Pure Bank

The concept of a pure bank can be useful in demonstrating lending behavior generally.

A pure bank provides only the following services:

• borrows money (debt from creditors)
• lends money (credit from debtors)
• hoards money (reserve)

The material differences from a real bank are:

• no state intervention (free bank)
• no cost of operation (perfectly efficient)

The bank is owned by its creditors in proportion to their credit, as is the case with any company. There are existing major banks that are owned by their account-holders, such as USAA and Vanguard, so this is not a distinction from a real bank. Neither a pure bank nor a real bank has "own capital" to lend, as all capital is borrowed from investors in one form or another. The objective of creditors is to maximize their rate of return. The objective of debtors is to minimize their interest expense.

Creditor accounts are money substitutes. This aspect distinguishes the bank from an investment fund. The money substitute may be either a demand deposit or a money market. The distinction is in the allocation of insufficient reserve (negative rate of return), with the former being "first come, first served" and the latter "breaking the buck".

The lack of state intervention is the common concept of free banking, where there is no statutory control, state insurance, discount capital, or seigniorage. The bank uses commodity money unless otherwise specified, which simplifies calculations by eliminating the need to offset price inflation or price deflation.

Perfect efficiency differs from a real bank only in the rate of return, as nothing is consumed in operations. All earning is a consequence of time preference. Uniform interest is assumed, as rate arbitrage is an expense. Demurrage is the expense of storing money. Expense and demurrage ratios are 1 for the pure bank.

Reserved capital is the money in which credit and debt are settled (zero maturity). Depreciation is the opportunity cost of it not being loaned, also known as "cash drag". Interest relations assume a single compounding period with the rate of interest over that period. This presentation simplification is inconsequential to implied relations.

Given the preceding definition of a pure bank, the following relations are absolute.

reserved     = borrowed - loaned
demurrage    = demurrage-rate * reserved
depreciation = interest-rate * reserved
interest     = interest-rate * loaned
return       = expense-ratio * interest

For the pure bank, the reserve ratio fully determines capital ratio, debt ratio, savings ratio, balance sheet and rate of return.

Reserve Ratio

reserve-ratio = reserved / borrowed
reserve-ratio = (borrowed - loaned) / borrowed

Capital Ratio

capital-ratio = reserved / loaned
capital-ratio = (borrowed - loaned) / loaned

Debt Ratio

debt-ratio = borrowed / reserved
debt-ratio = borrowed / (borrowed - loaned)

Savings Ratio

savings-ratio = loaned / reserved
savings-ratio = loaned / (borrowed - loaned)

Balance Sheet

The pure bank has no liabilities, only shareholder equity.

bank assets	shareholder equity
loaned + reserved	borrowed

Rate of Return

Creditor rate of return is additionally a function of the interest rate. The creditor's rate of return is less than the debtor's interest rate due to cash drag, the necessary expense of demand withdrawal. To reduce this expense, time constraints are typically included in real bank contracts. For example, by law any withdrawal from an interest-bearing U.S. bank account can be delayed for seven days. The creditor can only eliminate cash drag by holding the debt in an investment fund (i.e without settlement assurances) as opposed to a bank.

return-rate = interest-rate * loaned / borrowed

As shown in Savings Relation the capital ratio is the interest rate. The capital ratio includes present goods depreciation, which for money is demurrage. The pure bank demurrage is 1, so this drops out. Substituting capital ratio obtains a rate of return also in borrowed and loaned capital.

return-rate = (reserved * demurrage-ratio / loaned) * (loaned / borrowed)
return-rate = (reserved / borrowed) * demurrage-ratio
return-rate = reserved / borrowed

The rate of return on pure bank investment is the reserve ratio.

Real Banks

The independent capital ratios of all people, based on individual time preference, determine the market rate of interest. The above substitution for the bank's own capital ratio as the interest rate implies that the bank is setting the interest rate. However this is inherent in the concept of time preference. A bank can set any level of interest it prefers. There is no assumption for real banks that the market will oblige, so market interest and therefore market returns are assumed.

market-return-rate = market-interest-rate * (loaned / borrowed)
market-return-rate = market-capital-ratio * (loaned / borrowed)

The free bank also differs from the pure bank by operational expense, which directly reduces rate of return.

free-bank-return-rate = market-return-rate * expense-ratio

The real bank only differs from the free bank by tax (inclusive of regulatory expense), which directly reduces rate of return.

real-return-rate = free-bank-return-rate * tax-expense-ratio

The central bank (state) only differs from the real bank by taxpayer subsidy (inclusive of discounted borrowing), which directly increases rate of return.

central-return-rate = real-bank-return-rate * subsidy-income-ratio

Where tax includes seigniorage of the bank money, the Fisher Equation must be applied above to translate the interest rate from a nominal rate to a real rate. No other change is implied other than tax, which is accounted for by the real bank above. This tax is generally the source of subsidy, which is accounted for by the central bank above.

Every person, or company of people, is a real bank, and the state is a central bank.