Incentives from stock option grants: a behavioral approach

Transcription

1 Incentives from stock option grnts: behviorl pproch Hmz Bhji To cite this version: Hmz Bhji. Incentives from stock option grnts: behviorl pproch. 6th Interntionl Finnce Conference (IFC)- Tunisi, Mr 2011, Hmmmet, Tunisi. <hlshs > HAL Id: hlshs Submitted on 21 Mr 2012 HAL is multi-disciplinry open ccess rchive for the deposit nd dissemintion of scientific reserch documents, whether they re published or not. The documents my come from teching nd reserch institutions in Frnce or brod, or from public or privte reserch centers. L rchive ouverte pluridisciplinire HAL, est destinée u dépôt et à l diffusion de documents scientifiques de niveu recherche, publiés ou non, émnnt des étblissements d enseignement et de recherche frnçis ou étrngers, des lbortoires publics ou privés.

2 Incentives from stock option grnts: behviorl pproch Hmz BAHAJI Université de Pris Duphine, DRM Finnce (CEREG) Jnury 2011 Abstrct This pper exmines the incentives from stock options for loss-verse employees subject to probbility weighting. Employing the certinty equivlence principle, I built on insights from Cumultive Prospect Theory (CPT) to derive continuous time model to vlue options from the perspective of representtive employee. Consistent with growing body of empiricl nd experimentl studies, the model predicts tht the employee my overestimte the vlue of his options in-excess of their risk-neutrl vlue. This is nevertheless in strk contrst with common finding of stndrd models bsed on the Expected Utility Theory (EUT) frmework tht options vlue to risk-verse undiversified employee is strictly lower thn the vlue to risk-neutrl outside investors. In prticulr, I proved tht loss version nd probbility weighting hve counterviling effects on the option subjective vlue. In ddition, for typicl setting of preferences prmeters round the experimentl estimtes, nd ssuming the compny is llowed to djust existing compenstion when mking new stock option grnts, the model predicts tht incentives re mximized for strike prices set round the stock price t inception. This finding is consistent with compnies ctul compenstion prctices tht stndrd EUT-bsed models hve difficulties ccommodting their existence. The pper lso exmines the reltionship between risk tking incentives nd stock options nd finds tht n executive who is subject to probbility weighting my be more prompted thn risk-neutrl executive to ct in order to increse the firm s ssets voltility. JEL Clssifiction: J33, J44, G13, G32, M12 Keywords: Stock options, Cumultive Prospect Theory, Incentives, Subjective vlue. DRM-Finnce, Université de Pris Duphine, Plce du Générl du Lttre de Tssigny, , Pris Cedex 16, Frnce. E-mil: hbhji@yhoo.fr. The uthor is grteful to Clotilde NAPP, Jen-frnçois CASTA, Jcques HAMON, Frht SELMI nd Khlid ARDEM for their vluble comments. 1

3 Introduction Insted of n incresing interest for restricted stock nd performnce unit plns, the 2006 Hewitt Assocites Totl Compenstion Mesurement survey hs reveled tht stock options re still the most prevlent long term incentive vehicle 1. The stted rgument for the lrge use of executive stock options is tht they lign the interests of executives nd shreholders since they provide incentives for the mnger to ct in order to increse the firm vlue. The use of stock options hs even overtken the trditionl ren of executive popultion. Actully, firms compenstion prctices show tht stock options re issued to rewrd non-executive employees s well. This my seem puzzling t first sight in tht the gency rgument underlying the use of stock options is no more relible under the nonexecutive employees view becuse the formers re unlikely to influence the firm vlue by their decisions (Splt, 2008). Nevertheless, issunce of non-executive stock options results obviously from n greement between the firm nd the employee upon the terms of the compenstion involving stock options, which mens tht employees my be interested in stock options. In order to figure out the reson why stock options my be ttrctive to employees it is crucil to ssess the utility they receive from them. Moreover, understnding how employee evlutes its stock options (i.e. their subjective vlue) llows ssessing their incentive power nd the implied employee behvior in terms of risk tking. Most of the theoreticl literture on stock options relies on the Expected Utility theory (EUT henceforth) frmework to derive models of option vlue from the employee perspective. These models predict tht the nontrnsferbility of the options nd the hedging restrictions fced by the employee mke him vlue his options below their issunce cost born by the compny (i.e. their risk neutrl vlue). A seminl work by Lmbert et l (1991) hs introduced generl theoreticl frmework to nlyze the vlution of equity bsed compenstion contrct from mnger s perspective. The uthors studied stock option contrcts s specific exmple. They used the certinty-equivlence principle to derive the so-clled subjective vlue model. Their model ssumes tht the mnger is risk-verse gent with power utility function nd tht he integrtes his stock options in his globl welth ssessment process. They proved tht the subjective vlue decreses in risk version nd increses in the mnger diversifiction with respect to compny specific risk. Moreover, they showed tht stock options my not provide incentives for risk-verse mnger to select ctions tht increse the vrince of the compny stock price. Hll nd Merphy (2000, 2002) followed the sme pproch to nlyze the risk-djusted vlue of executive stock options long with their py-to-performnce sensitivity. Their mjor findings provide n economic rtionl to observed equity bsed compenstion prctices while perhps troubling from n efficiency perspective. Minly, they showed tht py-toperformnce incentives from stock options, when these re grnted s n dd-on to existing 1 80% of the responding prticipnt compnies to the survey hve reported tht stock option grnts represent in 2006 on verge bout 54% of their globl long-term incentives. 2

4 compenstion, re typiclly mximized by setting their exercise price round the stock price t inception. In contrst, when options grnts re ssumed to be ccompnied by reductions in csh compenstion, their model predicts tht restricted stocks provide much higher incentives for executives to increse the firm vlue nd my then be preferred to stock options. Henderson (2005) hs built on the previous works to exmine the effect on vlution nd incentives of mrket risk nd firm-specific risk. She proposed continuous time utility-bsed model to vlue stock options from the mnger s perspective. A prominent ssumption in his model is llowing the executive to invest the reminder of his welth (excluding stock options) in the mrket portfolio 2 nd risk-free sset. In prticulr, her results suggest tht stock options do not provide incentives to increse totl risk, but the bet of the compny insted. She lso finds incentives decrese in firm-specific risk whilst they my either decrese or increse in mrket risk depending on other fctors. In ddition, similr to Hll nd Murphy (2002), the uthor s model supports the grnt of restricted stocks rther thn stock options under efficient contrcting frmework. A common finding of the EUT-bsed models is tht the stock options vlue to risk-verse undiversified employee is strictly lower thn the vlue to risk-neutrl well diversified outside investors. This is nevertheless in strk contrst to severl surveys nd empiricl findings documenting tht employees my vlue their options bove their risk-neutrl vlue (i.e. the so-clled Blck Scholes vlue). Lcker s nd Lmbert s (2001) reserch bsed on survey of Knowledge@Whrton reders reveled tht mngers vlue their options substntilly bove the Blck Scholes vlue (BS henceforth). Moreover, the survey reveled tht younger mngers hve the most upwrd bis in the vlues they perceive from their options. According to the uthors, these results suggest tht mngers do not fully understnd the vlutions of stock options or possibly their incentive effects. Another rgument supporting mngers my tend to overestimte their options vlue is provided by Yermck (1997). He rgues tht to the extent tht compny executives hve superior informtion regrding compny prospects nd cn time their option grnts ccordingly, they my ctully vlue options higher thn would outside investors. Furthermore, relying on behviorl theories, recent reserches hve found evidence on employees drwing on heuristics nd subject to psychologicl bis while vluing their options, which my led them to overestimte the vlue of their options. For instnce, Devers et l.(2007) study exmines the effects of endowment nd loss version on mngers subjective stock-option vlution. Their min findings show tht mngers endow vlue from grnted unexercisble options in wy tht they vlue them in excess of their objective vlue (i.e. BS vlue). In the sme vein, Swers et l.(2006) drew conclusions from n experiment involving MBA students consistent with the view tht loss-verse mngers subjectively overvlue stock options reltive to 2 Prior works by Detemple nd Sundresn (1999) nd by Ci nd Vijh (2004) hve used trees pproch tied to specific numericl schemes to study utility bse vlution of executive stock options when the executive cn invest in the Mrket portfolio. 3

5 restricted stocks nd, therefore, will be less risk-seeking when wrded with stock options nd more risk-seeking when endowed with restricted stocks. Moreover, stndrd normtive models fil to predict stock options s prt of the compenstion contrct. Severl quntittive studies tking plce in principl-gent frmework showed tht EUTbsed models predict optiml compenstion contrcts which do not contin convex instruments like stock options (Holmstrom nd Milgrom, 1987; Dittmnn nd Mug, 2007). Consistent with these findings, studies focused on the effect of stock option nd restricted stock grnts on mngeril effort incentives (Jenter, 2001; Hll nd Murphy, 2002; Henderson, 2005) conclude tht stock options re inefficient tools for creting incentives for risk-verse mngers when they re grnted by men of n offset of csh compenstion. This pper nlyzes the vlution of stock options nd their incentives effect to n employee exhibiting preferences s described by Cumultive Prospect Theory (Tversky nd Khnemn, 1992). It ims to propose n lterntive theoreticl frmework for the nlysis of py-to-performnce sensitivity of equity-bsed compenstion tht tkes into ccount number of prominent ptterns of employee behvior tht stndrd EUT cnnot explin. Specificlly, the key ssumption underlying this work proceeds from Cumultive Prospect Theory (CPT herefter) nd sttes tht the employee choices under risk disply four min fetures. Firstly, insted of ssessing outcomes bsed on their contribution to his finl welth, the employee ssesses outcomes utility over gins nd losses determined reltive to reference level or benchmrk. The second feture is loss-version. It conveys the experimentl evidence tht people re more sensitive to losses thn they re to gins. In other words, losses loom lrger thn gins. The third feture consists in the principle ccording to which people choices disply diminishing sensitivity to incrementl gins nd losses. The lst piece is the nonliner probbility processing. It involves the well estblished tendency of individuls to overweight smll probbilities of lrge gins 3. I drew on this theoreticl frmework to derive continuous time model of the stock option subjective vlue using the certinty equivlence principle. I then performed sensitivity nlyses with respect to preferences-relted prmeters nd found tht loss version nd probbility weighting hve counterviling effects. In prticulr, I showed tht the more the employee is loss verse, the less would the option worth for him. In ddition, I proved tht the option subjective vlue is incresing in probbility weighting degree. My nlyses lso show tht, for given level of option moneyness, the subjective vlue of the option my lies strictly bove the BS vlue when the effect of probbility weighting tends to dominte tht of loss-version. This results led to the conclusion tht the lottery-like nture of stock-options, combining lrge gins with smll probbilities, my mke them ttrctive to employees subject to probbility weighting which is 3 There exists lrge literture comprising empiricl nd experimentl pplictions of overweighting smll winning probbilities. This literture include notbly Thler nd Ziemb (1988), Kchelmeier nd Sheht (1992), Cook nd Clotfelter (1993), Loughrn nd Ritter (1995) nd Jullien nd Slnie (2000). 4

6 consistent with the proposition tht employee option vlue estimte my exceed the BS vlue (Lmbert nd Lrcker, 2001 ; Hodge et l., 2006 ; Swers et l., 2006 ; Hllock nd Olson, 2006 ; Devers et l., 2007). Furthermore, this work elbortes on incentives from stock options nd on some implictions in terms of design spects. Following previous reserches, I defined incentives s the first order derivtive of the subjective vlue with respect to the stock price. A numericl nlysis of the incentive function shows tht stock option incentive effects re incresing in employee s degree of probbility weighting nd my even lie bove incentives for risk-neutrl individul. Moreover, I considered the incentive effects of setting the strike price of the option bove or below the stock price t inception. In this nlysis, I relied on Hll s nd Murphy s (2002) methodology in solving for the exercise price tht mximizes incentives holding constnt the compny cost of grnting the options. Similr to their pproch, I explored two situtions. In the first sitution, I ssumed tht the firm is precluded from chnging ny component of the employee compenstion pckge. In this cse, the model predicts tht, when the effect of probbility weighting previls (i.e. domintes tht of the loss-version), incentives could be infinitely incresed by grnting more nd more options t higher nd higher strikes, which suggests tht employee would be much more interested in premium options insted of discount options. However, when the loss-version effect domintes, the model predicts tht incentives re mximized within strike price rnges tht my include the grnt dte stock price depending on the level of loss-version. In the second sitution, the compny is ssumed to be llowed to djust existing compenstion when mking new stock option grnts. For typicl setting of preferences prmeters round the experimentl estimtes from CPT (Tversky nd Khnemn, 1992), the model predicts tht incentives re mximized for strike prices set round the stock price t inception, which is consistent with compnies ctul compenstion prctices. Additionl nlyses suggest lso tht loss-verse employees who re not subject to probbility weighting, or even with very week degrees of probbility weighting, receiving options t high exercise price would willingly ccept cut in compenstion to receive insted deep discount options or restricted shres for those of them displying more lossversion. This result is brodly consistent with Hll nd Murphy (2002) nd Henderson (2005) findings for non-diversified risk-verse employees. The lst item discussed in this pper is the risk tking incentives implied by stock option grnts. Similr to previous studies, I defined incentives for risk tking s the first order derivtive of the subjective vlue from the perspective of CPT-employee. The first essentil result from this nlysis is tht incentives for risk tking re incresing with probbility weighting degree. In fct, the probbility weighting leverges the positive effect of voltility on the subjective vlue through the convexity of the option pyoff. Another importnt finding from the model is tht n optioncompensted CPT-executive subject to probbility weighting my be more prompted thn riskneutrl executive to ct in order to increse ssets voltility. Moreover, consistent with findings from 5

7 EUT-bsed models, the CPT model predicts tht loss-verse executive who is not subject to probbility weighting my not prefer n increse in the vrince of the compny stock price return. This work is motivted by recent empiricl nd theoreticl reserches on employee compenstion incorporting CPT-bsed models. These models hve proved successful in explining some observed compenstion prctices, nd specificlly the lmost universl presence of stock options in the executive compenstion contrcts tht the EUT models hve difficulties ccommodting their existence. Therefore, they hve dvnced CPT frmework s promising cndidte for the nlysis of equity-bsed compenstion contrcts. This literture includes in prticulr Dittmnn et l. (2008) who developed stylized principl-gent model tht explin the observed mix of restricted stocks nd stock options in the executive compenstion pckges. It lso includes Splt (2008) who used the CPT frmework, nd especilly the probbility weighting feture, to prove tht stock options my be ttrctive to loss-verse employee subject to probbility weighting. He then explined the puzzling phenomenon tht riskier firms re prompted to grnt more stock-options to non-executive employees. This rticle proceeds s follows. The first section describes the fetures of the model of the stock option vlue from the perspective of representtive employee with preferences s described by CPT. Throughout this pper, we will refer to this employee s CPT employee. This section provides lso numericl nlyses on the model sensitivity to preferences-relted prmeters. The next section introduces incentive effects of stock options for CPT-employee nd exmines some design implictions in terms of strike price setting. The risk tking incentives question is explored in the third nd lst section. Appendices provide proofs of the propositions in the first section. 1. Stock option vlue from CPT-employee perspective. In this section, I develop bse-cse model for nlyzing the vlue of the stock-option contrct from the perspective of representtive employee with CPT-bsed preferences (subjective vlue henceforth). Specificlly, I ssume tht the employee is grnted Europen cll option on the compny s stock, denoted by S, with mturity dte t T nd strike price K. These re the trditionl fetures of executive stock options s reported in Johnson nd Tin (2000) nd used by prior studies focused on stock option incentives (Lmbert et l., 1991; Hll nd Murphy, 2002 ; Henderson, 2005). Often in prctice, stock options re Bermudn-style options. Thus, my model relies on nïve setting in tht it ignores complictions relted to erly exercise or forfeitures Theoreticl frmework Stock-option contrct. 6

8 The stock option contrct is issued in t 0. The contrct pyoff t expiry, t T, is h S K. T T I mke the ssumption tht the employee is not llowed to short-sell the compny stock nd tht he cn ern the risk-free rte r from investing in riskless sset. Moreover, the price dynmic of the stock is given by geometric Brownin motion represented by the following SDE: ds r q S dt S dz (1) t t t t Where Z is stndrd Brownin motion with respect to the probbility mesure IP. t respectively the totl vrince of the stock price returns nd the dividend yield Risk preferences. 2 nd q re Following Tversky nd Khnemn (1992), I consider tht, to ech gmble with continuous rndom outcome y ssigns the vlue: IR IR which the probbility density function is denoted by f y, the employee E y v y d f y (2) Note tht the expecttion E. is function of two distinct functions. The first function,. v, clled the vlue function, is ssumed of the form: v y y ; y, where 0 1nd 1 (3) y ; y This formultion hs some importnt fetures tht distinguish it from the stndrd utility specifiction. First, the utility is defined over gins nd losses ssessed bsed on the reference point denoted by. This ide ws proposed first by Mrkowitz (1952). As pointed out by Brberis nd Thler (2003), it fits nturlly the wy people perceive ttributes, in everydy life, reltive to their previous levels rther thn in bsolute fshion. The second importnt feture is the shpe of the vlue function. While it is convex over losses, it is concve over gins, which represents the observtion from psychology tht people re risk verse over gins nd become risk-seeking over losses. Moreover, the vlue function hs kink t the origin introduced by the prmeter 1. This feture, known s loss-version, gives higher sensitivity to losses compred to gins. Finlly, outcomes re 7

9 treted seprtely from other components of welth, which reflects the well-documented phenomenon of nrrow frming 4 (Thler, 1999). The second function, b., probbilities, represented by the cumultive probbility function F decision weights ccording to:, is clled the weighting function. It pplies to cumultive., in order to trnsform them into b, F y 1 F y F y (1 F y ) b F y F y (1 F y ) b b b 1 1 ; ; y y where nd 0.28 b 1 5 (4) This function stnds for nother piece of CPT, which is the nonliner trnsformtion of probbilities. Specificlly, it cptures experimentl evidence on people overweighting smll probbilities nd being more sensitive to probbility spreds t higher probbility levels (e.g. jump in probbility from 0.7 to 0.9 would be more striking to individul thn n equl-sized jump from 0.1 to 0.3). The degree of probbility weighting is controlled seprtely over gins nd losses by the weighting prmeters nd b respectively. The more these prmeters pproch the lower boundry t 0.28 the more the tils of probbility distribution re overweighted. For instnce, when b 1, probbility weighting ssumption is relxed. For simplicity, these prmeters re ssumed to be equl (i.e. b ) nd the weighting function will be denoted. in the rest of this pper. Finlly, it is worth mentioning tht this function (i) infinitely-overweights infinitesiml probbilities since p lim p 0 p 1 p nd (ii) infinitely-underweights ner-one probbilities in tht lim p 1 1 p 1.2. Stock option subjective vlue The model.. 4 Nrrow frming refers to people tendency to tret individul gmbles seprtely from other portions of welth. It stnds for n importnt feture of Mentl Accounting, the mentl process by which individuls formulte problems for themselves (e.g. the wy they code outcomes). Mentl ccounting is cptured in CPT vi the nonliner shpe of the vlue function. 5 The lower boundry t 0.28 is technicl condition to insure tht, b p is positive over ]0,1[ s required by the following first order condition: p p p p p 8

10 In order to estimte the subjective vlue of the stock option contrct described bove, I use the certinty equivlent pproch similr to tht dopted by Splt (2008). In prticulr, this vlue is defined s the csh mount, C,, tht leves the employee indifferent between this mount nd the uncertin pyoff of the contrct, h, irrespective of the composition of the reminder of his privte T welth. Formlly, C is the solution of the following eqution:, rt, IP T v C e E v h = IR T T v h d F S (5) The left-hnd side of the eqution bove represents the benefit for the employee from receiving the csh mount C, insted of the stock option contrct t the inception of the ltter. This mount is ssumed to be plced into the risk-free sset over the whole lifetime T of the stock option contrct. The other side of the eqution gives the expected utility to the employee from receiving the risky pyoff implied by the vlue function probbility mesure IP. v.. Here, the expecttion E IP relies on the trnsformed Let E, represent the expecttion in the right-hnd side of (5). It follows from (5) nd (3) tht C, 1 rt E, 0, e if E, 1 1 rt E,, e otherw ise (6) Where E, should write: (6.1),,,, E I I I With: x 1 I, g x K u du x dx (6.2) l1 l1 x 2 I K g x, u du x dx (6.3) l2 9

11 l2 3 I, x dx (6.4) g ( x) Se 2 r q T T x 2 (6.5) l 1 2 S ln r q T K 2 T (6.6) l 2 2 S ln r q T K 2 T (6.7) Here of. is the Gussin density function nd p. which is defined over 0,1 s follows: p p is the first-order prtil derivtive p p p 1 p 1 p ; p x dx 1 p (1 p) p 1 p l 1 p p p p ; p x dx 1 p (1 p) p 1 p l1 (6.8) Reference point set up. Although CPT specifies the shpe of the vlue function round the reference point, it does not provide guidnce on how people set their reference points. Neither does most of the psychologicl literture relying on the ssumption ccording to which the reference point is the Sttus quo. Insted, this literture dmits both the existence nd the importnce of non-sttus quo reference points since there re situtions in which gins nd losses re coded reltive to n expecttion or spirtion level tht differs from the sttus quo (Khnemn nd Tversky, 1979). Reserch on the Disposition Effect 6 hs pid greter ttention to non-sttus quo reference points (Shefrin nd Sttmn, 1985 ; Heisler, 1998 ; Oden, 1998 ; Gneezy, 1998). These studies hve found 6 Disposition effect is term coined by Shefrin nd Sttmn (1985) to refer to the tendency of individul investors to hold loser stocks nd sell winner stocks defined reltive to purchse price reference point. 10

12 strong evidence tht reference points re set in dynmic fshion. In prticulr, Gneezy (1998) documented tht mxim re more effective predictors thn re purchse prices. More specificlly, in studding exercise behvior of stock options holders, Huddrt nd Lng (1996), Heth et l (1999) nd Bhji (2009) hve shown empiriclly tht exercise ctivity is linked to shre price historicl mxim. These findings re consistent with employee exercise decisions tke into ccount the level of shre price with respect to non-sttus quo reference points. In principle, employee would updte the reference point in wy tht fits with his own expecttions regrding the underlying shre price t expiry. Intuitively, the employee could estimte the intrinsic vlue of the option bsed on his future shre price forecsts s he cn rely on the Blck Scholes vlue of the option disclosed by the firm. Following Splt (2008), I consider tht the reference point prmeter in the model,, is the BS vlue 7. Beyond the rgument of empiricl evidence on employee exercise behvior depending on non-sttus quo reference points, this ssumption is supported by firms common prctices in terms of stock option compenstion. Most of stock-option designers use the BS model in order to estimte the vlue of stock options s constituents of the totl compenstion pckge. This vlue is usully nnounced to the employee t the inception of the options. Moreover, the BS model is recommended in the FASB nd the IASC guidelines for determining the fir vlue of stock options (i.e. the mount n outside investor, with no hedge restrictions, would py for the option) tht needs to be disclosed in the finncil sttements. These sttements, comprising the BS vlue of the stock options, re provided to shreholders s well s stkeholders including employees The impcts of preferences-relted prmeters: numericl nlysis. To provide concrete outline on the profile of the subjective vlue yielded by the model reltive to the risk neutrl vlue profile, I performed numericl nlysis 8 of the vlue of 4-yer cll option ( T 4 ) with strike price K 100. For the remining option-relted prmeters, the figures were computed ssuming no dividend pyments ( q 0% ), 30% nd r 3%. Moreover, I set the curvture prmeter of the vlue function ( ) nd the loss version coefficient ( ) to respectively 0.88 nd 2.25 bsed on experimentl estimtes from CPT (Tversky nd Khnemn, 1992). 7 To be more precise, the vlue used here is the expecttion of the option pyoff t expiring yielded by the BS model (i.e. the nondiscounted BS vlue). Consistent with this specifiction, the probbility mesure IP used to derive the subjective vlue in (6) is the risk neutrl probbility mesure. Moreover, ignoring the probbility weighting feture (i.e. 1 ), this setting llows the subjective vlue implied by the model to converge towrds the risk neutrl vlue (i.e. BS vlue) when the preferences of the employee tend to risk neutrlity (i.e. 1 ). 8 The integrls in (6.2), (6.3) nd (6.4) were computed numericlly. 11

13 Furthermore, in order to clibrte the probbility weighting function, I used three different vlues of the prmeter within the rnge of vlues estimted in the experimentl literture 9. Figure 1 depicts the option vlue s function of the stock price. The three blue curves represent the vlue profiles from the perspective of three CPT employees with the sme vlue function nd different degrees of probbility weighting. At first sight, depending on the degree of probbility weighting nd the option moneyness ( S K ) the subjective vlue could lie either under or below the BS vlue. In contrst, stndrd EUT-bsed models predict tht the option vlue from risk-verse employee perspective is systemticlly lower thn the risk-neutrl vlue (Hll nd Murphy, 2002; Henderson, 2005). These preliminry results re consistent thought with some empiricl findings suggesting tht frequently employees re inclined to overestimte the vlue of their stock-options compred with the BS vlue (Lmbert nd Lrcker, 2001 ; Hodge et l., 2006 ; Swers et l., 2006 ; Hllock nd Olson, 2006 ; Devers et l., 2007). Figure-1: Option vlue ginst the stock price. This figure is plot of the subjective vlue computed under different probbility weighting prmeters (the blue curves). It illustrtes the profile of the subjective vlue compred to tht of the risk neutrl vlue (red curve). The Prmeters used re: T=4; K=100; σ=30%; r=3%; q=0%; =2.25; α=0.88. Figure-2: Sensitivity to probbility weighting. This figure is plot of the prtil derivtive of the subjective vlue with respect to ginst both nd the stock price voltility σ. It exhibits the locl effect of probbility weighting given the pyoff distribution skewness cptured by σ. The derivtive ws computed numericlly bsed on the following prmeters: T=4; K=S=100; r=3%; q=0%; =2.25; α= Tversky nd Khnemn (1992) got 0.65 on verge (=0.61 for gins nd b=0.69 for losses). These results re corroborted by Abdelloui s (2000) findings (=0.60 for gins nd b=0.70 for losses, hence n verge of 0.65). In ddition, Cmerer nd Ho (1994) obtined =0.56 for gins wheres Gonzles nd Wu (1996) nd Bleichrodt nd Pinto (2000) found =0.71 nd =0.67 respectively. 12

14 The results presented in Figure 1 show tht the option subjective vlue is incresing in probbility weighting degree (i.e. decresing in prmeter ). Actully, given the symmetric profile of the option pyoff, the expecttion E, in the subjective vlue formul (6) is positively ffected by the emphsis put on the til of the pyoff distribution which is governed by the prmeter : the lower the more overweighed will be smll probbilities nd the more underweighted will be medium to lrge probbilities. This lottery-like nture of n option, combining lrge gins with smll probbilities, my mke it ttrctive to CPT employee subject to probbility weighting. This preliminry outcome leds to proposition 1 which stts tht: Proposition 1: the vlue of the stock option contrct to CPT- employee is incresing with respect to his degree of probbility weighting (i.e. is decresing with respect to ). (Proofs re provided in ppendix A) Moreover, the effect of probbility weighting is expected to increse with the skewness of the distribution of the underlying stock price, which is cptured by the voltility prmeter given the Log-normlity ssumed in the model. In order to show tht, I performed numericl nlysis of the sensitivity of the subjective vlue to the probbility weighting degree s function of the voltility nd the prmeter. This sensitivity is defined s the prtil derivtive of subjective vlue with C respect to (, ). The results re reported in figure 2 in the form of grph. It shows tht the sensitivity to prmeter is negtive nd loclly decresing in voltility. Tht mens tht the more the shre price is voltile the more the option will be ttrctive for CPT employee subject to probbility weighting. This supports Splt s (2008) findings tht the effect of probbility weighting provides n economic rtionle to riskier firms (i.e. more voltile firms) for grnting more stock options to nonexecutive employees. Furthermore, I investigte the effect of loss version on the subjective vlue. The vrible of interest here is. In n nlogicl sense with the EUT frmework, the option vlue from the perspective of loss-verse employee is expected to decrese with his degree of loss-version. To C, verify this I computed numericlly the first order derivtive with respect to ( ) crossed over vrious levels of nd moneyness, rnging from 0.05 to 1 nd from 5% to 200% respectively. The outcome is reported in figure 3. It shows tht the sensitivity to loss version is negtive nd loclly decresing in moneyness. Tht mens tht the more the employee is loss verse, the less would the option worth for him. This conclusion is tken up in proposition 2 herefter: 13

15 Proposition 2: the vlue of the stock option contrct to CPT- employee is decresing with respect to his degree of loss-version (i.e. decresing function of λ). (Proofs re provided in ppendix B) Figure-3: Sensitivity to loss version. This figure is plot of the prtil derivtive of the subjective vlue with respect to λ ginst both λ nd the stock price S. It exhibits the locl effect of loss version given the option moneyness. The derivtive ws computed numericlly bsed on the following prmeters: T=4; σ=30%; K=100; r=3%; q=0%; =0.65; α=0.88. Figure-4: Sensitivity to the curvture prmeter. This figure is plot of the prtil derivtive of the subjective vlue with respect to α ginst both α nd the stock price S. It exhibits the locl effect of the curvture prmeter given the option moneyness. The derivtive ws computed numericlly bsed on the following prmeters: T=4; σ=30%; K=100; r=3%; q=0%; =0.65; λ =2.25. I performed similr nlysis in order to get view on the effect of the curvture prmeter. In the sme wy, I ssessed loclly the first order derivtive with respect to within rnge of vlues from 0.05 to 1 nd using shre prices rnging from 10 to 200. Figure 4 shows tht this derivtive is loclly incresing with the option moneyness for vlues of bove sy It lso shows brodly tht the subjective vlue is n incresing monotone function of over rnge of vlues round the experimentl estimte of 0.88 (from 0.7 to 1) irrespective of the option moneyness. 2. Incentives from stock-options. Stock-options re incentive tools used within principl-gent reltionship to lign the interests of the gent (employee) on those of the principl (shreholders). The shreholders grnt stock options in 14

16 order to provide the employee with incentives to mke efforts tht enhnce the vlue of the firm, nd thus their own welth. Indeed, ssuming tht employees re wre of how their ctions ffect the shre price, option holdings will prompt them to mke efforts tht increse shre price. Therefore, the incentive from single option grnt will depend on the degree of the sensitivity of the subjective vlue to the stock price The incentive mesure. Following Jensen nd Murphy (1990), Hll nd Murphy (2000, 2002) nd others, I defined the C, incentive effect s the first order derivtive of the subjective vlue with respect to shre price ( S which defines how the vlue from the employee perspective chnges with n incrementl chnge in the stock price. As we cn see in figure 5, which is plot of C, S for three vlues of, incentives re gretest for in-the-money 10 options. Also, the incentive effect is incresing with the degree of probbility weighting. Another observtion tht cn be mde from figure 5 is tht, for sufficiently high level of probbility weighting, the option cn give much more incentive to increse stock price thn is reflected by the BS delt. This is consistent with my previous finding tht the subjective vlue cn overstte the BS vlue. Note tht, consistently with the EUT-bsed models, when the employee process probbilities in liner wy ( 1 ) - which mens tht the probbility weighting ssumption is relxed nd only loss-version mtters - the incentives lie strictly under the BS delt whtever the level of the option moneyness. Tht is to sy tht the options re less ttrctive for both risk-verse employee nd loss-verse employee who re not subject to probbility weighting. The nlysis in figure 6 supports this. It shows tht, with hold constnt t 0.65, incentives re decresing in lossversion. Conversely, with loss-version prmeter set to 1, which mens tht the loss-version effect is neutrlized, incentives for representtive loss-neutrl employee, with degree of probbility weighting equl to experimentl estimte of 0.65, overstte the BS delt. ) 10 The terminology t-the-money is referring to stock-options with n exercise price equl to the stock price t inception. The expressions Out-of- the-money nd In-the-money re lso used throughout the pper to refer to options with strike price respectively below nd bove the grnt dte stock price. 15

17 Figure-5. The effect of probbility weighting on stock-option incentives. This figure is plots of the prtil derivtive of the subjective vlue with respect to stock price S ginst the option moneyness. It exhibits the combined effects of probbility weighting nd option moneyness on incentives. The derivtives were computed numericlly ssuming three different levels of probbility weighting (see the legend bove) nd holding the remining prmeters constnt t the following vlues: T=4; σ=30%; K=100; r=3%; q=0%; λ=2.25; α=0.88. Figure-6. The effect of loss-version on stock-option incentives. This figure is plots of the prtil derivtive of the subjective vlue with respect to stock price S ginst the option moneyness. It exhibits the combined effects of loss-version nd option moneyness on incentives. The derivtives were computed numericlly ssuming three different levels of loss-version (see the legend bove) nd holding the remining prmeters constnt t the following vlues: T=4; σ=30%; K=100; r=3%; q=0%; =0.65; α= Implictions for stock option design: optiml strike price. Setting the strike price of stndrd stock options boils down to defining the threshold ginst which the performnce is ssessed nd, consequently, to determining the likelihood of finl pyout. As we hve just noticed from Figure 5 nd Figure 6, incentives increses in the option moneyness nd, equivlently, decreses with strike price. In prllel, from the shreholders perspective, grnting inthe-money options is much more costly thn grnting out-of-the money or t-the-money options (recll Figure 1). This leds the firm to trde-off to mke when setting the exercise price of the options in the sense tht, holding his cost unchnged, she could either grnt fewer options t low strike price or increse the grnt size t higher exercise price. I relied on the methodology from Hll nd Murphy (2002) in figuring out the optiml exercise price stisfying the double purpose of mximizing incentives nd holding constnt the firm s cost of grnting options. I lso considered two situtions. In the first sitution, the firm does not djust the existing compenstion pckge when grting dditionl stock options. Thus, these dditionl options re considered s pure dd-on to existing compenstion. In the second sitution however, the employee nd the firm re llowed to brgin efficiently over the terms of the compenstion. Thus, the 16

18 firm is ssumed to fund dditionl options by n djustment to other compenstion components tht leves the employee indifferent between his initil pckge nd the new pckge including the dditionl grnt Sitution 1: Option grnt without compenstion offset. When the firm is not llowed to djust the existing compenstion pckge, the optiml strike price tht mximizes incentives for given mount of firm cost is the solution to the following constrined optimiztion problem: nc K, m x K S such tht n K c (7) Where K nc, S K denotes the incentives from receiving n options with strike price K nd the per-unit cost or the BS vlue of one option. The optimiztion problem in (7) is solved numericlly by vrying the prmeter K. First, the BS vlue is computed for given K which enble to determine the grnt size n tht leves the firm cost nc, constnt t c in ccordnce with the constrint in (7). Then, the cost function S K is ssessed bsed on n nd K. This procedure is reiterted recursively until the optiml vlue of K is found. I set the totl cost for the firm t c (hence grnt of n 1000 t-the-money options bsed on the retined option-relted prmeters). As reported in Figure 7, I vried the preferencesrelted prmeters nd within specific rnges in order to figure out the conditions under which n optiml strike my exist nd the robustness of the model predictions. The chrt on the left-hnd side exhibits the totl incentives from the grnt s function of the strike price nd the probbility weighting prmeter with the loss-version held constnt t 2.25 nc, the monotony of the totl incentives function ( S K. It shows tht the convexity nd ) shift with the degree of probbility weighting: while it is concve nd non-monotone over the rnge of strike prices for low levels of probbility weighting ( ) it becomes convex nd monotoniclly incresing for higher levels. Moreover, for higher level of voltility, probbility weighting is expected to ffect more deeply the shpe of totl incentives function (recll Figure 2). Specificlly, when the effect of probbility weighting previls (i.e. domintes tht of loss-version), the model predicts tht incentives could be infinitely incresed by grnting more nd more options t higher nd higher strikes. This result suggests tht, if the employee is subject to degree of probbility 17

19 weighting tht could potentilly led him to overstte the vlue of the option in excess of the BS vlue, he would sk for the mximum number of options (s much options s the compny could possibly grnt him) t their (highest) brekeven strike: the higher the strike price, the more smll probbilities of lrge gins there re, nd the more vluble become options to him. Furthermore, this mens tht such n employee would be much more interested in premium options insted of discount options. Actully, lthough this result is in strk contrst with the common stock-bsed compenstion prctices tht consist in grnting brodly t-the-money options, it shouldn t be rejected without tking into ccount the other fctors restricting the number of options the compny could issue irrespective of their strike prices, such like cpitl dilution limits. In ddition, s we cn notice from the plot in the middle of Figure 7, the effect of probbility weighting on totl incentives could be offset by n incresing loss-version. This time is vried nd held constnt t 0.65, which meets the experimentl estimte. When the loss-version effect domintes, the totl incentives function tends to non-monotone concve shpes, which mens tht it dmits finite extrems. In prticulr, when probbility weighting effect is neutrlized by setting 1 - in other words, the employee is not subject to probbility weighting - totl incentives curve tends to fltten round the optimum (see the green curves on the right-hnd side plot). This suggests tht there exists rnge of strike prices tht yield qusi-similr incentives. For instnce, with loss-version set t 2.25, optiml strike (s % of stock price t inception) is bout 142% nd lies within rnge of vlues from 121% to 153% tht yield totl incentives of bout 530 (rounded figure). With 6 the optiml strike drops by lmost 30% (to 112%) nd totl incentives re flt round the mximum vlue of 450 within the rnge from 91% to 111%. Note tht these results re consistent with Hll s nd Murphy s (2000, 2002) findings relying on the EUT frmework. Consistently, their work documented tht incentive-mximizing strike price exists when the options re considered s n dd-on to existing compenstion. In ddition, they showed tht totl incentives re reltively flt round the mximum nd tht optiml strike price decreses for more risk-verse nd less diversified employees. 18

20 Figure-7. Incentives from options grnt with totl cost of ssuming no djustment to existing compenstion pckge. This figure is plots of incentives (in K ) from options grnt for given strike price holding the totl BS vlue of the grnt constnt t Totl incentives re defined s the prtil derivtive of the totl subjective vlue of the grnt with respect to stock price S. The sub-figure on the left-hnd side depicts totl incentives s function of the strike price nd probbility weighting prmeter. Tht in the middle is plot of totl incentives ginst strike price nd loss-version. The sub-figure on the right-hnd side exhibits the combined effect of probbility weighting nd loss-version on the shpe of totl incentives s function of strike price. The derivtives were computed numericlly bsed on the following prmeters: T=4; σ=30%; S=100; r=3%; q=0%; α= Sitution 2: Option grnt with n djustment to existing compenstion. Now, let s considerer tht the compny is llowed to mke n efficient djustment to existing compenstion components (csh for exmple) in order to grnt dditionl options to the employee. The impct of this djustment should be neutrl with regrd to the totl compenstion cost for the compny. Moreover, ssuming this djustment involves csh compenstion, it must be ttrctive to the employee so tht he d be willing to give up some csh compenstion ginst extr options grnt. Therefore, it must leve the employee t his initil totl subjective vlue of the compenstion pckge. It follows tht the strike price tht mximizes totl incentives for given compny cost is the solution of this following optimiztion problem: m x nc K, S K subject to n K nc K c nd n 0 (8), Where c is fixed constnt. The constrint in (8) is the ggregtion of the compny cost constrint nd the employee vlue constrint used in Henderson (2005). 19

21 Similr to tht in (7), this optimiztion problem ws solved numericlly. In this nlysis, c ws chosen such tht for retined prmeters, the number of grnted t-the-money options n is round 1000, hence totl cost of Once c is set, n is determined from the constrint in (8) for given nc, strike price K S, then totl incentives S K re ssessed. These re depicted in Figure 8. The left-hnd side sub-figure exhibits totl incentives for different levels of K nd probbility weighting prmeter, with risk version held constnt t For ech combintion of K nd the constrint in (8) is solved for n, which llows to determine totl incentives. The plots indicte tht when the employee is deeply subject to probbility weighting ( ), strictly decresing throughout the depicted rnge of strike prices (see the curves in blue). In this cse, similr to EUT-bsed studies findings (Hll nd Murphy, 2002; Henderson, 2005), incentives re mximized through restricted stocks grnt rther thn stock options. The intuitive reson behind tht is tht, given tht the employee systemticlly vlues the options in excess of their BS vlue, efficient trde-off over compenstion lloction is mde vi the grnt of equity-bsed instruments tht employee vlues t their ctul cost, restricted stocks for instnce. nc However, for lower degrees of probbility weighting (typiclly ) the optiml strike price is not nil nd tends to increse with. Specificlly, the optimiztion problem is defined on the right (see the red solid curve) or on the left (see the red dshed curve) of the optiml strike K * nd nc, S K dmits n infinite brnch in K *, S (i.e. totl incentives re infinite t tht point). The technicl reson behind tht is tht the subjective vlue is equl to the BS vlue t the optiml strike, which yields ccording to the constrint in (8) n infinite grnt size ( lim n K * K K ). For instnce, K * rnges from 60% to 195% when tkes vlues within rnge of vlue round the experimentl estimtes ( ) 11. This suggests tht, depending on the extent to which employee weights probbilities, either grnts of discount options or premium options my result in Preto optimum to the contrct. As specific exmple, for n employee with 0.616, the optiml strike price K * is round 100%, which mens tht, for the previling setting, the model predicts t-the-money options s optiml in n efficient contrcting frmework since the compny s cost of grnting these options is equl to their vlue to the employee. In this cse, grnting options with strike price round the stock price t inception, which is consistent with observed prctices, results in perfect reconcilition of the compny nd the employee respective K is 11 Optiml strikes re expected to be lower for higher stock price voltility since probbility weighting effect tends to increse with voltility (recll Figure 2). 20

22 views. Note tht for similr setting in the previous sitution where the options re given s n dd-on to existing compenstion, the model indictes tht incentives from stock options could be infinitely incresed by grnting incresingly more options t greter nd greter strike price. Furthermore, for wek degrees of probbility weighting ( 0.85 ), the subjective vlue of the option lies strictly below the BS vlue becuse the effect of loss-version tends to dominte tht of probbility weighting. Thus the totl incentives function tkes non-monotone concve shpe nd shows finite extrems rnging from 35% to 81%. In prticulr, when probbility weighting ssumption is relxed ( 1 ), the function flttens round the mximum nd pproches monotone decresing shpe with incresing loss-version (see sub-figure on the right-hnd side). This result suggests tht loss-verse employees who re not subject to probbility weighting - even with very week degree of probbility weighting - receiving options t high exercise price would willingly ccept cut in csh compenstion to receive insted deep discount options or restricted shres for those of them displying more loss-version. Agin, this is brodly consistent with Hll s nd Murphy s (2002) nd Henderson s (2005) findings for non-diversified risk-verse employees. Lst but not lest, the sub-figure in the middle of the min figure is plots of totl incentives ginst the strike price; ech of them corresponds to given level of loss-version. The purpose of this nlysis consists in showing the effect of loss-version on K * by holding constnt t 0.65 nd vrying. It minly shows tht loss-version hs n effect opposite to tht of probbility weighting: when probbility weighting effect is dominnt 12 (i.e. the employee my potentilly put overstted vlue in the option, specificlly for high strikes) optiml strike increses nd totl incentives decresed with loss-version. Conversely, when loss-version effect is dominnt, s stted before, the model yields predictions comprble to tht of the EUT-bsed models. 12 See the limit cse of 1 where risk-version is ignored (the blue dshed curve). 21