1. The §482 presumption
The transfer pricing regulations hold two incompatible beliefs about the rate of return on assets. The comparable profits method (CPM) of §1.482-5 admits the rate of return on capital employed as a profit level indicator: operating profit divided by operating assets, where operating assets are measured at book value and include whatever fixed and intangible assets the tested party carries on its balance sheet. One ratio, one denominator, one rate. The residual profit split of §1.482-6(c)(3) holds the opposite belief: routine contributions, which the regulation defines by reference to what uncontrolled parties in similar activities contribute, receive a market return, and the residual profit is allocated to intangible property. The OECD Transfer Pricing Guidelines, Chapter VI, repeat the division under the DEMPE vocabulary: tangible capital earns a routine return, intangibles earn the residual.
In regression form the two beliefs are two restrictions on one equation. Write Y = \mathrm{OIADP}, K = \mathrm{PPENT} and I = intangible capital, and let the profit equation be:
\quad (1)\qquad Y(i) = c + \rho_1 K(i) + \rho_2 I(i) + u(i), \qquad i = 1, \ldots, N
The CPM rate-of-return indicator Y/(K+I) presumes c = 0 and \rho_1 = \rho_2. The residual profit split presumes \rho_2 > \rho_1, with \rho_1 pinned to a “routine” value and \rho_2 left to absorb the remainder. The same regulations treat intangibles as ordinary assets when the method is a ratio and as privileged assets when the method is a split. Neither presumption has been tested in a published transfer pricing report that I know of; both are asserted.
My previous post, Alpha, Partial Beta, and Partial Gamma (EdgarStat blog, November 6, 2026), wrote the rate-of-return equation under the CPM presumption as equation (5), \mathrm{OIADP} = c + \rho\,(\mathrm{PPENT} + \mathrm{INTANO}) + u, with one \rho for both asset classes. This sequel relaxes the restriction \rho_1 = \rho_2 and asks whether the inequality the pundits assert, \rho_2 > \rho_1, is a theorem or an industry-specific empirical finding. The critical point is made before any data are run: a presumption that appears in a regulation with both signs is not a presumption; it is a parameter to be estimated.
2. The nested hypothesis
Equation (5) of the previous post is the restricted model, and the pundits’ claim is a one-sided alternative within the unrestricted model. Stated as a pair:
\begin{aligned}
&(2)\quad
\text{Unrestricted:} && Y = c + \rho_1 K + \rho_2 I + u
\\[8pt]
&(3)\quad
\text{Restricted (5):} && Y = c + \rho\,(K + I) + u, \qquad \rho_1 = \rho_2 = \rho
\end{aligned}The restriction is a single linear constraint and is tested by the F statistic for nested models, or by the ratio of the estimated difference to its standard error:
\quad
(4)\qquad
\frac{\hat\rho_2 - \hat\rho_1}{\left[\operatorname{Var}(\hat\rho_1) + \operatorname{Var}(\hat\rho_2) - 2\operatorname{Cov}(\hat\rho_1,\hat\rho_2)\right]^{1/2}}The covariance term is not optional. K and I are correlated across firms, so the off-diagonal element of the coefficient covariance matrix is of the same order as the diagonal elements, and a report that compares the two estimates with separate confidence intervals understates or overstates the evidence depending on its sign (Weisberg, 2014, ch. 6; Davidson and MacKinnon, 2004, ch. 4).
Three remarks follow from writing the claim this way. First, the claim is testable, so it is prima facie empirical; a proposition that could be refuted by a sample is not a theorem. Second, the test has power only in samples where I varies: a comparables set in which most firms carry no intangible capital identifies \rho_2 from a handful of observations, and the standard error will say so. Third, the intercept c belongs in both models. Dropping it to recover the ratio form imposes a second restriction, c = 0, whose consequences for ratio-based indicators were the subject of the previous post and reappear in section 6.
3. Two meanings of I
The word “intangible” covers two economic objects that the regulations do not distinguish, and the sign of \rho_2 - \rho_1 depends on which one enters the regression.
Acquired intangible capital (INTANO). Compustat INTANO is the net book value of identifiable intangibles recognized in business combinations: patents, trademarks, customer relationships, technology, recorded at calculated “fair value” under the purchase price allocation (ASC 805) and amortized over estimated useful lives. Goodwill is carried in a separate item (GDWL) and is excluded. INTANO is therefore a stock purchased at a market price on a known date.
Self-developed intangible capital (PIM I). The perpetual inventory method cumulates expensed outlays into a stock at cost, the same construction that reproduces PPENT from CAPX:
\quad
(5)\qquad
K(t) = (1 - \delta_1)K(t-1) + \mathrm{CAPX}(t), \qquad I(t) = (1 - \delta_2)I(t-1) + E(t)where E = \mathrm{XRD} + \mathrm{XAD} + software development costs, and \delta_1, \delta_2 are the decay rates of the two stocks. This is the construction of my prior posts on capital measurement. PIM I is a stock no market has priced; it is the cumulated cost of the firm’s own research, advertising and software outlays, less decay.
The distinction is between a stock bought at a price and a stock built at a cost. Both K and PIM I are built at cost, so a regression on K and PIM I compares two stocks on one valuation basis. A regression on K and INTANO compares a cost-based stock with a price-based stock, and part of any difference between \rho_1 and \rho_2 is then a valuation artifact rather than a return differential. Section 4 shows that theory speaks to the two cases in different voices, and section 7 runs them as separate variants.
The methodological rule I apply to the comparables regression is that every regressor be a directly observable Compustat mnemonic; PIM I is constructed from observable flows and a chosen \delta_2. I treat it here as a research variable for testing the presumption, not as a regressor to be put before a tax authority, and section 5 shows why the \delta_2 dependence makes that caution necessary.
4. What theory says, and what it does not
Theory does not order \rho_1 and \rho_2. It constrains one of them in one case and is silent otherwise.
Equalization on capital advanced. The classical proposition, from Smith through Marx to Kalecki’s competitive sector, is that competition tends to equalize the rate of profit on capital advanced across its forms. Capital advanced is measured at cost, which is the PIM basis. On that basis the equilibrium condition is \rho_1 = \rho_2 = \rho, and any persistent gap is a rent accruing to one form of capital. The pundits’ inequality is then the claim that rents attach to self-developed intangible capital and not to installed capacity. My previous post found the reverse for 71 large corporations: the coefficient on PPENT, holding operating cost constant, behaves as a rent on installed capacity, unstable in sign across firms and significant where pricing power is strong. Nothing in theory privileges one form of capital as the residual claimant; oligopoly can locate the rent in either.
Acquired intangibles are priced at \rho. For INTANO theory is more specific. An intangible bought in a business combination is paid for at a price that capitalizes its expected royalty income at the acquirer’s discount rate. At the margin the acquisition has zero net present value, and the expected net return on the recorded asset is \rho: the seller captured the rent in the price. In transfer pricing vocabulary, acquisition converts a non-routine asset into a routine one on the acquirer’s books. Departures from \rho measure mispricing: unpriced synergies raise \rho_2 above \rho, overpayment lowers it, and the overpayment is carried in goodwill, which INTANO excludes. The empirical literature on acquirer returns finds them near zero or negative on average (Andrade, Mitchell and Stafford, 2001; Moeller, Schlingemann and Stulz, 2005). The theoretical prior for \rho_2 on INTANO is therefore \rho, not a premium over \rho_1.
The valuation basis matters for K. Book PPENT is historical cost less accumulated depreciation, and no market mechanism revalues it after installation. The measured \rho_1 is the economic return times the ratio of replacement cost to book value, plus any rent that the regression attributes to K because no other regressor carries it. The rents from self-developed intangibles, when I is absent from the equation, appear in c and in \rho_1. This is the omitted-variable mechanism by which a positive intercept and a high \rho_1 coexist in the restricted model, and it is a measurement result, not a law.
The summary is in two lines. On INTANO, theory predicts \rho_2 \approx \rho and leaves \rho_1 free; the pundits’ inequality is then the claim that tangible capital earns less than the cost of capital, which is not what they intend. On PIM I, theory predicts equality under competition and leaves the location of any rent to the data, industry by industry.
5. Expensing and the depreciation rate
Two measurement facts decide what an estimate of \rho_2 on PIM I means, and both can produce the pundits’ inequality without any rent existing.
Expensing. XRD and XAD are inside XOPR, so OIADP is already net of the whole current outlay E on intangible capital, not net of its decay \delta_2 I. The dependent variable consistent with a capitalized I is
\quad (6)\qquad Y^*(t) = Y(t) + E(t) - \delta_2 I(t-1)
which adds back the expensed outlay and deducts the amortization the PIM implies. Suppose the structural equation holds for Y^*:
\quad (7)\qquad Y^*(t) = c + \rho_1 K(t-1) + \rho_2 I(t-1) + u(t)
From the PIM identity, E = I - (1-\delta_2)I(t-1). For a stock growing at rate g, I(t) = (1+g)I(t-1), so E(t) = (g + \delta_2)I(t-1). Substituting into the definition of Y^*,
\quad (8)\qquad Y(t) = Y^*(t) - E(t) + \delta_2 I(t-1) = c + \rho_1 K(t-1) + (\rho_2 - g)\,I(t-1) + u(t)
A regression of unadjusted OIADP on K and PIM I estimates \rho_2 - g, not \rho_2. Expensing understates the return on self-developed intangible capital by the growth rate of that capital. For a firm whose R&D stock grows at 10 percent a year the understatement is ten points of return. A pundit who runs OIADP on PIM I without the adjustment biases the test against his own claim; one who adjusts the tested party’s Y and not the comparables’ biases it the other way.
The depreciation rate. PIM I is constructed, and \delta_2 enters the regressor and the adjustment alike. A higher \delta_2 shrinks the stock and inflates the estimate of \rho_2; a lower \delta_2 does the reverse. The estimate is a function of \delta_2, and the inequality \rho_2 > \rho_1 can be calculated or erased by the choice of \delta_2 alone. The conventional 15 percent for R&D is a convenience; industry estimates range from about 10 to 40 percent (Li and Hall, 2020). Any reported estimate of \rho_2 must be shown with the \delta_2 that produced it, and a conclusion that survives only at one \delta_2 is not a conclusion.
The same logic applies to \delta_1 for K, with one difference: for K the firm reports DFXA, so \delta_1 = \mathrm{DFXA}/\mathrm{PPENT}(-1) is observable firm by firm, whereas no filer reports the decay of its research stock. The asymmetry is itself an argument for the restricted equation (5) as the default: it needs no \delta_2.
The OIBDP form, Y gross of depreciation and amortization with E added back, estimates \rho_1 + \delta_1 and \rho_2 + \delta_2. Running it beside the OIADP form separates the decay differential from the return differential, so that a finding of \rho_2 > \rho_1 on the net equation is not merely \delta_2 < \delta_1 in disguise.
6. What the ratio does with the inequality
A positive \rho_2 does not raise the combined rate of return of a firm that holds intangible capital. The condition is stricter, and the intercept sets it.
Take two firms with the same K. One holds intangible capital I, the other none. Under the unrestricted equation, their ROA on K + I differ by
\quad
(9)\qquad
\frac{c + \rho_1 K + \rho_2 I}{K + I} - \frac{c + \rho_1 K}{K} = \frac{I\,(\rho_2 - \rho_1 - c/K)}{K + I}The firm with intangible capital shows the higher ROA if and only if
\quad (10)\qquad \rho_2 > \rho_1 + c/K
With c > 0, which the previous post found for about half of 71 large corporations, equality of the two rates lowers the ROA of the firm that carries I. The gap \rho_2 must clear is c/K, the intercept scaled by the firm’s tangible capital, and it is largest for the smallest firms. This is the ratio bias of the harmonic-mean theorem in a new place: the ratio Y/(K+I) equals \rho + c/(K+I) under the restricted model, so cross-sectional dispersion in ROA is dispersion in c/(K+I) even when \rho is one number for every firm.
Two consequences for practice. A comparables screen on ROA selects against acquisitive firms and against heavy R&D spenders whenever \rho_2 < \rho_1 + c/K, a composition effect that no comparability adjustment addresses. And a tested party whose ROA falls below the interquartile range of its comparables may do so because it carries more I at the same \rho, not because its profit is below arm’s length. The regression separates the two cases; the ratio cannot.
7. Test design
The test is run by industry on pooled annual observations, n firms over T years, N = nT, with lagged stocks:
\quad (11)\qquad Y(i,t) = c + \rho_1 K(i,t-1) + \rho_2 I(i,t-1) + u(i,t)
Three variants of I are estimated on the same sample:
| Variant | Regressors | What \rho_2 measures |
|---|---|---|
| A | K, INTANO | Net return on acquired intangibles priced at fair value |
| B | K, PIM I (Y^* as dependent variable) | Net return on self-developed intangible capital at cost, conditional on \delta_2 |
| C | K, PIM I, INTANO | Both, with the rent on self-developed capital separated from the priced return on acquired capital |
Reporting conventions follow EdgarStat practice: HC3 standard errors, coefficients reported as estimate \pm one standard error (68 percent interval), no p-values. The quantity of interest is the estimated difference \rho_2 - \rho_1 with its standard error, the covariance term included, and the F statistic for the restriction \rho_1 = \rho_2. For variant B the whole table is repeated over a grid \delta_2 = 0.10, 0.15, 0.20, 0.30, with \delta_1 taken from \mathrm{DFXA}/\mathrm{PPENT}(-1), and the \delta_2 at which the sign of the difference reverses, if any, is marked. The OIBDP form is run beside the OIADP form for the reason given in section 5.
Data limits to state in the report rather than discover under cross-examination:
- \rho_2 in variant A is identified only from firms with \mathrm{INTANO} > 0; report that count beside n.
- INTANO jumps at acquisition dates; the lag keeps the jump out of the contemporaneous profit.
- XAD is missing for most Compustat filers, so advertising enters E only where reported, and the estimate is best read as a return on research and software capital.
- Capitalized software already sits partly in INTANO; in variant C it must not enter both regressors.
- Firms with persistent negative OIADP are excluded before estimation, not after, on the going-concern grounds set out in my zombie-comparables memo.
The expected pattern, stated before the data are run so that it can be wrong: in capital-intensive oligopolies with pricing power, \rho_1 > \rho_2 in variants A and B; in pharmaceuticals, \rho_2 on INTANO near or below \rho_1 because acquired product rights are amortized over short lives; in software, \rho_2 on PIM I above \rho_1 at low \delta_2 and below it at high \delta_2. A pattern that varies by industry and by \delta_2 is the empirical answer to the question in the title.
8. Implications for transfer pricing
The residual is an assumption about \delta_2. The residual profit split assigns the residual to intangibles by fiat. Once intangible capital has a denominator, the residual is \rho_2 I, an estimate conditional on the depreciation rate used to build I. The split between routine and residual profit is then a depreciation-rate assumption presented as a finding, and a tax authority that disputes the split need only dispute \delta_2.
Acquired and self-developed intangibles are different objects under one legal word. The first earns \rho because it was bought at \rho; the second may earn a rent because no market priced it. A tested party that holds acquired IP, whether through a third-party acquisition or an intra-group transfer at a price supported by a valuation, cannot claim the return theory reserves for the developer. The arm’s-length return on INTANO is the discount rate the acquirer itself documented in the purchase price allocation, and the DEMPE residual is not owed to the holder of I by virtue of holding it.
Equation (5) is the default, not the simplification. If \rho_1 \approx \rho_2 survives the \delta_2 grid in an industry, a single \rho on K + I is the parameter, the restricted model is the efficient one, and the ratio still carries c/(K+I). If \rho_1 > \rho_2, the pundits’ doctrine inverts for that industry: rents accrue to installed capacity, and the “routine” tangible return was never routine. Only if \rho_2 > \rho_1 + c/K at every plausible \delta_2 does the doctrine hold, and then as a finding for one industry, not a presumption for all.
The ROA screen has a composition bias. Section 6 shows that firms carrying intangible capital show lower ROA than organic peers unless \rho_2 clears \rho_1 + c/K. A comparables set selected on ROA, or an interquartile range built from one, encodes that bias. The remedy is the one proposed throughout this series: estimate the regression with its intercept, report the coefficients with their standard errors, and let the restriction \rho_1 = \rho_2 be tested rather than presumed in either direction.
What the regulation should have said. The 1992 regulations introduced operating profit as the dependent variable and the arm’s-length range as an interval estimate. They stopped short of stating the profit equation whose parameters those intervals estimate. Had they written Y = c + \rho_1 K + \rho_2 I + u, the CPM and the residual profit split would have been recognized as two restrictions on one equation, and the question of this post would have been settled by estimation thirty years ago.
References
Andrade, G., Mitchell, M., and Stafford, E. (2001). New evidence and perspectives on mergers. Journal of Economic Perspectives, 15(2), 103–120.
Davidson, R., and MacKinnon, J. G. (2004). Econometric Theory and Methods. Oxford University Press.
Li, W. C. Y., and Hall, B. H. (2020). Depreciation of business R&D capital. Review of Income and Wealth, 66(1), 161–180.
Moeller, S. B., Schlingemann, F. P., and Stulz, R. M. (2005). Wealth destruction on a massive scale? A study of acquiring-firm returns in the recent merger wave. Journal of Finance, 60(2), 757–782.
OECD (2022). Transfer Pricing Guidelines for Multinational Enterprises and Tax Administrations, Chapter VI. Paris: OECD Publishing.
Silva, E. (2026). U.S. Corporate Profits, 1950–2024. Springer.
Silva, E. (2026). Alpha, partial beta, and partial gamma. EdgarStat blog, November 6, 2026. https://edgarstat.com/blog/alpha-partial-beta-and-partial-gamma/
Treas. Reg. §1.482-5 (comparable profits method); §1.482-6 (profit split method).
Weisberg, S. (2014). Applied Linear Regression (4th ed.). Wiley.