Crises and Rating Agencies

Transcription

1 Crses and Ratng Agences Petra Loerke Andras Nedermayer October 22, 2014 Abstract We consder a model wth a monopolstc proft maxmzng ratng agency, a contnuum of heterogeneous frms, and a compettve market of rsk-neutral buyers. Frms sell bonds, the value of a frm s bond s known to the frm and observable by the ratng agency, but not by buyers. Frms can choose to have the ratng agency rate the qualty of ther bonds. The ratng agency can reveal a sgnal of arbtrary precson about the qualty of the bond. We depart from the exstng lterature on ratng agences by addng aggregate uncertanty about the state of the world to the usual dosyncratc uncertanty. The well known result that havng one ratng class s optmal carres over from the setup wthout aggregate uncertanty. However, the optmal cutoff chosen by the ratng agency wll not be at the frst-best level any more: the ratng agency has more of an ncentve to be too lenent f the dstrbuton of aggregate uncertanty has a lower mean, a hgher varance, and s more left skewed. It has more of an ncentve to be too strct f the opposte holds. Keywords: Ratng agences, certfcaton, aggregate uncertanty JEL-Classfcaton: C72, D42, D82, G20 Economcs Department, Unversty of Mannhem, L7, 3-5, D Mannhem, Germany. Emal: ploerke@rumms.un-mannhem.de Economcs Department, Unversty of Mannhem, L7, 3-5, D Mannhem, Germany. Emal: anederm@rumms.un-mannhem.de.

2 1 Introducton Ratngs and other qualty certfcatons by thrd partes play an mportant role n today s economy. For nstance, the volume of rated debt ssues was over $8,000 bllon n Ratngs are used by nvestors to gude ther nvestment decsons. They are also crucal for fnancal regulaton: Basel III ncludes ratngs as one crteron for the calculaton of the captal adequacy requrements for banks. So does the Solvency II Drectve of the European Unon, passed on March 11, 2014, whch harmonzes nsurance regulaton n the European Unon and s scheduled to come nto effect on January 1, However, ratngs as a bass of regulaton have been vewed controversally, especally after the fnancal crss. The major concern s that the ratngs used for regulaton are gven by ratng agences, whch may have an ncentve to dstort ratngs n order to maxmze proft. As a reacton to ths concern, Secton 939A of the Dodd-Frank Act effectve snce 2010 requres that all federal agences must remove any reference to or requrement of relance on credt ratngs. The current artcle addresses the queston of ncentves to dstort ratngs by a proft maxmzng ratng agency under partcular consderaton of aggregate uncertanty. Aggregate uncertanty plays a major role n many markets. As an example, for subprme mortgages the queston was not only how good the subprme mortgages were that one partcular fnancal nsttuton nvested n. The queston was whether subprme mortgages as a whole were a suffcently safe nvestment. To nvestgate the effect of aggregate uncertanty on ncentves to dstort, we consder a model n whch all other possble ncentves to dstort are shut down. In partcular, we consder a monopolstc ratng agency that can credbly commt to a ratng strategy n a one perod model. Ths shuts down effects such as forum shoppng, renegng on the ratngs strategy, or reputatonal cycles. Besdes the ratng agency there s a contnuum of sellers sellng bonds. There s a contnuum of nvestors seekng to buy bonds. The mass of nvestors s larger than the mass of sellers, so that competton leads to prces beng bd up to the expected 2

3 value of a bond. The qualty of a seller s bond s perfectly known to the seller, but unknown to nvestors. The ratng agency has a technology to perfectly observe the seller s qualty. Sellers can decde whether they want to be rated. The aggregate dstrbuton of seller s types s ntally unknown to all market partcpants, except for a common pror about the dstrbuton of the aggregate states of the world. The states of the world dffer by a dfferent aggregate dstrbuton of sellers types. After sellers get rated, the aggregate state of the world s revealed to all market partcpants and nvestors buy the bonds. The prce depends on the expected qualty n a ratng class for the realzed aggregate state of the world. We show that n accordance to the exstng lterature, a proft maxmzng ratng agency wll choose a coarse bnary ratng: ether nvestment grade or junk bonds. However, n sharp contrast to the exstng lterature, aggregate uncertanty leads to the cutoff not beng at the frst-best level. Whether the ratng agency has an ncentve to be too lenent a negatve cutoff or too strct a postve cutoff s pnned down by three moments of the aggregate belef dstrbuton. The aggregate belef dstrbuton s defned as follows: Take for every state of the world the mean qualty of bonds that would be bought n frst-best. Market partcpants belef dstrbuton of these means s the aggregate belef dstrbuton. The ratng agency has more of an ncentve to be too lenent f the dstrbuton has a low mean, a hgh varance, and a low hgher order skewness defned as the sum of the thrd and hgher moments. A low hgher order skewness can be thought of as a left skewed dstrbuton,.e. wth a hgh probablty bonds have a mean qualty above average, but the dstrbuton has a fat tal at the bottom whch mples that wth a small probablty bonds have a very low mean qualty. The opposte result holds for a larger mean, lower varance, and a larger hgher order skewness. These results can be nterpreted as two opposte effects on the ratng agency s ncentve to dstort ratngs. One effect s pro-cyclcal: they have an ncentve to be too lenent before the outbreak of a crss nterpretng ths perod as a perod wth a large varance and left skewness of aggregate uncertanty and an ncentve to be too strct after the outbreak of the crss. The other effect s ant-cyclcal: a hgher mean n market 3

4 belefs about aggregate uncertanty lkely to occur before a crss gves the ratng agency an ncentve to be too strct and a lower mean after a crss to be too lenent. Whle anecdotal evdence suggests that the pro-cyclcal effect s stronger, 1 t s ultmately an emprcal queston, whch effect domnates. Ths sheds lght on a dsturbng aspect of usng credt ratngs for captal adequacy regulaton: they may ntroduce pro-cyclcalty nto the system. Captal adequacy requrements based on ratngs may be too lenent before and too strct after the crss. Our theory can be seen to justfy two possble polces to deal wth ths problem. One polcy, as n Secton 939A of the Frank-Dodd Act, s to remove any reference to or requrement of relance on credt ratngs from regulaton. Ths approach has the advantage of havng a clear unambguous rule. However, ths s also vewed controversally, snce t may be too costly for smaller banks to replace external credt ratngs wth nternal credt ratng systems. 2 An alternatve polcy would be to use credt ratngs, but take nto account ther cyclcalty n regulaton. In partcular, f one beleves that the pro-cyclcal element domnates, captal adequacy requrements based on ratngs should nclude ant-cyclcal elements to counterbalance pro-cyclcalty. We provde two extensons of our man result. Frst, we outlne an emprcal strategy to determne whether the pro-cyclcal or the counter-cyclcal effect domnates. Whle an emprcal analyss s beyond of the scope of ths paper, we show how the moments of the dstrbuton of aggregate uncertanty can be dentfed from the prces of fnancal dervatves. Second, we extend the model to a setup wth rsk averson. A model wth rsk averson explans why there are multple ratng categores and not just one.e. nvestment grade, and possbly a second, speculatve grade. The reason s that wth rsk averson, nvestors value more precse nformaton about the qualty of an asset to reduce rsk. We provde numercal examples to llustrate that a hybrd model of rsk averson and aggregate uncertanty preserves the key nsghts about 1 In hndsght, observers of fnancal markets consdered the ratngs of agences to have been too lenent before and too strct after the crss. 2 See, for example, 4

5 the ratng agency beng too lenent or too strct, but addtonally predcts multple ratng categores. Our paper relates to a large lterature on ratng agences, experts, and reputaton. We dffer from all papers mentoned below by havng market partcpants uncertanty about the aggregate dstrbuton of qualtes as the drvng force that determnes the ratng strategy. If one were to remove aggregate uncertanty from our model, t would reduce to the model n Lzzer 1999 s semnal contrbuton on certfcaton ntermedares. Lzzer 1999 shows the by now well known result that certfcaton ntermedares choose two categores correspondng to nvestment grade and junk bonds and set a cutoff at 0 whch s the frst-best level. Note that ths result can also be vewed as only one ratng category beng chosen nvestment grade and other assets not beng rated. Lzzer 1999 s work has been extended n a number of drectons, ncludng Doherty, Kartasheva, and Phllps 2012 s work on rsk-averse buyers. Wth rsk-averse buyers, t can be optmal to have more than two categores. Two papers allow for changes n the economc envronment n a dynamc model. In Bolton, Frexas, and Shapro 2012 the ratng agency trades off short term profts from consumers takng the ratng at face value and long term reputatonal concerns. They assume that n a boom the fracton of nave consumers s hgh and, together wth a low default rsk, ths gves the agency an ncentve to nflate ratngs durng booms. Bar-Isaac and Shapro 2013 nvestgate the qualty of ratngs when accuracy s costly for the agency. They combne reputatonal concerns wth the change of economc fundamentals whch affect, e.g., the costs for accuracy, possble profts and the default probablty. They fnd that the ratng qualty s lower n booms than n recessons. Our analyss s complementary to these artcles, snce we show that a ratng agency has an ncentve to dstort ratngs even f all nvestors are ratonal and t s costless for the ratng agency to assess the qualty of the ratng. Our results rely on the jont dstrbuton of aggregate and dosyncratc uncertanty. In a wder sense, our paper also relates to the lterature on experts and reputaton. Reputaton gves an ncentve to report truthfully. Strausz 2005 shows that 5

6 reputaton leads to monopolzaton and that honest certfcaton may requre a prce above that of a monopolst. Nevertheless, reputaton s often not enough to ensure accurate nformaton transmsson see Ottavan and Sørensen, 2006; Bouvard and Levy, 2009; Marano, Maths, McAndrews, and Rochet 2009 show that reputaton and confdence cycles may exst, because the certfer lkes to buld up reputaton so as to later nflate the grades and make larger profts. The paper s structured as follows. Secton 2 descrbes the model. Secton 3 shows that t s optmal to rate accordng to a smple cutoff rule and Secton 4 derves condtons under whch ths cutoff s postve or negatve. Secton 5 descrbes a stylzed emprcal dentfcaton strategy. Secton 6 shows that wth rsk-averse nvestors several ratng classes can be optmal but that the effects of aggregate uncertanty on the optmal cutoff reman. Secton 7 concludes. 2 Model There s one ratng agency, a contnuum of frms, and a contnuum of possble nvestors. Each frm sells a good of qualty t, where t s a random varable wth support [t, t] wth t < 0 < t. The frm has prvate nformaton about the qualty. Investors are rsk neutral and an nvestor s gross utlty from buyng the good s equal to the qualty t. There are N dfferent states of the world. The probablty of the world beng n state s ɛ. Havng a two dmensonal dstrbuton dfferent states of the world, dfferent dstrbutons of qualtes n each state of the world adds a consderable amount of complexty. To stll have a tractable model, we mpose a restrcton on ths two dmensonal dstrbuton. We assume that there s a mass κ of sellers whose qualty t s so low that one would never want to rate them we wll formalze ths later on. There s a mass µ of sellers whose qualty t s so hgh that one would always want to rate them. And then there s a mass λ of sellers wth ntermedate qualtes t t, t. We allow for arbtrary dstrbutons of κ, λ, µ wth the only restrctons that the sum κ + λ + µ s constant and Assumpton 1, but restrct the 6

7 dstrbuton condtonal on beng n t, t to be a dstrbuton F whch s the same for all states. We assume that F s contnuously dfferentable wth densty ft > 0 for all t n t, t. densty κ λ ft µ t 0 t t Fgure 1: κ and µ are the mass ponts at t and t n state. λ s the mass n state that s allotted to the types t t, t wth the dstrbuton F. Further, defne the expected masses on t, t as µ := ɛ µ and on t as λ := ɛ λ. Normalze λ to 1. The probabltes ɛ and the dstrbutons of qualty are known to all players. We assume that t s suffcently small: Assumpton 1. t < λ t tdf t + µ 0 t, = 1,..., N κ Assumpton 1 makes sure that we do not have to deal wth the unnterestng corner soluton n whch the ratng agency wants to rate all frms, ncludng t frms. A frm can choose to pay an upfront fee P to the ratng agency n order to get rated before the state of the world becomes known to market partcpants. agency rates frms that pad for a ratng accordng to a precommtted ratng strategy. 3 The tmng of moves s as follows: The The agency sets the ratng fee P and commts to a ratng strategy s, st = r, s : R R { }. Nature draws the state of the world and qualty t of each frm. 3 It does not matter n equlbrum whether the strategy s known at the begnnng or not. 7

8 The frms observe ther own qualtes, but not the state of the world, and decde whether to go to the agency to ask for a ratng or not. Ths decson depends on the own type t, the strategy of the agency s and the prce P. The agency observes the qualty of the frms askng for ratngs and gves ratngs accordng to ts strategy. The ratngs are publcly observable. However, nvestors do not observe whether a frm went to the ratng agency f the frm gets no ratng. Observng the state of the world, the buyers decde how much to bd n a second prce aucton for a good. Snce t s a second prce aucton, buyers bd ther own expected valuaton whch depends on ther belef about the expected qualty gven the nformaton s, P, r,. Assumng that there are more nvestors than frms, nvestors wll pay exactly the expected qualty n equlbrum. To solve the setup for equlbra we use Perfect Bayesan Equlbrum. We restrct the strategy of the frms to pure strateges and look at symmetrc equlbra. The profts of the agency n one state of the world s the ratng fee P tmes the mass of frms askng for a ratng. Ths mass depends on P and the ratng strategy s. The agency s rsk neutral and chooses s and P to maxmze expected profts before knowng the state of the world. The ratng agency s ratng strategy s parttons the set [t, t] nto M subsets, wth each subset m = 1,..., M beng the set of types T m = {t st = r m } wth M dstnct r m. 4 We wll call these subsets ratng classes n the followng. Snce n the end only the M dstngushable classes {T m } M m=1 matter and not the labels {r m } M m=1 attached to them, the followng analyss wll focus on {T m }. It s useful to defne the expected qualty n state condtonal on t beng above 4 Techncal speakng, there are M + 1 subsets because there can be types whch do not receve any ratng, st =. We wll show n the followng of ths paper that t cannot be optmal to have more than two ratng categores. Therefore, for the sake of notatonal smplcty, we do not consder an uncountable nfnty of ratng classes. To take nto account the possblty of an uncountable nfnty of ratng classes, e.g. full dsclosure, one could use the correspondence T r = {t st = r} wth r R { } nstead of the sets {T m } M m=1. 8

9 a threshold x > t as E x := λ t tdf + µ x t t λ df + µ. x A frm n t, t attaches probablty ˆɛ := ɛ λ / λ to beng n state. Consequently, from a t, t frm s perspectve, the expected qualty above a threshold x over all states s Ẽx := ˆɛ E x. In the followng, we wll assume that the vrtual valuaton functon attached to Ẽx s monotone n x for x t, t. Assumpton 2. Ẽx Ẽ 1 F x+ µ x s monotone n x for x t, t. fx Ths assumpton bascally ensures that the second-order condton s fulflled whenever the frst-order condton s fulflled and t excludes the corner soluton that t s optmal to only rate t. 3 Optmalty of Threshold Ratng Strategy In the followng, we wll show that t s optmal to rate all frms n an nterval [x, t] n one ratng class and not to gve a ratng to all frms wth t < x. Formally, st = 1 for all t x and st = for all t < x. 5 We wll show ths n four steps. Frst, we show that t cannot be optmal to exclude type t. Second, we show that the prce of a ratng s determned by frms wth t < t. Thrd, gven that t s ncluded, t s optmal to have only one ratng class rather than multple classes. Fourth, gven that there s only one ratng class, the set of types belongng to ths class has to be convex. Lemma 1. It cannot be optmal that t M m=1t m. It cannot be optmal that t M m=1 Tm. Part of the Lemma holds by Assumpton 1. The ntuton for part of the Lemma s that t should be ncluded n the ratng because t ncreases the mass of 5 Ths s equvalent to st = 1 for all t x and st = 0 for all t < x because frms wth t < x are not rated n equlbrum. 9

10 rated frms as well as, due to ts hgh type, other frms wllngness to pay for a ratng. Next, we state a lemma whch wll be useful throughout our analyss. The lemma states that f both frms wth t t, t and wth t = t are n the same ratng class, then frms wth t t, t have a lower wllngness to pay for a ratng then frms wth t = t. Lemma 2. Take an arbtrary ratng class T that ncludes frms wth t t, t and t = t. The wllngness to pay for a ratng s hgher for t than for t t, t. The reason s that frms update ˆɛ dfferently and we show that frms wth a type t assgn a hgher probablty to states wth hgher expected qualty than frms wth t t, t. Lemma 2 can be used to prove the next lemma, whch states that f there are multple ratng classes and the hghest type t s ncluded, then t s better to merge all ratng classes to one sngle class. Lemma 3. M = 1 wth T 1 = M m=1 Tm s better than t M m=1 Tm. { Tm } M m=1 wth M > 1 f Consderng types that the agency ntends to attract, the ratng fee s always determned by the type wth the lowest wllngness to pay for a ratng. Mergng the ratng class wth a lowest wllngness to pay wth classes wth a hgher wllngness to pay, the expected qualty and thus, also the mnmum wllngness to pay ncrease. The next lemma states that all frms above a threshold are rated whch means that no types n between are excluded. Lemma 4. If M = 1 and t T 1, then T 1 has to be convex. If the set s not convex, there s at least one unrated hole n the mddle and the agency can rate frms n the hole nstead of ratng some other types below wth the same mass. Ths ncreases the expected type n every state and, therefore, also the ratng fee the agency can charge from the frms. Lemmas 1, 3, and 4 together lead to the followng proposton. Proposton 1. It s optmal to choose M = 1 wth T 1 = [x, t] for some x. 10

11 Proposton 1 shows that the best equlbrum for the ratng agency s such that the agency offers the followng ratngs strategy: 1 f t x, st = otherwse, that s, all frms above some cutoff x get a postve ratng. Subsequently, all frms wth t [x, t] get rated and nvestors pay the expected qualty over [x, t]. As usual n such models, there s a multplcty of equlbra n the subgame followng the ratngs agency s announcement of ts prce P and ratng strategy s. For example, there s the trval equlbrum n whch no frm apples for a ratng and nvestors have the off-equlbrum belef that frms that do get a ratng are of the worst possble rated qualty x. Snce x s less than the prce of a ratng P, t s a best response for frms to stay unrated. The usual arguments for selectng the equlbrum we descrbed apply: The ratng agency has a frst-mover advantage, hence, t s reasonable that the equlbrum most favorable to the ratng agency wll be selected. Further, by a small perturbaton of ts strategy, the ratng agency can get rd of undesred equlbra. For example, f no frm gets a ratng, the agency mght ncentvze the frst frms who apply for a ratng n order to jump-start the market. 6 4 Optmal Threshold By Proposton 1 we can restrct our attenton to threshold rules whch consst of all types above a cutoff x beng pooled n one class and all types below not beng rated. If there were only one state of the world, the optmal threshold would be x = 0. To see ths, take a model wth only one state of the world, e.g. by settng µ = µ and 6 A smple, albet extreme example s the followng: As long as not all frms wth a qualty t [x, t] enter, frms get ther ratng fees refunded and get an addtonal small compensaton. Ths makes sure that any equlbrum n whch not all frms n [x, t] get rated breaks down, so that the refund never has to be pad n equlbrum. 11

12 λ = λ = 1 for all. Then the agency s proft s Π = 1 F x + µ t x tdf t + µt 1 F x + µ = t x tdf t + µt. whch s equal to welfare. The frst dervatve s Π = xfx, whch s equal 0 f x x = 0. Therefore, the optmal threshold for the agency s x = 0. 7 Ths specal case of our model corresponds to Lzzer 1999 s results. If there are N states of the world, the ratng agency s proft s N N Πx := λ 1 F x + µ ɛ E j xˆɛ j =1 =1 F x + µẽx j=1 where Ẽx s the expected value of a ratng from a frm s perspectve whch assgns the probabltes ˆɛ to dfferent states. The welfare wth N states of the world s N t W x := λ tdf t + µ t = =1 x N E xλ 1 F x + µ ɛ. =1 Defne the expected type on [x, t as E 0 x := Rearrange the expresson for the welfare to t x tdf t 1 F x. ɛ W x = λ 1 F xe 0 x + µ tɛ =1 F x + µêx wth Êx := 1 F xe 0x + µt, 1 F x + µ 7 It s easy to check that the second-order condton s also satsfed at x = 0. 12

13 whch can also be wrtten as Êx = ɛ λ 1 F x + µ E x. 1 F x + µ Êx s the expected value of a ratng from a welfare perspectve whch takes nto account that the quantty of frms beng rated λ 1 F x + µ s dfferent n every state. In the followng, we wll drop the argument x n E x, E 0 x, Ẽx, Êx when t s unambguous n order to smplfy notaton. the followng way. Ê and Ẽ compare n Lemma 5. The value of a ratng s larger from a welfare then from a frm s perspectve; Ê Ẽ for all x. Ths mples that W x Πx. For non-degenerate dstrbutons of the state of the world, the nequalty s strct and the ratng agency cannot extract the whole surplus, W x > Πx. 8 The dervatve of the proft wth respect to the cutoff s Π x = 1 F x + µ = fx Ẽx }{{} margnal effect ˆɛ E x fxẽx 1 F x + µ Ẽ. fx x }{{} nframargnal effect 1 and we wll show later that the frst order condton s suffcent for proft maxmzaton. Thus, the proft maxmzng cutoff s gven by Π x = 0. Changng the cutoff has two opposte effects on the agency s proft; ncreasng the cutoff decreases the mass of frms askng to be rated margnal effect but t also ncreases the expected qualty of frms beng rated and by ths t ncreases a frm s wllngness to pay for beng rated nframargnal effect. We call the expresson n the squared brackets n 1 the vrtual valuaton functon 8 Even f ˆɛ = ɛ for all, the nequalty s strct for non-degenerated dstrbutons. Besdes by the updatng of ˆɛ, the dfference between Ê and Ẽ s caused by the dfferent mass of frms beng rated n dfferent states of the world. 13

14 for Ẽ.9 By Assumpton 2 t s monotone and, thus, the frst order condton s suffcent to fnd an optmum. 10 Ths also mples that there s a unque soluton of the frst order condton. We are nterested n comparng the proft maxmzng cutoff wth the welfare maxmzng cutoff. Thus, we also have to determne the socally optmal cutoff. The dervatve of welfare wth respect to the threshold s W x = fx or wrtten n a smpler way Êx }{{} margnal effect 1 F x + µ Ê fx x. 2 }{{} nframargnal effect W x = ˆɛ λ xfx = xfx whch s the same as for one state of the world. The dervatve s 0 f x = 0 and thus, the welfare maxmzng cutoff s at 0. To derve the dfference between the proft of an agency and the welfare, we wrte the proft as Πx = λ 1 F x + µ ɛ E j xˆɛ j j = E j xˆɛ j λ 1 F x + µ ɛ j = E j x1 F x + µ j ˆɛ j E j xˆɛ j µ j j j µ ɛ =W x + Lx where Lx := j E jˆɛ j µ j E[µ] s the non-extractable part of the surplus loss compared to extractng total surplus. Remember that W 0 = 0 whch mples that 9 We can rewrte the vrtual valuaton n terms of E as ɛ λ E E 1 F + µ f. 10 The second order condton follows drectly from Assumpton 2. That we do not have a corner soluton at x = t can be seen by observng that the proft Πx s contnuous at x = t and lm x t Π x < 0. Assumpton 1 mples that there s no corner soluton at x = t see proof of Lemma 1. 14

15 L 0 = Π 0. Thus, the ncentve for the agency to dstort the ratng compared to the welfare maxmzng ratng s gven by the sgn of L 0. The optmal cutoff s postve f L 0 > 0 and t s negatve f L 0 < 0. Proposton 2. The dervatve L 0 s gven by L 0 = f0ẽ t Ê Ê Ẽ ˆɛ E 2 ˆɛ E 2. 3 Ẽ Snce the expresson before the parenthess s always postve, the sgn of L 0 and, therefore, the sgn of the proft maxmzng cutoff depends on the sgn of Ê Ẽ ˆɛ E 2 ˆɛ E 2. Ê Ẽ s postve and can be nterpreted as the dffer- Ẽ ence of the expected value of a ratng from a socal and a frm s perspectve. An ntuton for ˆɛ E 2 ˆɛ E 2 Ẽ s that t s the varance dvded by the mean of the posteror dstrbuton of E and t reflects the uncertanty about the state of the world: f ths uncertanty s suffcently large, the cutoff s negatve. The reason for ths s that frms care less about the effect of the cutoff x on the expected qualty of a rated frm f the expected qualty s to a large extent drven by uncertanty about the state of the world. Thus, the sgn of L 0 s determned by the dfference of the expected value of a ratng and the rato of varance to mean of the posteror dstrbuton of E. Whle the above expresson for L 0 provdes some nsghts on the determnants of the optmal cutoff, t s dffcult to use t for compartve statcs, snce a change of the mean and varance of E wll also change Ê. Therefore, n the followng, we wll express L 0 n terms of the moments of the posteror dstrbuton of E. From µ = λ 1 F x E E 0 t E, we can wrte µ = ɛ λ 1 F x E E 0 t E = 1 F x = 1 F x E t + t E 0 ˆɛ t E 1 ˆɛ t E t E 0 15

16 Pluggng 4 nto the defnton of Ê we get Ê = 1 F xe 0 + µt 1 F x + µ E t E 0 ˆɛ 1 t E t = t E 0 ˆɛ 1 =t 1 ˆɛ 1. t E t E Defne the scaled value of a ratng as e := E. Then t ˆɛ kth dervatve of 1 t1 e wth respect to e s k e k [ ] 1 1 = k! t1 e t1 e. k+1 1 t E = ˆɛ 1. The t e t Usng these dervatves one can construct a Taylor seres of 1 t1 e wth respect to e around e = 0. Ths yelds 1 t1 e = e k k! = k=0 k=0 e k t. k e k [ ] 1 t1 e e =0 Takng expectatons over the state of the world yelds where m k := ˆɛ e k mples that we can wrte [ ] 1 ˆɛ = m 1 + m 2 + t1 e t m k, s the kth moment of the posteror dstrbuton of e. Ths Ê = t Defne m 3+ := k=3 m k. Observe that 3 smplfes to L 0 = fẽ t Ê k=3 t 1 + m 1 + m 2 + k=3 m. 5 k Ê ˆɛ E 2. 6 Ẽ 16

17 Pluggng 5, Ẽ = m 1 t and ˆɛ E 2 = t 2 m 2 nto 6 yelds L 0 = fm 1 t t 1+m 1 +m 2 +m 3+ [ =fm 1 t m k 1 k=0 t t t2 m m 1 + m 2 + m 3+ m 1 t 1 m ] m 1 + m 2 + m 3+ m }{{ 1 } =:S The sgn of L 0 s gven by the sgn of S. Note that S only depends on the moments of e, more precsely, t depends only on the mean m 1, the second moment m 2 and the sum of all hgher moments m 3+. For example, let us start wth L 0 < 0. If we keep mean and second moment constant and ncrease the sum of hgher moments, S ncreases and L 0 can swtch sgn from negatve to postve. Ths means that we change the optmal cutoff from a negatve to a postve one by changng the hgher moments of the dstrbuton of e. We can calculate the threshold m 3+ for whch L 0 s 0. Set whch leads to m 1 + m 2 + m 3+ m 2 m 1 = 0 m 3+ = m2 2 + m 2 m 2 1 m 1 m 2. Observe that m 3+ s always postve because m 1 > m 2 and m 2 m 2 1 s the varance of e. Ths mples that for m 3+ < m 3+ the expresson S s negatve and thus L 0 = Π 0 < 0. Proposton 3. The optmal cutoff for the ratng agency s negatve f m 3+ < m 3+ and postve f m 3+ > m 3+. We also derve thresholds for m 1 and m 2. Frst, observe that S s ncreasng n m 1 and decreasng n m 2 gven that m 1, m 2, m 3+ > 0. Second, by settng the expresson n square brackets to zero and solvng for m 1 and m 2, respectvely, one gets thresholds for m 1 and m 2 that determne whether the cutoff of the ratng agency s postve or negatve. The thresholds are stated n the followng two Propostons. Proposton 4. The optmal cutoff for the ratng agency s negatve f m 1 < m 1 17

18 and postve f m 1 > m 1, where m 1 := 1 2 m m 2 + 2m 2 + m 3+ 2 Proposton 5. The optmal cutoff for the ratng agency s negatve f m 2 > m 2 and postve f m 2 < m 2, where m 2 := m m m 1 + m m Both thresholds, m 1 and m 2, are postve gven that m 1, m 2, m 3+ > 0. Propostons 3, 4, and 5 have a strkng mplcaton: the ratng agency has more of an ncentve to be too lenent f the dstrbuton of aggregate uncertanty s more left skewed n the sense of a smaller hgher order skewness or low m 3+, the mean s smaller, or the varance s larger. Left skewness and a hgh varance can be reasonably consdered as beng assocated wth a perod precedng the begnnng of a crss. For moments that can be reasonably assocated wth a perod shortly after a crss rght skewness, low varance, ncentve of the ratng agency move n the opposte drecton: the ratng agency has an ncreasng ncentve to be too strct. Ths gves the ratng agency an ncentve to rate pro-cyclcally: excessvely lenent ratngs expand nvestments durng booms, excessvely restrctve ratngs restrct nvestments durng recessons. Observe that the mean of aggregate uncertanty has a counter cyclcal effect; a small expected average, whch can be assocated wth a perod shortly after a crss, gves the ratng agency an ncentve to be too lenent. The opposte holds for a hgh expected average. 4.1 Example of Beta Dstrbutons It s llustratve to parametrze the posteror dstrbuton of E as a Beta dstrbuton wth support [E 0, t],.e. E havng a densty hy y α 1 1 y β 1, where y = E E 0 /t E 0. The dstrbuton of E /t s determned by the three parameters 18

19 α, β, and e 0 := E 0 /t. The upper bound of the support of E /t s 1. These three parameters determne m 1, m 2, and m 3+ the followng way: m 1 = α + βe 0 α + β, m 2 = 1 e 0 2 αβ α + β α + β + m2 1 m 3+ = α + β 1 1 e 0 β 1 1 m 1 m 2 It can be shown that ths s a one-to-one mappng from α, β, e 0 to m 1, m 2, m One can use ths one-to-one mappng for comparatve statcs wth respect to say m 3+ whle keepng m 1 and m 2 constant. Fgure 2 shows a Beta dstrbuton wth α = 3, β = 5 and e 0 = 0.1 dashed lne. For ths dstrbuton, L 0 = 0,.e. the ratng agences sets the cutoff at exactly the socally optmal level x = 0. For the dotted lne, m 1 and m 2 are kept constant and m 3+ s reduced by The dotted lne has a fatter lower tal whch means that t has a hgher mass at the bottom of the dstrbuton. The mean and varance reman the same, but f a crss hts, t s more lkely to be severe. For the dotted dstrbuton L 0 < 0 and hence the cutoff s negatve, x < 0, whch means that the ratng crtera are too loose compared to the socally optmal ones. For the sold lne, m 3+ s ncreased by 0.01 whle keepng m 1 and m 2 constant. For ths dstrbuton L 0 > 0 and hence x > 0, that s, the ratng 11 The mappng n the opposte drecton can be derved n closed form, but the resultng expressons are rather long and unnformatve and therefore omtted. m 1 and m 2 are the wellknown frst two moments of the Beta dstrbuton. m 3+ can be derved by observng that E[1 y 1 1 e 0 1 ] = E[ k=0 e e 0 y k ] = E[ k=0 ek ] = 1 + m 1 + m 2 + m 3+, where e = E /t = e e 0 y. For a Beta dstrbuton wth densty hy = y α 1 1 y β 1 /Bα, β the expected value s [ ] 1 1 E = 1 y 0 y α 1 1 y β 2 dy = Bα, β Bα, β 1 Bα, β = α + β 1, β 1 where the last equalty follows from the relaton of the Beta to the Gamma functon Bα, β = ΓαΓβ Γα + β, and the property Γx + 1 = xγx of the Gamma functon whch mply Bα, β = Puttng ths together yelds the expresson for m 3+. ΓαΓβ 1β 1 Γα + β 1α + β 1 = β 1 Bα, β 1. α + β 1 19

20 s too strct compared to the socally optmal one. Fgures 3, 4, and 5 llustrate the change of L 0 as m 3+, m 1, and m 2 are changed, respectvely, whle keepng the other parameters constant. The optmal cutoff for example can swtch from a negatve to a postve cutoff f the mean or the hgher order skewness ncrease or f the varance decreases. For all values of m 1, m 2, and m 3+, the parameters α, β, and e 0 are n permssble ranges Fgure 2: Densty of E /t for α = 3, β = 5, e 0 = 0.1 dashed lne. For the dotted lne, m 3+ s reduced by 0.01, for the sold lne, m 3+ s ncreased by 0.01, whle m 1 and m 2 are kept constant. The correspondng parameters are α = , β = , e 0 = for the dotted and α = 2.23, β = , e 0 = for the sold dstrbuton Fgure 3: Values of L 0 as m 3+ s changed and m 1 and m 2 are kept constant. Startng pont s α = 3, β = 5, e 0 = 0.1 whch corresponds to m 1 = , m 2 = , and m 3+ = for whch L 0 = 0. Further parameters are normalzed to t = 1 and f0 = The permssble ranges are α > 0, β > 0, and e 0 0, 1. 20

21 Fgure 4: Values of L 0 as m 1 s changed and m 2 and m 3+ are kept constant. Startng pont s α = 3, β = 5, e 0 = 0.1 whch corresponds to m 1 = , m 2 = , and m 3+ = for whch L 0 = 0. Further parameters are normalzed to t = 1 and f0 = Fgure 5: Values of L 0 as m 2 s changed and m 1 and m 3+ are kept constant. Startng pont s α = 3, β = 5, e 0 = 0.1 whch corresponds to m 1 = , m 2 = , and m 3+ = for whch L 0 = 0. Further parameters are normalzed to t = 1 and f0 = 1. 21

22 5 Emprcal Implcatons Our model shows how the ratng agency s ncentve to be too lenent or too strct depends on the moments of aggregate uncertanty. Snce these moments cannot be observed drectly, one may wonder about the emprcal content of our model. Frst, t should be noted that an emprcal estmate of the dstrbuton of aggregate uncertanty s non-trval, especally f the man concern s about the dstrbuton of aggregate uncertanty shortly before a crss. The reason s that only few crses occur, so t s dffcult to have larger amounts of data. However, an emprcal estmate of market partcpants belefs about the dstrbuton of aggregate uncertanty can be obtaned. We llustrate the basc dea of how to estmate these moments n a strongly stylzed setup contanng the core dea of the emprcal strategy. Consder the followng stylzed setup. There s an ndex for the bonds beng sold by the frms n the market. Further, there s a market for fnancal dervatves based on ths ndex. As an example, one can thnk of an ndex on securtzed assets backed by subprme mortgages. Call optons on the ndex can be bought n the frst perod of the model, before aggregate uncertanty s realzed. The optons expre n the second perod after aggregate uncertanty has realzed. Tme s dscrete and the optons are European optons. 13 Further, aggregate uncertanty s such that the md-qualty frms belefs are the same as the general market belefs, formally, ˆɛ = ɛ for all. 14 Suppose that the cut-off of the agency s close to 0 x 0, so that the value of the ndex E x s well approxmated by E 0. Further, assume that there exst a call opton wth strke prce y = E for each state of the world. Wthout loss of generalty, order the states of the world ncreasngly,.e. E j > E f j >. The second-perod value of a call opton wth strke prce y j n state s E y j f E > y j and 0 f E y j. Denote the frst-perod 13 In a dscrete two-perod model, t does not matter whether the opton s European or Amercan. In a contnuous tme model, calculatons for Amercan optons are somewhat more complex, but standard and well known n the lterature. 14 A suffcent condton s that λ = λ for all, that s, aggregate uncertanty enters through changes of κ and µ for dfferent states of the world. 22

23 prce of opton j wth strke prce y j as O j. O j s gven by the market s expected value of the second perod value gnorng dscountng: O j = N N ɛ max{e y j, 0} = ɛ E E j 8 =1 =j+1 where the second equalty follows from y j = E j. For = N, O j = 0. The next proposton shows that gven a set of call optons, the nformaton on ther strke prces y j and frst-perod prces O j dentfes the market s belefs about the dstrbuton of aggregate uncertanty; t dentfes the probablty ɛ for the expected qualty E. Proposton 6. Gven strke prces and frst perod prces {y j, O j } N j=1, the probablty mass functon of the dstrbuton of aggregate uncertanty s gven by ɛ j = O j O j+1 y j+1 y j O j 1 O j y j y j 1 for 1 < j < N and O N 1 ɛ N =, y N y N 1 N ɛ 1 = 1 ɛ. A smlar result can be obtaned for a contnuous dstrbuton of E. For the contnuous dstrbuton verson, drop the ndex n E and denote the dstrbuton of E as G. Assume that prces Oy for call optons wth a contnuum of strke prces y [t, t] are observed. Then Oy s gven by Oy = The frst dervatve s O y = t =2 max{e y, 0}dGE = t t y t y E ydge. 1dGc [y ygy] = 1 Gy, 23

24 and the second O y = gy. Ths s analogous to the dscrete dstrbuton result and the dstrbuton G s nonparametrcally dentfable gven data on call opton prces. In practce, one expects to observe less optons than there are states of the world, so parametrc assumptons are requred to be able to estmate the dstrbuton of E. In the followng, we make the parametrc assumpton that the dstrbuton G s a polynomal wth lower bound of support E 0 and upper bound t. As an example, consder a cubc functon GE = a 1 + 2a 2 E + 3a 3 E 2 + 4a 4 E 3. The prce of a call opton wll also be a polynomal functon of the strke prce y, snce Oy = where t y 1 GEdE = a 0 + a 1 y + a 2 y 2 + a 3 y 3 + a 4 y 4, 4 a 0 = a t. Suppose we observe data for fve call optons wth strke prces {y j } 5 j=1 =1 and opton prces {Oy j } 5 j=1. In ths case, the parameters {a } 4 =0 are gven by the lnear equaton system Oy j = a 0 + a 1 y j + a 2 y 2 j + a 3 y 3 j + a 4 y 4 j, j = 1,..., 5. 9 As long as the matrx [ y j ] j=1,...,5;=0,...,4 s non-sngular, the equaton system 9 yelds a unque soluton for the varables {a } 4 =0. Note that E 0 and t are unquely pnned down by the parameters {a } 4 =0 and by the equatons GE 0 = 0 and Gt = Gven the dstrbuton G of E, we can obtan the dstrbuton of e = E/t and 15 Whle the GE has multple roots due to G beng a polynomal, the soluton of GE 0 = 0 s unque nonetheless. Ths s because of the constrants G E > 0 for E [E 0, t] and y j [E 0, t] for all j. By the same reasonng, there s a unque soluton of Gt = 1. 24

25 the moments m 1, m 2, m 3+ of e. Ths n turn yelds S = m 1 + m 2 + m 3+ m 2 m 1 from expresson 7 and determnes the sgn of the margnal proft Π 0 at x = 0. Table 1 provdes examples of observed prces of call optons and correspondng estmated parameters, moments, and S. For the frst set of observatons frst lne, the ratng agency has an ncentve to choose the cutoff at the frst best level x = 0. For the second lne and thrd lne, the agency has an ncentve to choose a negatve and a postve cutoff, respectvely. observed prces estmated parameters moments S O 1 O 2 O 3 O 4 O 5 a 1 a 2 a 3 a 4 m 1 m 2 m Table 1: Example of parameter estmates for data on call opton prces O j = Oy j for strke prces y 1, y 2, y 3, y 4, y 5 = 80, 90, 100, 110, 120. We have llustrated the basc dea behnd an emprcal strategy to estmate the moments of aggregate uncertanty. To practcally apply ths strategy, several addtonal steps are requred, whch are beyond the scope of ths artcle. Frst, one needs to construct synthetc call optons for the ndex of the bonds beng rated. Second, the prcng of optons n a mult-perod envronment s more complcated than the smple two-perod setup we used for llustratve purposes. These problems are far from trval, but well studed n Fnance, see e.g. Hull Addtonally, one could use a dfferent parametrzaton for G nstead of the polynomal parametrzaton or, f suffcently many observatons are avalable, one could possbly even use a non-parametrc estmate of the functon Oy gven the observatons {y j, Oy j } j. Further, one would also want to estmate the confdence nterval for S. 25

26 6 Rsk-averson In the man part of ths paper we have assumed that nvestors are rsk neutral and we have shown that t s optmal for the agency to pool all types above a cutoff n one ratng class. Doherty, Kartasheva, and Phllps 2012 extend the model of Lzzer 1999 by allowng nvestors to be rsk-averse and they show that, f the level of rsk averson s suffcently hgh, the ratng agency rates types above a cutoff n several ratng classes. Frst, followng the paper of Doherty, Kartasheva, and Phllps 2012 we provde a smplfed hybrd model ncorporatng rsk averson and aggregate uncertanty. We show that ntroducng rsk averson n a model wth several states of the world can also yeld several ratng classes. In ths case our prevous analyss can be nterpreted as determnng the optmal cutoff of the lowest nvestment grade ratng class e.g. BBB. Second, we provde a numercal example to show that the effects of the moments of the expected qualty dstrbuton on the optmal cutoff have the same sgn as before even wth rsk averson and several ratng classes. We provde the smplest possble setup whch s rch enough to llustrate the dea. Assume that buyers are rsk-averse. Ther utlty of an asset s equal to t but ther expected utlty depends on both the mean and the varance of the qualty of the asset they buy. We nclude a second mass pont at t 2, t 2 t, wth mass γ n state. To avod confuson denote t by t 1. See Fg. 6. If buyers are rsk-averse, a welfare maxmzng ratng strategy needs to perfectly dsclose the type of all assets wth a postve value because any knd of poolng and beng vague about a frm s qualty leads to a welfare loss. However, such a strategy cannot be optmal for the ratng agency. 16 To analyze a general model wth rsk-averson s beyond the scope of ths artcle. In the followng, we compare two ratng strateges: poolng all types above a cutoff n one ratng class, whch s the optmal strategy wthout rsk-averson, and a strategy n whch the agency only pools low types and rates hgh types separately. Doherty, Kartasheva, and Phllps 16 To ensure that all frms wth t 0 are wllng to pay the ratng fee under full dsclosure, the ratng fee has to be 0. 26

27 κ densty λ ft µ γ multple class ratng strategy: sngle class ratng strategy: t x 0 t 1 t 2 junk BBB AAA junk nvestment grade t Fgure 6: κ, µ and γ are the mass ponts at t, t 1 and t 2 n state. λ s the mass n state that s allotted to the types t t, t wth the dstrbuton F show that strategy s optmal n a model wth one state of the world f the level of rsk-averson s suffcently hgh. Analogously to the case wthout rsk-averson, we derve the proft of the agency f t pools all types above a cutoff x n one class. The expected type above a cutoff x s Q x := E[t t x] and the varance s σ x := Var[t t x]. The buyer s valuaton for the asset of a seller n ths ratng class s Q x aσ x where a s a measure for rsk-averson. If a = 0, the buyers are rsk-neutral and the model s equvalent to before. The proft of the agency f t pools all types s ˆΠx := λ 1 F x + µ + γ ɛ ˆɛ Q x aσ x =1 F x + µ + γ ˆɛ Q x aσ x 27

28 where γ s the expected value of γ, γ = ɛ γ. Alternatvely, the ratng agency can pool t [x, t 1 ] and rate t 2 separately as shown n Fg. 6. If the agency rates types t 2 n a separate class, these sellers are wllng to pay a hgh ratng fee up to t 2 and therefore the ratng fee s determned by sellers n the class t [x, t 1 ]. Keepng the cutoff x constant, the mass of rated frms s the same for both strateges and the ratng fee decdes whch ratng strategy yelds hgher profts. If the agency pools types t [x, t 1 ], the expected type n ths ratng class s smaller than Q x but the varance s also smaller than σ x. Thus, t s not straght forward to see under whch strategy the ratng fee can be hgher. Now, we derve suffcent condtons such that the agency prefers to rate t 2 separately nstead of poolng all types above x n one ratng class. Defne z := γ t 2 and z as the expected value of z, z := ɛ z. Rewrte z as z = t 2 γ, whch can be nterpreted as the agency s proft f t charges a ratng fee of t 2 and rates only frms wth type t 2. Remember that Πx s defned as the proft f the agency rates only t [x, t 1 ] and pools them all n one class. Proposton 7. Take an arbtrary cutoff x. For any z wth z Πx there exsts a T 2 such that for all t 2 T 2 the ratng agency s better off poolng t [x, t 1 ] and ratng t 2 n a separate class than poolng all types above x n one ratng class. Snce nvestors are rsk-averse, ther expected utlty buyng an asset n a gven ratng class decreases f the varance nsde ths ratng class becomes larger. If the varance s suffcently large, nvestors are not wllng to pay any postve prce for an asset even f the expected qualty s postve. Thus, f the varance s large, the agency s better off splttng the types n several ratng classes n order to reduce the varance nsde one class and to ncrease nvestors wllngness to pay for an asset. The condton that z Πx ensures that the agency does not prefer to charge a ratng fee of t 2 and to exclude frms wth t < t 2 from the ratng. Rsk averson does not only have the effect of multple ratng classes becomng optmal, but t also has an addtonal effect on the optmal cutoff. Increasng the cutoff reduces the varance n a ratng class and ths can gve addtonal ncentves 28

29 to ncrease the cutoff. 17 In the followng we provde numercal examples n whch we show that the effects of the frst, second, and hgher moments are smlar to our analyss wthout rsk averson. 18 In the numercal example we have four states of the world. We take the Generalzed Pareto dstrbuton F t = 1 1 t/2 3 for t 1, 1 and fx t 1 = 1. Ths gves us E 0 = 1/4. We fx λ = 5, t 2 = 110, and ν = for all. The states only dffer n the weghts µ at the mass pont at t 1, wth µ 1 = 0.03, µ 2 = 0.2, µ 3 = 0.4, and µ 4 = 0.7. Changng the moments of the aggregate dstrbuton, we keep the dstrbuton nsde a state constant and therefore also the expected type and only vary the probabltes for the dfferent states. There s a one-to-one mappng from ɛ 1, ɛ 2, ɛ 3 to m 1, m 2, m 3+ and the fourth probablty s pnned down by ɛ 4 = 1 ɛ 1 ɛ 2 ɛ 3. For all values of the example, the probabltes are n [0, 1]. Fgures 7, 8, and 9 llustrate the change of the optmal cutoff as m 3+, m 1 and m 2 are changed whle keepng the other moments constant. The sold lne s the optmal cutoff for a = 0, the dashed lne for a = 0.01 and the dotted-dashed lne for a = If nvestors are rsk neutral, a = 0, the agency pools all types above the cutoff n one class. For a = 0.01 and a = 0.02, nvestors are rsk averse and the agency prefers to pool all types t [x, t 1 ] n one class and to rate t 2 separately. Note that ncreasng the level of rsk averson leads to an ncrease n the optmal cutoff x. The fgures show that our results from the man part of the paper carry over to a setup ncludng rsk-averson: Keepng the other moments constant, a hgher mean, a lower varance, or an ncrease n the hgher order skewness lead to an ncrease n the optmal cutoff. For changes wth the opposte sgn, the optmal cutoff decreases. 17 Doherty, Kartasheva, and Phllps 2012 show that the optmal cutoff can be postve even wth only one state of the world f the level of rsk averson s suffcently hgh. 18 In the man part of the paper the moments were defned for the dstrbuton of the expected type n [0, t] scaled by t. For the sake of comparson, n the numercal examples the moments are defned for the dstrbuton of the expected type n [x, t 1 ] and thus, the expected type s not nfluenced by changes n the mass on t 2. We devate from our prevous analyss by takng the threshold x as the lower bound of the nterval. In ths way we can determne the optmal cutoff explctly and not only ts sgn. 29

30 m Fgure 7: Values of the optmal threshold x as m 3+ s changed and m 1 and m 2 are kept constant. For the sold lne a = 0, for the dashed lne a = 0.01 and for the dotted-dashed lne a = The ratng strategy for the sold lne s to pool all types above x. For the dashed and dotted-dashed lne all types n [x, t 1 ] are pooled and t 2 s rated separately. The startng values are ɛ = 1/4 for all. Ths mples m 1 = , m 2 = and as a startng value m 3+ = m Fgure 8: Values of the optmal threshold x as m 1 s changed and m 2 and m 3+ are kept constant. For the sold lne a = 0, for the dashed lne a = 0.01 and for the dotted-dashed lne a = The ratng strategy for the sold lne s to pool all types above x. For the dashed and dotted-dashed lne all types n [x, t 1 ] are pooled and t 2 s rated separately. The startng values are ɛ = 1/4 for all. Ths mples m 2 = , m 3+ = and as a startng value m 1 =

31 m Fgure 9: Values of the optmal threshold x as m 2 s changed and m 1 and m 3+ are kept constant. For the sold lne a = 0, for the dashed lne a = 0.01 and for the dotted-dashed lne a = The ratng strategy for the sold lne s to pool all types above x. For the dashed and dotted-dashed lne all types n [x, t 1 ] are pooled and t 2 s rated separately. The startng values are ɛ = 1/4 for all. Ths mples m 1 = , m 3+ = and as a startng value m 2 = Conclusons We have consdered the proft maxmzng ratng strategy of a ratng agency n the face of aggregate uncertanty. We have shown that wth rsk neutral nvestors t s stll optmal for the ratng agency, as n a setup wthout aggregate uncertanty, to choose only one ratng class for rated frms and to not rate the remanng frms. The model s predctons about the cutoff for the ratng class strkngly dffer from the predctons of a model wthout aggregate uncertanty: the ratng agency has more of an ncentve to be too lenent f the expected average qualty s small, the varance large, and the hgher order skewness small. For larger averages, smaller varances, and larger hgher order skewness the opposte holds: the ratng agency has more of an ncentve to be too strct. These results can be nterpreted as ratngs havng ether a pro-cyclcal or an ant-cyclcal effect. We outlne an emprcal strategy to estmate the moments of aggregate uncertanty whch can be used to determne whch effect domnates. Our analyss dentfes one up to now unconsdered factor that affects the ratng strategy of an agency aggregate uncertanty and thereby sheds further lght n understandng the behavor of ratng agences. In lne wth our model, one dsturbng 31