We can determine a present *value* of identifiable assets, including a present value of specific intangible assets, from knowledge about their (a) initial investment, (b) comparable operating *rate of return*, and (c) estimated useful lives. For this purpose, CAPM (capital asset pricing model) is inapplicable to determine comparable operating rates of return for intangible assets because CAPM is designed to estimate *rates of* *return of traded equity shares, which reflect a stream of prospective dividends plus share price variation (capital gains or loss) during a certain period,* and not a prospective stream of

*operating profits attributed to specific intangibles*. For a given enterprise, streams of dividends and capital gains and their calculated risks (measured during a specified useful period) are unlikely to be discounted or capitalized by an operating rate of return attributed to intangible assets.

*Inter alia*, intangible assets are not frequently traded in ask-bid exchange markets and subject to speculative capital gains.

**Remembrance of Things Past**

We denote the present *value* of a stock of identifiable assets (hereafter assets, including intangible assets) at the end of the measured period by K_{t}, and the value of those assets at the *initial* period by K_{0}, and obtain in period *t *:

(1) K_{t} = K_{0} + K_{0} (1 + *r*) + K_{0} (1 + *r*)^{2} + ... + K_{0} (1 + *r*)^{t − 1}

for period *t* = 0, 1, 2, ..., *T *counting from the initial stock of assets*.*

This *recherche du temps perdu *formula (1) shows that the present value of certain identifiable assets (*e.g*., intangible assets transferred between two associated enterprises in period *t*) starts from the amount of the initial investment in the first period, K_{0}, second period it’s K_{0} + K_{0} (1 + r), third period it’s K_{0} + K_{0} (1 + r) + K_{0} (1 + r)^{2}, and this sequence follows until *T* (life span of the asset or average life span of a portfolio of identifiable assets).

Abiding by OECD transfer pricing rules, we must determine a discount rate (*r* ) from *comparable enterprises* and obtain K_{0} and *t = *0, 1, 2 , ...,* T *from *internal data* of the “tested party” (audited entity that has engaged in DEMPE activities self-developing the intangible assets transferred to an offshore affiliate).

We postmultiply equation (1) by a comparable operating profit markup factor (1 + *r*) to obtain (2):

(2) K_{t} (1 + *r*) = K_{0} (1 + *r*) + K_{0} (1 + *r*)^{2} + K_{0} (1 + *r*)^{3} + ... + K_{0} (1 + *r*)^{t }

Next, we subtract (2) from (1) to obtain (3):

(3) K_{t} − K_{t} (1 + *r*) = K_{0} − K_{0} (1 + *r*)^{t }

*r* K_{t} = K_{0} [(1 + *r*)^{t}* ** −* 1]

which we simplify further to obtain a present value equation (4) of the measured identifiable assets:

(4) K_{t} = K_{0} [(1 + *r*)^{t}* ** −* 1] / *r*, or stated in more parsimonious fashion:

(5) K_{t} = β(*t*) K_{0}, where the initial assets value multiplier is β(*t*) = {[(1 + *r*)^{t}* ** −* 1] / *r*} for a selected *t* period.

This time-function β(*t*) is posited (without derivation) in Davis (1941), formula (12), p. 293 and in Lange (1968), formula (1a), p. 102. The chart below shows the time behavior of the multiplier β(*t*) for *r* = 7.5%. The following table contains a few values of the multiplier at sequential values of *T* = the number of years after the initial (base year) creation of the assets being valued and *r* = 0.075:

*T* 1 2 3 4 5 6 7 8 9 10 11 12

β(*t*) 1.0 2.075 3.231 4.473 5.808 7.244 8.787 10.446 12.230 14.147 16.208 18.424

**Lange's Way**

For example, if an average investment such as CAPX, Research & Development, or Advertising of 850 thousand USD is made at the end of each of 12 years to create valuable assets expected to generate an operating profit return of *r* = 7.5% per year, the present value can be determined by using (4) or (5):

(4) K_{12} = 850 [(1 + 0.075)^{12}* ** −* 1] / 0.075 = 15,660 thousand USD

Given *r* = 0.075 (or 7.5%) and *T* = 12 years of past investments, the assets multiplier is β(*t*) = [(1 + r)^{t}* ** −* 1] / *r *= [(1 + 0.075)^{12}* ** −* 1] / 0.075 = 18.424, which we multiply by the initial stock of assets to determine the present value at *T* = 12.

We can use formula (4) to calculate a present *value* of any indentifiable asset (including the value of identifiable intangible assets) to satisfy tax compliance rules in the event of an associated enterprise asset purchase.

In conclusion, from knowledge about an intitial investment (K_{0}) to create certain identifiable assets subject to related party transfers, plus an estimated expected life of the created assets (*T*), we can perform *sensitive analyses* by varying the rate of return *within narrow limits* (0 < *r* ≤ *r*_{max}) and compute a defensible range of near-neighbor values to support the controlled assets purchase agreement.

**References**

Harold Davis, *Theory of Econometrics*, Principia Press, 1941.

Oskar Lange, *Theory of Reproduction and Accumulation*, Pergamon Press, 1968.

Published on Jan 29, 2019 9:22:32 AM

**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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