TIMELINESS, ACCURACY, AND RELEVANCE IN DYNAMIC INCENTIVE CONTRACTS

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1 TIMELINESS, ACCURACY, AND RELEVANCE IN DYNAMIC INCENTIVE CONTRACTS by Peer O. Chrisensen Universiy of Souhern Denmrk Odense, Denmrk Gerld A. Felhm Universiy of Briish Columbi Vncouver, Cnd Chrisin Hofmnn Universiy of Hnover Hnover, Germny Florin Sbc Universiy of Alber Edmonon, Alber July 3, 2003 Preliminry Snford Summer Cmp Version * We re hnkful o Sunil Du nd Sefn Reichelsein for shring heir unpublished noes on he nlysis of consumpion choice wih ime-ddiive preferences. Peer, nd Jerry nd Florin re hnkful for he finncil suppor of his reserch by he Dnish Socil Sciences Reserch Council, nd by he Socil Sciences nd Humniies Council of Cnd, respecively.

2 CFHS OUTLINE 1. Inroducion 1 2. Acions, Reporing Sysem, nd Compension Acions nd Compension Prior nd Poserior Noise Beliefs 8 3. Agen Preferences nd Choices Preferences Consumpion Choice Acion Choice Principl s Problem Under Full Commimen Principl s Preferences Firs-bes Conrc Full-commimen Conrcs Informiveness, Independence, nd Trnsformions Informiveness nd Independence Alernive Performnce Mesure Represenions Impc of Timing of Repors wih Full Commimen Impc of Timely Relese of Performnce Repors wih Exogenous Incenive Res Single Acion De Exmple Timely Relese of Purely Insurnce-Informive Repors Single Acion-De Exmple wih Opiml Incenives Timeliness Versus Accurcy Sequenil Acion-Informive Repors wih Incresing Precision Reporing Inervl Impc of Timing of Repors wih Limied Commimen Renegoiion-proof Conrcs The Desrucive Effec of Renegoiion on Delyed Repors Impc of Renegoiion in he Single-cion-de Exmple Timeliness Versus Accurcy wih Renegoiion Reporing Inervl wih Renegoiion 42 Appendix A: Proofs of Proposiions 44 References 47 Figure 1: Full Commimen (Renegoiion) Timeline Tble 1: Single Acion, Muliple Reporing Des Exmple wih Time-ddiive Preferences

3 This pper is under consrucion. Consequenly, our curren inroducion is very uilirin. I srkly describes he generl nure of he model nd he key resuls. There is only limied descripion of he moivion nd bckground for he nlysis, nd limied references o oher reled reserch. In fc, references re limied hroughou he pper, nd we invie he reder o provide us wih pproprie references. 1. INTRODUCTION The genesis for his pper goes bck o he mid 1960 s, when commiee of he AAA, mong ohers, chllenged ccouning reserchers o py greer enion o ccouning s role s source of informion for vrious decision mkers. In his response o his chllenge, Felhm (1968, 1972), exmined he economic impc of he imeliness, ccurcy, nd relevnce of informion in dynmic decision environmen, wih illusrions using n invenory conrol model. As ws common h ime, he focus ws on wh Demski nd Felhm (1976) (DF76) would ler cll he decision fciliing role of informion. Agency models begn o pper in he mid 1970 s (e.g., Demski nd Felhm, 1978 (DF78)), emphsizing wh DF76 clled he decision influencing role of informion. The mjoriy of he gency nlyses hve been single-period models nd, while he nlysis of dynmic models is sedily incresing, mos hve reed he des which repors re relesed s fixed. One of our objecives is o exmine he impc of he imeliness, ccurcy, nd relevnce of decision influencing informion, wih priculr emphsis on imeliness. (In he curren drf, ccurcy nd relevnce receive very limied enion.) In muli-period seing, he decision-influencing informion repored fer n cion hs been ken cn lso serve s decision-fciliing informion for selecing subsequen cions. We exmine principl-gen model in which h is no he cse. Rndomness is due solely o he noise in 1

4 1 performnce repors, nd h noise is ddiive. Hence, he gen s cions do no vry wih he repors received, lhough his cions do depend on he incenive res h will be pplied o his repored performnce. The principl chooses hose incenive res nd, in so doing, he considers he gross pyoff he will receive from he gen s cions nd he compension he mus py he gen. The compension mus cover he gen s personl cion coss plus wh we cll his consumpion risk premium. The decision influencing informion cn ffec which cions cn be induced (which perins o he relevnce or gol congruence issue), bu he primry effec exmined in his pper perins o he consumpion risk premium h mus be pid for given level of induced effor. Obviously, he moun of noise in repor nd he correlion cross repors ffecs he gen s risk premium (which perins o he ccurcy or precision issues). Ineresingly, he imeliness of he repors cn lso ffec he gen s consumpion risk premium, nd explorion of his effec is mjor componen of his sudy. As we demonsre, he impc of imeliness on he gen s consumpion risk premium depends significnly on he exen o which he principl nd gen cn commi o long-erm conrc nd on he gen s preferences. We iniilly ssume full commimen is fesible, nd hen, in he ls secion, we ssume full commimen is no fesible nd exmine he impc of limied commimen. To fcilie our nlysis, we consider muli-period exensions of he single-period LEN model (i ssumes liner conrcs, exponenil uiliy, nd normlly disribued performnce repors, see, for exmple, Felhm nd Xie, 1994). There re wo bsic wys of exending he gen s exponenil uiliy 2 funcion o encompss muliple consumpion nd cion des. The firs, which we erm PM, follows ppers such s Indjejikin nd Nnd (1999) (IN), Chrisensen e l. (2003, 2003b) (CFS, CFSb), nd Ôbc (2003), nd ssumes h he gen s uiliy is expressed s funcion of he NPV of his 1 I is lso imporn h we ssume preferences re represened by exponenil funcions in which he gen s disuiliy for effor is expressed s personl cos reducing his consumpion. These ssumpions remove ll welh effecs. Norml disribuions is noher imporn ssumpion h ensures h he poserior risk is independen of he relizion of prior repors. 2 PM is shor for muliplicely seprble preferences, for resons h re elbored upon in he pper. 2

5 consumpion. We explicily llow for posiive ineres re, while he ppers menioned ssume zero ineres re, nd s we shll see, his cn hve profound impc on how we cn inerpre resuls wih 3 PM. The second exension, which we erm PA, follows Du nd Reichelsein (1999, 2002) (DR99, DR02), nd ssumes h he gen s uiliy funcion is he sum of discouned sequence of exponenil funcions pplied o he consumpion ech de. The iming of consumpion nd compension cn differ, since he gen cn borrow or sve he mrke ineres re. Ineresingly, he gen s ceriny equivlen for given compension conrc hs he sme generl form for boh PM nd PA. The only difference is he effecive risk version used in clculing he consumpion risk premium. While he difference ppers smll, he impc is significn. Under PM, he iming of consumpion, compension, nd performnce repors re immeril when here is full commimen. On he oher hnd, under PA, while he iming of compension is immeril, he opiml iming of he gen s consumpion for given incenive conrc is unique. In priculr, he ddiive preferences resul in consumpion smoohing, which we chrcerize s going shor or long in nnuiies h spred he gen s bnk blnce plus his ceriny equivlen for fuure compension over his consumpion plnning horizon (which my be finie or infinie, if he considers his heirs). Consequenly, PA implies h he erlier he gen receives informion bou his risky compension, he lower is his consumpion risk premium, nd he lrger is he NPV of he principl s expeced ne pyoff. A repor is purely insurnce informive if i is no influenced by ny cion, bu is correled wih he noise in some cion-informive repor (e.g., he performnce of compeiors or informion bou economy- or indusry-wide evens h will influence he noise in he gen s performnce repors). The impc of he iming of his ype of repor highlighs he fc h incenive risk (i.e., he risk ssocied wih compension h vries wih cion-informive repors) is miiged by he gen s consumpion smoohing nd by he principl s provision of insurnce bsed on insurnce-informive repors. The 3 PA is shor for ddiively seprble preferences. 3

6 ler is less cosly o he principl hn he former, becuse he is risk neurl nd he gen is risk verse. Erly reporing would reduce he gen s risk premium if he principl inefficienly used he insurnceinformive repor. However, here is no benefi o erly reporing of pure insurnce informion if, fer i is repored, i is efficienly used by he principl, provided he repor is issued before or wih he firs cion-informive repor wih which i is correled. If issued fer he ler de, hen he erlier i is repored, he beer. The repors re defined o be sochsiclly independen if heir noise erms re independenly disribued cross periods, nd he repors re defined o be echnologiclly independen if only he curren period s cions influence he curren repor. The repors re fully independen if boh condiions re sisfied, nd deriving he opiml conrc is priculrly srigh forwrd in his cse. We demonsre h he noise erms cn lwys be rnsformed so h hey re sochsiclly independen. The rnsformion deducs he sr-of-period poserior men, which is liner funcion of prior noise erms. Using he sme liner rnsformion on he repors yields n equivlen represenion of he informion in erms of sochsiclly independen repors. We use his rnsformion in proving he resuls perining o he vlue of erly versus delyed reporing nd, in priculr, how his perins o purely insurnce-informive repors. Ineresingly, his rnsformion yields fully independen repors if he repors re genered by n uo-regressive process. Wih full commimen, if delying performnce informion permis chievemen of greer ccurcy wih no exr cos, hen he dely is sricly preferred wih PM, bu wih PA he improvemen in ccurcy mus be sufficien o offse he effec of reduced consumpion smoohing. Furhermore, if he more precise repor is preferred nd i is sufficien sisic for boh i nd he erly repor, hen he erly imprecise repor is redundn wih PM, bu is sill vluble wih PA. The key is h, for exmple, he erly relese of n imprecise forecs of he cul repor is vluble under PA becuse i fcilies ddiionl consumpion smoohing. 4

7 Mny of he resuls re illusred using simple single cion-de exmple. However, we lso use wo cion-de exmple o illusre he effec of wo- versus one-period reporing inervl, nd he impc of ggregion versus disggregion in wo-period reporing inervl. The ls mjor secion (Secion 7) explores he impc of he imeliness of repors when he principl nd gen re limied in he conrc commimens hey cn mke. We ssume hey cn commi o n incenive conrc for les one period (beginning before he cion for he period is ken nd coninuing unil fer ny repor for he period is issued (bu before he nex period s cion is ken see he imeline in Figure 1). We furher ssume he principl nd gen cnno leve he conrc unil i ermines. Rher hn model limied commimen s shor-erm conrcs, we model i s long-erm conrc h is subjec o renegoiion eiher he end of ech period, or fer repor is issued. The opiml liner conrc is hen represened by long-erm renegoiion-proof conrc (i.e., he principl nd gen cn mke muully greed o chnges, bu no such chnges will exis). The explici incenives under his pproch yield he sme resuls s he explici plus implici incenives h rise in shor-erm conrcing (wih renegoiion). A key feure of limied commimen is h he renegoiion de he principl mkes his ex pos opiml choice of he incenive conrc for he forhcoming period, ignoring he impc his hs on he gen s choice of prior cions. As we demonsre, reporing delys or long reporing inervls cn cuse he repors o hve zero, or even negive, vlue if here is renegoiion bsed on he clendr de. For exmple, wih full commimen, he le issunce of n insurnce-informive repor reduces is vlue relive o erly reporing, bu wih renegoiion, he repor hs vlue if issued when he cion is ken, bu hs zero vlue if issued ferwrds. Similrly, delys in issuing n cion-informive repor, e.g., o reduce is cos or o obin more precise repor, will be olly unrcive. Of course, he ler effec does no occur if renegoiion is riggered by he issunce of repor (s opposed o he ime on he clock). 5

8 Ineresingly, while he resuls for full-commimen differ significnly under PA versus PM, he desrucive effec of delyed reporing when here is renegoiion is very similr under PA nd PM. 2. ACTIONS, REPORTING SYSTEM, AND COMPENSATION Figure 1 provides ime line h highlighs he key des in our muli-period model, s well s he sequence of evens wihin seleced periods (period begins de -1 nd ends de ). The principl hires he gen = 0- (immediely prior o = 0) o commence work = 0 nd o coninue unil his reiremen de. The gen my receive pos-reiremen compension, which ceses he conrc erminion de T $. Furhermore, he gen my pln o consume (or provide for his heirs consumpion) beyond he conrc erminion de o he consumpion plnning horizon $T. 2.1 Acions nd Compension A he sr of ech pre-reiremen period = 1, 2,...,, he gen selecs cion 0, which represens he gen s effor in vecor of n sks. The personl cos of h effor is ssumed o be qudric, convex funcion represened by 6 = ½N' where ' is posiive semi-definie n n mrix (nd, herefore, inverible) nd 6 is expressed in de dollrs, = 1, 2,..., T. Observe h he mesure of civiy for ech sk cn be posiive or negive, nd 4 h he gen s personl cion coss cn be posiive for boh posiive nd negive civiy. The gen s curren nd ps cions re represened by N = ( N, N,..., N), nd his curren nd niciped fuure 1 2 cions re represened by N = (N, +1N,..., TN). A de, vecor of m (conrcible) performnce mesures, represened by y, is publicly repored. 4 This ssumpion voids he problems of corner soluions, which re ypiclly ssumed wy in he LEN model. 6

9 The repor = 0 is no influenced by he gen s cions, bu subsequen repors re poenilly influenced by he cions ken prior o de, s well s m 1 rndom noise vecor g. The relionship is liner nd ddiive, represened by y = MJ J + g, for ll = 1, 2,..., T, where M J is n m n J mrix of prmeers represening he expeced impc of he cions in period J on he repors de, J #. The beliefs bou he rndom noise g re discussed in he nex secion. The performnce mesures repored = 0 re no influenced by ny cion, bu cn be useful if he repored noise (i.e., y 0 = g 0) is correled wih he noise in subsequen repors, nd hose repors re influenced by subsequen cions. In generl, we ssume y 0 is repored subsequen o signing he iniil conrc, bu we will lso consider pre-conrc repors. As in he sndrd LEN model, we resric he compension pid ech de o be liner funcion of he performnce mesures repored up o de. We ssume h he principl nd gen cn borrow nd lend consn riskless ineres re 4. Hence, he iming of he compension is flexible, e.g., he principl s nd gen s preferences re he sme wheher given compension w is pid de or J- wr is pid de J, where R = 1+4. Consequenly, he opiml liner conrc is no unique, unless we resric he iming. If boh pries cn commi o long-erm conrc (he full-commimen cse), hen i is srighforwrd o prove h for ny se of opiml liner conrcs, here lwys exiss n equivlen conrc in which ll fixed compension (which we denoe s f) is pid he conrc de = 0-. We le w represen he vrible wge pid de, for = 0,1,..., T. Is iming is lso flexible, nd we ler ssume h eiher he vrible compension ech de depends only on he repor for h period (i.e., w = vny, for = 0, 1,..., T), or ll vrible compension is pid de T (i.e., w = 0, for = 0, 1,..., T-1, nd w T = ), wih n pproprie djusmen in he incenive res for he differences in he ime-vlue of money. In boh cses we represen he conrc offered = 0-, denoed z, in erms of he 7

10 fixed wge nd he incenive res pplied o he repors ech de, i.e., z = (f, ), where = (v 0,v 1,...,v T). A de > 0-, he conrc for he remining periods is z = = (v +1,...,v T). 2.2 Prior nd Poserior Noise Beliefs The relized noise hisory, represened by = (g N, g N,..., g N ), is joinly normlly disribued wih n 0 1 T ex ne men vecor of zeros nd n ex ne vrince/covrince mrix Vr 0-( 5 ). A de he gen knows he relized noise hisory N = (g N, g N,..., g N ), since is period componen cn be inferred by 0 1 he gen: g = y - MJ J. Since he noise erms re joinly normlly disribued wih zero mens, he poserior disribuion for g J given ny hisory, < J, is normlly disribued wih poserior men h is liner funcion of (wih no inercep) nd poserior vrince h is he sme for ll relized hisories. More specificlly, he poserior men of he m J 1 vecor g J, given, is 6 E[g J] / E[g J* ] = : Jk g k, -1 where = [: J1N,..., : JN] = Cov 0-(g J, )Vr 0-( ), nd he kh elemen : Jk is n m k m mrix of covrince coefficiens. The corresponding m J mj poserior vrince/covrince mrix is -1 E J / Vr [g J] = Vr 0-(g J) - Cov 0-(g J, )Vr 0-( ) Cov 0-(,g J) = Vr 0-(g J) - Cov 0-(,g J). 5 The subscrip 0- indices h he vrince is bsed on he informion vilble = See Secion in Chrisensen nd Felhm (2003) (CF) for generl expressions for deriving poserior beliefs from he prior join norml disribuion. 8

11 3. AGENT PREFERENCES AND CHOICES In his secion we inroduce he gen s preferences nd hen chrcerize his sequenil choice of consumpion nd cions for n exogenously specified reporing sysem nd incenive conrc. 3.1 Preferences The gen s preferences re represened by uiliy funcion defined over he sequence of consumpion, ne of effor cos, c = (c 0, c 1,..., )N. We consider wo bsic exponenil uiliy funcions. Our primry focus is on ime-ddiive funcion wih eiher finie or n infinie consumpion plnning horizon, bu we lso consider ime-muliplicive uiliy funcion. The ler hs been frequenly used in he lierure becuse i voids he complicions resuling from he smoohing of consumpion hrough gen borrowing nd lending. This pper seeks, in pr, o shed ligh on he impc of differences in preferences on he relive rnking of lernive reporing sysems. PA ddiively-seprble preferences: In his seing, he gen s uiliy funcion is 7 u (c 0,c 1,..., ) =, where r is he gen s risk version wih respec o he consumpion in ech period, $ is he discoun -1 fcor, (1+4), nd is he gen s consumpion plnning horizon, which my be finie or infinie. PM muliplicively-seprble preferences: In his seing, he gen s uiliy funcion is m u (c 0,c 1,..., ) = =. Under PM, i is immeril wheher is finie or infinie nd, o simplify noion, we gin le r represen he gen s risk version even hough he PM prmeer my differ from he PA prmeer. 7 The uiliy funcion for ech period is ssumed o be equivlen, bu when ggreged he gen hs ime preference such h he uiliy funcion for ech period is discouned using he mrke discoun re. This is no essenil, bu i simplifies he nlysis nd produces inuiively ppeling resuls. 9

12 Virully ll he ppers h use PM ssume zero ineres re, ofen wih he semen h his is wihou loss of generliy. Th is, ny problem wih posiive ineres re cn be represened s problem wih zero ineres re merely by mesuring ll monery mouns in, for exmple, = 0 dollrs, e.g., consumpion de cn be represened s = $ c. However, his ends o msk he implicions of he ime-vlue of money when ssuming PM. Therefore, we explicily include he ineres re nd mesure monery mouns in he nominl dollrs of ech period. 3.2 Consumpion Choice We now chrcerize he gen s sequenil consumpion choice given n exogenously specified incenive conrc z = (f, ), which induces priculr sequence of cions. As we demonsre, he consumpion choice in he PA seing cn be chrcerized s he moun genered by n nnuiy purchsed wih he gen s curren bnk blnce plus funds borrowed gins his ceriny equivlen from fuure compension minus fuure personl coss. Hence, wih PA, we mke exensive use of he nnuiy morizion fcor, A /, which specifies he moun per period h cn be pid des, +1,...,, given he invesmen of one dollr in n nnuiy de. The gen s consumpion plnning horizon cn exend beyond he gen s expeced life, reflecing his preference o leve n endowmen o his heirs. This llows for he possibiliy h he gen s consumpion plnning horizon is infinie, resuling in n nnuiy morizion fcor of * A = A / = 1 - $ = 4$, for ll. Afer he gen hs received his compension de nd pid his period personl coss, he hs cum-consumpion bnk blnce of B 0 = f + w 0, = 0, nd B = R(B -1 - c -1) + w - 6, for = 1,2,...,, 10

13 wih w / 0 for ll > T nd 6 / 0 for ll >. This is he moun currenly vilble o he gen for consumpion de or for sving, ineres re 4, for consumpion in subsequen periods. Curren consumpion cn be greer hn he curren bnk blnce (which cn be negive) he difference in h cse represens borrowing gins fuure compension. The gen s expeced uiliy de for consumpion, +1,...,, given he informion vilble o he gen before he chooses his de consumpion, is denoed by U (B ) for PA nd U (B ) for PM. m The cum-consumpion bnk blnce, B, is explicily recognized, while oher informion de, such s, is reed s being implicily represened by he subscrip on U, i =,m. For simpliciy of i noion, we lso re he niciped incenive conrcs nd induced cions s implici. A de, he gen (or his heir) consumes he remining bnk blnce, i.e., =, nd, hence, = - exp[- r ], i =,m. (1) A ll preceding des = 0, 1,..., -1, he gen s consumpion decision problem fer y hs been repored is PA: U (B ) = {- exp[- rc ] + $ E[ (R(B - c ) + w )]}. (1b) PM: = exp[- r$ c ] E [ (R(B - c ) + w )]. (1c) The following proposiion chrcerizes he gen s consumpion choice nd expeced uiliy ech consumpion choice de, for PA nd PM. As we discuss below, he difference in he wo preference funcions mnifess iself in differences in consumpion choice ech de nd differen mesures of risk version wih respec o unceriny bou he NPV of fuure compension ech consumpion de. In his chrcerizion we do no explicily differenie mong he pre-reiremen, pos-reiremen, nd pos-compension periods hose re implicily hndled by recognizing h 6 = 0 for > nd w = 0 for > T. 11

14 8 Proposiion 1: The opiml consumpion choice nd expeced gen uiliy ech = 0, 1,...,, for given incenive conrc nd given niciped sequence of cions, re chrcerized s follows, for i = nd m represening preferences under PA nd PM, respecively. i c = A[B + CE ], i =, (2) B = w - 6, i = m, (2b) U (B) = i - exp[- (B + CE )], i =, / ra, (2c) - exp[- (B + CE )], i = m, / r$ (2d) m CE = $ E[W - K ] - $CRP, i =, m, (2e) i i i J - ( +1) CRP / ½ Vr [E +1[W +1]] + $ = ½ $ Vr J-1[E J[W J]], i =, m, (2f) wih W / w + $ W +1 nd K / 6 + $K +1. A key feure of he PA resul is h curren consumpion equls he moun h would be pid by n nnuiy h begins immediely, ssuming he moun invesed in he nnuiy equls he curren bnk blnce plus he ceriny equivlen of fuure compension minus he gen s fuure personl coss. The nnuiy will chnge from period o period s informion bou fuure compension chnges. There is no new compension fer de T nd, hence, consumpion is consn fer h de, i.e., c = ATB, T for $ T. On he oher hnd, while PA implies he gen prefers o smooh consumpion, PM implies he is indifferen wih respec o h iming. Consequenly, he opiml consumpion choice for PM is no unique. We simplify he nlysis by exogenously seing he gen s consumpion equl o his bnk 8 The PA componen of his proposiion is similr o Lemm 1 in DR99, he wo-period Lemm 1 in DR02, nd resuls in unpublished lecure noes by Reichelsein (2003). However, s in Fudenberg e l. (1990), hey resric heir nlyses o he infinie consumpion plnning horizon cse. In n unpublished noe, Du (2002) exmines wo-period consumpion plnning horizon model nd briefly exends i o T periods. While some of he Du nd Reichelsein resuls re similr o ours, our formulion emphsizes he role of he nnuiy fcor in represening he implemenion of consumpion smoohing, nd his is subsequenly useful in highlighing he difference beween he ceriny equivlens under PM versus PA wih eiher finie or n infinie consumpion plnning horizon. 12

15 blnce, which implies h ech period he consumes his compension minus his personl cos. The expeced uiliy expressions in (2c) nd (2d) inroduce effecive risk version prmeers / ra nd / r$. They consis of he consumpion risk version prmeer r (which is ssumed o be consn over ime) nd n djusmen prmeer h poenilly vries from period o period nd depends on he form of he gen s preferences. Under PA, he djusmen prmeer is he nnuiy fcor A, reflecing he fc h he gen smoohs his consumpion hrough he implici use of nnuiies. In essence, ra convers r, he gen s risk version wih respec consumpion, ino mesure of risk version h is pplied o mesure of he gen s risk wih respec o nex period s compension informion. Under PM, on he oher hnd, he djusmen fcor is $, reflecing he fc h he gen expresses his preferences in erms of compension (nd consumpion) mesured in de 0 dollrs. Observe h he effecive risk version cn be incresing, consn, or decresing, depending on wheher he gen s preferences re represened by PA wih finie consumpion plnning horizon, PA wih infinie, or PM, respecively. i Expression (2e) inroduces W, K, nd CRP. The firs, W, represens he NPV of he gen s compension for periods hrough T. Observe h E +1[W +1] will vry wih he new informion received de +1 nd, hence, i is rndom vrible from he perspecive of de. The second, K, is similr expression for he gen s personl cos, excep i is no influenced by he informion, i.e., K +1 is i no rndom vrible. The hird, CRP, is he NPV of he gen s consumpion risk premi for periods +1 nd beyond (mesured in +1 dollrs). The firs wo elemens re he sme for boh PA nd PM, bu he consumpion risk premium for ech de J > is bsed on he effecive mesure of risk version, which is pplied o he vrince of he NPV of fuure compension s mesured by E [W ]. The personl 9 coss re no rndom, so hey do no conribue o he risk. Also, s is well known, while he poserior J J 9 FIN provide model in which cion choices nd, hence, he gen s personl coss, vry wih he informion received. In h seing here is personl cos risk s well s compension risk. 13

16 vrince of normlly disribued rndom vrible, such s W J, depends on he informiveness of he repors received up o de J-1, he poserior vrince is iself independen of he specific repors received, e.g., Vr [E [W ]] is consn, so h he fuure risk premi CRP re no rndom vribles. i 10 J-1 J J J 3.3 Acion Choice We now consider he cion choices he sr of ech period for given reporing sysem nd n exogenous conrc z = (f, ) = 0. The fc h he noise is normlly disribued nd ddiive implies h he gen s cions do no influence his consumpion risk premium under eiher PA or PM. Hence, for boh PA nd PM, -1 he gen chooses, for = 1,...,, so s o mximize E -1[W ] - K, where J- E -1[W] = $ vjn M Jk k + : J,-1,k g k, (3) J- K = $ ½ JN' J J (3b) Differeniing (3) - (3b) wih respec o immediely provides he following resul. Proposiion 2: The gen s opiml cion choice, for boh PA nd PM, given incenive res v,..., v T, re chrcerized by -1 J- = ' $ MJNv J. œ = 1,...,. (4) A key feure of (4) is h, since curren effor my influence boh curren nd fuure performnce mesures, he induced curren cion poenilly depends on he incenive res pplied o ll hose mesures. Consequenly, cions cn be influenced by incenive compension pid in he posreiremen period, even hough he gen does no ke ny cions in h period. 10 The consumpion risk premium is rndom vrible in he FIN model, i.e., here is risk premium risk! 14

17 4. PRINCIPAL S PROBLEM UNDER FULL COMMITMENT We now consider he principl s conrc choice given his nicipion of he gen s rionl response o he conrc. The principl s firs-bes choice highlighs he fc h, in his pper, he reporing sysem does no ply decision-fciliing role if here is no decision-influencing role, hen he repors re ignored. The nlysis of he second-bes opiml conrc ssumes h boh he principl nd he gen cn fully commi o dhere o he conrc for he enire T periods. Ler we exmine how limied commimen ffecs he impc of performnce mesure chrcerisics. 4.1 Principl s Preferences The principl s expeced gross pyoff from cions ken in period is represened by b N, for = 1, 2, 11...,. This pyoff is reflecs he NPV o de of ll curren nd fuure effecs of. If ny pr of he pyoff is conrcible, hen he conrcible elemens re lso included mong he performnce mesures, wih explici recogniion of he iming of he repors. The principl is ssumed o be risk neurl wih he sme discoun re s he gen. The NPV de = 0, 1,..., of his expeced fuure ne pyoffs (given he repors up o de, nd his conjecures wih respec o he fuure incenive res nd gen cions) is J- A = 7 - $ E[W +1], where 7 / $ bjn J. A he conrc de, he principl s preferences re represened by A 0- = 70 -{f + E 0-[W 0]}. Since he fixed wge does no ffec he gen s decisions nd only serves o direcly increse his ceriny equivlen, he principl chooses f o be jus sufficien o induce he gen o ccep he conrc 11 The expeced gross pyoffs of he gen s cions re fixed, or in oher words, he performnce mesures re no informive bou fuure expeced gross pyoffs. The key feure in he FIN model is h ps performnce mesures re correled wih fuure gross pyoffs. 15

18 i o h is offered, i.e., f = W + K E [W ], i =, m. The reservion wge does no hve ny o subsnive effec on he nlysis, so we le W = 0. Hence, in selecing he conrc offered = 0-, he principl seeks o mximize = 70 - {K 0 + }, i =, m. (5) 4.2 Firs-bes Conrc The firs-bes cse ssumes he gen s cions re conrcible, so h he gen cn be induced o underke he desired cions wihou imposing ny incenive risk (i.e., riskless penly hres re sufficien). Hence, he principl s objecive is o selec he cions he gen will be required o implemen so s o mximize A 0- = 70 - K 0 = $ { bn - ½ N' }. The firs-order condiions chrcerizing he opiml cion choices re b - ' = 0, œ = 1,...,, which implies h he firs-bes cion for period nd he principl s ne pyoff re * -1 = ' b, * * * -1 A 0- = 70 - K 0 = ½bN' b. Observe h he ssumpions of our model re such h he gen s opiml cions re independen 12 of he performnce repors nd, hence, he gen does no ber ny personl-cos risk. The ler resuls becuse he gen s mrginl produciviy nd mrginl personl coss re known consns he only rndomness is in he ddiive noise of he performnce mesures. Hence, he performnce repors 13 hve no vlue (no mer when repored) if he gen s cions re conrcible. As is common in 12 See FIN for n exmple of model wih personl-cos risk. 13 Of course, s is well known, here re oher condiions besides conrcible cions under which firs-bes cn be chieved. See CFS, for exmple. 16

19 single-period models, his implies h i is opiml o py he gen fixed wge h covers his personl coss. 4.3 Full-commimen Conrcs We now chrcerize he opiml incenive conrc ssuming boh he principl nd he gen cn fully commi o be bound by he iniil conrc for he enire T periods. In priculr, hey cnno chnge (i.e., renegoie) he conrc some subsequen de even hough hey would boh gree o chnge. Hence, he principl chooses he incenive res = (v, v,..., v ) so s o mximize (5) subjec o (4), which 0 1 T specifies he cions h will be induced by he chosen incenive res. Subsiuing (4) ino (5) nd differeniing which respec o ech incenive re provides he following implici chrcerizion of he opiml full-commimen incenive res. Observe h he incenive res direcly ffec he gen s consumpion risk premium, bu only ffec he principl s gross pyoff nd he gen s personl coss hrough is impc on he gen s cion choices. Proposiion 3: The opiml full-commimen incenive res re chrcerized by he following firs-order condiions for ech v, = 0, 1,..., T, nd i =, m: 14-1 J-h c [70 - K 0]N h - % bhn' h MhN - $ v J = 0 (6) -1 where / M h ' h MJhN + : hh E h,h-1 : JhhN, : JJJ is n ideniy mrix, nd h is given by (4). Firs-order condiion (6) expresses v s n implici funcion of he incenive res for he oher 14 Keep in mind h M J = 0 for J = 0, J >, nd J >, s well s for > T. Similrly, : JJ = 0 for J > nd > T, nd b J = 0, ' J = 0 for J = 0. Furhermore, summions over null ses re equl o zero nd we hve divided hrough by $ since i is common o ll erms. 17

20 periods nd, hence, he combined firs-order condiions for ll des define se of N = m mt equions in N unknows. The firs erm in (6) represens he mrginl gross pyoff due o he impc of n increse in he incenive re for y on induced curren nd prior cions h ffec y. The second erm, which conins, represens he mrginl cos of he induced cions plus he mrginl consumpion risk premium, king ino ccoun he direc effec on he vrince of incresing v plus is impc on he covrince wih he incenive compension in boh ps nd fuure periods. 5. INFORMATIVENESS, INDEPENDENCE, AND TRANSFORMATIONS 5.1 Informiveness nd Independence The following definiions nd clssificions re inroduced o fcilie our subsequen discussion of he impc of performnce mesure chrcerisics. () A repor y is cion-informive wih respec o J if M J 0, nd i is no cion-informive if M J = 0 for ll J <. (b) A repor y is insurnce-informive wih respec o n cion-informive repor y J if Cov 0-[y,y J] 0, nd i is no insurnce informive if Cov 0-[y,y J] = 0 for ll J. ( c) A repor is purely insurnce informive if i is no cion-informive, bu is insurnceinformive wih respec o some cion-informive repor. 15 (d) The performnce mesures in reporing sysem re echnologiclly independen if he repor relesed ech de = 1,..., = 0 for ll J., is only cion-informive wih respec o, i.e., M 0, bu M J (e) The performnce mesures in reporing sysem re sochsiclly independen if he noise erms g 0, g 1,..., g T re independenly disribued cross ime, i.e., : hhh = I nd : Jhh = 0 for J h. (f) The performnce mesures in reporing sysem re fully independen if hey re boh echnologiclly nd sochsiclly independen. 15 This erm is used by Fudenberg, Holmsrom, nd Migrom (1990) (FHM) o refer o he srucure of he principl s pyoff. However, h is becuse hey ssume he pyoff is conrcible informion. In conrcing seing, he condiion is more ppropriely pplied o he srucure of he relion beween he performnce mesures nd he cion. 18

21 Obviously, he simples cse o solve is one in which here is full independence nd he repor ech de is cion-informive wih respec o. In h cse, he opiml conrc cn be solved s sequence of one-period problems. The ineresing feure of his specil cse is h idenicl problems in ech period do no resul in idenicl soluions unless he ineres re is zero or he gen s preferences re represened by PA wih n infinie consumpion plnning horizon. Corollry 1: If he performnce mesures re fully independen, hen for ll = 1,..., T nd i =, m, he fullcommimen incenive res, nd induced cions re chrcerized s follows: c -1-1 v = [ ] M ' b (7) c = ' MN[ ] M ' b, (7b) i -1 where Q / M ' MN + E, Furhermore, if he periods re nominlly idenicl, i.e., b = b 1, M = M 11, ' = ' 1, nd E,-1 = E 10, for ll = 2,...,, nd he ineres re is sricly posiive, hen *v +1* <, =, > *v * if he gen s preferences re represened by PA wih finie, PA wih infinie, or PM, respecively. 5.2 Alernive Performnce Mesure Represenions In generl, he performnce mesures provided by reporing sysem re neiher echnologiclly nor sochsiclly independen. However, we cn lwys genere n lernive represenion of he repors h consiss of sochsiclly independen sufficien sisics. Sochsiclly Independen Sufficien Sisics: Noe h if he noise vecors in he performnce mesures re correled cross periods, he correlion beween performnce mesure repored de nd prior performnce mesures reflecs he fc h pr of he informion in y is lredy prly 16 Ppers h use PM commonly ssume zero ineres re (e.g., see IN99, CFS, nd CFSb), in which cse $ equls one for ll, nd he induced cions re consn over ime under he ssumed condiions. 19

22 reveled by he noise vecors in prior mesures. The new informion provided by he noise vecor g given he hisory of noise vecors cn be represened by : * / g - E -1[g ], for ll = 0,1,..., T. (8) Observe h his is liner rnsformion of he noise vecors, wih E -1[g] = :,-1,J g J. Consequenly, using his liner rnsformion we cn define n lernive represenion of he performnce mesures (referred o s sisics) h hve he following srucure: s = y - :,-1,J y J = J + *, for = 0,1,...,T, (8b) where = 0, = M, nd / - : M. (8c),-1,k kj s Le L 0, L 1,..., L T be he incenive res in conrc bsed on he sisics, i.e, reporing sysem 0. The hisory of sisics ny de is sufficien o recover he hisory of he originl repors, nd his fc llows us o deermine he incenive res for he originl performnce mesures from he incenive res for he sisics. In priculr, J- v = L - $ : J,J-1,L J. (9) wih 17 The key chrcerisic of he rnsformion is h he noise erms for he sisics re independen, Vr 0[* ] = Vr -1[* ] = Vr -1[g - E -1[g ]] = Vr -1[g ] = E,-1. Hence, he sisics re sochsiclly independen nd he gen s ex ne consumpion risk premium given incenive res L 0, L 1,..., L T is 17 This follows from he fc h E -1[g ] is n orhogonl projecion of g ono. 20

23 i s s CRP 0- (0 ) = ½ $ LNE,-1L, i =, m. (2) Wih his rnsformion, he opiml full-commimen incenive res re chrcerized by (6) wih M J replced wih, nd boh : nd : se equl o 0 for h < nd h < J, nd equl o I for h = = J, i.e., hh Jhh -1 s J-h s -1 s c c s bhn' h MhN - $ M h ' h MJhNL J - E,-1L = 0. (6 ) Two specil cses provide ineresing insighs ino he srucure of he opiml incenive res. The firs occurs in seings in which he repors follow n uo-regressive process, he oher occurs in seings in which se of performnce mesures re purely insurnce-informive. The former produces very simple resuls, while he ler is used ler o explore he impc of informion iming. Auo-regressive Process: We define he performnce mesures s following n uo-regressive process if here exis exogenous mrices of weighs 8 J for ll = 1,..., T nd J = 0,..., -1, such h y = M + 8Jy J + *, (10) where he * 0,..., * T re independenly, normlly disribued wih zero mens. Obviously, he following rnsformion yields fully independen performnce mesures, s = y - 8Jy J = M + *. (10b) This llows he pplicion of (7) o deermine he opiml full-commimen incenive res for he sisics, which cn hen be used in (9) o derive he opiml incenive res for he originl performnce mesures. In pplying (9) i is imporn o noe h (10) implies h : = 8, so h rnsformion,-1,j J (10b) is specil cse of rnsformion (8). 18 Purely Insurnce-informive Mesures: Repors = 0 cnno be cion informive since hey pre-de ll cions, which implies = M = 0. Consequenly, firs-order condiion (6) for L Noe h (10) implies h he noise vecor for y is g = 8Jg J + *. Since * is independen of ll prior noise vecors, i hs zero condiionl men vecor given he informion -1 nd, hus, : = 8.,-1,J J 21

24 simplifies o E0,0- L 0 = 0, which implies L 0 = 0 (since E 0,0- is posiive definie). Of course, his does no imply h v 0 equls zero, since y 0 cn be correled wih subsequen cion-informive mesures. Informion used in consrucing budges provides n exmple of his ype of insurnce-informive repor, nd compuion of he difference beween repored ernings nd budgeed ernings (which represens is expeced vlue) illusres rnsformion (8). Of course, here cn be oher purely insurnce-informive mesures, such s he performnce of compeiors h cn be used o consruc relive-performnce mesures. Assume, for exmple, h for some $ 1, y is no influenced by ny cion (i.e., M = 0 for ll J), bu i is informive wih respec o o he noise in some cion-informive repor. If y is informive bou some prior cion-informive J repor, hen he rnsformion resuls in sisic s h is cion-informive nd L is non-zero. However, if y is only informive bou fuure cion-informive repors, hen he sisic s is no cion-informive, nd L equls zero. In he ler cse, he sisic is merely whie noise. Lemm 1 Assume h for some y, M J = 0 for ll J = 1,...,, nd :,-1,J = 0 for ll J = 0,..., -1, hen = 0 for ll J = 1,...,, nd L = 0. On he oher hnd, if :,-1,J 0 nd M Jk 0 for some J = 1,..., -1(nd some k = 1,..., J), hen is non-zero for some J = 1,..., -1, nd L is non-zero (excep in knifeedge cses). 6. IMPACT OF TIMING OF REPORTS WITH FULL COMMITMENT We now exmine he impc of chnges in he iming (bu no he conen) of performnce repors. Since he gen cn borrow nd lend on he sme erms s he principl, he iming of compension is immeril s long he ne presen vlue is unchnged (see FHM, for exmple). However, he iming of he relese of informion my no be irrelevn since i my ffec he principl s expeced compension cos hrough he ex ne consumpion risk premium pid o he gen. We firs demonsre his for exogenous incenive res, nd hen we exmine he iming of insurnce-informive performnce 22

25 mesures for opiml full-commimen incenive res. 6.1 Impc of Timely Relese of Performnce Repors wih Exogenous Incenive Res Consider he impc of delying he repor of y from de o de +1 on he gen s ex ne con- i sumpion risk premium, CRP 0-. Since he ex ne consumpion risk premium hs very simple form in he sochsiclly independen sufficien sisic represenion of he performnce mesures, i is insrucive o exmine he impc of delying he repor of he sisic s from de o de +1. The so sd originl sysem is denoed 0 nd he sysem wih delyed repor is 0. The incenive res re ssumed o be exogenous nd held consn cross reporing sysems, excep h L is replced by RL since he incenive pymen for s is now pid one period ler. Noe h s nd s +1 re sochsiclly independen. Hence, we cn direcly compue he difference in he ex ne consumpion risk premi from s (2 ), +1 - = ½ {$ - $ } = ½ $ {R - }, Using he definiions of he effecive risk version, = Ar for PA nd = $ r for PM, nd he fc h RA +1 > A, proves he following proposiion. Proposiion 4: For ny exogenously given incenive re L, he difference in he gen s consumpion risk premi for so sd he originl reporing sysem 0 versus he delyed sysem 0, re - = 0, - = ½ $ r{ra +1 - A } $ 0, where he ler is sricly posiive if, nd only if, 0. For PM, he iming of he relese of informion does no mer. In his cse, he gen s preferences 23

26 depend only on he ol NPV of his compension, nd his is no ffeced by he iming of he sumpion choice. If here is new informion in y or equivlenly in s, i.e., non-zero covrince mrix E,-1 informion relese. The key for PA is h he iming of informion relese ffecs he gen s con, nd h informion ffecs he gen s compension, i.e., L 0, hen he gen s consumpion risk premium is reduced by hving he informion erly rher hn ler. Hving he informion erlier rher hn ler llows he gen o smooh his consumpion over one more period (or o sr n infinie nnuiy one period erlier). Of course, if here is loss from delying he informion one period wih PA, he sme rgumen used sequenilly esblishes h he loss from delying he informion increses wih he number of periods he informion is delyed. 6.2 A Single Acion De Exmple In he following exmple, he finie-lived gen kes n cion 1 he sr of period 1, cos 61 = 2 ½ 1, nd hen reires. There re wo repors: y = m 1 + g nd y i = g i, wih -. The firs performnce mesure is cion-informive nd he second is purely insurnce-informive provided D 0. Le v nd v represen he incenive res for he wo performnce mesures expressed in i = 2 dollrs (for convenience of comprison). Observe h (4) implies h 1 = $mv, i.e., only he incenive re pplied o he cion informive repor ffecs he gen s cion choice. Pnel A of Tble 1 clcules he gen s consumpion risk premium under PA given rbirry J incenive res. Reporing sysem 0 repors y de = 1,2, nd repors y de J = 0-,0,1,2. The cion-informive mesure cnno possibly be repored prior o = 1, since h is he de he cion is ken. However, here is no such resricion on he insurnce-informive repor. The erlies de we i 24

27 consider is pre-conrc repor represened by J = 0-. There is no incenive re ssocied wih preconrc repor, bu, since i ffecs poserior beliefs, i does ffec boh he fixed wge nd, more significnly, he cion informive incenive re. 12 The gen s consumpion risk premium is minimized (for ll bu 0 ) if v = - D v. If h knife- edge cse holds, hen he gen s consumpion risk premium only depends on when y is repored. This is due o he fc h his se of incenive res provides excly he sme liner rnsformion of he performnce mesures s he sochsiclly independen sufficien sisics. Th is, if y i is repored 19 prior o y hen s i = y i = g i nd s = y - Dy i. If v i = - Dv, hen L = v implies h L i = 0 Hence, he compension risk is exclusively due o he reporing of s. Since s i nd s re independen, n erly repor of s (or y ) does no chnge he gen s beliefs bou his fuure compension implying h he i i gen cnno use n erly repor of y i o smooh consumpion. However, n erly repor of y (or s ) = 1 is sricly preferred o reporing i = 2 since i llows he gen o smooh he uninsured risk 12 v (g - Dg i) over one more period. If y i is repored subsequen o y s wih 0, hen s = y nd s i = y i - 2 Dy. In h cse, v i = - Dv implies h L i = - Dv nd L = (1-D )v. Hence, he incenive res re non- zero on boh sochsiclly independen performnce mesures implying h here is compension risk ssocied wih boh s nd s i. Hence, he gen prefers n erly rher hn ler repor of s i. The following relions summrizes he impc of informion iming for he knife-edge cse v = - Dv (0 ) = (0 ) = (0 ) = (0 ) > (0 ) > (0 ) = (0 ) = (0 ), On he oher hnd, if v i Dv, hen L i 0 nd he iming of y i mers (since i ffecs he beliefs bou fuure compension), wih erly reporing preferred: (0 ) > (0 ) > (0 ) > (0 ) i i 19 We cn use he sme rnsformion o produce sochsiclly independen performnce mesures wihin given period, if he wo performnce mesures re repored he sme de. 25

28 (0 ) > (0 ) > (0 ) > (0 ) > (0 ). The key in ech cse is he number of periods over which he rndom compension cn be spred. This is refleced by he fc h he effecive risk version, ra, increses over ime, i.e., he pplicion of he nnuiy morizion fcor A 0 is preferred o A 1 which is preferred o A Timely Relese of Purely Insurnce-Informive Repors The principl mus compense he gen for his effor cos nd his consumpion risk premium. Proposiion 4 esblishes h, wih exogenous incenive res, he gen s consumpion risk premium wekly decreses he erlier repors re relesed. The nlysis in his secion idenifies condiions under which erlier reporing does or does no hve sricly posiive benefi o he principl given h he selecs he opiml incenive res h will be implemened when he repors re received. Keep in mind h (4) implies h he induced cions depend on he incenive res for he cioninformive repors, bu hey re no influenced by he incenive res for purely insurnce informive repors. Consequenly, we cn exmine he principl s demnd for erly reporing of insurnce informive repors king he iming of nd incenive res for he cion-informive repors s exogenous. If he repor y is purely insurnce-informive, i.e., y = g, nd uncorreled wih he noise in ny prior repor given he informion de -1, hen he sochsiclly independen sufficien sisic c represenion s of y is whie noise, i.e., i is no cion-informive, nd L equls zero. Hence, Proposiion 4 implies h delying he reporing of y one period does no ffec he gen s consumpion risk premium nd, hus, i does no ffec he principl s expeced compension cos. More generlly, le p denoe he erlies de which repor y is issued wih which y is correled given he informion p de p-1, i.e., : p,p-1, 0, nd : J,J-1, = 0, for J < p. Hence, s p will depend on y, while s J will no depend on y for ll J < p. Hence, he erlies de which he informion in y migh be used (in n opiml 26

29 s full-commimen conrc bsed on 0 ) is de p. This esblishes h even wih PA, here is no vlue for reporing purely insurnce-informive performnce mesure unil, possibly, he relese of n cion-informive repor conining mesure h is correled wih he purely insurnce-informive mesure. Now consider purely insurnce-informive repor y = g h is correled wih prior cion- informive mesures given he informion de -1. Ineresingly, even hough y is purely insurnce- informive, i follows from he definiion of he sochsiclly independen sufficien sisics h s is cion-informive, i.e., here will be some p < wih :,-1,p 0 where y p is cion-informive. Since s is c cion-informive, Lemm 1 ses h L 0 (excep in knife-edge cses). Wih PA, Proposiion 4 hen implies h he principl cn reduce he consumpion risk premium pid o he gen by hving n erly rher hn ler repor of y. Proposiion 5: Suppose y is purely insurnce-informive bou fuure cion-informive performnce mesures. Then here is no vlue o he principl of hving n erly repor of y. On he oher hnd, if y is purely insurnce-informive bou prior cion-informive performnce mesures, hen i is sricly vluble for he principl o hve n erly repor of y wih PA, bu no wih PM. 6.4 Single Acion-De Exmple wih Opiml Incenives In Secion 6.1 we esblish h under PM, he iming of he repors is immeril. However, under PA, reporing dely increses he gen s consumpion risk (since less consumpion smoohing is fesible), excep for wh ppers o be knife-edge cse (which is v /v = - D in he exmple in Secion 6.2). Pnel B of Tble 1 repors he opiml incenive re for he insurnce-informive repor (relive o v ) nd he resuling consumpion risk premium for he gen (which forms pr of he principl s expeced i cos of compension). Here we see h i is opiml for he principl o se v = - Dv, if he insurnceinformive repor is relesed before or he sme ime s he cion-informive repor. This is 27 i

30 precisely he knife-edge cse from Pnel A. I represens no-cos insurnce, i.e., he gross incenive risk 2 represened by vy (wih vrince v ) is divided ino insurble nd uninsurble componens vdy ind vs = v (y - Dy i), respecively (wih vrinces v D nd v (1-D )). The insurble componen is efficienly borne by he risk neurl principl (by seing L i = 0, see Lemm 1). This elimines he risk verse gen s use of consumpion smoohing o miige h risk, nd leve him o use consumpion smoohing o miige he uninsurble risk wih he sme cion incenives (by seing L = v ). Consequenly, erly issunce of he insurnce-informive repor (relive o he repor de for y ) hs no vlue o he principl, i.e., (0 ) = (0 ) = (0 ) = (0 ), (0 ) = (0 ) = (0 ). Given h y is relesed wih or before y, he ler resolves he uninsured risk. The uninsured risk i mus be borne by he gen for incenive resons, bu he gen cn miige is cos hrough consumpion smoohing. In he firs se, he incenive risk is no resolved unil = 2, wheres in he second se i is resolved = 1, llowing he gen o smooh his consumpion risk over one more period. Hence, he differences in he risk premi for he wo ses re ribuble enirely o differences in he sr de nd he lengh of he nnuiy he gen cquires. The ler difference is lso inerpreble s difference in he effecive risk version, i.e., = ra 2 versus, = ra Compring 0 o 0, 0, nd 0 is lso insrucive. Erlier reporing of y i does no reduce he risk premium if i is repored wih or before y, bu here is reducion if y i is repored wih or before y insed of fer, i.e., (0 ) = (0 ) > (0 ) > (0 ). The gen s cion choice is only influenced by v, so h v is seleced solely o reduce he gen s consumpion risk premium. The mximum insurnce, i.e., v = - Dv, is implemened wih 0 (nd 0 ) since y i is uninformive bou he uninsurble risk (g - Dg i). However, less insurnce is implemened i i 28