Again the restless orb (orphan, blind) his toil renews, and sweat descends in dews. Homer, Odyssey, Book 11, 740-741.
The capital asset pricing model (CAPM) is widely used to calculate the expected return of equity shares, considering their risk relative to a stock market portfolio. The CAPM is ill-suited to valuing assets that lack stock’s spot market price volatility. Thus, I argue that the CAPM should not be used to determine the arm’s length remuneration for the intra-group transfer of intangibles.
The CAPM is postulated in a simple linear fashion:
(1) R = μ + β (M – μ)
where R denotes the one-period return of the selected company stock and M denotes the one-period return of the reference (benchmark) stock market index, such as the Standard & Poor’s 500 Index. The intercept μ denotes a country specific default risk-free sovereign interest rate, and the slope β is regarded as an equity risk-coefficient.
The (“capital gain”) stock price return can be defined by R = LN(P/P(−1)), where P is the selected company stock price; and the market index return M is defined in similar logarithm calculation.
The CAPM Equation
Equation (1) is better represented as a regression equation by combining the μ parameter:
(2) R = α + β M
in which α = (1 – β) μ.
Inter alia, equation (2) is found in Lorrie & Hamilton, 1973, pp. 199, 223-226 and Modigliani & Pogue, 1974, pp. 76-77. However, Lorrie & Hamilton, 1973 and Modigliani & Pogue, 1974, do not recognize that the variable R in equation (2) is a weighted average of μ and M. Equation (2) is not sensible because dividends (the prime mover of stock prices) are omitted.
A random term can be added to equation (2) to estimate the intercept and slope coefficient of this simple regression model using OLS (ordinary least squares). Equation (1), or its regression formula (2), appears misspecified because the primary source of stock price variations (i.e., stock specific dividends) are omitted. Thus, the slope coefficient of (2) may be biased upward.
Omitted variable bias is explained in Colonescu, 2018, § 6.3, pp. 104-106. See Gordon, 1959 and Friend & Puckett, 1964, about a discount rate specification before the CAPM became fashionable in financial economics.
If this coefficient bias is corrected by using two-stage least squares (TSLS), good (robust) linear fit between R and M may still be elusive. Else, if TSLS is applied, it is not clear what is the proper instrumental variable to be used because dividends (the most appropriate proxy variable) are issued quarterly and not with the same daily or monthly frequency of quoted stock prices.
The acronym CAPM refers to asset pricing model, so equation (2) is related to a well-recognized stock price equation.
Ergo, redefine R by the special formula that is true when R > 0.
(3) R = [P – P(−1)]/P(−1) = [(P/P(−1)) – 1]
Next, substitute (3) into (2) and obtain:
(4) P = φ P(−1)
in which φ = (1 + R) > 0.
The variable P is the sale price (payoff) and P(−1) is the purchase price per equity share.
Equation (4) is a random walk process if φ ≤ 1. Unit-roots are obtained if φ = 1, which would imply that R = 0. See Dickey, 1986.
Consider Stock Dividends
In long term positions, stock prices cannot be measured without reference to a dividend income-generating stream. In theory, a stock (or equity share) derives its price from the underlying dividends, and not e.g. from royalties (consideration from licensing intangibles or natural resources). As expressed in the hymn quoted by Williams, 1938, p. 58 (see also Lorrie & Hamilton, 1973, p. 116):
A cow for her milk
A hen for her eggs
An orchard for fruit
Bees for their honey
Stocks for dividends.
Instead of (3), the return of stock investing can include dividends per share:
Z = R + D/P(−1)
(5) Z = [P/P(−1) – 1] + D/P(−1)
where Z is the total stock price return and D is dividend per stock share.
The total stock return variable is measured by Z = LN(P + D) – LN(P(−1)).
Solve (5) for P and obtain Z from the slope coefficient of an autoregressive equation, including an intercept:
P = (1 + Z) P(−1) – D
(6) P = λ P(−1) – D
in which λ = (1 + Z) > 0 is the slope coefficient of the autoregression (6) from which Z = (λ – 1) is obtained.
CAPM and Transfer Pricing
In transfer pricing, much bloodletting occurres between the IRS and certain MNEs (multinational enterprises) over equation (2). See, for example, the Veritas, Medtronic, and Amazon cases.
However, setting aside the problem of the misspecification of equation (2), the CAPM is inapplicable to discount forecasted royalties in transfer pricing because the underlying stream of dividend income may be unrelated to the time-behavior of intangible-generating royalties.
In contrast to stock prices, royalties are not traded in stock exchanges (they are not subject to hedge and mutual fund raids), and thus royalties represent more stable streams of income.
The discount rate of royalties must reflect these more stable facts because they are immune to the speculative spot price variations that dominate stock price returns.
The CAPM (1) is misspecified and regression estimates of beta (excluding dividends per share) are biased.
For a licensor of intangibles or mineral rights, royalties represent a flow of prospective income that is not expected to behave with the high volatility of R, measured by the CAPM.
Intangibles are not subject to high-volume spot trades like stocks (traded equity shares); thus, royalties must be discounted by a less volatile rate of return more similar to the dividend yield [D/P(−1)], excluding R.
As a result, an intra-group transfer of intangibles based the CAPM is under-valued because the discount rate is high and volatile; and high volatility produces inefficient ranges of intangible values. Thus, the CAPM is inappropriate to discount much more stable (than the spot price of equity shares) royalty income streams associated with production (technology or trade) intangibles, marketing intangibles, or mineral rights.
David Dickey, “Time Series and the Stock Market,” in Richard Brook, Gregory Arnold, Thomas Hassard & Robert Pringle (editors), The Fascination of Statistics, Marcel Dekker, 1986.
James Lorrie & Mary Hamilton, The Stock Market (Theory and Evidence), Richard Irwin, 1973.
John Williams, Theory of Investment Value, Harvard University Press, 1938.
In the seminal book (which was based on his Ph.D. dissertation at Harvard University), John Williams’ (1900-1989) developed a theory of “intrinsic” stock value based on discounted dividends. Williams was ambitious by claiming in the Preface of his book: “To outline a new sub-science that shall be known as the Theory of Investment Value and that shall comprise a coherent body of principles like the Theory of Monopoly, the Theory of Money, and the Theory of International Trade, all branches of the larger science of Economics.” Williams (who became a Professor of Economics at Harvard) expressed gratitude to two towering émigré economists, Joseph Schumpeter and Wassily Leontief. Emphases added.
The novelty claimed by Williams must be compared with Irving Fisher, The Theory of Interest, Macmillan, 1930 and Ernest Brown, The Mathematical Theory of Investment, Ginn & Compnay, 1913.