Someone puts K_0 euro on a savings account and adds into the following n year in each case at the year end r euro. On the savings account the bank at the end would enter afterwards each yearly the interest of the past capital and would credit the new insert. Instead of determining in n steps the assets according to n years, we compute the value with the Zinseszinsformel, on which the initial a mark insert increases:

K_n=K_0 \ cdot q^n

(q stands for p for the Aufzinsfaktor q=1+ \ frac {} {100}, whereby p is the interest rate in per cent.)

We interpret the annual payments as pension, whose final value we determine with the pension formula for pensions:

E=r \ cdot \ frac {q^n-1} {q-1}

Thus the total assets amount to

S=K_0 \ cdot q^n+r \ cdot \ frac {q^n-1} {q-1}.

This formula is called 1. Savings bank formula.

If the annual amounts are not installation-paid but are taken off, one receives according to the 2. Savings bank formula:

S=K_0 \ cdot q^n r \ cdot \ frac {q^n-1} {q-1}.

If the periodic deposits are transacted respectable removals not at the year end but at the yearly beginning, one counts on the final value formula for pensions. (The situation is somewhat artificial: Who deposits already K_0 euro and takes then immediately again r off One receives then the variants

S=K_0 \ cdot q^n+r \ cdot \ frac {q (q^n-1)}{q-1}

and

S=K_0 \ cdot q^n r \ cdot \ frac {q (q^n-1)}{q-1}.

See also: Mathematics of finance, pension calculation, compound interest, Zinseszinsformel

Related links

• http://www.bennoehr.com/zinseszins_mit_raten.htm
  • Javascript for the solution from tasks of standard to the calculation of compound interest: Computation of initial capital, final capital, time, rate, interest rate

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