Annual percentage rate

The annual percentage rate, or more precisely, the annual percentage rate estimates the annual and related to the nominal amount of the loan costs of loans. It is expressed as a percentage of payout. For loans, the interest rate or other price-determining factors may change during the term, it is referred to as the initial annual percentage rate.

The effective interest rate is essentially determined by the nominal interest rate, the disbursement rate ( discount), the principal and the interest rate fixed duration.

• 8.1 installment loan 8.1.1 Redemption at maturity
• 8.1.2 repayment in equal installments periodically
• 8.1.3 Repayment by k grace periods in periodic equal installments

• 8.2.1 Equivalence equation
• 8.2.2 Existence and Uniqueness

Framework for the calculation of the annual percentage rate

By using the effective interest only loan quotes with the same fixed interest rate period can be compared.

If factors such as particular redemption -free years, eradication replacement, type of redemption clearing, handling charges and loan fees were included mathematically correct in the effective interest rate determination, then they can be quite different in the compared loans, because the most important task of the effective interest rate calculation is precisely to make different shaped loans comparable.

The effective interest rate no valuation fees ( Taxkosten or valuation fees), provisioning rates, partial payment surcharges and account management fees are included. This must be considered when powered picked deals are to be compared objectively. The effective interest rate takes into account, in contrast to the nominal interest rate all other price-determining factors from the regular credit history, ie, the effective interest rate is the total cost of the loan per year in percent. Price determining factors are nominal interest rate, processing fees, disbursement rate, repayment rate, the beginning and height, interest and redemption settlement dates.

Comparison of several characteristics of loans and investments

In addition to the statutory calculation, there are universal mathematical methods, the judge all the payments into an investment and withdrawals from the system regardless of the type and designation of such payments an effective interest rate as a measure of comparison, which can be expressed as an annual percentage rate as well. The independence of the nature and wording of the considered deposits and withdrawals is an advantage of this classical method known as pension bill. The mathematics behind the geometric series is relatively old, but it was for previous price information regulations do not sufficiently simple to implement. However, the necessary computer Implements ( iterative ) computation methods can be used without any problems today. For the mere cost or yield calculation such calculations can even use to compare very differently designed loans and investments, especially in a subsequent review. This is especially important if loans and investments were designed so that a comparison with other financial products in the market is difficult. As a decision aid in the selection of loans and investments also measures the effective interest rate but only one aspect of a loan or an investment. Other aspects such as risk, safety, pricing, etc. must also be evaluated.

Price regulation

According to § 6 para 1 PAngV the total cost of loans as the price to be specified as an annual percentage of the loan and to call the annual percentage rate.

The prescribed in the Regulation now corresponds to the calculation of the internal rate of return. It is in the pension bill long known method and was in Germany in 2000 under pressure from the EU Council of Ministers for Consumer Affairs (1996 ) introduced. Unlike the old method can be not so easy manipulated by the distribution of credit costs on differently weighted categories of costs with today's method of calculating the effective interest rate.

Due to the weaknesses of the former PAngV method there was at that time at the banks programs where the internal effective interest calculation method of the former AIBD (Standard Method of Calculating Yields for International Bonds, Association of International Bond Dealers, from 1969 to 1992, Zurich ) was used. Compared to the customers but had to be given to the effective interest rate after that time PAngV. In addition to the interest of the banks to be manipulated through product design rate of charge was another reason for their opposition to the use of the internal rate of return, that this process is iterative: A computer needs to calculate the internal rate of multiple passes until the required accuracy is achieved. But already in the 80s was the method is also used in calculators and spreadsheet programs.

For consumer loans, the indication of the annual percentage rate of 2 BGB in connection with § 492 Para consulted in accordance with Article 247 § 3 para 1 No. 3 BGB necessarily the content of the Treaty in order to enable the consumer interest comparisons. To protect consumers in § 494 paragraph 3 of the Civil Code is also determined:

If the annual percentage rate set too low, then the consumer loan agreement underlying borrowing rate is reduced by the percentage by which the annual percentage rate is set too low.

Calculate the annual interest by the uniform method

The simplest way to calculate the approximate annual percentage rate is the uniform method:

Credit cost = ( total repayment - payment amount ) or ( number of installments × rate amount - amount paid )

This information can include:

• Processing fee
• Interest
• Any payment protection insurance or credit life insurance (if required in offer)

Net loan amount = loan principal amount - cost of credit

Scope

The uniform method allows for certain types of loans, an estimate of the effective interest rate. Expel legally valid and of financial services is more complicated to be calculated, but more accurate effective interest rate after PAngV. The uniform method should be regarded as rough calculation, with which you can quickly get especially when with equal monthly installments to be repaid loans an impression of the expected actual effective interest rate. The result may differ from the PAngV - effective rate.

Example

A consumer loan of 10,000.00 EUR is taken. The interest rate is 0.5 % per month and is during the entire term of the original sum of EUR 10,000.00, the term is 60 months. To provide 3 % handling fee may be charged. The processing fee is paid when borrowing, provided that full 10,000.00 EUR.

Note: Interest and principal payments are monthly, but the credit amount is only at the end of the term as fully repaid, ie a monthly rate of

The eff. Interest rate ( annual interest rate) is calculated by discounting all income and expense on an interest date with the result " zero ". Then correspond discounted revenue and expenditure.

Assuming the disbursement of € 10,000 and the fee € 300 the time of payment as well as 60 monthly installments in the amount of € 221.67, beginning with the time of payment results in an eff. Interest rate of 14.5950 %. If the mortgage payments start one month later, the eff. Interest rate 14.0221 %. The each to be determined gezinste amount will be calculated as follows: Amount x (1 interest rate) ^ " in years " where " in years " may be a fractional value and a negative value.

Calculation of eff. Annual interest rate on bonds

The effective annual interest rate for bullet bonds that run over several years and the interest rate again mitverzinsen ( compound interest ) is calculated with interest factors:

Example: A bond runs three years and bears interest at 1.5 % in the first, 2 % in the second and 3 % in the third year.

For bonds with the approximation formula with up (premium) and discounts (discount ) is often calculated:

( This formula deviates at small values ​​for runtime and premium / discount hardly the correct result from, but failed to sufficiently large values. )

However, there is the factor by only a further interest factor with which then a more precise formula can be constructed:

Calculation of eff. Annual interest rate for loans with fixed monthly installments

The following calculation formula is derived for loans for which neither unique Supplements ( processing fees ) nor surcharges ( discount) are agreed.

The usually specified by the bank interest rate is actually absolutely no annual interest rate, but twelve times an "effective interest rate month ". After each month - and after each twelfth of the year - is therefore expected offset and new. The compounding effect thus occurs after the first month, and this means that under the conditions specified in this section, the effective annual interest rate is always higher than said from the bank interest rate.

To derive the calculation rule we illustrate the formation of monthly or annual account finite amounts for monthly and annual finite netting against each other. The following variables play a role:

For the interest rate quoted by the Bank applies after i months:

For the effective annual interest rate applies after one year ( in comparison to the above formula for i = 12):

The subtrahend results partly from the rates that do not include interest payments within the year - as an effective annual interest rate is applied only at the end of the year - and the interest on these payments made before year-end rates until the end of the year: The first is eleven months to the second ten, etc., and the last one is done exactly at the end and therefore scored no interest. This interest rate must also be credited to the repaid.

Equating the two formulas for G12 and replacement of R by x G0 finally delivers for zeff following formula:

The calculation of the annual percentage rate is not only from the bank interest rate, but also on the speed of repayment dependent, ie the ratio between R and G0. Is greatly simplified the formula if no redemption occurs, but pay the rates, the interest due only. Then z = x / 12, and it follows:

This formula may seem implausible because they = 24/11 is not defined for z and provides for even higher values ​​of z even nonsensical negative results. One must realize, however, what it means to get a bank rate of interest presented by 218 %. Within six months will be paid on monthly amounts, a larger sum than the amount owed represents the beginning of the year. According to the procedure of the effective annual interest rate, this means that the debtor who yes during the year does not pay interest, but only wipes out, forms a deposit at the bank from the sixth month. This credit must also pay interest on the debt as before, the Bank of course - but the other way. Both interest rates are offset against each other and must evaluate to zero without any repayment. But this can not work if the amount owed has been repaid by the beginning of the year before the half. Consequently, the interest on the payments made in the course rather than at the end of the year must be decisive. And she 'll just have to be very high - indefinitely in extreme cases.

For interest rates in normal orders of magnitude, however, the formula not only delivers (again ) correct, but also plausible results. When a liability sum of 100 euros repaid within a year, with a bank interest rate is set to 10%, so the effective annual interest rate is about 10.65%. If the debt with the same bank interest rate will only be maintained, so the effective annual interest rate amounts to 10.48 %

Effective interest rate on construction loan

On 11 June 2010 a new Consumer Credit Directive came into force. As part of the implementation of new rules for the calculation of the annual percentage rate on real estate loans are: Provides for a contract that the loan continues with variable interest when the debtor and creditor does not agree by the end of the fixed interest rate on a new fixed-rate period, the Commission shall request the price regulation that the bank for the remaining term of their current interest rate for variable rate loans based on sets. This is generally less than the interest rate during the fixed interest period. This often results in an effective interest rate below the target rate.

Effective interest rate for discount

When discount is a discount to face value or at a pre- paid interest for the duration of the fixed interest rate, which is reflected in a lower payment of the loan amount The calculation of the effective interest rate by the nominal interest rate ( and the other parameters such as maturity, redemption ). Conversely, can be determined on the basis of an effective interest rate of the corresponding nominal interest rate.

The effective interest rate in the case of discount acts as a measure that makes a payment based on lower and consequently offered a differing nominal interest rate variant of a loan with the variant of the full payout comparable. Basically, it is in each of the two versions to a different packaging of the same product, so that one calls for equivalence of the two variants and performs the mathematical modeling for calculating the effective interest rate the purposes of this equivalence.

There are different approaches for calculating the effective interest rate resulting from different interpretations of the concept of discount and therefore not necessary lead to the same result. One approach assumes that the respective calculated at nominal or effective interest rate interest amounts during the loan period must be equal. (Discount as interest in advance of advance, see below installment loans ). Another, according to PAngV used approach requires that the relevant nominal interest rate cash flows (annuity, residual debt ) must comply with the Endwertberechnung to Effetivzinssatz (see below annuity ).

The effective interest rate on discount will be higher than the nominal interest rate, as spoken clearly fail the Raten-/Annuitätenzahlungen based on the lower payout is smaller and thus correspond to compensate for a higher interest rate must

For simplicity, one assumes a loan of one unit of money with a duration of Interest Periods (about years ), which is added to a nominal interest rate of. This amounts to the Disagiosatz. For example, to mean that 90 % of the capital is paid. A Disagiosatz of 0 means no discount, while a Disagiosatz of 1 means that no payment will be made. Since the second case does not make practical sense, the Disagiosatz between 0 and only one will be.

Below some loan types are exemplified with the corresponding effective interest rate calculation.

Installment loan

Redemption at maturity

In this amortization loan only the interest is paid during the term. The repayment will be made only at the end of the term.

The following must apply:

It follows for the effective interest rate:

A Disagiosatz of and a nominal interest rate of for the duration of periods results in an effective interest rate, ie 11.58%.

Repayment in equal installments periodically

The remaining loan in the - th period, and is the sum of the interest in both cases:

The following must apply:

It follows for the effective interest rate

The term can be regarded as the average run time.

The variant of the bullet loan is this a special case, namely when the number of installments equal to 1, so.

A Disagiosatz of and a nominal interest rate of for the duration of periods results in an effective interest rate, ie 12.28%.

Repayment by k grace periods in periodic equal installments

This is a variation of the combination of the previous two. This grace period will be accepted. The amount of interest in both cases is:

From the condition for the effective interest rate follows:

One can see that this variant is identical to the case of repayment in equal installments periodically.

A Disagiosatz of and a nominal interest rate of for the duration of periods with two grace periods results in an effective interest rate of, ie 11.84%.

Annuity

In contrast to the above discussed amortization loan is an annuity loan one, in which periodically a composite of interest and repayment constant rate to be paid during the agreed interest period. The fixed interest rate period may be different from the repayment duration that is required to the loan at the agreed terms would be fully redeemed.

The approach in determining the effective interest rate is to weight the emissions from the nominal interest rate cash flows at the effective interest rate, which leads to the solution of the equivalence equation.

Equivalence equation

First, the remaining debt is determined at the end of the interest period. The following applies:

The individual cash flows are in the respective period corresponding weights ended annuities.

The equivalence equation is then:

Or reformulated:

The effective interest rate is then the rate of interest at which the sum of the discounted annuities is equal to the final value of the payout minus outstanding debt.

A Disagiosatz of and a nominal interest rate of for the duration of periods results in an effective interest rate, ie 11.37%.

Existence and uniqueness

The fixed point equation can not always be solved directly, so that in particular for larger Iterative methods are required to approximate the solution. A possible approximation is the fixed-point iteration using the Iterationsfunkton

In the application of this process is going on so that you can find an interval that is imaged by the iteration function in itself and satisfies the requirements of, for example, the following theorem (see theorem on the existence and uniqueness in fixed-point iteration ).

Below is shown a paradigmatic approach, which examines the conditions for existence of a unique solution. In this case is assumed.

From the points 2.5, 3.5 and 4 it follows that the fixed point in the intervals or a unique fixed point has, therefore there exists (see theorem on the existence and uniqueness in fixed-point iteration ).

Examples