Recognition of Non-Controlling Interest in Consolidated Financial Statements Based on Property Rights

André Casajus; Helfried Labrenz

doi:10.1515/rle-2015-0050

Article

Recognition of Non-Controlling Interest in Consolidated Financial Statements Based on Property Rights

André Casajus and Helfried Labrenz

Published/Copyright: April 22, 2017

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From the journal Review of Law & Economics Volume 13 Issue 3

Abstract

We suggest a recognition of non-controlling interest in consolidated financial statements that takes into account the property rights structure within subsidiary companies, in particular, majority requirements on restructurings, which may differ between countries. Our approach rests on a property rights index based on cooperative game theory. This index captures a parent company’s ability to acquire future gains of the subsidiary.

Keywords: consolidation; majority requirements; property rights; Shapley value

MSC 2010: 91A12; 91B99

JEL Classification: C71; G32; G34; M41; M44

Funding statement: We are grateful to Matthias Herfert, Frank Huettner, Esther Pittroff, and a number of anonymous referees for valuable comments on this paper. Financial support for André Casajus by the Deutsche Forschungsgemeinschaft (grant CA 266/4-1) is gratefully acknowledged.

Appendix A. Majority requirements for restructurings in selected countries

Table 3 shows the majority requirements for restructurings in some countries that are economically important. Superscripts at the majority quotas indicate whether the quota refers to the share of capital (qs), to the voting rights (qv), or to the so-called show of hands (qh)

A majority of three-fourth is required in Germany (GER) and in Great Britain (GB). In France (FRA), Switzerland (CHE), the Netherlands (NLD), Spain (ESP), and in the US state of California (CA), a quorum of two-thirds applies. In Italy (ITA), the quota depends on the legal form. While a majority of three-fourth is required in the società per azioni (s.p.a.), a simple majority applies to the società a responsabilità limitata (s.r.l.). In the Netherlands, the quota depends on the presence of the shareholders. If at least one-half of the share capital is present, then a simple majority is required. Otherwise, the quota is two-thirds. In the US states of Delaware (DE) and of New York (NY), a simple majority is required.

Appendix B. Game theoretic foundations

A (TU) game is a pair N,v consisting of a non-empty and finite set of players N and a coalition functionv∈VN:=f:2N→ℝ|f∅=0, where 2Ndenotes the power set of N. Subsets of N are called coalitions; vS is called the worth of coalition S indicating what the players in S can “achieve” if they cooperate. A value on N is an operator φ that assigns a payoff vector φN,v∈ℝN to any game N,v, where φiN,v denotes player i’s payoff.

A rank order on N is a bijection ρ:N→1,…,n, where ρi is the position of i∈N in ρ. The set of all rank orders on N is denoted by RN. The set of players up to and including i in ρ is denoted by Piρ=j∈N:σj≤σi. The marginal contribution of i in ρ is defined as MCiρ,v:=vPiρ−vPiρ∖i. Finally, the Shapley value, Sh, gives player i∈N the average marginal contribution over all rank orders,

(3)ShiN,v=RN−1⋅∑ρ∈RNMCiρ,v,v∈VN,i∈N.

Appendix C. Uniformly free-floating minor interests

Fix q∈0.5,1 and w1∈0,1. For (large) n>1, we consider the player sets ℕn:=1,2,…,n and share vectors wn=w1,w2n,…,wnn,win=1−w1n−1 for i∈ℕn∖1. Let the coalition function vn be given by ^[15]

(4)vn(S)={1,wn(S)>q and 1∈S,wn(S)q≥wn(S) and 1∈Swn(S),wn(S)≥1−q and 1∉S0,1−q>wn(S) and 1∉S, S⊆n.

One can easily check that vn fits the definition given in eq. (1). From eq. (3) and eq. (4), one obtains

(5)limn→∞Sh1(ℕn,vn)= {1,1 >w1 >q,w1+(1−q)⋅1−q1−w1,q≥w1 >0

as follows.

For ρ∈Rℕn, set ρ1:=ρ1. By eq. (4), we have

MC1ρ,vn=

(6)(i)1,w1+ρ1−1⋅1−w1n−1>q,1−q>ρ1−1⋅1−w1n−1,(ii)1−ρ1−1⋅1−w1n−1,w1+ρ1−1⋅1−w1n−1>q,ρ1−1⋅1−w1n−1≥1−q,(iii)w1+ρ1−1⋅1−w1n−1,q≥w1+ρ1−1⋅1−w1n−1,1−q>ρ1−1⋅1−w1n−1,(iv)w1,q≥w1+ρ1−1⋅1−w1n−1,ρ1−1⋅1−w1n−1≥1−q,

The ranges in eq. (6) can be rewritten as

(7)(i)ρ1>q−w11−w1⋅n−1+1,1−q1−w1⋅n−1+1>ρ1,(ii)ρ1>q−w11−w1⋅n−1+1,ρ1≥1−q1−w1⋅n−1+1,(iii)q−w11−w1⋅n−1+1≥ρ1,1−q1−w1⋅n−1+1>ρ1,(iv)q−w11−w1⋅n−1+1≥ρ1,ρ1≥1−q1−w1⋅n−1+1,

If w1>q, then MC1ρ,vn=1 for all ρ∈Rℕn. By (3), we already obtain the first line of eq. (6). Let now q≥w1. Since q≥1−q, we have to consider two cases.

Case A: q−w1≥1−q. Non-empty ranges in eq. (7): (ii), (iii), (iv).

Case B: 1−q>q−w1. Non-empty ranges in eq. (7): (i), (ii), (iii).

We deal with Case A only; Case B can be handled analogously. The ranges in eq. (7) become

(8)(ii)n≥ρ1>q−w11−w1⋅n−1+1(iii)1−q1−w1⋅n−1+1>ρ1≥1(iv)q−w11−w1⋅n−1+1≥ρ1≥1−q1−w1⋅n−1+1,

As n→∞, the probabilities that a rank order fits one of these ranges are (ii) 1−q1−w1, (iii) 1−q1−w1, and (iv) 2q−1−w11−w1.

We show this for range (iv); the other ranges can be handled analogously. The probability that player1 is in one of the positions 1 to n in some order ρ∈Rℕn is 1/n. Approximately, ^[16] the number of positions in range (iv) is

q−w11−w1⋅n−1+1−1−q1−w1⋅n−1+1=2q−1−w11−w1⋅n−1.

Hence, the approximate probability that player 1’s position in an order is in range (i) is

2q−1−w11−w1⋅n−1n.

Taking the limit n→∞ gives the probability 2q−1−w11−w1 as claimed.

Moreover, as n→∞, the average marginal contributions MC1ρ,vn for rank orders fitting these ranges are (ii) w1+1−q2, (iii) w1+1−q2, and (iv) w1. By eq. (6), this is clear for range (iv). Now, we show this for range (ii); range (iii) can be handled analogously. By eq. (6), the marginal contribution of player 1 depends linearly on her position in range (ii). Hence, in order to approximate the average marginal contribution in this range, we can take the arithmetical mean of the boundary positions according to eq. (8),

12⋅1−n−1⋅1−w1n−1+1−q−w11−w1⋅n−1+1−1⋅1−w1n−1=w1+1−q2.

Combining the average marginal contributions and the probabilities determined above, we obtain

limn→∞Sh1ℕn,vn=1−q1−w1⋅w1+1−q2+1−q1−w1⋅w1+1−q2+2q−1−w11−w1⋅w1

=w1+1−q⋅1−q1−w1

as claimed.

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Published Online: 2017-4-22

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https://doi.org/10.1515/rle-2015-0050

Keywords for this article

consolidation; majority requirements; property rights; Shapley value