A magic number

Anyone can try out the follow simple program (any language or spread sheet applications you like):

N = 100
for i = 1:N
loan(i) = .9^i;
res(i) = loan(i)/9;
end
sum(loan)+sum(res);

You will get an answer to the last line, 9.997. Now if you change N from 100 to 1000. You will get the answer 10 to the last line.You can make N even bigger, say one million, but the magic number 10 is still the answer. Of course, this is not magic at all because this is the convergent answer to an infinite series. The interpretation, on the other hand, is more interesting that the number itself. So let's see what the variables mean in the above program.

If the number N represents the number of banks all founded at the same time, and assuming none of them has capital, they just are called to be "banks". In order for them to do business, they must somehow get funding from somewhere. Then one day, one source appears and brings $1 to the first bank. The source tells the first bank that he can loan out only 90% of his asset and must maintain 10% of asset as reserve. The requirement must be kept for the sequential loans for all the banks. For example, the second bank can only loan out 90% of his asset (can be loan from the first bank) and keep 10% reserve. That explains the loan and reserve ratio to be 9.

Since no bank has initial funding except the first one, so the first bank pipes down his $1 to all N banks. Every bank follows the same lending rules; 90% lending and 10% reserve. Then when we calculate how much capital is circulating in the all banks, we find there is $10 total. Ten times larger than the initial $1. This is the multiplying factor in economy and used many times to stimulate liquidity.

This magic number will not exceed 10 no matter how many banks are allowed in the business. It will be converging to 10. However, in order to reach 10 all banks must follow the same sequence: borrow and lend. Otherwise this is a broken chain.  When this capacity is maximized, we feel there is much more money than we think, $10 versus $1, in the market. When some banks only borrow but not lend, combined capital is slightly smaller in a few digits after the decimal point. But if the initial fund is in trillions not $1, the a few digits would end up a significant amount. Nonetheless the model is still the same.

The 90% and 10% are what the Federal Reserve requires banks to maintain their loan ratio. These two numbers were typical number and the Fed actually has been adjusting these numbers according to economical conditions. This is a tool to control fund liquidity in the market many central banks use to macro-manage.

An example is that when the 2008 crisis struck, there was no liquidity in the market. So central banks around the globe worked together to inject capital to largest banks in the world. As a result, market flew again. Of course, the banks didn't loan out continuously to smaller borrowers because there were few qualified. That was another story that related to a more mysterious part of economy: inflation.