French system
The most common model is the French model. In this system the installments are fixed, unless the interest rate is variable. In which case, the calculations will need to be repeated each time the interest value is reviewed, usually once a year.
Most of the interest on the loan is paid to the lender during the first installments of the loan, since the accrual of interest at each maturity is calculated only based on two variables: interest rate applied and capital pending amortization. Therefore, by owing a greater amount at the beginning of the loan, more interest is paid at each installment maturity, and as capital is amortized, the monthly interest accrual will be lower.
This does not mean that with the French system, only interest is paid at the beginning of the loan (a fairly widespread myth), nor is it always true that at the beginning most of the installment is allocated to interest (if the repayment period is short and the interest rate is low, more amortization than interest would be paid from the first installment).
In Spain, for mortgage loans, the permitted offer is with the French amortization system and interest based on an annual APR. Next, an example for a fixed interest rate, which results in the same monthly payment throughout the amortization:
To obtain the total cost of the mortgage loan, you normally have to add the prices of: appraisal and its registration, contract and its registration, Documented Acts Tax and its management, insurance associated with the property, notary and its registration, required life insurance, and Property Transfer Tax (calculated by the price of the property). Lending entities are obliged to record all these concepts in the written offers, concepts that are paid at the beginning of the contracting and purchase.
To calculate the fee that must be paid periodically to the bank, the following formula is used:.
Note: The -term factor in the formula is an exponent, not a subtraction.
The interest must be the value (in %) that is applied during each period.
For example, to calculate the installment of a mortgage loan of 100,000 capital units, with a 15-year term and a fixed interest rate of 1.605% per year in which payments are made monthly, we use the following calculations:
Of the total capital of the mortgage (), a fixed installment () is paid in each period, composed of:.
It will be taken as:
Keep in mind that since a calculation is made for each payment period, since the indicated interest is normally annual, it is necessary to convert it into an interest rate equivalent to the payment frequency (usually months), given that the interest rate always has to be expressed in the same unit of time as the payment frequency (if paid once a year, the interest rate will be annual, if paid twice, the interest rate will be semiannual...). For 1.605% compounded annually, the equivalent monthly interest rate would be:.
Where k is the frequency of capitalization, that is, the number of times that interest is accumulated to capital to produce new interest in a period of time. If we consider the time period of one year, with annual interest k will be equal to 1, with semi-annual interest, k will be equal to 2, with quarterly interest k will be equal to 3 and so on.
In the event that it is a simple interest rate (not used in banking practice) the monthly interest rate equivalent to our interest rate of 1.605% would be:.
Initially, the mortgage principal (), the interest (), and the total period in which the mortgage must be paid (, corresponding to the number of periods; months, normally) are known.
It is interesting to determine the quota () for each period (month).
In this way, in each period the bank is owed:.
Therefore, at the end of the first period, the bank is owed:.
Subsequently, calculations are made on what is owed up to that point:
And continuing like this, at the end of the last period (), and therefore having settled the debt with the bank (the last payment corresponds to what was owed at that time plus interest ()), nothing is no longer owed:.
Since the last term corresponds to a geometric progression, it can be reduced to:.
And clearing up, the payment per period, which is what we were interested in finding out:.
This formula can be reduced further by dividing numerator and denominator by , and substituting the first formula:.
Remembering that it is the periodic installment, the capital, the number of installments, and the interest for each period:.
Increasing quota system
In the increasing installment amortization system, the return of capital is established by applying an annual increase according to a geometric ratio.
The result is that the part of capital to be returned increases as the maturity or natural end of the operation approaches.
The increasing amortization system can be combined with both a fixed rate interest and a variable rate interest. If the interest is at a variable rate then the fee can have two increase factors: first, that of the increasing amortization system and second, if the reference index with which the variable rate is calculated increases.
In Spain, after the French amortization system, which is the most widespread, it is the second most present amortization system in financing consumer loans with mortgage guarantees to purchase a residence.
This system has been used by regional and state Public Administrations to promote and facilitate access to protected housing, both in the form of Officially Protected Housing and in Housing at an Agreed Price.[10][11].
FACUA summarizes the essentials of the increasing fee model indicating that "The advantage lies in paying more affordable fees at the beginning of the loan, these fees increase considerably over the years. The drawback is that this increase is around 1% - 2% annually, to which must be added the increases that the Euribor (or other chosen reference index) registers annually. This would mean paying about 200 euros more per month than the previous year (page 7, Market Study). mortgage loans, December 2007)”.[12].