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Estimating transactions given balance sheets and an income statement Arya Anileelliugham John CSchrocder Douglas A JAMES in Accounting Education Aug 2000 15 3 ProQuest pg 393 Issues in Accounting Education Vol 15 No 3 August 2000 Estimating Transactions Given Balance Sheets and an Income Statement Anil Arya John C Fellingham and Douglas A Schroeder ABSTRACT One way to develop an appreciation tor the power and beauty of double entry is to derive consistent transaction vectors ie start with financial statements and derive the transaction amounts that could have generated the statements Working backward to uncover transactions the inverting exercise complements the more tra ditional approach of working forward from transactions to financial statements In addi tion the approach highlights a fundamental accounting activity aggregation Financial statements summarize a firms transactions using only a relatively small number of account balances A consequence of aggregation is that there are infinite consistent transaction vectors However because these infinite solutions are all prepared in ac cordance with doubleentry they are linked to each other in a systematic fashion We show how a directed graph representation of the accounting system can be used as a parsimonious means of characterizing all consistent transaction vectors As the num ber of accounts and transactions are increased the inverting exercise becomes te dious if one does not make use of the directed graph To emphasize this point the inverting exercise is conducted using a set of published financial statements We also discuss the issue of picking the most likely transaction vector from the set of consistent transaction vectors The authors have used this note in undergraduate MBA and PhD classes For the PhD class the note has been supplemented by the more rigorous analyses in the linear algebra literature and the accounting literature INTRODUCTION Reproduced with permission of the copyright owner Further reproduction prohibited Without permission ggregation is a pervasive theme in accounting Managerial ac counting texts routinely talk about collecting costs ofinputs in various categories or cost pools Some cost pools are assigned to specic products prod uct costs while others are assigned to the period in which they arise period costs Tax accountants regularly aggre gate at least implicitly by calculating taxes as a fraction of a bottomline num ber The preparation of nancial state ments arguably the most basic task an accountant undertakes is itself an ex ercise in aggregation Information re garding the rms activities several transactions is conveyed as best as Anil Arya is an Associate Professor John C Fellingham is a Professor and Dou glas A Schroeder is an Associate Profes sor all at The Ohio State University We thank Rick Antle Joel Dernski John Hill associate editor Jonathan Glover Yuji I n Brian Miendnr David E Stout editor Gil bert Strong Richard Young students in the undergraduate and MBA programs at The Ohw State University workshop participants at Case Western Reserve University The Ohio State University Purdue University Univer sity of Florida Yale University and two anonymous referees for helpful comments Professor Arya acknowledges support from the Deans Research Professorship Professor Fellingham acknowledges support from the H P Wolfe Foundation 394 possible using a relatively small num ber of account balances In this paper we present a tech nique for starting with the aggre gated nancial statement data and working backward to the disaggre gated transaction data The technique exploits the students training in doubleentry their understanding of basic linear algebra concepts and their prociency in computer technology Students are introduced to ideas from several elds of study not all of which are familiar to traditional accountants However as the AICPA Vision Project 1998 4 reports mlany of the tradi tional essential skills of CPAs are be ing replaced by new technologies that are increasing in number and being rapidly developed often from unex pected sources Important pedagogi cal objectives arise First the method allows students to take a published set of nancial statements and infer something about transactions from the account bal ances The students apply their ac counting knowledge to realworld situations Typically asking the stu dents to work backward to transactions with a real set of statements leads to frustration since there is not a unique answer With the method presented in the paper answers are always acces sible Studying problems with multiple solutions is consistent with the recom mendation in Accounting Education Change Commission 1990 1992 For example the Accounting Education Change Commission 1992 250 rec ommends students should plossess enhanced analytical skills and the abil ity to confront unstructured prob lemsthat is problems with more than one defensible solution Second Rebele et a1 1998 42 ex plicitly ask Should instructors order the presentation of information in a Issues in Accounting Education manner consistent with business ac tivities financial statements or some other order We speculate that there may be benets to reversing the order on occasion By doing the inverse op eration students may be better able to understand the fundamental account ing aggregation process of transform ing transactions into financial statements A common and often effec tive way to learn about any process is reverse engineering ire do the pro cess backward The different perspective generates new insights and forges con nections The method presented in the paper always allows the student to go backward for any accounting problem whether it is in a textbook or encoun tered in a job situation eg when an auditor analyzes nancial statements Third by studying the technique students learn a variation of linear re gression Students have seen applica tions of regression in management accounting courses and perhaps in empirical capital markets research The standard regression problem typi cally has no exact solution since there are more equations observations than variables However in our setting there are multiple solutions since there are more variables than equations Relating our technique to regression analyses is consistent with Chois 1993 recommendation for a stronger foundation in quantitative analysis in accounting education Finally another reason to study a broad issue like aggregation is that it allows us to make connections between accounting and other elds such as lin ear algebra Recently there have been calls for fundamental changes in ac counting education both regarding what we teach and how we teach For example the Bedford Committee report AAA 1986 185 states alccounting faculties should be receptive to an expansion of Reproduced with permission of the copyright owner Further reproduction prohibited Without permission Arya Fellingham and Schroeder educational requirements in the liberal arts and sciences that aim to develop the students capacities for analysis synthesis problemsolving and com munication We think highlighting the links in academic ideas from other dis ciplines and basic concepts in account ing points to the richness of account ingl Accounting is academic and practical at the same time The theme of linear aggregation the use of straight lines is more prevalent in ac counting than in any other business discipline The entire eld of linear a1 gebra is devoted to similar ideas It seems somewhat satisfying that the language developed in the two disci plines is connected A historical note is in order here Given its underlying math ematical structure it is perhaps not sur prising that the rst published work de scribing the doubleentry bookkeeping system and its underlying rationale Summa de Arithmetica Geometrica Proportioni et Proportionalito by Luca Pacioli in 1494 was primarily a trea tise on mathematics Having motivated the study of the invertibility problem we now return to the theme of aggregation Aggregation in the accounting process is achieved through simple linear aggregation rulesl The linear aggregation rule we study in this paper is doubleentry book keeping The hookkeeping process is lin ear if all transaction amounts are doubled then so are all the account bal ancesl This permits a matrix represen tation The bookkeeping process is doubleentry every transaction affects two account balances by an equal and opposite amount one is debited and the other is credited This permits use of a special kind of matrix and a visual rep resentation of how accounts are linked through transactions The special matrix is referred to as an incidence matrix 395 and the visual representation is a di rected graph We highlight aggregation proper ties of bookkeeping by studying an es timation problem presented with a set of financial statements consisting of beginning and ending balance sheets and an income statement what is the readers best guess of the underlying transactions72 The reader may be an auditor an analyst or any other in formed user of nancial statements Solving the estimation problem re quires the reader to understand the invertibility processthe process of identifying all sets of transaction amounts that generate the same nan cial statements We dene transaction vectors to be consistent if they gener ate the given nancial statements The directed graph representation of the accounting system proves convenient in identifying all consistent transac tion vectors All consistent transaction vectors differ only to the extent they rely on the loops of the directed graph That is the loops of the graph represent the indeterminate part of transactions The nonuniqueness in the invertibility solution emphasizes that accounting aggregates which results in information 1055 The loss in infor mation is captured by the degrees of freedom available in writing the invertibility solution The extent of aggregationhow many accounts should be used to represent how many transactionsis not exogenous as is usually the case in classroom prob lems but is a choice variable for the designer of the accounting system Understanding the link between the l See folexample Demski 1994 or Ijiri 1975 for discussions oilinear aggregation in accounting 2 We thank Joel Demski for suggesting this question Reproduced with permission of the copyright owner Further reproduction prohibited Without permission 396 number of accounts and the loss in information can presumably help the designer make better choices regard ing the accounts to use to summarize transaction data Finally we would like to point to a specic incident in class that moti vated this paper One of the authors was teaching an introductory nancial accounting class and was preparing ii nancial statements for a set of trans actions that were listed in a textbook problem As a homework assignment the instructor assigned a simple prob lem He gave the students a new set of transaction amounts and asked them to redo the class problem The instruc tor chose the transaction amounts quite arbitrarily It was only in the next lecture he realized that the bal ance sheet and income statement for the homework problem were identical to the class problem Other than say ing this was a result of accounting aggregation he was unable to give a more satisfying answer nor was he able to explain how other such transaction amounts could be characterized This paper is a consequence of that embar rassing situation3 The remainder of this paper is or ganized as follows In the context of a stylized example the next two sections explain the invertibility process and dis cuss the estimation problem The esti mation problem refers to picking the most likely transaction vector from the set of consistent transaction vectors The penultimate section repeats the entire sequence with a set of published nan cial statements Coldwater Creek Inc 1997 The nal section discusses as sumptions of our technique and suggests directions for future research Issues in Accounting Education THE NVERTIBILITY PROCESS In this section we start with a set of nancial statements and a list of the seven transactions the firm undertook during the period The goal is to solve for the amounts of the seven transac tions that could have generated the nancial statements We refer to this as the invertibility process Numerical Example A manufacturing rm recognizes the following seven transactions or ac counting events 1 Purchase of raw materials inven toty for cash 2 Plant and administrative buildings acquired for cash Cash period expenses Cash sales Cost of goods sold Product cost depreciation Period cost depreciation4 soeew 3 The authors have used this note in undergradu ate MBA and PhD classes Given the non traditional nature of the subject the authors have been pleasantly surprised by the effective ness of communicating this material using Ex oel The invertibility prooess is emphasized inthe undergraduate and MBA classes Unlike the estimation problem the invertibility step re quires no discussion of probability distributions and is a mechanical exercise that highlights ac counting structure For the PhD class the mate rial in this note is complemented by the more rig orous analyses in Arya et a 2000 Butterworth 1972 Liiri 1967 1971 Mattessich 1964 and Strang1998 Depreciation associated with plant is treated as a product cost We credit plant and debit inventory A more complete journal entry would debit manu facturing overhead and close overhead to inven tory Depreciation associated with administra tive buildings is treated as a period cost We credit plant and administrative buildings and debit general and administrative expenses Reproduced with permission of the copyright owner Further reproduction prohibited Without permission Alya Fellingham and Schroeder 397 The rms assumed income statement for the period along with beginning and ending balance sheets are as follows Balance Sheet Ending Balance Beginning Balance Cash 2 10 Inventory 4 0 Net plant and administrative buildings 6 0 Owners Equity 12 10 Income Statement Sales 10 Cost of goods sold 5 General and administrative expenses 3 1 Income 2 The invertibility question is what were the amounts of the seven transactions events that generated these nancial statements The answer is that there are an innite number of consistent transaction vectors however they are all related to each other in a systematic fashioni Below are examples of two sets of transactions events that are consistent with the nancial statements Journal Entries Set 1 Set 2 1 Inventory 7 8 Cash 7 8 2 Net plant and administrative buildings 8 9 Cash 8 9 3 General and administrative expenses 3 1 Cash 3 1 4 Cash 10 10 Sales 10 10 5 Cost of goods sold 5 5 Inventory 5 5 6 Inventory 2 1 Net plant and administrative buildings 2 1 7 General and administrative expenses 0 2 Net plant and administrative buildings 0 2 Reproduced with permission of the copyright owner Further reproduction prohibited without permission 398 Issues in Accounting Education It is easy to verify that each set results in the same statements by posting to Taccounts in the regular way For example the Taccounts below book the en tries in Set 2 Net plant and Cash Inventory administrative buildings bb 10 bb 0 bb 0 8 1 1 8 5 5 2 9 16 4 10 9 2 61 2 7 1 3 eb 4 eb 6 eb 2 General and Sales Cost of goods sold administrative expenses 10 4 5 5 31 10 5 7 2 3 Except for journal entries 4 and 5 which are the same for both sets there does not appear to be any discernible pattern connecting the two sets Further more trial and error to nd all the allowable sets does not really seem feasible What we need is a systematic approach In the next section we study a method for combining transactions into nan i cial statements that is for going forward through the accounting cycle The method is matrix multiplication and is equivalent to the Taccount formulation used for preparing financial statements The matrix representation however is a more systematic approach for going backward from nancial statements to transac tions and helps in characterizing all consistent transaction vectors Step 1 Directed Graph Construct a matrix that contains all the information about which accounts are connected by journal entries We call this the Amatrix The matrix has a row for each account in the financial statements Each column of the matrix has two nonzero entries a 1 and a 1 to represent journal entries that affect two ac countsr5 We will use the convention that 1 is the debit part and 1 is the credit part The Amatrix for the example is given below Note the Amatrix is the same for both Set 1 and Set 2 since it depends only on the nature of the journal entries and is independent of the amounts 6 Compound journal entries can be incorporated in this framework by expressing them as a combination of simple journal entries Reproduced with permission of the copyright owner Further reproduction prohibited Without permission Arya Fellingham and Schroeder 399 Cash 1 1 71 1 o o o 1 Inventory 1 0 0 0 1 1 0 Net plant and administrative buildings 0 1 0 0 0 1 1 Sales 0 0 0 1 0 0 0 Cost of goods sold 0 0 0 0 1 0 0 General and administrative expenses 0 0 1 0 0 0 1 The rst column is the journal entry that debits inventory 1 and credits cash 1 the other columns are constructed similarly The dimensions of the Amatrix are m x 11 ms 11 where In is the number of accounts and n is the number of trans actions In the example m 6 and n 7 A matrix that has a 1 and a 1 in each column is referred to as an incidence matrix Because of doubleentry accounting is a natural example of such a matrix Denote by Ax the vector of changes in account balances adjusted for the debit credit convention The convention we use is to multiply accounts that have a credit balance by minus one and debit balances by plus one This is the same conven tion that is used when recording the Amatrix In addition since income state ment accounts are reset at the start of each period the beginning balance in these accounts is zero In the example A Cash 1 2 10 s l A Inventory 4 0 4 A Net plant and administrative buildings 6 7 0 6 and applying A Sales 10 0 10 A Cost of goods sold 5 0 S A General and administrative expenses l 3 L 0 L 3 the convention results in Ax 8 4 6 10 5 3 That is during the period cash declined net credit by 8 inventory increased net debit by 4 and so forth There are two things to note One the elements in Ax sum to zero This repre sents the basic accounting identity that assets equals liabilities plus owners eq uity Two Ax can be prepared by multiplying A the doubleentry matrix times the vector of transaction amounts Denote the vector of transaction amounts by y y yly7 The following illustrates matrix multiplication with the transac tions vector y from Set 2 oooopH oooio ecocoa I ooiOOH aoatco I oooiwo For an excellent discussion on properties of incidence matrices and more generally on linear algebra see Strong 1993 Reproduced with permission of the copyright owner Further reproduction prohibited Without permission 400 There are different ways to accom plish matrix multiplication One way is to multiply each row in A times the col umn vector of transaction amounts For example consider the rst row cash times the transaction vector y 718719 711110 0X5 01 02 8 This calculation adds up all the transactions affecting cash the sum is 8 indicating a decline in the cash bal ance of 8 beginning balance of 10 and ending balance of 2 The individual el ements added up correspond to the en tries in the cash Taccount with debits Issues in Accounting Education positives on the left and credits nega tives on the right A directed graph is a convenient visual representation of the Amatrix A directed graph is a network of edges connecting nodes The nodes corre spond to accounts and edges between two nodes correspond to transactions The graph is directed when we assign an arrow to each edge We adopt the convention that the arrow points to ward the account that is debited The directed graph for the example is shown in Figure 1 The directed graph representation is possible because the doubleentry matrix is an incidence matrix Loops FIGURE 1 The Directed Graph for the Example Inventory 4 Net Plant and Administrative Buildings 6 Cost of Goods Sold 5 Sales 10 General and Administrative Expenses 3 Reproduced with permission of the copyright owner Further reproduction prohibited Without permission Arya Fellingham and Schroeder are an important part of the directed graph Loops are easy to identify visu ally Start from a node travel along edges and try to return back to the starting node If you can do so you have identied a loop Traveling around a loop may involve reversing one or more arrows In our example there are two loops One loop starts from the cash node moves along edge 2 to Net Plant and Administrative Buildings then moves along edge 6 to Inventory and nally proceeds along edge 1 in the opposite direction of the arrow back to the starting node The sevenele ment vector representing this loop has 1 in the second position 1 in the sixth position 1 in the rst position and zero elsewhere 1 1 0 0 0 1 0 Similarly a second loop connecting Cash Net Plant and Administrative Buildings and General and Adminis trative Expenses can be expressed as 0 l 1 0 0 0 1 These loops are denoted L1 and L2 respectively The loops of the graph have a spe cial role The Amatrix times each loop is equal to zero ie ALl AL2 0 In linear algebra terms loops are referred to as nullspace vectors A nullspace vector is equivalent to a sequence of transactionsevents that leaves the account balances unchanged The graphical trick will always work the loops of the directed graph span the nullspace That is every vector y that satises Ay 0 can be written as a lin ear combination of the loops in our example Ll and L2 The number of linearly indepen dent loops is n m 1 the number of transactions less the number of accounts plus one The characterization of the number of vectors required to span the nullspace comes from an understanding of the fundamental subspaces of any matrix eg Strang 1998 155160 In 401 the accounting case the row space of A is spanned by m 1 vectors Know ing m 1 account balances is sufcient to infer the mth account balance be cause Assets equal Liabilities plus Owners Equity Since the vectors in the row space and the nullspace have n elements n Vectors are required to span both spaces This leaves 11 a m 1 n m 1 vectors to be supplied by the nullspace For our example n m 1 2 Knowing the dimension ofthe nullspace is useful since we then know when to stop looking for loops The other part of any solution is a particular y vector denoted yP where y is any solution to the matrix equa tion Ay Ax All consistent y vectors are of the form y yP lel k2L2 where k is any arbitrary constant Note Ay Ayl kALl WZ which by denition is Ax 0 0 Each consis tent y differs only in the choice of k The ks represent the degrees of free dom in the invertibility process We next explain how a reader can nd yP from a set of nancial statements Step 2 Finding a Particular Solution Although there are a number of ways to derive a particular solution using a spreadsheet like Excel is a con venient approach The following steps will work 1 Open an Excel worksheet 2 Write the m X n Amatrix For the example the matrix is entered in cells B22H7 see Figure 2 3 Write the choice variable y Any ar bitrary y vector of length n will do For our example enter a vector of all ls in cells J22J8 4 Write the constraint Ay Ax The LHS of the constraint Ay is calcu lated from the Amatrix and the y vec tor using the matrix multiplication in Reproduced with permission of the copyright owner Further reproduction prohibited Without permission 402 Issues in Accounting Education Excel To do this cells B13 through and constraints For our example these B18 are shaded and the following fbr are entered as shown in Figure 24 Note mulais pasted MMULTB2H7J2J8l that no objective function is specied Remember to hit ShiitControlRe since we are interested in nding a fea turn since it is a matrix operation sible solution not optimization Click The RHS of the constraint Ax is en the Options box and choose Assume tered in cells E13E18 Linear Model This assumption ap plies since the doubleentry process is The Excel optimization package lineart The output from running Solver Solver is accessed from the Tools menu is presented in Figure 3 The vector yP The Solver display has cells to accommo appears in cells J22J8 and is 8 9 1 date objective function choice variables 10 5 1 2 FIGURE 2 Spreadsheet to Compute a Particular Solution Using Solver Variable g I aoaLo D D I I I D I 0 I I I I Reproduced with permission of the copyright owner Further reproduction prohibited without permission Arya Fellingham and Schroeder 403 FIGURE 3 The Output Spreadsheet Variable 9 Cash 1 1 l 1 o 0 0J 3 Inventory 1 0 0 D l l CI 9 Net Plant 0 l 0 CI 0 1 1 1 Sales 0 o o l g o o 10 CBS 0 O D 0 I D 01 J 5 l3 A Exp 0 D 1 D D D 1 l i J 2 Constraint L l T LIIS Ag RHS Ax 8 8 i 4 J 4 s e i0 i0 i 5 5 i 3 3 The complete invertibility so ution is We now have all the consistent vec as follows tors an innite set However only one of them corresponds to the readers yl 8 1 0 best guess In the next section we iden y 9 1 1 tify this vector 2 ya 1 o 1 THE INFERENCE PROBLEM Suppose prior to observing the y4 10 k1 0 12 o nancial statements the reader believes y5 5 0 O the transactions are independent and y 1 1 o normally distributed The question is 6 how can the reader best use the infor y7 2 0 1 mation in the nancial statements to Verify two things One A times the rst vector is Ax and A times the sec ond vector as well as A times the third vector is 0 Two since any consistent y vector can be expressed as above Set 1 and Set 2 must correspond to differ ent choices of ks An astute student will note that this is indeed the case Set 1 corresponds to k1 1 and k2 2 while set 2 corresponds to k1 0 and 11207 revise hisher prior beliefs A plausible revision would use both the prior be liefs and the nancial statements and this is exactly what happens The best guess of the nullspace component of transactions comes directly from the pnors The best guess of the row space 7 In the example we entered a vector of all ls in cells 1243 and then ran Solver on get y if we had used a diiferent starting point Solver may come up with a different y Of course any 3 that Solver yields will satisfy the invertibility So 1mm for some choice of k and k1 i Reproduced with permission of the copyright owner Further reproduction prohibited Without permission 404 component of transactions comes di rectly from the nancial statements The intuition for this result is straightforward Since all information in the nullspace is lost during the ag gregation process of preparing finan cial statements the observability of Ax does not lead to any updating of the readers beliefs regarding the nullspace component However the observability of Ax does result in belief revision of the row component of the transaction vector This updating is crisp Every Ax vector is associated with a unique coun terpart in the row space of transac tions That is there is only one transaction vector residing entirely in the row space that is consistent with the nancial statements Continuing with the example as sume the readers priors are that trans actions are normally distributed with mean i 7 9 1 10 5 1 2 and identity variancecovariance matrix Notice that y is not consistent with the observed Ax The change in cash for example is 7 9 1 10 7 while the cash component ofo is 48 Formally the best guess posterior mean is made up of two additive parts a nullspace component and a row space component The nullspace component of the best guess is the same as the nullspace component of the prior mean vector The row space component of the best guess depends only on the nan cial statements8 A regression program will nd the nullspace component of y Find the combination of nullspace vectors that minimizes the distance between itself and y In other words get as close as possible to 37 without leaving the nullspace Regression problems can be thought of in terms of projections A projection matrix projects a vector the dependent variable in a regression Issues in Accounting Education into the columns of a matrix the inde pendent variables We wish to project y into a matrix whose columns are the nullspace vectors of A Denote this matrix by N For our example the two columns in N are L1 and L2 The pro jection matrix is P NNTN1NT which can be found in any discussion of regression9 P is an n x n matrix P can be calculated in an Excel spread sheet since Excel can accomplish the required transposition inverse and matrix multiplication operations For the example the nullspace component of y is FY 0125 325 3375 0 0 O125 3375 Now nd the row space component This component does not rely at all on the specication of the prior beliefs but only on the nancial statements themselves So for this part we start with any consistent solution say yl The unique row space component can be found using the same projection matrix P If we were interested in the nullspace component we would mul tiply by P however what we want is the component in the row space How ever if a vector is not in the nullspace it must be in the row space there is no other alternative So to nd the row space part simply nd yP PyP ie premultiply yP by I P where I is the n x n identity matrix 5 The choice of the identity variancecovariance matrix rules out the possibility oilearning about the nullspace component of the best guess mm the row space component 9 The projection matrix is also easy to derive The problem is in nd regression weights A which solve the equation NA y Unless y is already in the columns of N there is no solution To nd the best weights multiply both sides ofthe equation by w and take the inverse to obtain A NNWNTy The solution NA is NNNYNTy P Reproduced with permission of the copyright owner Further reproduction prohibited Without permission Arya Fellingham and Schroeder The row space component of the best guess is I PyP 75 6 45 105 15 1510 It is easy to verify that this com ponent has the following two properties One it lies entirely in the row space since it is orthogonal to the two nullspace vectors 1 1 0 0 0 l 0 and 0 1 1 0 0 0 1 Two it is a consis tent solution that is AI Pyp Ax Now that the two components have been solved for the best guess of trans actions is fully specied The best guess is I PyP Py For our ex ample the best guess is 7625 75 0125 925 6 325 1125 45 73375 10 10 0 5 5 0 1375 15 e0125 1875 15 3375 The transaction vector 7625 925 1125 10 5 1375 1875 is the best guess because it has the following properties 1 it is consistent with Ax and 2 from among all consistent vec tors it is the most likely ie it is the posterior mean COLDWATER CREEK In this section we conduct the invertibility and updating exercises on the scal year 1997 nancial state ments supplied in the Appendix of Coldwater Creek a publicly traded company11 We repeat the same steps as before Recall the three steps 1 Draw a directed graph to nd the loops that make up the nullspace N where the loops are in the columns of N 405 2 Use Excel to find a consistent vec tor of transactions yP 3 Compute the projection matrix P NNTN1NT from the nullspace identied in step 1 From prior be liefs y derive the best guess of transactions as I Py Py Step 1 Directed Graph The rst step isto construct a directed gral presented here as Figure 4 Most of the transactions are straightforward and follow directly from the nancial statements A couple deserve comment In the footnotes it is explained that Coldwater Creek temporarily capitalizes the costs of ac quirirlg catalogs in the account Pre paid Catalog Costs transaction 23 in the Appendix When the catalogs are shipped the account Deferred Cata log Costs is used transaction 24 in the Appendix Deferred Catalog Costs are amortized to the income statement transaction 12 in the Appendix Also a transaction connects interest on the income statement with Selling Gen eral and Administrative Expenses since Coldwater Creek offers loans to its executives at less than a market rate of interest transaction 26 in the Appendix The difference is debited to Selling General and Administrative Expenses The nullspace for Coldwater Creek can be identied from its directed graph in Figure 4 There are 19 accounts In and 26 transactions 11 Therefore m This solution uses the yv vector trim the previ ous section ie y s 9 1 10 5 1 2 Ofoourse we could have used any consistent vector as yv For example recall k 1 and k 2 yields an other canister solution 7 8 3 10 5 2 0 this is Set 1 Premultiplying this man by l s P yields the same row space component Every Ax corresponds to a unique consistent transaction vector in the low space n This example is based on a case written by Antle et al 1993 We thank the authors for permis sion to adapt their case Reproduced with permission of the copyright owner Further reproduction prohibited Without permission 406 Issues in Accounting Education FIGURE4 The Directed Graph for Coldwater Creek Inc Fiscal 1997 ruseavabicsA Sales 1617 046597 Accoums payable Y 92m Y1 memory yom ufsaies 27772 12026 Prtpaidlriglxngog cosrYLDcengsfalalog itllEh cm 7 3573 adminismuve 8764 Acliablllliis 7mg Pmpaidzxpenses 2273 Properly equlymenl 6581 mo emxpnynble h1451 m Defemd income yn Income in uxcnmn expense 1 343 m 7357 Dercmd income y m nannian NS Execunve 1m m 620 VI Inlzltsl 7m 50 Revolving 1m 57 ofcmdil 102547 there are n m 1 8 linearly inde loops are put in the rows ie this ma pendent nullspace vectors Examina trix is the transpose of N the nullspace tion of the directed graph yields the matrix fol owing nu lspace vectors where the f r 0100000010010000000000000 00 110000000011000000000000 00 000011000000000011000000 00 000000000000001100100000 01 000001010000000001000001 00 011000000001100000000000 00 000000000000011000010000 00 00000000011000000OOOILJOO Reproduced with permission of the copyrighi owner Further reproduction prohibited wikhout permission Arya Fellingham and Schroeder Step 2 Finding a Particular Solution The second step is to nd a particu lar solution yP that satises the con sistency requirement Ay Ax A convenient method is to write the A matrix and Ax in an Excel spreadsheet and then use Solver to find a solution One solution for y is reported in the third column in Table 1 Step 3 Estimating the Best Guess The third step can also be accom plished in an Excel spreadsheet Con struct the projection matrix P from the nullspace basis vectors N The pro 407 jection matrix is the same as in a regu lar regression problem The best guess is constructed from the prior beliefs and the particular solution W The ex pression for the best guess is I PyP P and is presented along with its components in Table 1 C ONCLUSION Linear representations have proved illuminating in analyses of many funda mental accounting problems The ac counting literature has a rich tradition of using a matrix a linear transformation to represent the doubleentry system TABLE 1 The Best Guess of Transactions Amounts for Goldwater Creek Inc Fiscal 1997 Transaction y y I Pb Pi 1 By P 1 250000 245020 24502000 0 24502000 2 150000 138684 10438585 60125 16451085 3 65000 544 3903111 28000 5703111 4 5000 109356 3618035 27375 880535 5 4000 6531 3533435 28375 995935 6 6000 7320 732000 0 732000 7 0 1563 1723717 166875 54967 8 1000 10254 1129517 156575 439233 9 250000 246697 24669700 0 24669700 10 35000 0 43429815 60125 2582685 11 125000 120126 12012600 0 12012600 12 60000 0 743053 50750 5818053 13 5000 0 3105658 22750 830658 14 5000 107053 3390735 27375 653235 15 4000 0 3175335 28375 337835 16 6000 0 452200 2250 677200 17 0 0 217500 1500 67500 15 0 7857 116000 750 41000 19 2000 57 1561717 166875 107033 20 1000 0 2155917 155875 587167 21 0 7712 101500 750 26500 22 0 6869 234700 2250 9700 23 60000 5092 12 52253 50750 6327253 24 60000 3673 1110353 50750 6155353 25 125000 147398 14789800 0 14789800 26 1000 0 3723335 32375 435835 Reproduced with permission of the copyright owner Further reproduction prohibited Without permission 408 See for example Butterworth 1972 Ijiri 1967 1971 and Mattessich 1964 The advantages of formally representing the accounting system are highlighted in this paper by ask ing a simple question to what extent can a reader of nancial statements determine the underlying transaction amounts The answer follows from a mere inspection of a directed graph in which nodes are denoted by ac count balances and edges are denoted by transaction amounts The directed graph representation is possible only because accounting is doubleentry Since there are potentially innite transactions that yield the same nan cial statements the next issue is one of choosing the most likely transaction vector the estimation problem This step depends not just on the observed nancial statements but also on the readers priorsl The best guess of the transactions is composed of two or thogonal components One component is uniquely determined by the nan cial statements the prior beliefs play no role here The second component comes from the readers priors alone the actual realization of nancial state ments plays no role here In addition the two components are easy to deter mine using the directed graph and a spreadsheet package Two assumptions of our approach deserve mention First we solve for the amounts of the transactions The choice of which transactions to model in the rst place ie the choice of the Issues in Accounting Education Amatrix comes from the readers un derstanding of the rms activities As is often the case the readers judgment complements his or her mechanical skills Second the best estimate assumes that transactions have identity vari ancecovariance structure We do not mean to imply that transactions for a rm eg Coldwater Creek have the same variance or are independent but rather we use this as a benchmark case The general covariance case is a straightforward extension as discussed in Arya et al 2000 A substantial portion of recent re Search in accounting theory has stud ied the role of accounting institutions and practices in alleviating problems associated with information asymme try in decentralized organizations However as Demski 1992 19 writes the research has often ignored charac teristics of information that make it uniquely accounting We have learned a great deal in the abstract about information We have learned to use the tools of economics of uncertainty in market and nonmarket settings The next step in our evolution must be a return to accounting We have to rediscover accounting structure and ac counting institutions Unless we moderns get on with the task we will be properly labeled the artisans of cacophony Our hope is that formally model ing doubleentry will enable current and future researchers teachers and students to incorporate its rich struc ture in decision and control problems Reproduced with permission of the copyright owner Further reproductlon prohibited Wlthout permlssion Arya Fellingham and Schroeder 409 APPENDIX Coldwater Creek located in Idaho operates a directmail catalog business Items marketed through these catalogs include womens and mens apparel jew elry and household items The beginning and ending balance sheets and income statement in thousands for scal year 1997 are as follows Balance Sheet 22898 31097 CURRENT ASSETS Cash 33 1 9095 Receivables 40 19 2342 Inventories 53051 25279 Prepaid expenses 2729 456 Prepaid catalog costs 2794 1375 TOTAL CURRENT ASSETS 62924 38547 Deferred catalog costs 7020 3347 Property and equipment 26661 20080 Executive loans 1620 TOTAL ASSETS 98226 61974 CURRENT LIABILITIES Revolving line of credit 10264 Accounts payable 27275 18061 Accrued liabilities 10517 5969 Income taxes payable 451 Deferred income taxes 919 76 TOTAL CURRENT LIABILITIES 48975 24557 Deferred income taxes 375 230 TOTAL LIABILITIES 49350 24787 STOCKHOLDERS EQUITY Preferred Stock Cornnmn Stock 101 101 Additional Paidin capital 38748 38748 Retained earnings 10026 1662 TOTAL STOCKHOLDERS EQUITY 48875 37187 TOTAL LIABILITIES AND STOCKHOLDERS EQUITY 98225 61974 Income Statement Net Sales 246697 Cost of sales 120126 GROSS PROFIT 126571 SGA 107083 INCOME FROM OPERATIONS 19488 Interest net and other 57 INCOME BEFORE TAX 19545 Provision for income taxes 7857 NET INCOME 1 1688 Reproduced with permission of the copyright owner Further reproduction prohibited without permission 410 Issues in Accounting Education The following 26 transactionsevents represent the economic activities of Coldwater Creek during scal year 1997 Collection of accounts receivable Payment of accounts payable Payment of accrued liabilities Cash payments for prepaid expenses Acquire property and equipment for cash Cash payment to reduce income taxes payable Cash loaned to executives Cash received on revolving line of credit Sales on account i Recognize SGA expenses credit accounts payable i Decrease inventory and recognize cost of goods sold i Amortize deferred catalog costs and recognize SGA Recognize SGA expenses credit accrued liabilities Amortize prepaid expenses to SGA i Amortize property and equipment to SGA i Recognize tax expense credit taxes payable i Recognize tax expense credit shortterm deferred taxes i Recognize tax expense credit longterm deferred taxes Recognize interest revenue on executive loans Recognize interest expense on the revolving line of credit Reclassify longterm deferred taxes as shortterm i Reclassify shortterm deferred taxes as taxes payable Apply accrued liabilities to prepaid catalog cost i Reclassify prepaid catalog cost as deferred catalog cost when catalogs mailed i Acquire merchandise inventory on account i Recognize SGA for the difference between the market interest rate and the amount charged to executives for executive loans PgPPPFN NNNNNNNHHHHHHHHHH wwwNHmeqamBWNHO Reproduced with permission of the copyright owner Further reproduotlon prohibited wwthout permlssion Arya Fellingham and Schroeder 411 REFERENCES Accounting Education Change Commission AECC 1990 Objectives of educa tion for accountants Position statement number one Issues in Account ing Education Fall 307312 1992 The rst course in accounting Position statement number two Issues in Accounting Education Fall 249251 American Institute of Certied Public Accountants AICPA 1998 CPA Vision Project httpwwwcpavisionorgproject American Accounting Association Committee on the Future Structure Con tent and Scope ofAccounting Education The Bedford Committee 1986 Future accounting education Preparing for the expanding profession Is sues in Accounting Education 68195 Antle R S Garstka and F Hein 1998 Articulation and analysis Coldwater Creek Inc Working paper Yale University Arya A J Fellingham J Glover D Schroeder and G Strang 2000 Inferring transactions from nancial statements Contemporary Accounting Research forthcoming Butterworth J E 1972 The accounting system as an information function Journal of Accounting Research Spring 127 Choi F S 1993 Accounting education for the 2lst century Meeting the chal lenges Issues in Accounting Education Fall 423430 Demski J 1992 Accounting theory Working paper Yale University 1994 Managerial Uses afAccounting Information Boston MA Kluwer Academic Publishers Ijiri Y 1967 The Foundations ofAccounting Measurement A Mathematical Economic and Behavioral Inquiry Englewood Cliffs NJ Prentice Hall 1971 Fundamental queries in aggregation theory Journal of the Ameri can Statistical Association 66 766782 1975 Theory of Accounting Measurement Sarasota FL AAA Mattessich R 1964 Accounting and Analytical Methods Homewood IL Rich ard D Irwin Rebele J E B A Apostolou F A Buckless J M Hassell L R Paquette and D E Stout 1998 Accounting education literature review 19911997 Part 1 Curriculum and instructional approaches Journal of Accounting Education 16 151 Strang G 19981ntraductinn to LinearAlgebra Wellesley MA WellesleyCam bridge Press Reproduced with permission of the copyright owner Further reproduction prohibited Without permission Reconciling Financial Information at Varied Levels of Aggregation ANIL ARYA The Ohio State University JOHN C FELLINGHAM The Ohio State University BRIAN MITTENDORF Yale School of Management DOUGLAS A SCHROEDER The Ohio State University Abstract Financial statements summarize a firms fiscal position using only a limited number of accounts Readers often interpret financial statements in conjunction with other information some of which may be aggregated in a different way or not at all This paper exploits properties of the doubleentry accounting system to provide a systematic approach to reconciling diverse financial data The key is the ability to represent the doubleentry system by network flows and thereby access wellrecognized network optimization techniques Two specific uses are investigated the reconciliation of audit evidence with managementprepared financial statements and the creation of transactionlevel financial ratios Keywords Aggregation Auditing Double entry Ratio analysis JEL Descriptors C61 M41 M42 Le rapprochement de linformation financire divers niveaux dagrgation Condens Les utilisateurs des tats financiers reoivent des informations dont le niveau dagrgation varie Les rapports comptables contiennent des donnes relatives aux flux priodiques dans ltat des rsultats et des donnes qui rsultent de cumuls dans le temps refltant la situation financire dans le bilan De plus linformation comptable que renferment les tats financiers est ellemme lexpression synthtique de nombreuses oprations sousjacentes Et le processus ne sarrte pas l puisque les ratios touchant les principales activits reprsentent encore un autre chelon dagrgation Accepted by Steve Huddart We thank Rick Antle Joel Demski Ron Dye Jon Glover Karl Hackenbrack Steve Huddart Yuji Ijiri Pierre Liang Paul Newman Eric Spires Shyam Sunder Rick Young David Ziebart and especially an anonymous referee and workshop participants at Carnegie Mellon Florida North Carolina Ohio State and Yale for helpful comments Anil Arya acknowledges financial assistance from the John J Gerlach Chair John Fellingham acknowledges financial assistance from the H P Wolfe Chair Contemporary Accounting Research Vol 21 No 2 Summer 2004 pp 30324 CAAA 304 Contemporary Accounting Research Lorsque le niveau dagrgation des informations prsentes varie les lments de donnes individuels importent non seulement pour les informations nouvelles quils fournissent directement mais aussi pour les cascades dinformations indirectes susceptibles den dcouler Grosso modo ces lments de donnes se comparent aux pices dun cassette le simple fait den placer une correctement peut rvler lemplacement de plusieurs autres de ces pices Par surcrot le fait de placer une pice correctement peut aussi nous apprendre que dautres pices ne sont pas leur place De faon analogue les auteurs cherchent dterminer si les informations dont le niveau dagrgation varie se raccordent correctement ce quils assimilent un exercice de rapprochement ou de contrle de cohrence Lorsque les informations sont cohrentes il faut ensuite se demander si leur combinaison pourrait engendrer des renseignements pertinents utiles Plus prcisment les auteurs modlisent lagrgation en fonction de la tenue des comptes en partie double Les motifs du choix de ce principe particulier dagrgation linaire sont vidents Si les activits des entreprises et les degrs de complexit de linformation prsente ont vari au fil des ans la mcanique sousjacente de la tenue des comptes en partie double a perdur Pour mieux apprcier les proprits distinctives du systme de tenue des comptes en partie double les auteurs se posent deux questions Premirement les informations contenues dans un ensemble dfini de donnes financires portant la fois sur les oprations et les soldes des comptes sontelles rciproquement cohrentes Deuximement dans laffirmative quels autres renseignements peuvent tre tirs de la combinaison des donnes relatives aux oprations et aux comptes La premire question revt la forme dun problme de vrification les lments probants recueillis dans le cadre de la vrification concordentils avec les tats financiers tablis par le client La deuxime question revt la forme dune analyse indiciaire compte tenu de ce que nous apprennent sur lentreprise ses tats financiers et les autres informations quelle produit que peuton affirmer au sujet de certains de ses ratios financiers La tenue des comptes en partie double joue un rle indniable dans ltude des auteurs Elle permet lapplication dune mthode danalyse apparente aux techniques bien connues doptimisation de rseau Les rseaux se caractrisent par des parcours orients reliant divers lieux nuds Les parcours sont sujets des contraintes de capacit les nuds des contraintes de demande et lon peut attribuer un cot prcis aux dplacements ou expditions le long de chaque parcours Loptimisation de rseau est utilise pour rsoudre une foule de problmes itinraires de livraison optimaux attribution efficiente des tches programmation de lutilisation des machines et programmes de prestation des services publics pour nen nommer que quelquesuns Le lien entre les rseaux et la tenue des comptes en partie double vient du parallle entre les comptes et les nuds et entre les oprations et les parcours reliant les nuds En dautres termes une opration consiste dbiter un compte ou nud et en crditer un autre Dans le cas prsent les contraintes de demande du rseau sont simplement les soldes leurs fluctuations figurant dans les tats financiers ou les dbits diminus des crdits ports aux comptes Lintgration dinformations lchelon de lopration est simple elle aussi Les donnes relatives aux oprations imposent des contraintes de capacit aux parcours qui y sont associs Une fois dtermin le circuit permettant loptimisation du rseau il est relativement facile de rpondre aux deux questions poses par les auteurs CAR Vol 21 No 2 Summer 2004 Reconciling Financial Information at Varied Levels of Aggregation 305 Dans le premier cas il sagit de dterminer si les tats financiers ou les contraintes de demande concident avec les lments probants ou les contraintes de capacit En dautres termes les auteurs se demandent sil existe un ensemble doprations qui satisfont les deux ensembles de contraintes Une variante de lalgorithme standard du flot maximum et de la coupe minimum permet de traiter cette question de manire efficace et mme dobtenir une rponse concluante Sil existe une solution ralisable au problme de vrification lalgorithme de flot maximum et de coupe minimum produit un vecteur doprations conforme tant aux lments probants quaux tats financiers Dans le cas contraire lalgorithme rvle un hyperplan sparateur qui tablit la distinction entre les tats financiers et lespace explor grce la pondration des colonnes de la matrice partie double les pondrations tant limites par les lments probants Dans le second cas les auteurs vont audel du contrle de cohrence et sintressent loptimisation En supposant que les informations dont on dispose relativement au solde de compte et lopration soient cohrentes que peuton dire au sujet des autres donnes financires lgard desquelles on ne dispose gure dinformation directe Les auteurs se penchent plus particulirement sur le calcul des limites relatives aux ratios financiers fonds sur les oprations Ils utilisent pour ce faire une proche variante dune autre technique de rseau bien connue lalgorithme du plus court chemin par ajustement successif Une faon systmatique daborder le problme des ratios financiers peut contribuer corriger les inexactitudes des ratios financiers traditionnels Ainsi le calcul des ratios de rotation des comptes clients et des comptes fournisseurs exige la connaissance des ventes et des achats crdit Ces donnes lies aux oprations sont habituellement obtenues par approximation les ventes totales sont substitues aux ventes crdit et les achats crdit sont remplacs par la variation des stocks majore du cot des marchandises vendues Plutt que dutiliser des approximations imparfaites il se peut que lon prfre calculer les limites possibles des ratios ce qui permettrait en outre damalgamer aisment au calcul dautres informations Les problmes prcis sur lesquels se penchent les auteurs mettent en relief un thme sousjacent la tenue des comptes en partie double suppose un rseau de comptes relis entre eux qui ncessitent une vision holistique de linformation financire Mme lorsque les oprations sont indpendantes le simple fait de dbiter et de crditer simultanment des comptes engendre une corrlation dans les soldes des comptes Pour reprendre lanalogie du cassette cest cette corrlation qui conduit la concordance des lments dinformation et lapprentissage indirect Heureusement les diagrammes de rseau et les techniques doptimisation de rseau facilitent la comprhension des liens que renferme le systme comptable et les progiciels commerciaux contenant des sousprogrammes doptimisation de rseau prtablis simplifient la rsolution des problmes comptables denvergure raisonnable 1 Introduction Financial statement users encounter information at varied levels of aggregation Accounting reports include periodic flow numbers in the income statement and timeaggregated stock financial status numbers in the balance sheet Further the account information in the statements is itself a summary of a large number of CAR Vol 21 No 2 Summer 2004 306 Contemporary Accounting Research underlying transactions Even this is not the end of the aggregation process Ratios representing key activities embed yet another layer of summary When information is presented at varied levels of aggregation individual pieces of data are important not just for new information they directly provide but also for indirect information cascades they potentially create A rough analogy can be made with a jigsaw puzzle where correctly placing a single piece can reveal how several other pieces fit Importantly correctly placing a piece can also reveal some incorrectly placed pieces In a similar fashion we look to see whether or not information at varied levels of aggregation fits together We view this as a reconciliation or a consistency check exercise If the data are consistent we then ask whether information at different aggregation levels can be combined in a meaningful useful manner More precisely we model aggregation via double entry The reasons for focusing on this particular linear aggregation rule are obvious1 While firm activities and reporting complexities have varied over the years the underlying doubleentry mechanics have endured In order to obtain a better appreciation for the unique properties of the doubleentry system we address two questions First is a given set of financial data concerning both transactions and account balances mutually consistent Second if so what other information can be gleaned by joint consideration of the transactions and accounts data The first question is couched in terms of an auditing problem is gathered audit evidence consistent with the financial statements presented by a client The second question is couched in terms of a ratio analysis exercise given what is known about a firm from its financial statements and other disclosures what can be said about some of its financial ratios The role of double entry in our inquiry is unmistakable Double entry allows an analogue to wellknown network optimization techniques Networks are characterized by directed paths connecting various locations nodes2 The paths are subject to capacity constraints the nodes are subject to demand constraints and travel or shipment along each path may be associated with a specific cost Such networks have been used to answer a myriad of questions about such issues as efficient shipping routes task assignments machine schedules and utility delivery plans The connection between networks and doubleentry bookkeeping comes from viewing accounts as nodes and transactions as paths connecting nodes That is a transaction consists of debiting one account or node and crediting another In this case the network demand constraints are simply the change in financial statement balances or account debits less credits Incorporating information at the transaction level is also straightforward Transactionsrelated data place capacity constraints on the associated path With the network optimization connection in hand answering the two questions we raised above turns out to be rather straightforward The first question asks whether the financial statements or demand constraints are consistent with the audit evidence or capacity constraints In other words does there exist a set of transactions that satisfies both sets of constraints A variant of the standard max flow min cut algorithm addresses the question efficiently Further the algorithm provides a conclusive answer If the audit problem CAR Vol 21 No 2 Summer 2004 Reconciling Financial Information at Varied Levels of Aggregation 307 has a feasible solution the max flowmin cut algorithm yields a transactions vector that is consistent with both the evidence and the financial statements Otherwise the algorithm reveals a separating hyperplane the hyperplane that separates the financial statements from the space spanned by weighting the columns of the doubleentry matrix where the weights are bounded by the audit evidence The second problem we address goes beyond a consistency question to one of optimization That is assuming that the account balance and transaction information at ones disposal are consistent what can be said about other financial data for which there is little or no direct information Specifically we look at the calculation of bounds on transactionsbased financial ratios In this case a slight variation of another wellknown network technique the successive shortestpath algorithm proves helpful3 A systematic approach to the financial ratio problem may help fix inaccuracies in traditional financial ratios For example credit sales and credit purchases are needed to compute the receivables turnover and payables turnover ratios These transaction amounts are typically approximated credit sales is replaced by total sales and credit purchases by change in inventory plus cost of goods sold Instead of crude approximations one may prefer to calculate feasible bounds on a ratio Such an approach also allows other information to be easily assimilated into the calculation The specific problems we address underscore an underlying theme double entry introduces an interconnected network of accounts which necessitates a holistic view of financial information Even when transactions are independent the mere act of simultaneously debiting and crediting an account introduces a correlation in account balances Ijiri 1975 Sunder 1997 Returning to our jigsaw analogy it is this correlation that leads to interlocking pieces of information and indirect learning Fortunately network diagrams and network optimization techniques simplify the process of understanding linkages in the accounting system And the availability of commercial software packages with preprogrammed network optimization routines facilitates solving realistically sized accounting problems 2 Representation of the doubleentry system Financial statements linearly aggregate information in several transactions using only a few account balances Let A denote this transformation matrix A has m rows and n columns where m is the number of accounts and n is the number of transactions m n There are two nonzero entries in each column corresponding to the accounts that are connected by a journal entry We adopt the following sign convention for the nonzero entries the account that is debited is assigned 1 and the account that is credited is assigned 1 For obvious reasons we refer to A as the doubleentry matrix The financial statement preparation process is represented mathematically by Ay x where y is an nlength vector of transaction amounts and x is the resulting mlength vector of changes in account balances4 The change in balance in account i is denoted xi i 1 2 m Also the transaction amount corresponding to the journal entry that credits account i and debits account j is denoted yij CAR Vol 21 No 2 Summer 2004 308 Contemporary Accounting Research A basic premise of the paper is that a reader of financial statements observes x but not y This is not to say that the reader may not have some other sources of information about y for example footnote disclosures acquired expertise about the firm or other gathered evidence While interpreting the financial statements in light of this other information the reader can take advantage of the fact that the aggregation process is double entry Double entry allows a directed graph representation of the accounting system in which nodes correspond to accounts and edges correspond to journal entries Arya Fellingham Glover Schroeder and Strang 2000 The graph is directed in that the arrow of each edge points to the account that is debited by that journal entry As we will shortly see it is this representation that opens the door to exploiting network flow algorithms Before turning to such an analysis we present parameters for an example that we will use throughout the paper to provide intuition Consider the following financial statements Balance sheet Cash Receivables Inventory Plant Total Assets Payables Owners equity Total liability and equity Ending balance Beginning balance 11 8 3 11 33 10 23 33 8 7 4 10 29 7 22 29 Income statement Sales CGS GA Income 7 3 3 1 Though the specific transaction amounts are unknown the reader of the financial statements knows about the firms main operations and hence the transactions it could undertake in the normal course of business These transactions and the matching journal entries are listed below The journal entries involve Cash Receivables Inventory Plant Payables Sales Cost of Goods Sold CGS and General and Administrative GA These accounts are labeled 18 respectively Activity Credit account Debit account Collection of receivables Cash purchase of plant Payment of payables Bad debt expense Credit sales Depreciation period cost Recognition of CGS Accrued expenses Purchase inventory on credit Depreciation product cost Receivables 2 Cash 1 Cash 1 Receivables 2 Sales 6 Plant 4 Inventory 3 Payables 5 Payables 5 Plant 4 Cash 1 Plant 4 Payables 5 GA 8 Receivables 2 GA 8 CGS 7 GA 8 Inventory 3 Inventory 3 CAR Vol 21 No 2 Summer 2004 Amount y21 y14 y15 y28 y62 y48 y37 y58 y53 y43 Reconciling Financial Information at Varied Levels of Aggregation 309 The A matrix representing the firms journal entries and the x vector representing the financial statement account balances are presented below x 3 1 1 1 3 7 3 3 A 1 1 1 0 0 0 Cash Receivables 1 0 0 1 1 0 Inventory 0 0 0 0 0 0 Plant 0 1 0 0 0 1 0 0 1 0 0 0 Payables Sales 0 0 0 0 1 0 CGS 0 0 0 0 0 0 0 0 0 1 0 1 GA 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 The equivalent directed graph for A in our example is presented in Figure 1 3 Consistency of information Reconciling transactionsbased and accountsbased information involves checking whether they are mutually consistent A setting in which this issue is at the forefront is auditing5 While an auditor does not express an opinion regarding the Figure 1 The directed graph representation of A for the example y37 3 7 y43 4 y53 y14 y15 5 1 y48 y58 y21 8 y28 6 y62 2 CAR Vol 21 No 2 Summer 2004 310 Contemporary Accounting Research underlying transactions a principal means of establishing the validity of a balance sheet and income statement is to trace the statement figures to the accounting records and back through the records to the original evidence of transactions Pany and Whittington 1994 40 The emphasis on gathering transactionsrelated evidence is most apparent in the cycles approach Arens and Loebbecke 20006 Auditing the sales and collection cycle for example entails checking credit sales sales returns cash collections and chargeoffs to confirm the balance in accounts receivable To elaborate an auditor gathers evidence regarding both account balances and transactions Accountsrelated evidence can of course be directly compared with the account balances presented by management In this section we focus on incorporating transactionsrelated evidence in the verification exercise Audit evidence limits permissible transaction amounts Let vectors yL and yU denote the lower and upper bounds imposed by audit evidence on the n transaction amounts These bounds may reflect the fact that the auditor employs interval estimates confidence intervals when evidence is gathered through sampling techniques Alternatively the bounds can reflect tolerable misstatements or represent bounds of reasonable industry benchmarks in the analytical review phase Precise evidence in the form of point estimates is a special case of this framework The auditors problem is to determine whether or not there exist transaction amounts that are consistent with management prepared financial statements Ay x and audit evidence yL y yU Denoting y yL by y this reconciliation problem is equivalent to finding a y that satisfies Ay x and 0 y yU yL where x x AyL Assume for simplicity that y is not bounded from above that is yU 7 Then the reconciliation problem reduces to finding a y that solves Ay x and y 0 Because any reconciliation exercise can always be rewritten in this form and upper bounds can be incorporated there is no loss of generality in studying the problem with yL 0 and yU as we do from here on Below we present an algorithm to solve the reconciliation problem At the heart of the algorithm is the max flow min cut method The max flow min cut method is an efficient graphical approach for solving network flow problems8 The method can be viewed as a simplification of the longstanding linear programming simplex technique where the simplification arises because network flow problems can be represented by an incidence matrix An incidence matrix is one in which each column contains a single 1 and a single 1 and all remaining elements are 0 eg Strang 1988 1027 The doubleentry matrix A is clearly an incidence matrix and hence permits the following algorithm to determine consistency see Ahuja Magnanti and Orlin 1993 166206 and in particular 169709 OBSERVATION 1 The following consistency algorithm an application of the max flow min cut algorithm reconciles transactionsbased evidence with account balance information by yielding either a consistent transactions vector or a separating hyperplane 1 Construct graph G with node set V vertices and arc set E edges as follows CAR Vol 21 No 2 Summer 2004 Reconciling Financial Information at Varied Levels of Aggregation 311 a Add two artificial nodes a source node s and a sink node t to the directed graph representation b Connect s to all accounts with a credit balance and connect all accounts with a debit balance to t 2 Select an initial flow in G say yij 0 for all i j E 3 Specify the capacity maximum permissible flow of each arc i j by uij uij is for arcs not involving s or t If yU placed a nontrivial upper bound on some transaction amount then the capacity of the corresponding arc would reflect the bound For arcs involving s or t the capacity is the absolute value of the associated nodes balance or usj xj and uit xi 4 Construct a new graph G as follows a G has the same nodes as G b If yij uij in G place arc i j in G The capacity of this arc in G is the remaining capacity uij yij c If yij 0 in G place the reverse arc j i in G This arcs capacity in G is yij 5 In G find a directed path P from s to t Determine the maximum amount that can flow along the path that is the minimum capacity along the path 6 In G add to yij if i j is in P subtract from yij if j i is in P and leave yij unchanged otherwise 7 Repeat steps 4 5 and 6 until there are no paths left in G flowing from the source node to the sink node 8 If all the arcs in G connected to the source and sink nodes are saturated that is ysj usj and yit uit the financial statements and audit evidence are consistent In this case the yij in G is a consistent transactions vector 9 If all arcs in G connected to the source and sink nodes are not saturated the financial statements and audit evidence are inconsistent In this case construct a vector with m positions one for each node in the audit problem network other than s and t that is one for each account For each node in G that can be reached from s assign a one to the corresponding position in Assign a zero to all other positions in The resulting mlength vector confirms inconsistency as it identifies a separating hyperplane To see the algorithm at work return to the example Figure 2 shows the initial G and G The graphs represent the original directed graph from Figure 1 with CAR Vol 21 No 2 Summer 2004 312 Contemporary Accounting Research added arcs and nodes labeled with dashes For simplicity the uncapacitated arcs uij are left unlabeled in G Note that G in Figure 2 has 10 possible paths from s to t Step 5 allows any of those paths to be chosen And as the algorithm unfolds some of those paths will be shut off as certain arcs become saturated Hence the algorithm may take fewer than 10 iterations and there may be multiple solutions to the algorithm at the end of step 7 Fortunately each possible solution will lead to the appropriate conclusion in steps 8 and 9 For this example one implementation of the algorithm is shown below as identified by path P and flow Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5 Iteration 6 P s 6 6 2 2 t 1 P s 6 6 2 2 1 1 t 3 P s 6 6 2 2 1 1 4 4 t 1 P s 6 6 2 2 1 1 4 4 3 3 7 7 t 2 P s 5 5 8 8 t 3 P s 3 3 7 7 t 1 At the end of these iterations G and G are as in Figure 3 In Figure 3 there are no paths in G following the direction of arrows that emanate from the source node and finish in the sink node Hence we are at a stopping point To see whether step 8 or 9 applies we check for saturation Since in G all arcs from the source and to the sink are saturated in G there are no paths departing s and no paths arriving at t the transaction evidence and account balance information are consistent A consistent transactions vector immediately follows from G y y21 y14 y15 y28 y62 y48 y37 y58 y53 y43 6 3 0 0 7 0 3 3 0 2 The reader can verify that Ay is indeed x in the example Now let us consider a slight change to the example that results in a conclusion that the transaction evidence is inconsistent with the account balance information Assume x 2 1 1 1 2 3 1 1 With this x the algorithm yields the following iterations Iteration 1 Iteration 2 Iteration 3 Iteration 4 P s 6 6 2 2 t 1 P s 6 6 2 2 1 1 t 2 P s 5 5 8 8 t 1 P s 3 3 7 7 t 1 At the end of these iterations G and G are as in Figure 4 From Figure 4 there are no paths left in G that start at the source node and finish at the sink node Hence we are at a stopping point But in G the arcs s 5 and 4 t are not saturated so there is an inconsistency The nodes that can be reached from s in G are 3 5 7 and 8 so 0 0 1 0 1 0 1 1 CAR Vol 21 No 2 Summer 2004 Reconciling Financial Information at Varied Levels of Aggregation 313 Figure 2 Initial G with flows and G with capacities for x 3 1 1 1 3 7 3 3 Initial G 0 3 7 0 4 0 0 0 0 s 0 0 5 0 0 1 t 0 0 0 0 0 0 8 0 6 0 2 Initial G 3 7 4 1 3 1 s 3 5 1 3 t 3 7 1 8 6 2 CAR Vol 21 No 2 Summer 2004 314 Contemporary Accounting Research Figure 3 Final G with flows and G with capacities for x 3 1 1 1 3 7 3 3 Final G 3 3 7 2 4 1 3 0 3 s 3 0 5 1 3 1 t 3 0 3 7 6 1 8 0 6 7 2 Final G 3 3 7 2 4 1 3 3 s 3 5 1 3 1 3 3 7 6 8 6 CAR Vol 21 No 2 Summer 2004 7 2 1 t Reconciling Financial Information at Varied Levels of Aggregation 315 Figure 4 Final G with flows and G with capacities for x 2 1 1 1 2 3 1 1 Final G 1 3 7 0 4 1 1 0 0 s 1 0 5 0 2 1 t 1 0 1 3 2 1 8 0 6 3 2 Final G 1 3 7 4 1 1 3 s 1 1 5 1 2 1 t 1 1 3 2 1 8 6 3 2 CAR Vol 21 No 2 Summer 2004 316 Contemporary Accounting Research This coupled with the theorem of the separating hyperplane confirms there is an inconsistency The theorem of the separating hyperplane see Strang 1988 41920 states that either there is a nonnegative solution to Ay x or there exists a such that Tx 0 and AT 0 For the example Tx 1 0 and AT 0 0 1 1 0 1 0 0 0 1 0 In geometric terms the vector in the theorem essentially identifies the hyperplane that separates x from the space spanned by placing nonnegative weights on the columns of A also known as the cone of A In particular the separating hyperplane is orthogonal to the vector Because our running example is in 8space involves 8 accounts visualizing the hyperplane and hence appreciating the theorem is not an easy task Fortunately the doubleentry system with its zero one characterization of the hyperplane lends itself to an intuitive schema divides the accounts in two account groups those labeled with a zero and those labeled with a one Figure 5 lists the groups along with the sum of balances in each group for the example The figure also lists all transactions that link an account in any group with an account in the other group In the example all such transactions point to the oneaccount group involve debits to the oneaccounts How then can the oneaccount group have a net credit balance It is this inconsistency that proves the financial statements must have been generated using another transaction in particular a transaction involving a credit to a oneaccount and a debit to a zeroaccount The two conditions in the separating hyperplane conduct precisely the exercise in Figure 5 AT 0 means the only permissible transactions connecting the two account groups involve credits to a zeroaccount and debits to a oneaccount Tx 0 indicates that the financial statements specify a net credit balance in the oneaccount group Besides providing the separating hyperplane that confirms an inconsistency the algorithm may also provide some evidence about the root of the problem For example suppose an auditor determines audit evidence and financial statements are inconsistent and suspects a firm may be incorrectly capitalizing an expense say research and development RD Before investigating such suspicions further the auditor may wish to revisit the consistency question that is add the extra transaction of incorrectly classifying RD debit the appropriate asset and credit the appropriate expense With this additional transaction in play does the inconsistency remain If there remains an inconsistency one can be sure that misclassification of RD is not the sole reason the financial statements and evidence cannot be reconciled If the additional transaction removes the inconsistency suspicions that the firm is misclassifying RD are bolstered 4 Drawing conclusions from consistent information Although the accounting process aggregates transactions by a few account balances this often is not the end of the summarization process For example users of financial statements commonly summarize balance sheet and income statement numbers by meaningfully constructed ratios that are calculated for a variety of purposes including analytical review The traditional ratios make use of account balance CAR Vol 21 No 2 Summer 2004 Reconciling Financial Information at Varied Levels of Aggregation 317 Figure 5 Representation of the separating hyperplane theorem for x 2 1 1 1 2 3 1 1 Zerogroup Account Onegroup Balance 1 2 2 1 4 1 6 3 sum 1 y15 y28 y48 y43 Account Balance 5 2 8 1 3 1 7 1 sum 1 information because that is the information directly available to users Even if a ratio depends on transaction information it is estimated by approximating the transaction amount using financial statement balances Turnover ratios are a case in point where for example credit sales in the receivables turnover ratio is approximated by total sales The previous section suggests an alternative the user can calculate the range of feasible values for the ratio rather than approximate the transaction component of the ratio using account balances The calculation of ratio bounds necessitates solving an optimization problem Specifically we minimize a weighted linear combination of transaction amounts while ensuring that transaction amounts are consistent with the financial statements and with any gathered evidence Finding the maximum of a weighted combination can also be couched as a minimization problem by reversing the sign on each weight Formally the problem is Min w Ty st Ay x y0 In the statement of the problem wij is the weight associated with transaction yij The weights are determined by the transaction or set of transactions for which a range of values is sought For example if the lower bound on yij ykm is sought then wij 1 wkm 1 and all other weights are zero The optimality question can be addressed with only slight modifications to the consistency algorithm The optimality algorithm follows two stages see eg Ahuja et al 1993 3204 Bazaraa Jarvis and Sherali 1990 61920 OBSERVATION 2 The following twostage optimization algorithm a variant of the successive shortestpath algorithm can be used to determine bounds on any linear combination of consistent transaction amounts CAR Vol 21 No 2 Summer 2004 318 Contemporary Accounting Research Stage 1 Checking boundedness From the directed graph for A identify any loop of transactions in which all arrows point in the same direction either all clockwise or all counterclockwise Suppose that there is such a loop Without loss of generality say the loop is ykl ylm and ymk If wkl wlm wmk 0 then the problem is unbounded that is the minimum is If there is no such negative unidirectional loop only then proceed to stage 2 Stage 2 Finding a minimum This stage simply conducts the consistency algorithm with two changes First in each iteration of step 4 label each arc in G with both its capacity and weight The weight on arc i j is wij the weight on arcs added to the source and sink nodes that is s j or i t is zero and the weight on a reversed arc is the negative of the weight on the original arc that is wji wij Second in each iteration of step 5 instead of choosing any path from s to t find a specific path the one where the sum of arc weights in G is the minimum the leastweight path Stage 1 makes use of the notion of a transactions loop to address boundedness The presence of such a loop implies that there is no effect on account balances when each transaction in the loop is changed by an equal amount Clearly if the sum of weights on the transactions in a loop is negative there is no limit to minimization On the other hand if there is no such negative loop the problem is bounded It is only in the latter case that we move to stage 2 to find the bounded minimum Stage 2 augments the consistency algorithm by identifying a leastweight path in each iteration Just as the max flowmin cut algorithm was central in determining consistency the successive shortestpath algorithm is key in determining optimality For a simple setup with only one wij 0 finding the leastweight path in each iteration is rather straightforward For complex problems however this task could be more onerous In such cases procedures such as Dijkstras shortestpath algorithm can ensure proper identification of the leastweight path for this and other shortestpath approaches see Ahuja et al 1993 chapters 45 As an application of Observation 2 consider the construction of an accounts payable turnover ratio inventory purchases on credit divided by average payables for the papers running example From the financial statements average payables equal 0510 7 85 Inventory purchases on credit is typically approximated by CGS plus change in inventory which equals 2 in our example This approximation yields a turnover ratio of 285 024 Alternatively the optimization algorithm can be used to determine the range of values for inventory purchases on credit y53 Finding the minimum value of y53 can be accomplished by conducting the optimization with w53 1 and wij 0 for the remaining transactions Applying the algorithm to the example with the goal of minimizing y53 yields precisely the iterative process listed following Figure 2 Hence Min y53 0 Repeating the process with w53 1 to determine the upper CAR Vol 21 No 2 Summer 2004 Reconciling Financial Information at Varied Levels of Aggregation 319 bound reveals Max y53 2 Hence the range for the accounts payable turnover ratio is 0 85 0 to 2 85 024 Note that the standard approximation corresponds to the maximum value the ratio can take10 The range representation alerts us to the fact that the approximation used in traditional ratio analysis may be crude In the case of the turnover ratio the standard approximation assumed zero product cost depreciation As a financial statement reader gathers more consistent evidence on transactions this point estimate for the ratio is unaffected because it relies only on account balances However the range calculation properly incorporates the new evidence Suppose that a careful reading of the financial statement footnotes reveals bad debt writeoffs y28 were 15 This information alters the bounds for y53 Min y53 15 and Max y53 2 The updated range for the accounts payable turnover ratio is 1585 018 to 285 024 Though bad debt writeoffs may seem to be unrelated to accounts payable turnover the joint consideration of financial statements and additional evidence highlights the interdependency inherent in the financial statements one that goes unnoticed using standard ratio analysis The interdependency is a natural consequence of double entry The exercise of debiting an account and crediting a different account simultaneously introduces a dependency in account balances even when transactions may themselves be independent Taking a network approach to ratio analysis also opens doors to the creation of other ratios based on transactions For example instead of using timesinterestearned as a measure of ability to pay debt obligations one may wish to calculate an interest coverage ratio of cash inflowsinterest expense here for all transactions involving a debits to cash wij 1 and for all others wij 0 Also to measure collection success one may wish to compute a collections flow ratio of receivable collections average receivables after all a favorable receivables turnover ratio may simply imply large bad debts For our example the firms accounts receivable turnover sales average receivables is 093 This is not particularly encouraging A further investigation by way of a collections flow ratio serves to heighten concerns of the firms ability to collect from customers The receivable collections y21 bounds are Min y21 4 and Max y21 6 yielding bounds on the collections flow ratio of 053 and 08 After including the footnote information on bad debts y28 15 the bounds are updated to Min y21 Max y21 45 yielding a collections flow number of 06 There are of course many other conceivable transactionsbased ratios that can be calculated As with standard ratio analysis the determination of which ratios are of use is surely a contextual exercise The point is not that a particular ratio is of importance but that systematically incorporating both accountsbased and transactionsbased information adds a new dimension to the usual financial statement analysis toolkit 5 Implementation issues Though the graphical approach presented in previous sections works it does raise concerns about applicability to realistically sized accounting systems that are presumably much larger As it turns out the strength of network flow algorithms is CAR Vol 21 No 2 Summer 2004 320 Contemporary Accounting Research their adaptability to large problems In fact it is precisely when there are a large number of nodes and arcs that the network flow algorithms really outshine standard linear programming approaches In discussing standard network problems in particular shortest path maximum flow and minimum cost flow Ahuja et al 1993 2 make this point eloquently In the sense of traditional applied and pure mathematics each of these problems is trivial to solve It is not very difficult but not at all obvious for the later two problems to see that we need only consider a finite number of alternatives So a traditional mathematician might say that the problems are well solved Simply enumerate the set of possible solutions and choose the one that is best Unfortunately this approach is far from pragmatic since the number of possible alternatives can be very large more than the number of atoms in the universe for many practical problems So instead we would like to devise algorithms that are in a sense good that is whose computational time is small or at least reasonable for problems met in practice As alluded to previously the reason efficient algorithms for network flows can be designed is because of the incidence property of the associated matrix Not only does an incidence matrix have entries restricted to the values 1 0 and 1 so do all its four fundamental subspaces each of which can be identified graphically see Observation 1 in Arya et al 2000 and Strang 1988 102 This property means any basis of the system of equations represented by an incidence matrix can be put in a triangular form In terms of linear programming this allows the problem to be addressed without the use of inverse operations instead solving equations by backward substitution one variable at a time eg Luenberger 1989 124 By exploiting such features and the simple structure of the problems network flow algorithms can improve upon the standard simplex approach As far as practical implementation goes many software packages have been developed to exploit the beneficial properties of network flow algorithms eg CPLEX GENOS LNOS NETFLOW Because our algorithms are an amalgam of standard network flow approaches these packages can readily be accessed either to run the consistency algorithm or to calculate bounds on ratios In fact the SAS software package with which many accountants are familiar contains network tools as part of the SAS OR package for example Proc Netdraw to draw directed graphs and Proc Netflow to implement network flow algorithms The appendix demonstrates the use of Proc Netflow in SAS OR for our accounting examples 6 Conclusion The doubleentry aggregation process that is ubiquitous in financial reporting creates an interconnected network of financial accounts The interconnectivity means some financial information that is seemingly unrelated to a financial statement readers primary focus could prove useful We underscore the importance of such interdependencies using two examples one from the field of auditing where interdependencies CAR Vol 21 No 2 Summer 2004 Reconciling Financial Information at Varied Levels of Aggregation 321 play an explicit role and one from the realm of financial statement analysis where consideration of interdependencies is less apparent The tasks are simplified thanks to close connections between accounting systems network flow diagrams and graphical optimization techniques Taking a network view of the accounting system can also shed light on other questions For example a financial statement user or auditor may wonder what information he should gather to have the greatest impact on his understanding of the firm The natural tendency to seek out information about a companys core operations or about the largest most material transactions may turn out to be incorrect Instead the user may get a greater understanding by uncovering information about a relatively small transaction Though the direct impact of such information is negligible it could have a substantial indirect impact by narrowing the range of possible values for several other transactions This indirect impact arises from the interconnectedness of the accounting system Appendix Below we present the SAS code and output that runs the consistency algorithm for our original example Program title Audit Problem title2 Path Consistency data acctg input ffrom tto capac cards Rec Cash 999999 Cash Plant 999999 Cash Payables 999999 Rec GA 999999 Sales Rec 999999 Plant GA 999999 Inv CGS 999999 Payables GA 999999 Payables Inv 999999 Plant Inv 999999 s Sales 7 s Inv 1 s Payables 3 Cash t 3 Rec t 1 Plant t 1 CGS t 3 GA t 3 CAR Vol 21 No 2 Summer 2004 322 Contemporary Accounting Research proc netflow maxflow sourcenodes Quotes for case sensitivity sinknodet arcdataacctg arcoutpath tail ffrom head tto proc print datapath var ffrom tto capac flow status run Output Audit Problem Path Consistency OBS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 FFROM TTO CAPAC Inv CGS 999999 Rec Cash 999999 Rec GA 999999 Plant GA 999999 Payables GA 999999 Payables Inv 999999 Plant Inv 999999 s Inv 1 Cash Payables 999999 s Payables 3 Cash Plant 999999 Sales Rec 999999 s Sales 7 Cash t 3 Rec t 1 Plant t 1 CGS t 3 GA t 3 FLOW 3 6 0 0 3 2 0 1 2 3 1 7 7 3 1 1 3 3 STATUS KEYARC BASIC KEYARC BASIC LOWERBD NONBASIC LOWERBD NONBASIC KEYARC BASIC KEYARC BASIC LOWERBD NONBASIC UPPERBD NONBASIC KEYARC BASIC UPPERBD NONBASIC KEYARC BASIC KEYARC BASIC UPPERBD NONBASIC UPPERBD NONBASIC UPPERBD NONBASIC UPPERBD NONBASIC UPPERBD NONBASIC KEYARC BASIC From log NOTE NOTE NOTE NOTE NOTE NOTE NOTE NOTE NOTE Number of nodes10 Number of arcs18 Number of iterations performed neglecting any constraints16 Of these 8 were degenerate Maximal flow11 Optimum neglecting any constraints found Minimal total cost0 The data set WORKPATH has 18 observations and 13 variables PROCEDURE NETFLOW elapsed time was 073 seconds with 1300mb available memory CAR Vol 21 No 2 Summer 2004 Reconciling Financial Information at Varied Levels of Aggregation 323 The SAS output confirms consistency because all arcs emanating from s and those linked to t are saturated Capacity equals Flow The same conclusion was reached when the graphical algorithm in Observation 1 was used to solve the example The log indicates that the maximal flow is 11 which is what we had obtained earlier the sum of s was 11 Further the flows presented in the above output provide a consistent transactions vector as specified by the flows unrelated to s and t in the SAS output report Also as mentioned previously there is not a unique consistent transactions vector so the graphical approach and the computer package may provide different solution vectors depending on the iterations conducted in the algorithm Such is the case in this example The optimization bounds calculated in Observation 2 can be accomplished in SAS using the same code as above with one change A cost variable cost is added to the input data with each cost entry corresponding to the appropriate wweight Endnotes 1 There is a long tradition in accounting of formally modeling the doubleentry accounting system eg Butterworth 1972 Ijiri 1971 Mattessich 1964 Williams and Griffin 1964 The emphasis on linear aggregation is a pervasive theme in Demski 1994 and Ijiri 1975 2 For excellent overviews of network flow problems see Ahuja Magnanti and Orlin 1993 Bazaraa Jarvis and Sherali 1990 and Luenberger 1989 11763 3 The connection to network optimization also means that the problems can be framed as transportation problems In this case the graphical approach is sidestepped in favor of a set of tableaus The tableaus require linking all accounts nodes not just accounts connected with sensible journal entries This makes the transportation approach more tedious than this papers graphical approach We thank the referee for pointing us to the appropriate graphical algorithms 4 x is also recorded using the same convention as in A Each entry in x equals the change in an account balance multiplied by 1 if the account is an asset or an expense accounts with debit balance and 1 otherwise The beginning balance of income statement accounts is zero 5 As a tantalizing observation we note that an audit industry has arisen around the production of doubleentry financial statements and there appears to be no comparable audit activity in the absence of double entry 6 Arens and Loebbecke 2000 list five basic transaction cycles sales and collection acquisition and payment payroll and personnel inventory and warehousing and capital acquisition and repayment 7 Incorporating a nontrivial upper bound in our setup is straightforward The details are spelled out in Observation 1 8 As an aside we note that standard presentations of network algorithms make a simplifying integrality assumption capacities costs and demands are integers Ahuja et al 1993 6 Integrality guarantees that solutions are integers and thus simplifies the search for a solution In our setup where account balances are rounded up to the nearest dollar or nearest thousand the integrality assumption is automatically CAR Vol 21 No 2 Summer 2004 324 Contemporary Accounting Research satisfied And financial data presented to the penny can simply be multiplied by 100 to allow for integral network flow data 9 More specifically the algorithm is an application of the augmenting path algorithm to a feasible flow problem Ahuja et al 1993 1801 which rests on the notion of a residual network Ahuja et al 1993 177 The algorithm is easier to follow in the context of our example the reader may wish to see Figures 2 and 3 alongside Observation 1 10 One may argue that a more appropriate definition of the turnover ratio in our setting would include all credit purchases not just inventory purchases in the numerator This corresponds to y53 y58 in the numerator Our approach applies to any linear combination and thus can also be used here The approach yields 3 and 5 as bounds on y53 y58 and hence 035 to 059 as the redefined turnover ratio bounds Note in this case the standard approximation is not even feasible when the consistency of the entire system is kept in mind References Ahuja R T Magnanti and J Orlin 1993 Network flows Theory algorithms and applications Upper Saddle River NJ PrenticeHall Arens A and J Loebbecke 2000 Auditing An integrated approach Upper Saddle River NJ PrenticeHall Arya A J Fellingham J Glover D Schroeder and G Strang 2000 Inferring transactions from financial statements Contemporary Accounting Research 17 3 36585 Bazaraa M J Jarvis and H Sherali 1990 Linear programming and network flows New York John Wiley Butterworth J 1972 The accounting system as an information function Journal of Accounting Research 10 1 127 Demski J 1994 Managerial uses of accounting information Boston Kluwer Academic Press Ijiri Y 1971 Fundamental queries in aggregation theory Journal of the American Statistical Association 66 336 76682 Ijiri Y 1975 Theory of accounting measurement Sarasota FL American Accounting Association Luenberger D 1989 Linear and nonlinear programming Reading MA AddisonWesley Mattessich R 1964 Accounting and analytical methods Homewood IL Richard D Irwin Pany K and O Whittington 1994 Auditing Burr Ridge IL Richard D Irwin Strang G 1988 Linear algebra and its applications Orlando FL Harcourt Brace Jovanovich Sunder S 1997 Theory of accounting and control Cincinnati OH International Thomson Publishing Williams T and C Griffin 1964 The mathematical dimension of accountancy Cincinnati OH SouthWestern Publishing Company CAR Vol 21 No 2 Summer 2004