Banking – equations – for geeks

In the modern world, a bank operates based on core capital categorized as tier 1 and tier 2 ( “BIS-Basel II” 2006). In this tier 1 and 2 capital is the money invested by stockholders or other investors along with other instruments. Net earnings from the difference between the interest paid to depositors (or borrowed from another institution) and the interest paid by borrowers is accounted as primary profits for investors in the bank. Tier 1 and 2 capital usually is greater than or equal to capital reserve requirements and is supposed to be secondary in position to the needs of demand depositors. A bank generally loans more money than it has in deposits by maintaining sufficient tier 1 and 2 capital as reserves and thus optimizes profits.

A side effect of modern banking is that it makes it possible for banks to charge what we now consider reasonable interest rates while being more profitable than money-lenders are and carrying lower overall risk. This is due to the difference between the borrower’s loan repayment income streams relative to core investor capital and the income of the simple system of money-lending. In banking, overall risk is lowered dramatically vis-à-vis money-lending both because lower interest rates are less likely to precipitate default and because a much larger pool of borrowers exists relative to invested capital.

Money multiplier in reserve banking

The standard formula for the banking money multiplier, m is:

                                                                                     (1)

where R = capital reserve fraction

The primitive equation is below. At the limit as n → ∞, it renders to the simple one above.

                                                    (2)

where R = capital reserve fraction

            i = iteration number on loans/deposits

n = iteration limit

This equation has an asymptote at equation 1.

In figure 1 is a curve relating the number of iterations of equation 2 with the multiplier achieved for that iteration. In a hypothetical system, if 30 days are required to approve each loan after acquiring new capital, then in one year 12 iterations are possible in that system. However, the number of iterations of loans for a real world banking system is variable with regard to time and this 30 day hypothetical system is simply a model system. Find the 12 iteration point (roughly) on the graph.

Figure 1 – Iteration (x axis) versus multiplier (y axis) for a 5% reserve banking system.

In the real world, a bank’s lending is dependent on both capital availability and borrower creditworthiness. The iteration period could be short if a backlog of approved borrowers is present, making the multiplier high in a short time period. Conversely, if a bank lacks creditworthy borrowers then there is no multiplier at all. So, in the real world, the multiplier can be extremely variable versus time; there is no rule that adding X amount of capital to banks will result in Y amount of new money created in any fixed time period.

In addition, in the real world, money is also taken out as circulating cash, loan losses, etc. so the true multiplier is always less than the theoretical values given by equation 1 or 2.

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