire. The nebulous definitions provided by the OECD *Transfer Pricing Guidelines* (2017), ¶¶ 2.103 and 2.014 create more pain than relief. Likewise, the vague definitions provided by US 26 CRF 1.482-5(b)(4)(i) and (d)(6) are misconceived because they aggregate heterogeneous elements disrespecting short- versus long-term assets vintages (acquisition dates), assets associated and those not associated with interest deductibility, economic cycle dynamics, and different depreciation schedules.

*Prima facie*, we notice several conceptual problems with assets as a denominator to profits (understood to mean *operating profits* calculated before or after depreciation):

First, the OECD and the US regulations prescribe aggregating short-term (inventories, accounts receivable) with long term (property, plant and equipment (net of depreciation)) assets. Also, they prescribe aggregating short-term (accounts payable) with long-term liabilities. However, short-term liabilities don't incur interest expenses, and according to accounting practices interest expenses covering long-term liabilities are deducted *after* operating profits are calculated.

Second, the OECD and the US regulations prescribe aggregating non-depreciable (inventories, accounts receivable) and depreciable (property, plant and equipment) assets. These separate and distinct assets are subject to different economic cycles, which in economics are recognized by the authors Kitchin ([1923] inventory 3-5 year cycles measured from peak to peak or measured from trough to trough), Juglar ([1862] fixed-investment 7-12 years cycle), and Kuznets ([1930], infrastructure or building 12-25 year cycles).

These different business cycles result from different processes; thus, their underlying economic dynamics and their dependent and independent variables are distinct. Aggregating these separate and distinct assets into a putty-clay lump-sum is heroic. Please consult Joseph Schumpeter, *Business Cycles*, two volumes, McGraw-Hill, 1939, for an early taxonomy of different business cycles. A more recent and more rigorous survey is found in Günter Gabisch and Hans-Walter Lorenz, *Business Cycle Theory* (A Survey of Methods and Concepts), Springer-Verlag, 1989, including coverage of Kalecki’s [1935] fixed-investment model.

Third, we question if long-term liabilities should be part of operating assets because the matching interest expenses are deducted after operating profits (EBITDA or EBIT) are calculated.

Four, unlike using sales or costs from the income statement as denominators to operating profits, assets are an *endogenous variable*. This means that we must conform to an explicit asset accumulation equation, such as (3) below. To be consistent with accounting and economic principles, equation (3) restricts our measure of assets to property, plant and equipment, which is different from the OECD and IRS arbitrary definitions.

Academic research can’t rescue a fact-checker from regarding assets as an accounting inferno. To understand that academics can’t provide rescue, read *inter alia* Franklin Fisher (ed.), *Antitrust and Regulation*, MIT University Press, 1985. We suggest to abandon “net profits weighted by assets” as a transfer pricing profit indicator in favor of using (depending if the tested party is an importer from related parties, exporter or service provider) operating profit margin or operating profit markup on total operating costs.

Hereafter, we shall focus on an important but neglected issue of assets being *endogenous* to the returns on assets system of equations. For this purpose, P(*t*) denotes net operating profits (P = OIADP = EBIT) for a selected enterprise (whose index is implicit for simplification), A(*t*) denotes *net* Assets, G(*t*) denotes CAPX, which is *gross *investment in added property, plant and equipment, and *t* = 1, 2, 3 … are indices of historical years in which financials for these accounting variables are available:

(1) P(*t*) = β A(*t*) + U(*t*),

which posits that net operating profits are proportional to net assets (defined below) plus a random error. This structural linear equation (1) prescribed by the OECD and US regulations is estimated in transfer pricing under an opaque theory (without a corresponding dynamic specification) that assets gravitate across-industry to form an equilibirum or uniform returns on assets.

(2) U(*t*) is Normal(0, σ),

which is an assumption about the time-behavior of the regression residuals that needs testing using *e*.*g*. Durbin-Watson statistcs. See G. Maddala, *Econometrics*, McGraw-Hill, 1977, sections 7-5 (Analysis of Residuals) and 12-7 (Tests for Serial Correlation).

(3) A(*t*) = (1 – δ) A(*t* – 1) + G(*t*),

which is a consensus asset accumulation definition. See e.g. Charles Hulten and Frank Wykoff, “The Measurement of Economic Depreciation,” in Charles Hulten (ed.), *Depreciation, Inflation & The Taxation of Income from Capital*, The Urban Institute Press, 1981, equations (6) and (12).

Above, the coefficient beta denotes returns on assets and delta denotes an applicable depreciation rate. Two comments are in order. First, structual equation (1) needs an intercept that for now (following convention) we set aside *cum* *grano salis*. Second, the error terms are not likely to be random, so (2) must be replaced by another equation that recognizes serial correlation among the residuals. Third, we hold that considering (3) is necessary to well-specify a *return on assets* final or estimating equation.

We substitute (3) into (1) and obtain:

(4) P(*t*) = β [(1 – δ) A(*t* – 1) + G(*t*)] + U(*t*)

(5) P(*t*) = λ_{1} A(*t *– 1) + λ_{2} G(*t*) + U(*t*)

where λ_{1} = β (1 – δ) and λ_{2} = β.

In final equation (5), the partial regression coefficient λ_{1} = β (1 – δ) is an estimated *returns on assets*, and it differs from β in structural equation (1) by the depreciation coefficient of property, plant and equipment—that is, beta is displaced by an adjustment when the depreciation of property, plant and equipment is not zero, which we expect to be the usual case. See Maddala, *ibidem*, section 9-5 (Omission of Relevant Variables), pp. 190-191 (Dropping Variables), and pp. 461-462 (Specification Errors).

In econometrics, we don't estimate structural equations such as (1); instead, we estimate reduced-form or final equations such as (5) in which the right-hand side contains exogenous variables or lagged dependent variables (but it doesn't contain endogenous variables). Among others, Ronald Wonnacott and Thomas Wonnacott, *Econometrics* (2nd edition), John Wiley & Sons, 1979, pp. 258, 274, 291 contain a schematic explanation that in econometrics we employ a regression system in which exogenous variables determine endogenous variables. See also G. Maddala, *ibiden*, pp. 220-221.

Now, from (3) or (5) we note that G(*t*) is the gross investment flow reported on the cash-flow statement that accumulates over time into an assets stock A(*t*). This is what we mean by saying that assets forming the base of operating profits can’t be defined willy-nilly; instead, the gross investment flow variable must be consistent with the matching stock of assets accumulation. In accounting principles, G(*t*) = CAPX(*t*) represents gross investment in added property, plant and equipment, excluding inventories and other assets such as accounts receivable and excluding liabilities. In short, the period gross investment *flow* G(*t*) engenders the beginning-of-period assets *stock* A(*t *– 1), which is a partial base of operating profits. We say *partial base* because the same period gross investment, or G(*t*) = CAPX(*t*), is the other partial base of operating profits as visible in reduced-form equation (5).

In contrast to *structural* equation (1), *final* equation (5) includes two determining variables to operating profits: beginning-of-period property, plant and equipment and (the same period as net operating profits are measured) gross investment in additional property, plant and equipment. To be precise, gross investment in depreciable property and plant should be separated from gross investment in depreciable equipment because they are subject to distinct depreciation schedules and distinct business cycles. However, accounting practices under GAAP and IFRS report a single aggregate figure under PPENT (property, plant and equipment (net)).

Thus, the OECD and IRS definitions of operating assets (cited above) are misconceived. We recall hearing Charles Berry during the drafting of the US transfer pricing regulations insisting that assets are “dirty” numbers to be avoided. If we wish to use “net profits weighted by assets,” where assets include items other than property, plant and equipment net of accumulated depreciation, then (3) must be changed in a consistent fashion. However, we have difficulties obtaining the accounting flows matching the stock of inventories and the matching flows for other hard-to-measure assets.

In practice, the problems of using “net profits weighted by assets” are even more difficult to establish reliable returns on assets because we would need to replace (2) by another expression, such as the usual first-order correlation among the residuals, and this recognition would modify the estimating return on assets equation (5). See G. Maddala, *ibidem*, section 12-6 (Estimation procedures when residuals are AR(1)). Since the non-controversial asset accumulation equation (3) is recursive, model (5) can be converted into an equation in which A(*t * – *k*), where *k* is the adopted terminal lag period, becomes negligible and would disappear from (5) being replaced by one-period lagged operating profits typical of distributed-lag econometric models. Again see Maddala, *ibidem*, sections 9-3 (Lagged Dependent Variables) and 16-2 (Infinite Lag Distributions (coverage of Geometric or Koyck lag)) and Hulten and Wykoff, *opus citatum*, equation (12).

We draw a conclusion that using *returns on assets*, where “net profits are weighted by assets,” is perilous because aggregating short-term with long-term assets, and aggregating non-depreciating with depreciating assets, can lead to an unreliable measure of arm's length returns on assets. Moreover, after we recognize that assets are endogenous, estimating structural equation (1) leads to an omission bias by excluding relevant variables (such as gross investment or G(*t*) = CAPX(*t*)), making returns on assets according to (1) unreliable. Thus, using assets as a base to operating profits is not worth the pain that this contentious denominator may provoke in tax controversy. Operating profit margin or operating profit markup are more reliable, and they are pure numbers and thus are not dependent on the different units or the vintages in which *heterogenous assets* are aggregated in accounting practices.

**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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