The Modigliani–Miller theorem is false

The Modigliani-Miller_theorem states that dividend policy doesn't matter.  The basis for that is that a $50/share company that pays a $1 dividend, might become worth $49, but the investor got $1, so the investor still has $50 total worth of economic value.

If that same $50/share company gives an absurdly high non-cash dividend of 1 share to every shareholder, then the share value drops to $25/share, but shareholders have twice the shares, so still have $50 of value.  The offer is a complete waste of time to all involved because the outcome is identical to offering no dividend at all.

The same offer as the last, but this time the shareholder is offered the choice between a $25 cash dividend or 1 bonus share.  If 80% of investors choose the cash option, then the company will have paid $20 per share in dividends.  Its new capitalization per share is still (300/12) $25, and both those who chose to accept cash or shares have $50 of value.  However, the choice itself is of extremely high value to shareholders, and at least of obvious greater value than no choice at all.

So dividend policy obviously does matter.  Modigliani–Miller theory is management serving.  It is used to preserve their right to run the company into the ground instead of ever paying the investor back.  Their mandate to serve shareholders is a minimum satisficing one, instead of a passionate one.  A foundation of natural finance is that the only ones who should want ultimate purpose-holding (right to unlimited profits) authority is executive management, because their agenda subjugates any mandates they pretend to have.

The specific real world flaw of Modigliani–Miller theory is that all investors necessarily want to continue investing, and if given cash and a decision would reinvest in the same company at current market prices.

Note that any firm (worth $50) that offered shareholders the more reasonable (from management's greedy perspective) regular option of a dividend of 1 share per 10 shares owned, or $4.55 per share, roughly a 9% (1 - 10/11)  yield, would even if maintaining theoretical shareholder value,  actually significantly improve shareholder value by making share holding more attractive (only real source of share price increases) to buyers.  The share price increase would be even more beneficial for those that take share dividend over cash dividend.

Proof that Dividends matter
The choice of a stock or cash dividend, under Canadian tax law, offers in the stock option, 0 tax, liquidity and transaction cost issues.  If everyone chooses the stock dividend, there is no impact on the market value which management cares about, but is irrelevant to investors.  Even though there are tax cost from accepting cash dividend, there is also a corresponding future tax bonus from the reduction in market cap.  The first proof that Dividends matter is that most rational people would choose the cash dividend despite the theoretical inferiority of receiving it.  Public companies sometimes approximate this choice with DRIPS which are inferior to the choice because they may (do everywhere I know of) incur non-deferred tax costs for those that choose the option closest to a stock dividend.  50%-85% (believe higher end) of investors choose not to reinvest.  This proves a market appetite for dividends.

In the case of banks, future earnings expectations are based principally on book value, as it makes money from its money.  When it issues dividends, it reduces both its book and market value by the dividend amount, and so affects its future earnings projections.  Since most healthy banks trade above book value, the second proof that Dividends matter, apparently in favour of not paying them, is that market price as a function of book value and earnings optimism, drops by more than the cash dividend payment amount.  Yet this again proves a market appetite for dividends.

If the dividend choice (cash or stock) is significantly above earnings, for example both of the above ($50) examples, then book value and market value will both converge to each other, and converge to 0 if the dividend is continuous. For a $50 stock, and a 1 share for share dividend choice, the capitalization per share in each successive period halves [25, 12.5, 6.25, 3.125...], excluding  continued earnings contributions.  When companies trade below book value, then counter-intuitively, paying a high dividend increases shareholder value even if it decreases corporate survivability.  Trading below book value, is the market telling the corporation it is incompetent in managing its assets, and telling it, it should return investor capital.  The third proof that dividends matter is that since corporate capitalization goes to zero as a result of high dividend choice, Those who choose the dividend realize full theoretical capitalization payments in cash, while those who choose shares risk losing the full capitalization if bankruptcy results.  Whenever management chooses not to pay dividends it asserts its privilege to bankrupt the company without ever returning capital to shareholders.  Proof that traditional corporate structures are corrupt is that management asserts its privilege to power, salaries and longevity of the company over its mandate to maximize shareholder value.  Minority shareholders do not hold the democratic power to direct purpose of the corporation, and so are merely passive beneficiaries who are deceived by illusions of possible future dividends.

Note that the above offer of 1 share or equivalent cash dividend per share does not necessarily lead to a decrease in capitalization, much less bankruptcy!  A $50/share company with $5 in earnings (10%) maintains its capitalization if only 20% of investors choose the cash dividend.  The shareholder that chooses dividends each period will have collected (1023/1024) $49.95 over 10 periods, while his percentage of ownership will have decreased by 1/1024th.  Excluding the timing of tax payments, can be a net 0 tax benefit as well: capital losses offset dividend gains.

Very high dividend choices provide a natural, mathematical achievement of ideal perfect competition models for capital transfers, which imperfect markets corrupted by limited demand, supply, and information, and further corrupted by power, privilege and policy cannot achieve.