An annuity loan is a loan with constant amounts of repayment (rates). Contrary to the redemptions the height of the rate which can be paid remains the same over the entire running time. The annuity rate or briefly annuity consists of an interest and to a repayment portion. Since with each rate a part of the balance of debt is erased, the share of interest is reduced in favor of the repayment portion. At the end of the running time the credit debt is completely erased.

The interest rate is fixed at the time of conclusion of an annuity loan during a contractually agreed upon period. This period can extend also over the complete credit period. The repayment should amount to per year at least 1% of the credit sum.

Determination of the annuity

The height of R of the annuity of a credit with the credit sum S_0 with a interest rate of i and a running time of n years can be computed with the following formula (q = 1 + i):

R = S_0 \ cdot \ frac {i \ cdot (1+i) ^n} {(1+i) ^n-1} = S_0 \ cdot \ frac {i \ cdot q^n} {q^n-1}

\ frac {i \ cdot q^n} {q^n-1} is called thereby annuity, and/or recovery factor (ANF^n_ {n, i}, and/or WGF^n_ {n, i}) and is equal to the reciprocal value of the pension bar value factor.

Further formulas

The balance of debt S_t after t years lets itself through to compute

S_t = S_0 \ cdot \ frac {q^n - q^t} {q^n - 1}.

During annuity repayment the balance of debt decreases exponentially.

The interest payment that t-ten period (Z_t) result from the balance of debt at the end of the preceding period multiplied by the interest rate i:

Z_t = S_ {T-1} \ cdot i = S_0 \ cdot \ frac {q^n - q^ {T-1}} {q^n - 1} \ cdot i

The instalment in that period (T_t) are given by the difference between annuity R and interest payment Z_t t-ten:

T_t = R - Z_t = S_0 \ cdot \ frac {q^ {T-1}} {q^n - 1} \ cdot i

annuity repayment

With the formulas of the under-year old annuity repayment also the cases of loan can be computed, at which the payment of the annuity takes place instead of once at the year end several times annually, for example quarterly or monthly. The number of dates of payment per year is named m. The m - 1 payments within the yearly are regarded thereby only as repayment and contain no interest component, only the last payment at the year end in this year the interest accumulated one slams shut.

The individual annuity r, which is paid m-times annually, amounts to with a interest rate of i per annum:

r = \ frac {R} {m + \ frac {i} 2 \ cdot (m - 1)}.

The yearly annuity R computes itself as during the annual annuity repayment as product made of credit sum and annuity factor. The formula subordinates the rule of the linear interest charges with under-year old running times.

Areas of application

Private loans of banks and savings banks are often assigned as annuity loans, since remaining the same rate offers a good calculation basis for the customer.

The annuity loan is a form of the real estate financing. In Germany the interest rate is usually fixed for five, ten or fifteen years. Afterwards the contract can be quit and/or a new interest rate for the continuation of the contract must be negotiated.

Alternatively also a variable interest rate can be agreed upon, which is updated in regular intervals, approximately in dependence of the EURIBOR or another indices. A further option has it to replace the annuities through equal lasting monthly instalments with which a twelfth of the nominal annual interest set is to be paid in each case. This combination (monthly repayment with equal lasting rates, which can be affected however annually by changes of interest) is for instance in Spain the most usual form. Since here the customer more risk carries, by far lower interest rates are required (2005: under 3% annual percentage rate).

See also mortgage and land charge.

See also

Savings bank formula, pension calculation, finaldue loan

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