Topic 11: Fieldwork ? Audit Sampling for Tests of Details of

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					Topic 11: Fieldwork ― Audit Sampling
for Tests of Details of Balances

   BMGT 422 - Auditing
   Arens 12e Ch. 17
   James J. McKinney
     Topic 1: Overview                                  Topic 18: Legal Liability
     Topic 2: Accounting Information System                       Fraud Detection
                                                                  IT Auditing
                                                                  Other Engagements – NOT AUDITING!

       Topic 11: Sampling - Tests of Details of Balances
(these topics affect fieldwork and   Topic 9: Sampling in Tests of
completion)                                    Controls and Substantive
Topic 3: Cycles                                Tests of Transactions
Topic 4: Management Assertions       Topic 10: Tests of Controls
         and Audit Objectives        Topic 11: Sampling - Tests of Details   Topic 16: Audit Completion
                                               of Balances
Topic 5: Evidence                                                            Topic 17: Reporting
                                     Topic 12: Sales and Collections Cycle
Topic 6: Materiality                 Topic 13: Inventory and Warehousing
Topic 7: Audit Risk                            Cycle
Topic 8: Test Types and              Topic 14: Acquisition and Payment
         Audit Plan                            Cycle
                                     Topic 15: Cash

          Pre-Acceptance Client
                                         Test of       Substantive
                                                                      Complete the Audit:
          Review                      Controls and      Testing:         Contingent
          Understand Business        Reassessment      Substantive       Liabilities
          Initial Analytical                                             Subsequent
          Procedures                 of Control Risk   Tests of          Events
          Engagement Letter                            Transactions      Final Analytical
    Make Preliminary                                   Analytical        Procedures
    Assessments:                                                         Make
          Controls                                     Procedures        Reassessments
          Risk                                         Tests of             Risk
          Materiality                                  Details of           Materiality
    Determine preliminary                                                   Evidence
    procedures based on previous
    assessments and:                                                     Audit Report on
          Management Assertions                                          Financial
          Audit Objectives                                               Statements
          Evidence Availability,                                         Audit Report on
          Persuasiveness, and Cost                                       Internal Controls
    Prepare:                                                             (Public Co’s)
          Audit Plan                      Sampling                       Management
          Audit Program                                                  Letters
!              Comparison of the 14 Steps Between
                 TOC/STOT and TDB Sampling
                                      Audit sampling for tests of
           Audit sampling for tests    controls and substantive
    Step    of details of balances       tests of transactions
                          State the objectives
                           of the audit test.
                         Decide whether audit
                          sampling applies.
           Define misstatement        Define attributes and
                conditions.           exception conditions.
    4                    Define the population.
    5                  Define the sampling unit.
!                Comparison of the 14 Steps Between
                   TOC/STOT and TDB Sampling
                                         Audit sampling for tests of
             Audit sampling for tests     controls and substantive
    Step      of details of balances        tests of transactions
           Specify Tolerable            Specify the Tolerable
           Misstatement.                Deviation Rate.
                                    Specify Risk
           Specify Risk
    7                               of Assessing Control
           of Incorrect Acceptance.
                                    Risk Too Low.
           Estimate Anticipated     Estimate the Estimated
           Misstatement in the pop. Pop. Deviation Rate.
                            Determine the initial
                               sample size.
!              Comparison of the 14 Steps Between
                 TOC/STOT and TDB Sampling
                                      Audit sampling for tests of
           Audit sampling for tests    controls and substantive
    Step    of details of balances       tests of transactions
    10                     Select the sample.
                            Perform the audit

    12                   Generalize from the
                       sample to the population.
              Analyze the                   Analyze the
             misstatements.                 deviations.
                        Decide the acceptability
                          of the population.
                             Sampling Risk - Substantive Testing

                             Conclusion Based on Entire Population
                                                 Accept       Believe
Conclusion Based on Sample

                                               Balance as    Balance is
                                              Fairly Stated Unacceptably        Note: Whereas with
                                                             Misstated           RACRTL, failure had
                                 Accept                     Risk of Incorrect    only a potential for
                               Balance as                     Acceptance         audit failure since
                              Fairly Stated    CORRECT                           insufficient
                                                             Beta or Type II
                                                                 Error           substantive testing
                                                                                 was done, RIA
                               Believe        Risk of                            implies that a
                              Balance is     Incorrect                           balance is misstated
                                             Rejection                           and therefore more
                             Unacceptably                     CORRECT
                              Misstated   Alpha or Type I
                                                                                 likely to result in
                                                 Error                           audit failure.
                                                                                RIR results in loss of
                Risk of Incorrect Acceptance
One way to think of determining RIA is to modify the Audit Risk
 model from:
  New definitions:
    PDRAP: : Risk that analytical procedures would not detect the misstatements not
     prevented by internal controls.
          Other TDB : Risk that other tests of details of balances (beside the sampling test

     being performed) would not detect the misstatements not prevented by internal controls.
  We can rewrite the formula as: RIA=AR/(IRxCRxPDRAPx PDROther TDB)
  Suppose we have the following:


    PDR =20%

    RIA=.05/(1.0 x 1.0 x .2) = .25 or 25%
!   Terminology with Test of Controls Comparison

     RIA (Risk of Incorrect Acceptance):
        AKA Acceptable Risk of Incorrect Acceptance – Similar to RACRTL for TOC
        (Need for PPS and CVS) [In ACL Confidence Level = One minus RIA]
     RIR (Risk of Incorrect Rejection):
        Similar to RACRTH for TOC (Need for CVS only)
     AMR (Anticipated Misstatement [$] Rate [%]):
        AKA Expected Error Rate or Expected Misstatement Rate – rates are
        determined by dividing AM by population $ – Similar to EDR for TOC [In
        ACL is called Expected Total Errors]
     TMR (Tolerable Misstatement [$] Rate [%]):
        AKA Materiality or Tolerable Error – Similar to TDR for TOC [In ACL is
        called Materiality]
     PM (Projected Misstatement):
        Similar to Sample Deviation Rate for TOC [In ACL is called Most Likely
     ULM (Upper Limit on Misstatement):
        AKA Worst Case Error, Upper Misstatement Limit, or Misstatement Bounds
        – Similar to UPL for TOC [In ACL is called Upper Error Limit]
                 Impact on Sample Sizes

    RIA                      ↓   SS   ↑
    AMR                      ↓   SS   ↓
    TMR                      ↓   SS   ↑
    RIR                      ↓   SS   ↑ Only used for CVS
    Population Variability   ↓   SS   ↓ Only used for CVS
       PPS Sampling – Other Names

Monetary-Unit Sampling (MUS)
Probability-Proportional-to-Size Sampling (PPS)
Dollar-Unit Sampling (DUS)
Cumulative Monetary Amount (CMA)
Sampling-Proportional-to-Size (SPS)
 In Test of Controls - document was the sample item.
 When testing a control on a document, a control was 100% not
  working (a deviation) or working.
 In PPS, think of each $ being the item to be selected and tested.
 However if a $ is selected within a $4,000 dollar invoice, which is
  your selected $? If cannot isolate then test the entire invoice. If
  find a misstatement, consider the misstatement to be
  proportionately spread throughout the invoice's $. For example if
  you find a $1,000 misstatement, then assume that each $ in the
  invoice is misstated by 25 cents ($1,000/$4,000). Thus say if you
  selected dollar 200 in the $4,000, invoice, you conclude that $ is
  misstated by 25 cents.
 This $ is considered to have a 25% tainting (note you can sum
  tainting %’s unlike regular %’s).
                               PPS Sample Size
Note: In past exams we determined sample size and evaluated samples using tables
 similar to those used for test of controls. This method is valid but other sources
 emphasize the approach below. Required: only RIA, AMR, TMR, and Table.
SS=(population book value x RF0) / (TM- (AM x EF) )
  RF is the reliability factor for 0 misstatements and the appropriate RIA (see following page)
  EF is the expansion factor for the appropriate RIA (see following page)
SS = ($500,000 x 3.0) / ($15,000 - ($3,000 x 1.6)) = 147.05
  TM or materiality for assertion = $15,000
  Population book value=$500,000
Note: ACL does not round when computing SS.
The sample size may not equal the number of items actually tested. SS here actually
 equals the number of intervals selected. Amounts over the interval might actually
 consist of one or more intervals from the SS. See examples on pages that follow.
!                                        Reliability and Expansion Factors
                                         Reliability Factors for Misstatements
                                                   Risk of Incorrect Acceptance
                                                 1%            5%            10%            20%            25%            50%
        Number of Misstatements

                                     0          4.61          3.00           2.31           1.61           1.39            .70
                                     1          6.64          4.75           3.89           3.00           2.70           1.68
                                     2          8.41          6.30           5.33           4.28           3.93           2.68
                                     3         10.05          7.76           6.69           5.52            5.11          3.68
                                     4         11.61          9.16           8.00           6.73           6.28           4.68
                                     5         13.11         10.52           9.28           7.91           7.43           5.68
                                     6         14.57         11.85          10.54           9.08           8.56           6.67
                                     7         16.00         13.15          11.78          10.24           9.69           7.67
                                     8         17.41         14.44          13.00          11.38          10.81           8.67

                                  Expansion Factors for Anticipated Misstatements
                                                   Risk of Incorrect Acceptance
                                         1%            5%            10%            20%            25%             50%
                                         1.9           1.6            1.5            1.3           1.25             1.0

    AICPA, Audit and Accounting Guide: Audit Sampling
!                               PPS Selection

    To evaluate statistically, you need a random sample. Could select random $'s
     throughout, however traditionally firms have selected PPS samples using
     systematic random selection. This avoids having an invoice less than the
     interval being selected twice.
    1) Easy way: Set up a list of cumulative $. Should exclude zero and negative
    2) Next first determine the interval: Population $ divided by sample size.
         Round down if necessary (forces sample size to be higher). If you have to
         estimate population $, estimate high amount.
    3) Pick a random $ from 1 up to and including the interval amount. This is your
         random start.
    4) You will skip through the cumulative $ until you hit that random start.
    5) Add the interval to the random start and find that $. Add the interval again a
         find that $. Keep adding intervals until you have gone through the entire
!                         PPS Selection - Example

     Population = $5,050 with only 6 items (to make it easy)
     Sample Size =5
     Thus interval is? Round down to $1,000 – (note it is unacceptable to round for tests
      or projects unless directed to do so).
     Determine that random start is $499.
                                                      Note: that #2,#3,#4 and #6 were
    Invoice     Invoice Cumulative      Dollar        Note any amount that is greater than
                Amount     $           Selected       the interval will always be selected
      1            $300        $300                   once.
                                                      Our Original Population is 6 items
      2             700       1,000          $499     totaling $5,050.
      3             900       1,900        $1,499     100% Examined Items is the 1 item
                                                      totaling $2,100
      4           2,100       4,000        $2,499
                                                      Sample Population is the original
                                           $3,499     population minus 100% examined
      5             400       4,400                   items - in this case 5 items totaling
                                                      $2,950 (or $5050-2100).
      6             650       5,050        $4,499
                                                      Our sample size would then be 3 items
     Total       $5,050               5 $ selected    related to the Sample Population with 1
                                                      additional item tested 100%.
!                        PPS Selection - Example
                                                  Invoice 1
                                                              $1,000   $499
                                                  Invoice 2
                                                                       ($499 cumulative $)

                                                 Invoice 3
    Invoice   Invoice    Cumulative    Dollar                 $1,000   $499
              Amount        $         Selected                         ($1,499 cumulative $)
      1          $300          $300

      2           700         1,000       $499
                                                              $1,000   $499
      3           900         1,900     $1,499                         ($2,499 cumulative $)
                                                 Invoice 4
      4          2,100        4,000     $2,499    $2,100
      5           400         4,400
                                                              $1,000   $499
      6           650         5,050     $4,499                         ($3,499 cumulative $)

     Total     $5,050                      5$    Invoice 5
                                      selected     $400

                                                              $1,000   $499
                                                 Invoice 6             ($4,499 cumulative $)

    1. Resolve and document misstatements just as you did for deviations in Test of controls.
    2. Compute misstatement amount (in testing for existence or performing vouching it would be
       book - audited and in testing for completeness or performing tracing it would be audited -
    3. Segregate items in Original Population over the interval or 100% Examined Items. These
       items are tested completely. For these items: misstatement found=PM=ULM.
    4. For items in the Sample Population (Original Population items< interval):
      a)Compute tainting %'s for all misstatements in Sample Population (in testing for existence or performing
         vouching it would be misstatement / book and in testing for completeness or performing tracing it would be
         misstatement / audited). Individual taintings should range fro m 0%-100%. Example if book amount is
         $5,000, audited amount is $3,750. Then misstatement is? $1,250. Tainting % is? ($1,250/$5,000=25% or
      b)Rank misstatements from largest tainting % in magnitude to smallest %.
      c) Compute PM by multiplying tainting %'s by the interval. Summing the products to compute PM for the
         Sample Population.
      d)For the number of misstatements obtain the reliability factors for the given RIA.
      e)Compute the differences. Multiply the differences by the respective PM to compute ULM's.
      f) Sum the products and add the reliability factor for zero misstatements x the SI, this is the ULM for the
         Sample Population.
      g)Misstatement <= PM (= if 0$ misstatement found) < ULM .
    5.Should add PM from the Sample Population and misstatement amounts
      from 100% Examined Items (items over the interval) to determine overall
      PM (projected misstatement) for the Original Population.
    6.Should add ULM from Sample Population and misstatement amounts from
      100% Examined Items (items over the interval) to determine overall ULM
      (upper limit on misstatement) for the Original Population.
    7.Allowance for sampling risk (ASR) or precision = ULM - PM.
    8.Decision rule if ULM>TM (Materiality) then reject population, if ULM <TM
      then accept.
    Note that PM is the point estimate like sample mean, ULM is like the
    confidence bound.
                                     Evaluation Example
                               Original Population - The one on which we want to draw a conclusion
      1.   RIA=25%, Original Population=$600,000, AM=$6,500,TM=$36,000  RF=1.39, EF=1.25  S.S. = ($600,000 x 1.39) /
           ($36,000 - ($6,500 x 1.25)) = 29.92 will round up to 30. Note these are intervals.
      2.   $600,000 / 30 = $20,000 interval.
      3.   Select all 30 intervals regardless if over interval ($20,000) or not - for this example assume 20 invoices< interval and 4 > interval
            the 4 invoices total $200,000.
      4.   Perform testing
      5.   Segregate population into two groups (only when doing final evaluation)

100% Examined Items - $200,000                                                Sample Population (Less than Interval ) - $400,000
(For this class we mean over Interval)
Assume only 100% item misstatement found
totaled $3000.                                         0$            $1,000               $8,000                                 $37,910
                                                              Sample Misstatement          PM              $29,910                ULM
                                                            Tested 20 items, found 3 (.20+.15+.05) x        ASR or        [$4,000x(2.70-1.39) +
                                                             misstatements totaling $20,000 interval       Precision      $3,000x(3.93-2.70) +
                                                             $1,000. Misstatement       = $4,000 +                        $1,000x(5.11-3.93)] +
                                                             taintings are 5%, 20%, $3,000 + $1,000=                        [1.39x$20,000] =
                                                                    and 15%.             $8,000.                            $5,240+$3,690 +
                                                                                                                            $1,180+$27,800 =

Then total misstatement for test would by $1,000+$3,000=$4,000. PM $8,000+$3,000=$11,000. ULM
ULM ($40,910) > TM (materiality) [36,000]  would fail test.
What happens to PM if doubled sample size Interval should decrease in half, Tainting sum double  No change
What happens as SS ↑  Sample Miss. ↑ PM (no change) ULM ↓ Tainting Sum ↑Interval ↓ Precision ↓
So if increase SS here, need precision to go down to $36,000-$11,000=$25,000 in order to pass. Might make sense to do.
What if TM was $12,000, need precision to get to $1,000 - does not make sense to do.
                 ULM > TM, Now What?
    (assuming you think results may be incorrect and the population is not actually
          misstated – i.e., you thought the sample was non-representative)
    Increase sample size. Makes sense if there is lot of precision (large ASR), and it is the precision that is
     causing you to fail. Using systematic random sampling would have to double SS. Using unrestricted
     random (if population entered on computer then not a problem) then could pick smaller S.S. Still may
     be impossible. Could be a problem of turn-around if doing confirms or can't extend sample if doing a
     physical inventory selection.
    Segregate misstatements based on a qualitative difference - example particular clerk makes a series of
    Expand other substantive procedures within cycle.

           (assuming you think results are correct and the population is misstated)
     Have client go through entire population and correct all potential problems and then
      reperform test with new sample. Usually impractical.
     Make AJE for PM (leaving precision) or in extreme cases possibly ULM. Debit : bad sales
      or bad debt expense credit A/R (unspecified customer) or allowance for bad debts. As find
      actual misstatements in A/R, then credit customer and debit unallocated.
     Allocate materiality from other cycles.
     Issue a qualified or adverse opinion – if client does not make adjustment.
     You usually need to choose one logical choice that will depend on your population.
                                               PPS vs. CVS
                           PPS                                                       CVS
          Probability-Proportional-to-Size Sampling                       Classical Variables Sampling
Low error populations with errors all in one direction.    Usually need many misstatements. Can have both over
                                                           and understatements.
Used on tests like A/R testing and search for unrecorded   Used on tests like inventory counts and price testing.
Cannot handle zero or negative balances well. Need to      Handles zero and negative balances.
know book values -usually not an issue.                    One method does not require book values.
Requires less auditor sophistication to use.               Requires greater auditor sophistication.

Requires no computer to perform.                           Usually requires a computer.
Each $ is evaluated.                                       Documents are treated as selected items.

SS and evaluations based on attribute sampling theory      SS and evaluations based on Normal Distribution.
(Hypergeometric Distribution).
PM determined by multiplying tainting total by interval.   PM computation varies. Example: Ratio Approach: ratio
                                                           of audited sample total to book sample total is 90%.
                                                           Would multiply by total population $ to get PM.
PPS method handles sample variation indirectly through     Must know variation, either by past experience or
selection method.                                          through use of pilot samples.
Method indirectly stratifies by $ amount.                  To reduce sample sizes, should usually stratify by $ or
                                                           qualitative characteristic.
Usually smaller SS than CVS.                               Usually larger SS than PPS.
              Additional Resources
AICPA Audit Guide: Audit Sampling (obtain through RIA

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