There is a theory that fractional reserve banking inherently gives banks the ability to create money at no material cost. The theory does not depend on the state privilege of seigniorage. It is considered a consequence of the accounting practices of free banking. This is sometimes referred to as creating money ex nihilo or "out of thin air".

Banks do not, as too many textbooks still suggest, take deposits of existing money from savers and lend it out to borrowers: they create credit and money ex nihilo – extending a loan to the borrower and simultaneously crediting the borrower’s money account."

Lord Turner, Chairman of the UK Financial Services Authority until its abolition in March 2013. Stockholm School of Economics Conference on: “Towards a Sustainable Financial System". 12 September 2013

Adherents describe two competing views on money creation. The traditional understanding is naive in relation to their more practical view, as implied by Lord Turner. The theory states that banking inherently creates not only credit, but also money.

Money is created by miners at a material cost, potentially sold to people, and eventually lent to people. This theory holds that the lender is lending only money he owns. As such the lender is operating at full reserve and cannot engage in the practice of fractional reserve, which is considered fraudulent. As an honest lender he is only able to issue claims (representative money) against money in his possession, preventing credit expansion and therefore persistent price inflation.

Money substitutes are created by banks, at no material cost, as a consequence of fractional reserve lending. The supply of these substitutes expands with every loan, contracting only as loans are settled. Given the implied lack of constraint on credit expansion, overall debt grows without bound, creating persistent price inflation.

In a free market people can perform the same operations as banks, without necessarily calling themselves banks. Therefore the distinction between these two possibilities must be based on obscuration of the supposed fraud. The theory holds that this obscuration is accomplished using an accounting trick that is not widely understood. So let us investigate the difference. Any money will suffice in this investigation of the money substitutes created in either case, including Gold, Bitcoin or monopoly money.

In the naive view, the potential lender has saved both the liquidity required for personal consumption (hoard) and the amount intended for earning interest (investment). All lending in this scenario originates from savings, such as gold accumulated from panning. Savings includes the sum of the hoard (money) and the amount that credit exceeds debt: savings = money + (credit - debt). Money is gold and credits are money substitutes:

In this view of personal lending, Person hands over 81oz of gold to Borrower. Borrower accepts an obligation to repay Person with interest at loan maturity. To simplify the accounting we will assume zero interest and no accounting (i.e. discounting) for repayment risk:

Person has actually lent to his own enterprise (e.g. lending business) a fraction of his savings, which is accounted for below. Let us assume that Person hoards 10% of his savings for the liquidity required for near-term consumption and his Business hoards 10% for the same reason:

Person's business is operating with 10% reserve, as 90% of his deposited money is at risk of default. Projecting this into the naive view of banking requires only renaming "Lender" to "Depositor" and "Business" to "Bank". There is no need to assume that these are distinct individuals:

By properly accounting for Person having money at risk (i.e. a depositor) we can see that all lending is fractionally reserved. There are two loans in this scenario reserved at 10%, resulting in monetary substitutes (credit) of 171% of money. Given the assumption of uniform time preference, Borrower will lend 90% of his savings, as will all subsequent borrowers. Assuming a minimum practical loan of 1oz, after 43 loans credit expansion terminates at 8.903 times the amount of money.

Where r is the uniform level of individual reserve and m is the amount of money, the total amount of credit c for any number of loans n is given by the following partial sum:

c = ∑(n=1..n)[m * (1 - r)^n] =
(m * (r - 1) ((1 - r)^n - 1))/r =
(100oz * (10% - 1) ((1 - 10%)^43 - 1))/10% = 890.3oz

The reserve ratio rr is given by the ratio of money to credit:

rr = m/c = 100oz/890.3oz = ~11.23%

The money multiplier is given by the inverse of the reserve ratio:

1/rr = 1/(100oz/890.3oz) = 8.903

It is only because a single dollar is considered the smallest lendable unit that the series is limited to 43 iterations. A continuous function produces a money multiplier of 9 at 10% hoarding.

Iteration yields the following table:

Notice that, at full expansion, for any person to spend from his hoard while maintaining his time preference, a loan must be settled to offset the spending. The settlement process moves the money from the former borrower to its lender, and cancels the note. The person in receipt of the spent money must lend it in order to satisfy his time preference, and so on.

No further expansion is possible without an increase in the amount of money or an overall reduction in time preference. An increase in money increases the absolute amount of credit and a reduction in time preference increases the proportion of credit to money. Given that money and credit evolve together, there is never any actual increase in money substitutes apart from these changes.

In the typical practice of bank accounting, Bank does not hand over the money. Instead it creates account entries in a process referred to as "credit creation". It creates offsetting ledger entries for Depositor's proceeds and loan ("credit" and "debt"), and offsetting balance sheet entries for itself ("asset" and "liability"). At the time of loan issuance, the accounts are as follows:

This is where explanations of the theory tend to terminate. The offsetting accounts of both Bank and Borrower balance, but Borrower has 81oz of gold to spend, and Bank has not had to turn over any gold to Borrower. There is still only 100oz of money, but Borrower has 81oz of money substitute and Bank has 81oz more in assets. The theory proclaims that Bank has thus created not only credit, but also money. Notice that everything still balances, and all accounts can be settled, seemingly validating the theory as espoused by Lord Turner, that: "...they create credit and money ex nihilo – extending a loan to the borrower and simultaneously crediting the borrower’s money account."

This however demonstrates no actual spending of either the loan credit or the bank asset. Let us take this a bit further by assuming Borrower clears his account, and therefore the corresponding Bank asset and liability entries.

Notice that the this is identical to the outcome of the naive view. There is no distinction between these supposedly-competing views on money creation, invalidating the theory. This resolves the centuries-old debate, apparently begun between Plato and Aristotle, regarding whether money is based on mining or credit. The theories are identical, as money and credit are a duality.

According to Joseph Schumpeter, the first known advocate of a credit theory of money was Plato. Schumpeter describes metallism as the other of "two fundamental theories of money", saying the first known advocate of metallism was Aristotle.

Adherents of the two theories are merely talking past each other. Bitcoin, as fiat (i.e. non-use-value money) without state support, has finally made observable both the logical errors of metallism, which attempted to show the necessity of use value to money, and chartalism, which attempted to show the necessity of state support to fiat.

Recall that each loan is reserved at 10%, so Bank can lend up 8.903 times the amount of money on reserve, or 890.3oz of money substitute against 100oz money reserved. If Bank reserves each loan at 0%, credit expansion would be infinite. However this implies zero time preference, or the idea that time has no value, implying that all money is lent indefinitely. In the case of Bank, 0% reserve implies no liquidity to satisfy any withdrawal (i.e. immediate failure). Yet given zero time preference there could never be any withdrawals, making the scenario irrelevant. Credit expansion is necessarily finite.

So let us revisit the scenario where Bank creates credit at negative reserve (i.e. out of thin air), this time considering spending. For example, on deposits of 0oz Bank intends to issue a loan of 1000oz. Instead of relying on reserved money to eventually settle the loan, Bank "creates money" on its balance sheet. Bank then increases Borrower's credit and debt accounts, representing the borrowed money and the obligation to repay respectively:

When Borrower trades 1oz (from his credit account) for a car, his credit account is decreased by 1oz and Merchant's is increased by 1oz. Note that Borrower now owes Bank 1oz, as anticipated by the loan agreement.

All looks good until Merchant attempts to withdraw from his account. At that point Bank has defaulted and Merchant is unpaid. If Merchant's account is with another bank, the payment fails as soon as the two banks attempt to settle accounts. With a hypothetical negative reserve, the accounts balance as follows, indicating Bank's demise (negative money):

The money must actually be moved from the control of Bank to Merchant or Merchant's bank, which is not possible. A simpler example is the failure of any attempt by Borrower to withdraw from his account. Bank may create as much money substitute as it wants, but negative reserve is just an empty promise. In this example Bank has created 1000oz of promises that it cannot keep.

The failure to recognize these principles likely results from failure to consider the settlement process. This likely stems from the failure to recognize the inherent duality of money and credit, as the former must always exist to settle the claims implied by the latter. This likely stems from the habit of referring to money (e.g. gold) in the same terms as money substitutes (e.g. credits for gold).

The offsetting asset and liability entries served only to account for loans issued and outstanding, which are the basis of Bank's balance sheet. Bank similarly did not create the offsetting credit and debt entries to obscure fraudulent money creation. Bank created these accounts for two reasons:

• Preclude physical transfer just to redeposit the money into Bank.
• Encourage redeposit into Bank as opposed to a competitor (or Borrower hoard).

When Bank has insufficient reserve to satisfy withdrawals, either due to loans in default or a bank run, it has only two options, default or borrow. To prevent the former, central banking exists to provide the latter. This is the meaning of the term "lender of last resort". State Banking Principle provides a detailed explanation of this actual source of monetary inflation.

In summary, it has been shown that:

• Banks have no ability to create money.
• Fractional reserve is inherent in lending.
• The fraction of reserve is an expression of time preference.
• Zero reserve eliminates any chance of being able to settle accounts.
• No distinction exists between naive and practical theories of money creation.