(Version 1.5.1)

Collection of formulas

This page con­tains some use­ful for­mu­las nee­ded to cal­cu­late the most im­por­tant pa­ra­me­ters of an annuity credit. (At least they were helful when we planned the con­struc­tion of our house. But please, use them at your own risk.) Please use the credit cal­cu­la­tor to ge­ne­rate an amor­ti­zation plan.

Please note that your browser needs to support MathML in order to display the formulas correctly.

Definitions

Nominal interest rate per period:	$p$	Interest factor:	$q\equiv 1+p$
Number of interest periods:	$n$	Initial capital/amount of loan:	$K_0$
Account balance after n periods:	$K_{\mathrm{n}}$	Annuity:	$r$
Initial am­or­ti­za­tion rate:	$t_0$

Formulas

To be calculated	In arrear	In advance	Marginal case: q = 1
Account balance after n interest periods	$K_n=K_0{q}^n-r\frac{q^n-1}{q-1}$	$K_n=K_0{q}^n-r\frac{q(q^n-1)}{q-1}$	$K_n=K_0-nr$
Total number of interest periods	$n=\frac{\ln (\frac{r}{r-K_0(q-1)})}{\ln (q)}$	$n=\frac{\ln (\frac{r}{rq-K_0(q-1)})}{\ln (q)}+1$	$n=K_0/r$
An­nu­ity (for com­plete am­or­ti­za­tion after n periods)	$r=\frac{K_0{q}^n(q-1)}{q^n-1}$	$r=\frac{K_0{q}^{n-1}(q-1)}{q^n-1}$	$r=K_0/n$
An­nu­ity (cor­re­spon­ding to the initial am­or­ti­za­tion rate)	$r=K_0(p+t_0)$	$r=K_0\frac{p+t_0}{q}$	$r=K_0(p+t_0)$

Remark: The formulas listed here are likewise valid for annuity credits and for a capital which is con­sumed by constant pay­ments. In the latter case the "amount of the loan" (the capital) as well as the annuity are po­si­tive in­stead of ne­ga­tive.

If a capital is con­sumed two types of payment need to be dis­tin­guished: In ad­vance and in arrear. Payment in ad­vance means that pay­ments are made at the be­gin­ning of every inte­rest period. Si­mi­larly, payment in arrear means that pay­ments are always per­formed at the end of an interest period. In case of an annuity credit, payment in advance would en­force the first pay­ment di­rect­ly after the payout of the loan. This is equi­va­lent to a credit payed back in arrear having an amount which is by one annuity lower. There­fore, the credit cal­cu­la­tor always assumes pay­ments in arrear.

The marginal case (q=1) means that no interest is payed at all. In this case there is no dif­fer­ence be­tween pay­ments in advance and in arrear.

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