Incomplete Transfer Pricing Reg Needs an Update

The economic substance regulation mandates all risks to be clearly allocated, preferably in a written agreement, and priced at arm’s length on an ex ante basis.¹ Once a deal is struck, there is no revisiting it on an ex post basis. Ex ante in this context means “on the first day of the fiscal year”; ex post means “on its last day.”² This concept applies to all controlled transactions within the meaning of reg. section 1.482-1(i).

A transaction with economic substance is priced by the selection of a specified or unspecified method. The “best method” requirement of reg. section 1.482-1(c) provides that the relative reliability of methods must be evaluated. The reliability of assumptions and data are important considerations in such evaluation.

With the exception of the income method of reg. section 1.482-7(g)(4), all methods specified in the transfer pricing regulation of reg. section 1.482 are comparable methods. All comparable methods are deemed to satisfy the realistic alternative principle (RAP) of reg. section 1.482-1(f)(2)(ii)(A). The income method is the result of an application of the RAP, and therefore satisfies the RAP. That regulation provides some guidance on the selection of a discount rate for an application of the discounted cash flow (DCF) method.

A taxpayer relying on an unspecified method must use a method that satisfies the RAP. The RAP requires the equalization of net present values (NPV), properly risk-adjusted.

The organization of the methods specified in the regulation is property-based. Methods for tangible property are grouped together, and so are methods for intangible property and services. We do not have a section focused on financial transactions.

The purpose of this article is to explain that a significant subset of controlled transactions falling under the scope of the economic substance regulation are left unpriced because the transfer pricing regulation is incomplete.

Grouping transactions by the nature of the property transacted is not the most productive method as far as the valuation of the transactions is concerned. This observation is very important. The statement that the regulation is incomplete is not to be construed as saying, “It would be nice to have more methods specified in regulation, or methods that address items of property not yet addressed (for example, financial property).” It certainly would be nice to have that, but that is not my point.

Instead, my statement should be construed as saying, “The regulation is asking taxpayers and the IRS economists to do something that, no matter how compliant a taxpayer wants to be, or how diligent an IRS economist wants to be, he or she cannot accomplish based on current guidance.” Completing the regulation means addressing that serious issue.

Reg. section 1.482 is incomplete. It fails to provide any guidance necessary to correctly value that subset of controlled transactions involving nonlinear payments. Yet the economic substance regulation does not provide an exception to the requirement that all risks in a transaction be allocated and priced on an ex ante basis. There is no exception provided for nonlinear payments.

The Reg’s Incompleteness

All controlled transactions cause a payment function to exist. The transfer pricing valuation exercise is the valuation of that function.

That payment function takes as argument one or more variables and returns the value of the payment. Below are four examples of controlled payment functions:

• P_t = 0.5Q_t — [1] Linear (tangible property).

• R_t = 0.02S_t — [2] Linear (plain-vanilla).

• Y_t = I* - I_t — [3] Affine-linear (year-end adjustment to a point, any item of property).

• R_t = max(0.04S_t - $20 million; 0)S_t — [4] Nonlinear (IP license with “floor”).

Equation [1] is the payment function for buying Q_t widgets at the per unit price p = $0.5. The function P_t is linear in Q_t. Equation [2] is the royalty payment function for a plain-vanilla license with royalty rate α = 2 percent levied on all licensee’s sales S_t. The function R_t is linear in S_t. Equation [3] is an ex post “business restructuring” payment function bringing the year-end operating income I_t of a tested party to I^* (a preset value). The function Y_t is affine-linear in I_t. Finally, equation [4] is the royalty payment function for a complex license with royalty rate α = 4 percent, but also with a floor excusing the first $20 million of royalty payments, irrespective of the value of licensee’s sales S_t.

All controlled payment functions are either linear, affine-linear, or nonlinear. These are mutually exclusive; a payment function cannot be at the same time linear and affine-linear, linear and nonlinear, or affine-linear and nonlinear. It falls in one and only one category. Every payment function must belong to one of these three categories.

Figure 1 graphs the royalty payment (payment function) in a plain-vanilla license, such as the one in equation [2], and in a license with a floor, a ceiling, and three royalty rates applicable to different tranches of sales (complex license).

The complex license yields, obviously, a seriously nonlinear payment function; the plain-vanilla license’s payment function goes through the origin and has the same slope everywhere, which makes it linear.

Ask yourself the following question: “How would you set the three royalty rates in the complex license, knowing the royalty rate in the plain-vanilla one, such that the NPV of these two licenses are equal, and knowing that, in the complex license, the royalty rate on the middle tranche of sales is zero, and two of the three royalty rates that are neither the floor nor the ceiling must be non-zero?”

And by the way, you cannot discount the payments in the complex license into an NPV; they are nonlinear in licensee’s sales. See Figure 1.

The nonlinearity of the royalty payment in the complex license concerns intangible property. It is the design of the license that causes these nonlinearities, not the nature of the property.

As indicated, the set of all controlled payment functions can be divided into subsets that are mutually exclusive. There are three of these.

Each of these subsets contains a number of elements that correspond to different ways to structure intercompany transactions. Thus, when structured one way, a payment concerning the same item of property could yield an affine-linear payment (for example, year-end adjustments to a point using equation [3]) and a nonlinear one when structured another way (for example, a year-end adjustment to a range). See the table.

Figure 2 illustrates the classification of controlled transactions as disjoint subsets of a set, according to the analytical form of the payment function they cause.

Linear functions go through the origin and have constant slope everywhere. Affine-linear functions have constant slope everywhere but do not go through the origin. Nonlinear functions do not have constant slope everywhere.

Figure 2 reflects the classification of controlled transactions relevant to the selection of the technically correct valuation method. It is definitely not property-based.

It is well known that nonlinear payments cannot be discounted into an NPV. The proportionality between the argument of the function and its value is broken by its nonlinearity. Affine-linear functions can be discounted, but with a very specific adjustment to the discount rate of the argument controlled by an equation. The affine displacement of the function preserves proportionality between the argument of the function and its value, but it requires making an adjustment to the discount rate of the argument to account for that affine displacement of the function. See Figure 3.

In other words, if y = bx (linear), you can discount y at the discount rate used to discount x. If y = a + bx (affine-linear), you can discount y at the discount rate used to discount x adjusted to account for the affine displacement caused by a. If y = 0 for and y = bx for (nonlinear), you have lost your ability to di-scount y at the discount rate used to discount x and there is no adjustment to that discount rate to help you out. You have lost the ability to calculate an NPV by discounting payments.

For example, a total cost function is often written C_t = Ψ_t + ψS_t. In that equation, the intercept Ψ_t is called fixed costs (in U.S. dollars), as it does not co-vary with sales. The term ψS_t is called the variable costs and 1 - ψ the contribution margin. In the presence of fixed costs, total costs cannot be discounted at the discount rate for S_t without an adjustment to that rate to account for the affine displacement of the variable costs caused by the presence of fixed costs. The discount rate for C_t is strictly lower than the discount rate for S_t, which makes the discount rate for S_t - C_t strictly greater. That is the economics of the income method of reg. section 1.482-7(g)(4) with intangible development costs (IDC) Ψ_t. The only technical challenge of that regulation is the calculation of the correct adjustment to the discount rate when going from the licensing alternative (no fixed costs IDC) to the cost-sharing alternative (fixed costs IDC). This adjustment is controlled by a specific equation. In other words, affine-displacement is a mathematical issue between the argument of a function and its value; it has nothing to do with functional analysis or finding different comparables to benchmark discount rates.

As a legal matter, the transfer pricing regulation compels taxpayers and IRS economists to turn nonlinear payments into an NPV satisfying the economic substance regulation (compelling ex ante pricing) and the RAP (compelling an NPV calculation).

If you cannot calculate the NPV of nonlinear payments because you do not have a discount rate, how are you supposed to meet your legal obligation?

This issue is neither new nor specific to transfer pricing. Economists and mathematicians have faced it for centuries in the context of fairly pricing options written on equity. Options’ payoffs are nonlinear as surely as certain controlled payments are.

Ancient societies traded options. The earliest record of options trading dates to 332 B.C., when Aristotle reported that Greek mathematician and astronomer Thales of Miletus, who lived from about 624 B.C.-548 B.C., bought the rights to buy olives prior to a harvest, reaping a fortune in the process. The 1634-1637 Tulipmania episode in Holland involved significant options trading. The first serious attempt at solving the technical valuation issues caused by nonlinearity in payoffs dates to the PhD dissertation of French mathematician Louis Bachelier. That was in 1900.

A final answer was discovered by a number of U.S. mathematicians and economists, including Paul Samuelson (1965), who leveraged advances that had been made in the first half of the 20th century in stochastic calculus to correct an error made by Bachelier in his dissertation, and Robert Merton (1969, 1973), who is often credited alongside Fischer Black and Myron Scholes. The fair pricing of options issue was then settled in 1973 with the Black and Scholes European option pricing equation.

My point here is that we do not have the excuse that the problem is somehow new or unsolved or “too hard for transfer pricing people.”

The Black and Scholes (1973) equation calculates the NPV of the nonlinear payoff function of a European option (put or call) without discounting.

A Striking Analogy

Borrowing an idea from finance, what if we could perfectly replicate controlled payment functions by judiciously assembling options into a synthetic? Replication implies that you are indifferent between entering into the controlled transaction involving nonlinear payments or holding the replicating synthetic involving the exact same nonlinear payments. The approach sounds very much consistent with the idea behind the RAP. And we do know how to price the synthetic.

The table presents six of the most common controlled transactions yielding either an affine-linear or a nonlinear payment function. The combination of contingent claims (options) listed in the table perfectly replicates the affine-linear or nonlinear payments of the controlled transactions they are associated with.

Note that both a business restructuring transaction with ex post adjustment to a point (for example, 1.5 percent operating margin) and the income method of reg. section 1.482-7(g)(4) yield affine-linear payments that can be discounted by DCF, or can be priced without discounting, with the contingent claims listed in the table. In both cases, an adjustment to the discount rate of the argument of these functions is necessary if DCF is used.

The table identifies the specific contingent claims (options) that perfectly replicate the payment function caused by the controlled transactions they are associated with in the table. These can be used to measure nonlinear controlled payment functions without discount rates.

This pricing method can be called the contingent claim method, as it is called in finance.

The analogy between options’ payoff functions and controlled payment functions is striking. It is obvious when we graph them side-by-side and even more obvious when we write down the equations that control their values.

Replication of Payment Functions by Contingent Claims

Transaction	Payment Function	Contingent Claims Replication	Tested Party Position Replication	Counterparty Position Replication
Year-end tested party adjustments to LQ, UQ	Nonlinear	- One Put (K = LQ) - One Call (K = UQ)	- Long Put (K = LQ) - Short Call (K = UQ)	- Short Put (K = LQ) - Long Call (K = UQ)
Business restructuring to a point Q	Affine-linear	- One Put (K = Q) - One Call (K = Q)	- Long Put (K = Q) - Short Call (K = Q)	- Short Put (K = Q) - Long Call (K = Q)
License with floor	Nonlinear	One Call (K = )	Short Call (K = )	Long Call (K = )
License with ceiling	Nonlinear	One Call (K = )	Long Call (K = )	Short Call (K = )
License with royalty steps to for i>2	Nonlinear	- One Call (K = ) - One Call (K = )	- Short Call (K = ) - Long Call (K = )	- Long Call (K = ) - Short Call (K = )
The income method (platform contribution transaction)	Affine-linear	- One Put (K = IDC) - One Call (K = IDC)	- Short Put (K = IDC) - Long Call (K = IDC)	- Long Put (K = IDC) - Short Call (K = IDC)

For example, the nonlinear payoff function of a European call option striking at K and written on underlying E_t is:

The royalty payment in a license with a floor set at (third row of the table), like the one I discussed earlier, is:

Set E_t = βS_t and K = β and you cannot distinguish the equations describing the payoff of the option and the payment of the royalty. The problem of pricing one is the exact same problem as pricing the other. Their graphs are exactly the same. In other words, no matter what value S_t takes, the option pays off exactly the royalty due under the license, C_t = R_t for all S_t.

Two assets that yield the exact same payoff in all possible market conditions (S_t) must have the same price.

If you can calculate the NPV of the option, then clearly you can use that measure as the NPV of the royalty payments and as a measure of an arm’s-length result.

You can calculate the NPV of the option, courtesy of Black and Scholes. Therefore, you can calculate the NPV of the royalty payments. No discounting is involved in any of that.

Nonlinear Controlled Payments

An obvious question is that of the severity of the problem caused by a regulation that does not address a significant subset of controlled transactions. Functional analysis will not help you price a nonlinear payment, nor will any of the regulatory guidance.

This observation raises the question: How often do we encounter nonlinear controlled payments in transfer pricing?

I would argue that we encounter them all the time.

The table listed six controlled transactions that are probably the most common.

I could have added to that list: (1) any contingent clawback provision (inserted into cost-sharing arrangements, for example, after the tax court’s decision in Altera) and reverse clawback provisions (inserted into cost-sharing arrangements, for example, after the Ninth Circuit’s Altera decision and before the U.S. Supreme Court declined to hear the case);³ (2) financial and performance guarantees; (3) rights of first refusal; and (4) rights to early termination of a commitment. And the list goes on.

Any contractual provision that is at the same time contingent and derivative is likely to yield a nonlinear payment. With luck, they are affine-linear as in the income method. IDCs in the cost-sharing alternative cause an affine displacement of variable costs present in the licensing alternative.

As to the severity of the issue, most taxpayers administer some of their controlled transactions with year-end adjustments when the profit-level indicator of the tested party (TP) falls outside an arm’s-length range. Figure 4 shows the graph of the payment function caused by these ex post adjustments, from the point of view of the TP. A positive payment is a payment received; a negative one is a payment made. The nonlinearity of these payments is obvious.

These year-end adjustments are typically left unpriced in the ex ante written agreement.

Yet taxpayers have the realistic alternative of not performing ex post adjustments, and instead expose themselves to reg. section 1.482-1(e)(3). The fact that a taxpayer does not want to be exposed to that section by way of ex post adjustments, which is a realistic alternative as well, does not excuse compliance with the economic substance regulation and the RAP. That is the price to pay to avoid reg. section 1.482-1(e)(3).

However, what does excuse a lack of compliance is that tax folks are not experts at derivative pricing; neither are transfer pricing advisers nor IRS economists. It is certainly unreasonable to ask taxpayers, advisers, or IRS economists to go back to graduate school to learn derivative pricing so they can satisfy a transfer pricing regulation that is utterly silent about a rather important technical valuation issue that, until 1973, left actuaries to use DCF to price options despite everyone’s knowing that these prices were not fair prices. A little guidance, in that respect, would go a long way and serve the transfer pricing community well.

Note that once you know which contingent claims to assemble, and in what position (long or short), applying the method is straightforward. Anyone at a tax department, the IRS, or advisory firms knows how to plug numbers into an equation and can understand the approach.

In fact, the only challenge of the method is to come up with the combination of contingent claims (synthetic) that perfectly replicates the payment function in any given controlled transaction. The table does just that. Any reliable Black and Scholes online calculator can then be used to get the options’ premia. Options’ premia are linearly additive. A long position carries a positive sign (add) and a short position carries a negative sign (subtract). Asking people to add and subtract numbers that are easy and cheap to obtain (options’ premia) is reasonable, but these people need to know what numbers they are supposed to add and subtract. That requires guidance.

Providing such guidance and fair notice of how the regulation is supposed to be applied in specific cases is the purpose of the regulation.

Conclusion

The contingent claim method I discussed in this article is easy to apply once the correct contingent claims that perfectly replicate a nonlinear controlled payment function have been identified.

The role of the regulation is to provide guidance insofar as: (1) the identification of a nonlinear payment function is concerned (how do you know a payment is nonlinear?); and (2) the selection of the correct contingent claims is concerned (see the table) for a specific nonlinear payment (given a nonlinear payment, what synthetic of contingent claims perfectly replicates it?). Item (1) requires somewhat advanced mathematical skills (most people will call an affine-linear function linear); item (2) requires specialized derivative pricing skills. These are not skills we can reasonably assume most transfer pricing professionals possess; a few may, but most do not. Transfer pricing is a multidisciplinary field with practitioners coming from a variety of educational backgrounds. Assuming some shared understanding of basic legal and valuation principles is fine, but for more specialized technical issues, it is the role of the regulation to level the playing field by providing meaningful, practical guidance.

We have none of that when it comes to nonlinear controlled payments.

With that guidance, taxpayers, IRS economists, and advisers can focus on the measurement of the NPV of the nonlinear payments.

This is not optional; you cannot discount nonlinear payments into an NPV by way of DCF.