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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…

1. A physician reads the bank’s balance sheet This note is a digression from transfer pricing. I spent a morning with Clément Juglar’s Des crises commerciales et de leur retour périodique en France, en Angleterre et aux États-Unis, in the…

1. The equation The Springer book U.S. Corporate Profits, 1950–2024 estimates, for each of 71 corporations, a bivariate relation between total revenue (REVT) and total operating expense before depreciation (XOPR = COGS + XSGA). The expanded specification adds net property,…

Abstract An inverted U is not established by naming a scatterplot or finding a significant squared term. The shape, the location of its turning point, and the economics behind the relationship all require scrutiny. A scatterplot is evidence, not an…

Summary Anthropic recorded a $94.3 billion loss on its convertible notes in the first nine months of 2026, 2.8 times its revenue, because its shares became more valuable.[1][2] Ednaldo Silva, writing on LinkedIn, also pointed out that the loss on the notes was…

1. The 1989 argument and the rule In 1989, in my second year at the IRS in Manhattan, I wrote a research paper on Kitchin’s inventory cycles. Its thesis was that year-by-year transfer pricing audits are indefensible. Operating profits (then…

Summary The inventory cycle is the recurring alternation between periods when the growth rate of businesses’ stocks of goods rises and periods when it falls. The phase of the cycle affects comparables’ operating margins through inventory accounting. For instance, a…

Introduction The inventory-based theory of price stickiness which was pioneered by Alan Blinder in 1982 proposes that firms use stock levels as a primary buffer against demand fluctuations, choosing to let inventory accumulate or deplete rather than continuously adjusting prices.…

1. Introduction The AI buildout expands capital expenditure and contracts employment at the same time. Commentary treats the two as separate stories, one about investment and the other about labor markets. In a Leontief–Solow price system they are one event:…

1. Why not industry averages The 1968 regulations restricted the use of industry statistics under the resale price and cost plus methods to situations in which data on comparable uncontrolled transactions could not be found (Reg. §§ 1.482-2(e)(3)(vii) and 1.482-2(e)(4)(iv)). The…

1. Purpose This memorandum assembles the documentary support for Ednaldo Silva’s contributions to the Section 482 transfer pricing regulations proposed in 1992, issued in temporary form in 1993, and finalized in 1994. Its principal exhibit is an Internal Revenue Service…

1. What the regulation says Treas. Reg. § 1.482-1(f)(2)(iii) prescribes no count of years. Paragraph (A) admits data for “one or more years before or after the year under review.” Paragraph (B) ties the extent of multiple year data to the…

1. Purpose Kalecki’s single-product price equation, , has a uniform markup over unit prime cost . The uniform case is amenable to one regression across many products. Non-uniform markups , one per firm, are handled in my book U.S. Corporate…

Summary Before Accounting Standards Update (ASU) 2014-15, U.S. generally accepted accounting principles established by the Financial Accounting Standards Board (FASB GAAP) did not contain any guidance on management’s responsibility to evaluate whether there is substantial doubt about an entity’s ability…

EdgarStat excludes zombie firms from the comparables used to benchmark a tested party’s operating profits. A zombie is a mature firm whose operating profits fail to cover its interest expense for consecutive years; it survives on creditor forbearance, not on…

ASC 842 and the depreciation of owned property make asset-based comparables under § 482 unreliable: a history, a one-store example, and data on 1,324 registrants.

1. Corporate Tax Planning A common transfer pricing plan assigns a fixed role to each member of a multinational group. A contract manufacturer, a limited risk distributor, or a retailer is labeled a “routine” entity and is promised a low…

Introduction Consider the uniform distribution of a random variable Y with support from 0 to b. Its mean is 0.5b yet the mean is just a moment of the distribution –  it’s a measure its central tendency. Yet, to get…

This is the first post in a series on reported research and development (R&D) as evidence in a transfer pricing functional analysis. In this post, we describe the components of the various R&D amounts. Three definitions determine the reported R&D…

1. The issue: a ratio is a restricted model Financial ratios are the lingua franca of transfer pricing and credit analysis. Profit indicators such as and coverage indicators such as , leverage covenants expressed as net debt/EBITDA, and distress scores…

1. The IQR does not shrink with sample size (n) For a normal population (Abramowitz and Stegun 1964, ch. 26; Hoaglin, Mosteller and Tukey 1983), and . Therefore: The half-width is the classical “probable error” (der wahrscheinliche Fehler), which the…

Tariffs imposed under the International Emergency Economic Powers Act (IEEPA) were collected on goods imported into the United States from February 2025 until February 2026. On February 20, 2026, the Supreme Court held that IEEPA did not authorize the tariffs.…

1. The dispute A recurring objection from some practitioners holds that a corporate transfer pricing audit must proceed transaction-by-transaction: one adjustment for tangible goods, another for royalties, a third for services, each priced on its own. The objection sounds rigorous,…

In U.S. Corporate Profits, 1950–2024 (Springer, 2026), I estimated the operating profit markup of 71 leading U.S. corporations with a bivariate regression, . The slope is the markup on operating costs, and the operating margin follows from it by indirect…

1. The question you will be asked A reviewer can ask why we run a regression when every other consulting firm reports the median and the interquartile range of a profit level indicator (PLI). The PLI (or ratio statistics) answers…

1. The IRS set carries everything The Commissioner’s expert tested six concentrate manufacturers, the Supply Points (tested parties), in Brazil, Chile, Costa Rica, Ireland, Mexico and Swaziland. A seventh, in Egypt, was adjusted in the taxpayer’s favor and drops out…

1. Two objects, one name A reader noted that ratio is among the oldest constructions in mathematics, and takes my article to argue that ratios are invalid or biased. My article makes no such claim. My issue focus on the…

Abstract Let for and let denote the ratio indicator. We show that the arithmetic mean of overstates the slope by exactly , where is the harmonic mean of ; that the -th quantile of overstates by exactly , where is…

Adjusted taxable income (ATI) is an IRC §163(j) calculation on the federal corporate return that is filer-specific and not reported in public filings. The equation below, derived using Compustat data fields, is the closest approximation to ATI. The sections that…

Thin capitalization rules ask a single question in two different ways. The older way looks at the balance sheet and asks whether the borrower carries more debt than its equity supports. The newer way looks at the income statement and…

“A trillion here, a trillion there, pretty soon you’re talking real money.” — with apologies to Everett Dirksen On August 17, 2026, the Wall Street Journal reported that nine large technology companies had approximately $3 trillion of off-balance-sheet commitments, most…

For applied transfer-pricing work, I prefer the conventional econometric formulation because every step is visible, reproducible, and open to challenge. Regulatory support for using regression analysis is found in Section 1.482-1(e)(2)(iii)(B), which says that reliability may be increased through application…

Introduction A prior post on this blog recommended HC3 standard errors for the reduced-form markup equation estimated on transfer pricing benchmark sets. It stated the estimator and left two questions unanswered, both of the kind that surface under examination rather…

Introduction Another post traced the ownership of transfer pricing data from Disclosure, Inc. in Bethesda through Primark and Thomson to Ryan in Dallas. That was a history of corporations. This one is a history of a practice, and it is…

A transfer pricing comparables set is small. Thirty independent companies is a generous count; eight to fifteen comparables are common. Each contributes three fiscal years, so the pooled regression runs on fewer than 45 annual observations and often on fewer…

Ordinary least squares (OLS) produce two things: an estimate of the intercept and slope coefficients, and an estimate of their uncertainties. The coefficient estimates are sound, based on matrix algebra inversion. The standard errors rest on an assumption that is…

1. Reliable PLI The interquartile range is not an estimator. A benchmark set of n comparables produces a distribution of ratios : operating margin, markup, return on assets. If the underlying relation is , then and the ratio carries a…

1. The Apparent Contradiction Two positions I have defended appear, at first sight, to pull in opposite directions. First: corporate profits and asset bases are understated because research and development (XRD), advertising (XAD), and software development outlays are expensed as…

1. The System in Three Equations Start from the GDP expenditure identity, collect autonomous expenditure as and decompose it into its mean and its fluctuation, . Close the identity with two behavioral equations, both linear in lagged income: consumption with…

Introduction This article shows that the reported profit level indicator OMAD used in transfer pricing is always lower when intangible-producing expenses such as R&D are expensed rather than capitalized and the firm is growing over time while adopting a constant…

Introduction This article shows that reported profit level indicators (PLIs) used in transfer pricing are distorted when intangible-producing expenses such as R&D are expensed rather than capitalized. For margin-based PLIs the distortion falls on the numerator alone; for asset-based PLIs…

1. The Complaint A commentator objected that stating unit price is proportional — or log-proportional — to unit labor cost (ULC) confuses correlation with causation. The objection deserves a serious answer, because the aphorism it invokes is usually deployed as…

1. The Apparent Contradiction Two positions I have defended appear, at first sight, to pull in opposite directions. First: corporate profits and asset bases are understated because research and development (XRD), advertising (XAD), and software development outlays are expensed as…

1. Two Ratios, One Regime Change Plot two ratios from FRED over 1950-2025. The first is private nonresidential equipment investment as a share of GDP: it has oscillated within a narrow band, roughly 5 to 7 percent, for four decades,…

Discussions of crude oil’s forward curve start the same way — plot , call it contango or backwardation. Contango is , and backwardation is the reverse . This ratio construction is a mistake, and it is the same PLI (profit…

Separate-entity state tax base exposure to international transfer pricing if federal enforcement priorities shift from § 482 to attribution, and why many states already possess the statutory authority to respond without new legislation. “Plurality must never be posited without necessity.”…

Purpose This note derives the combined profit split method when each member of a controlled group has its own profit-per-payroll coefficient. The purpose is to show that variable entity-level coefficients do not defeat a simple payroll-based profit split. They collapse…

1. Introduction The combined profit split method requires an allocation key, and the choice of key determines the reliability of the entire exercise. We propose relative payroll as the allocation key for splitting the combined operating profits of controlled enterprises…

1.  Introduction In transfer-pricing analysis and corporate-profit studies, the objective of regression analysis is estimating a meaningful slope coefficient rather than predicting the dependent variable. This distinction has implications for how rival operating profit specifications are selected and evaluated. A…

A controlled party’s reported operating profit is the residual of the intercompany prices under audit examination. Where transfer prices have been set to move income across borders, the reported operating profits are distorted. I’ve developed an internal benchmark for each…

Introduction In certain cases, the data generation process for operating profit (Y) given Net Sales (X) for a company is given by the following equation: This could be the case, for example, when the data is rightly skewed. In this…

1. Introduction A Form 8-K is the filing companies use when an event does not fit the timetable of the next scheduled report. A 10-Q or 10-K follows a reporting cycle. An 8-K follows a trigger. The trigger can be…

Two regressions are implicit in transfer pricing benchmarking: the linear regression of operating profits on revenue, and the double-log regression of the same two variables. Each produces a slope coefficient. The two coefficients have different meanings, different units, and answer…

Ednaldo Silva’s blog post “Measurements and Uncertainties in Transfer Pricing” shows that uncertainty propagates through mathematical operations. This principle is directly relevant for econometric models in which variables are transformed, differenced, and estimated over time. Starting from error propagation, we…

The contrived linear function , with the zero intercept specified in the U.S. and OECD transfer pricing guidelines, is a special case of a much more general structural form: the power function, Two questions arise the moment one fits (1)…

The term prewhitening sounds arcane and likely worse. The word belongs to a long-standing optical metaphor in spectral analysis that predates its econometric usage. This note sets out the etymology, the formal statistical content, and the specific procedure introduced by…

1. Preliminaries The geometric distributed-lag specification for revenue is with Rt revenue, Xt advertising expense, α the intercept (base revenue), γ the impact multiplier, λ the carryover rate, and ut a stochastic disturbance. Two assumptions deserve naming before any algebra.…

Based on Ednaldo Silva’s LinkedIn post “Debt-Driven GDP Growth”, we examine the relationship between government debt and GDP growth. The analysis starts from a simple GDP identity, derives a structural equation for government spending, and transforms it into an empirical…

Inventory markdown disclosure in U.S. public filings runs on two regulatory tracks. GAAP under ASC 330 sets a narrow trigger that applies only to a defined subset of markdown activity. The SEC overlay under Regulation S-K is where the substantive…

1. Introduction Transfer pricing benchmarking under the TNMM (or CPM in the U.S.) depends on a credible comparables set. The standard workflow is familiar: screen a database by SIC code or industry keyword, apply financial filters, and document why each…

1. The Geometric Distributed-Lag Model Like research and development, advertising is a risk-taking investment. The economic rationale is straightforward: a campaign launched in period t builds brand awareness and consumer loyalty that generate revenue and operating profit in the current…

This post builds on my previous one, “Impairment and Amortization: Where Transfer Pricing Analysis Meets GAAP Income Statement Classification,” assuming the Comparable Profits Method (CPM) is chosen under 1.482-1(c) and focusing on building the comparable set. The purpose of this…

1. Introduction Transfer pricing analysis for intangible-intensive corporations requires structural econometric models, not ratio-based profit level indicators (PLIs). The standard operating margin and the Berry ratio are quotients of accounting aggregates. When the underlying structural relationship has a nonzero intercept…

Laying the Groundwork This post assumes the Comparable Profits Method (CPM) is the best method under 1.482-1(c), with operating margin after depreciation (OMAD) as the profit indicator (PLI). The key question addressed in this post is whether the operating income…

Swanson-Baumol Prior Theoretical Work The Swanson-Baumol 2005 work is a seminal paper in antitrust and patent law. Swanson and Baumol argue that once a technology is adopted into a standard, the patent holder gains immense market power. This is not…

Compared with the Additive and Multiplicative Rivals

The Harmonic Mean Bias in Ratio-Based Profit Level Indicators

Why Induced Heteroscedasticity Compounds the Intercept Bias

This article shows that ratio-based profit level indicators (PLIs)—including return on assets (ROA), Berry ratio, and operating margin—are biased estimators of the structural profit rate when the underlying relationship between...

Abstract The perpetual inventory method (PIM) for estimating capital stock (PPENT) is attributed to Raymond Goldsmith (1951). Goldsmith provided conceptual descriptions without explicit algebraic derivations—a ceremonial citation. However, the extended PIM formula with geometrically declining weights is the standard solution…

1. Introduction Transfer pricing practice relies on ratio analysis. Operating profit divided by revenue, operating profit divided by total cost, operating profit divided by operating assets—these profit level indicators (PLIs) are ratios of the form m(i) = Y(i) / X(i).…

Introduction Extending algebraic developments from Ednaldo Silva, I formalize a truncation of the perpetual‑inventory method (PIM) equation with explicit and tractable error bounds. This treatment of finite‑sample PIM truncation does not appear in standard asset‑valuation cookbooks. Framework Consider the case…

A complete statistical procedure for transfer pricing practitioners. Transfer pricing practice usually determines the arm’s length range using the interquartile range of the chosen profit indicator, defined as a ratio m(i) = Y(i)/X(i) across comparable companies.

Return on Assets and the Harmonic Mean

On May 23, 2017, the OECD released a draft of Action 8 of the Base Erosion and Profit Shifting (BEPS) plan, providing taxpayers and tax administrations with a new transfer pricing approach to measure hard-to-value-intangibles (HTVI):

Transfer pricing industry practice traditionally employs descriptive ratio analysis under 26 CFR 1.482-5(b)(4)(ii) within the Comparable Profits Method (CPM) to determine an arm’s length range.

In this part of the article, I follow closely the exposition of the valuation formulas in the Financial Theory and Corporate Policy textbook by Copeland, Weston and Shastri.

Interest stripping is a tax technique where a company uses high-interest, intercompany debt to shift profits from high-tax jurisdictions to low-tax jurisdictions, by creating significant deductible expenses for the borrowing entity of the MNE.

From Equation Counting to Class Conflict: A Mathematical Reading of Political Economy

Introduction When applying the Comparable Profits Methods (CPM), Return on Assets (ROA) is often used as a Profit Level Indicator (PLI) to establish, in principle, the arm’s length price for a related party transaction. In general, this PLI is given…

Analysts often work with ratios. Whether examining price-to-earnings ratios in finance or profit indicators in transfer pricing, the ratio m(i) = Y(i) / X(i) is often reported. However, this simple ratio masks a systematic bias that affects quartile estimates and other location parameters.

Controlling producer price inflation (PPI) doesn’t require austerity.

Some basic rules of differential calculus are useful for understanding economic discourse, such as discussions in business or financial media about U.S. economic performance.