11. Financial sector, interests |
1999-07-08 To Chap 10 |
| Model D1 |
(This part of the model is preliminary. It assumes one
account with the net balance per borrower/lender. A future model
will have separate accounts for assets and liabilities).
The financial sector holds the mechanisms that makes it possible to borrow and save money. It includes all banks. The central bank is a special case that is not treated yet, it will be modeled in a coming model.
Picture 11.1:1. Financial sector, payment flows and accounts, part of Model D1, picture 12.1:1.
This simple model has the only function to bring about savings and loans. All accounts hold the net balance between assets and liabilities. The accounts are drawn as capacitor symbols (they are charged with money).
Savings and loans have the same interest rates. The model of the financial sector has no employees and it pays no salaries. The employees are included in the private sector (the enterprises).
The numbering of the flows X(.) is the same as for Model D1 which is shown in chapter 12. The balances on the accounts are denoted by Y(.) . The balances vary with time and depend upon the values that they had at a previous time. The balances are the core the dynamic model and constitute some of the state variables of the system. The notion of state variables is explained in the control theory (1). If the state variables (e.g. the number of employees and the account balances) are known and the parameters (e.g. tax rates and interest rates) are given, then it is possible to calculate the dependent variables X(.) (e.g. tax incomes).
Public borrowing X(1), the saving of the households X(12), the financial saving of the companies X(29) and the saving abroad X(18) are all net flows, i.e. the difference between saving and borrowing. These flows are accumulated as the borrowers liabilities to the saver. For the public sector, the flow is denoted as borrowing instead of saving, because the state has mostly borrowed during the last years.
Demands accumulate in the sector from which the money comes and debts accumulate in the receiving sector. The financial sector (the banks) has a demand Y(3) on the public sector, which in turn has a debt of the same amount. Because it is a debt, the balance on the public account is -Y(3).
If the debt at the beginning of the year is Y(3)(t) then it will be Y(3)(t+1) = Y(3)(t) + X(1)(t) at the end of the year, where X(1)(t) is the public borrowing during the year. The equation describes the change during the year, i.e. one part of the dynamic behavior of the system. The other dynamic equations are designed in the same manner. The net balances for the four sectors at the end of the year are:
| Public sector | Y(3)(t+1) = Y(3)(t) + X(1)(t) | Eq. (11:1) |
| Households | Y(4)(t+1) = Y(4)(t) + X(12)(t) | Eq. (11:2) |
| Private sector (enterprises) | Y(5)(t+1) = Y(5)(t) + X(29)(t) | Eq. (11:3) |
| Abroad | Y(6)(t+1) = Y(6)(t) + X(18)(t) | Eq. (11:4) |
Table 11.1:1. Dynamic equations, net balances (assets and liabilities) at the end of the year.
These assets and liabilities generate payments of interests. The interests are not accumulated on the accounts, but are paid at the end of the year. The interest is usually calculated day by day, the model uses the balances on the accounts at the beginning of the year as the base for the interest calculation. Different interest rates are used for the different sectors, otherwise it will not be possible to fit the data to the statistics. On the other hand, the differences between interest rates on assets and liabilities are neglected. These differences belong to the business part of activities of the bank and are included in the profits of the private sector. The interest rates are or, hr, pr and ur where the first letters stand for the different sectors. The interest rate is given as a factor, i.e. or =0,06 means the public sector pays 6 percent interest on this net debt. The interest payments of the public sector will be X(36) = or * Y(3)(t). We get the following equations for the yearly interests paid:
| Public sector | X(36) = or * Y(3)(t) | Eq. (11:5) |
| Households | X(37) = hr * Y(4)(t) | Eq. (11:6) |
| Private sector (enterprises) | X(38) = pr * Y(5)(t) | Eq. (11:7) |
| Abroad | X(39) = ur * Y(6)(t) | Eq. (11:8) |
Table 11.1:2. Yearly interest payments.
The balance of payments for the financial sector is formulated from figure 11.1:1 and the rules that were described in chapter 5.
| Balance of payments | -X(1)+X(12)+X(29)-X(18)+ X(36)-X(37)-X(38)+X(39)=0 | Eq. (11:9) |
Table 11.1:3. Payment balance for the financial sector.
In order to understand how saving, interests and balances on accounts interact, two cases are sketched as an illustration. First we have to recognize that saving occurs as a result of a deposit on a savings account and due to the mortgage of a debt. Negative saving results due to borrowing in a bank and when withdrawing money from a savings account.
The Svensson family saves 20 000 kronor which they deposit in the bank during the year. The balance on the account was 100 000 at the beginning. Thus X(12) = 20 000 kr and Y(4)(t) = 100 000 kr. At the end of the year, they have Y(4)(t+1) = 120 000 kr on the account (eq. 11:2). If the interest rate is 5 percent, then the interest will be X(37) = 0,05 * 100 000 kr = 5 000 kr (eq. 11:6). Now they can choose to withdraw the interest from the account and they will have 120 000 kr at the beginning of the next year. If they choose not to withdraw the interest money, a saving will occur which we can calculate by the equation 11:9, X(12)-X(37)=0. All other flows are zero in this example. The additional saving is X(12)=X(37)= 5 000 kr at the end of the year. The family has saved a total of 25 000 kr during the year and at the beginning of the next year, they have 125 000 kr on the account. The bank has a debt to the family of an equally big amount 125 000 kr.
Picture 11.2:1. Saving and interests if the family does not withdraw the interest from the account, balances at the beginning of the next year. (kKr = 1000 Kr = 1000 Swedish kronor).
The Karlsson family has a loan of 500 000 kr for a house. They pay a mortgage of 50 000 kr and pay 5 percent interest on the loan. Their balance at the beginning of the year is Y(4)(t) = -500 000 kr and their saving X(12) = 50 000 kr. At the end of the year the balance is Y(4)(t+1) = -500 000 kr + 50 000 kr = -450 000 kr. Their interest income is X(37) = 0,05 * (-500 000 kr) = -25 000 kr (which means interests are paid, the minus sign indicates that the flow goes in the opposite direction to the arrow in the figure). Because they immediately pay the interest, their debt at the beginning of the next year will be 450 000 kr. They have paid 50 000 + 25 000 kr = 75 000 kr during the year to the bank. That amount was taken from their income. The bank has received the same amount which may be lent to the state.
Picture 11.2:2. Mortgage and interest payments for a loan to a house, balances at the beginning of the next year.
The financial sector can have net assets which are not zero. The balance of payments (eq. 11:9) can be written as The total saving = -X(1)+X(12)+X(29)-X(18) = Total interests = X(36)-X(37)-X(38)+X(39). If the sum of all interest payments is not zero, then a net balance is formed in the financial sector. The sum of all interests will be different from zero if the interest rates are different for different customers. The model accumulates the assets as balances on the accounts, not as "a pile of money". The model assumed that the sum of all payment to and from the sector is zero (eq. 11:9). A model with a central bank that issues money can have assets in the form of cash.
The Excel calculus that describes Model D1 can be downloaded here.
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