Savings bank formula

In financial mathematics, savings bank formulas are differential equations that establish a relationship between the initial capital and the final capital after a certain number of periods in years, a rate and an interest rate (each per period). It is a combination of the final value calculation for compound interest and the pension calculation .

formula

If the final capital is searched for with an initial capital , an interest rate (with an interest factor ), a term in years and an annual rate , then the following formulas result for ${\displaystyle K_{n}}$ ${\displaystyle K_{0}}$ ${\displaystyle i}$ ${\displaystyle q=1+i}$ ${\displaystyle n}$ ${\displaystyle R}$ ${\displaystyle q\neq 1}$

• arrears installment (payment of the installment on December 31 of each year):

${\displaystyle K_{n}=K_{0}\cdot q^{n}\pm R\cdot {\frac {q^{n}-1}{q-1}}}$

• Advance installment (payment of the installment on January 1st of each year):

${\displaystyle K_{n}=K_{0}\cdot q^{n}\pm \ R\cdot q\cdot {\frac {q^{n}-1}{q-1}}}$

In both cases, the initial capital is available for interest on January 1st of the first year. ${\displaystyle K_{0}}$

When adding the rate, capital is built up and when subtracting capital is reduced. The formula also applies to loans with constant rates , with the initial capital then being negative. ${\displaystyle K_{0}}$

Derivation

Payment in arrears

At the end of the first year, interest is paid on the initial capital using the interest factor and the first installment is paid (installment in arrears). The net present value is then ${\displaystyle q}$ ${\displaystyle R}$

${\displaystyle K_{1}=K_{0}\cdot q\pm R}$.

In the 2nd year, the existing capital is again subject to interest using the interest factor and the installment is paid. The net present value is thus in the 2nd year ${\displaystyle q}$

${\displaystyle K_{2}=K_{1}\cdot q\pm R=K_{0}\cdot q^{2}\pm R(1+q)}$.

In the 3rd year is the net present value

${\displaystyle K_{3}=K_{2}\cdot q\pm R=K_{0}\cdot q^{3}\pm R(1+q+q^{2})}$.

The annual capital value is obtained in the same way ${\displaystyle n}$

${\displaystyle K_{n}=K_{n-1}\cdot q\pm R=K_{0}\cdot q^{n}\pm R(1+q+q^{2}+\dots +q^{n-1})}$.

If you replace the sum in brackets on the right-hand side with the formula for the geometric series , you get the above savings bank formula for the subsequent installment payment.

Advance payment in installments

In the case of advance payment, both the previous year's capital value and the installment paid at the beginning of the year are subject to interest using the interest factor . The first year is then ${\displaystyle q}$

${\displaystyle K_{1}=(K_{0}\pm R)\cdot q=K_{0}\cdot q\pm R\cdot q}$.

The same derivation as for the payment in arrears with the replacement instead provides the savings bank formula for the payment in advance. ${\displaystyle qR}$ ${\displaystyle R}$

Individual evidence

1. ↑ Alexander Karmann: Mathematics for economists . Oldenbourg, Munich 2008, ISBN 978-3-486-58706-7 , p. 255 ff .