interest
The calculation of interest describes a mathematical method for calculation of interest that are levied as payment on borrowed amounts of money.
In principle, the calculation of interest divided into the " Simple Interest Calculation ", will not be added to the accumulating and unpaid interest and the interest-bearing amount of money, such as credit, loans or savings, and the compound interest calculation in which unpaid interest to the basic amount are added and taken into account in the further interest.
Furthermore, one ( interest rates ) may differ ( repeated interest), as well as the special case of continuous compounding of interest on the number of compounding periods per year between annual (one-time interest) and under -standing interest. Standard case is the annual interest rate: The capital is once a year, usually bearing interest at year-end. The interest rate is referred to as in arrears, the preliminary interest rate than on an accrual following the Interest Period.
If paid within the interest period on a savings account or lifted it, so the mixed interest rate is used by financial institutions in general. Therefore, this type of interest also comes with all facilities with a term that is not a multiple of the Interest Period (for example, 3.5 years at an annual interest rate ) on the application. This is known as broken runtime.
While the calculation of interest in general from an absolutely paid, or borrowed amount to be released ( initial capital ), the branch of pension bill deals with regular deposits and withdrawals. For calculations of the repayment of loans, the repayment of Directors.
- 2.2.1 Examples
- 3.1 Simple Interest 3.1.1 example
- 3.2.1 Example 1
- 3.2.2 Example 2
- 4.1 Example
Preliminary remarks
The charges listed in this article formulas for the calculation of interest using the following symbols:
- Initial capital K0 ( capital after 0 years )
- Final capital: Kn ( capital after n years )
- Duration (number of years ): n input in years
- Duration (days): t input in days
- Interest rate: p ( per interest period)
- Interest rate than Dezimalangabe: (per interest period)
- Interest rate as an interest factor: (per interest period)
Varies depending on the method of calculating the year 360-366 days, the month 28 to 30 to 31 days. For example, 7% interest rate for the term of 360 days.
Annual interest rate
Simple interest without compound interest ( interest -linear )
With annual compounding applies to the final capital
By rearranging formulas are obtained for calculating the necessary for a particular final capital start-up capital, interest rate or maturity:
Example
A starting capital of € 1,000 is invested at an interest rate of 5 percent over 2 years. In case of simple interest, a final capital of would
Compound interest ( exponential interest)
The formula for the capital after years with annual interest and compound interest is:
The formula can adapt itself to determine the start-up capital, the interest rate or the term for a given end value:
Examples
A starting capital of € 1,000 is invested at an interest rate of 5 percent over 2 years. With annual compounding a final capital of would
FS / final capital
A starting capital of € 1,000 is at an interest rate of 5 % pa created over 2 years. With compound interest results in a final capital of
Generally, where the duration sought, after which the seed capital has doubled, then:
This value can be estimated by the 72 rule.
During the year interest rate
In the year income investments with interest earned several times a year. The period of interest is thus less than one year. Are usual for example, periods of:
- Half a year,
- A quarter or
- A month or
- Daily basis at rest months.
The number of compounding periods in a year is expressed in formulas by the symbol m. In quarterly interest m would, for example, 4 ( 4 quarters per year). Often a so-called nominal annual interest rate ( inom ) is specified.
The relative period irel rate is then:
The formulas of the intra-year interest rates are then used as described above, the Rate of Interest applies only not per year but per interest period. The term is also specified not in years, but in interest periods.
Simple interest
For the final capital Kn, k after n years and k Periods:
This represents the total number of interest periods after n years and k periods shown (term in Interest Periods ).
Example
A capital of € 1,000 is invested with monthly compounding (m = 12 ) at a nominal annual interest rate of 6 percent.
The relative period interest rate is 0.5%. After 2 years and 4 months results with simple interest of a final capital
Compound interest
For the final capital Kn, k after n years and k Periods:
The term that is calculated analogously with the simple interest calculation.
In addition to the relative and nominal interest rate can be the case of compound interest, the effective interest rate ieff determine. An annual interest rate on the effective interest rate leads to the same result as a mid-year rate of return relative to the interest rate. The following applies:
Multiplying the clamp and allows the higher powers of inom ( the small inom almost nothing to contribute to the sum ) away, one can estimate the effective interest well:
The additional interest income at a rate of interest during the year compared to the annual interest rate can be estimated by:
If only the effective interest rate is given, the relative period interest rate is (in this case also matched interest ikon called ) from the following formula:
Some textbooks (eg Fischer: Finance for beginners, Oldenbourg ) define the conformal annual interest rate than full-year interest rate in less than one year compound interest.
Example 1
A capital of € 1,000 is applied as above ( m = 12; inom = 6%, irel = 0.06 / 12 = 0.005).
After 2 years and 4 months, the capital of compound interest
The effective interest rate is 6.1678 %:
Calculated using the effective interest rate
Example 2
A capital of 10,000 € is applied to inom = 3% per year.
At an annual interest rate ( m = 1) is the capital with interest after one year:
The effective interest rate is ieff = inom = 3.00%.
In an intra-year quarterly interest rate ( m = 4) is the capital with interest after one year:
The additional interest income at a quarterly rate of return over the annual interest
And can be estimated by:
In an under -standing monthly interest rate ( m = 12), the capital amounts with interest after one year:
The additional interest income at a monthly rate of return over the annual interest
And can be estimated by:
With a year under continuous compounding (m = ∞, see below) is the capital with interest after one year:
The additional interest income at a constant rate of return over the annual interest
And can be estimated by:
An investment with an annual interest rate of 3.05% unique thus results in a higher interest return than an investment with a nominal interest rate of 3.00 % and a ( any ) under -standing interest. Many banks advertise with the higher interest income during the year at a quarterly rate of return, without the higher interest income to quantify exactly. On the above example is easy to see that the periodic quarterly interest only a minimal additional interest income provides: € 3.39 with an investment of 10,000 €.
Mixed interest
Usually Write Banks and other financial companies on current accounts and savings accounts interest rates at the end of the Interest Period well. For savings accounts, and other current accounts, this is usually the end of the year, at contractually determined investments often another time.
Although it is actually moved by compound interest, is capital that was not on the last interest settlement date and are therefore not the entire interest period applied via an interest rate of simple interest, as well as on a payment date within the interest period incurred up to that point in the year.
The following diagram shows a typical installation is: the system falls on any day of the year, the capital will interest a few years and finally paid on any day within the year again.
The entire investment period is made up as follows:
First, the capital over the remaining period 1 (t1 days) bears interest at simple interest. The capital thus obtained accrue interest over the n years after the compound interest formula. The rest period 2 (t2 days) is then simply bear interest from the capital at the end of the nth year. In summary, the following formula is for the capital on the payment date:
When broken system run times, the value date practice of banks is to be noted: When savings in Germany the Anlagetag is usually included in the calculation, the date of payment but will not earn interest. Otherwise - for example in demand and time deposits - While the payment date, but not the Einzahlungstag bears interest vice versa.
In less than one year interest rate one proceeds analogously and varied according to the reference period (eg, n in quarters, 90 instead of 360 in the denominator ).
Example
On 25 June 2008 € 1,000 be invested at an interest rate of 2.5 % in a savings account. What is the payout on termination of the passbook on April 12, 2013?
By the end of 2008 to pass after German interest calculation method days. The capital is the total annual fixed 2009-2012 ( n = 4). In 2013, be paid for daily interest.
The capital is on the payment date so
The calculation of simple interest favors the investor: if compound interest would be calculated over the entire period, one would obtain in the present case
Continuous compounding
The constant interest rate is a special case of under -standing interest with compound interest, in which the number of periods tends to infinity (even instantaneous rate and continuous compounding). The period of each Interest Period therefore approaches 0
For the final capital after n years at an interest rate i applies:
A starting capital of € 1,000 is invested at an interest rate of 5 percent over 2 years. With continuous compounding a final capital of would
One of the advantages of continuous compounding is that you have to worry about the interest capitalization, there is virtually anytime capitalized. Thus, the continuous compounding is often based on mathematical models, as is particularly easy to handle this type of interest. A well-known example is the Black-Scholes model.