Faculty Working Papers A STOCHASTIC MODEL OF THE INTERNAL CONTROL SYSTEM Seongjae Yu #106 College of Commerce and Business Administration University of Illinois at Urbana-Champaign FACULTY WORKING PAPERS College of Commerce and Business Administration University of Illinois at Urbana-Champaign April 17, 1973 A STOCHASTIC MODEL OF THE INTERNAL CONTROL SYSTEM Seongjae Yu #106 A STOCHASTIC MODEL OF THE WTERKAL COWTROL SYSTEM BY Seongjae Yu, Dai vers tty of Illinois John Meter, University of Minnesota March, 1973 Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/stochasticmodelo106yuse ROLE OF INTERNAL CONTROL IK AUDITING The primary purpose of incorporating a set of Internal controls in the accounting system is to maintain a high probability of preventing and/or eliminating errors, Irregularities and fraud in the financial information process. The reliability of the system of internal control not only provides evidence as to the bona fides of the output of the sys- tem, but also influences the nature and extent of the auditor's examin- ation. The first step the auditor undertakes, therefore, is to establish to what extent the financial information system is supported by internal controls and to what degree such a system is reliable. It has become axiomatic that the effectiveness of internal controls must be taken into account in determining the extent and nature of the audit procedures appropriate in a given examination. The more reliable the system Is, the less extensive the tests the auditor need conduct* Recognizing this inverse relation between effectiveness of internal controls and audit scope, the American Institute of CFA's requires all auditors to first evaluate the reliability of internal controls as a matter of audit stem- 2 dards . Recently, the Committee on Auditing Procedures of the AICPA released several statements on the subject of internal controls which re -emphasize the importance of the study of the reliability of internal 3 controls. Despite the emphasis on evaluating the reliability of Internal 1 R. K. Mautz and H. A. Share f, The philosophy of Auditing (Ameri- can Accounting Association, 1961), p. 140. 2 The American Institute of CPA's, Statements on Auditing Procedure No. 33 (New York: The AICPA, 1963), p. 16. 3 For example, see Statement on Auditing Procedure Nos.49. 52, 5£ (New York: The AICPA, 1971-2). -2- controls, the auditor has bean devoid of means that enable him to quan- titatively evaluate the reliability of the Internal control system. Conventionally, the auditor uses questionnaires, flow charts, and tests of transactions for evaluation purposes. While these r*toods have merit, they do not result in an objective, quantitative evaluation. Trueblood and Cycrt stated one and one -half decades ago, The auditor typically deals with many subjective evalu- ations such as tlu appraisal of the overall functioning of the system o* internal control. . . , perhaps *>nly if the fabric of internal control has been formulated into mathematical models will the complete chain of reasoning be demonstrated by objective means. . . .^ The purpose of this paper is to propose a model that might be able to serve as the basis for an objective, quantitative evaluation of the reliability of the internal control system. Tying In this evaluation with the evaluation of the bonafides of the accounts Is not considered in this paper. NEED FOR STOCHASTIC SYSTEMS EVALUATION The financial information system consists of many operating ele- ments, Including Internal controls, which are methodically connected so that the system can produce* wheneve Inputs are given, some type of out- puts . The quality of the output depends on the quality of performance of the individual elements, but, generally, the probability that such per- formance la perfect is less than one. That Is, each element in the sys- tem has some propensity for Introducing errors and/or for not eliminating them. Four aspects of the financial Information system are of key impor- tance in developing a model describing the system of internal control: it. M. Trueblood and R. M. Cyert, Sampling Techniques In Accoun - ting (Englewood Cliffs: Prentice -Hall, 1957), p. 53. -3- 1. Internal control measures and devices are Integrated and meshed Into the financial information system « This implies that the evaluation of the system of internal controls cannot be done by itself in a vacuum but should be done in relation to the o erall system's reliability. 2. A system's reliability depends on the quality of performance of the operating elements. Mo matter how many internal controls are employed, if the persons who operate the controls do not perform their duties pro- perly, the internal controls do not assure high system reliability. 3* The quality of the accounting documents changes as they are processed through the system. Bach operating element can alter the quality by either adding or correcting errors. In a payroll system, for example, a time clock attendant makes an error in recording work hours (i.e., hour error only at this point); next, the error -ridden time card is used to compute gross wage (i.e., hour error and gross wage error at this point); on the basis of the computed gross wage, taxes are deducted (i.e., hour error, gross wage error, and deduction error at this point). Next, the internal auditor reviews the original time card and happens to detect the hour error and rectifies all the related errors (i.e., no error at this point), This situation not only shows how ;h^ quality o* the documents may change from one step to the next , but also indicates that the quality of the accounting output may be strongly affected by the quality of the input. 4. Since the operating elements add or correct errors typically in a fashion that may be viewed as probabilistic, the quality of the accoun- ting data may be regarded as a stochastic variable and the movement of error states of the accounting data may be described as a stochastic -4- process . In this paper* the term reliability is defined as the probability of the internal control system eliminating errors* irregularities, or fraud in the financial information process under the operating conditions present during a given period. MODELING OF INTERNAL CONTROL SYSTEM In order to facilitate the presentation* we will discuss the proposed modeling process and the nature of the problems encountered for a small segment of the accounting system, namely in terms of the payroll system. The reason the payroll system is chosen is: (1) it is a system found in every business organization; (2) it is a self-contained system that has limited interactions with other segments of the accounting sys- tem; and (3) the payroll is usually a significant portion of total business expenses . A hypothetical manual payroll system is presented in the form of a flow ehart in Figure 1. This payroll system has the basic payroll con- trol measures: time card punching, foreman's review, payments by checks, review by the controller* the use of the imprest account, and monthly bank reconciliation. Definition of Operating Elements Since we shall view the quality of accounting data processing as a stochastic process* we first need to define the operating elements of the payroll system. For our purposes* an operating element will be a per- formance unit that can affect the state of error or quality of documents. Operating elements may be classified according to various criteria that the auditor considers most appropriate for his purposes . The criteria -5- that appear to be useful for our purpose are "by people" and "by function," Using these criteria, sixteen separate operating elements are obtained for the payroll system, and they are shown in the flow chart of Figure 1. The rationale for the two criteria are: (1) the performance of the elements can be easily observed and evaluated when they are classified by people and function because many documentary evidences in accounting are easily identifiable by these classifications; (2) in a financial information sys- tem where possibilities of intentional ab well as unintentional errors and irregularities are of major concern, the arrangement of people and functions can have a significant Impact on the system reliability. A constraint, however, must be taken into consideration in defining the payroll system. opera ting elements by the two criteria: independent relations between defined elements must be maintained. The reason is that the mathematical model to be suggested requires independent relations. This restriction is not a serious one, however, because independent rela- tions are typically found in financial systems to help reduce and control errors, intentional end others. Definitions of Errors and Error Starts Kext, we need to define the kinds of errors the operating elements may commit during the operating processes. Errors in payroll accounting are made typically in work hours, employee names, deductions, over-or- under -payments, pay rates, and payments to non-existent employee. Errors may arise singly or in combination. Furthermore, an error of one kind may induce other kinds of errors. For example, an error in employee name may lead to an error In pay rate and deductions for another employee. To make the demonstration of the model simple, we will define just - 5a - -> Initial) & A: Dept. X Time card punohing <- Initial) * B: Dept. Y Time card punching K Foreman' ir approval- Foreman's* -> Ei Payroll clerk A, Time cards processing last period's earnings reoord cards, "~Y~ V V 0: Payroll clerk B Operation of acctg machine: paychecks; earnings stmt*; payroll register; updated earnings record cards F: Payroll c lerk A' Adding machine control tape T: Controller Bank account reconciliation ( Term inal ) Figure 1 -6- two kind* of errors: (1) monetary errors, and (2) non-monetary errors. Monetary errors Include any error that is represented by a $ sign. For example, errors in pay rate, tax deductions, net pay, and gross pay fall in this category. Non-monetary errors include all other errors such as errors in name, social security number, work hours, etc. We can there- fore enumerate four distinct categories that describe the quality state of an accounting document after the processing by any operating element: Category 1: No error of either kind Category 2j Monetary error only Category 3 : Non -monetary error only Category 4: Monetary as veil as non-monetary errors As the payroll operation proceeds from one element to the next, the pay- roll documents will move between these four error states. Consider a vector of the form (r^, r~) where r. is an indicator variable taking on the values 1 and to represent the presence or absence, respectively, of monetary errors, and r„ is defined likewise for non-mone- tary errors. We can therefore describe the four categories as follows: ». * (0, 0) : Absentee of any errors s ? « (1, 0); Presence of monetary error only «* ■ (0» *) 5 Presence of non-monetary error only 8/ •» (1, 1): Presence of monetary and non-monetary errors In general, we may define as many relevant kinds of errors as the audit situation warrants. We simply need to enumerate, according to the above rules, all possible combinations of error states, one of which each accounting document must occupy at any point in the processing. Operation Frobability Matrix 5 Operating clesteate perform basically two types of functions: (1) transformation operations, and (2) decision operations. A transfor- mation operation converts input data into output in accordance with certain rules. For example, time card punching, preparation of paychecks from punched time cards, and posting to payroll ledger are transformation oper- ations. These are shown in rectangular boxes in the flow chart of Figure 1. A decision operation, on the other hand, is basically a sorting oper- ation. It distinguishes one input from another and takes a separate action for each, like a sieve or gravel grader. For example, a foreman reviews time cards and rejects any unusual ones as incorrect; or the con- troller reviews employees' earnings statements and rejects for correction those he judges as incorrect statements . Decision operations are shown in diamonds in the flow chart of Figure 1. While each operating element is supposed to follow a set of deter- ministic rules, its performance is not always perfect, resulting in some deviations that are commonly called errors. Each operator has some pro- pensity to introduce errors, as well as a propensity to change and elim- inate errors. These propensities are a function of the operator's skill, the quality of the input, and the characteristics of the transformation or decision operation. One way of describing the propensities is by use of the probability concept . \. C. Hare, Jr., Systems A nalysis; A Diagnostic Approach (Hew York: Harcourt, Brace & World, 1967), pp. 27, 41. ft. Leasing, "A Per former -Oriented Approach to Systems Quality," Mana gement Science. . 16, 1969, p. B-264.- •8- To illustrate how probabilities can be uaad to model those pro- penalties, consider a payroll clerk who prepares employees • earnings statements and paychecks from time cards and earnings record cards which are assumed to have no errors. In other words, the payroll clerk performs a trana formation operation, and the documents reach him in state s« . The payroll clerk may produce error "free paychecks and earnings statements (state e ) most of the time, say with a probability of .95. Occasionally, he produces outputs containing monetary errors only (state s 2 )> non-monetary errors only (state s~) or both (state s^) , say with probabilities of .02, .01, .02, respectively. The situation can be shown by s tree diagram: ■^ s«: Error -free input Input state of error a.: Error -free output 8-: Output with monetary error only s_: Output with non-monetary error only s : Output with monetary and non -monetary errors Output stste of error We indicated earlier that the operator's propensities typically depend on the quality of the input. Thus the probability that the output document occupies any specific error atate may be different from the one shown in the tree diagram if the input state is not s^. Thus, we need four different tree diagrams, as there ere four different input error states In our example. The presentation of four different trees csn best be summarised by a matrix: -9- s x - (0, 0) « 2 « (1, 0) » 3 - (0, 1> t 4 - (1, I) • x - (0, 0) p ll p 12 P 13 p 14 » 2 - < l » °> p 21 P 22 p 23 p 24 •3 * (0, 1) p 31 p 32 p 33 p 34 « 4 - (i, 1) \ 41 P 42 P 43 P 44 / tatrix P is called e IL* M formation Probability Matrix. Tb • p 1 of p. . represents the input state of error, corresponding to a,, •«» •*» 8/, respectively. The second subscript, J, represents the output state of. error. The symbol p is the probability, given an input state a*, that the transformation operator will produce an output state s.. The symbol pj4 is called a transition probability, and must satisfy the conditional O^Pij^l. s J p iJ - 1. A decision operation may be described in a similar way. Consider a foreman who reviews the time cards at the end of each period and clas- sifies them as either approved or rejected before they are forwarded to the payroll department. His decision will not always be perfect, but sub- ject to some probabilistic pattern. If he receives time cards reflecting over-stated work hours (state s~), he may classify them as incorrect with a probability of, say, .96, and as correct with probability .04. This may be shown as follows: ~> s^i overstated time cards 83: classified as incorrect S3: classified as correct -10- or, in vector notation: classified at incorrect classified as correct •1 • 3 : (0, "2 0, "3 .96, ■ 4 0) 3 * 3 J |0, 0> 3 .04, 0) A similar expression can be obtained for the other incotamg quality states Combining these, we may obtain a Decision Probability Matrix Q as follows: Incorrect category Correct category •l ■2 «i <»2 q 3 *4 6 o «i *2 <3 •Q / where qj + q? - 1, and q» q" $ 1 for every j, (J- 1, 2, 3, 4). We shall denote the first and second components of Q as Q* and Q", respectively. Basic Operation on Input Vector To illustrate how the elements discussed so far are combined, suppose the punched time cards forwarded to Payroll Clerk A (Figure 1) are distributed among the error state? according to the following probab- ility distribution: s, 8. W, 3 .034 "4 0). e l j ' (.966 This vector signifies that the time cards are free from errors with prob- abilities of .966; have non-monetary error only with probability .034; and have no errors otherwise . This vector is called an input vector. Payroll Clerk A performs his payrolls processing on this input. Suppose -11- Payroll Clerk A has the following transformation probability matrix: s B, s, B, S <94 .44 E ' «3 .06 .56 Then, the output quality resulting from Payroll Clerk A*s operation is obtained by matrix multiplication W- * P on 8 1 8 2 8 3 8 4 W - Wj * P - (.925 .075 0). W is called the output vector. It shows that Payroll Clerk A has a ten- dency to increase the non-monetary error rate from .034 to .075. This output vector is then used as an input vector for the next operation (Payroll Clerk B's transformation) to obtain another output vector. This process is continued until the last operation. As a result, we obtain a sequence of output vectors showing how the quality of the documents changes between the four error states as they are processed through the system. However, the basic operation linking the input vec- tor and the transformation probability matrix: 1 o iss subject to some modification because not all operations are connected in one series. Sometimes, the documents are branched for different pro- cessing routes and then merged , sometimes merging of various types of documents takes place, and sometimes documents are returned for correc- tion. We consider now how these more complicated cases are handled. -12- 'te§2£hkBX>^BS£&l23&.* Branching generally arises due to internal controls such as reviews,, approvals, or comparison* that, result in sorting between acceptable and unacceptable documents. In our payroll example, the Controller reviews employees' pa checks, payroll register and updated earnings record cards and rejects any erroneous items before be signs the paychecks • Suppose his decision probability matrix Q has the following form j Q ' i Approved as oorrect Q"? Rejected as Incorrect S "2 s 3 a .99 8 3 .87 .92 .65 .01 •2 •3 '4 N 13 .08 .35 \ / 4 Also assume the documents the Controller receives from the payroll clerks have the error distributions 2 lj • (.894 .011 .046 .049). Then, by multiplying « and Q, we obtain, X w i * q " w i " L Ql * Q ") , W, • Q M 1 ■ *"0* "Ol -[(.885 01 .042 .032), (.009 .001 .004 .017)) , where the first and second component vectors are denoted &b W* and W", ' o o* respectively, JSOte that the Controller, asaong the 89.4% of perfect docu- ments, classifies 99% (i.e., .885/. 894) as correct, and 1% (i.e., .009/. 894) as incorrect, and similarly for documents in the other error categories, thus, a set of payroll documents is branched, with certain probabilities, "13- for different routes and actions. There Is another type of branching that arises when there are multi -copies of the same document for a given transaction. An example amy be found outside the payroll syrfcem: the sales department prepares three copies of a sales transaction; one copy is routed to the customer, the second to the accounts receivable clerk, and the third to the inven- tory clerk. This type of branching is not real branching, as each party receives the identical documents of the identical quality. In this case, the basic operation of the form Wj • F is applicable for each route. Me r Ring operations . Documents that are branched or processed separately may merge together again. There are two types of merging operations. With the first type of merging, documents representing sep- arate transactions of different departments are merged together. In the payroll example, the time cards of Department X employees are merged with those of Department Y employees in order for Payroll Clerk A to process them together. This kind of merging is handled by adding the two out- put vectors from each department, with suitable normalization. Suppose 50 time cards from Department X find 100 time cards from Department Y are merged » and that the output vectors are: V « (.938 .062 0) x W - (.980 .020 0), respectively. The merging of these two sets is obtained by adding the vectors with appropriate normalization, which is necessary to maintain the characteristic of the probability vector after merging: W, / 3 + 2W / 3 « W v „ • (.966 .034 0). -14- Thia direct addition is justified because each vector represents different transactions. The second type of merging, which corresponds to the second case of branching discussed earlier, arises when the input vectors coming from different routes represent the identical set of transactions or when different documents pertaining to the same transaction are combined to form one complete document. For example, Payroll Clerk B receives two sets of documents for a given employee *s payroll processing: the earnings record card showing the accumulated payroll information up to the last pay per- iod and the current period's time card. These are combined to constitute one record. One important problem with this kind of merging is that the error state of a given transaction document coming from one route may be dif- ferent from the error state of the transaction document from another route. When merged, the outcome state may be still another type of error state. For example, consider an employee's earnings record card that con- tains some monetary error: ^ * (*» Q ) • Also suppose that the same em- ployee's time card has omiy non-monetary error: s~ «■ (0, 1). When Pay- roll Clerk B puts these two documents together for the employee's paycheck preparation, he processes with an input document having both monetary and non-monetary errors: s^ * (1, 1). Let W^ and W r denote the input vectors representing the qualities of th£ earnings record cards and the time cards, respectively; and let their components be W t « (w x w 2 w 3 w 4 ) ®r-<"{ w 2 w 3 W i>' -15- H0) s 3 -(0,l) %"-(l»l> (0,0) (1,0) (0,1) (1,1) v l* w l v 2'*i *3' w l v ¥ i (1,0) (1,0) (l.D (1,1) w l' w 2 v 2' w 2 V w 2 V w 2 (0,1) (1,1) (0.1) (1,1) V w 3 v 2' w 3 w 3' w 3 V w 3 (l.D (1,1) (1,1) (1,1) "l ' w 4 ' w 2' w 4 1 w 3' w 4 w 4- w 4 States from the last period ** earnings record cards Figure 2 k mechanical method of preparing the table is to follow the rule: (i,j) ♦ - (m,n), where i, j, h, k, n, n ■ 0, or 1; and b, n ■ 1 if the vector addition results in a number greater than or equal to one; otherwise m, n ■ 0. Since the operating elements are defined to have independent -16- relations, the table of Figure 2 can be used as a joint probability table, each cell occurring with probability v^ • w r , (i,r * 1, 2, 3, 4). Wow grouping the joint probabilities w • w* according to the newly formed error states in the table, the following merged vector W^ ■ (wj wj wj w?) is obtained: Si - (0,0): w£ -(yy w«) s 2 * (1,0) « w£ - (w x . «•) 4- (w 2 • w{) + (w 2 . wp s 3 » (0,1): *» * (w x -,«j) + (v 3 .n{) + (w 3 .w$) s 4 - (1,1) J W4 * (v 2 • wj) + (w 3 • w£) + (w x • wp + (* 2 ' w 4^ + < w 3 * w 4> + < w 4 * w 4> + 0* 4 • w J) + (w 4 * w£) + (w 4 . wp . Feedba ck operation for correction . A feedback operation typically arises when an internal control element detects errors and returns the error-ridden document back to an appropriate operator for correction* This feedback operation always requires branching (i.e., separation of error documents) and merging (i.e., reunion with the corrected documents). Returning to the example used for explaining a branching operation, suppose the Controller reviews paychecks and payroll register and separates between correct and incorrect documents as follows: Correct documents: W^ * (.885 .01 .042 .032) Incorrect documents: WJJ * (.009 .001 .004 .017). Since in accounting data processing, most of the errors detected by inter- nal control measures receive special attention, we assume that all detected errors emerge eventually as correct documents. To reflect this, we use a -17- spfttctal transformation matrix Ki *l 1 a, 1 1 1 8, * R to which the rejected output vector W£ is multiplied. For our example, we obtain: W" • R - (.009 .001 .004 .017) ( 10 10 10 - (.031 0). Now, thla vector la rejoined to W Q : W o +W o * R " <- 916 * 01 - * 2 - 32>. This merged output vector reflects the quality of the documents passed through the controller's inspection. When the documents returned ?or correction receive less than per- fect attention and consequently still contain errors, the problem becomes more complicated, involving repeated feedback loops. An overall view of the system. We have explained how accounting data processing, including systems of internal control, can be described by the basic operations of branching, merging, and feedback, and have discussed how to model these processes. These processes have been com- bined to model the payroll system used as an example in this paper, with -18- hypothetical trans forms t ion and decision matrices provided in the appen- dix. The results are shown in Figure 3. We discuss the implications of this model next. USES OP THE MODEL The auditor can use the proposed model for internal control for at least two purposes: (1) for the probabilistic evaluation of the sys- tem reliability; and (2) for assisting in the design of the internal con- trol system. Probabilistic Evaluation of the System Reliability The immediate purpose of reviewing the internal control system is to obtain information that will show how much confidence can be placed in the system. This information is summarized in the terminal output vector produced from the model. In the payroll example, we have two terminal output vectors : •l *2 8 3 % W fc « (.992 .002 .004 .002) W - (.871 .056 .042 .031), m where W\ aad W denote the output vectors from the bank reconciliation t m a ad the journal entries to the payroll expense accounts, respectively. The closer the first component of each vector is to 1, the higher the probability that the syscea produces reliable results. If the first com- ponent it; not of high probability, we may wish to examine the third com- ponent of the vector, which is associated with non-monetary errors and thus not related to the monetary bona fides of the payroll expense, to see if most of the final errors Are of the non-monetary type. 19- Initial)""^ A; Sept. X Time card punching Initial) ~> (..qs8 o .o«l o) B ; Dep t. Y Time card punching (.966 ,0?f of B: Payroll cleric A Time cards processing AID .030 o) llTT^i^T lCWz M M\ > o°$) (Ms o >o7S *) (.9^ earnings reoord cards, "V (.?10 M] .o«fa.do9J .OK 0) - •* — Qi Payroll clerk B Operation of aoctg machine} paychecks; earnings stmt.; payroll register; updated earnings record cards I I I I I i I ! V P: Payroll c lerk A Adding machine oontrol tape CHI .036 .[06 .023) ffii .on ,m .eg*) (.881 .044 ,ooi r&tfQ ^~ — ~.^ — ( .ff*i .oejj^o^r .oi z) (.00$ MS/.bty ■ / Comparison V ■>! < Controller's review J f/6 .©fa .0^2. .o?2) V —7 K; Controller Preparation of payroll voucher / (/1"7g ,0ft C .0AT ,#o2) (.9/7 ,0/0 .0*2 .030 i — j£ •Y .<'#'• M: Journal clerk Posting to ledger (Terminal) ^ Lj Treasurer Payroll fund ' transferring check [H'B JOS M? ^n^Bal* 0#8"? /fi/6 .07/ ,02 Prob(s 4 ) Figure 4 T Operating elements Suppose we want to see the impact of a non-monetary error Intro- duced at a given, point in the process on the system's reliability. The .reduction of such an error into the system may be simulated by using a probability vector of the form: s l s 2 s 3 s 4 W(s 3 ) = ( 1 ), which signifies the existence of an error of the non-monetary type (s~) -22- with probability one. To study the Impact of such an error at the begin- ning of the ays tea, we use this vector as the input vector for the first operating element of the system and go through the aeuai process to come out with a terminal outpv tor. #e can study the impact of a non-mone- tary error at any other paint in the system in similar fashion. Such an- alyses can be most helpful. For instance , auppor.e that introducing a non- monetary error in the prep-* ratios* of the payroll register leads to low terminal quality. Hence, a clos® examination of such errors at the stage of the payroll register preparation is most Important. Effect of changes in operating probability matrices . The model can also be used for investigating the effect of unusual performance of operating elements. For example, hiring of new employees, breakdown of accounting machines, intentional disturbance of operation by dishonest employees, intensive employee education, etc., would result in different performances. To see the impact of such changes on the quality of the terminal outputs, we would need to obtain different operation probability matrices reflecting the new situation and replace the old matrices with the nos8iblllty of analyzing the (tern of internal control in objective, quantitative terms. It enables the fabric of operating and control elements to be mathematically struc- tured and quantitatively evaluated. As a result, the auditor may be bet- ter able to j (i) analyse the internal control system and explicitly find out the weak and strong areas, (2) assess the impact of the weaknesses and strengths on the quality of accounting data in probabilistic terms, -24- (3) advise his el lent about system problems, and (4) properly adjuat his audit programs to meet the situation. A consequence of the use of the proposed model in auditing is that it provides a new purpose to statistical sampling methods in auditing. Conventionally, statistical sampling methods are usud in a direct attempt to evaluate the system reliability. However, the applications are on piecemeal bases, and the objective Integration of the statistical tests applied to various segments of the internal control systera has been diffi- cult. With the stochastic process model, otatistical sampling methods largely will be used for estimating the probability matrices. The proposed model is airio applicable to a computerized accounting system. Such a system typically involves processes puch as preparation of source documents, transmission of documents to the EDF department to be merged with other documents, input conversion to machine -readable form, disposition of error messages from the computer, and distribution of out- 7 puts to appropriate users. During each of these operating processes, the quality of documents may change, so that such a process may be described by the kind of model discuss&d in this paper. Th« use of the proposed model may open a new road toward the quan- tification of the auditor's judgment as to the bone fides of account balances, for account balances are directly influenced by the system's reliability. Like many new models , the proposed one is not free of implementation difficulties. Problems like definitions- of operating elements and errors nfc-ad careful consideration, because too detailed definitions might greatly pp. 103-116. 7 6ordon B. Davis, Auditing and EDP (Hew York: The AICPA, 1968), -25- increaae the complexity of the flow chert end error statea, while too coeree definitions might feil to reveel significant information. For example, if we define just three kinds of errors ins teed of two, the number of error states according to our error states definition would 1 2 be 8 (2 instead of 2 ) . Since the Increase of the number of error stetes is exponential, a detailed definition of error kinds would easily result in e prohibitively large number of error states end consequently too large a probability matrix size. A large probability matrix size not only causes computational problems, but also creates various estimation problems. Even though the g statistical estimation of transition probabilities is conceptually clear, the cost and feasibility of estimation may be a practical constraint. We do not believe these problems are Insurmountable, but exten- sive work will be required to Implement our proposed approach. 8 For example, see T. W. Anderson and L. A. Goodman, "Statistical Inference About Markov Chains," The Annals of Mathematical Statistics . 28, 1957, pp. 89-110. -26- APPENDIX The following hypothetical data were used in generating the se- quence of output vectors shown in Figure 4 in accordance with the model discussed in this paper. The order of the elements in the vectors and matrices corresponds to the definition of the error states, i.e., s,, V V V 1. The initial input to time punching operation -(1000) 2. Transformation and Decision Probability Matrices: A * Time card punching in Dept. X (50 employees) P(A) - .94 .06 .97 .03 *s B ■ Time card punching in Dept Y (100 employees) / \ P(B) / .92 .08 .98 S .02 / C • Dept X foreman's approval of time cards (Approve as correct) Q.(C) .92 .45 .37 (Reject as incorrect) .08 .55 .63 .64 ■27- D " Dept. Y foreman's approval of time cards (Approve as correct) (Reject as Incorrect) Q(D) .96 .41 .70 .19 .04 .59 .30 .81 V E * Entering hours, rates, deduc- F ■ Preparing an adding machine tions to time cards by Payroll control tape by Payroll Clerk Clerk A A P(E) - / *N .94 .06 1 .44 .56 1 F(F> <95 / G ■ Operation of accounting ma- chine by Payroll Clerk B, producing paychecks and ear- nings statements, updated earnings record cards, and payroll register. .04 .01 1 N .03 .09 .88 / P(C) ^93 .03 .02 .0 .02 .89 .01 .08 .01 .02 .94 .03 P00 K * Preparation of payroll voucher by controller 1 .95 .05 .971 .01 .04 H ■ Sorting spoiled checks by Payroll Clerk B .95 / f (Accept as correct) (Reject as spoiled) Q(H) • .99 .62 .40 .15 .01 .38 .60 .85 \ -28- I « Comparison of payroll register with the controlling tape by Payroll Clerk A Q(X) ■ f (Accept ae agreed) (Reject as discrepant) \ 1 .18 .98 .82 .02 .69 .31 J ■ Review of the paychecks and earnings statements, updated earnings record cards, and payroll register by controller (Approve *b correct) Q(J) .99 .87 .92 V .65 L ■ Preparation of the fund trans- fer check by treasurer to transfer fund to the payroll bank account *(*) f» .01 °1 .01 .99 1 . Loi c .99 Processing of the paychecks bv che bank P(M) r; 1 1 1 1 !S / (Reject as incorrect) \ .01 .13 .08 .35 If * Journal entries of the pay- roll expenses by the general ledger bookkeeper P(M)- [.95 .05 .01 .95 .02 .02 .99 .01 .01 .01 .01 .97i T » Monthly reconciliation of the payroll bank account by controller P