# Banking/Tutorials

## The arithmetic of leverage

Suppose, as an example, that an entrepreneur with capital of £10 thousand, borrows £100 thousand and uses it to make an investment - thus creating a leverage of 10. That puts him in a position to make profits that are vastly in excess of what would have been possible had he only £10 thousand to invest. But it also greatly increases his risk of insolvency. If his investment were to make a loss of over 10 per cent - say, £11 thousand, that would leave him with an investment worth £89 thousand. Together with his retained capital of £10 thousand, he would then have only £99 thousand with which to repay the £100 thousand loan, and he would therefore be insolvent.

If his borrowing had created a leverage of 20, a loss of over 5 per cent would have made him insolvent, and so on.

## The arithmetic of money creation

The statement in the article that a fractional-reserve banking system creates money may seem hard to believe, but it can easily be established by a simple example involving nothing more than simple arithmetic.

Suppose, as an example, that banks are required to hold ten per cent of their deposits as reserve. Then suppose that someone with a thousand pounds/*dollars* at their disposal decides to deposit it in a current/*checking* account. The thousand pounds/*dollars* remains available, so putting it in the bank does not reduce the amount of money in the economy that is available for spending or investment.

Then - as shown in the table below - the bank adds the required £/$100 to its reserves and lends the remaining £/$900 to one of its customers, which adds that amount to the total of money that is available for spending or investment. In the second transaction shown in the table, that customer deposits it in a bank, and a repetition of the same sequence releases a further 810 into the money supply, transforming the initial 1000 from the previous 1900 to 2710 - and so on.

As the table shows, the amount added to the money supply diminishes with each transaction, suggesting that there is a limit to the total addition to the money supply that is possible. In fact the mathematics of series reveals what that limit is. The series: 1 + r + r^{2} + r^{3} + . . is a geometric series, and it can be shown that if r is less than 1, and if the series could be continued until it had an infinite number of terms, the total of all its terms would be 1/(1 - r).

In the example shown in the table, since the reserve ratio is assumed to be 10 per cent, every loan is for 90 per cent of the preceding deposit, and the appropriate value of r in the algebra column is therefore 0.9. The hypothetical bottom row shows the final amount of the addition that would be made the money supply after an infinite number of transaction would be 1,000 divided by one minus 0.9, which is 10,000. That amount could not be reached in reality, but it is approached more closely with every succeeding transaction.

Transaction Deposit Reserve Loan Available money *Algebra*1,000 P First 1,000 100 900 1,900 P+Pr Second 900 90 810 2,710 P+Pr+Pr ^{2}Third 810 81 729 3,439 P+Pr+Pr ^{2}+Pr^{3}Infinite number 10,000 P/(1-r)

## Money creation in reverse

The process of money creation depends upon the existence of a stable banking system in which banks retain a stable fraction of their deposits as reserves. If a banking crisis leads to bank failures, or prompts banks to "deleverage" by using deposits mainly to build reserves, the process can go into reverse, creating a rapid reduction in the money supply, often referred to as a "credit crunch".