Journal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997 EXECUTIVE STOCK OPTIONS: RISK AND INCENTIVES

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1 Journal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997 EXECUTVE STOCK OPTONS: RSK AND NCENTVES Socorro M. Quintero *, Leslie Young ** and Micael Baur *** Abstract We perform comparative analysis on an optimal compensation contract comprised of stock options and a fixed salary for two types of managers. n a principal-agent framework, we obtain te pareto optimal solution for te level of stock options in a manager s compensation contract. We ten analyze te sensitivity of tis optimal contract to canges in te firm s performance, te volatility of te firm s payoffs, and te manager s risk attitudes. n an uncertain environment, we find tat sareolders can reduce te volatility of teir ownersip claim by including stock options in a manager s contract. We also find tat a compensation contract wit options remains near pareto optimality in te face of canging business conditions. Sareolders are spared te costs of continuously recontracting wit managers to keep teir incentives properly aligned wit sareolders. NTRODUCTON Ever since te paper of Jensen and Meckling [11], attention as been increasingly given to agency teoretic implications to modern financial teories. Similar issues are also addressed in principal-agent problems spun by te important works of Savell [17], Harris and Raviv [9], and Holmstrom [10] wic recognize te asymmetry regarding wat te principal and te agent bring into or invest in te firm. Bot veins of researc are interested in increasing te firm owner s welfare tru incentive alignment. n te former, one would be interested in incentive alignment by reducing agency costs, and in te latter, by designing an optimal incentive contract. Managerial compensation in te form of fractional ownersip in te firm is an appealing and a feasible alternative for eiter scool. Te relevance of using ownersip is underscored by te fact tat te majority of existing executive compensation packages ave a significant proportion of teir remuneration in te form of stocks and stock options. Unlike stocks, stock options allow fractional ownersip in te firm only in te event tat some favorable future states occur. Altoug it is still a matter of debate, some empirical works suc as tose of Aggrawal and Mandelker [1] and Morck, Sleifer and Visny [15], ave found tat managerial security oldings on te firm ave reduced agency costs and increased firm value. Tese empirical results togeter wit te results of McConnell and Servaes [14] suggest tat managerial firm ownersip and firm value do not follow a monotonic relationsip. Teir results suggest tat tere is some optimal level of fractional ownersip in te firm, after wic te principal or sareolder does not benefit from compensating te manager wit more ownersip claims to te firm. One of te objectives of tis paper is to obtain a closed-form solution for te optimal level of stock options provided to te manager in a principal-agent framework. t goes furter and does a comparative analysis on te sensitivity of te optimal contract to small perturbation to te value of parameters in te model, suc as te firm s expected pay-offs, te volatility of te firm s pay-offs, and te manager s risk attitutdes. Te sensitivity of te contract are differentiated and compared for two types of managers wo are distinguised from eac oter depending on weter or not stock options elicit managerial effort to improve firm performance. Stock options will subsequently affect te distribution of ownersip claims and inferences are drawn regarding te benefits to sareolder ownersip. *Oklaoma City University **Cinese University of Hong Kong ***Ball State University 59

2 60 Journal Of Financial And Strategic Decisions Tis paper suggests tat wile te manager, wose effort is motivated by stock options, requires a iger level of stock options, iring tese managers to run a firm in a dynamic environment promotes more stability in te firm s ownersip structure. Similarly, wile te manager wose effort is invariant to stock options requires lesser amount of stock options, te ownersip structure of firms run by tese manager will be more unstable. Tat is, te level of original sareolder s ownersip in a firm operating in a dynamic environment as more uncertainty. Realizing te more dynamic environment under wic firms operate, te optimal level of stock options granted at te signing of te contract could easily be far from te optimum prior to its maturity five to ten years later, tereby defeating te initial incentive objectives of te contract at te time it was signed. As suc, undesirable incentive effects may result. To maintain te proper incentives, contracts need to be periodically adjusted. t comes as no surprise tat we frequently observe Board of Directors making adjustments to te long-term compensation contracts of teir executives. Among tose compensation items tat are frequently adjusted are te stock options. Te later part of te paper provides comparative analysis on te sensitivity of a compensation contract wit stock options. t compares te necessary level of adjustment made in response to different canges in te operating parameters of te contract so tat pareto optimality is maintained. Te analysis in tis paper suggests tat te manager wose effort is motivated by contingent ownersip claim (i.e., te stock options) requires lesser adjustments as a result of parametric canges. For te same cange in expected firm cas flow, te manager wose effort is motivated by stock options will require less adjustment to is original level of stock options. t is also conceivable tat te manager s attitude can cange over te term of te contract. For te same increase in risk aversion by bot kinds of managers, te manager motivated by stock options will require less of an increase in te level of stock options. Moreover, and rater interesting, wen te volatility of firm earnings increases due to some externality, te manager wose effort is motivated by stock options requires a decrease in is stock options but te manager wose effort is invariant to stock options requires an increase in is stock options. Given tat te contract periods are often over extended periods of time and canging parameters are likely te norm rater tan te exception, te sensitivity analysis suggests tat over an extended period of time, iring te manager wose effort are motivated by te level of stock options, toug initially requiring a iger level of stock options, benefits te sareolders by reducing te volatility of is sare ownersip in te firm. n some instances, te sareolder ends up giving less of is ownersip in te firm tan was initially anticipated. Tat te levels of adjustments are less, or tat it takes fewer and less amount of adjustments to maintain pareto optimality for contracts drawn wit managers wit incentive effects, also suggests tat suc contracts are more likely to remain close to pareto optimality over a longer period of time tan a contract drawn wit a manager wit no incentive effects. Tis suggests tat tere will be a difference in te frequency of recontracting for tese two types of managers. Recontracting and adjustments in te executive compensation contract will be more frequent for executives wose effort is indifferent to stock options. Section sets up te model. Managers are assumed to ave positive marginal utility for income but ave a dislike for effort. Stock options are provided as an incentive for te manager to perform at or above a certain level of performance as measured by te firm s cas flow wic as a random component and a component resulting from managerial effort. A common performance measure is firm value. Wen te desired firm value is acieved or exceeded, te manager gets rewarded wit a fraction of te firm ownersip. Generally speaking, in te event of favorable firm performance, a manager wit stock options will experience a jump in is income wen e becomes a fractional owner of te firm and is enabled to particpate in watever te firm s upside gains end up to be by te end of te contract period. Certainly, tere are managers wose effort to increase firm performance can be motivated by te possibility of firm ownersip or large upside gain, particularly if tey believe teir effort can improve te cance of favorable firm performance or event to occur. Tis kind of manager is identified as a Type manager. On te oter and, tere can be managers wose effort dedicated to increase performance are ardly influenced by stock options. Suc indifference migt be present wit managers or executives about to retire or, for oter reasons, among managers wo just do not find value in te incremental earnings from compensation in te form of ownersip. Tis latter case can exist if regulations and/or te negative signalling effect of selling of is sares preclude te manager from te economic benefit of firm ownersip gained troug te exercise of is stock option at te end of te contract period. Tis managers is identified as a Type manager. Te analysis in tis paper, te pareto optimal contracts and te comparative analysis, focuses on tese two types of managers. Tese managers are restricted from edging and trading teir options during te contract period. Te obtained closed-form solutions derived in Section consist of te stock option and fixed salary comprising te optimal contract so tat te manager spends effort to maximize sareolder s welfare. Te solution depends on

3 Executive Stock Options: Risk And ncentives 61 parameters, mentioned earlier, suc as te expected firm cas flow and volatility wic are easily measured. Oter parameters of te contract, suc as te sareolder s and manager s risk attitude, are endongeous to te model and are not directly quantifiable but for wic teir direction of cange are generally discernable. As suc, our result is able to furter provide useful insigt towards a better understanding of compensation contracts using executive stock options. Beck and Zorn [2] investigated te efficient allocation of stocks between te principal and te agent. Tis paper is distinguised from teirs by studying options in wic, unlike stocks wic award firm ownersip from te time te contract is signed, fractional firm ownersip is contingent upon performance. As suc, fractional ownersip in te firm is not a certainty wit stock options. THE MODEL To simplify our analysis, we allow one sareolder to represent all te sareolders of a firm wit uncertain cas flows. At time 0, te representative sareolder and a manager design an executive compensation package tat will maximize te expected utility of te sareolders wile maintaining te reservation price of te manager. Te compensation package is made up of a fixed salary and an option to purcase a fractional sare of te firm at some future time, t=1, only if te firm s net cas flow exceeds a certain level. Tus, a riskless and a risky income sources comprise te managerial income portfolio and are endogeneous in our model. Bot te sareolder and te manager ave positive marginal utility for wealt, and bot ave concave utility functions. Tus, risk bearing will play a major role in te design of te compensation package and te consequent allocation of time 1 ownersip in te firm via te stock option. Aside from being risk averse, te manager as an increasing dislike for effort. Te manager is restricted from trading is stock options prior to its maturity wic occurs at t=1. Tis trading restriction precludes te manager from engaging in riskless edging strategies using is executive stock options. Altoug most executive stock options ave staggered maturity dates, lumping te maturity dates of all executive stock options to a single future date does not affect te general outcome of our results. At time 0, te representative sareolder of te firm and te manager negotiate a contract wic relates te manager s time 1 compensation to te firm s time 1 cas flows. Te firm s time 1 cas flows ave a random component and a component tat is directly te outcome of managerial effort. For simplicity we denote te firm s time 1 casflow as te sum of tese components, θ + e. θ is te random variable. t is te component of te firm s casflow wic is beyond te control of te manager. Te level of managerial effort, e = e(w,c), is a function of te compensation package. Tis compensation package comprises a state-independent salary, w, plus call options on a fraction, c, of te total sares of te firm were 0 c 1. Te sareolder can not give more tan e owns. As suc, c cannot exceed 1 nor be less tan zero. Te sare constitute a claim on te firm s time 1 cas flows, θ + e, net of te manager s state-independent salary, so its total price is θ + e-w. Let X be te time 1 exercise price of te option on te firm. Wen te time 1 net firm cas flow is equal to or exceeds X, te manager can exercise is stock option and claim a sare c of te firm upon payment of te exercise price of cx to te sareolder. Tus wen c(θ + e-w)-cx > 0, te manager gains from exercising is option. Let θ 0 be a borderline state. Let l denote te set of low states were te firm s net cas flow, θ + e -w < θ 0 so tat e does not exercise is options and terefore receives an income i l (θ) = w. Let denote te set of ig states wen net firm cas flow, θ + e -w θ 0, so tat e exercises is option and receives i (θ) = w + c(θ + e-w) - c X = w + c(θ + e-w - X). n oter words, te manager s total income, contingent on te state θ and te level of e, is given as follows: (l) f θ + e-w < θ 0, ten i l (θ) = w () f θ + e-w θ 0, ten i (θ) = w + c(θ + e-w - X ) = (1 - c)w + c(θ + e - X) At te borderline state, θ 0 =X-e+w, exercising te option gives te manager i (θ 0 ) = i l (θ 0 ) = w. Te Manager s utility function is composed of is expected utility for income in te low states, π l u(w,e), and is expected utility in te ig states, E [u((1-c)w + c(θ+e-x),e)]. π l is te probability of te low state and E [.] is our expectation operator for te ig state. For any given w,c, te manager s optimal coice of effort level is te solution to is expected utility maximization problem given by: Equation 1 MAX e R(w,c) = π l u(w,e) + E [u((1-c)w + c(θ+e-x),e)]

4 62 Journal Of Financial And Strategic Decisions were E l [u e ]+E [u e ] +E [u i c] = 0 and E l [u ee ] + E [u ee ] + E [u ii c 2 ] < 0 are te first and second order conditions, respectively. Applying te mplicit Function Teorem at te optimal level of e, it follows tat: Equation 2 de dc Similarly, Equation 3 de dw Fc E [ uiicz ] + E [ ui ] = = > 0 F 2 E [ u ] + E [ u ] + E [ u c ] e l ee ee ii Fw E[ uiic( 1 c )] = = < 0 2 F E [ u ] + E [ u ] + E [ u c ] e l ee ee ii Since w and c are substitute sources of income, an increase in c will be accompanied by a decrease in w. Hence te net cange in managerial effort from an increase in c will depend on te incentive effects from te increase in stock options and from te decrease in te fixed salary tat was substituted for te additional stock options. Furtermore, it will depend on te manager s marginal rate of substitution of te fixed salary received in all states for te contingent form of compensation. Teorem 1 proves tat te manager s compensated effort supply elasticity to an increase in stock options is positive. Tis suggests tat te net effect of an increase in te level of stock options, including te conditions stated above, is to increase te level of managerial effort. Tis confirms te positive net effect of stock options on te level of managerial effort. Teorem 1: Given a risk averse manager wit an increasing dislike for effort, is compensated effort supply elasticity to increases in stock options is positive and is given by: c θ + ε = > e e E[( e w X ) ui ] c ew 0 E [ u ] + E [( 1 c ) u ] l i i Proof: Consider te effect of a small cange w and c on te manager s expected utility. A small cange in te fixed salary, δw, on te manager s expected utility is given by: Equation 4 R( w, c ) δw = {El [u i ]+E [(1-c)u i ]}δw > 0 w Similarly 2, te effect of a small cange in te level of stock option, δc, is: Equation 5 R( w, c ) δc = {E [(θ+e-w-x)u i ]}δc > 0 c Substituting bot equations (4) and (5) into te definition for te slope of a compensated effort supply function, we obtain: Equation 6 R( w, c ) δc e e c E θ e w X ui ec c w R w c δw E u E c u e w [( + ) ] (, ) l( i ] + [( 1 ) i ] w were upon multiplying bot sides by c/e and applying te results of equations (2) and (3), it follows tat te compensated effort supply elasticity, ε, due to an increase in c is:

5 Executive Stock Options: Risk And ncentives 63 Equation 7 c θ ε = e e E [( + e - w - X)u i ] c E [u ]+ E [(1 - c)u ] l i i e w > 0 Q.E.D Managers wo ave positive effort supply elasticity are referred to as Type managers. Tese are managers wose level of efforts are motivated by te level of stock options and fixed salary in is compensation package. t is assumed tat te manager will exercise is stock options if te ig states occur. As suc, te sareolder s income wen teir firm is managed by a Type manager is contingent on state θ at time = 1 and te managerial effort. Te sareolder s income, (θ), is as follows: (l) f θ + e-w < θ 0, ten l (θ) = θ + e-w () f θ + e-w > θ 0, ten (θ) = (1-c)(θ + e-w) + cx = (1 - c)(θ + e) + cx - (1-c)w Te sareolder s problem is: MAX w c subject to: R(w,c) = u * H(w,c) = E l [U(θ + e-w)] + E [U((1 - c)( θ + e-w) + cx)] were R(w,c) = π l u(w,e) + E [u((1-c)w + c(θ + e-x),e)] is te manager s utility function given earlier by equation (2) and u * is te manager s reservation price. u * represents te manager s expected utility from te best alternative employment elsewere. Te closed form solution to te sareolder s optimization problem gives te optimal level of stock options, c, and fixed salary, w, comprising te optimal compensation contract. t is given by Proposition 1. THE PARETO OPTMAL CONTRACT Proposition 1: f compensation as an incentive effect on te level of managerial effort, ten to a second order approximation, te Pareto optimal level of stock option c satisfies: Equation 8 z ( 1 c ) AV z cav = B + ( 1 c ) b + ( 1 c ) eε c were V is te upper semi-variance of te firm s revenue in state, z is te expected incremental firm cas flow in te ig state, a and A are te manager s and sareolder s level of risk aversion, b and B are te marginal rate of substitution of low and ig income state incomes for te manager and te sareolder, respectively; e is te level of managerial effort, c is te fractional claim of te manager on te firm from is options, and ε is te manager s compensated effort supply elasticity. Proof: Te proof to Proposition 1 is available in Appendix. Remarks: Te optimal level of c is exact for managers and sareolders wose utility functions are quadratic. V E [(θ - θ ) 2 ] is te upper semi-variance of te firm s revenue in state. t is a measure of te volatility of te firm s earnings; z = θ + e-w-x is te expected incremental firm cas flow in te ig state. (Recall tat at te borderline state, X = θ 0 - e+w). t is te difference between te firm s value in te ig state and te exercise price. a -[u ii (i ( θ ))]/u i (i ( θ )) and A -[U ( ( θ ))]/U ( ( θ )) are te manager s and sareolder s Arrow-Pratt measures of absolute risk aversion, respectively; and B = E l [U ]/E [U ] and b = π l u i /E [u i ] are te marginal rate of substitution between ig income state and low income state for te sareolder and manager, respectively. cav and (1-c)AV can be interpreted as te marginal risk premia assigned by te manager and sareolder to te firm s returns, respectively. So tat z - cav and z - (1-c)Av are like risk-adjusted cas flows of te firm from te point of view of te manager and te sareolder, respectively.

6 64 Journal Of Financial And Strategic Decisions f stock options ave no incentive effects so tat manager s level of effort is unaffected by te level and combination of stock options and te fixed salary w in te compensation package, ten te pareto optimal compensation package for te manager of a risky firm is given by Proposition 2. We sall identify tese kinds of managers wose effort supply elasticity is equal to zero as Type managers. Tese are managers wose level of effort is invariant to te level of stock options and fixed salary in is compensation package. Tis lack of incentive effect can exist if regulations and/or te negative signalling effect of selling of is sares after te contract period preclude te manager from te economic benefit of firm ownersip gained troug te exercise of is stock option at te end of te contract period. Proposition 2: f compensation as no incentive effects, ten, to a second order approximation, te Pareto optimal level of c satisfies: Equation 9 z ( 1 c ) AV z cav = B + ( 1 c ) b + ( 1 c ) were te firm s incremental cas flow in te good states, z = θ -w-x, and te firm s volatility, V, ave no component attributed to managerial effort. Te parameters B, b, A and a are as identified earlier and carries a similar interpretation as tose noted in Proposition 1. Proof: Te proof to Proposition 1 is available in Appendix. Remarks: cav and (1-c)AV are te risk premia for te manager and te sareolder. Hence, z - cav is te risk-adjusted incremental firm cas flow for state as viewed by te manager. Wile z - (1-c)AV is te riskadjusted incremental firm cas flow for state as viewed by te sareolder. Tese risk-adjusted firm cas flows suggest tat giving ownersip in te form of a stock option is a means for risk sifting. Te optimality condition, equation (9), clearly sows tat as c increases, cav, te risk premium of te manager increases and tat of te sareolder, (1-c)AV, decreases. Given tat fixed salary and stock options are subtitutes, wen te manager receives a lower fixed salary in excange for more stock options, te fixed cost to te sareolder in all states is reduced. Tis reduced cost increases te sareolder income seltered from uncertainty in all states. Te risky component of te sareolders income in all state decreases, tereby reducing te risk borne per unit of income. On te oter and, te increase in te level of stock option in te managers compensation packages increases te risk per unit income borne by te manager. COMPARNG THE OPTMAL STOCK OPTON FOR THE TYPE AND TYPE MANAGERS Wen ε 0, te expression for te closed form solution for c is given by equation (8). t reduces to equation (9) wen ε = 0. Using te same parametric values for bot, te optimal level of c for a Type manager will differ from a Type manager. n te discussion below, we sow tat te optimal level of c for a Type manager wen ε 0 will be greater tan tat for a Type wen ε = 0, ceteris paribus. However wen te comparative analysis is done inte nest section, altoug te Type manager needs a iger level of stock options tan a type manager, te Type manager will require less frequent or lesser amount of compenation adjustments wen operating in a dynamic environment were parameters are likely to cange. Let C=1-c, were C is te fraction of te firm retained by te sareolder 3. Tis simplifies equation (9) into a more algebraically manageable function. Consider te case of a Type manager wose effort is invariant to te level of stock options, i.e., ε= 0. As a function of C, te optimal solution for tis manager is embodied in a familiar quadratic function wose solution can be illustrated grapically. t is te equation of a parabola: Equation 10 0 = C 2 (a+a) + C(Ba+bA-a) + (B-b)z/V-Ba were z z,v V (For notational convenience, te subscript are dropped). ts vertical intercept is (B-b)z/V- Ba <

7 Executive Stock Options: Risk And ncentives Te relevant 4 solution to tis quadratic function is te positive orizontal intercept given by C * in Figure 1. Te optimal level c * can be readily obtained by noting tat 1-C * =c *. On te oter and, te optimal level of C wen stock options motivate positive managerial effort (ε 0), as te case for te Type manager, is given by te solution to a nonlinear function made up of (i.) a non-linear function wic is asymtotic to C=1, is a decreasing function of C and as a negative vertical intercept -eεbb < 0 and, (ii.) a parabolic function wose vertical intercept is (B-b) z/v - Ba < 0. Tis was obtained by rewriting equation (8) as a function of C. Were upon substitution, (i.) is embodied on te left-and side and (ii.) is embodied in te rigtand side in te function below Equation 11 2 eε Bb C( B + b) C + + = C 2 (a+a)+ C(Ba+bA-a) + (B-b) z/v- Ba V 1 C 1 C 1 C Te solution to tis non-linear function can be grapically illustrated by te intersection of two lines: one line represents a parabola wit a vertical intercept of (B-b) z/v- Ba < 0; and te oter is a line asymtotic to C=1, is a decreasing function of C, and as a negative vertical intercept -eεbb < 0. Wen ε 0, te solution is te intersection of tese two lines and is identified in Figure 1 by C e*. Using te same parametric values, for te quadratic function in equation (10) and te non-linear function in equation (11), it follows from te grapical representation of teir solutions tat C e* <C *. Tat is, te optimal fractional level of te firm retained by te sareolder is less wen stock options ave incentive effects. Since C=1- c, it follows tat te optimal level of executive stock options will be iger for te manager wose effort is motivated by te level of stock options, Type manager. Furtermore, it is iger te greater te incentive effects of te stock options. Tis follows since wit iger compensated effort supply elasticity, te vertical intercept of te asymptotic line sifts down tereby reducing te value of C e* were te two lines intersect. Given te same value for z in bot (11) or (10), te z produced by te Type wit te stronger incentive effects will suggest tat a iger portion of tis cas flow and firm value is due to managerial effort. Tis suggests tat at time 1 te firm being divided up via te manager s exercise of is stock option as a portion coming from te manager s contribution to its size. On te oter and, te z produced by te Type manager as no effort contribution. Te Type manager appears to be compensated for risk bearing alone. Tus it is not inconsistent tat te level of stock option for tis type of managers sould be less tan tat for te manager wose efforts are affected by te stock option level. COMPARATVE STATC ANALYSS deally one would wis tat te operating parameters on wic te Pareto optimal contract was based upon would remain te same trougout te contract period. n reality tis is ard to acieve, given tat tere are events and market developments tat will permanently cange te operating parameters of te firm wic are beyond te control of eiter te manager or te sareolder. Tese unexpected market developments can appen midstream of te duration of te compensation contract and cange te firm s expected cas flow and volatility. Furtermore, risk attitudes generally cange troug time. As suc, a long-term executive contract wic awards executive stock options could easily be far from its optimal at some later time, tereby defeating te incentive effectiveness of te contract at te time it was signed. n te discussion following, te paper analyzes te direction and relative magnitude of adjustment to te initial pareto optimal contract wen tere are some perturbation in te value of some firm parameters, suc as an increase in te expected firm cas flow, te volatility of te firm s random cas flows, and te risk attitude of te manager. n oter words, it addresses te sensitivity of te pareto optimal contract to parametric canges. Sensitivity of te contract, and consequently, its frequency of adjustment are related to te type of managers running te firm. Te analysis below compares level of adjustment needed to restore pareto optimality or te relative degrees of sensitivity of contract between a Type manager and a Type manager. Te comparative analysis of te sensitvity of te optimal level of C to parametric canges is obtained by applying te mplicit Function Teorem (FT) to te optimality conditions (11) and (10) (wic are alternative expressions for equations (8) and (9), respectively). Comparing tese results, one obtains an expression wic tells

8 66 Journal Of Financial And Strategic Decisions us ow te optimal stock option canges wit perturbations in one of te parameter. 5 Te sareolder can use tese results to determine te direction and relative amount of adjustments to te level of executive stock option to maintain pareto optimality. Knowing te sensitivity of te contract enables te sareolder to determine te frequency of renegotiation or adjustment of long term executive contracts. Sensitivity Of Contract To Canges n Risk Attitudes Suppose tat events and market conditions causes a permanent increase in a, te absolute risk aversion of te manager. Tis will result in te manager s assigning a iger risk premium for risk bearing. One solution is to modify te contract wit more stock option wit an accompanying decrease in te fixed salary, since it is assumed tat c and w are substitutes. Anoter is to decrease te level of stock option and increase te fixed salary. t suggests tat to maintain Pareto optimality, te optimal fraction of te firm wic te manager receives troug is stock options sould decrease for eiter type of manager. Applying te mplicit Function Teorem (FT) to te optimality conditions (10) and (11) and comparing te results using te same values for te oter parameters in bot expression: Equation 12 dc da Fa V ( 1 C )( C + B ) V(1 - C)(C + B) = = > F C a + A + ab + Ab a V [2C(a + A)+ ab+ Ab - a]v + > 0 C [ 2 ( ) ] γ no incentive effects Type wit incentive effects Type eε eε were b > B 1, C 1 and γ = [-Bb+cB+Cb+C] + [B+b+2z] > 0. 2 ( 1 C ) ( 1 C ) (11) sows tat an increase in te manager s risk aversion requires an increase in te ownersip sares retained by te sareolder. n oter words, an increase in te risk aversion of te manager will require a smaller amount of is income in te form of stock options. Type managers will want less stock options as part of teir compensation package as tey become more risk averse. Tis comes as no surprise because as managerial risk aversion increases, te manager will prefer to ave less of is income subject to uncertainty tereby increasing te relative proportion of is fixed salary, w. Tis increases te proportion of te manager s riskless income and reduces is risk bearing. Relative to a Type manager, te Type manager needs smaller or less adjustments to is stock options for te same cange in risk aversion because tis manager can mitigate te effect of increased risk aversion on te risk premium, cav, by producing iger expected firm cas flow troug increased effort, tereby restoring te te risk-return trade-off of is compensation package. Sensitivity To Canges n Te Level Of Te Firm Expected Cas Flow Firm parameters tat are often monitored are performance measures associated wit te firm s cas flow suc as sales, net income and stock price. However, structural canges in te market can cange permanently te distribution of te firm s cas flow and its expected value. As suc, an increase in te expected value of te firm value wile keeping te volatility constant will result in an increase in value of te manager s stock option due to te better cange of iger cas flow in te good states. Tere is now a iger cance tat is stock options will be in-te-money at expiration. For te Type manager wose effort is invariant to te level of stock options so tat stock options appears solely to serve as compensation for risk bearing, te adjustment to is package is obvious. To maintain pareto optimality, te level of te stock options as to be decreased. n so doing, te increased expected payoff to te manager is offset by te reduced number of stock options. Tis solution may not be too obvious for te Type manager wose effort partly depends on te level of stock options, because decreasing te level of stock option may decrease is level of effort terefore defeating te effectives of stock options as an incentive tool to improve long run firm performance. Applying te mplicit Function Teorem to te Pareto optimality conditions (10) and (11) and comparing te magnitude of teir absolute values, equation (15) sows tat te fraction of te firm retained by te sareolder, C,

9 Executive Stock Options: Risk And ncentives 67 increases for bot te Type and Type managers. However, te magnitude of te cange in ownersip retained by te sareolder is larger wit a Type manager and lesser wit a Type manager. Equation 15 dc dz Fz = = F C 2 B b B - b C a + A + ab + Ab a > [2C(a + A)+ ab+ Ab - a]v + > 0 ( ) γ no incentive effects Type wit incentive effects Type n terms of te manager s stock options, tere will be less of a decrease in te contingent ownersip claim of te Type manager wose effort depends on stock options. Type managers will experience a greater decrease in stock options tan Type managers for te same amount of increase in firm s expected cas flow z, because suc an increase in z on te part of te Type manager amounts to a windfall increase for firm cas flow, and consequently te value of is stokc options, wit out te benefit of incentive effects or te supply of effort. On te oter and because Type manager effort is affected by te level of stock options, dropping te level of stock options as muc as tat for a Type manager can ave an undesirable incentive effect on te level of effort tat a Type manager will expend to increase furter firm cas flow. Hence te adjustment will not be as drastic as tat for a Type manager. Assuming expected cas flows affect directly te market value of te firm, tis inverse relationsip between canges in te firm performance and te executive s total option value is consistent wit te inititally puzzling empirical results found in Tables 6 and 8 of Murpy (1985) wic suggest tat te executive s total option value ave a negative time series coefficient wit stock price and sales. Effect Of An ncrease n Te Volatility Of Te Firm s Cas Flow Consider te manager wose income comes from a state-independent fixed salary and a state contingent income, te stock option wic is te risky component of is income. Te manager can estimate is total expected income if te distribution of te firm s cas flow is given. Te pareto optimal compensation package corresponds to a certain acceptable risk borne per unit expected income. Suppose expected cas flow is to remain te same, but V, te upper semi-variance of te firm s cas flow, increases due to some events in te market. Tis increase will result in te manager experiencing a iger risk borne per expected income. Tis decreases te manager s welfare and te original pareto optimal compensation package will no longer be optimal. Unlike an increase in risk attitude, an increase in te volatility of te firm s cas flow increases te risk premia of bot te sareolder s, (1- c)av, and tat of te manager,cav. Looking at equation (9), tis statement is self-evident. For a Type manager wose effort is invariant to stock options and for wom stock option serves as compensation for risk-bearing, one would guess tat, if te volatility of firm cas flow increases due to some externalities and te manager is not motivated by te possibility of improved income from options oldings, te level of stock options sould be increased to maintain pareto optimality. Tis can be verified by applying te mplicit Function Teorem to equation (10). t indicates tat as te volatility of firm cas flow increases, te ownersip retained by te sareolder sould decrease. Equation 13 dc dv = FV = ( B b) z / V F [ 2C( a + A) + ab + Ab a ] V C 2 < 0 Q.E.D. Tis suggests tat additional stock options be provided to a Type manager to compensate im for increased volatility of firm cas flow and for more risk bearing. Te opposite adjustment is needed for a manager wose effort increases wit te level of stock options and wit wom stock options ave strong incentive effects. An increase in te volatility of earnings requires a decrease in te level of stock options to maintain pareto-optimality. Altoug te risk per unit expected return increases wit increased firm volatiltiy, an increase in te volatility of te firms earnings does not necessary make im worst off,

10 68 Journal Of Financial And Strategic Decisions if e can maintain a certain risk return trade-off wit te efforts expended to increase firm expected cas flows. Tis is true if te manager s effort supply elasticity is ig suc tat eε(b+c)(b+c)/v 2 > - (B-b)/V 2 or, te increase in income generated from effort compensates for te te incremental risk borne, tereby keeping is risk-return trade-off at te same level. Tis conclusion can readily be obtained by applying te FT to equation (11) and realizing tat an increase in te ownersip retained by te sareolder translates to fewer stock options for te manager. Equation 14 dc dv = FV = ( B b) z / V eε( B + C )( b + C ) / V F [ 2C( a + A) + ab + Ab a ] V + γ C 2 2 > 0 Q.E.D. To maintain proper incentives under volatile market conditions, tis result suggests tat te sareolder appears better off iring managers motivated by contingent sare ownersip via stock options. n bot cases, we let b > B so tat it follows tat te sareolder s preference for favorable states of nature, or for income in tese ig states, is relatively greater tan tat of te manager. Tis is probably not an unrealistic assumptions, because sareolders as principal owners of te firm sould be inerently more interested in te favorable performance of te firm tan teir agents, te managers. CONCLUSON Te analysis of executive stock options provided in tis paper limits itself to a principal-agent framework wen edging and trading of executive stock options are restricted during te contract period. Te closed form solutions for te optimal contract comprise a compensation package wit only a fixed salary and stock options. Not surprisingly, we find tat te optimal level of executive stock options is greater for managers motivated by te incentive effects of stock options. For managers wose effort is invariant to te level of teir stock options, te sole function of options is risk bearing and te level of stock options is less. Wile te sareolder s fraction of ownersip retained will be lower wen tey ire managers wo are motivated by stock options, te sareolder can benefit by aving a less volatile ownersip claim. Tis effect is especially relevant if we ave sareolders wo are concerned over te certainty of firm ownersip and control for firms operating in dynamic environments wit uncertainties about firm cas flows and risk attitudes. An interesting result wic follows from te analysis is tat wen te volatility in te firm s cas flow increases, pareto optimality in te compensation contract for a manager wit incentive effects requires a decrease in is stock options. Tis increases te fractional ownersip of te sareolder. Wereas for a manager wit no incentive effects, an increase in te volatility of te firm s cas flow requires an increase in stock options. Anoter interesting result is tat compensation contracts wit stock options stay near pareto optimality. Sareolders benefit from tis in two ways. First, sareolder wealt increases since managerial incentives remain continuously aligned wit tose of teir owners. Secondly, sareolders avoid te costs associated wit frequent recontracting wit managers. Similarly, for eiter canges in managerial risk aversion or expected firm cas flows, te adjustment to maintain a pareto optimal compensation contract is smaller if stock options are present. Tese are true for contracts drawn wit managers wose efforts are motivated by stock options. Tus wile tese manager motivated by stock options (Type ) inititally requires a iger level of stock options tan oter managers, tere are tree benefits to te sareolder to iring tem. One, te sareolder s ownersip is less volatile in a igly dynamic environment wen parameters are constantly canging. Two, under certain conditions, te sareolder s retained ownersip may actually be iger tan wat e anticipated. Tree, if stock options motivates positive effort supply elasticity, te value of te firm increases and, figuratively speaking, te sareolder ends up retaining, wit less uncertainty, a larger sare of a bigger pie.

11 Executive Stock Options: Risk And ncentives 69 ENDNOTES 1. At te border: i (θ 0 ) =(1-c)w +c(θ 0 -X). Since θ 0 = X+w, it follows tat i (θ 0 ) = (1-c)w + c(x+w-x) = w. 2. Tese two expressions ave been considerably simplified by applying te first order condition, E l[u e]+e [u e] +E [u ic] = By replacing c by 1-C, we are able to express te solution in terms of familiar functional forms and tus facilitate our analysis. 4. Since bot sareolder can ave only long positions in te firm, we disregard solution wit negative values of C or c. 5. Anoter way to determine te cange in te optimal level of C wen stock options ave incentive effects is to trace te sift of te point of intersection of te two lines (given by te LHS and RHS of (11) wen a parameter is canged. Tis can be illustrated by using Figure 1, as well. For te case wen tere are no incentive effects, one needs only to note te cange in te positive orizontal intercept of te parabola given be (10). 6. E [θ(θ - θ )] = E [(θ - θ )(θ - θ )] + θ E [(θ - θ )] = E [(θ - θ ) 2 ] + 0 REFERENCES [1] Agrawal, Anup and Gerson Mandelker, Managerial ncentives and Corporate nvestment and Financial Decisions, Journal of Finance 42, 1987, pp [2] Beck, Paul and Tomas Zorn, Managerial ncentive in te Stock Market Economy, Journal of Finance 37, 1982, pp [3] Berle, Adolp and Gardiner Means, Te Modern Corporation and Private Property 1932, (McMillan,New York, NY). [4] Brickley, James, Sanjai Bagat and Ronald Lease, Te mpact of Long-Range Managerial Compensation Plans on Sareolder Wealt, Journal of Accounting and Economics 7, 1985, pp [5] Couglan, Anne and Ronald Scmidt, Executive Compensation, Management Turnover, and Firm Performance, Journal of Accounting and Economics 7, 1985, pp [6] Fama, Eugene, Agency Problem and te Teory of te Firm, Journal of Political Economy 88, 1980, pp [7] Grossman, Sanford and Oliver Hart, An Analysis of te Principal-Agent Problem, Econometrica 51, 1983, pp [8] Haugen, Robert and Lemma Senbet, Resolving te Agency Problems of External Capital Troug Options, Journal of Finance 36, 1981, pp [9] Harris, Milton and Artur Raviv, Optimal ncentive Contracts wit mperfect nformation, Journal of Economic Teory 20, 1979, pp [10] Holmstrom, Bengt, Moral Hazard and Observability, Bell Journal of Economics 10, 1979, pp [11] Jensen, Micael and William Meckling, Teory of te Firm: Managerial Beavior, Agency Costs and Ownersip Structure, Journal of Financial Economics 3, 1976, pp [12] Lambert, Ricard A., William N. Lanen and David F. Larcker, Executive Stock Option Plans and Corporate Dividend Policy, Journal of Financial and Quantitative Analysis 24, 1989, pp [13] Lewellen, W., C. Loderer, and A. Rosenfeld, Merger Decisions and Executive Stock Ownersip in Acquiring Firms, Journal of Accounting and Economics 7, 1985, pp [14] McConnell, Jon and Henri Servaes, Additional Evidence on Equity Ownersip and Corporate Value, Journal of Financial Economics 27, 1990, pp

12 70 Journal Of Financial And Strategic Decisions [15] Morck, Randall, Andrei Sleifer and Robert Visny, Management Ownersip and Market Valuation an Empirical Analysis, Journal of Financial Economics 20, 1988, pp [16] Murpy, Kevin, Corporate Performance and Managerial Remuneration: An Empirical Analysis, Journal of Financial Economics 7, 1985, pp [17] Savell, Steven, Risk Saring and ncentives in te Principal and Agent Relationsip, Bell Journal of Economics 10, 1979, pp [18] Stiglitz, Josep, ncentives, Risk, and nformation: Notes Towards a Teory of Hierarcy, Te Bell Journal of Economics 6, 1975, pp FGURE 1 Te Optimal Level Wen Tere Are ncentive Effects, C e*, s Te ntersection Of Te Two Non-Linear Functions f(c) εebb V (B-b)z/V-Ba C e* C* C 2 (a+a)+c(ba+ba-a)+(b-b)(z/v)-ba C C=1 2 Bb C( B + b) C εe C 1 C 1 C V

13 Executive Stock Options: Risk And ncentives 71 APPENDX Proof to Proposition 1 Forming te Lagrangean for te sareolder s maximization problem wit a Type manager. Equation a.1 L = H(w,c) + λ(u* - R(w,c)) ts first order conditions are: Equation a.2 L c = E l[u e c ] + E [(1-c)U e c - E [(θ+e-w-x)u ] + λ{-e l [u e e c ] - E [(θ+e-w-x)u i - E [u e e c ] - E [cu i e c ]} = 0 Equation a.3 L w = E l[u e w ] + E l [-U ] + E [-(1-c)U + E [(1-c)U e w ] + λ{-e l [u e e w ] - E l [u i ] - E [(1-c)u i ] - E [u e e w ] - E [cu i e w ]} = 0 Applying te Envelope Teorem we can simplify equations (a.2) and (a.3) to Equation a.4 L c = {E l[u ] + E [(1-c)U ]}e c - E [(θ+e-w-x)u ] - λe [(θ+e-w-x)u i ] = 0 Equation a.5 L w = {E l[u ] + E [(1-c)U }e w -E l [U l ] - E [(1-c)U ] - λ{e l [u i ] + E [(1-c)u i ]} = 0 Dividing equation (a.4) by equation (a.5) yields: Equation a.6 {E l [U ]+ E [(1 - c)u ]}e c - E [( θ+ e - w - X)U ] = E [U ] + E [(1 - c)u }e - E [U ] - E [(1 - c)u ] l w l Rearranging terms: E [( θ+ e - w - X)u i ] E [u ]+ E [(1 - c)u ] l i i Equation a.7 {E l [U ] + E [(1-c)U ]}e c - E [(θ+e-w-x)u ] = + E [( θ+ e - w - X)u i ] E [u ]+ E [(1 - c)u ] l i i E [( θ+ e - w - X)u i ] E [u ]+ E [(1 - c)u ] l i i {E l [U ] + E [(1-c)U }e w {- E l [U ] - E [(1-c)U ])}

14 72 Journal Of Financial And Strategic Decisions and dividing by {E l [U ]+E [(1-c)U ]} Equation a.8 E [( θ+ e - w - X)U l ] e c - E [U ]+ E [(1 - c)u ] Hence, Equation a.9 l E [( θ+ e - w - X)u ] E [u ]+ E [(1 - c)u ] e w -1 i = ( ) l i i e c - E [( θ+ e - w - X)u i ] E [u ]+ E [(1 - c)u ] l i i e w = E [( θ+ e - w - X)U ] E [U ]+ E [(1 - c)u ] l - E [( θ+ e - w - X)u i ] E [u ]+ E [(1 - c)u ] l i i Applying Teorem 1 to te LHS, we obtain: Equation a.10 eε E = [( θ+ e - w - X)U ] c E [U ]+ E [(1 - c)u ] l E [( θ+ e - w - X)u i ] - E [u ]+ E [(1 - c)u ] l i i were te first expression (actually, its inverse) on te RHS upon dividing and multipying by E [U ] becomes: Equation a.11 E l [U ]+ E [(1 - c)u ] = E [U ( + e - w - X] θ E l [U ] / E [U ] +(1 - c) E [ θu ] E [U ] + e - w - X Applying a Taylor expansion to E [ θ U ] about θ we obtain: E [U ] Equation a.12 E l [U ]+ E [(1 - c)u ] B+(1 - c) = E [( θ+ e - w - X)U ] z - (1 - c)av Tese are parameters associated wit te sareolder. V = E [(θ- θ ) 2 ] is te measure of uncertainty of firm output and is income in ig income state ; A - U ( (θ))/u ( ( θ )), te sareolders absolute risk aversion; z = θ +e-w-x is te mean net gain in te ig income state, unadjusted for risk and ownersip sturcture; and B E l [U ]/E [U ] is a measure associated wit te sareolders preference for ownersip income. Te parameters associated wit te manager can likewise be obtained by multiplying and dividing te second term in equation (a.10) by E [u i ], and applying a Taylor expansion about θ to obtain: Equation a.13 E l [u i ]+ E [(1 - c)u i ] b+(1 - c) = E [( θ+ e - w - X)u i ] z - cav were a -u ii (i ( θ ))/u i (i ( θ )) is te manager s absolute risk aversion at ig incomes and b π l u i /E [u i ] is associated wit te manager s preference for ownersip income. Applying te results in equations (a.12) and (a.13) into equation (a.10), we obtain an equation for te optimal level of c, wen performance pay as an incentive effect on te te level of effort. Equation a.14 e c ε z - (1- c)av = B+(1 - c) - z - cav b+(1 - c) Q.E.D.

15 Executive Stock Options: Risk And ncentives 73 Proof to Proposition 2 Te lagrangean of te sareolder maximization problem wit a Type manager: Equation a.15 L = H(w,c) + λ(u * - R(w,c)) wit te first order conditions being: Equation a.16 and L w = -E l[u ] - E [(1-c)U ] + λ{-π l u i - E [(1-c)u i ]} = 0 Equation a.17 L w = -E [(θ-w-x)u ] + λ(-e [(θ-x-w)u i ]) = 0 Eliminating λ by dividing equation (a.16) by equation (a.17) we obtain a measure for te sareolder s and manager s marginal rate of substition given by: Equation a.18 E l [U ]+(1 - c)e [U ] = π lu i +(1 - c)e [u i ] E [( θ - w - X)U ] E [( θ - X - w)u ] i were te RHS (LHS) gives te manager s (sareolder s) expected marginal utility of income in all state to expected marginal utility of income in te ig state, given te contractual arrangement. Dividing and multiplying te LHS by E [U ], and te RHS by E [u i ], and using te identitiy E [(θ-w-x)u ] E [θu ]-wu -XU, we obtain: Equation a.19 E l [U ] / E [U ] +(1 - c) E [ θu ] / E [U ] - w - X = πlu i / E [u i ]+(1 - c) E [ θu ] / E [u ] - w - X i i Te expressions E [ θ U ] and E [ θ u i ] in te denominator of equation (a.19) can be simplified to more familiar E [U ] E [u ] economic terms as follows: Using a Taylor expansion of U U ( (θ)) about θ Equation a.20 U [ (θ)] = U [ ( θ )] + (θ- θ )(1-c)U [ ( θ )] i were θ = E [θ] and ( θ ) (1-c) θ + cx - (1-c)w is te sareolder s mean income in ig states. Applying te expectation operator E [.] to bot sides and using te fact tat E [θ- θ ] = 0, equation (a.20) becomes: Equation a.21 E [U ] = U ( ( θ ))

16 74 Journal Of Financial And Strategic Decisions Multiplying equation (a.20) by θ, and ten applying te expectation operator to bot sides, we get: Equation a.22 E [θu ] = E [θu ( ( θ ))] + E [θ(θ- θ )(1-c)U ( ( θ ))] Upon dividing by equation (a.21), we obtain: Equation a.23 E [ θu ] = θ E [U ] + E [θ(θ- θ ](1-c) U ( ( θ )) U ( ( θ )) were E [θ(θ- θ )] = E [(θ- θ ) 2 ] is te variance 6 of te ig outcome states wic we will denote by V. Tis U ( ( )) measures te degree of uncertainty of income in state. - θ = A is te sareolder s absolute risk aversion U ( ( θ )) for state income. Tus, equation (a.23) becomes: Equation a.24 E [ θu ] = θ - (1-c)AV E [U ] Since θ is always greater tan zero in state and te marginal utility of income is always positive, it follows tat te RHS of equation (a.24) is also greater tan zero, or: Equation a.25 θ - (1-c)AV > 0 Following a similar approac, E [ θ u i ] from equation (a.19), associated wit te manager s utility in te ig E [u ] income states, can be sown to be: Equation a.26 E [ θu i ] = θ - cav > 0 E [u ] were a = - i i u ii(i ( θ )) is te manager s absolute risk aversion associated wit state income. u ( i ( θ )) i Substituting equations (a.26) and (a.24) into equation (a.19) and rearranging terms, we obtain a closed form solution determining te optimal level of c: Equation a.27 θ - w - X - (1 - c)av E [U ] / E [U ]+1 - c = l θ - w - X - cav u / E [u ]+1- c πl i i Q.E.D Tis is a quadratic equation in c were te LHS contains parameters relating to te sareolder wile te RHS contains parameters relating to te manager.