We can determine the value of certain identifiable assets (including the value of intangible assets) from knowledge about the associated (a) initial investment, (b) comparable growth rate of investments (g), and (c) estimated longevity of annual investment flows. Intangible producing investments include research & development, software development expenses, marketing and advertising expenses.

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 (loss).

**Remembrance of Things Past**

We denote the estimated value of the 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 define in period *t*:

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

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

This formula (1) shows that the 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 (the *ovum*)) in the

a) first period, K_{0},

b) second period it’s K_{0} + K_{0} (1 + g),

c) third period it’s K_{0} + K_{0} (1 + g) + K_{0} (1 + g)^{2}, and

d) this sequence follows until *T* (the expected longevity of the annual investments).

See Patterson & Schott (1979), pp. 160-162 and Hulten (1990), equations 1, 32 and 38, pp. 121, 133, 138, including comments by Ernst Berndt, p. 154.

We must determine a growth rate of investments (g) and obtain K_{0} and *t = *0, 1, 2 , ...,* T *from *internal data* of the “tested party” (the audited entity that has engaged in DEMPE activities leading to the creation of self-developed intangible assets that are subsequently transferred to an offshore affiliate).

We postmultiply equation (1) by the growth factor (1 + g) to obtain (2):

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

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

K_{t} − K_{t} (1 + g) = K_{0} − K_{0} (1 + g)^{t }

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

We simplify (3) to obtain the value equation (4) of the measured identifiable assets:

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

(5) K_{t} = β(*t*) K_{0},

where the initial assets value multiplier is β(*t*) = {[(1 + g)^{t}* ** −* 1] / g} for the selected *t* period.

This time-function β(*t*) is posited (without derivation) in Davis (1941), formula (12), p. 293, Allen (1963), pp. 176, 184 and Lange (1968), formula (1a), p. 102.

The chart below shows the 30-year time behavior of the multiplier β(*t*) for g = 3.25% per year. The following table contains several values of β(*t*) (= beta multiplier) at the sequential values of *T* = the number of years after the initial (base year), with inflation adjusted investments growing at g = 3.25% per year:

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

β(*t*) 1.0 2.033 3.099 4.199 5.336 6.509 7.721 8.972 10.263 11.597 12.974 14.395

**Oskar Lange's Way**

For example: If the average annual investments in CAPX, Research & Development, Software Development Cost, or Advertising is 245 USD million, at the end of five years (*T* = 5 years), with expected g = 3.25% per year, the year-end assets value can be determined by using equation (4) or (5):

(4) K_{5} = 245 USD million {[1 + 0.0325)^{5}* ** −* 1] / 0.0325} = 245 USD million (5.336) = 1,307 USD million

Given g = 0.0325 (or g = 3.25% per year) and *T* = 5 years of stable investments, the assets multiplier is β(*t*) = {[(1 + g)^{t}* ** −* 1] / g} = {[(1 + 0.0325)^{5}* ** −* 1] / 0.0325} = 5.336, which we can multiply by the *initial stock* of assets to determine the assets value at *T* = 5.

This g = 3.25% is our estimated growth rate of U.S. “Real Gross Private Domestic Investment: Fixed Investment: Nonresidential: Intellectual Property Products: Research and Development (Y006RX1Q020SBEA)” measured from 2002 to 2019. See https://fred.stlouisfed.org/series/Y006RX1Q020SBEA

We can use formula (4) to calculate the value of any identifiable asset (including the value of identifiable intangible assets) to satisfy tax compliance rules in the event of an associated enterprise asset transfer or related party asset purchase.

We can perform *sensitive analyses* by varying the growth rate of investments *within narrow limits* (0 < g ≤ g_{max}) and compute a defensible range of near-neighbor assets values to support the controlled assets transfer license or purchase agreement.

**References**

Roy Allen, *Mathematical Economics* (2^{nd} edition), Macmillan, 1963.

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

Charles Hulten, “The Measurement of Capital” in Ernst Berndt & Jack Triplett (editors), *Fifty Years of Economic Measurement* (NBER studies in income and wealth, Vol. 54), University of Chicago Press, 1990.

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

K. Patterson & Kerry Schott (editors), *The Measurement of Capital* (Theory & Practice), Macmillan, 1979. Papers from the conference on capital measurement held at Southampton University (UK) in the Summer of 1976.

**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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