Fixed-rate mortgage

An annuity is a loan with constant repayment amounts ( rates). In contrast to the amount of the loan repayment payable rate over the entire term remains the same (unless it was agreed a fixed rate period over the entire term ). The annuity or annuity briefly consists of an interest rate and a repayment component. As with any rate a part of the remaining debt is repaid, the interest portion decreases in favor of the redemption share. At the end of the term of the loan debt is fully repaid.

The interest rate will be written at the conclusion of an annuity loan over a period contractually agreed. This period may extend over the entire loan term. The repayment should be in the first year at least 1 percent of the loan (rest) total. Then increases with increasing rates of speed up to theoretically 100 % of the loan balance in the last year.

Example calculation

Amortization schedule for an annuity of 100,000 euros with an interest rate of 5.00% p. a and a term of 5 years.

Determination of annuity

The height of the ( annual ) annuity for a loan with the loan amount at an interest rate of (eg, 5 percent) and a term of years can be set via

Charge, provided. this is called recovery, or annuity ( or ) and is equal to the reciprocal value of the annuity value factor.

The Annuitätenformel in words says:

Determining the travel time

If you want to the runtime as a function of, and calculate, so you have only the above formula for the annuity according to dissolve. Obtained here

Alternatively, the run-time calculate in the following way:

Substituting in the formula for the term of the annuity by the amount resulting from interest rate and repayment term, with the repayment rate corresponds to (), is obtained by transforming the simple relationship

To determine the maturity in years.

Find the payment of installments place several times a year, there is the slightly modified formula

For the total number of rates ( not years ). This corresponds to the number of installments per year.

The calculations are for an assumed over the entire duration constant, rate. The actual running time may therefore differ in practice, perhaps materially, from the predicted.

Determination of principal payable

When analyzing an amortization table can be seen that the principal payments form a geometric sequence with the interest factor:

Thus, the eradication rates of all periods can be traced back to the first repayment installment. This can easily be determined in two alternative ways:

Therefore applies to the first repayment of principal: in which

The sum can be converted using the sum formula for geometric series in the following closed expression: After dissolved, finally results

Other formulas

The remaining debt after periods can be calculated by

If instead the term of the annuity R n is known, then can the remaining debt after periods calculated by:

The interest payment of the t- th period () arising from the residual debt at the end of the previous period multiplied by the interest rate:

Also interesting is the sum of the interest payments made by Periods:

Hence the sum of the interest payable results until the redemption of the annuity loan (periods):

The repayment rate in the -th period () is given by the difference between annuity and interest payment:

In the annuity repayment increases exponentially.

During the year annuity

The formulas for the interim annuity Also the loan cases can be calculated, where the payment of the annuity instead of once a year takes place several times a year, for example, quarterly or monthly. The number of payment dates per year will be denoted by.

Early payments are an interest advantage to the lender. The total of these payments will bear interest only half right. Payments during the year are regarded only as a principal and not contain an interest component, until the final payment at the end of this year accrued interest will be added.

The individual annuity that is paid once a year, is at an interest rate of pa and a reversionary payment by installments

In a vorschüssigen installment applies:

The Jahresannuität is also referred to as conforming replacement pension years and is always a reversionary annuity annuity. It is calculated as the annual annuity as the product of the loan amount and annuity. The formula assumes the normal case of linear interest rate in less than one year maturities.

Areas of application

Personal loans from banks are often awarded as annuity, since the constant rate provides a good basis for calculating the customer.

The annuity is a form of real estate financing. In Germany the interest rate is usually fixed for five, ten or fifteen years. Thereafter, the contract may be terminated or a new interest rate for the continuation of the contract must be negotiated.

Alternatively, a variable interest rate to be agreed, which is updated at regular intervals, for example, depending on the EURIBOR or another index. Another option is to replace the annuities through fixed monthly installments, in each of which one-twelfth of the annual nominal interest rate is payable. This combination ( monthly repayment in equal installments, which can be affected annually by changes in interest rates, however ) is about in Spain the most common form. Since in this case the customer will bear more risk, much lower rates are required (2005: effective under 3 % annual interest ).

See also mortgage and mortgage.

Comparison with other loan types

Repayment schedules for the three most common types of loans: Capital: 100,000 euros, interest rate: 5.00% p. Others, duration: 5 years