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Can a company A hold 100% of shares in company B and can this company B have 100% of shares in company A

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If this happens, the pair of companies is worthless and might as well not exist.

Proof 1:

  • No shares of either company are owned, directly or transitively, by any natural person (human). The ownership chain bounces back and forth endlessly between A and B.
  • There is no way for the companies to do anything, because there is no way to legally exercise control or make decisions, which can only be done (directly or transitively) by human owners.

Proof 2:

  • Call "external assets" the net assets other than shares of A or B themselves.
  • A owns 100% of B, so Value of A = External assets of A + Value of B.
  • B owns 100% of A, so Value of B = External assets of B + Value of A.
  • By adding the two equations above, External assets of A + External assets of B = 0.
  • So A and B do not own any net external assets.
  • No value can be created by making the two companies own each other, so the construct is an empty shell.
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    ad 1: The CxO people don't have to be identical to the owners. But with your #2, you make a point.
    – glglgl
    Oct 1, 2019 at 16:39
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    I agree it makes no sense, but what is actually preventing it from happening?
    – DJClayworth
    Oct 1, 2019 at 17:03
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    As formulated by OP I agree there can be no value, but is P1.1 necessarily true? Could a company somehow come to be owned by nobody, but still have holdings an e.g. a board of directors that maximized shareholder (the company) value? I can't think of a way it would actually come to be, even if it were theoretically possible though.
    – John K
    Oct 1, 2019 at 18:30
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    Also, I disagree on your math... either External assets of A or External assets of B or both could demonstrably be > 0, which actually leads to the value of A and B diverging.
    – user12515
    Oct 1, 2019 at 23:18
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    Also, "proof" #2 is classic proof by contradiction - if one of these companies has any legal assets (e.g. 1$ in their bank account), as soon as you reach the "conclusion" of "So A and B do not own any net external assets." then the contradiction simply demonstrates that some of your assumptions (e.g. your method of recursively calculating value of companies with nonzero mutual ownership) are wrong.
    – Peteris
    Oct 16, 2019 at 20:23

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