It has become very common to consider that market prices, and even the application of “fair” asset valuation methods, are widely accepted and sufficient to comply with the arm’s length principle. However, this is not necessarily the case, as there are economic phenomena, such as synergies, that have peculiar implications.
Synergies imply that the interaction of assets, which may occur due to their presence in the same organization, results in a joint value generation greater than the sum of the value generated by each asset individually without interacting with each other.
In terms of an asset transaction, this could mean that the value contribution for the entity purchasing the asset is not the same as the amount lost by the entity selling it.
Regarding the valuation of the asset in question, a widely accepted approach is the present value of discounted future cash flows, where the main challenge is identifying the profits of the company that would be attributable to the analyzed asset.
To simplify this point, the Shapley value tool can help determine the profits that would be “fairly” attributable to a specific asset located within a company.
The Shapley value is a wealth distribution method in cooperative game theory, assuming that all participants collaborate to form a grand coalition. It is a “fair” distribution in the sense that it is the only distribution with certain desirable properties (efficiency, asymmetry, linearity, and null player).
Implications of Arm’s Length
However, entity A should not be willing to sell asset Δ at a price lower than
f(v(a, Δ) – v(a)), since otherwise, it would not compensate for the loss of profits for A from ceasing to exploit asset Δ.
Thus, if it were true that v(Δ) < v(a, Δ) – v(a), there would be no impediment for this to happen, but w_Δ would not be arm’s length, despite being a “fair” price, since it is not a price at which the asset owner would be willing to sell.
Furthermore, if v(a, Δ) – v(a) > v(b, Δ) – v(b), the contract curve between A and B would be empty (∅). That is, there would be no possible price at which the transaction could take place since, at any price, either A, B, or both would worsen their initial situation, which would not comply with the arm’s length principle.
The aforementioned analysis leads us to reflect that, although the priority of the transfer pricing framework is compliance with the arm’s length principle, the lack of information in the analysis often leads to the risk of not having a way to demonstrate or determine arm’s length prices. In such cases, theoretical “fair” prices can be an acceptable alternative, especially when dealing with transfer pricing analysis.
Shapley Value and Fair Utility Distribution
Where n is the total number of players, and the summation extends over all subsets of N that do not contain player i.
Example:
Suppose we have three players ({a, b, Δ}), where a represents the set of assets held by A (except for Δ), b represents the set of assets held by B, and Δ is a difficult-to-value asset.
Initially, asset Δ is owned by A, and its sale to B is analyzed. Therefore, the estimation of the “fair” utility corresponding to asset Δ is: φΔ=12(v(Δ)+v(a,Δ)−v(a))φ_Δ = \frac{1}{2} \left(v(Δ) + v(a,Δ) – v(a)\right)φΔ=21(v(Δ)+v(a,Δ)−v(a))
Where f(φ) = w, and f: R → R is a function where φ is the annual utility of an asset, and w is the present value of the asset.
Thus, the price of asset Δ would be: f(φΔ)=wΔf(φ_Δ) = w_Δf(φΔ)=wΔ
Given a group N (of n players) and a function v: 2^N → R with v(∅) = 0, where ∅ denotes an empty set, the function v, which assigns subsets of real actors, is called the characteristic function. The function v has the following meaning: if S is a coalition of players, then v(S), called the coalition value, describes the total sum of payments to the members of S that can be obtained from such cooperation.
According to the Shapley value, the amount that player i obtains during a coalition game (v, N) is: φi(v)=∑S⊆N\i∣S∣!(n−∣S∣−1)!n!(v(S∪i)−v(S))φ_i (v) = \sum_{S \subseteq N \backslash {i}} \frac{|S|! (n – |S| – 1)!}{n!} \left(v(S \cup {i}) – v(S)\right)φi(v)=S⊆N\i∑n!∣S∣!(n−∣S∣−1)!(v(S∪i)−v(S))
