A Priori Bounds on Individual Transaction Amounts In this section we calculate upper and lower bounds on all individual transactions using only the financial statements. This will allow comparing additional evidence about one transaction at a time - when it is acquired - with the financial statements. If the additional evidence about a transaction indicates the transaction falls between the upper and lower bounds for the transaction calculated from the financial statements, we will say the evidence is consistent with the statements. As before, we first present aggregate account diagrams which offer the answer, and then connect to a theorem (in this case, duality) which allows a method of finding the appropriate aggregate account diagram.

Directed Graph Representation

(Figure 2 from the introductory section)

Consider the problem of finding the upper and lower bounds on y1 prior to acquiring any additional evidence. The directed graph of interest is from the original financial statements as presented above. The following circle picture is helpful.

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Figure 1

A Priori Bounds

An upper bound on y1 and y4

Figure 1 provides an upper bound for both y1 and y4, that is 6. If either transaction carries more than 6 to the "one" account, then the other must necessarily be negative. However, Figure 1 does not, by itself, resolve whether 6 is the least upper bound (called the sup) for either transaction. To do that, all possible aggregate account diagrams must be confronted to find the one indicating the sup. As it turns out, 6 is the sup for y1, but not for y4. An upper bound for y4 (and, indeed, the sup) is seen from Figure 2. When the accounts are aggregated as in Figure 72, it is seen that y4 can be no greater than 2.

Figure 2

An upper bound for y4

Lower bounds can also be determined from aggregate account diagrams. Figure 3 specifies a lower bound for y1.

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Figure 3

A Priori Bounds

A lower bound for y1

Since y1 is the only arrow going into the right hand aggregate account, it must carry at least 4. The aggregate account diagrams presented to this point are suggestive, but not conclusive. In order to verify they supply least upper bounds (sups) and greatest lower bounds (infs), we require a connection to linear programming, in particular to the theorem of duality. The problem can be written algebraically as a linear program; the following program will yield the least upper bound on yi. Maximize

yi

Subject to

Ay = x y≥0

Rewriting the objective function as vTy, where v is a vector of all zeros except for a one in position i, allows a convenient representation of the dual formulation. The dual formulation is much easier to solve; in fact, the solution can be read off the directed graph. Primal:

Maximize

vT y

Subject to

Ay = x

Dual: Minimize Subject to

λTx ATλ ≥ v

y≥0

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The duality theorem of linear programming: If either of the programs has a finite optimal solution, then so does the other. Furthermore, the values of the objective functions are equal. The duality theorem allows us to pick which of the two formulations we wish to solve. In this case, the dual is the easier one; its solution is attainable from the follow the arrows algorithm. The only difference between the dual and the problem solved in the previous section is that v is on the right hand side of the constraints rather than a vector of all zeros. But that constraint is automatically satisfied, since the transaction arrow under consideration must start outside the "one" account and go in. Therefore, the transaction position in ATλ is +1, as the debit part is included and the credit is not.

Directed Graph Representation

(Figure 2 from the introductory section)

To find the maximum amount of transaction y1, first attach a one to cash and a zero to receivables. Then follow the arrows out of cash, assigning ones to accrued liabilities, inventory, CGS, plant, and G & A. That determines a feasible aggregate account with balance 3 + (-3) + (1) + 3 + 1 + 3 = 6. It is also the optimal aggregate account, since there are no negative account balances which can be added Note at this point that this aggregate account is the one represented 4

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A Priori Bounds

in the aggregate account diagram of Figure 1, and we have verified the diagram supplies the least upper bound for y1. It is worth noting that when searching for extra accounts to add to decrease λTx, one need only look at accounts which are connected by some path of transaction arrows to accounts already in the "one" account. If it were feasible to add unconnected accounts with a net negative balance, that would be equivalent to a negative λTx with all arrows in; hence, no feasible solution exists to the constraint set. This connectedness feature is what enables employing the follow the arrows algorithm in relatively large directed graphs. The same technique can be used for large firms without a significant increase in difficulty. The maximum y2 is found in a similar fashion. For a feasible aggregate account include plant, inventory, CGS, and G&A. To further minimize the dual objective function, also include accrued liabilities. Maximum y2 = 1 + (-1) + 3 + 3 + (-3) = 3. Transaction minima are calculated in a parallel fashion. Primal:

Minimize

vT y

Subject to

Ay = x

Dual: Maximize Subject to

λTx ATλ ≤ v

y≥0 The key is to find a λ vector which specifies an aggregate account. Since λ = 0 is feasible in the dual, the minimum transaction value may be zero. If the following procedure yields a feasible solution with strictly positive objective function value, then the minimum transaction is greater than zero. Assign ones and zeros in the λ vector so as to maximize the balance in the one-aggregate account subject to 1.

transaction arrow i is going into the one-aggregate account.

2.

no other arrows are going into the one-aggregate account.

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A Priori Bounds

Since we wish to restrict the arrows going into the one-aggregate account, we are effectively following the arrows as before, but this time in the opposite direction. To find the minimum possible value for y1 assign a 1 to cash and a zero to receivables. The aggregate account consisting only of cash is feasible in the sense that no other arrows (besides y1) are going in. Adding plant to the aggregate account maintains feasibility and increases the balance, hence, the minimum value of y1 is 3 + 1 = 4. And we have reproduced the aggregate account diagram in Figure 3. When considering transaction y3, accrued liabilities is a feasible aggregate account, but the balance is negative. Hence the minimum y3 is zero. Another transaction which has a minimum of zero, but for a slightly different reason is y4. First a one is assigned to G&A. But then y8 is going into the aggregate account, so accrued liabilities must be included. By the same logic cash and receivables must also be included. But once receivables is added, y4 is no longer going into the aggregate account. Hence there is no feasible aggregate account and minimum y4 is zero. The maxima and minima for all the transactions, as well as the associated aggregate accounts, are tabulated below.

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Transaction

A Priori Bounds

Maximum

Minimum

y1

Cash + Pbles + Inv + G&A + CGS + Pl = 6

Cash + plant = 4

y2

Plant + inv + CGS + G&A + Pbles = 3

Plant = 1

y3

Pbles + inv + CGS + G&A = 2

0

**

y4

G&A + Pbles + inv + CGS = 2

0

*

y5

-sales = 7

-sales = 7

y6

G&A + Pbles + Inv + CGS = 2

0

y7

CGS = 3

CGS = 3

y8

G&A = 3

G&A + Plant + cash + AccRec + Sales = 1

y9

Inv + CGS = 2

- (G&A + Pbles) = 0

y10

Inv + CGS = 2

0

Table 1

**

** *

No feasible solution

**

No positive solution

Transaction maxima and minima

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