Why is the Modigliani-Miller theorem logical?

Question

Quoting this SE question, who put it nicely, the M&M theorem basically states that:

...in a no-frictions world, two identical firms have the same enterprise value, regardless of their financial structure.

Given a firm A, whose liabilities are 50% equity, and 50% debt, and a firm B which is financed 100% by equity, the value of firm A and the value of firm B are the same.

But I'm not sure if this is logical in the real world, even under the assumptions of "absence of taxes, bankruptcy costs, agency costs, and asymmetric information, and in an efficient market":

Let's look at the example of a company that's originally controlled by a single person who needs to find a way to finance the company, and let's look at two different options:

1. Going public, keeping 75% to himself, and selling the remaining 25% to the public thus securing the required funds.
2. Going public, keeping 87.5% to himslef and selling the remaining 12.5% to the public, and the "rest" of the money that needs to be raised (the equivalent of 12.5% in the other universe of the first option) will come from debt with some interest rate.

In the first option, it seems to me like the company should have a higher value: both companies are public so they pay fees to their stock exchange and need to take out money on making reports, but in the second option the company also has a debt that it will actually need to return, while in reality the money that came from selling shares (the equity) isn't a debt that needs to be repaid so the money will stay inside the company.

From my understanding stocks get their value (and therefore the motivation for people to buy shares, a.k.a the expectation to earn money) from a few things, and non of them make a difference between the two options:

1. Value - Theoretically the value of each share is derived from the value of the company - the total assets (and potential) minus the debt (and risk). This isn't real money that comes out of the company's pocket.
2. Dividends (or expectation of future dividend) - these are not a factor here since in both cases they are the same amount (it's just that in the 2nd option the original owner takes more money to his pocket since his stakes are higher). Also there is no legal obligation to pay dividends, so the owner doesn't have to ever pay dividends.
3. Share buybacks - The company isn't legally obligated to buy shares back, so again money won't come out of the company's pocket.
4. Share appreciation - the expectation that someone else (simply another player on the market or someone trying to acquire the company) will buy your share gives it value.

All of these, in my opinion, shouldn't change wether I give 25% to the public or 12.5% percent, and the only thing that's different here, is that in the 2nd option the company also has debt, which should decrease the company's value since this is actual money that needs to be paid to someone, and it will come out of the company's pocket.

Therefore, as the different distribution of shares (25% to the public vs 12.5% to the public) doesn't make a difference in the amount of money that the company loses (or earns), all companies shouldn't take debt, unless the money from the IPO isn't enough. (and under the assumption that the owner has more than 50% of the shares to have complete control of the company)

Summarizing in one sentence - the two companies have the same capital and the same potential, but the latter also has a debt it must pay in a few years.

Where is the flaw in my logic?

---

Remark: I wanted to ask why is this theorem correct, but as this isn't math, I think the word "logical" is better suited. This is just a side note, that sadly got most of the spotlight in the comments, to let people know that I understand that this isn't a mathematical theorem with a proof like in math courses. If this prevents you from understanding my question, imagine I'm using the word "correct".

Answer 1

Putting some hypothetical numbers: pre IPO let’s say the assets are worth 75mm and therefore the founder has equity worth 75mm. Now in option 1 we sell 25mm of new shares to the public so now we have assets = 75mm+25mm cash =100mm, and on the other side we have equity =100mm. In option 2 we sell 12.5mm of equity and 12.5mm of debt, so have again assets =100mm, versus now 87.5mm equity and 12.5mm debt.

The MM theorem says that the value of the company is determined by the assets which are generating the income, which is the same in both Options. However it’s still useful to look at the income statement. Let’s say the EBIT generated by the assets is 15mm. Then in Option 1 you simply have 15mm income versus 100mm equity for a 15% ROE. In Option 2 the income available to the equity holders is 15mm - interest of (at 8% say) 1mm = 14mm, on equity of 87.5mm which is a ROE of 16%. So you have a higher ROE because the company is more levered and therefore the equity is more risky in Option 2. The MM theorem argues that the correct discount rate for the equity cashflow in Option 2 results in the equity being worth 87.5mm, so that the increased ROE is exactly offset by the increased discount rate.

You also make the point that the company is somehow worth more in Option 1 because there’s no debt to be repaid. But think about Option 2 after repayment of the debt. You just have 87.5mm assets (75mm +12.5mm cash) and 87.5mm equity. Ok this company has then issued fewer external shares than option 1 but the founder’s equity is still worth 75mm in both cases.

Hope that provides a clear example.