Indexing Execuive Sock Opions Relaively Jin-Chuan Duan and Jason Wei Joseph L. Roman School of Managemen Universiy of Torono 105 S. George Sree Torono, Onario Canada, M5S 3E6 jcduan@roman.uorono.ca wei@roman.uorono.ca Sepember 2003 Absrac We propose and analyze a relaive indexing scheme for execuive sock opions. In conras o he absolue indexing scheme of Johnson and Tian (2000) which bases he opion s payoff on he dollar difference beween he sock price and he indexed exercise price, our scheme defines he opion s payoff on he raio of he sock price over he indexed exercise price. The absolue indexing is shown o sill yield an opion value moving in andem wih he index, which is undesirable for i ends up rewarding or penalizing he execuive for he facors ouside his/her conrol. Besides correcing he misalignmen beween reward and performance, relaive indexing also possesses many desirable incenive properies. On a per dollar basis, i has a much higher dela (i.e., sensiiviy o sock price) and vega (i.e., sensiiviy o firm volailiy) han absolue indexing, implying ha relaive indexing has a sronger performance incenive and is more effecive in encouraging he CEO o underake more risky projecs which oherwise may be shun due o his/her less-diversified personal wealh. Boh auhors graefully acknowledge he financial suppor from he Social Sciences and Humaniies Research Council of Canada.
Indexing Execuive Sock Opions Relaively Absrac We propose and analyze a relaive indexing scheme for execuive sock opions. In conras o he absolue indexing scheme of Johnson and Tian (2000) which bases he opion s payoff on he dollar difference beween he sock price and he indexed exercise price, our scheme defines he opion s payoff on he raio of he sock price over he indexed exercise price. The absolue indexing is shown o sill yield an opion value moving in andem wih he index, which is undesirable for i ends up rewarding or penalizing he execuive for he facors ouside his/her conrol. Besides correcing he misalignmen beween reward and performance, relaive indexing also possesses many desirable incenive properies. On a per dollar basis, i has a much higher dela (i.e., sensiiviy o sock price) and vega (i.e., sensiiviy o firm volailiy) han absolue indexing, implying ha relaive indexing has a sronger performance incenive and is more effecive in encouraging he CEO o underake more risky projecs which oherwise may be shun due o his/her less-diversified personal wealh. 0
1. Inroducion Execuive sock opions have received unprecedened aenion in he recen years from academic researchers, policy makers, boards of direcors, and he invesmen public in general. One of he issues being debaed is wheher / how o design compensaion schemes based on relaive performance. Alhough he vas majoriy of sock opions have fixed exercise prices and hence reward absolue performance, aenion has been drawn o sock opions whose exercise prices are linked o cerain benchmark indices. Rappapor (1999) discussed he general idea and relaed issues of indexing. Johnson and Tian (2000) designed and analyzed an indexing scheme for sock opions. The logic behind indexing he exercise price is boh appealing and simple. To illusrae, suppose an a-he-money call opion is graned o a CEO wih an exercise price of $100. A mauriy, he CEO would pocke $20 if he sock price has gone up by 20%. Ye over he same period, he overall sock marke reurn may be, say, 40%. The firm s sock has under-performed he marke, and he $20 compensaion o he CEO is hus no jusified. Likewise, if he firm s sock and he marke have declined by 10% and 30% respecively, he CEO would no have received any compensaion since he opion expires ou-of-he-money, ye he CEO has ouperformed he marke. Had he exercise price been indexed o he marke, he above undue compensaion or penaly would have been avoided. Indexing according o Johnson and Tian (2000) would give rise o an exercise price of $140 in he firscaseifhefirm s sock has a bea of one, which in urn renders he opion ou-of-he-money. In he second case, he indexed exercise price would become $70 and lead o a opion payoff of $20. Such indexing appears o have offered a compensaion commensurae wih he firm s relaive performance. There is a serious problem wih such an indexing scheme, however. Meulbroek (2001) poined ou ha he opion value is homogeneous of degree one wih respec o he sock price and he exercise price. The opion s value would increase even when he sock price and he indexed exercise price increase by he same facor. For insance, suppose an opion is graned 20% in-he-money wih he curren sock price and he exercise price being $120 and $100 respecively and he sock 1
has a bea of one. If neiher he sock nor he marke index has changed (relaive o heir iniial levels) by he opion s mauriy, hen he payoff is $20, he same as he inrinsic value a he graning ime. If he sock and he marke have boh increased by 10%, he opion s payoff should sill be $20 since he sock did no ouperform he marke. However, he opion s payoff is acually $120(1.1) - $100(1.1) = $22, which has also increased by 10%. I is easy o see ha a similar effec exiss when he sock and he marke decline by he same percenage poin. In fac, such effec exiss a any ime before he opion s mauriy: holding oher hings consan, he opion s fair value would increase or decrease in andem wih he simulaneous movemens in he sock price and he exercise price. In shor, such an indexing scheme does no achieve is inended purpose. Amomenofreflecion reveals he roo of he problem: he indexing scheme proposed by Rappapor (1999) and Johnson and Tian (2000) adjuss he absolue level of he exercise price, bu i is he relaive level ha ruly capures he relaive performance. Hereinafer, we will refer o heir scheme as absolue indexing, and he scheme proposed in his paper relaive indexing. Absolue indexing focuses on he dollar value of he exercise price, and herefore can deermine wheher and by how much he sock has under-performed or ouperformed he marke. As a resul, he CEOmayberewardedforpoorrelaiveperformance and penalized for good relaive performance. There are wo key aspecs in indexing - marke volailiy effec and marke level effec. Wha absolue indexing acually achieves is he removal of he marke volailiy from compensaion. As a resul, he opion conrac is effecively linked o he volailiy of he sock relaive o he marke. Simply pu, absolue indexing has succeeded in handling he volailiy effec bu failed o address he marke level effec. We propose a relaive indexing scheme which is a simple modificaion of he Johnson and Tian (2000) indexing approach. Insead of focusing on he absolue level of he sock price and he exercise price, we ake as he underlying variable he raio of he sock price over he index level, i.e., reaing he marke index as a numeraire asse. The opion conrac is hen srucured around his variable inending o measure he relaive performance of he sock in relaion o he marke. Relaive indexing is shown o simulaneously deal wih boh he marke volailiy and level effecs, 2
in he sense ha he incenive conrac no longer rewards or punishes he execuive for an oucome driven by he overall marke. We find ha absolue indexing eiher rewards oo generously or penalizes oo harshly. The only siuaion in which i presens a fair compensaion is when he sock price and he index never change during he life of he opion, he leas ineresing siuaion. In addiion, he misalignmen beween performance and compensaion is bigger when he firm s bea or sysemaic risk is high. We show ha relaive indexing can correc he above deficiencies. Relaive indexing is also more desirable han absolue indexing in erms of incenive effecs. To begin wih, on a per dollar basis, relaive indexing s dela (i.e., he opion s sensiiviy o he sock price) is much higher han is absolue indexing counerpar, reflecing a sronger performance incenive. Relaive indexing also leads o a higher vega (i.e., he opion s sensiiviy o firm s volailiy), which means i is more effecive han absolue indexing in inducing CEO s o underake risky projecs. This is useful since CEO s end o shun projecs ha are risky bu beneficial o he firm, due o heir less-diversified personal wealh. Finally, relaively indexed opion is also superior o is absoluely indexed counerpar in preserving is value agains he passage of ime, i.e., i has a lower rae of ime decay. Our heoreical findings are backed by a case sudy based on real marke daa. The res of he paper is organized as follows. Secion 2 briefly reviews he absolue indexing scheme of Johnson and Tian (2000), and hen presens he relaive indexing scheme. A comparison of he wo schemes is also given in his secion. Secion 3 examines he incenive effecs of he wo indexing schemes. Secion 4 presens a case sudy based on real marke daa. Secion 5 concludes he paper. 2. Execuive Sock Opions: Relaive versus Absolue Indexing 2.1. Absolue Indexing A Brief Review In his secion, we review he absolue indexing scheme of Johnson and Tian (2000). Suppose here exiss an index which can capure he sysemaic risk of he firm s sock. The sock price S and 3
he index price I are assumed o follow a join geomeric Brownian moion: and wih ds S =(µ S q S )d + σ S dz S,, (2.1) di I =(µ I q I )d + σ I dz I,, (2.2) dz S, dz I, = ρd, where µ S,q S and σ S are respecively he expeced reurn, dividend yield and volailiy of he firm s sock, µ I,q I and σ I are he index counerpars, and z S, and z I, are sandard Wiener processes wih correlaion ρ. Furhermore, we define β ρ σ S σi. Johnson and Tian (2000) defined he indexed exercise price a ime as µ β I H = S 0 e η, (2.3) I 0 where η =(r q S ) β(r q I )+ 1 2 ρσ Sσ I (1 β), and r is he risk-free ineres rae. The indexed exercise price is he expeced sock price (wih he expecaion aken a he iniial graning ime) condiional on he realized index level and zero expeced excess reurn. In oher words, his specific choice of indexing is moivaed by E 0 [S I ]=H when α µ S r β(µ I r) =0,i.e.,hesock is no expeced o yield an abnormal reurn. The payoff of a European syle, indexed opion a mauriy is max[s T λh T, 0]. Parameer λ deermines he moneyness of he opion a he iniial graning ime. Since S 0 = H 0, λ = 1 corresponds o he opion being graned a-he-money, λ > 1 ou-of-he-money, and λ < 1 in-he-money. By reaing he indexed opion as an exchange opion and applying he formula derived by Margrabe (1978), he value of he indexed opion a ime can be wrien as C (A) = e q S(T ) [S N(d 1, ) λh N(d 2, )], (2.4) 4
where N( ) is he cumulaive disribuion funcion of a sandard normal random variable, and d 1, = ln( S λh )+0.5σ 2 a(t ) σ a T, d 2, = d 1, σ a T, σ a = σ S p 1 ρ 2. The opion value in (2.4) is homogeneous of degree one wih respec o he sock price S and he indexed exercise price H. Moreover, he criical volailiy o he opion value, i.e., σ a,measures he sock price swing relaive o he indexing numeraire H. In oher words, σ a is effecively a relaive volailiy. Holding oher hings consan, when S and H change by a facor x, he opion value will also change by he same facor, an undesirable feaure as far as rewarding he execuive for his/her relaive performance is concerned. 2.2. Relaive Indexing An Alernaive Design Absolue indexing only focuses on he level of he exercise price, and he opion s payoff depends on he dollar difference beween he sock price and he exercise price. In order o reward genuine superior performance, we need o link he opion s payoff o he relaive change beween he sock price and he benchmark level. To his end, we propose he following payoff srucure: H 0 e (r q S)T max[ S T H T λ, 0], where H 0 = S 0 and H 0 e (r q S)T deermines he dollar size of he conrac. By design, he relaive performance variable S T H T has removed he ne growh facor in he sock and he benchmark which is he risk-free ineres rae minus he sock s dividend yield. I is herefore naural o include e (r qs)t in he dollar size definiion o compensae for he discouning effec due o making he coningen paymen a he fuure dae, T. We refer o his indexing scheme as relaive indexing. I is sraighforward o show ha he value of his opion a ime is C (R) = H 0 e r q ST S N(d 1, ) λn(d 2, ), (2.5) H where d 1, and d 2, have been defined earlier. The relaionship beween he opion values under he wo indexing schemes is 5
C (R) = H 0e (r q S) C (A). (2.6) H I is apparen from (2.6) ha, he wo opions have equal values a he iniial graning ime. Aferwards, apar from he deerminisic componen, he opion value under relaive indexing equals ha under absolue indexing scaled by he indexing benchmark H. This feaure is aracive because i removes he marke level effec so ha he opion value becomes homogeneous of degree zero in I, i.e., he opion value is unaffeced by he movemen of he marke. This asserion will be formally esablished in he following subsecion. 2.3. Comparisons Beween Absolue Indexing, Relaive Indexing and No-indexing Using (2.1) and (2.2), we have he following relaionship: ln S = α + r(1 β) (q S βq I ) 1 ³ σs 2 βσi 2 + β ln I + ε (2.7) S 0 2 I 0 where ε has mean 0 and sandard deviaion σ a. Recall ha α µs r β (µ I r), defining he sock s abnormal performance. I is well known ha in he equilibrium of a single-index economy, α should equal 0, i.e., here should be no abnormal performance. In general, however, abnormal performance may exis, i.e., α need no be zero. Thus, ln S = ln S β ln I η H S 0 I 0 = α 1 2 σ2 S(1 ρ 2 ) + ε. (2.8) The above expression and (2.5) imply ha he value of he relaively indexed opion, C (R) is independen of he marke index level I becauseiisafuncionof S H of he firm specific riskε. Since C (R) is increasing in S H whichisinurnafuncion whichisinurnincreasinginα, he relaively indexed opion rewards or penalizes abnormal performance depending on α > 0 or α < 0. Since d 1, and d 2, ake he same form under boh indexing schemes, he effec of removing marke volailiy is idenical for boh schemes. The difference hough is on he marke level effec. In he case of absolue indexing, C (A) is an increasing funcion of I. In oher words, he execuive 6
is rewarded or penalized for somehing beyond his/her conrol. successfully deals wih he marke level effec; ha is, C (R) In conras, relaive indexing does no respond o he movemen of I, and hus no undue compensaion or penaly is imposed for facors beyond he execuive s conrol. To illusrae he above poin, we simulae he bivariae sysem in (2.1) and (2.2) o obain 5, 000 pairs of I and S and use hem o compue he corresponding opion values a ime. We hen plo he opion values agains he realized index and sock prices. The simulaion parameers are: S 0 = I 0 =$100,=0.05, T =9.95, σ S =0.20, σ I =0.15, ρ =0.75, r=0.08, q S = q I =0.02, µ I =0.12, λ =1.0. 1 The sock s expeced reurn is calculaed via µ S = r + β(µ I r) by seing α =0. We choose a smaller so ha he plos are no overwhelmed by he randomness in I and S. Figure 1 presens he plos. Panel A plos he opion values agains he sock prices. The plo reveals ha higher sock prices are generally associaed wih higher opion values, as should be. I is also apparen ha he slope for he absoluely indexed opion is more pronounced han is relaively indexed counerpar. Moving o Panel B which plos he opion values agains he index prices, i is clear ha he relaively indexed opion does no depend on he index level. Deviaions from he horizonal line are simply due o he idiosyncraic risk. In conras, we see a posiive associaion beween opion values and he index realizaions for he absoluely indexed opion. The above observaions illusrae he drawback of absolue indexing: he opion conrac awards or penalizes he execuive for marke-wide facors. Noice ha we have produced plos a differen correlaions or beas and differen abnormal performances (i.e., α 6= 0). Oher han upward or downward locaion changes, he plos mainain he same profile. In summary, Figure 1 clearly demonsraes how relaive indexing can correc he undesirable level effec inheren in absolue indexing. To furher demonsrae he key poin of he paper, we repor in Table 1 how indexed opion values respond o insananeous changes in he sock price and he index. Specifically, we assume ha here are no changes up o ime, bu he sock price and he exercise price undergo insan- 1 Unless oherwise specified,weusehesameparameervaluesfor subsequen numerical analyses. 7
aneous changes a ime. This design of he numerical exercise will avoid he complicaion of ime decay in opion values. Table 1 repors resuls for opions which were graned five years ago and have anoher five years o go before mauriy. 2 We also examined oher combinaions and he quaniaive conclusions are he same. Themosellingparofheableishemiddlepanelwihβ =1. Here, he benchmark and he index change by he same percenage poin. As a resul, he sock price and he benchmark or he exercise price will change by he same percenage poin as long as he index also change by he same amoun. I is seen ha when he sock and he index change by he same percenage poin (increase or decrease), he opion value under absolue indexing would change by he same amoun. In conras, he opion wih relaive indexing does no experience any change in his case. This means ha relaive indexing can accomplish he inended purpose of relaive performance compensaion while absolue indexing canno. I is clear from he able ha, wih absolue indexing, he larger he changes in he sock price and he index, he bigger he unjusified change in he opion value. When he sock price goes up chiefly due o a bullish marke, he CEO is rewarded alhough he/she did no conribue o he sock s performance. When he firm s sock loses value simply due o a bearish marke, he CEO is penalized for his/her normal or perhaps beer performance. More generally, we can see he deficiencies of absolue indexing hrough oher combinaions of changes. When he sock price increases by less han he index does (e.g., 8% vs. 10%), he opion value under absolue indexing even increases slighly, bu he opion value under relaive indexing goes down, as a well-designed incenive opion should be. When he sock price increases by more han he index does (e.g., 10% vs. 8%), he opion value under absolue indexing increases much more han he opion value under relaive indexing does. Bu his magniude of increase is no warraned. On he oher hand, when he sock price decreases by less han he index does (e.g., -8% vs. -10%), he opion value under absolue indexing even decreases slighly, bu he opion value under relaive indexing goes up. This oucome is desirable because he firm has ouperformed he marke. When he sock price decreases by more han he index does (e.g., -10% vs -8%), 2 Following Johnson and Tian (2000), we vary firm volailiy σ S o obain differen bea s. 8
absolue indexing would bring he opion value down much more han he relaive indexing does, which penalizes he CEO unfairly. In summary, absolue indexing eiher rewards oo generously or penalizes oo harshly. The only siuaion in which i represens a fair compensaion is when he sock and he marke say pu (i.e., 0% change in boh), a siuaion which is unlikely o occur and is of no ineres o any compensaion design. When he sock s bea is no uniy, absolue indexing exhibis similar biases. I is seen ha, for he same percenage change in he index, a larger bea leads o a bigger percenage change in he benchmark. Meanwhile, a bigger divergence beween he changes in he sock price and he benchmark corresponds o a bigger valuaion bias by absolue indexing. As a resul, he higher he bea, he bigger he bias associaed wih absolue indexing. We can herefore infer ha firms having a higher sysemaic risk will suffer from he wors misalignmen beween compensaion and performance if hey use absolue indexing for sock opions. For insance, when he sock and he index decline by 8% and 10% respecively, he CEO s opion package would suffer a 3.11% loss under relaive indexing if he sock s bea is 0.75. If he firm uses absolue indexing, he loss would be 10.47%, in which case he CEO suffers an addiional 7.36% (=10.47% 3.11%) loss over and above he fair loss. This number becomes 14.58% (=18.15% 3.57%) if he sock s bea is 1.25. Lasly, as demonsraed by Johnson and Tian (2000), he correlaion is a key parameer deermining he value of an indexed opion. For boh ypes of indexed opions, a he ime of graning, maximum value is achieved when he correlaion beween he sock and he index is zero. Once he opions are graned, a higher correlaion will lead o a lower value for boh ypes of opions due o p he lower effecive volailiy σ a = σ S 1 ρ 2. We omi he plos for breviy. 3. Incenive Effecs In addiion o he valuaion effecs discussed in he preceding secion, relaive indexing generaes differen incenive effecs. As shown by Johnson and Tian (2000), under absolue indexing, he comparaive saics are 9
C (A) S = e q S(T ) N(d 1, ), C (A) I = λβ S 0 I 0 ³ I I 0 β 1 e η q S (T ) N(d 2, ), C (A) σ S C (A) σ I C (A) ρ C (A) ³ = λh e q p h ³ S(T ) (1 ρ 2 )(T )N 0 (d 2, ) ρ ln I σi I 0 + ³ µ = λh e q S(T ) ρσ S ln I ρσ I 0 S 2 +(r q I ) ρσ S N(d 2, ), = λh e q S(T ) = q S C (A) σ 2 I σ ae q S 2 T µ ρσ S T 1 ρ 2 N 0 (d 2, )+ (T ) h σs σi ln σ 2 I ³ I I 0 + λh N 0 (d 2, ) ληh e q S(T ) N(d 2, ), C (A) r = (β 1)λH e q S(T ) N(d 2, ), where N 0 ( ) sands for he sandard normal densiy funcion. 3 ³ i ρσi 2 ρ2 σ S (r q I ) ρ σ I N(d 2, ), ³ σs σ I 2 ρ 2 σ S (r q I ) σ S σi i N(d 2, ) By equaion (2.6), we have he following corresponding comparaive saics under relaive indexing: C (R) S = H 0e (r q S ) H C (R) I = H 0e (r q S ) H C (R) σ S = H 0e (r q S ) H C (R) σ I = H 0e (r q S ) H C (R) ρ = H 0e (r q S ) H C (R) = H 0e (r q S ) H C (R) r = H 0e (r q S ) H C (A) S, C (A) I C (A) σ S C (A) σ I C (A) ρ C (A) C (A) r β I C (R), C (R) h ³ ³ i ρ ln I σi I 0 + ρσi 2 ρ2 σ S (r q I ) ρ σ I, C (R) ³ µ ρσ S ln I ρσ σi 2 I 0 + S 2 +(r q I ) ρσ S, σi 2 C (R) h ³ ³ σs σi ln I I 0 + σs σ I 2 ρ 2 σ S (r q I ) σ S + C (R) + βc (R). h β(r q I ) 1 2 ρσ Sσ I (1 β) Some of he parial derivaives are easy o sign while ohers are no. Insead of focusing on he sign, we will conras numerically he comparaive saics across he hree ypes of opions: i, σi i,, absolue indexing, relaive indexing and no-indexing. Following Johnson and Tian (2000), we adjus he number of indexed opions so ha all opions have he same value a he graning ime. We hen muliply he indexed opion s parial derivaives by he value adjusmen facors. Figure 2 presens he plos for sock dela, he opion value s sensiiviy o he sock price. Here, he curren index level is assumed o be $100, and he opion is five years old and has a remaining ime o mauriy of five years. When he indexed opions are iniially graned, he 3 We have added C(A) I which is no in Johnson and Tian (2000). In addiion, we repor he opion s sensiiviy o ime by keeping he mauriy ime consan whereas Johnson and Tian (2000) repored he opion s sensiiviy o mauriy by keeping he currren ime consan. 10
delas of he wo ypes of opions are equal. However, as shown in Figure 2, once he opions are graned, he relaively indexed opion has a higher dela for all moneyness siuaions, meaning ha his compensaion conrac is more responsive o he sock price movemen when oher facors are fixed. The more he opions are in-he-money, he bigger he difference in dela s. Alhough no shown, figures wih oher ime combinaions (i.e., varying, while keeping T equalo10years) reveal similar paerns. The observaions lead o an imporan implicaion: relaive indexing no only reflecs he rue spiri of compensaion based on relaive performance, bu also brings abou sronger incenives han absolue indexing. The laer is especially rue for deep in-he-money opions. For compleeness, we presen in Figure 3 he plos for index dela, he opion value s sensiiviy o he index price. As expeced, index dela is negaive for boh ypes of indexed opions. In addiion, he relaively indexed opion is much more sensiive o he index, especially when he opion is deep-in-he-money. Figure 3 is essenially an alernaive presenaion of Figure 2, since he index consiues he exercise price. We have wo vega s o presen: he opion s sensiiviy o he firm volailiy and ha o he index volailiy. We will discuss he former firs. Figure 4 conains wo plos, wih Panel A for newly graned opions ( =0,T = 10), and Panel B for previously graned opions ( =5, T =5). I is seen in Panel A ha he wo indexing schemes lead o he same vega iniially, and a-he-money opions have he highes vega. However, as shown in Panel B, he wo schemes have drasically differen vega s afer he opions are graned. When he opions are ou-of-hemoney, vega increases wih moneyness; when he opions are in-he-money, vega sill increases wih moneyness under relaive indexing, bu remains unchanged under absolue indexing. Moreover, a all levels of moneyness, vega is larger for relaive indexing. We can draw wo implicaions from Figure 4. Firs, he risk incenive of relaive indexing is higher han ha of absolue indexing, boh of which are higher han he risk incenive of no indexing. This means ha relaive indexing is he mos effecive in erms of inducing he CEO o ake risky projecs o enhance he firm value. (As discussed by Johnson and Tian (2000), CEO s end o avoid risk aking due o heir less diversified 11
personal wealh.) Second, for previously graned opions, risk incenive of relaive indexing is higher when he opion is in-he-money, in conras wih ha of absolue indexing. This again is a desirable feaure of relaive indexing since CEO s, in order o proec heir wealh, will have a endency o avoid risky projecs once he opions are deep in-he-money (i.e., o proec one bird in hand insead of chasing wo birds in he bush). The higher vega would induce adequae risk aking on he par of he CEO when he firm is successful. As for he index volailiy, as long as he opion is no oo new (i.e., is no oo small) and he index has no increased subsanially (i.e., I is no much higher han I 0 ), vega is negaive under boh indexing schemes. This is easy o undersand since σ I only affecs he benchmark level H, and for mos siuaions, H is posiively relaed o σ I. Figure 5 is an index volailiy counerpar of Figure 4. I is seen ha he sensiiviy o index volailiy or he index vega becomes smaller (i.e., less negaive) as he opions become deep ou-of-he-money. In addiion, he index vega is larger (i.e., more negaive) for relaive indexing. Alhough he index volailiy is no under he CEO s conrol, Figure 5 reveals ha an opion wih relaive indexing is more sensiive o he index volailiy han is absolue indexing counerpar. Nex, we plo in Figure 6 he opion s sensiiviy o correlaion agains he sock prices. The plos are for wo levels of correlaion: ρ =0.5 andρ =0.9, corresponding o a bea of 0.67 and 1.2 respecively. A ime = 0, he plos are similar o Figure 4 of Johnson and Tian (2000), and are omied for breviy. We plo he sensiiviy in Figure 6 for =5andT =5. Iisseenha he correlaion sensiiviy is generally posiive for in-he-money opions, and is higher when he correlaion iself is high. Moreover, as he opion becomes deep in-he-money, he sensiiviy levels off for absolue indexing bu coninues o increase for relaive indexing. The above observaions generally hold rue for oher combinaions of and T (as long as is no oo small) and oher levels of I. 4 There are wo imporan implicaions. Firs, in conras o wha Johnson and Tian (2000) have concluded (viz, he CEO has an incenive o reduce he correlaion o zero), he CEO will have an incenive o increase he correlaion (or equivalenly, he sysemaic risk) when he opion 4 Specifically, when is small or when I is high, he general paerns in Figure 6 sill obain, excep ha we see a negaive sensiiviy in a wider range of moneyness, and he sensiiviy becomes posiive a a higher sock price. 12
is several years old and is deep in-he-money. This incenive is much higher wih relaive indexing. Johnson and Tian s conclusion only applies when = 0, whereas our conclusion is applicable mos of he ime wihin he life of he opion. Second, unlike absolue indexing, relaive indexing is associaed wih a higher correlaion sensiiviy when he opion becomes deep in-he-money. 5 This means ha he CEO should be under a more wachful eye when he sock opions are several years old and are deep in-he-money. The incenive o pursue a higher correlaion means a higher sysemaic risk of he sock (holding oher hings equal), somehing he firm may no desire. We now examine hea, he opion s sensiiviy o he passage of ime. Panel A of Figure 7 plos he sensiiviy agains he sock price by assuming =5,T = 5, while Panel B plos he sensiiviy agains ime o mauriy by assuming he same curren ime = 1. In Panel B, he adjusmen facors for indexed opions are calculaed by equaing opion values iniially for each mauriy (i.e., =1,T =1, 2, 3, 4,...). These facors are hen applied o hea s of he indexed opions. 6 As seen in Panel A, hea is negaive for non-indexed opions, reflecing he usual ime decay effec. However, for deep in-he-money indexed opions, he passage of ime will acually increase he opion value, and his effec is much more pronounced for relaively indexed opions. Inuiively, his is similar o he ime decay of a deep in-he-money European pu opion. When an indexed opion is deep in-he-money, waiing can be cosly because he index may go up, leading o a higher exercise price. As seen in Panel B of Figure 7, he plos for he non-indexed opion and he absoluely indexed opion are basically mirror image of hose in Johnson and Tian (2000) who examined he opion s sensiiviy o ime o mauriy. The plo for he relaively indexed opion exhibis some ineresing feaures. To begin wih, he ime decay is smaller han he oher wo opions. Moreover, when 5 A closer examinaion of he comparaiive saics for he correlaion reveals he inuiion behind he differen paerns. To sar wih, H is generally negaive, which means a higher correlaion should be associaed wih a ρ higher opion value due o a lower exercise price. However, a higher correlaion also means a lower overall volailiy in he pricing formula, which leads o a lower opion value. When he opion is deep in-he-money, he volailiy impac diminishes, and he exerice price impac remains. This is why he correlaion sensiiviy levels off for absolue indexing. Wih relaive indexing, here is also a division effec, i.e., H is a divisor of he opion value wih absolue indexing. This effec makes he sensiiviy increases wih he moneyness. 6 We also generaed plos by assuming differen values of. Since hey all share he same paern, we omi hem for breviy. 13
he mauriy is long enough (longer han 6 years in his case), he opion s value acually increases wih he passage of ime. 7 These observaions lead o an imporan pracical implicaion: in erms of preserving he opion value agains he passage of ime, relaively indexed opions are he bes choice. Finally, wih respec o he opion s sensiiviy o ineres rae, we know ha he value of a non-indexed opion is posiively relaed o he ineres rae. For opions wih absolue indexing, as long as is no zero, he ineres rae sensiiviy is posiive / negaive when β is greaer / less han one. For opions wih relaive indexing, he ineres rae sensiiviy is posiive when β is greaer han one, and mosly posiive when β is less han one. We omi he plos for breviy. In summary, absolue indexing and relaive indexing can creae quie differen incenive effecs. In many ways, relaive indexing leads o more desirable incenive effecs. Firs of all, compared wih absolue indexing, relaive indexing is associaed wih a much sronger performance incenive since he opion has a much bigger dela per dollar of graning cos. Second, relaive indexing has a sronger risk incenive han absolue indexing due o he opion s higher vega. Insofar as CEO s end o shun risky projecs due o heir non-diversified personal wealh, sock opions wih relaive indexing can be an effecive ool o couner heir risk-aversion and enhance he value of he firm. Third, relaive indexing can beer preserve he conrac s value agains he passage of ime hanks o is smaller hea or sensiiviy o ime. 4. A Case Sudy To illusrae how he wo indexing schemes would fare in he real world, we race he values of ficiious index opions based on marke daa. We idenify a period in which he general marke experienced a cycle of upward and downward movemens. Taking he Nasdaq index as he benchmark, he period from January 2, 1998 o December 31, 2001 presened such a cycle, as shown in Figure 8. The index was on he upward rend unil March 2000 when i peaked a 5,049, and hen 7 We have also produced plos for oher values of including =0, and hey all share he same feaure. 14
ook a downward urn. The company we selec is Inel Corp. which graned 600,000 convenional sock opions o is CEO, Mr. Craig R. Barre, on January 2, 1998. These incenive opions have a en-year mauriy and an exercise price equal o he sock s fair marke value on he graning day. We assume away he vesing and early exercise feaures. In addiion o he graned non-indexed opions, we examine ficiious opions ha are indexed absoluely and relaively in he manner discussed in his paper. Using daily closes of he index and he sock for he aforemenioned period, we obain he following esimaes: σ S =0.565, σ I =0.379, ρ =0.718. 8 To faciliae calculaions, we assume a risk-free rae of 6% and dividend yields of 0% and 2%, respecively for he sock and he index. Corresponding o he realized sock and index values, we calculae he daily heoreical opion values and plo hem ogeher wih he index and he sock prices. For scaling purposes, we muliply he sock price and opion values by a facor of 50. Figure 8 conains he resuls. 9 To begin wih, we see general co-movemens beween he sock and index values. The relaively high correlaion coefficien of 0.718 aess o his observaion. This means ha, mos of he ime, he sock s movemens are aribuable o he marke, no he CEO s performance. In he early period afer he iniial gran, he opion values under he wo indexing schemes were indisinguishable. During he firs half of 1999, Nasdaq was on an upward rend while he Inel sock headed downward. Boh indexing schemes were able o reflec he relaively poor performance of he sock, as indicaed by he downward rend in he opion values. The same observaion can be made for he laer par of 1999. For he early par of 2000, boh he sock and he index climbed significanly in a more or less parallel fashion. In his period, he value of he absoluely indexed opion appreciaed much more han is relaively indexed counerpar, reflecing is overshooing level effec. Then, when i came o he marke correcion in he fall of 2000, he value of he absoluely indexed opion dropped much more han is relaively indexed counerpar, again indicaing is overshooing level effec. Even more ineresing is he period around July 2001 during which he sock hovered 8 Daily closing prices are downloaded from hp://finance.yahoo.com. The sock prices are adjused for dividends and splis. The adjused price for January 2, 1998 is $17.76. 9 We also calculaed he heoreical values of he non-indexed opion. As expeced, hey are much higher han is indexed counerpars and closely mimick he rend of he sock price. Since hey do no convey any ineresing insighs, we omi hem in he plo for clariy. 15
around $1450 (or $29 per share) while he Nasdaq index was on a down urn, a case of superior performance in he relaive sense. How did he wo indexing schemes fare? The relaively indexed opion saw a corresponding upward rend as desired, bu he value of he absoluely indexed opion remained more or less unchanged. This simple case sudy demonsraes he drawback of absolue indexing and shows how relaively indexing can be used o correc i. 5. Conclusion Among he many issues surrounding execuive compensaion, he quesion of wheher and how o design relaive performance compensaion has begun o arac much aenion. Johnson and Tian (2000) designed and sudied an indexing scheme which enables he exercise price of he sock opion o floa wih he marke. As poined ou by Meulbroek (2001), such an indexing scheme, referred o as absolue indexing, suffers a key drawback: i can reward CEO s for no effors (when he sock marke is on he upward rend) and penalize CEO s for poenially good effors (when he sock marke is on a downward rend). In oher words, absolue indexing, by focusing only on he exercise price level, fails o address he undesirable consequence of he opion s inheren propery: he opion s value is homogeneous of degree one wih respec o he sock price and he exercise price. The opion s value would increase or decrease even if he sock price has gone up or down purely because of he overall marke movemen, a siuaion ha indexing is supposed o correc in he firs place. In his aricle, we modify he scheme by Johnson and Tian (2000) o correc he aforemenioned drawback. In deermining he opion s payoff, raher han looking a he dollar difference beween he sock price and he indexed exercise price, we examine he raio of he sock price over he indexed exercise price, hence he erm relaive indexing. We compare he wo indexing schemes in erms of valuaion and incenive effecs. We find ha absolue indexing ends o reward oo generously and penalize oo harshly, espe- 16
cially when he firm s bea or sysemaic risk is high. Relaive indexing can correc his deficiency and a he same ime leads o many desirable incenive effecs. For one hing, on a per dollar basis, i has a much higher dela, which is equivalen o a higher performance incenive. For anoher, relaive indexing has a sronger risk incenive, which induces CEO s o underake more risky projecs which may be beneficial o he firm bu oherwise avoided due o CEO s less diversified personal wealh. Finally, relaively indexed opions have anoher advanage over non-indexed or absoluely indexed opions: hey can beer preserve he opion s value agains he passage of ime, i.e., hey have he smalles ime decay. 17
References [1] Johnson, Shane A. and Yisong S. Tian, 2000, Indexed Execuive Sock Opions, Journal of Financial Economics, 57(1), 35-64. [2] Margrabe, W., 1978, The Value of An Opion o Exchange One Asse for Anoher, Journal of Finance, Vol 33, 177-186. [3] Meulbroek, Lisa K., 2001, Execuive Compensaion Using Relaive-Performance-Based Opions: Evaluaing he Srucure and Coss of Indexes Opions, Working Paper, HarvardBusi- ness School. [4] Rappapor, Alfred, 1999, New Thinking on How o Link Execuive Pay wih Performance, Harvard Business Review, 91-101. 18
Table 1. Changes in Indexed Opion Values In Response o Changes in Sock Price and Index β =0.75 β =1.00 β =1.25 Change in Change in Change in Change in Change in Change in Change in Change in Change in Change in Change in sock index benchmark opion value: opion value: benchmark opion value: opion value: benchmark opion value: opion value: price level price abs. indexing rel. indexing price abs. indexing rel. indexing price abs. indexing rel. indexing (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%). -10-10 -7.60-23.92-17.66-10.00-10.00 0.00-12.34-3.89 9.63-8 -10-7.60-10.47-3.11-10.00-0.25 10.84-12.34 3.57 18.15-10 -8-6.06-31.83-27.43-8.00-17.26-10.06-9.90-10.26-0.40-5 -5-3.77-12.36-8.93-5.00-5.00 0.00-6.21-1.88 4.61-4 -5-3.77-5.39-1.68-5.00-0.19 5.07-6.21 1.75 8.49-5 -4-3.02-16.68-14.08-4.00-8.69-4.89-4.97-5.06-0.09 0 0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 2 1.50 5.17 3.62 2.00 2.00 0.00 2.51 0.72-1.74 8 10 7.41 11.71 4.01 10.00 0.70-8.45 12.65-3.31-14.17 10 8 5.94 37.67 29.95 8.00 17.71 8.99 10.10 9.75-0.32 10 10 7.41 27.07 18.31 10.00 10.00 0.00 12.65 3.42-8.19 Noe: 1. This able repors changes in indexed opion values in response o changes in he sock price and he index. The opions were graned five years ago ( = 5) and have a remaining ime o mauriy of five years (T =5). To avoid he complicaion of ime-decay in opion values, we assume ha he sock and he index remain a heir iniial level (S 0 = I 0 = $100) for he firs five years, and hen undergo a one-ime change a ime = 5 years indicaed in he firs wo columns. The percenage changes in he benchmark price (i.e., indexed exercise price) and he opion values correspond o his one-ime change. 2. We fix he correlaion ρ a 0.75 and he index volailiy σ I a 0.15. For each level of β, weimplyhefirm s volailiy σ S. 3. Oher inpus: H 0 = $100, r=0.08, q S = q I =0.02, λ =1.0.
Opion value ($) Figure 1: Opion Values vs. Sock and Index Prices (coninued) Panel A: Indexed Opion Values vs. Sock Prices 45 40 Absolue indexing Relaive indexing 35 30 25 20 15 10 5 0 90 95 100 105 110 115 Sock price ($) 20
Opion value ($) Figure 1: Opion Values vs. Sock and Index Prices Panel B: Indexed Opion Values vs. Index Prices 45 40 Absolue indexing Relaive indexing 35 30 25 20 15 10 5 0 90 95 100 105 110 115 Index price ($) Noe: 1. This figure plos opion values agains simulaed sock and index prices. The iniial prices for he sock and he index are S 0 = I 0 = $100. The opions are 0.05 years old wih a remaining ime o mauriy of 9.95 years (i.e., =0.05, T =9.95). The sock price and he index price a ime are simulaed by assuming he dynamics in (2.1) and (2.2), while assuming µ S = r + β(µ I r). The opion values are calculaed based on he realized S and I. Each panel plos 5,000 realizaions. 2. For beer visual effecs, we muliply he value of he absoluely indexed opion by 1.5. 3. Oher inpus for calculaions: H 0 = $100, σ S =0.20, σ I =0.15, ρ =0.75, r =0.08, q S = q I =0.02, µ I =0.12, λ =1.0. 21
Dela (sensiiviy o sock price) Figure 2: Opion s Sock Price Dela vs. Sock Price 4 3.5 3 2.5 2 1.5 1 Non-indexed Absolue indexing Relaive indexing 0.5 0 40 70 100 130 160 190 220 250 Sock price ($) Noe: 1. This figure plos opion s sensiiviy o he sock price agains he sock price iself. The index level is fixed a $100. The opions are five years old wih a remaining ime o mauriy of five years (i.e., =5,T = 5). To ensure comparabiliy, we adjus he number of indexed opions so ha all opions have he same value iniially. These adjusmen facors are hen applied o he dela s of he indexed opions. 2. Oher inpus for calculaions: S 0 = I = I 0 = H 0 = $100, σ S =0.20, σ I =0.15, ρ =0.75, r =0.08, q S = q I =0.02, λ =1.0. 22
Dela (sensiiviy o index price) Figure 3: Opion s Index Price Dela vs. Index Price 0 40 70 100 130 160 190 220 250-5 -10 Absolue indexing -15 Relaive indexing -20-25 Index price ($) Noe: 1. This figure plos opion s sensiiviy o he index price agains he index price iself. The sock price is fixed a $100. The opions are five years old wih a remaining ime o mauriy of five years (i.e., =5,T = 5). To ensure comparabiliy, we adjus he number of indexed opions so ha all opions have he same value iniially. These adjusmen facors are hen applied o he dela s of he indexed opions. 2. Oher inpus for calculaions: S 0 = S = I 0 = H 0 = $100, σ S =0.20, σ I =0.15, ρ =0.75, r =0.08, q S = q I =0.02, λ =1.0. 23
Vega_s (sensiiviy o firm volailiy) Vega_s (sensiiviy o firm volailiy) Figure 4: Opion s Firm Volailiy Vega vs. Sock Prices Panel A: = 0 years, T- = 10 years 400 350 300 250 200 150 100 50 0 Non-indexed Absolue indexing Relaive indexing 40 70 100 130 160 190 220 250 Sock price ($) Panel B: = 5 years, T- = 5 years 1800 1600 1400 1200 1000 800 600 400 200 0 Non-indexed Absolue indexing Relaive indexing 40 70 100 130 160 190 220 250 Sock price ($) Noe: 1. This figure plos opion s firm volailiy vega agains he sock price. The index level is fixed a $100. Opions in Panel A are newly graned (i.e., =0,T =10); opions in Panel B are five years old wih a remaining ime o mauriy of five years (i.e., =5,T = 5). To ensure comparabiliy, we adjus he number of indexed opions a each level of firm volailiy σ S so ha all opions have he same value iniially. These adjusmen facors are hen applied o he vega s of he indexed opions. 2. Oher inpus for calculaions: S 0 = I = I 0 = H 0 = $100, σ S =0.25, σ I =0.15, ρ =0.75, r=0.08, q S = q I =0.02, λ =1.0. 24
Vega_I (sensiiviy o index volailiy) Vega_I (sensiiviy o index volailiy) Figure 5: Opion s Index Volailiy Vega vs. Index Prices Panel A: = 0 years, T- = 10 years 1000 500 0-500 -1000-1500 -2000-2500 -3000-3500 -4000 40 70 100 130 160 190 220 250 Absolue indexing Relaive indexing Index price ($) Panel B: = 5 years, T- = 5 years 1000 0-1000 -2000-3000 -4000-5000 -6000-7000 -8000 40 70 100 130 160 190 220 250 Absolue indexing Relaive indexing Index price ($) Noe: 1. This figure plos opion s index volailiy vega agains he index price. The sock price is fixed a $100. Opions in Panel A are newly graned (i.e., =0,T = 10); opions in Panel B are five years old wih a remaining ime o mauriy of five years (i.e., =5,T = 5). To ensure comparabiliy, we adjus he number of indexed opions a each level of index volailiy σ I so ha all opions have he same value iniially. These adjusmen facors are hen applied o he vega s of he indexed opions. 2. Oher inpus for calculaions: S 0 = S = I 0 = H 0 = $100, σ S =0.25, σ I =0.15, ρ =0.75, r=0.08, q S = q I =0.02, λ =1.0. 25
Sensiiviy o correlaion Figure 6: Indexed Opion s Sensiiviy o Correlaion vs. Sock Prices 420 380 340 300 260 220 180 140 100 60 20-20 Absolue indexing, correlaion = 0.5 Relaive indexing, correlaion = 0.5 Absolue indexing, correlaion = 0.9 Relaive indexing, correlaion = 0.9 0 50 100 150 200 250 Sock price ($) Noe: 1. This figure plos indexed opion s sensiiviy o correlaion agains he sock price All opions are five years old wih a remaining mauriy of five years (i.e., =5, T =5). 2. Oher inpus for calculaions: S 0 = I = I 0 = H 0 = $100, σ S =0.20, σ I =0.20, ρ =0.75, r=0.08, q S = q I =0.02, λ =1.0. 26
Thea (sensiiviy o ime) Thea (sensiiviy o ime) Figure 7: Opion s Thea (Sensiiviy o Time) vs. Sock Prices and Mauriies Panel A: Thea vs. Sock Prices 50 45 40 35 30 25 20 15 10 5 0-5 Non-indexed Absolue indexing Relaive indexing 40 70 100 130 160 190 220 250 Sock price ($) Panel B: Thea vs. Time o Mauriy 2 0-2 0 1 2 3 4 5 6 7 8 9 10-4 -6-8 Non-indexed Absolue indexing Relaive indexing -10 Time o mauriy (years) Noe: 1. This figure plos opion s hea (sensiiviy o ime) agains sock prices and mauriy. In Panel A, we assume ha all opions are five years old wih a remaining ime o mauriy of five years (i.e., =5,T =5). To ensure comparabiliy, we adjus he number of indexed opions so ha all opions have he same value iniially a =0. In Panel B, all opions are one year old, wih a remaining o mauriy being ploed (i.e., =1, T =1, 2, 3...). Again, we adjus he number of indexed opions so ha all opions havehesamevalueiniiallya = 0 wih a ime o mauriy T =2, 3, 4....These adjusmen facors are hen applied o he indexed opions hea for each S or T. 2. Oher inpus for calculaions: S 0 = I = I 0 = A = $100, σ S =0.20, σ I =0.20, ρ =0.75, r=0.08, q S = q I =0.02, λ =1.0. 27
Price Level Figure 8: Tracing he Opion Values for Inel Corp s CEO Nasdaq Absolue indexing Inel*50 Relaive indexing 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 Jan-98 Jul-98 Jan-99 Jul-99 Jan-00 Jul-00 Jan-01 Jul-01 Jan-02 Dae Noe: 1. This figure plos daily closes of he Nasdaq index and he Inel sock price muliplied by 50 (he upper wo graphs). I also plos he daily values of indexed opions (absolue and relaive indexing) from Jan. 2, 1998 o December 31, 2001 (he lower wo graphs). Again, for scaling purposes, he opion values are muliplied by 50. We are racing wo conracs graned on Jan. 2, 1998. The sock price on he graning dae was $17.76 (afer adjusing for dividends and splis according o hp://finance.yahoo.com). All opions were graned a he money. The CEO, Mr. Craig R. Barre, was graned 600,000 unis of sock opions wih a ime o mauriy of 10 years. We assume away vesing and early exercise feaures. Indexing based on he Nasdaq index is ficiious. 2. The following parameer esimaes are obained using daily closes of he index and he sock: σ S =0.565, σ I =0.379, ρ =0.718. We assume a consan ineres rae of 6%, a zero dividend yield for he sock, and a 2% dividend yield for he index. 28