Annuity loan

Repayment and interest portion of an annuity (€ 100,000, entire term at 2.5% interest). As the dashed green line shows, the annuity remains the same over the entire term. Alternatively, this can also be determined by adding the blue (corresponds to the interest) and red bars (corresponds to the repayment) for any year.

An annuity loan is a repayment loan with constant repayment amounts (installments). In contrast to the installment loan , the amount of the installment to be paid remains the same over the entire term (provided a fixed interest period has been agreed over the entire term). The annuity or shortly annuity consists of an interest - and a repayment component together. Since part of the remaining debt is repaid with each installment, the interest portion is reduced in favor of the repayment portion. At the end of the term, the loan debt is fully repaid.

The interest rate is fixed for a contractually agreed period when an annuity loan is concluded. This period can also extend over the entire loan term. The repayment should be at least 1 percent of the (remaining) loan amount in the first year. It then increases as the number of installments progresses up to theoretically 100% of the remaining loan amount in the last year.

Determination of the annuity

The amount of the (annual) annuity of a loan with the loan amount at an interest rate of (e.g. 3 percent ) and a term of years can be calculated using ${\displaystyle R}$ ${\displaystyle S_{0}}$ ${\displaystyle i}$ ${\displaystyle \Rightarrow i=0{,}03}$ ${\displaystyle n}$

${\displaystyle R=S_{0}\cdot {\frac {(1+i)^{n}\cdot i}{(1+i)^{n}-1}}=S_{0}\cdot {\frac {q^{n}\cdot i}{q^{n}-1}}}$

calculate, where . is called the recovery or annuity factor ( , or ) and is equal to the reciprocal of the pension present value factor . ${\displaystyle q=1+i}$ ${\displaystyle {\frac {q^{n}\cdot i}{q^{n}-1}}}$ ${\displaystyle WGF_{n,i}^{n}}$ ${\displaystyle ANF_{n,i}^{n}}$

The annuity formula in words says:

${\displaystyle \mathrm {Annuit{\ddot {a}}t} ={\text{Kreditsumme}}\cdot {\frac {(1+{\text{Zinssatz}})^{\text{Laufzeit}}\cdot {\text{Zinssatz}}}{(1+{\text{Zinssatz}})^{\text{Laufzeit}}-1}}}$

Example with an interest rate of 3% and a term of 5 years:

${\displaystyle \mathrm {Annuit{\ddot {a}}t} ={\text{Kreditsumme}}\cdot {\frac {1{,}03^{5}\cdot 0{,}03}{1{,}03^{5}-1}}}$

Determination of the running time

If you want to calculate the running time as a function of , and , you only have to solve the above formula for the annuity after . One obtains here ${\displaystyle i}$ ${\displaystyle S_{0}}$ ${\displaystyle R}$ ${\displaystyle n}$

${\displaystyle n=-{\frac {\ln(1-{\frac {i\cdot S_{0}}{R}})}{\ln(q)}}.}$

If the installments are paid several times a year, the formula is slightly different

${\displaystyle n={\frac {\ln(1+{\frac {i}{t}})}{\ln(1+{\frac {i}{m}})}}}$

for the total number of installments (not years). This corresponds to the number of installments per year and is the so-called initial repayment, which indicates the rate of reduction of the loan after the first installment payment. It results from the formula ${\displaystyle m}$ ${\displaystyle t}$

${\displaystyle 1-t={\frac {S_{0}(1+i)-R}{S_{0}}}}$,

from what

${\displaystyle t={\frac {R}{S_{0}}}-i}$

results.

The calculations apply to an assumed interest rate that remains the same over the entire term. The actual running time can therefore in practice differ considerably from the one calculated in advance.

Determination of the repayment installments

When analyzing a repayment plan, it can be seen that the repayment installments form a geometric sequence with the interest factor : ${\displaystyle T_{t}}$ ${\displaystyle q}$

${\displaystyle T_{t}=T_{1}\cdot q^{t-1}}$

This means that the repayment installments for all periods can be traced back to the first repayment installment. This can easily be determined using two alternative options: ${\displaystyle T_{1}}$

If the annuity is known ${\displaystyle R}$	If the term is known ${\displaystyle n}$
Annuity is defined as the sum of the repayment installment and interest, therefore applies to the first repayment installment: where ${\displaystyle \!\,T_{1}=R-Z_{1},}$ ${\displaystyle Z_{1}=S_{0}\cdot i}$	The sum of all repayment installments over the term must correspond to the loan amount , i.e.: ${\displaystyle T_{t}}$ ${\displaystyle n}$ ${\displaystyle S_{0}}$ ${\displaystyle S_{0}=\sum _{t=1}^{n}T_{t}=\sum _{t=1}^{n}T_{1}\cdot q^{t-1}=T_{1}\sum _{t=1}^{n}q^{t-1}}$ The sum can be converted into the following closed expression with the help of the sum formula for geometric series : After resolving, it finally results ${\displaystyle S_{0}=T_{1}{\frac {q^{n}-1}{q-1}}}$ ${\displaystyle T_{1}}$ ${\displaystyle T_{1}=S_{0}\cdot {\frac {q-1}{q^{n}-1}}}$
You can now replace in the above formula with the respective expression: ${\displaystyle T_{1}}$
${\displaystyle T_{t}=(R-S_{0}\cdot i)q^{t-1}}$	${\displaystyle T_{t}=S_{0}\cdot {\frac {q-1}{q^{n}-1}}q^{t-1}}$

More formulas

The remaining debt by period can be calculated by ${\displaystyle S_{t}}$ ${\displaystyle t}$

${\displaystyle S_{t}=S_{0}\cdot {\frac {q^{n}-q^{t}}{q^{n}-1}}.}$

If the annuity is known instead of the term , the remaining debt can be calculated according to periods using: ${\displaystyle n}$ ${\displaystyle R}$ ${\displaystyle S_{t}}$ ${\displaystyle t}$

${\displaystyle S_{t}=S_{0}\cdot q^{t}-R\cdot {\frac {q^{t}-1}{i}}.}$

The interest payment of the -th period ( ) results from the remaining debt at the end of the previous period multiplied by the interest rate : ${\displaystyle t}$ ${\displaystyle Z_{t}}$ ${\displaystyle i}$

${\displaystyle Z_{t}=S_{t-1}\cdot i=S_{0}\cdot {\frac {q^{n}-q^{t-1}}{q^{n}-1}}\cdot i.}$

The sum of the interest payments made by period is also interesting : ${\displaystyle t}$

${\displaystyle Z_{cum,t}=\sum _{x=1}^{t}Z_{x}=S_{0}\left(t{\frac {q-1}{q^{n}-1}}q^{n}-{\frac {q^{t}-1}{q^{n}-1}}\right)=t\cdot R-\left(R-S_{0}\cdot i\right)\cdot {\frac {q^{t}-1}{q-1}}.}$

This results in the sum of the interest payments to be made until the annuity loan is repaid ( periods): ${\displaystyle n}$

${\displaystyle Z_{cum,n}=\sum _{x=1}^{n}Z_{x}=S_{0}\left(n{\frac {q-1}{q^{n}-1}}q^{n}-1\right).}$

The repayment installment in the -th period ( ) is given by the difference between the annuity and the interest payment : ${\displaystyle t}$ ${\displaystyle T_{t}}$ ${\displaystyle R}$ ${\displaystyle Z_{t}}$

${\displaystyle T_{t}=R-Z_{t}=S_{0}\cdot {\frac {q^{t-1}}{q^{n}-1}}\cdot i.}$

With annuity repayment, the repayment increases exponentially .

Annuity calculation of the banks

When using the above formulas, you will often notice differences compared to offers from a bank or online annuity calculators. This section describes how these differences come about. In order to describe this fact as clearly as possible, we will limit ourselves to the case of monthly installment payments. All other cases, such as quarterly or half-yearly installments, are to be considered analogously. The following table shows all the names of the sizes that are used in the following.

designation	meaning	unit
${\displaystyle S}$	Loan amount	EUR
${\displaystyle R}$	Annual rate	EUR
${\displaystyle i}$	Annual interest	-
${\displaystyle t}$	Initial repayment	-
${\displaystyle n}$	running time	Years
${\displaystyle S_{k}}$	Remaining debt after months ${\displaystyle k}$	EUR
${\displaystyle r}$	Monthly Rate	EUR
${\displaystyle i_{\text{M}}}$	Monthly interest	-
${\displaystyle {\widehat {i}}}$	Nominal interest rate as defined by the banks	-
${\displaystyle {\widehat {t}}}$	Initial repayment as defined by the banks	-
${\displaystyle {\widehat {R}}}$	Annual rate as defined by the banks	EUR

Deviation from interest, installment and initial repayment

The banks often advertise an interest rate, the so-called nominal interest rate. However, this nominal interest rate of the banks only corresponds to the actual interest rate if the installments are not paid during the year. When calculating the annual to the monthly rate, the bank divides the annual rate by 12. It is neglecting the fact that it is now getting the annual rate earlier over the year. This increases the actual interest compared to the nominal interest specified by the bank. We now denote the values ​​as determined by the bank with a hat if they do not match the actual values. The values ​​that match in each case are the loan amount , the monthly interest and the monthly installment . The bank assumes the loan amount , the annual interest rate and the initial repayment . She calculates the annual rate ${\displaystyle i}$ ${\displaystyle {\widehat {\ }}}$ ${\displaystyle S}$ ${\displaystyle i_{\text{M}}}$ ${\displaystyle r}$ ${\displaystyle S}$ ${\displaystyle {\widehat {i}}}$ ${\displaystyle {\widehat {t}}}$ ${\displaystyle {\widehat {R}}}$

${\displaystyle {\begin{aligned}{\widehat {R}}&=({\widehat {i}}+{\widehat {t}})S\end{aligned}}}$

and the monthly rate

${\displaystyle {\begin{aligned}r&={\frac {\widehat {R}}{12}}.\end{aligned}}}$

From this we now calculate the actual values. First of all, the monthly interest results from

${\displaystyle {\begin{aligned}i_{\text{M}}&={\frac {\widehat {i}}{12}}.\end{aligned}}}$

The actual interest results from the equation ${\displaystyle i}$

${\displaystyle {\begin{aligned}(1+i)&=\left(1+i_{\text{M}}\right)^{12}\\i&=\left(1+i_{\text{M}}\right)^{12}-1.\end{aligned}}}$

According to the Price Indication Ordinance, this actual interest must be taken into account and shown in the effective annual interest . If there are no further fees, corresponds to the APR. For the actual annual rate applies ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle R}$

${\displaystyle {\begin{aligned}R&={\frac {i}{i_{\text{M}}}}r.\end{aligned}}}$

The actual interest and the actual annual installment are thus higher than those shown by the bank when paying monthly installments. For small deviations there are small deviations, for large but very large deviations, as the following examples show: ${\displaystyle i}$ ${\displaystyle i}$

• At results . The actual interest rate is 0.5% higher than the bank's nominal interest rate. ${\displaystyle {\widehat {i}}=0{,}01}$ ${\displaystyle i=0{,}01005}$
• At results . The actual interest rate is 34025% higher than the bank's nominal interest rate. ${\displaystyle {\widehat {i}}=12}$ ${\displaystyle i=4095}$

Now we use known formulas to determine the actual initial repayment (still related to the year) and the term in years. ${\displaystyle t}$ ${\displaystyle n}$

${\displaystyle {\begin{aligned}t&={\frac {R}{S}}-i\\n&={\frac {\ln \left(1+{\frac {i}{t}}\right)}{\ln(1+i)}}.\end{aligned}}}$

A repayment plan based on monthly sizes is therefore correct, whereas a repayment plan based on the annual sizes of the bank would not be correct if monthly installment payments were agreed in the manner described.

Calculation of the term and the last installment

The term is usually a crooked number. In practice, the term is calculated in months, rounded up and a lower rate agreed in the last month. The initial repayment in the first month is defined by ${\displaystyle t_{\text{M}}}$

${\displaystyle {\begin{aligned}1-t_{\text{M}}&={\frac {S(1+i_{\text{M}})-r}{S}}\\\Longrightarrow t_{\text{M}}&={\frac {r}{S}}-i_{\text{M}}.\end{aligned}}}$

Let it be the term in months. Then the rounded term in months is how many full monthly installments have to be paid. The actual term in months is rounded up by, whereby the last installment is usually lower. We now calculate the last monthly installment. The remaining debt before the last installment is ${\displaystyle m=12n}$ ${\displaystyle \lfloor m\rfloor }$ ${\displaystyle m}$ ${\displaystyle S_{\lfloor m\rfloor }}$

${\displaystyle {\begin{aligned}S_{\lfloor m\rfloor }&=\left(S-\left({\frac {1}{i_{\text{M}}}}-{\frac {1}{i_{\text{M}}(1+i_{\text{M}})^{\lfloor m\rfloor }}}\right)r\right)(1+i_{\text{M}})^{\lfloor m\rfloor }\\&=\left(S-{\frac {r}{i_{\text{M}}}}\right)(1+i_{\text{M}})^{\lfloor m\rfloor }+{\frac {r}{i_{\text{M}}}}\end{aligned}}}$

and the last installment can therefore be determined using

${\displaystyle {\begin{aligned}r_{\lceil m\rceil }&=S_{\lfloor m\rfloor }(1+i_{\text{M}}).\end{aligned}}}$

In addition, the following useful relationships apply:

${\displaystyle {\begin{aligned}{\frac {i}{t}}&={\frac {i_{\text{M}}}{t_{\text{M}}}}={\frac {\widehat {i}}{\widehat {t}}}\\{\frac {R}{i}}&={\frac {r}{i_{\text{M}}}}={\frac {\widehat {R}}{\widehat {i}}}\\(1+i)^{n}&=1+{\frac {i}{t}}\\m&={\frac {\ln \left(1+{\frac {i_{\text{M}}}{t_{\text{M}}}}\right)}{\ln(1+i_{\text{M}})}}={\frac {\ln \left(1+{\frac {i}{t}}\right)}{\ln(1+i)^{\frac {1}{12}}}}={\frac {\ln \left(1+{\frac {i}{t}}\right)}{{\frac {1}{12}}\ln(1+i)}}=12{\frac {\ln \left(1+{\frac {i}{t}}\right)}{\ln(1+i)}}=12n.\end{aligned}}}$

If you use to denote the number of installment payments during the year, this generally results in the term in years: ${\displaystyle \mu }$

${\displaystyle {\begin{aligned}n&=\mu {\frac {\ln \left(1+{\frac {\widehat {i}}{\widehat {t}}}\right)}{\ln(1+{\frac {\widehat {i}}{\mu }})}}.\end{aligned}}}$

Annual annuity repayment

The formulas for annuity repayment during the year can also be used to calculate loan cases in which the annuity is paid several times a year, for example monthly or quarterly, instead of just once at the end of the year.

If the number of payment dates is per year, the first payments within the year are usually only regarded as repayments, i.e. they do not yet contain any interest component that is only added in full to the last payment at the end of the year for the entire previous year. ${\displaystyle m}$ ${\displaystyle m-1}$

The amount of the individual annuities to be paid annually is calculated according to the formulas for the linear interest rate for terms of less than one year from the annual annuity, which in turn is the product of the loan amount and the annuity factor, as is the case with the annual annuity repayment, i.e. an annual annuity that is always in arrears . ${\displaystyle m}$ ${\displaystyle r}$ ${\displaystyle R}$

Is the effective interest rate p. a. and the total term of the loan in years, with advance payment in installments ${\displaystyle i}$ ${\displaystyle n}$ ${\displaystyle r}$

${\displaystyle r={\frac {R}{m+{\frac {i}{2}}\cdot (m+1)}}=S_{0}\cdot {\frac {(1+i)^{n}\cdot i}{(1+i)^{n}-1}}\ /\ (m+{\frac {i}{2}}\cdot (m+1))}$.

In the case of subsequent installments, however, the following applies:

${\displaystyle r={\frac {R}{m+{\frac {i}{2}}\cdot (m-1)}}=S_{0}\cdot {\frac {(1+i)^{n}\cdot i}{(1+i)^{n}-1}}\ /\ (m+{\frac {i}{2}}\cdot (m-1))}$

If the interest factor on which the lender is based is to be determined from what may be of interest for the comparison of different loan offers, this results for a subsequent payment of the installments after converting the last-mentioned formula as the maximum of the solutions of the following polynomial (this always as a trivial solution has the value ): ${\displaystyle q}$ ${\displaystyle r}$ ${\displaystyle q=1}$

${\displaystyle (S_{0}-r\cdot {\frac {m-1}{2}})\cdot q^{n+1}-(S_{0}+r\cdot {\frac {m+1}{2}})\cdot q^{n}+r\cdot {\frac {m-1}{2}}\cdot q+r\cdot {\frac {m+1}{2}}=0}$.

Percentage annuities amortization

A special form of annuity repayment is the so-called. Percentage annuity, in which the amount of the first repayment installment is not defined as the difference between annuity and debit interest, but rather - regardless of the latter - as a fixed percentage of the loan amount.

application areas

Private loans from banks are often granted as annuity loans, as the constant rate provides a good basis for calculation for the customer.

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

Alternatively, a variable interest rate can be agreed which is updated at regular intervals, for example depending on the EURIBOR or another index. Another option is to replace the annuities with constant monthly installments, for which one twelfth of the nominal annual interest rate has to be paid. This combination (monthly repayment with constant installments, which, however, can be affected by interest rate changes every year) is the most common form in Spain.

See also mortgage and land charge .

Comparison with other types of loans

Repayment plans for the three most common types of loan: Capital: 100,000 euros, interest rate: 3.00% pa, term: 5 years, interest and repayment annually in arrears

Repayment Loans

year	Remaining debt	interest	Repayment	rate
1	€ 100,000	€ 3,000	€ 20,000	€ 23,000
2	€ 80,000	€ 2,400	€ 20,000	€ 22,400
3	€ 60,000	1,800 €	€ 20,000	€ 21,800
4th	€ 40,000	€ 1,200	€ 20,000	€ 21,200
5	€ 20,000	600 €	€ 20,000	€ 20,600
to hum		€ 9,000	€ 100,000	€ 109,000

Annuity loan

year	Remaining debt	interest	Repayment	rate
1	€ 100,000	€ 3,000	€ 18,835	€ 21,835.46
2	€ 81,165	€ 2,435	€ 19,401	€ 21,835.46
3	€ 61,764	€ 1,853	€ 19,983	€ 21,835.46
4th	€ 41,781	€ 1,253	€ 20,582	€ 21,835.46
5	€ 21,199	€ 636	€ 21,199	€ 21,835.46
to hum		€ 9,177	€ 100,000	€ 109,177

Maturity loans

year	Remaining debt	interest	Repayment	rate
1	€ 100,000	€ 3,000	0 €	€ 3,000
2	€ 100,000	€ 3,000	0 €	€ 3,000
3	€ 100,000	€ 3,000	0 €	€ 3,000
4th	€ 100,000	€ 3,000	0 €	€ 3,000
5	€ 100,000	€ 3,000	€ 100,000	€ 103,000
to hum		€ 15,000	€ 100,000	€ 115,000

Individual evidence

1. ↑ Bernd Luderer, Uwe Würker; Entry into business mathematics ; 9th edition, Springer-Verlag, 2014; Pp. 112-113.
2. ^ Manfred Precht, Karl Voit, Roland Kraft; Mathematics 2 for non-mathematicians ; Oldenbourg Verlag, 2005, p. 114.

Please note the information on legal issues!