3.3 Case II: No Fraud is Detected in Period One
3.4.3 Expected Second Period Fraud Detection Probability in the
In the first period, the expected probability of detecting fraud conditional on fraud existing is given by Pr(Z)F ,|/F ,) = <!> 5, under any regime, and the probability of detecting fraud, Pr(Z)F1) is given by:
Pr(£>F.i) = <I)s e- Yx + ( l -0>)-<&£- e - Y i (3-22)
To determine the expected second period fraud detection probability in the deregulated audit market, we must first determine what kind of auditor-client match we expect to observe in the second period. The second period engagement behavior will depend on a number of factors as determined in the previous sections.
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Summarizing the relevant events we have that the expected auditor type in the second period will depend on:
a) The expected outcome of the first period audit: DF l versus DMFX,
b) The beliefs all auditors share regarding opportunities to commit fraud in period two, y2 , and how these beliefs compare with the cutoff values y and y 2 in the case that fraud is detected or not in period one respectively,
c) The incumbent’s expected type, and d) The client’s expected second period type.
Points a) and b) lead to four possible scenarios that are presented below:
Feasible Second Period Scenarios
for DF l for DNF l
Scenario 1
Scenario 2 y2 ^ r 7 r 2 >7 2
Scenario 3 y2 >y7
Scenario 4 Yi > r 2 Yi > Y 2
At the beginning o f period one, all auditors share common beliefs about the values of the basic variables Q), <&H, , 6, yx, and y2. This implies that all auditors can compute y and y2 , and thus, determine which of these four scenarios will hold in the second period. They also know that their expected period one fraud detection
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technology is given by <I>S, and that the expected probability o f detecting fraud in period one is given by 9 - y x. These four scenarios are represented schematically in Figure 3.4. of the previous section.
Combining all the prior information auditors share at the beginning o f period one, the beliefs regarding points a) through c) above, and the second period engagement behavior determined in the preceding sections, we can construct the following table of expected second period events.
Table 3.2 summarizes the “branches” o f the decision tree that all auditors face at the beginning of period one before any auditor has been hired, together with the ensuing second period engagement opportunities.
The likelihood of any one of these events occurring will be the same for each of the four scenarios described above, but with different second period engagement opportunities depending on which scenario holds. The auditor-client match will either be high or low, with the corresponding probability of fraud detection; the auditor will either detect fraud or not depending on the client type that engages her and the outcome of the audit; and the second period client she faces will either be fraudulent or not.
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Triplet (Auditor- Client Match / First
Period Report / 2nd. Period Fraud)
Joint Probability o f triplet (Match/Report/2nd. Period Fraud)
Scenario
1 2 3 4
1 H ! ^ F . l / t FJ. t o ^ H - e r r Y z E E OS OS
2 F / D „ / t „ < y O V 0 - j v ( l - y 2 ) E E E E
3 H I D F l / t F'2 tu -(l E OS E OS
4 H I D NF l / t NF2 E E E E
5 L ! H Fl f t F2 ( l - t y ) - ( < t >£ - 0 ? V r 2) E E IR IR
6 L I H F l f t NF2 ( l- < y ) - [ < V 0 - ) v ( l- y 2)] E E IR IR 7 L / D NFl Jt f 2 (l-a))\(l-<&L - Y f ) ' 6 Y i ] E IR E IR
8 L ! DNF l f t NF 2 {i-co) [ \ - e - Y z- Q > L e-Yx ( i - r 2)] E ER E IR
Where: E = Incumbent auditor is retained and Engaged in second period IR = Low type Incumbent Resignation
OS = Client switches high type incumbent (Opinion Shopping)
Table 3.2: Joint probability of triplet Match / Report / 2nd Period Fraud
The second column of Table 3.2, (Auditor-Client Match / First Period Report / Second Period Fraud), presents the eight feasible situations that an auditor can find herself in at the end of period one, as viewed from the beginning o f the game. The third column, (Joint Probability of triplet: Match/Report/2nd. Period Fraud), is the joint probability that the auditor should find herself in that position at the end o f period one.
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The remaining columns present the second period engagement possibilities according to the scenario that holds. The incumbent auditor will either be retained, replaced or will resign.
As an example, let us walk through row 1 of Table 3.2. Here the auditor-client match has resulted in a high match, the auditor has detected fraud in period one, and the client will commit fraud in period two. 81 The probability of ending up in this position at the end of period one is given by the joint probability that the auditor be a high type, given by CD, that the client be a opportunist given by 0, that opportunities to commit fraud in period one exist given by yx, that the auditor detects fraud given by and that opportunities to commit fraud in the second period exist given by y 2. That is, the joint probability is given by: co <f>H -0 y x y 2. The second period engagement possibilities will depend on the scenario in place. In the case of scenario 1, for example, we have that y 2 < y^ and we know that fraud has been detected in period one.
Therefore, the result of part i) of Proposition 3.1 applies. That is, that a high type incumbent retains her client. In the case of scenario 3, for example, part ii) o f Proposition 3.1, which states that a high type incumbent will be replaced by a fraudulent second period client, applies. The remaining cases follow through in the same manner.
81 The incumbent auditor obviously does not know that the client has or will commit fraud in period two, and in that sense at the end o f period one the auditor cannot distinguish between the situation o f row 1 or row 2. At the beginning o f period one, however, the auditor can compute the joint probability o f being in any o f the eight situations depicted in Table 3.2.
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Given Table 3.2, it is now possible to derive the expected fraud detection probability in the four second period scenarios when the audit market is deregulated.
The following proposition states that the expected probability of detecting fraud in the second period differs across the four scenarios, and that it can be higher, equal, or lower than the first period expected fraud detection probability
Let Sc 1, through Sc4, denote the four possible second period scenarios in the deregulated audit market.
Proposition 3.11: The expected second period fra u d detection probabilities in the deregulated audit m arket w ill differ across the fo u r possible scenarios and can be ranked as follows:
Pr(lV 2 |rF 2 , 5c2) > Pv(pF21 t F:2, Scl) = Pr(Z)F21^.2, & 4 ) > Pr (DF2 \tF2, Sc3) where:
(3.23) Pr( ^ k F . 2 . ‘S'c2) = O s + m - ( l - m ) - r 1 •(<!>„ -<Dt )2
^{DF2\tFZ,Sc3) = ^ s -Q} {\-a)) r l (<P„ - O j 2 (3.25) (3.24)
(3.26)
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The interpretation of Proposition 3.11 is one of the key results of Chapter 3 as it ties together issues related with low type incumbent resignations, fraudulent client opinion shopping and the desirability of mandatory rotation or retention rules.
The main difference between scenarios 2 and 3, where the fraud detection probabilities are highest and lowest respectively, is that in the first case auditor switches only occur after a no-fraud report, while in the second, auditor switches only occur after a fraud report. Recall that auditor switches can be caused either by a low type incumbent resignation or by client opinion shopping, and therefore, Proposition 3.11 can be interpreted from two different points of view.
The result of Proposition 3.11 implies that low type incumbent resignations, on average, produce an efficient realignment of auditors -a s measured by an increase in the expected second period fraud detection probability- for parameter values such that resignations occur only after a clean no-fraud report (Scenario 2). Note that such behavior by the low type incumbents allows fraudulent second period clients to engage in opinion shopping if the incumbent auditor is a high type, but that on average, opinion shopping is not successful.
An interesting fact that seems quite counterintuitive is that scenario 2 is only feasible for auditor-client pre-alignments below a given value, that is for the lower tail o f values of 00. Thus, after the low type incumbent resigns, the probability that a client engage an auditor with a high fraud detection technology is not necessarily very high, yet despite this fact, the expected second period fraud detection probability increases.
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Formally stated, scenario 2 is only feasible for go’s such that y ^ (a ) < y 2 (a ) as can be seen in Figure 3.4.
When parameter values are such that low type incumbent resignations only occur after a qualified fraud report (Scenario 3), then low type incumbent resignations do not produce an efficient realignment o f auditors given that the expected second period fraud detection probability decreases. That is, the fact that low type incumbents are crowed out of the market by the pool of potential successors with superior fraud detection technology, is not always efficient.
Note that the increase (decrease) in the fraud detection probability given by a) ( l -0}) y l -(<0W - ® L) 2, is concave in © and that it obtains its maximum value for a> = \ . Therefore, for low and high values of © where scenarios 2 and 3 are most likely to hold, the stated difference in fraud detection probability may actually be quite small.
Note also that co• ( 1 - to)- y x • (O w - <&L)2 increases with the spread between the high and low auditor fraud detection technology. Therefore, in markets with homogeneous auditor technologies, fluctuations in the probability of fraud detection will be small regardless o f the amount or frequency of auditor switches.
In scenario 4 where low type incumbents always resign, the expected fraud detection technology is the same as in period one. Note from Figure 3.4, that when the opportunities to commit fraud in the second period are high, scenario 4 is the most probable of the four scenarios.
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Viewed from the perspective of client opinion shopping, Proposition 3.11 implies that opinion shopping, is on average successful only when it occurs exclusively after a qualified fraud report given that the expected fraud detection probability decreases. Thus, one would anticipate that room for improvement may exist with rotation or retention rules if they can actually increase the expected fraud detection probability. Recall, that when clients opinion shop after a clean no fraud report, then, opinion shopping is on average unsuccessful because the expected fraud detection technology increases, leaving the fraudulent clients in a worse position. In this case, rotation / retention rules may actually decrease the average fraud detection technology which would be inefficient.
It appears from the above discussion that rotation / retention rules can produce conflicting effects, especially if these rule have to be specified before the audit takes place and cannot be made contingent on the outcome o f the first period audit.