# German amortization system

The German repayment system is a loan repayment method characterized by constant repayment terms and anticipated interest, calculated on the outstanding capital of the previous period.

This form of loan repayment has two aspects. In one of them it is equated with the Italian system, of constant capital repayment installments (A). In the other, the amortizing term (a) is constant. In Economipedia we have chosen to show the second, since it is the one taught in most university curricula.

Therefore, we can say that it is a variant of the French amortization system. The difference between the two is due to the fact that in the German method the interest is calculated in advance, on the capital outstanding at the beginning of the year, in the French it is calculated on the capital outstanding at the end of the year. This method is, after the French system, the most used in mortgages.

## Installment to pay, interest and principal in the German amortization system

We will use some simple formulas that can be automated in a spreadsheet. As we said, it must be borne in mind that the interest (Ik) is always calculated on the outstanding capital at the beginning of the year (Ck-1) and that a fee (Io) must be paid at the beginning, calculated on the entire loan granted ( Co).

- a: constant periodic fee
- Co: borrowed capital
- i: annual interest rate of the loan
- n: number of periods

Once the installment is calculated, the same is done with the amortized capital, the interest, the accumulated amortized capital and the outstanding capital, all of them in each period. We can use the following formulas. As we can see, to calculate the interest Ik, the outstanding capital of the year in question k is used, not the previous year, as was the case in French:

- Ak: Capital amortized in year k
- a: Constant amortization term (sum of principal plus interest)
- Ik: Interest of year k
- Ck and Ck-1: Capital alive in year k and (k-1)
- mk: Amortized capital accumulated in year k
- Co: Capital granted in the loan

## Example of a loan amortized by this method

Let's imagine a loan of € 10,000, 3% and 5 years. At the time of the concession we will have to pay the interest (anticipated) calculated on the capital granted. From here, it works similar to the French method.

We start by obtaining the constant amortization term (a) with its formula and then the rest of the variables. You can see the formula at the top of each column:

We can see that in the German amortization system the rate (a) is constant. On the other hand, interest decreases every year and is paid in advance, so that in year 0 we pay the first (€ 300). In addition, the amortization of capital (A) increases in each period. As we can see, it has quite a few similarities with the French system.

Amortization table Annuity

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