3.2 Case I: Fraud is Detected in Period One
3.2.4 Client Opinion Shopping versus Incumbent Resignation
From Proposition 3.1 we know that auditor switches may be observed in equilibrium when beliefs regarding second period opportunity to commit fraud y2 are greater than y 2. We have also determined that the cause of an auditor switch can be due to a client-initiated change (opinion shopping) or an auditor-initiated change (resignation).
In this section, I analyze under what conditions an auditor switch after a qualified fraud report in period one is more likely to occur due to client opinion shopping as opposed to a low type incumbent auditor resignation. This distinction is of interest for a number o f reasons: It is often argued that one o f the reasons that triggers auditor switching is opinion shopping.49 Furthermore, a positive association between a client’s propensity to switch auditors and the receipt of a qualified report has been found empirically.50 No consensus exists, however, concerning the degree of association between switching and the receipt of a qualified report.51
49 See, for example. Financial Reporting Release No. 3 1, SEC [1988].
50 See, for example, Chow and Rice [1982], and Sarhan et al. [1991].
51 See, for example, Krishnan [1994].
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Formally stated we have that the probability of observing an auditor switch after a fraud report in period one conditional on y 2 > y ^ is given by:52
Pr(Su|z)F !, y2 > y 2) :
4>< ■7i + (3.12)
The first term o f the above expression, co-<b H
■7i , is the ex-ante probability that an auditor switch will occur after a fraud report due to client opinion shopping,
' ( l - a > )0 L
denoted P r(o s|z )F , , y 2 > 72)> and the second term , the ex-ante
probability that an auditor switch will occur after a fraud report due to a low type incumbent resignation, denoted Pr^/i?|z)F 1, y 2 > 7 2) •
The posterior probabilities, conditional on a switch having been observed after a fraud report, are given by:
- 7 2 Pr(0S|Sw, DF l) =
?t(IR\Sw, Df a ) =
(3.13) V Y2+(\-G>)'<S>L
52 N ote that both the fraudulent client and the incumbent auditor will know the cause o f the auditor switch with certainty. If the incumbent is a low type she resigns by submitting a fee larger than the bid made by the pool o f successors, fact that is observed privately by both the client and the incumbent. If the
incumbent is a high type, she is simply replaced by the client. These details, however, are not observed by the other players. Therefore, the probability o f observing an auditor switch as given is (3.12) is for an external observer such as the pool of potential successor auditors or the government.
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Given that all parties share common beliefs about y2, it is trivial to determine which o f the two possible causes of auditor switching, opinion shopping or low type incumbent resignation, is more likely to have taken place when a switch is observed.
We have that:
Pr(<9S|iSw, DF l) > Pr(/i?|iSW, DF , ) , whenever co • <l> H - y 2 > (1 - G)) - 4> L .
Note that the above inequality can go in either direction depending on specific parameter values.
No conclusion, however, can be drawn from above regarding the general propensity towards client opinion shopping when fraud has been detected in period one, or if in general, opinion shopping is more likely to occur than low type incumbent resignation.
A better picture of the overall second period engagement behavior, together with its causes, is needed in order to later analyze the benefits or costs of retention and rotation rules.
These rules need to be efficient over a range o f possible parameter values and cannot be judged based on a the results of a specific case given by a determined combination of parameter values. This issue is addressed next.
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We have that Pr(os|l>F,, y 2 > y 2) > Pr^Z/?j£ ) F1 , y 2 > Y2), whenever
( l - t y ) - 3 > , (1 - a ) ® ,
Yn --- . Note, however, that an interval Y < Y i < --- may exist
CQ-Q>h - 2 a
where the contrary is true.53
This seems to suggest that the cause of an observed auditor switch will always depend on the second period beliefs y2, as opposed to, for example, the characteristics
( l - d A O ,
of the auditors. Note, however, that for small values of go, --- will be greater
than one. Given that Y i *s restricted to the (0,1) interval, small values of co would (l-ty)-<I>,
imply that y , - --- would never hold. On the other hand, for large values of 0)<S>H
(1-©)•<&, ( l- ty ) < I > ,
(D,---will be close to zero, which would imply that r , > --- would co-<S>H
tend to hold. Thus, it appears that causes for auditor switching can be represented in terms o f auditor characteristics, given by and co, as opposed to market variables such as the second period opportunity to commit fraud.
Proposition 3.2, states that if auditor-client pre-alignment, represented in the current setup by go, is greater than a given threshold (which is determined by
$>H and O t ) then auditor switches after fraud has been detected and reported in period
53 The analysis is done using the ex-ante probabilities, but the same results can be derived using the conditional probabilities given in (3.13).
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one are more likely to occur due to client opinion shopping than due to low type incumbent auditor resignation, regardless of the second period beliefs y 2 > y concerning the opportunities to commit fraud.
Proposition 3.2: For any combination o f auditor fra u d detection technologies O H > L, there exists a W l e (0,1) such that fo r any co>Wr, we have that:
P r(os|l> F ,, y, > y 2) > P t(z r|/)f ,, y 2 > y z) fo r any y 2 > y z,
The result that opinion shopping is the most probable cause for auditor switches when fraud has been detected in period one and auditor-client pre-alignment is high is not surprising given that fraudulent clients will want to switch auditors precisely when the incumbent is a high type. What is interesting, however, is that it holds for any beliefs y 2 > y i .
At first, this result might appear to contradict with some empirical findings, such as Smith [1986], who found that only five cases o f auditor switches, in a sample o f 139 clients that changed auditors after a qualified report, suggested the possibility of opinion shopping. Note, however, that only “successful” opinion shopping, defined as the increased likelihood of receiving a clean opinion in the year after an auditor switch, can be observed empirically. There may be many unobserved instances in which clients have engaged in opinion shopping that reported no benefits.
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Proposition 3.2 states that client opinion shopping is the most probable cause for auditor switches when fraud has been detected and auditor-client pre-alignment is high, as is commonly assumed, but it does not state that fraudulent clients will be successful in engaging a successor auditor with a lower fraud detection technology. Note that auditor switches due to low type auditor resignations will also occur in equilibrium when conditions of Proposition 3.2 hold, but that they w ill be less likely than auditor switches due to client opinion shopping.
An interesting fact, that may seem like a contradiction, is that the probability that a fraudulent client engages a low type successor by switching decreases as auditor-client pre-alignment increases, yet the likelihood that a fraudulent client engage in opinion shopping increases.
Proposition 3.3 states that when auditor-client pre-alignment is lower than a given threshold, then, auditor switches after fraud has been detected and reported in period one are more likely to occur due to a low type incumbent resignation, regardless of the second period beliefs y2 > y 2 concerning the opportunities to commit fraud.
Proposition 3.3: For any combination o f auditor fra u d detection technologies H ><& L, there exists a Q)r e (0,l), with Q)1 < W l , such that for any 0 0<0) ! , we have that:
P r(o s|£ > F>1 , r 2 > r 2) < >7i > r 2) f o r any Yi > 72,
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Proposition 3.3 does not imply that client opinion shopping is precluded, but rather that the increased probability of low auditor-client matches implies a higher probability o f low type incumbent resignations due to expected second period unprofitable engagements.
Propositions 3.2 and 3.3 are not terribly insightful in isolation, but will result of interest when comparisons are made across cases (fraud detected in the first period and fraud not detected), and when the efficiency of rotation and retention rules are explored.
When the auditor-client pre-alignment is such that a)1 < Q ) < al , then the most probable cause for an observed auditor switch can go in either direction depending on the specific values of CD and as stated in Corollary 3.1. That is, there exists an interval of auditor-client pre-alignment values for which the cause of an observed auditor switch depends both on auditor characteristics and second period beliefs.
Corollary 3.1: When 6)' < Q) < W1, then there exists a y \ (to) e (jr2 ,l) such that:
P t ( o s |d f j, / 2>r2)<Pt(zr|£)f>i,r2>r2) V y 2 e ( y 2,y\(a>))
Pr(<3s|z)F I, y 2 > r 2) z Pr(//?|z)F I,yz >r2) v y 2 e[ y ‘2(eo),l)
Propositions 3.2 , 3.3, and Corollary 3.1, are represented schematically in Figure 3.2.
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0 — ---
0 ©■ H? 1
CO
Figure 3.2: Client Opinion Shopping versus Incumbent Resignation in Case I
Where: Area A is where Pr^/i?joF1 , y 2 > 7 2)> Pr^OsjDF1,y 2 > y ^ j ,
Area B is where Pr^(9.s|.DF ,, y 2 > y^) > Pr^Z/?|z>F1, y 2 > y^ j , Area C is where no auditor switches are observed because y 2 < y ^ , L I is the line that represents y (to), and L2 represents y\(eo) .
It is clear from Propositions 3.2 and 3.3 that auditor-client pre-alignment plays a key role in determining the most probable cause of auditor switches after fraud has been
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detected in the first period, given fixed fraud detection technologies. This result will be o f key importance in the interpretation of the effectiveness of forced retention or rotation rules: Offsetting effects between curbing client opinion shopping and allowing efficient auditor realignment due to low fraud detection technology exist. For example, if the intended effect of these rules is to curb opinion shopping, then forced retention will only be efficient in markets where high auditor-client pre-alignment exists, because when the pre-alignment is low, forced retention would hinder auditors who are trying to exit an expected unprofitable second period engagement. These issues will be formally addressed in Section 3.4.
As stated earlier, the value of y i e (0,1) is determined solely by auditor characteristics. Specifically, by the likelihood that an auditor be a high type 0), and the respective fraud detection technologies <£>w and d>t . Comparative statics on y i show that this threshold value will decrease both with increases in the likelihood to, and the high fraud detection technology $>w. That is, as the high fraud detection technology becomes more powerful and/or the likelihood o f a high auditor-client match increases, we will tend to observe more auditor switches after fraud has been detected in period one. Comparative statics with respect to the low fraud detection technology depend on the likelihood that an auditor be a high type and on the spread between the high and low
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fraud detection technology.54 However, as the spread between the high and low fraud detection technology is reduced, the threshold increases.55
Table 3.1 presents a summary of events and results of Case I: Fraud is detected in period 1.
0 Y i ^ r 2
Pool o f potential successor auditors bid: FS = K + L - q - ( l — <PL) - y 2 Low type incumbent bids Fl = Fs, and is retained
High type incumbent bids Fh = Fs, and is retained ii) Y2 > Y2:P °°l of potential successor auditors bid:
F ' = K + L q < 1 * , ) - r t ,
Low - Incumbent resigns by bidding FL > F$
- Successor engaged
High U ^ i n c u m M h t |^ ^ - Incumbent bids FH = F$
- Client retains incumbent
- Incumbent bids FH = F£
- Client switches to successor
Table 3.1: Summary of events and results of Case I
54 For co<[( 1-<&#)-( l-O J 2]/! ( l- 0 ô ) 2-( I-O t)2], we have that d ^ /d <t>L < 0. The cutoff point for go depends directly on the spread that exists between the fraud detection technologies (<bH - <t>L). The greater the spread, the higher the cutoff go (as defined in the above inequality) w ill be.
55 It is easy to see that the limit o f £ as approaches is equal to one.
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