Explaining the "BMNRs" money game with mathematics

Author: Theclues Source: X, @follow_clues

The core mechanism of equity dilution: Issuance will change the distribution of equity per share, leading to a transfer of value from existing shareholders to new shareholders, unless certain ideal conditions ( such as the market fully accepts the issuance without adjusting the valuation ) continue to hold. Below, I will illustrate mathematically why this effect cannot be avoided in reality, and ultimately undermines the logic of the "eternal cycle."

1. Example Assumption

• Initial State:

Company assets: 10 billion USD ETH ( net assets = 10 billion USD, assuming no liabilities ).

Market capitalization: $11 billion ( means the market is offering a 10% premium, potentially based on growth expectations or speculation ).

Assuming the total share capital is S shares, then:

Net Asset Value per Share ( NAV ) = 100/S billion USD, Price per Share = 110/S billion USD ( Premium = 10% )

• Action: Raise $5 billion ( to issue new shares ), fully purchase $5 billion ETH.
• In order to keep the stock price unchanged, the additional issuance must be priced at the current stock price of 100/S. This is a "market price issuance."

2. Calculate the situation after the increase in issuance

If the market price is increased without changing the stock price (

• Additional shares issued: Raising 5 billion dollars requires new shares N=0.4545S.
• New total share capital: 1.4545S.
• New total assets: 100 + 50 = 150 billion USD ETH.
• New net asset value per share: 150/1.4545S billion ) increased by approximately 3.13% from the initial 100/S (.
• New market cap ) assuming the market accepts the stock price unchanged (: 16 billion USD.
• New premium: 10/150=6.67%) decreased from 10% to 6.67%(.

On the surface, the stock price remains unchanged at 110/S, and the net asset per share has even slightly increased.

But there is a hidden dilution effect here:

• Value Transfer Occurred: An additional $5 billion in assets has been added, shared by all shareholders ) on the new (. The equity ratio of existing shareholders has decreased from 100% to 68.75%. They originally owned the entirety of $10 billion in assets, now only owning 68.75% of $15 billion ), which is $10.313 billion (, a net increase of $313 million. However, without the issuance of new shares, they could have owned $11 billion; here, new shareholders have shared part of the appreciation at a "discount" ) due to the premium compression (.
• Not Real Value Increase: The additional $5 billion is external capital injection, not value created internally by the company. Its "value increase" is merely an accounting illusion—similar to depositing someone else's money into your own bank and then claiming that your family's wealth has increased.
• Premium compression is a warning: An initial 10% premium reflects the market's optimism about "growth potential" ) as it anticipates more issuance cycles (. However, each issuance dilutes this potential, causing the premium to gradually decrease ) from 10% to 6.67%, and lower next time (.
• Why? Because the company is essentially an "ETH holding shell" with no unique business, the market will gradually view it as an ETH ETF ) with a market cap ≈ net asset value, premium → 0(. If the premium is 0 for a day, further issuance cannot be done at a price higher than the net asset value; otherwise, no one will buy.

3. Continuous cycle, the effect will be amplified and destroy the model

Assuming the example is repeated several times ) each financing is equivalent to 50% of the current assets, issued based on the stock price at that time, assuming the stock price remains unchanged (:

• After Round 1: Assets 15 billion, Market Value 16 billion, Premium 6.67%.
• Round 2: Financing 7.5 billion ) 50% of 1500 (, the number of new shares ≈ 0.46875 S' ) S' is the current equity (, new assets 22.5 billion, new market value 23.5 billion, premium ≈ 4.44%, net asset per share increases but the premium continues to decrease.
• Round 3: Similar, the premium dropped to ≈3%.

After several rounds, the premium approaches 0. At this point:

• The issuance price is forced to equal the net asset value per share ) with no premium space (.
• Net asset per share does not increase: For example, Asset A, Capital T, issuing 0.5A) priced at A/T(, new shares = 0.5T, new assets 1.5A, new per share = 1.5A/1.5T = A/T) remains unchanged(.
• Cycle failure: Without the "stock price increase" driving the next issuance, the model shifts from "value addition" to "zero-sum" – new funds merely dilute old shares, with no net benefit.

This is exactly the manifestation of the dilution effect: initially covered by premiums, later exposed, leading to value transfer ) new shareholders entering at low cost, old shareholders' equity diluted (.

4. If it is not a market price increase, the dilution is more obvious

) closely related to the "affordable issuance" scenario (

5. Why is this effect unavoidable in practice?

The market is not infinitely rational or optimistic: Your assumption relies on the market always accepting "stock prices remain unchanged," but investors will calculate dilution ) using EV/EBITDA or NAV discount models (. Once they realize that the model has no intrinsic cash flow ) with no dividends, relying solely on holding ETH (, FOMO turns into panic, and stock prices collapse in advance.

Nature of Mathematics: Dilution is an arithmetic inevitability. Unless the growth rate from issuance > dilution rate ) Gordon model: value = frac{D}{r-g}, where g is growth, but g depends on external ETH rise, not perpetual (, otherwise value does not increase.

In summary, the new shareholders of BMNR are constantly eroding the rights of old shareholders through the issuance of new shares, merely masked by the rise of ETH. Other cryptocurrencies also show similar patterns; the larger the proportion of new issuance to current market value, the faster the dilution effect!