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)
• uniform interest (efficient market)
• no cost of operation (efficient operations)

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 operational efficiency differs from a real bank only in the rate of return, as nothing is consumed in operations. A perfectly efficient market implies uniform interest, and that all earning is a consequence of time preference. Uniform interest is ultimately an operational efficiency, as rate arbitrage incurs an expense.

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". Demurrage is the cost of storing the reserve, which, as an operational expense, is zero for the pure bank. Interest relations assume a single compounding period with the rate of interest over that period. This presentation simplification is inconsequential to implied relations. The efficiency rate is 1 for the pure bank.

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 (inclusive of present good depreciation ratio) is the interest rate in the case of uniform interest. 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

Given that the demurrage rate is zero, the demurrage ratio is 1. Therefore the rate of return on pure bank investment is the reserve ratio.

return-rate = reserved / borrowed

Real Banks

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

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

The real bank differs from the free bank only by the presence of tax, inclusive of regulatory expense.

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

The central bank (state) differs from the real bank only by the presence of taxpayer subsidy (negative expense), inclusive of discounted borrowing.

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.