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# Accuracy of the computation of the internal rate of return (IRR)

## Method and accuracy of the computation of the IRR

For the calculation of effective interest, the so-called ISMA-Method (International Securities Market Association, Rule 803, Standard Maturity Yield Definition) is the widely accepted standard.
The major regulations:
```- use of the internal interest rate method
- daily accounting of the payments
- daily interest capitalization```
Nevertheless there are regional differences. For example the Federal Republic of Germany regulated in §4 of the so-called "Preisangabenverordnung"(PAngV) how banks have to calculate an publish the net yield and interest. For usability with monthly disbursements the PAngV ruled out that months are in general 30.416 days, and not the exact number of days.

## How is IRRQ.com calculating?

with the "best" methode: internal rate of return, day-exact, year-exact and if one year is not complete a day as a 365.25th part of a year. You may designate the as act/365.25 method.
The reason for this proceeding?

The result should describe the success of an investment as accurately as possible.
```1.) day-exactly, so that the different length of the months (28-31 days) don´t matter.
2.) year-exactly, because one year is one year, no matter how long it is.
3.) 365.25 days, because this is the most accurate value.```
to 1.) If you decide to count months you must tell your software that 28. Feb. to 31. March is one month or 1/12th part of a year or 30.416 days. Who would like to program this: Please announce yourselves.

to 2.) If one invests on 2004-01-10 \$ 100 and receives on 2005-01-10 \$ 110 back, one want this to be exactly 10% and not 9.99% or 10.01%.

to 3.) 365.25 days is the average number of days for a year.

## Accuracy of the calculations

In the appendix of the german quotation regulation ("Preisangabenverordnung") some arithmetical examples are mentioned, which I checked with the Microsoft excel spreadsheet software and the IRRQ.com software. Please keep in mind that the PangV method is calculating with generalized months.
```
yield acc.   PAngV      Excel     IRRQ.com
example 6.1                        12,92%     12,96%    12,97%
example 6.2                        16,85%     16,90%    16,91%
example 6.3                        13,07%     13,05%    13,07%
example 6.4                        13,19%     13,18%    13,20%
example 6.5                         9,96%      9,96%     9,96%
example 6.6                         6,17%      6,14%     6,14%
```
The excel formula for the calculation was:
``` =(1+IKV(SUMMEWENN(A:A;ZEILE(INDIREKT(MIN(A:A)&":"&MAX(A:A)));B:B);0))^365,25-1
```
For all examples which did not have a particularly date 2004-01-01 was selected as starting date. Complete identical results were obtained in the case of the example 6.5 and 6.6. Here all payment dates were on the same day of the year and the investment duration covered several years. Otherwise, deviations may arise as a result of different counting of the days, especially with February with only 28 days. In general, the error is not higher than 0.02 between Excel and IRRQ.com.

Another example taken from the "Handbuch der Renditeberechnung" written by Dr. Manfred Frühwirth was also recalculated. This cashflow:
```01.03.1997  -99,4
13.05.1997    2,0
13.08.1997    2,0
13.11.1997    2,0
13.02.1998    2,0
13.05.1998    2,0
13.08.1998   12,0
13.11.1998   11,8
13.02.1999   11,6
13.05.1999   11,4
13.08.1999   11,2
13.11.1999   11,0
13.02.2000   10,8
13.05.2000   10,6
13.08.2000   10,4
13.11.2000   10,2
```
is producing the following yields according to the corresponding method:
```  Yield acc. ISMA Method:                8,709%(p.a.)
Yield acc. Moosmüller Method:          8,707%(p.a.)
Yield acc. Amerikanischen Method:      8,436%(p.a.)
Yield acc. Braeß/Fangmeyer Method:     8,699%(p.a.)
Yield acc. PangV Method:               8,728%(p.a.)
my computations:
Yield acc. IRRQ.com Method:            8,70%(p.a.)
```
It has to be considered that only the method of the International Securities Market Association (ISMA) is in use all over the world.

Summary: The deviations between the relevant computations (ISMA, Excel) and the results on this web page are usually smaller than 0.01.

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