Effective interest rate

The effective annual interest rate quantifies the annual costs of loans related to the nominal loan amount . It is given as a percentage of the payout. In the case of loans, the interest rate of which or other price-determining factors can change during the term, it is referred to as the initial APR .

The effective interest rate is essentially determined by the nominal interest rate , the payment rate ( discount ), the repayment and the fixed interest period.

Framework conditions for calculating the effective annual interest rate

With the help of the effective interest rate, only loan offers with the same fixed interest period can be compared.

If factors such as, in particular, repayment free years , repayment replacement, type of repayment offsetting, processing fees and loan fees have been mathematically correctly included in the effective interest rate calculation, then they can be quite different for compared loans, because the most important task of the effective interest rate calculation is to make differently structured loans comparable.

The effective interest rate does not include any estimate fees (tax costs or appraisal fees ), commitment interest , partial payment surcharges, notary fees and account management fees. This must be taken into account if offers obtained are to be compared objectively. In contrast to the nominal interest rate, the effective interest rate takes into account all other price-determining factors from the regular credit history, i.e. In other words , the effective interest rate indicates the total cost of the loan per year as a percentage. Factors that determine the price are nominal interest rate, processing fees, payout rate, repayment rate, start and amount, interest and repayment clearing dates.

Comparison of differently structured loans and investments

In addition to the calculation required by law, there are also universally applicable financial mathematical methods that determine an effective interest rate as a comparative measure from all payments into an installation and payments from the installation, regardless of the type and name of these payments , which can also be expressed as the effective annual interest rate. The independence of the type and designation of the payments in and out under consideration is an advantage of this classic method of so-called pension calculation . The underlying mathematics of the geometric series is relatively old, but was not sufficiently easy to implement for earlier price information regulations. However, the necessary (iterative) calculation methods can now be used without any problems when implemented on a computer. For the pure calculation of costs or returns, such calculations can even be used to compare very differently structured loans and investments , especially in the case of a subsequent valuation. This is particularly important when loans and investments have been designed in such a way that it is difficult to compare them with other financial products on the market. As a decision-making aid when selecting loans and investments, this effective interest rate also measures only one aspect of a loan or an investment. Other aspects such as risk, security, price development, etc. must also be assessed.

German price indication regulation

Pursuant to Section 6 (1) of the German Price Indication Ordinance , the total cost of loans must be stated as the annual percentage of the loan and denoted as the effective annual interest rate .

The procedure prescribed in the regulation now corresponds to the calculation of the internal rate of return . It is a long-known procedure in pension accounting and was introduced in Germany in 2000 under pressure from the EU Council of Ministers for Consumer Affairs (1996). In contrast to the old method, with today's calculation method the effective interest rate can no longer be manipulated so easily by distributing credit costs to differently weighted cost categories.

Because of the weaknesses of the PAngV method at that time, there were programs at the banks in which a method of the then AIBD (Standard Method of Calculating Yields for International Bonds, Association of International Bond Dealers, 1969–1992, Zurich) was used for internal effective interest calculation . However, the effective interest rate according to the PAngV at that time had to be indicated to the customer. In addition to the interest of the banks in interest rates that can be manipulated through product design, another reason for their resistance to the application of the internal rate of return was that this process is iterative: a computer needs several runs to calculate the internal rate of return until the required accuracy is achieved. But as early as the 1980s, the method was also available in pocket calculators and spreadsheet programs.

In the case of consumer loans , the indication of the effective annual interest rate according to § 492 Paragraph 2 BGB in conjunction with Art. 247 Paragraph 3 Paragraph 1 No. 3 EGBGB is mandatory in order to enable the consumer to compare interest rates. In order to protect the consumer, Section 494 (3) BGB additionally stipulates:

If the APR is indicated too low, the borrowing rate on which the consumer loan agreement is based is reduced by the percentage by which the APR is indicated too low.

Calculation of the annual percentage rate using the uniform method

The easiest way to calculate the approximate APR is the Uniform method:

${\ displaystyle {\ text {eff. Annual interest rate}} = {\ frac {\ text {Credit costs}} {\ text {Net loan amount}}} \ cdot {\ frac {24} {({\ text {Term in months}} + 1)}}}$

Borrowing costs = (total repayment - payout amount) or (number of installments × installment amount - payout amount)

this includes:

• Handling fee
• interest
• Possibly residual debt insurance or credit life insurance (if included in the offer)

Net Loan Amount = Nominal Loan Amount - Borrowing Cost

scope of application

The uniform method enables the effective interest rate to be estimated for certain types of loan. Legally valid and to be shown by financial service providers is the more complicated to calculate but more accurate effective interest rate according to PAngV. The uniform method is to be regarded as a rough calculation with which one can quickly get an impression of the actual effective interest rate to be expected, especially in the case of loans to be repaid with the same monthly installments. The result can deviate from the PAngV effective interest rate.

Example of an installment loan with constant monthly payments

A consumer loan of 10,000.00 EUR is taken out. The interest rate is 0.5% per month and refers to the original sum of 10,000.00 EUR during the entire term, the term is 60 months. A 3% processing fee is charged for provision. The processing fee is paid when the loan is taken out, the full 10,000.00 EUR are provided.

${\ displaystyle {{\ text {Interest}} = 0 {,} 5 \% {\ text {per month}} \ cdot 10,000 {,} 00 \ mathrm {\ euro} \ cdot 60 {\ text {months}} = 3,000 {,} 00 \ mathrm {\ euro}}}$
${\ displaystyle {{\ text {processing fee}} = 3 \% \ cdot 10,000 {,} 00 \ mathrm {\ euro} = 300 {,} 00 \ mathrm {\ euro}}}$
${\ displaystyle {{\ text {Credit costs}} = {\ text {Interest}} + {\ text {Processing fee}} = 0 {,} 005 \ cdot 10000 \ cdot 60 + 0 {,} 03 \ cdot 10000 = 3,300 {,} 00 \ mathrm {\ euro}}}$

Note: Interest and repayment are paid monthly, but the loan amount is only considered fully repaid at the end of the term, i.e. H. monthly is a rate of

${\ displaystyle {\ frac {3,300 \ mathrm {\ euro}} {60}} + {\ frac {10,000 \ mathrm {\ euro}} {60}} = 55 \ mathrm {\ euro} +166 {,} 67 \ mathrm {\ euro} = 221 {,} 67 \ mathrm {\ euro}}$ due.
${\ displaystyle {\ text {eff. Annual interest rate}} = {\ frac {3,300 {,} 00 \ mathrm {\ euro}} {10,000 {,} 00 \ mathrm {\ euro}}} \ cdot {\ frac {24} {60 + 1}} = 12 {,} 9836 \%}$

The eff. Interest rate (annual interest rate) is calculated by adding interest to all income and expenses at an interest point in time with the result “zero”. Then all income and expenses are equal to interest.

Compared to the annuity method , the approximation using the uniform method results in a lower interest rate for short terms (or a higher rate for long terms). In the example above, the annuity method results in an effective annual interest rate of 12.5115%.

Calculation of the eff. Annual interest rate for bonds

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

${\ displaystyle {\ text {eff. Annual interest rate}} = \ left [\ left (\ prod _ {i = 1} ^ {\ text {term}} \ left (1 + {\ frac {\% {\ text {interest rate}} _ {i}} { 100 \%}} \ right) \ right) ^ {\ frac {1} {\ text {Runtime}}} - 1 \ right] \ cdot 100 \%}$

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

${\ displaystyle {\ text {eff. Annual interest rate}} = \ left [\ left (\ left (1 + {\ frac {1 {,} 5 \%} {100 \%}} \ right) \ cdot \ left (1 + {\ frac {2 \% } {100 \%}} \ right) \ cdot \ left (1 + {\ frac {3 \%} {100 \%}} \ right) \ right) ^ {\ frac {1} {3}} - 1 \ right] \ cdot 100 \%}$
${\ displaystyle = \ left [(1 {,} 015 \ cdot 1 {,} 02 \ cdot 1 {,} 03) ^ {\ frac {1} {3}} - 1 \ right] \ cdot 100 \%}$
${\ displaystyle = \ left [{\ sqrt [{3}] {1 {,} 06636}} - 1 \ right] \ cdot 100 \%}$
${\ displaystyle = \ lbrack 1 {,} 02165-1 \ rbrack \ cdot 100 \%}$
${\ displaystyle = 0 {,} 02165 \ cdot 100 \%}$
${\ displaystyle = 2 {,} 165 \%}$

In the case of bonds, the approximate formula with surcharges ( agio ) and discounts ( disagio ) is often used :

${\ displaystyle \% {\ text {Yield pa}} \ approx \% {\ text {Interest rate pa}} \ cdot {\ frac {\ text {Nominal value}} {\ text {Purchase price}}} \ pm {\ frac {\ text {Agio or Disagio}} {\ text {Duration}}}}$

(This formula hardly deviates from the correct result for small values ​​for term and premium / discount, but fails for correspondingly large values.)

However, the factor is just another interest factor with which a more precise formula can then be constructed: ${\ displaystyle {\ frac {\ text {Nominal value}} {\ text {Buy price}}}}$

${\ displaystyle {\ text {eff. Annual interest rate}} = \ left [\ left ({\ frac {\ text {nominal value}} {\ text {purchase price}}} \ prod _ {i = 1} ^ {\ text {term}} \ left (1+ { \ frac {\% {\ text {Interest rate}} _ {i}} {100 \%}} \ right) \ right) ^ {\ frac {1} {\ text {Term}}} - 1 \ right] \ cdot 100 \%}$

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

The following calculation rule is derived for loans for which neither one-off surcharges (processing fees) nor discounts (disagio) have been agreed.

The so-called. "Nominal" annual interest rate is strictly speaking not a real year interest rate, but only the - if the loan in monthly to be paid off rate - twelve times the actually coming to apply "effective month interest rate" (the one with a view of the "nominal annual interest rate" in the interest calculation also referred to as the "relative period interest rate"), which means that after each interest period - in this case after every month or twelfth of the year - the balance is balanced and recalculated. In contrast to real annual interest, the compound interest effect takes effect after the first month, which means that the “effective annual interest rate” is always higher than the “nominal” interest rate reported by the banks.

To derive the calculation rule, we compare the formation of the monthly or year-end account amounts for monthly (exponential) and year-end (linear) balancing. The following variables play a role:

 G 0 = Debt at the beginning of the year R. = monthly installment, the interest and u. U. also contains repayment z = nominal bank interest rate z eff = effective annual interest rate

For the nominal annual interest rate specified by the bank, the following applies after 12 months and thus 12 additional installments:

${\ displaystyle G_ {1} = G_ {0} \ left (1 + {\ frac {z} {12}} \ right) -R}$
${\ displaystyle G_ {2} = G_ {1} \ left (1 + {\ frac {z} {12}} \ right) -R = G_ {0} \ left (1 + {\ frac {z} {12 }} \ right) ^ {2} -R \ left [\ left (1 + {\ frac {z} {12}} \ right) +1 \ right]}$
${\ displaystyle G_ {3} = G_ {0} \ left (1 + {\ frac {z} {12}} \ right) ^ {3} -R \ left [\ left (1 + {\ frac {z}} {12}} \ right) ^ {2} + \ left (1 + {\ frac {z} {12}} \ right) ^ {1} + \ left (1 + {\ frac {z} {12}} \ right) ^ {0} \ right]}$
${\ displaystyle \ vdots}$
${\ displaystyle G_ {12} = G_ {0} \ left (1 + {\ frac {z} {12}} \ right) ^ {12} -R \ left [\ left (1 + {\ frac {z}} {12}} \ right) ^ {11} + \ cdots + \ left (1 + {\ frac {z} {12}} \ right) ^ {0} \ right]}$
${\ displaystyle G_ {12} = G_ {0} \ left (1 + {\ frac {z} {12}} \ right) ^ {12} -R {\ frac {\ left (1 + {\ frac {z } {12}} \ right) ^ {12} -1} {\ left (1 + {\ frac {z} {12}} \ right) -1}}}$
${\ displaystyle G_ {12} = G_ {0} \ left (1 + {\ frac {z} {12}} \ right) ^ {12} - {\ frac {12R} {z}} \ left [\ left (1 + {\ frac {z} {12}} \ right) ^ {12} -1 \ right]}$
${\ displaystyle G_ {12} = \ left (G_ {0} - {\ frac {12R} {z}} \ right) \ left (1 + {\ frac {z} {12}} \ right) ^ {12 } + {\ frac {12R} {z}}}$

For the effective annual interest rate, on the other hand, after one year, i.e. again 12 subsequent installments:

${\ displaystyle G_ {12} = G_ {0} \ left (1 + z _ {\ mathrm {eff}} \ right) -R \ left [12+ \ left ({\ frac {1} {12}} + \ cdots + {\ frac {11} {12}} \ right) z _ {\ mathrm {eff}} \ right] = G_ {0} \ left (1 + z _ {\ mathrm {eff}} \ right) -R \ left [12+ \ left ({\ frac {11 \ cdot 12} {2 \ cdot 12}} \ right) z _ {\ mathrm {eff}} \ right]}$

The subtract results on the one hand from the installments that do not contain any interest payments within the year - since an effective annual interest rate is only applied at the end of the year - and from the interest on these installments made before the end of the year up to the end of the year: the first is eleven months on, the second ten, etc. and the last is paid exactly at the end of the year and therefore does not earn any interest. This interest must also be credited to the repayment.

Equating both formulas for G 12 and replacing R by x G 0 finally yields the following formula for z eff :

${\ displaystyle z _ {\ mathrm {eff}} = 2 {\ frac {\ left (1 + {\ frac {z} {12}} \ right) ^ {12} (z-12x) + 12x (z + 1 ) -z} {z (2-11x)}}}$

The calculation of the effective annual interest rate is therefore not only dependent on the bank interest rate, but also on the speed of repayment, i.e. on the ratio of the installments R to the initial debt G 0 . The formula is greatly simplified if there is no more repayment at all, but the installments only pay the interest due (so-called "perpetual loan"). Then x = z / 12, and from this it follows:

${\ displaystyle z _ {\ mathrm {eff}} = {\ frac {24z} {24-11z}}}$

This formula may seem implausible because it is not defined for z = 24/11 and even gives nonsensical negative results for even higher values ​​of z . However, one has to understand what it means to be presented with a bank interest rate of 218%. Within six months, a greater sum than the debt amount at the beginning of the year will be paid out of the monthly amounts. According to the effective annual interest rate approach, this means that the debtor, who does not pay interest during the year, but only repays it, forms a credit balance with the bank from the sixth month on. Of course, the bank has to pay interest on this credit just as it did on the debt - only with the opposite sign . Both interest rates are offset against each other and must result in zero without any repayment. But that cannot work if the debt amount from the beginning of the year has already been paid off before the half of the year. As a result, the interest on the installments paid during the year rather than at the end of the year must be decisive. And for this they have to be very high - in extreme cases, infinite.

For interest rates of normal magnitude, however, the formula delivers not only (also) correct, but also plausible results. If a debt of 100 euros is repaid in monthly installments within one year, with a nominal bank interest rate of 10%, the effective annual interest rate is 10.65%. If the debt is only to be held at the same bank interest rate, the effective annual interest rate is 10.48%.

Effective interest rate on construction loans

On June 11, 2010, a new consumer credit directive came into force. As part of this implementation, new rules apply to the calculation of the effective annual interest rate for real estate loans: If a contract provides for the loan to continue with variable interest rates if the debtor and creditor do not agree on a new fixed interest rate by the end of the fixed interest rate, the Price Indication Ordinance requires that the bank uses its current interest rate for floating rate loans for the remaining term. This is generally below the interest rate during the fixed interest period. This often results in an effective interest rate below the borrowing rate.

Effective interest rate with discount

A discount is a discount on the nominal value or interest paid in advance for the duration of the fixed interest rate, which is expressed in a lower payment of the loan amount. The effective interest rate is calculated using the nominal interest rate (and other parameters such as term, repayment). Conversely, the associated nominal interest rate can be determined on the basis of an effective interest rate.

The effective interest rate in the case of discount acts as a measure that makes a loan variant based on a lower payout and therefore offered with a different nominal interest rate comparable to the full payout variant. Basically, each of the two variants is a different packaging of the same product, so that an equivalence of the two variants is required and the mathematical modeling to calculate the effective interest rate is carried out in the sense of this equivalence.

There are different approaches to determining the effective interest rate, which arise from different interpretations of the concept of discount and therefore do not necessarily lead to the same result. One approach assumes that the interest amounts calculated at the nominal or effective interest rate must be the same for the duration of the loan. (Disagio as interest paid in advance, see installment loans below). Another approach used in accordance with the PAngV requires that the payment flows relevant to the nominal interest rate (annuity, residual debt) must correspond to the final value calculation at the effective interest rate (see annuity loan below).

The effective interest rate in the case of discount will be higher than the nominal interest rate, since, to put it clearly, the installment / annuity payments on the basis of the lower disbursement are lower and must therefore correspond to a higher interest rate to compensate.

For the sake of simplicity, a loan of 1 monetary unit with a term of interest periods (about years) is assumed, which is taken out at a nominal interest rate of . The discount rate is here . E.g. should mean that 90% of the capital is paid out. A discount rate of 0 means no discount, while a discount rate of 1 means that no payment will be made. Since the last case does not make any practical sense, the discount rate will be between 0 and 1 only. ${\ displaystyle n}$${\ displaystyle r _ {\ text {nom}}}$${\ displaystyle d}$${\ displaystyle d = 0 {,} 1}$

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

Installment loan

Repayment on maturity

With this amortization loan, only the interest is paid during the term. The repayment is only made at the end of the term.

The following must apply:

${\ displaystyle no _ {\ text {nom}} + d = (1-d) no _ {\ text {eff}}}$.

From this it follows for the effective interest rate:

${\ displaystyle r _ {\ text {eff}} = {\ frac {r _ {\ text {nom}} + {\ frac {d} {n}}} {(1-d)}}}$.
example 1

A discount rate of and a nominal interest rate of for the duration of periods results in an effective interest rate of 11.58%. ${\ displaystyle d = 0 {,} 05}$${\ displaystyle r _ {\ text {nom}} = 0 {,} 1}$${\ displaystyle n = 5}$${\ displaystyle r _ {\ text {eff}} = 0 {,} 1158}$

Repayment in equal installments

The remaining loan is in the - period and the sum of the interest in both cases is: ${\ displaystyle i}$${\ displaystyle {\ frac {(ni)} {n}}, i = 0, \ ldots, n-1}$${\ displaystyle Z}$

${\ displaystyle Z _ {\ text {eff}} = (1-d) {\ frac {r _ {\ text {eff}}} {n}} \ sum _ {i = 0} ^ {n-1} ni = (1-d) {\ frac {(n + 1)} {2}} r _ {\ text {eff}}}$ or.
${\ displaystyle Z _ {\ text {nom}} = {\ frac {(n + 1)} {2}} r _ {\ text {nom}}}$.

The following must apply:

${\ displaystyle Z _ {\ text {nom}} + d = Z _ {\ text {eff}}}$.

From this follows for the effective interest rate

${\ displaystyle r _ {\ text {eff}} = {\ frac {2 (Z _ {\ text {nom}} + d)} {(1-d) (n + 1)}}}$.

The expression can be viewed as the mean running time. ${\ displaystyle {\ frac {n + 1} {2}}}$

The variant of the bullet loan is a special case here, namely when the number of installments is 1, that is. ${\ displaystyle n = 1}$

Example 2

A discount rate of and a nominal interest rate of for the duration of periods results in an effective interest rate of 12.28%. ${\ displaystyle d = 0 {,} 05}$${\ displaystyle r _ {\ text {nom}} = 0 {,} 1}$${\ displaystyle n = 5}$${\ displaystyle r _ {\ text {eff}} = 0 {,} 1228}$

Repayment after k repayment-free periods in periodically equal installments

This is a variant of the combination of the two preceding. Here are grace periods adopted. The sum of the interest in both cases is: ${\ displaystyle k}$

${\ displaystyle Z _ {\ text {eff}} = (1-d) [kr _ {\ text {eff}} + {\ frac {r _ {\ text {eff}}} {nk}} \ sum _ {i = 0} ^ {nk-1} {nki}] = (1-d) \ left [{\ frac {(n + k + 1)} {2}} \ right] r _ {\ text {eff}}}$ or.
${\ displaystyle Z _ {\ text {nom}} = \ left [{\ frac {(n + k + 1)} {2}} \ right] r _ {\ text {nom}}}$.

From the condition it follows for the effective interest rate: ${\ displaystyle Z _ {\ text {nom}} + d = Z _ {\ text {eff}}}$

${\ displaystyle r _ {\ text {eff}} = {\ frac {2 (Z _ {\ text {nom}} + d)} {(1-d) (n + k + 1)}}}$.

It can be seen that this variant is identical to the case of repayment in periodically equal installments . ${\ displaystyle k = 0}$

Example 3

A discount rate of and a nominal interest rate of for the duration of periods with 2 grace periods results in an effective interest rate of 11.84%. ${\ displaystyle d = 0 {,} 05}$${\ displaystyle r _ {\ text {nom}} = 0 {,} 1}$${\ displaystyle n = 5}$${\ displaystyle r _ {\ text {eff}} = 0 {,} 1184}$

Annuity loan

In contrast to the amortization loan discussed above, an annuity loan is one in which a constant rate composed of interest and amortization is periodically paid during the agreed fixed interest period . The fixed interest period can differ from the repayment period , which is required to fully repay the loan on the agreed terms. ${\ displaystyle t}$${\ displaystyle n}$${\ displaystyle N}$

The approach to determining the effective interest rate consists in weighting the cash flows arising from the nominal interest rate with the effective interest rate that leads to the solution of the equivalence equation.

Equivalence equation

First, the remaining debt at the end of the fixed interest period is determined. The following applies: ${\ displaystyle R}$

${\ displaystyle R = (1 + r _ {\ text {nom}}) ^ {n} - (r _ {\ text {nom}} + t) {\ frac {(1 + r _ {\ text {nom}}) ^ {n} -1} {r _ {\ text {nom}}}} = 1-t {\ frac {(1 + r _ {\ text {nom}}) ^ {n} -1} {r _ {\ text {nom}}}}}$.

The individual cash flows are the weighted annuities in the respective period . ${\ displaystyle (r _ {\ text {nom}} + t)}$

${\ displaystyle (1-d) (1 + r _ {\ text {eff}}) ^ {n} = (r _ {\ text {nom}} + t) {\ frac {(1 + r _ {\ text {eff }}) ^ {n} -1} {r _ {\ text {eff}}}} + R}$

or reformulated:

${\ displaystyle r _ {\ text {eff}} = {\ frac {r _ {\ text {nom}} + t} {1-d}} {\ frac {(1 + r _ {\ text {eff}}) ^ {n} -1} {(1 + r _ {\ text {eff}}) ^ {n} - {\ frac {R} {1-d}}}}}$

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

Example 4

A discount rate of and a nominal interest rate of for the duration of periods results in an effective interest rate of 11.37%. ${\ displaystyle d = 0 {,} 06}$${\ displaystyle r _ {\ text {nom}} = 0 {,} 1}$${\ displaystyle n = 7}$${\ displaystyle r _ {\ text {eff}} = 0 {,} 1137}$

Existence and uniqueness

The fixed point equation cannot always be directly solvable, so that, in particular for larger iterative processes, methods are necessary to approximate the solution. One possible approximation method is the fixed point iteration using the iteration function ${\ displaystyle n}$

${\ displaystyle r_ {i + 1} = F (r_ {i}) = {\ frac {r _ {\ text {nom}} + t} {1-d}} {\ frac {(1 + r_ {i} ) ^ {n} -1} {(1 + r_ {i}) ^ {n} - {\ frac {R} {1-d}}}} =: {\ frac {r _ {\ text {nom}} + t} {1-d}} {\ frac {f_ {1} (r_ {i})} {f_ {2} (r_ {i})}}}$.

When using this method, one proceeds in such a way that one finds an interval which is mapped in itself by the iteration function and which meets the requirements z. B. the following sentence (see sentence on existence and uniqueness in fixed point iteration ) is sufficient.

In the following, a paradigmatic approach is presented that examines the conditions for the existence of a unique solution. It is assumed here. ${\ displaystyle r _ {\ text {nom}}> 0}$

1. If . then is .${\ displaystyle {\ frac {R} {1-d}} = 1}$${\ displaystyle r _ {\ text {eff}} = {\ frac {r _ {\ text {nom}} + t} {1-d}}}$
2. If then applies ${\ displaystyle {\ frac {R} {1-d}} <1}$
1. ${\ displaystyle {\ frac {f_ {1} (r)} {f_ {2} (r)}} <1}$for everyone .${\ displaystyle r> 0}$
2. ${\ displaystyle F}$is monotonously increasing for everyone .${\ displaystyle r> 0}$
3. ${\ displaystyle F (r)> {\ frac {r _ {\ text {nom}}} {1-d}}}$for all ( complete induction over with and 2.2).${\ displaystyle r \ geq {\ frac {r _ {\ text {nom}}} {1-d}}}$${\ displaystyle n}$${\ displaystyle r = {\ frac {r _ {\ text {nom}}} {1-d}}}$
4. ${\ displaystyle F (r) <{\ frac {r _ {\ text {nom}} + t} {1-d}}}$for everyone (because of 2.1.).${\ displaystyle r> 0}$
5. ${\ displaystyle F}$maps the interval in itself (because of 2.3 and 2.4).${\ displaystyle I_ {1} = \ left [{\ frac {r _ {\ text {nom}}} {1-d}}, {\ frac {r _ {\ text {nom}} + t} {1-d }} \ right]}$
3. If so , then ${\ displaystyle {\ frac {R} {1-d}}> 1}$
1. ${\ displaystyle {\ frac {f_ {1} (r)} {f_ {2} (r)}}> 1}$for everyone .${\ displaystyle r> 0: f_ {2} \ left ({\ frac {r + t} {1-d}} \ right)> 0}$
2. ${\ displaystyle F}$is monotonously decreasing for everyone .${\ displaystyle r> 0: f_ {2} \ left ({\ frac {r + t} {1-d}} \ right)> 0}$
3. ${\ displaystyle F (r)> {\ frac {r _ {\ text {nom}} + t} {1-d}}}$for all and for (because of 3.1).${\ displaystyle r \ geq {\ frac {r _ {\ text {nom}} + t} {1-d}}}$${\ displaystyle f_ {2} \ left ({\ frac {r _ {\ text {nom}} + t} {1-d}} \ right)> 0}$
4. ${\ displaystyle F (r) for all and (because of 3.2).${\ displaystyle r> {\ frac {r _ {\ text {nom}} + t} {1-d}}}$${\ displaystyle f_ {2} \ left ({\ frac {r _ {\ text {nom}} + t} {1-d}} \ right)> 0}$
5. ${\ displaystyle F}$maps the interval in itself (because of 3.3 and 3.4).${\ displaystyle I_ {2} = \ left [{\ frac {r _ {\ text {nom}} + t} {1-d}}, F \ left ({\ frac {r _ {\ text {nom}} + t} {1-d}} \ right) \ right]}$
4. ${\ displaystyle F '\ neq 1}$for everyone .${\ displaystyle r \ geq {\ frac {r _ {\ text {nom}}} {1-d}}}$

From points 2.5, 3.5 and 4. it follows that the fixed point iteration in the intervals or has a clear fixed point, i.e. it exists there (see theorem on the existence and uniqueness in fixed point iteration ). ${\ displaystyle I_ {1}}$${\ displaystyle I_ {2}}$${\ displaystyle r _ {\ text {eff}}}$

Examples

${\ displaystyle d}$ ${\ displaystyle n}$ ${\ displaystyle t}$ ${\ displaystyle r _ {\ text {nom}}}$ ${\ displaystyle R}$ ${\ displaystyle I_ {1}}$ ${\ displaystyle I_ {2}}$ ${\ displaystyle r _ {\ text {eff}}}$
0.06 5 0.02 0.1000 0.8779 [0.1064; 0.1277] 0.1172
0.06 7th 0.02 0.1000 0.8103 [0.1064; 0.1277] 0.1137
0.05 1 1.00 0.0070 0.0000 [0.0074; 1.0600] 0.0599
0.05 5 0.01 0.0342 0.9465 [0.0359; 0.0465] 0.0458
0.10 7th 0.01 0.0342 0.9224 [0.0491; 0.0523] 0.0521
0.10 15th 0.02 0.0342 0.6165 [0.0379; 0.0602] 0.0450
0.15 5 0.01 0.0325 0.9360 [0.0524; 0.0803] 0.0699
0.25 5 0.01 0.0300 0.9469 [0.0533; 0.4637] 0.0695
0.30 2 0.07 0.0250 0.8583 [0.1357; 0.6168] 0.2368