Effective Interest Methods

Use

The system uses a specific effective interest method to calculate the effective interest for a financial transaction.

The various effective interest methods differ with regard to the following criteria used for the comparison account in the system:

Interest calculation type (linear or exponential)

For more information, see Linear Interest Calculation and Exponential Interest Calculation.
Interest calculation method (for example, 30/360, act/365)

For more information, see Interest Calculation Methods.
Interest capitalization event

The following table provides an overview of all the effective interest methods supported by financial mathematics.

Overview of Effective Interest Methods

Number	Effective Interest Method	Interest Calculation Type	Interest Capitalization	Interest Calculation Method	Use
1	PAngV	Linear	Annual, starting from the start of the comparison account	30/360	Obsolete (previously Germany)
2	AIBD/ISMA	Exponential	On each payment date	Can be defined freely	International
3	Braess	Linear	Annual, starting from the end of the comparison account	30/360	Germany
4	Moosmueller	Linear	On each payment date	Can be defined freely	Germany
5	US Treasury	Linear	On each payment date	Can be defined freely	United States
6	EU Act/365	Exponential	On each payment date	act/365	EU
7	Bond Formula	Exponential	On each payment date	act/actP	International
8	EU 30.42/365	Exponential	On each payment date	30.42/365	EU
9	Linear	Linear	Once, at the end of the comparison account	Can be defined freely	Technical
10	U.S. APR	Linear	On each payment date	Dependent on unit period	United States

The following example illustrates the comparison accounts for all the effective interest methods supported by financial mathematics and outlines the differences between them.

A bank's customer takes out a loan for EUR 100,000 with a discount of 2%. Interest is calculated at a rate of 5.5% per year using the interest calculation method 30/360. The customer pays interest in arrears every six months and repays the loan in full at the end of the term. This results in the following nominal cash flow:

Nominal Cash Flow for Underlying Transaction

Due Date	Operation	Amount	Remaining Capital
February 12, 2000	Nominal capital	100,000	100,000
February 12, 2000	Discount	-2,000	100,000
July 1, 2000	Interest	-2,123.61	100,000
January 1, 2001	Interest	-2,750	100,000
July 1, 2001	Interest	-2,750	100,000
July 1, 2001	Repayment	-100,000	0

1. Effective Interest Method: PAngV

This effective interest method was defined by the German regulation on price transparency ("Preisangabenverordnung" – PAngV). This regulation specifies which costs have to be included in the effective interest rate calculation as factors affecting the price.

If the fixed interest period differs from the end of the term, the effective interest rate is referred to as the initial effective annual interest rate.

The criteria for management of the comparison account are:

Linear interest calculation
Interest calculation method: 30/360
Interest capitalized after one year of the contract term, starting on the day of the first turnover on the comparison account (such as the loan disbursement amount). The broken period is at the end of the determination period.

PAngV states that as of September 1, 2000 the effective interest rate has to be calculated using the effective interest method EU-30.42/365.

Comparison Account: Effective Interest Rate 7.12895% PAngV

Date	Operation	Amount	Balance	Days	Interest
February 12, 2000	Discount	-2,000	-2,000
February 12, 2000	Incoming payment	100,000	98,000	139	2,697.52
July 1, 2000	Interest payment	-2,123.61	95,876.39	180	3,417.49
January 1, 2001	Interest payment	-2,750	93,126.39	41	756.10
February 12, 2001	Interest capitalization	6,871.11	99,997.50	139	2,752.50
July 1, 2001	Repayment	-100,000	-2.50
July 1, 2001	Interest payment	-2,750	-2,752.50
July 1, 2001	Interest capitalization	2,752.50	0.00

2. Effective Interest Method: AIBD/ISMA

This effective interest method was introduced by the Association of International Bond Dealers (AIBD) and is known as rule 603 of the AIBD. The association is now known as the International Securities Market Association (ISMA), so the method is referred to today as the ISMA method.

The criteria for management of the comparison account are:

Exponential interest calculation
Interest calculation method: 30/360, for example
Interest capitalized on each payment date

Comparison Account: Effective Interest Rate 7.17016% AIBD/ISMA

Date	Operation	Amount	Balance	Days	Interest
February 12, 2000	Discount	-2,000	-2,000
February 12, 2000	Incoming payment	100,000	98,000
February 12, 2000	Interest capitalization	0	98,000	139	2,655.60
July 1, 2000	Interest payment	-2,123.61	95,876.39
July 1, 2000	Interest capitalization	2,655.60	98,531.99	180	3,471.30
January 1, 2001	Interest payment	-2,750	95,781.99
January 1, 2001	Interest capitalization	3,471.30	99,253.29	180	3,496.71
July 1, 2001	Repayment	-100,000	-746.71
July 1, 2001	Interest payment	-2,750	-3,496.71
July 1, 2001	Interest capitalization	3,496.71	0.00

3. Effective Interest Method: Braess

The Braess effective interest method was developed originally for the area of securities. As opposed to the PAngV method, this method puts the broken period at the start of the comparison account.

The criteria for management of the comparison account are:

Linear interest calculation
Interest calculation method: 30/360
Interest capitalized after one year of the contract term, starting on the day of the last turnover on the comparison account (such as the last coupon for a bond). The broken period is at the start of the determination period.

Comparison Account: Effective Interest Rate 7.12665% Braess

Date	Operation	Amount	Balance	Days	Interest
February 12, 2000	Discount	-2,000	-2,000
February 12, 2000	Incoming payment	100,000	98,000	139	2,696.65
July 1, 2000	Interest payment	-2,123.61	95,876.39
July 1, 2000	Interest capitalization	2,696.65	98,573.04	180	3,512.48
January 1, 2001	Interest payment	-2,750	95,823.04	180	3,414.49
July 1, 2001	Repayment	-100,000	-4,176.96
July 1, 2001	Interest payment	-2,750	-6,926.96
July 1, 2001	Interest capitalization	6,926.97	0.01

4. Effective Interest Method: Moosmueller

The Moosmueller effective interest method was developed for the area of securities.

The criteria for management of the comparison account are:

Linear interest calculation
Interest calculation method: 30/360, for example
Interest capitalized on each payment date
The interest rate resulting from the comparison account is the periodic interest rate per year. The system converts the period interest rate into the annual interest rate, which is used to represent the effective interest rate.

Comparison Account: Effective Interest Account 7.16187% MOOSM, Periodic Interest Rate Per Year 7.03803%

Date	Operation	Amount	Balance	Days	Interest
February 12, 2000	Discount	-2,000	-2,000
February 12, 2000	Incoming payment	100,000	98,000	139	2,663.11
July 1, 2000	Interest payment	-2,123.61	95,876.39
July 1, 2000	Interest capitalization	2,663.11	98,539.50	180	3,467.62
January 1, 2001	Interest payment	-2,750	95,789.50
January 1, 2001	Interest capitalization	3,467.62	99,257.12	180	3,492.87
July 1, 2001	Repayment	-100,000	-742.88
July 1, 2001	Interest payment	-2,750	-3,492.88
July 1, 2001	Interest capitalization	3,492.87	0.01

The periodic interest rate per year is 7.03803%. The system uses the following formula to convert this into the annual interest rate (IS represents the number of interest settlements per year).

5. Effective Interest Rate: US Treasury

This effective interest method is used primarily in the United States.

The criteria for management of the comparison account are:

Linear interest calculation
Interest calculation method: 30/360, for example
Interest capitalized on each payment date

Comparison Account: Effective Interest Rate 7.03803% U.S. TR

Date	Operation	Amount	Balance	Days	Interest
February 12, 2000	Discount	-2,000	-2,000
February 12, 2000	Incoming payment	100,000	98,000	139	2,663.11
July 1, 2000	Interest payment	-2,123.61	95,876.39
July 1, 2000	Interest capitalization	2,663.11	98,539.50	180	3,467.62
January 1, 2001	Interest payment	-2,750	95,789.50
January 1, 2001	Interest capitalization	3,467.62	99,257.12	180	3,492.87
July 1, 2001	Repayment	-100,000	-742.88
July 1, 2001	Interest payment	-2,750	-3,492.88
July 1, 2001	Interest capitalization	3,492.87	-0.01

6. Effective Interest Method: EU Act/365

This effective interest method is based on EU Guideline 98/7/EC of the European Parliament and European Council of February 16, 1998, which stipulates that the effective interest rate is to be calculated using the internationally recognized AIBD/ISMA method and the interest calculation method act/365.

The criteria for management of the comparison account are:

Exponential interest calculation
Interest calculation method: act/365
Interest capitalized on each payment date

Comparison Account: Effective Interest Rate 7.18358% EU Act/365

Date	Operation	Amount	Balance	Days	Interest
February 12, 2000	Discount	-2,000	-2,000
February 12, 2000	Incoming payment	100,000	98,000
February 12, 2000	Interest capitalization	0.00	98,000	140	2,642.66
July 1, 2000	Interest payment	-2,123.61	95,876.39
July 1, 2000	Interest capitalization	2,642.66	98,519.05	184	3,506.32
January 1, 2001	Interest payment	-2,750	95,769.05
January 1, 2001	Interest capitalization	3,506.32	99,275.37	181	3,474.63
July 1, 2001	Repayment	-100,000	-724.63
July 1, 2001	Interest payment	-2,750	-3,474.63
July 1, 2001	Interest capitalization	3,474.63	0.00

7. Bond Formula

This effective interest method is used internationally to evaluate bonds.

The criteria for management of the comparison account are:

Exponential interest calculation
Interest calculation method: act/actP
Interest capitalized on each payment date

Comparison Account: Effective Interest Rate 7.05398% Bond Formula

Date	Operation	Amount	Balance	Days	Interest
February 12, 2000	Discount	-2,000	-2,000
February 12, 2000	Incoming payment	100,000	98,000
February 12, 2000	Interest capitalization	0.00	98,000	140	2,648.14
July 1, 2000	Interest payment	-2,123.61	95,876.39
July 1, 2000	Interest capitalization	2,648.14	98,524.53	184	3,474.95
January 1, 2001	Interest payment	-2,750	95,774.53
January 1, 2001	Interest capitalization	3,474.95	99,249.48	181	3,500.52
July 1, 2001	Repayment	-100,000	-750.52
July 1, 2001	Interest payment	-2,750	-3,500.52
July 1, 2001	Interest capitalization	3,500.52	0.00

As opposed to the AIBD/ISMA method, this effective interest method uses exponential interest calculation with periodic interest settlement during the year. The abbreviation "IS" stands for the number of interest settlements per year.

The interest amount of EUR 2,648.14 on February 12, 2000 is calculated as follows:

The interest amount of EUR 3,474.95 on July 1, 2000 is calculated as follows:

The interest amount of EUR 3,500.52 on January 1, 2001 is calculated as follows:

8. Effective Interest Method: EU 30.42/365

The effective interest method is based on EU Guideline 98/7/EC of the European Parliament and European Council of February 16, 1998.

The criteria for management of the comparison account are:

Exponential interest calculation
Interest calculation method: 30.42/365
Interest capitalized on each payment date

The effective interest rate according to EU-30.42/365 is calculated using the present values. Interest is calculated at the same calculation time for all individual payments (t0); in this case, on February 12, 2000. The system uses iteration to calculate the effective interest rate at which the total of all present values is zero. This interest rate is also known as the actuarial return or internal rate of return (IRR).

We are looking for the interest rate for which the following mathematical requirements are met:

The net present value (NPV) of all discounted payments is zero.

The present value (PV) is the discounted final value of a payment on the present date.

In the example, the system determines that all the above mathematical requirements are met if the interest rate is 7.17413. The net present value of 0.01 is the result of rounding differences. The following table shows the history of the present value determination.

Present Value History: Effective Interest Rate 7.17413% EU 30.42/365

Event	Operation	Amount	Days	Present Value	Net Present Value = Σ Present Values
t0 February 12, 2000	Incoming payment	98,000	0.00000	98,000	98,000
t1 July 1, 2000	Disbursement	-2,123.61	140.66667	-2,067.66	95,932.34
t2 January 1, 2001	Disbursement	-2,750	323.16667	-2,586.37	93,345.97
t3 July 1, 2001	Disbursement	-2,750	505.66667	-2,498.31	90,847.66
t3 July 1, 2001	Disbursement	-100,000	505.66667	-90,847.65	0.01

The following figure shows the present value determination using a time stream:

For example, how does the system calculate the present value of EUR 2,067.66 if t1 is July 1, 2000 and the interest rate is 7.17413%?

A payment of EUR 2,123.61 is made on July 1, 2000. How much is this payment worth on February 12, 2000?

First, the system determines the number of days between February 12, 2000 and July 1, 2000 using February 12, 2000 as the starting point. This means that the broken period is at the end of the time period. The system counts four complete months from February 12, 2000 to June 12, 2000 and 19 calendar days from June 12, 2000 to July 1, 2000. A full month is 365/12 days. This results in the following quotient:

The system assumes the following values:

End value = 2,123.61

Interest rate = 7.17413

Days = 140.66667

Base days = 365

If you enter these values in the formula for the present value calculation, the resulting present value is EUR 2,067.66.

As with other effective interest rate methods, the system displays the results of the effective interest method EU-30.42/365 in a comparison account:

Comparison Account: Effective Interest Rate 7.17413% EU 30.42/365

Date	Operation	Amount	Balance	Days	Interest
February 12, 2000	Discount	-2,000	-2,000
February 12, 2000	Incoming payment	100,000	98,000
July 1, 2000	Interest payment	-2,123.61	95,876.39	140.66667	55.95
July 1, 2000	Interest capitalization	55.95	95,932.34
January 1, 2001	Interest payment	-2,750	93,182.34	323.16667	163.63
January 1, 2001	Interest capitalization	163.63	93,345.97
July 1, 2001	Repayment	-100,000	-6,654.03	505.66667	9,152.35
July 1, 2001	Interest payment	-2,750	-9,404.03	505.66667	251.69
July 1, 2001	Interest capitalization	9,404.04	0.01

9. Effective Interest Method: Linear

The criteria for management of the comparison account are:

Linear interest calculation
Interest calculation method: 30/360, for example
Interest capitalized once at the end of the comparison account

Comparison Account: Effective Interest Rate 7.27187% Linear

Date	Operation	Amount	Balance	Days	Interest
February 12, 2000	Discount	-2,000	-2,000
February 12, 2000	Incoming payment	100,000	98,000	139	2,751.59
July 1, 2000	Interest payment	-2,123.61	95,876.39	180	3,486
January 1, 2001	Interest payment	-2,750	93,126.39	180	3,386.02
July 1, 2001	Repayment	-100,000	-6,873.61
July 1, 2001	Interest payment	-2,750	-9,623.61
July 1, 2001	Interest capitalization	9,623.61	0.00

10. Effective Interest Method: U.S. APR

The effective interest method is based on the American Truth in Lending Act (TILA) of 1968, which is also known as Regulation Z.

The criteria for management of the comparison account are:

Linear interest calculation
Interest calculation method depends on unit period
Interest capitalized on each payment date

The system calculates the effective interest rate according to U.S. APR using the present values. Interest is calculated at the same calculation time for all individual payments (t0); in this case, on February 12, 2000. The system uses iteration to calculate the effective interest rate at which the total of all present values is zero.

As with other effective interest rate methods, the system displays the results of the effective interest method U.S. APR in a comparison account:

Comparison Account: Effective Interest Rate 7.05261% U.S. APR

Date	Operation	Amount	Remaining Capital	Days	Interest
February 12, 2000	Discount	-2,000	-2,000	0	0.00
February 12, 2000	Incoming payment	100,000	98,000	0	0.00
July 1, 2000	Interest payment	-2,123.61	95,876.39	138	55.90
July 1, 2000	Interest capitalization	55.90	95,932.29	0	0.00
January 1, 2001	Interest payment	-2,750	93,182.29	318	163.59
January 1, 2001	Interest capitalization	163.59	93,345.88	0	0.00
July 1, 2001	Interest payment	-2,750	90,595.88	498	251.69
July 1, 2001	Interest capitalization	251.69	90,847.57	0	0.00
July 1, 2001	Repayment	-100,000	-9,152.43	498	9,152.43
July 1, 2001	Interest capitalization	9,152.43	0.00	0	0.00

For more information, see Effective Interest Rate Calculation According to U.S. APR.

More Information

Comparison Account

Effective Interest Rate Calculation