16 November 2010

Compound Interest and Loan Growth

The following might be seen as a continuation of the basic principle of money creation discussed in November 13, 2010. So far we haven’t said much about interest rates and interest payments. However, these are inseparable from banking business and also from money creation logic. Normally banks pay depositors for their deposits a low rate (for simplification reasons, we might assume the customer’s rate for deposits to be zero) and require borrowers to pay a higher rate.

The well-known issue is that money for paying interests has never been created. Therefore it’s theoretically not possible for society to pay back all the loans and interests, and get fully out of the debt. If this would be sought, someone had to default. When assuming no “printing” of money by central banks and no frictions in transferring money from one person to another, then purely mathematically the defaulted loans should be more-less equal to the interest payments due for loans minus interests payable for deposits for the same period (with the small difference coming from the fact that part of the banks’ assets are in liquidity reserves that do not earn or almost do not earn interest). This would also mean that banks make zero profits. As we now shall see, compound interest is what changes the game.

According to Wikipedia, compound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also earns interest by itself. As far as lending is concerned, when does this occur in practice? When interest amounts are added to the loan principal? Well, in several cases. For example, if a customer is in trouble with loan repayments, he/she might be offered an option to restructure her/his loans so that already accumulated interests are added to the principal and enabled to be paid back later together with the original principal amount (i.e. interests are capitalised). Of course, interests added to the principal in this way will in turn earn interest. Alternatively, a “good” customer may be offered to refinance her/his loan (e.g. for paying a bigger house or in the form of renewing her/his student loan) and pay the interests of the existing loan from the amount of newly granted loan. In later case, speaking about the compound interest may not be correct formally, but the effect is the same: loan principal increases without new deposits being created, and starts earning interests. The number of possible ways for restructuring and refinancing activities is only limited by fantasy.

What concerns bank’s profits, then interests added to the principal amounts are recorded as profits. Profits are bigger also because defaults have artificially lowered via restructuring and refinancing activities.

Now let’s play a little more “money game” in Excel.  For this purpose we make the following hypothetical assumptions:
* There is only one commercial bank.
* At the outset, there are 1,000 units of money in cash; all this money is in the hands of the bank: 900 units as deposits from the public and 100.00 as bank’s equity capital.
* The bank keeps 10% of deposits as liquidity reserves in central bank. Reserves earn no interest.
* The bank pays no dividends to its owners.
* No interest is paid to deposits.
* All loans are with the maturity of one year.
* All loans are bullet loans that are taken at the beginning of the year and according to the schedule shall be paid back at the end of the year together with interests.
* All loans bear the same interest rate, let’s say 5%.
* No loans are provisioned or written off – every troubled loan is restructured and unpaid interests are added to the principal amount (i.e. are capitalised).
* Central bank doesn’t “print” any new money.
* The bank has no other funding sources than deposits and equity.
* Required capital adequacy ratio is 8%, and is calculated simply as equity divided by total assets (this means that the bank’s equity is not allowed to fall below 8% of its total assets).
* People are willing to borrow as much as the bank is ready to lend.
Sure, these assumptions are oversimplifications. However, they enable to illustrate the essence of the problem.

Bank’s balance sheet to begin with looks as depicted below. On the asset side, there are the following items:
* Cash – cash available for granting new loans
* Reserves with central bank – liquidity reserves, according to the chosen reserve requirement 10% of deposits
* Loans – loans to the public divided into original (new) loans and restructured loans; restructured loans also include capitalised interests.
On the liability side we see:
* Deposits – deposits from the public
* Share capital (part of common equity) – capital paid in by bank owners
* Retained earnings (the other part of common equity) -- profits from the previous periods that are not paid out to the bank owners.
At starting point, the capital adequacy ratio is 10% (100.00 in equity divided by 1,000.00, the total value of assets).

Now the bank starts lending out its available cash – exactly as described in one of our previous posts “Money Creation – Basic Principle”. If there would be no restrictions in the form of required capital adequacy ratio, the bank would be able to lend to the customers up to 8,100 units of money (assuming that people do not bury the cash and all of it ends up in bank deposits). Taking into account the restriction to the capital, the maximum amount of loans that can be granted at this stage, is 250.00 (the available capital enables 1,250.00 in total assets; this is found as the division of equity in the amount of 100.00, and the required capital adequacy ratio of 8%). Resulting from this stage of money creation process, the bank’s balance sheet looks as follows:

Now let’s allow one year to go by. As all loans had the maturity of one year, people have to start paying back all their debts. Principal to be repaid is 250.00, interest payable amount to 12.50 (5% of the principal amount). Repayments are done from deposits. Paid interests end up in the retained earnings of the bank’s balance sheet. Thus, the resulting bank’s balance sheet looks like the one below. The bank is back in a situation very similar to the starting point depicted in Figure 1 above. The difference is that now it is able to lend more, as its equity capital has increased by the amount of retained earnings. The same has happened with available cash for lending: as there are less deposits, reserves in central bank are smaller too. Deposits have declined from 900.00 to 887.50 – which is by 12.50 units, i.e. exactly the amount of interest paid. Thanks to the retained earnings, bank’s capital adequacy ratio has improved from 10.0% (equity of 100.00 at the beginning divided by the total assets of 1000.00 at the beginning) to 11.25% (112.50/1000.00).


Next, the bank again lends out as much money as it can given the constraint of capital adequacy ratio. This time the total amount of loans will be 406.25 units of money (equity capital of 112.50 divided by the minimum capital adequacy ratio of 8% gives the total maximum amount of assets 1,406.25; this means that deposits can amount up to 1,406.25-112.50=1,293.75, the respective required reserves in central bank are 129.38 and remaining available cash 1,000.00-129.38=870.63; maximum loan amount is then 1,406.25-129.38-870.63=406.25). Again, at the end of the year customers pay back all the loans and interests for those loans; repayment is done from deposits. In result, deposits decline by the amount of interest payment when compared to the starting point situation depicted in Figure 3 above. Interests are recorded as bank’s retained earnings, which increases bank’s equity capital.

The very same process continues until at some point people’s deposits are not enough for repaying all the loans plus interests. In our example, this point arrives at the end of year eight, when the loan principal to be repaid amounts to 7,480.21 and interest accordingly to 374.01 units of money. In total this makes 7,854.22 units. The available deposits for loan repayment at the same time are 7,801.79 units of money. Let’s assume that part of the customers, so-called good customers, will repay all of their loans and interests; the others, so-called bad customers, will default with both principal and interests. Total repaid loan principal plus interests cannot exceed the total amount of deposits. Thus, the remaining part of 52.43 units (7,854.00-7,801.79) will end up in restructured loans. It’s important that from this point on, the capitalised interests included in the restructured loans, start earning interests by themselves. The other important thing to note is that accrued, but not received earnings (i.e. capitalised interests) increase bank’s equity capital – thus the bank is no more restricted by capital adequacy requirement in its lending activity. Instead, the restriction comes from the minimum reserve requirement: the deposits cannot exceed 10,000 units of money (because available cash is 1,000 units, and all of it has to be kept in liquidity reserves), which in turn implies that the newly granted loans in each year are limited to 10,000. The total amount of loans however continues to grow as more and more loans need to be restructured because compounding interests make loan repayments more and more difficult. Figures 4 and 5 below illustrate this abstract discussion.



From Figure 4 we see how capitalised interests make bank’s balance sheet to grow. We also see that retained earnings consist of two types of earnings: 1) those that are actually received, i.e. interests that customers have paid, and 2) those that are accrued but not yet received, i.e. interests that customers have not been able to pay but that are added to the loan principal amounts and show in that way up in the asset side. Furthermore, 1,000.00 units of cash are now available for granting new loans.

Figure 5 reveals that even after granting the maximum amount of new loans permitted by reserve requirement, bank’s capital adequacy ratio is still 9.52%. Thus, capital is not a limiting factor to the growth of loans and deposits. This is thanks to the compound interest and accrued, but not actually received earnings.

Figure 6 below shows how bank’s loan portfolio continues to grow within the period of 100 years. As it turns out, the vast majority of this growth comes from restructured loans that largely consist of capitalised interests.


As we have seen by now, loan growth is exponential. There is no money for repaying everything and as time goes on, the gap between deposits (customers’ assets) and loans (banks’ assets) only goes worse. One can continue playing the above money game by making different assumptions about loan loss provisions, dividend ratios, required reserve ratios etc. – the outcome is more-less the same, except only when very extreme assumptions are taken.

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