The OECD is enamored with the present value of intangibles. To check this paramour, suppose that we don’t have historical (past) data and must rely on projections of sales attributed to certain identifiable intangibles that need to be valued. The *present value* of projected taxable profits expected from identifiable intangibles can be determined using two formulae, depending on available information. For intangible assets, the stream of taxable profits is called *royalties*.

Suppose that the projected royalties have a *definite* time-horizon T. In such circumstances, present value can be determined by using a well-known formula:

_{0}= ρ ∑(0, T) Ŝ

*/ (1 + π)*

_{t}^{t}

where *t* = 0, 1, 2, …, T and Ŝ_{t}*≥* 0 are projected sales. The two unknown parameters are ρ > 0 (comparable royalty rate obtained from RoyaltyStat) and π > 0 (risk-adjusted discount rate). For a given period, projected royalties are proportional to the same period projected sales: R* _{t}* = ρ Ŝ

*.*

_{t}Given T, we must forecast the sales time-series Ŝ* _{t}* plus estimate two parameters: ρ and π. Among the two unknown parameters, estimating π is more difficult. CAPM (capital asset pricing model) is inappropriate to calculate π, but dilettantes propagate this practice.

Suppose instead that the projected royalties have an *indefinite* time-horizon, where *t* → long-run (many years ahead). In such circumstances, present value can be determined by using a similar formula:

_{0}= ρ ∑(0, ∞) Ŝ

*/ (1 + π)*

_{t}^{t}

where π > 0 and *t* is long towards infinity. The difference between (1) and (2) is the time-horizon of the royalties stream.

Present value calculations are perilous (they are not for beginners). In recent US transfer pricing litigations covering the intra-group transfer of intangibles (*e*.*g*., *Medtronic*), the IRS confused formula (1) and (2). The IRS used (2) but the fact-checker opined that (1) fitted better the facts and circumstances. The fact-checker also rejected the IRS discount rate. Present value calculations require a *test of reasonableness* because (*inter alia*) results are very sensitive to variations in the discount rate. Like in *Veritas*, present value (also called “income method”) can lead to an incubus tragedy of errors.

We shall expand formula (2) and elicit certain useful implications that can illuminate intangibles royalties cases like *Amazon*, *Medtronic *or *Veritas*. Our objective is to simplify (2) by avoiding uncertain forecasts of sales. The key to our novel development is that we can truncate the time-horizon and obtain a simpler expression for (2).

Suppose that we *project sales* to grow at a fixed long-run rate (gamma) per year. *E*.*g*., suppose that this growth rate of projected sales is equivalent to the growth rate of the country’s GDP (or better yet it’s equivalent to the long-run growth rate specific to the applicable industry in which the intangibles are employed):

*= Ŝ*

_{t}_{0 }(1 + γ)

^{t}where γ > 0 and T is long term.

We substitute (3) into (2) and obtain:

(4) V_{0}= ρ ∑(0, ∞) Ŝ

_{0 }(1 + γ)

*/ (1 + π)*

^{t}^{t}

To simplify (4), we define a mixed parameter:

(5) λ* ^{t}* = (1 + γ)

*/ (1 + π)*

^{t}^{t}→

(6) λ = (1 + γ) / (1 + π) where π > γ and thus λ < 1.

We substitute (5) into (4) and obtain:

(7) V_{0} = ρ ∑(0, ∞) Ŝ_{0 }λ^{t}

(8) V_{0} = ρ Ŝ_{0} (1 + λ + λ^{2} + λ^{3} + … )

_{0}= ρ Ŝ

_{0}/ (1 – λ)

(10) V_{0} = ρ Ŝ_{0} μ

where (as reminder) ρ is a comparable royalty rate obtained from RoyaltyStat and μ = 1 / (1 – λ) is our valuation multiplier.

Equation (10) implies that present value can be determined by a simple formula consisting of the base-year sales (Ŝ_{0}) multiplied by a convergence parameter (μ) and by a comparable royalty rate. In economics, it's customary (*e*.*g*., Kahn's investment multiplier or Leontief's input/output matrix inverse) to truncate an indefinite time-series λ* ^{t}* (where λ < 1) to a few period terms. As stated before, among the underlying parameters, π is the most challenging to compute partly because we must first destitute a dogma in which risk-adjusted discount rates are calculated by rote using CAPM as

*deus ex machina*.

In a prior paper, “Lump-Sum Royalty Payments for Intangibles” (1999), we converted this asset valuation multiplier μ = 1 / (1 – λ) > 1 into an expression in which the growth rate of projected sales and the risk-adjusted discount rate are explicit parameters. We replicate this exercise on the Addendum below. Truncated (10) appears innocuous, but intractable disputes over risk-adjusted discount rates can spoil tax settlements between tax administration and corporate taxpayers.

**Addendum**

We know that the time-series ∑(0, ∞) λ* ^{t}* = (1 + λ + λ

^{2}+ λ

^{3}+ …) = 1 / (1 – λ) = μ if λ < 1. See Dahlquist and Bjöck (1974), p. 66, equation 3.1.4. Else see Lang (1986), pp. 474-475.

Knowing this well known time-series convergence formula, we expand (6) as follows:

(A1) μ = 1 / (1 – λ) = 1 / [(1 − (1 + γ) / (1 + π)]

Next, we multiply the numerator and denominator of (A1) by the discount factor (1 + π), and obtain:

(A2) μ = (1 + π) / (1 + π) [(1 − (1 + γ) / (1 + π)], which we simplify further:

(A3) μ = (1 + π) / [(1 + π) − (1 + γ)]

(A4) μ = (1 + π) / (π − γ) > 1 because π > γ.

As a result of π > γ, a long-term stream of royalties can be converted into a lump-sum monetary amount using several equivalent formulae:

(A5) V_{0} = ρ Ŝ_{0} / (1 – λ) = ρ Ŝ_{0} [(1 + π) / (π − γ)] = ρ Ŝ_{0} μ

The parsimonious equations (A5) are the same as equation (10). It follows from (A4) or from (A5) that π > γ such that V_{0} > 0.

This means that the discount rate must exceed the growth rate of projected sales. Again, *without access to past data* (which seems abnormal), we can rely on future projections, and as such the valuation of certain identifiable assets, including so-called hard-to-value intangibles, can be reduced to calculating three parameters (comparable royalty rate, risk-adjusted discount rate, and growth rate of the projected sales influenced by the identifiable assets being valued).

A central message of this exegesis is that *we can't consider only the future* (rely on uncertain projections) and forget the past. A fatal defect of present value is that calculations are wobbly depending on the discount rate. Another fatal defect is that the reliability of present value calculations can be ascertained only *ex-post*, after we have data about actual sales such that we can compare them with *ex-ante* or projected sales. Only *ex-post* can we determine if present value calculations are reliable. *Ex-ante* we are subject to blarney emasculated as science. Projections without considering errors are inane. A serious treatment of testing projections versus actual data, including a description of his novel reliability U^{2} statistic, can be found in Theil (1966), p. 28, equation 4.1.

**References**

Germund Dahlquist and Åke Bjöck, *Numerical Methods*, Prentice-Hall, 1974.

Serge Lang, *A First Course in Calculus* (5th edition), Springer, 1986.

Ednaldo Silva, “Lump-Sum Royalty Payments for Intangibles,” *Tax Management Transfer Pricing*, Vol. 8, No. 2, May 19, 1999.

Henri Theil, *Applied Economic Forecasting*, Rand McNally, 1966.

**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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