COMMISSION DELEGATED REGULATION (EU) 2023/67
of 20 October 2022
supplementing Regulation (EU) 2021/1060 of the European Parliament and of the Council by establishing standardised off-the-shelf sampling methodologies and modalities to cover one or more programming periods
Article 1
Subject matter and scope
Article 2
Definitions
Article 3
Audit population
Article 4
Multi-period sampling and stratification
Article 5
Selection of a random statistical sample
Article 6
Selection of a random non-statistical sample
Article 7
Subsampling
Article 8
Calculation of the total error rate
Article 9
Entry into force
ANNEX I
SAMPLING PARAMETERS
1. Materiality threshold
2. Confidence level
3. Parameter
z
Confidence level |
90 % |
80 % |
70 % |
60 % |
z value (bilateral) |
1,645 |
1,282 |
1,036 |
0,842 |
z' value (unilateral) |
1,282 |
0,842 |
0,524 |
0,253 |
4. Anticipated standard deviation of errors or error rates and anticipated error
ANNEX II
FORMULAS FOR SAMPLE SIZE CALCULATION AND EXTRAPOLATION OF ERRORS
1. MUS STANDARD APPROACH
1.1.
MUS standard approach – one period
NON-STRATIFIED |
STRATIFIED |
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Sample size calculation |
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where - weighted mean of the variances of the error rates for the whole set of strata, with the weight for each stratum equal to the ratio between the stratum book value (BV h ) and the book value for the whole population (BV)
and is the variance of error rates in each stratum |
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where
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Extrapolation of errors |
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Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive stratum, the projected error is the sum of the errors found in the units belonging to the stratum:
For the non-exhaustive stratum, i.e. the stratum containing the sampling units with book value smaller than the interval, the projected error is
The projected error at the level of population is the sum of the two components above:
|
Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive groups, the projected error is the sum of the errors found in the units belonging to those groups:
For the non-exhaustive groups, i.e. the groups containing the sampling units with book value smaller than the interval, , the projected error is
The projected error at the level of population is just the sum of these two components above:
|
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Sampling precision:
where is the standard-deviation of error rates in the sample of the non-exhaustive stratum (calculated from the same sample used to extrapolate the errors to the population) |
Sampling precision:
where is the standard-deviation of error rates in the sample of the non-exhaustive group of stratum h (calculated from the same sample used to extrapolate the errors to the population) |
1.2.
MUS standard approach – two periods
NON-STRATIFIED |
STRATIFIED |
Sample size calculation |
|
First period
where
|
First period
where
|
Second period
|
Second period
where
|
Notes: Whenever different approximations for the standard-deviations of each period cannot be obtained/are not applicable, the same value of standard deviation may be applied to all periods. In such a case is just equal to the single standard-deviation of error rates .The parameter σ refers to the standard-deviation obtained from auxiliary data (e.g. historical data) and s refers to the standard-deviation obtained from the audited sample. In the formulas, whenever s is not available, it may be substituted by σ. Formulas under the heading “First period” are used to calculate the sample size after the first sampling period of the accounting year in the case of a standard recalculation of the sample size referred to in Article 5(6), point (a). In the case of the global recalculation of the sample size referred to in Article 5(6), point (b), these formulas are used after the first sampling period and if needed also after the second sampling period in order to adjust to updated sampling parameters. Formulas under the heading “Second period” are applicable only in the case of a standard recalculation of the sample size referred to in Article 5(6), point (a). They are used to recalculate the sample size of the second period in order to adjust to updated sampling parameters. If the formula results in a negative number, the formula and consequently the standard approach to recalculation of the sample size cannot be applied based on the established set of the updated parameters. |
|
Extrapolation of errors |
|
Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata:
For the non-exhaustive strata, i.e. the strata containing the sampling units with book value smaller than the interval, the projected error is
The projected error at the level of population is the sum of the two components above:
|
Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata:
For the non-exhaustive strata, i.e. the strata containing the sampling units with book value smaller than the interval, the projected error is
The projected error at the level of population is the sum of the two components above:
|
Sampling precision:
where is the standard-deviation of error rates in the sample of the non-exhaustive strata of period t (calculated from the same sample used to extrapolate the errors to the population) |
Sampling precision:
where is the standard-deviation of error rates in the sample of the non-exhaustive group of stratum h in period t (calculated from the same sample used to extrapolate the errors to the population) |
1.3.
MUS standard approach – three periods
(1)
NON-STRATIFIED |
STRATIFIED |
Sample size calculation |
|
First period
where
|
First period
where
|
Second period
where
|
Second period
where
|
Third period
|
Third period
|
Notes: Whenever different approximations for the standard-deviations of each period cannot be obtained/are not applicable, the same value of standard deviation may be applied to all periods. In such a case is just equal to the single standard-deviation of error rates .The parameter σ refers to the standard-deviation obtained from auxiliary data (e.g. historical data) and s refers to the standard-deviation obtained from the audited sample. In the formulas, whenever s is not available, it may be substituted by σ. See also notes above for the two-period MUS standard approach as regards the use of the standard approach to the recalculation of the sample size and the global approach referred to in Article 5(6). |
|
Extrapolation of errors |
|
Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata:
For the non-exhaustive strata, i.e. the strata containing the sampling units with book value smaller than the interval, the projected error is
The projected error at the level of population is the sum of the two components above:
|
Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata:
For the non-exhaustive strata, i.e. the strata containing the sampling units with book value smaller than the interval, the projected error is
The projected error at the level of population is the sum of the two components above:
|
Sampling precision:
where is the standard-deviation of error rates in the sample of the non-exhaustive strata of period t (calculated from the same sample used to extrapolate the errors to the population) |
Sampling precision:
where is the standard-deviation of error rates in the sample of the non-exhaustive group of stratum h in period t (calculated from the same sample used to extrapolate the errors to the population) |
2. SIMPLE RANDOM SAMPLING
2.1.
Simple random sampling – one period
NON-STRATIFIED |
STRATIFIED |
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Sample size calculation |
||||||||||||||||
where is the standard-deviation of errors in the population |
where - the weighted mean of the variances of the errors for the whole set of strata:
and is the variance of errors in each stratum |
|||||||||||||||
where
|
||||||||||||||||
Extrapolation of errors |
||||||||||||||||
In the framework of application of the off-the-shelf methodologies laid down in this Delegated Regulation, a single extrapolation method, ratio estimation, applies for SRS referred to in Article 5(1), point (b), and equal probability selection referred to in Article 6(1), point (b), for the purpose of simplification and legal certainty. This does not limit the application of other extrapolation methods by the audit authorities under Article 79 of Regulation (EU) 2021/1060. |
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Projected/extrapolated error (SRS/equal probability selection): If an exhaustive stratum is used, the projected error in this group is the sum of the errors found in the units belonging to the stratum:
For the random stratum the projected error is
The projected error at the level of population is the sum of the two components above:
|
Projected/extrapolated error (SRS/equal probability selection): If an exhaustive stratum is used, the projected error in this group is the sum of the errors found in the units belonging to those groups:
For the random strata the projected error is
The projected error at the level of population is just the sum of these two components above:
|
|||||||||||||||
Sampling precision:
where s q is the sample standard deviation of the variable q:
Precision is exclusively calculated with data pertaining to the non-exhaustive strata. |
Sampling precision:
where
is a weighted mean of the sample variances of the variable q h , with
Precision is exclusively calculated with data pertaining to the non-exhaustive strata. |
2.2.
Simple random sampling – two periods
NON-STRATIFIED |
STRATIFIED |
Sample size calculation |
|
First period
where
|
First period
where
|
Second period
|
Second period
|
Notes: Whenever different approximations for the standard-deviations of each period cannot be obtained/are not applicable, the same value of standard deviation may be applied to all periods. In such a case is just equal to the single standard-deviation of errors .The parameter σ refers to the standard-deviation obtained from auxiliary data (e.g. historical data) and s refers to the standard-deviation obtained from the audited sample. In the formulas, whenever s is not available, it may be substituted by σ. Formulas under the heading “First period” are used to calculate the sample size after the first sampling period of the accounting year in the case of a standard recalculation of the sample size referred to in Article 5(6), point (a). In the case of the global recalculation of the sample size referred to in Article 5(6), point (b), these formulas are used after the first sampling period and if needed also after the second sampling period in order to adjust to updated sampling parameters. Formulas under the heading “Second period” are applicable only in the case of a standard recalculation of the sample size referred to in Article 5(6), point (a). They are used to recalculate the sample size of the second period in order to adjust to updated sampling parameters. If the formula results in a negative number, the formula and consequently the standard approach to re-calculation of the sample size cannot be applied based on the established set of the updated parameters. |
|
Extrapolation of errors |
|
In the framework of application of the off-the-shelf methodologies laid down in this Delegated Regulation, a single extrapolation method, ratio estimation, applies for SRS referred to in Article 5(1), point (b) and equal probability selection referred to in Article 6(1), point (b), for the purpose of simplification and legal certainty. This does not limit the application of other extrapolation methods by the audit authorities under Article 79 of Regulation (EU) 2021/1060. |
|
Projected/extrapolated error (SRS/equal probability selection): If an exhaustive stratum is used, the projected error in this group is the sum of the errors found in the units belonging to the strata:
For the non-exhaustive strata, the projected error is:
The projected error at the level of population is the sum of the two components above. |
Projected/extrapolated error (SRS/equal probability selection): If an exhaustive stratum is used, the projected error is the sum of the errors found in the units belonging to the strata:
For the non-exhaustive strata, the projected error is:
The projected error at the level of population is the sum of the two components above. |
Sampling precision:
Precision is exclusively calculated with data pertaining to the non-exhaustive strata. |
Sampling precision:
Precision is exclusively calculated with data pertaining to the non-exhaustive strata. |
2.3.
Simple random sampling – three periods
(2)
NON-STRATIFIED |
STRATIFIED |
Sample size calculation |
|
First period
where
|
First period
where
|
Second period
where
|
Second period
|
Third period
|
Third period
|
Notes: Whenever different approximations for the standard-deviations of each period cannot be obtained/are not applicable, the same value of standard deviation may be applied to all periods. In such a case is just equal to the single standard-deviation of errors .The parameter σ refers to the standard-deviation obtained from auxiliary data (e.g. historical data) and s refers to the standard-deviation obtained from the audited sample. In the formulas, whenever s is not available, it may be substituted by σ. See also notes above for the two-period simple random sampling as regards the use of the standard approach to the recalculation of the sample size and the global approach referred to in Article 5(6). |
|
Extrapolation of errors |
|
In the framework of application of the off-the-shelf methodologies laid down in this Regulation, a single extrapolation method, ratio estimation, applies for SRS referred to in Article 5(1), point (b), and equal probability selection referred to in Article 6(1), point (b), for the purpose of simplification and legal certainty. This does not limit the application of other extrapolation methods by the audit authorities under Article 79 of Regulation (EU) 2021/1060. |
|
Projected/extrapolated error (SRS/equal probability selection): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata:
For the non-exhaustive strata, the projected error is:
The projected error at the level of population is the sum of the two components above. |
Projected/extrapolated error (SRS/equal probability selection): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata:
For the non-exhaustive strata, the projected error is:
The projected error at the level of population is the sum of the two components above. |
Sampling precision:
Precision is exclusively calculated with data pertaining to the non-exhaustive strata. |
Sampling precision:
Precision is exclusively calculated with data pertaining to the non-exhaustive strata. |
ANNEX III
ADJUSTMENTS RELATED TO SINGLE AUDIT ARRANGEMENTS
Sampling design |
MUS standard/PPS: Exclusion of sampling units |
MUS standard/PPS: Replacement of sampling units |
Population used for sample selection |
Reduced (adjusted) population (i.e. population excluding operations/other sampling units affected by Article 80 of Regulation (EU) 2021/1060) |
Original population(1) |
Parameters used for sample size calculation |
Correspond to the original population |
|
Recommended approach to projection/extrapolation of errors and precision calculation |
Projection of error and precision calculation is carried out in the first stage for the reduced population. In the next stage it is adjusted to reflect the original population. Such adjustment may be performed by multiplying the projected error and precision by the ratio between expenditure BV (h) original of the original population and the expenditure BV (h) reduced of the reduced population. In the case of units of high-value stratum affected by Article 80 of Regulation (EU) 2021/1060 (or any other exhaustive stratum), there could be a need to calculate the error for the high-value stratum and to project this error to the units which were not audited in this stratum using the formula (where EE e reduced represents the amount of error in the sampling units of the high-value stratum audited, BV e original refers to book value of the original high-value stratum and BV e reduced refers to the book value of units in the high-value stratum which were subject to audit.) |
Projection of error and precision calculation is carried out for the original population. The units of high-value stratum (or units of any other exhaustive stratum), which are excluded from the audit procedures due to provisions of Article 80 of Regulation (EU) 2021/1060 should be replaced by the sampling units of the low-value stratum. In such a case there could be a need to calculate the error for the high-value stratum and to project this error to the units which were not audited in this stratum using the formula (where EE e reduced represents the amount of error in the sampling units of the high-value stratum audited, BV e original refers to book value of the original high-value stratum and BV e reduced refers to the book value of units in the high-value stratum which were subject to audit). |
Sampling design |
Simple Random Sampling/equal probability selection: Exclusion of sampling units |
Simple Random Sampling/equal probability selection: Replacement of sampling units |
Population used for sample selection |
Reduced (adjusted) population (i.e. population excluding operations/other sampling units affected by Article 80 of Regulation (EU) 2021/1060) |
Original population(2) |
Parameters used for sample size calculation |
Correspond to the original population |
|
Recommended approach to projection/extrapolation of errors and precision calculation |
Projection of error and precision calculation is carried out for the reduced population. In the next stage it is adjusted to reflect the original population based on the following approaches: The adjustment may be performed by multiplying the projected error and precision by the ratio between expenditure BV (h) original of the original population and the expenditure BV (h) reduced of the reduced population. Projection of error may also be performed directly for the original population. Precision should not be calculated directly for the original population. The precision calculated for reduced population should be adjusted for the original population by multiplying the precision of the reduced population by the ratio .In the case of units of high-value stratum (or any other exhaustive stratum) subject to Article 80 of Regulation (EU) 2021/1060, there could be a need to calculate an error for the high-value stratum and to project this error to the units which were not audited in this stratum. It would be performed using the formula , where EE e reduced represents the amount of error in the sampling units of the high-value stratum audited, BV e original refers to book value of the original high-value stratum and BV e reduced refers to the book value of units in the high-value stratum which were subject to audit. |
Projection of error is carried out for the original population. Precision has to be calculated for the reduced population (population from which all sampling units subject to Article 80 of Regulation (EU) 2021/1060 were deducted). Subsequently, it should be in the next stage adjusted to reflect the original population. It may be performed by multiplying the precision of the reduced population by the ratio between expenditure BV (h) original of the original population and the expenditure BV (h) reduced of the reduced population. It should be also noted that even if the audit authority did not select any sampling units affected by Article 80 of Regulation (EU) 2021/1060 in its sample, the precision will also have to be calculated to the reduced population and subsequently adjusted using the above mentioned formula. In the case of units of high-value stratum (or any other exhaustive stratum) subject to Article 80 of Regulation (EU) 2021/1060, there could be a need to calculate an error for the high-value stratum and to project this error to the units which were not audited in this stratum. It would be performed using the formula , where EE e reduced represents the amount of error in the sampling units of the high-value stratum audited, BV e original refers to book value of the original high-value stratum and BV e reduced refers to the book value of units in the high-value stratum which were subject to audit. |