### Definitions

Initial asset: | K_{0} | (asset after 0 Interest Periods) |

Final asset: | K_{n} | (asset after n Interest Periods) |

Duration: | n | number of compounding periods |

Interest Rate: | p | per Interest Period |

Initial asset: | K_{0} | (asset after 0 Interest Periods) |

Final asset: | K_{n} | (asset after n Interest Periods) |

Duration: | n | number of compounding periods |

Interest Rate: | p | per Interest Period |

The interest calculation includes calculation procedures for the interest on borrowed asset. E.g. in the form of loans or savings. The interest is paid in fixed periods e.g. yearly. That is, A savings interest is credited annually to an interest. The interest calculation gives formulas to calculate the asset development as a function of the interest rate, asset and initial asset. A linear and an exponential interest rate is distinguished according to whether interest is added to the asset or not.

To calculate interest for a specified period with an equal rate, the general interest rate formula:

${K}_{n}={K}_{0}\left(1+\frac{p}{100}n\right)$

The interest formula specifies the final asset after n interest periods. If the interest rate is annual, n is the number of years. This means that the initial asset is n-times interest and the final asset is the sum of initial asset and interest. The interest rates are not part of the asset.

The time (in interest rates) until the capital invested doubles can be calculated using the formula for capital doubling, depending on the interest rate:

${n}_{T}=\frac{100}{p}$

After n _{ T } Interest Periods, the invested asset has doubled.

${K}_{n}={K}_{0}\left(1+\frac{p}{100}n\right)$

$=1000\left(1+\frac{5}{100}10\right)$

$=1500$

1000 € will be invested 10 years with 5 percent interest. This results in a final asset of 1500 €.

${n}_{T}=\frac{100}{p}=\frac{100}{5}=20$

In the example, the initial asset after 20 years doubled.

In the compound interest rate formula, interest on the previous periods is interest-bearing in the following interest periods. This means that interest income is added to the interest-bearing capital. The compound interest rate formula is mathematically an exponential function. Generally described by f(x) = e^{x}. The exponential function also describes growth processes in nature (e.g., the multiplication of bacteria).

${K}_{n}={K}_{0}{\left(1+\frac{p}{100}\right)}^{n}$

The compound interest is the final asset at after n interest periods. If the rate of interest payable on an annual basis n the number of years. Interest is added to this the investment asset and interest in subsequent periods.

The time (in interest rates) until the capital invested doubles can be calculated using the formula for capital doubling, depending on the interest rate:

${n}_{T}=\frac{\mathrm{log}2}{\mathrm{log}\left(1+\frac{p}{100}\right)}$

After n _{ T } Interest Periods, the invested asset has doubled.

${K}_{n}={K}_{0}{\left(1+\frac{p}{100}\right)}^{n}$

$=1000{\left(1+\frac{5}{100}\right)}^{10}$

$=1628.89$

1000 € are invested 10 years with 5% interest and reinvestment of interest. This results in a final capital of € 1628.89.

${n}_{T}=\frac{\mathrm{log}2}{\mathrm{log}\left(1+\frac{p}{100}\right)}$

$=\frac{\mathrm{log}2}{\mathrm{log}\left(1+\frac{5}{100}\right)}$

$=14.2$

In the example, the initial capital is doubled after 14.2 years. With linear interest in the first example only after 20 years.

Calculated values for the capital according to the interest rate and compound interest rate formulas