VALUING THE RELOAD FEATURES OF EXECUTIVE STOCK OPTIONS

Transcription

1 VALUING THE RELOAD FEATURES OF EXECUTIVE STOCK OPTIONS Jane Saly, Unversty of Mnnesota Rav Jagannathan, Northwestern Unversty and Steven Huddart, Pennsylvana State Unversty SYNOPSIS: For optons wth a reload feature, the holder s automatcally enttled to new optons when the ntal opton s exercsed. Under Statement of Fnancal Accountng Standards No. 123, the grant date value of executve stock optons excludes the value of a reload feature because the Fnancal Accountng Standards Board beleves t s not feasble to value a reload feature at the grant date. We show how the Bnomal Opton Prcng Model can be used to value optons and the reload feature at the grant date. Ignorng the reload can substantally understate the value of the opton. Accordngly, the Fnancal Accountng Standards Board may wsh to reconsder the accountng for reload features. JEL Classfcaton: C80, D84, J33, M12, M41 Keywords: compensaton, valuaton, bnomal opton prcng model, restoraton, replacement, contnuaton, replenshment, accelerated ownershp Ths draft: Aprl 30, 1999 An earler verson of ths paper, by Rav Jagannathan and Jane Saly was ttled ÒIgnorng reload features can substantally understate the value of executve stock optons.ó Steven Huddart extended the analyss to consder cases of multple and unrestrcted reloads. Rav Jagannathan and Jane Saly thank Bob McDonald and Judy Rayburn for helpful dscussons. Steven Huddart partcularly thanks Bob Whaley for helpful nsghts and encouragement. Send correspondence to: Professor Jane Saly Carlson School of Management Room th Avenue South Mnneapols, MN telephone: facsmle: e-mal: saly@csom.umn.edu

2 INTRODUCTION Stock optons are an ncreasngly popular method for compensatng executves. In addton, executve stock optons often nclude unque features that are not found n exchange-traded optons. One such feature, the reload, enttles the holder to automatcally receve new optons when the orgnal opton s exercsed. 1 As we show, a reload feature can add consderably to the optonõs value, whch may be one explanaton for the ncreasng frequency of ther use. It s mportant to measure the value that the feature adds to an opton. Under SFAS No. 123, the reload feature s not valued at the ntal grant date, but when the orgnal opton s exercsed and the new reload optons are ssued. In 1995, the Fnancal Accountng Standards Board (FASB) recommended ths delay n recordng the value: The Board contnues to beleve that, deally, the value of an opton wth a reload feature should be estmated at the grant date, takng nto account all of ts features. However, at ths tme, t s not feasble to do so. Accordngly, the Board concluded that the best way to account for an opton wth a reload feature s to treat both the ntal grant and each subsequent grant of a reload opton separately. (Statement of Fnancal Accountng Standards No. 123, 186, p. 61) As Arnason and Jagannathan (1994) show, t s feasble to value the executve stock optons ncludng ts reload feature at the date of grant usng the Bnomal Opton Prcng method. In ths paper, we examne how the value added by the reload feature depends on characterstcs of the frm and the executve stock opton. We then provde an algorthm 2

3 for valung executve stock optons wth reload features. Our man example s a Norwest Corporaton opton grant wth a sngle reload. The number of new optons granted under the provson equals the number of shares tendered to pay the exercse prce plus any taxes due on exercse. In ths case, the reload feature adds 24 percent to the value of a comparable opton wthout ths feature. The value added by the reload provson vares wth dvdend yeld, volatlty, and terms of the feature. For example, f the reload feature for the Norwest opton were changed to enttle the holder to new optons equal to the number of shares tendered to pay the exercse prce alone, then the feature would add 15 percent to the value of a comparable opton wthout a reload feature. The mpact of other factors s shown below. Accordngly, the Fnancal Accountng Standards Board may wsh to reconsder ts recommendaton for reload optons. The next secton reports on the prevalence and terms of reload features. We then present the value of our example opton wth a reload and an analyss of how the value of the reload feature vares wth dvdends, volatlty, term to expraton, number of reloads allowed, and number of optons granted at reload. After a dscusson of some lmtatons of opton prcng models, we conclude. For readers nterested n applyng the valuaton technque, we provde three appendces. Appendx 1 descrbes how to value a conventonal opton n a smple 3-perod example and how to add a sngle reload feature to that value. Appendx 2 descrbes the general mathematcal model and extends t to the case of multple reloads ncludng an unlmted number of reloads. Appendx 3 provdes Mathematca computer code for valung reload optons. 3

4 PREVALENCE AND TERMS OF RELOAD OPTIONS A reload feature automatcally grants new optons to an executve when the orgnal optons are exercsed. The executve must relnqush shares of stock, nstead of cash, to meet the exercse prce. The exercse prce of reload optons s typcally set equal to the frmõs market prce when the orgnal optons are exercsed. They expre at the same tme as the orgnal optons. A conventonal opton granted wth a reload feature s called a frst-generaton opton to dstngush t from the reload opton granted when the frst-generaton optons are exercsed, whch we label a second-generaton opton. The number of reload optons granted when frst-generaton optons are exercsed vares by frm. Commonly, one second-generaton opton s granted for each share tendered by the executve n payment of the exercse prce on the orgnal optons. 2 Sometmes the second-generaton optons themselves may be reloaded; we refer to ths case as multple reloads. Hence, a comprehensve valuaton procedure must ncorporate both the number of tmes a reload s allowed and the number of second-generaton optons granted per frst-generaton opton exercsed. An opton wth a reload feature (frst-generaton opton) s more valuable than a comparable conventonal opton (.e. an opton wthout a reload feature). The holder of an opton wth a reload feature benefts from the ablty to exercse exstng optons and lock n a gan, and stll hold optons for future exercse. Frms clam that reload optons encourage stock ownershp and help keep executvesõ compensaton ted to frm performance (Gay, 1999). The popularty of reload features s ncreasng n many ndustres. As reported n Table 1, from 1992 to 1997, a rsng fracton of all opton grants to executves are grants 4

5 of reload optons (second- or later generaton grants). These fgures are a lttle msleadng snce frms are only requred to report reload fgures for second-generaton grants. The number of optons wth a reload feature (frst-generaton grants) s ncluded wth conventonal optons n number of opton grants. Hence, there must be more opton grants wth reload features that have not yet been trggered. Table 2 reports that executves who have second-generaton optons are employed by larger frms. In addton, grants of reload optons are more common n fnancal servces than n other ndustres. A fnancal servces frm, Norwest Corporaton, was the frst to offer optons wth a reload feature (Gay, 1999). [Table 1] [Table 2] The terms for reload features also vary across frms. Norwest Corporaton allows only one reload, whereas Frst Bank System allows up to three reloads, and Frst Chcago places no lmt on the number of tmes an opton may be reloaded. The number of second-generaton optons granted under the reload feature also vares substantally across frms. Amgen offers new optons equal n number to the shares tendered n payment of the exercse prce, whereas Frst Bank System and Norwest Bank offer new optons granted equal n number to the shares tendered to pay both the exercse prce and any taxes that become due on exercse. Both the number of reloads allowed and the number of new optons granted per opton exercsed, affect the executveõs choce of when to exercse, and the value of the 5

6 reload feature. Hemmer et al. (1998) show that, f the number of reloads allowed s unrestrcted and a new opton s granted for every share tendered n payment of the exercse prce, then t s optmal to exercse whenever the current stock prce rses above the exercse prce. Ths s a useful observaton on the optmal exercse polcy n ths specal case, but practcal questons regardng valuaton and optmal exercse strategy cannot be addressed usng ther approach. Hemmer et al. do not address exercse polcy when the optons may only be reloaded a fxed number of tmes, or when the number of new optons granted dffers from the number of shares tendered n payment of the exercse prce. In ths paper, we provde an algorthm that determnes the grant date value of an opton wth a reload feature for any number of reloads (ncludng an unrestrcted number), and for any prescrbed formula of new optons granted per opton exercsed. Computatonally, ths method dffers from the one proposed n Hemmer et al., but provdes the same result n the cases they consder. VALUING THE RELOAD FEATURE OF AN EXECUTIVE STOCK OPTION In ths secton, we examne a partcular executve stock opton wth a reload feature to llustrate our pont that gnorng the reload feature can substantally understate the value of the opton. We then demonstrate how dvdends, volatlty, term to expraton, the number of reloads allowed, and number of optons granted at reload affect the value added by the reload feature. We adopt the Bnomal Opton Prcng Model (Cox, Ross, and Rubensten, 1979) rather than the Black-Scholes Model (Black and Scholes, 1973) because the Bnomal 6

7 Opton Prcng Model s more versatle n ncorporatng early exercse and addtonal features. Snce a full explanaton of how to value optons wth reload features s necessarly techncal n nature, we have chosen to llustrate and explan t n the Appendces. A study of the materal n the appendces s not necessary for the dscusson that follows. A Reload Feature Adds Substantally to the Value of the Opton To llustrate how the reload feature affects the value of an opton, we use the example of Mr. R. Kovacevch, CEO of Norwest Corporaton, who was granted 138,000 optons to buy stock at $14.53 per share n May The optons became exercsable n 1994 and have a one-tme reload opton f Norwest stock s tendered n payment of the exercse prce. Norwest grants reload (second-generaton) optons equal to the number of shares tendered to pay the exercse prce plus any taxes owed n connecton wth the exercse. 3 These reload optons have the same expraton date as the orgnal optons and an exercse prce equal to the stock prce on the day the reload feature s trggered. Snce only one reload s allowed, the reload optons themselves do not have a reload feature. If Kovacevch chooses to exercse the optons upon vestng n 1994 when the stock prce s $26, he wll pay to the corporaton 138,000 _ $14.53 = $2,005,140 and receve 138,000 shares, each worth $26 or $3,588,000 n total. Assumng that the margnal ncome tax rate for Kovacevch s 48.1%, the tax payable upon exercsng the optons wll be _ ($26-$14.53) _ 138,000 = $761,356. If Kovacevch pays the exercse prce and taxes wth shares he already owns (each worth $26), he wll have to 7

8 pay a total of $2,005,140 + $761,356 = $2,766,496 wth 106,404 shares (.e., $2,766,496 dvded by $26 per share). Thus, Kovacevch gves up hs 138,000 optons and 106,404 shares of stock to the company, and receves n return 138,000 shares of stock and 106,404 new optons wth an exercse prce of $26 and 7 years to expraton. We can wrte the formula for the number of second-generaton optons receved per frstgeneraton opton exercsed as Z = [X+.481(S - X)] / S (where X s the exercse prce and S s the current stock prce at the tme of exercse). We need the followng nputs for usng our method (descrbed n Appendx 2) to value the opton at grant date: the current stock prce at the date of the ntal (frstgeneraton) grant, $14.53; the exercse prce, $14.53; the tme to expraton, 10 years; the dvdend yeld, 3%; the annual stock prce volatlty, 27.3%; the rsk-free rate, 7% smple nterest; and KovacevchÕs margnal tax rate, 48.1% (Appendx 2 n Arnason and Jagannathan, 1994). We represent stock prce movements over the lfe of the opton usng a bnomal tree wth one step per month. The grant-date value of a conventonal opton wth the above nputs s $5.23. Addng the reload feature ncreases the opton's value by 24% to $6.49. If the executve only receves X/S new optons per orgnal opton exercsed (.e., the number of new optons granted s not ncreased by the amount of taxes the executve on exercse of the orgnal grant), the reload feature adds 15% to the value of a conventonal opton. Table 3 presents the value of a reload opton granted-at-the-money for stock volatltes rangng between 20 and 50 percent per year, annual dvdend yelds rangng from zero to fve percent, up to fve reloads, and 5- or 10-year maturtes. The holder s assumed to receve one second-generaton opton for each share of stock used to pay for 8

9 the exercse prce (.e. as descrbed above, X/S second-generaton optons for each frstgeneraton opton exercsed). An nterest rate of 7% s assumed for all calculatons. The values n the table are standardzed by the grant-date market value of the stock. Thus, readng from the table, the grant-date value of a 5-year conventonal opton on a stock payng no dvdend wth a market value of $17 and a volatlty of 20% s $17 _.335, or $5.70. Smlarly, the grant-date value of an opton that may be reloaded fve tmes s $17 _.400, or $6.80, whch s 19% more than the otherwse dentcal opton wthout reload. [Table 3] A Reload Feature Adds More Value When The Stock Pays More Dvdends Dvdends have a substantal mpact on the value of the reload feature. Whle the value of the conventonal opton falls wth ncreases n dvdend payout, the value of the reload feature ncreases as a percentage of the value of the conventonal opton. The rato of reload opton value to conventonal opton value s useful n assessng the condtons for whch a reload feature s most valuable. Compare the value of addng a reload to an opton when the dvdend yeld s 5% versus when there are no dvdends. If the volatlty s 20%, the ncrease n value from addng fve reloads s [(.226/.177) Ð 1] = 28% when dvdend yeld s 5% and [(.400/.335) Ð 1] = 19% when no dvdends are pad. Thus, the reload feature s worth more for optons on hgh dvdend yeldng stocks. 4 A Reload Feature s More Valuable When The Underlyng Stock s Volatle The table also reveals that reload features are more valuable for hgh volatlty stocks. From the example above, for an opton on a stock wth volatlty of 20% and 9

10 payng no dvdend, the ncrease n value from addng fve reloads s 19%. Ths ncrease n value s less that t would be for a hgher volatlty stock. For example, for a 5-year opton wth no dvdends and underlyng stock prce volatlty of 50%, the ncrease n value from addng fve reloads s [(.640/.520) Ð 1] = 23%. The Reload Feature s Relatvely less Attractve for Longer Lved Optons Next, the table shows that the reload feature adds more to a conventonal opton's value for short-maturty optons. Agan as calculated above, for a 5-year opton on a stock that pays no dvdend and has a volatlty of 20%, the ncrease n value from addng fve reloads s 19%. For a 10-year opton on the same stock, the ncrease n value from addng fve reloads s [(.582/.523) Ð 1] = 11%. The Value Of A Reload Opton Is Greater If More Optons Are Granted At The Tme Of Reload As mentoned earler, frms dffer n how many reload optons are granted. In our example, the holder receves optons equal to the number of shares surrendered to pay for the exercse prce and any ncome taxes owng due to the exercse. In that case, the value of the opton wth a reload feature s $6.49 or 24 percent hgher than an otherwse smlar conventonal opton. If Kovacevch receved reload optons equal n number to the shares requred ust to pay the exercse prce, the value of such optons would be $5.99, only 15 percent hgher than the conventonal opton. 5 10

11 The Value Of An Opton Increases As More Reload Grants Are Allowed Fgure 1 shows that the value of an opton ncreases wth the number of tmes an opton may be reloaded. Fgure 1 s based on the stock parameters of Norwest and the reload factor, Z, of Kovacevch's optons. An opton that may reloaded once s worth $49 - $5.23 = $1.26 more than a conventonal opton. Each addtonal opportunty to reload adds value to the opton but at a decreasng rate: a second opportunty to reload s worth an addtonal $0.47 and a thrd opportunty to reload s worth $0.22 more. An opton that may be reloaded an arbtrary number of tmes s worth $7.37, or ust $0.20 more than an opton that can be reloaded only three tmes. [Fgure 1] Exercse Behavor Fgure 2 plots the optmal exercse regon assumng a sngle reload s allowed on a stock that pays no dvdends. The fgure shows that exercse depends on both the stock prce level and tme remanng to expraton. The longer the tme to expraton, the hgher the threshold stock prce at whch exercse s optmal. [Fgure 2] In summary, the reload feature adds 24% to the value of a conventonal opton n our example. In other cases, ths amount may be more or less dependng on the characterstcs of the underlyng stock and the terms of the opton. The ncremental value depends the sze of the dvdend payout, the volatlty of the underlyng stock, the number 11

12 of years the opton s outstandng, the number of tmes reloads are allowed, and the number of reload optons. LIMITATIONS OF THE USE OF OPTION PRICING MODELS We have made several assumptons n usng that the Bnomal Opton Prcng Model to value an executve stock opton: that the executve's rsk averson s the same as that of an average trader n the market who holds the stock, that the executve wll reman employed wth the frm throughout the opton lfe, and that nether lqudty needs, rsk averson, nor behavoral decson bases wll cause the executve to exercse the opton earler than our model determnes s optmal. 6 Thus, our method values a hypothetcal tradable opton that has the reload feature. The value to the executve need not be the same as the value of a hypothetcal tradable opton snce the executve cannot trade hs optons; nstead he must exercse whle the holder of a traded opton could sell. In addton, an executve typcally has a large porton of ther wealth ted to the stock of ther frm and cannot dversfy away the unsystematc rsk n these optons as an ndependent trader could do. These dfferences mply that theoretcal models desgned to value traded optons yeld valuatons strctly greater than the value to the executve (Huddart, 1994). On behalf of the corporaton, the net cost of the optons may also be less than the value to an ndependent trader. If the executve s lkely to exercse the opton for any of 12

13 the reasons lsted above, then the far value of the opton to the corporaton s reduced. In addton, the purpose of grantng optons often s to provde ncentves to the executve to ncrease the value of the frm. In essence, the shareholders gve up some share of the pe n order to ncrease the overall value of the pe. Ths paper s slent on these ssues. Instead, we calculate the value of an opton wth a reload feature gnorng potental feedback from ncentves provded by optons to the stock prce process. The method can readly be modfed to handle vestng and stock performance restrctons on exercse. 7 For nstance, n the case of tme-based vestng restrctons, n those perods that the optons are not exercsable, the value of the opton s ust the holdng value; the value from exercsng s gnored. It s also possble to modfy the model to account for the executve's rsk preferences, lqudty needs, and the probablty of employment termnaton. The dffcultes presented by these latter three factors le n relably estmatng parameters that capture rsk averson, lqudty needs, and the lkelhood of termnaton, not n mplementng the valuaton when these parameters are known. These lmtatons to analytcal valuaton methods apply to reload optons and conventonal optons alke. SFAS 123 suggests estmatng the tme to expraton of optons based on hstorcal patterns of exercse. 8 However, as Kulatlaka and Marcus (1994) pont out, ths approach can lead to based estmates of the value. Hemmer, Matsunaga and Shevln (1994) show that use of an expected tme to exercse can mpart substantal upward bas n the estmated opton value. The Bnomal Opton Prcng Model can be adapted to ncorporate optmal early exercse n the valuaton, avodng the 13

14 bas ntroduced by estmatng expraton dates (Carpenter, 1998 and Cuny and Joron, 1995). CONCLUSION SFAS 123 states, ÒÉ deally, the value of an opton wth a reload feature should be estmated at the grant date, takng nto account all of ts features ÉÓ ( 186, p. 61). However, the FASB beleves that valuaton s not feasble and recommends delayng recognton of the value of the reload feature untl a reload (second-generaton) grant s ssued (.e., when the orgnal opton s frst exercsed). Ths paper demonstrates a feasble method to value both the opton and the reload feature at the tme of the ntal grant. We also show that the reload feature s ganng n popularty and potentally adds consderable value to the underlyng opton. In vew of ths, the Fnancal Accountng Standards Board may wsh to reconsder ts recommendaton regardng the tmng of valuaton for the reload feature. Our algorthm also characterzes the optmal exercse polcy for optons wth the reload feature, as we show n fgure 2. Knowledge of the optmal exercse polcy obvously has value to the holders of reload optons. Compensaton consultants may wsh to understand how the value of the reload feature vares wth the terms of the optons and the characterstcs of the underlyng stock. Consultants could use the algorthm to assst frms n customzng the terms of a reload feature and n decdng whether reload features are attractve to ther executves. Frms can ncrease the relatve value of a reload feature by reducng the term to maturty, by ncreasng the number of second-generaton optons granted, by ncreasng the number 14

15 of opportuntes to reload, and by reducng the exercse prce of the frst-generaton optons. In addton, reload features are relatvely more valuable for executves n hgh dvdend or hgh volatlty frms. 15

16 APPENDIX 1. USING THE BINOMIAL OPTION PRICING MODEL The appendx llustrates the applcaton of the Bnomal Opton Prcng Model to the valuaton of reload optons. The man steps are: (1) calculate the stock prce tree, (2) value a conventonal opton, (3) value the opportunty to reload the opton one tme by scalng the values calculated n step (2), and (4) value the opton wth a reload feature by addng the value of a reload opton to the proceeds from exercse at each node. For expostonal convenence, we value an opton to buy one share at an exercse prce of $10.00 wth three years to expraton. The grant date stock prce s $10.00, the annual volatlty s 30%, the frm pays no dvdends, and N = 3 perods. Followng Cox et al. (1979), the stock prce tree s constructed from an up factor, u = exp(.30 _ 1) = 1.35, a down factor, d = 1/1.35 =.741, and the rsk-neutral probablty of an uptck, p =. 54. Calculate the Stock Prce Tree Table A1 presents the array of possble stock prces for the 3 perods. The stock prce perods before expraton gven net uptcks snce the grant date s S = $10.00 _ 0 u. At node A, the current stock prce, 3 S s $10.00 per share. Thus, stock prce at node B, 1 S 2, s $10.00 _ 1.35 = $13.50$ and at node C, so on. S [Table A1] 1 2, prce s $10.00 _.741 =$7.41, and 16

17 Value a Conventonal Opton Now assume that you have an opton to purchase one share for $10.00 wthn the next 3 tme perods. The value of ths opton, C, at a gven node, descrbed by the remanng tme to maturty,, and the net number of uptcks snce the grant date,, s determned by computng the value at expraton, = 1 and then workng backward to = 1, = 2, and so on. At each node, the holder has the choce of exercsng or holdng the opton. The value of exercsng, V x, s the greater of 0 and the dfference between the market prce and the exercse prce. Thus, at expraton node G, the exercse value of the opton, V x, s $ $10.00 = $14.60 and at expraton node I, the exercse value of the opton, V x, s $0.00 snce exercsng would result n a loss of $ $10.00 = -$2.59. The value of holdng the opton s determned by the expected payoff from holdng the opton for one more perod V h = pc (1 1 + r p) C 1 1. where p =.54 s the probablty of an uptck and r = 7% s the rsk-free nterest rate. The calculatons at node D n table A1 are V h.54 $ $3.50 = = $8.87, 1.07 and V x = $ $10.00 = $ h x Thus, C 1 = $8. 87 because V > V. Hence, t s optmal to hold rather than exercse the opton at node D. Exercse s worthwhle only at expraton nodes G and H. In the 17

18 absence of dvdends, the value of exercsng a conventonal opton before expraton s always less than the value of holdng. Value the Opportunty to Reload the Optons Now assume that the holder gets a reload grant of one share for each orgnal opton when he exercses the opton pror to expraton. Usng the notaton n the body of the paper, these assumptons correspond to Z = 1 and m = 1. The value of the reload opton at node D s the same as the value of a conventonal opton granted at node D wth an exercse prce of $18.23 and 1 perod to expraton. Table A2 shows the value trees for the reload optons ssued at nodes D and B. The calculatons for the holdng value of a reload opton at node D (see panel 1) are V h.54 $ $0.00 = = $ V x = $18.23 $18.23 = 0, and C 0 1 = $3.21. [Table A2] Snce the holder receves one opton for each opton exercsed, the value, V the reload opton at node D s $3.21. If the holder receved a fracton of a reload opton for each orgnal opton exercsed, then the value of the reload would be that fracton tmes the value of one reload opton. The calculatons are smlar for the value of the R, of 18

19 reload opton ssued at node B. Table A3 shows the value of a reload opton ssued at each node. [Table A3] Value of the Opton wth a Reload Feature To value the orgnal opton wth the reload feature, the value of a reload opton, r V, s added to the exercse value of the orgnal opton for the same node, consstent wth equaton (2). The value of the opton at that node s stll the maxmum of the holdng value and the exercse value ncludng the value of a reload opton. One stll works backward from the value at expraton. Thus, the holdng value of an opton wth a reload feature dffers from the holdng value of a conventonal opton because the former depends on the value of reload optons at successor nodes. Table A4 presents the value of an opton that may be reloaded once. The values at expraton do not change from the value of a conventonal opton because both the orgnal and the reload optons expre at I = 0. At node D, the exercse value of the opton s now $ $3.21 = $ Thus, the value of exercsng s greater than the value of holdng and the opton value at node D s $ Ths ncreases the holdng value at node B from $5.24 to $6.53. [Table A4] The value of the opton wth a reload feature s $3.68 at node A. The value of the same opton wthout a reload feature s $3.03 at node A. The dfference s $.65 or a 21.5% ncrease n value due to the addton of a reload feature. Also note that the addton 19

20 of a reload feature leads to early exercse at node D whereas early exercse was never optmal wthout the reload feature. 20

21 APPENDIX 2. MATHEMATICAL MODEL AND EXTENSION TO MULTIPLE RELOADS Overvew Our algorthm s a varaton of the standard Bnomal Opton Prcng Method. 9 Fundamental to Bnomal Opton Prcng Model s the dea that stock prce movements are well approxmated by assumng the stock prce can only move to two possble values n a short nterval of tme. The frst step s to construct a prce tree that probablstcally descrbes future stock prce movements over tme. The tme from the grant date to the expraton of the optons s dvded nto short perods. Over each perod, the stock prce s assumed to ether rse or fall by a fxed factor wth a fxed probablty. Every node n the tree corresponds to a partcular tme to expraton and stock prce level. Each node n one tme perod s connected to two nodes n the next tme perod, representng a rse or fall n the stock prce by a fxed factor. Next, the value of the opton s calculated at each node, workng backwards recursvely from the expraton date. At expraton, valung the opton s straghtforward. At each earler node, the value of the opton can be determned from a partcular recursve equaton that depends only on the (already computed) values of successor nodes and parameters used to descrbe the stock prce tree. The value of the opton at every node s determned by computng the value at expraton, and then workng backward to nodes one perod pror to expraton, then two perods pror to expraton, and so on. The recursve equaton compares the value of holdng the opton for one more perod to the value of exercsng the opton mmedately and sets the value at that node to the larger of these quanttes. The value from mmedate exercse s the dfference 21

22 between the market prce and the exercse prce, plus the expected value of reload optons receved on exercse. The value of holdng the opton untl the next perod s the dscounted and weghted sum of the value of the opton f held one more perod n the case the stock prce rses and, the value of the opton f held one more perod n the case the stock prce falls. The weghts reflect the lkelhood the stock wll rse (or fall). The dscount factor s one plus the rskless nterest rate. The value of the opton on the grant date s the value at the startng node of ths tree. Optmal early exercse s ncorporated by assumng that the holder wll exercse whenever the value of exercsng s hgher than the value of holdng. Snce the decson to exercse or hold s made at each node, optmal early exercse s embedded n the valuaton. By ncludng the value of reloaded optons n the exercse value, the reload feature s embedded n the valuaton as well. Model The Bnomal Opton Prcng Model uses the Black-Scholes (1973) opton valuaton assumptons. In partcular, the rskless rate of nterest, r *, s assumed to be constant and asset prces are assumed to be lognormally dstrbuted wth a constant volatlty rate, σ (Cox and Rubnsten, 1985). Certan notaton and defntons facltate the descrpton of the bnomal method. Let X be the opton's exercse prce. Defne N to be the total number of tme steps durng the lfe of the opton, and T to be the opton's tme to expraton, n years. At each step, the asset return s ether u = exp( σ T / N ), wth probablty p = (1+r- d)/(u-d), or d = 1/u, wth probablty 1-p, where r = (1+r * ) T/N -1. Let S denote the stock prce tme steps before expraton when the stock prce has rsen tmes (net) snce the grant date. 10 For a stock that pays no dvdends, the stock prce at node (, ) s S 0 0 = S N u where N S s the stock prce at node (N,0), whch 22

23 corresponds to the grant date. For a dvdend-payng stock, ths expresson generalzes to 0 S N f (, ) where f (, ) u (1 y) d ( ), y s the quarterly dvdend expressed as a constant fracton of the stock prce, and the exponent d() s the number of dvdend payments made snce the grant date up untl tme. Let C be the value of a conventonal call opton value at node (, ). Workng backward from the end of the call opton's lfe, C s the maxmum of the proceeds from mmedate exercse and the expected present value of the possble opton values at Ð 1. C = max S pc X, (1 1 + r p) C 1 1. (1) By movng backward through tme and repeatng these computatons, the current value of an Amercan-style opton s determned. Ths basc method has been used to value conventonal optons for twenty years. To generalze the valuaton method to handle reloads, one determnes the total value from exercse by addng the value of second-generaton optons granted as a result of the reload provson to the proceeds from exercse. The recursve equaton then values an opton at tme as the greater of the total value from exercse and the expected payoff from holdng the opton untl the next perod, Ð 1. 0 Elaboratng on the notaton above, let C ( m, S N, X ) be the value of an opton 0 that may be reloaded m tmes, has grant date stock prce S and strke prce X, at node (, N ) n a bnomal tree. The value of the an opton at node (, ) s the maxmum of the value of the opton f exercsed, ncludng the value of any reload optons; and, ts value f held 23

24 for one more perod, whch s a weghted dscounted sum of the opton's value gven ether an uptck or downtck. Ths value can be expressed recursvely as C ( m, S 0 N, X ) = max S pc X + ZC ( m, S ( m 1, S, S 0, X ) + (1 p) C 1+ r ( m, S, X ) N 1 N, ) (2) where Z s the number of second-generaton optons granted per frst-generaton opton exercsed. When the number of new optons granted equals the number of shares needed to pay the exercse prce, Z = X / S. Snce m = -1 mples that no more reloads are allowed, t s understood that C ( 1, S, S ) = 0 for all and. At expraton, the opton must be exercsed, so for all, k, m, and X ( 0 S X) 0 C0 ( m, S, X ) = max,. 0 k To reduce the number of bnomal trees that must be constructed and evaluated, t s computatonally effcent to standardze by the strke prce X. Defne c ( m, x) C ( m, S 0 N / X,1) = C ( m, S X 0 N, X ) where 0 x SN / X, the rato of the stock prce at the date of grant to the exercse prce. In practce, the exercse prce of most grants s equal to the stock prce on the date of grant; hence, x = 1. For premum optons (.e., those that are out-of-the-money on the date of grant), x s less than one. In ths notaton, the grant date value of an opton struck at-the- 0 money that cannot be reloaded (.e., a conventonal opton) s wrtten Xc (0,1). The functon c s nterpreted as the value of an opton per dollar of the strke prce. Analogous to equaton (2), the opton's value, per dollar of the ntal strke prce, s convenently rewrtten as N 24

25 c ( m, x) = max x f (, ) 1+ Z * c p c ( m, x) + (1 p) c 1+ r ( m 1,1), 1 1 ( m, x) (3) where Z* ZS X s the number of new optons granted per old opton exercsed / multpled by the rato of the stock prce at tme to the strke prce of the exstng opton. In the case where the number of new optons granted s equal to the shares tendered to pay the exercse prce, Z* = 1. Extenson to an Unrestrcted Number of Reloads If the number of tmes the opton can be reloaded s unrestrcted, a modfcaton of the recursve equaton (3) s requred. Let c ( A, x) denote an opton that may be reloaded an arbtrary number of tmes. Smply substtutng c ( A,?) n (3) wherever c ( m,?) or c ( m 1,?) appears yelds a system that cannot be solved recursvely. Ths s because c 0 (,1) s expressed as a functon of tself, as are opton values at other nodes n 1 A the tree. The key smplfcaton comes from observng that 0 c ( A,1) = pc 1 1 ( A,1) + (1 p) c 1+ r 1 1 ( A,1) snce the executve s ndfferent between holdng and exercsng an at-the-money opton that can be reloaded an unrestrcted number of tmes. Substtutng the rght hand sde of ths equalty wherever the left-hand sde appears n equaton (3) yelds 25

26 c ( A, x) = max 1 pc x f (, ) 1+ Z * pc ( A, x) + (1 p) c 1+ r ( A,1) + (1 p) c 1+ r 1 1 ( A, x), 1 1 ( A,1), whch can be solved recursvely snce c ( A, x) s expressed n terms of successor nodes n the tree for all and. So, the grant date value can be determned by computng the values of nodes n a sngle bnomal tree, workng backwards from the expraton date. Ths means the valuaton of an opton that may be reloaded an arbtrary number of tmes s no more complex computatonally than valung a conventonal opton usng the bnomal method. Frequency of Steps n the Algorthm The accuracy of the valuaton ncreases wth the number of tmes each year that the bnomal tree allows the stock prce to vary. All calculatons n ths paper are based on bnomal steps of one month,.e., the stock s modeled as varyng twelve tmes per year. Ths represents a good tradeoff between accuracy of the valuaton and the processng resources requred to compute values. As the number of tmes the opton may be reloaded grows large, the computaton tme requred to value the opton also grows because each tme the opton can be reloaded requres an addtonal tree to be generated. Even for 10 reloads the computatonal demands are not excessve Ð all values presented n ths paper were computed on a desktop computer. Appendx 3 provdes the Mathematca computer code to mplement ths algorthm. 26

27 APPENDIX 3. MATHEMATICA PROGRAM A Mathematca notebook for ths program s avalable from the authors. Valuaton of Reload Stock Optons Program descrpton Ths program values reload optons assumng the value of the frm's stock evolves accordng to a bnomal process. Wth probablty p, frm value ncreases by factor f each perod. Wth complementary probablty, frm value falls by factor 1/f. The ntal value of the stock s s and the strke s x. Dvdends are assumed to be a constant fracton of the stock prce, pad quarterly. Intalzaton ClearAll[d,dv,f,n,m,mts,p,s,sx,t,tax,v,x,OptonValue] Parameters Intal stock prce s=14.53 Dvdend yeld dv=.03 Strke prce x=14.53 Tme to expraton of the opton, n years t=10 Number of opton reloads allowed Choose m=99 f the number of reloads s unrestrcted. m=1 Tax rate on whch optons are reloaded tax=.481 Annual volatlty of stock returns v=.273 Number of perods n bnomal tree per year n=12 27

28 Dscount factor, per perod d=(1+.07)^(1/n) Factor by whch stock prce ncreases f=exp[v*sqrt[1/n]] Probablty of an uptck p=(d-1/f)/(f-1/f) Recursve Formula for Opton Value Prce dynamcs Market-to-strke rato at node (, ) gven the market to strke rato at the grant date was sx mts[sx_,_,_]:=mts[sx,,]=sx*f^*(1-dv/4)^floor[4*(t-/n)] Value of an opton wth m reloads remanng and market to strke rato sx at node (, ) OptonValue[m_,sx_,_,_]:= OptonValue[m,Round[10000*sx]/10000,,]= If[==0,If[mts[sx,,]>1,mts[sx,,]-1,0], Max[( p *OptonValue[m,Round[10000*sx]/10000,-1,+1] + (1-p)*OptonValue[m,Round[10000*sx]/10000,-1,-1])/d, mts[sx,,]-1+(1+(mts[sx,,]-1)*tax)*optonvalue[m-1,1,,0]]] Termnaton condtons The reload value of an opton wth no reloads left s zero. OptonValue[-1,sx_,_,_]:=OptonValue[-1,sx,,]=0 Unrestrcted reloads Ths code handles the case when an arbtrary number of reloads are allowed. Such cases are coded by settng m=99. OptonValue[99,sx_,_,_]:= OptonValue[99,Round[10000*sx]/10000,,]= If[==0,If[mts[sx,,]>1,mts[sx,,]-1,0], Max[(p*OptonValue[99,Round[10000*sx]/10000,-1,+1]+ (1-p)*OptonValue[99,Round[10000*sx]/10000,-1,-1])/d, mts[sx,,]-1+(1+(mts[sx,,]-1)*tax)* (p *OptonValue[99,Round[10000*sx]/10000,-1,1]+ (1-p)*OptonValue[99,Round[10000*sx]/10000,-1,-1])/d]] Opton Value x*optonvalue[m,s/x,t*n,0] 28

29 References Arnason, S., and R. Jagannathan Evaluatng executve stock optons usng the bnomal opton prcng model. Teachng Note, Unversty of Mnnesota. Black, F., and M. Scholes The prcng of optons and corporate labltes. Journal of Poltcal Economy, 81 (May-June): Carpenter, J. N The exercse and valuaton of executve stock optons. Journal of Fnancal Economcs 48 (May): Cox, J. C., S. A. Ross, and M. Rubnsten Opton prcng: a smplfed approach. Journal of Fnancal Economcs 7 (September): Cox, J., and M. Rubnsten Opton Markets. New Jersey: Prentce-Hall. Cuny, C. J., and P. Joron Valung executve stock optons wth a departure decson. Journal of Accountng and Economcs 20 (September): Fnancal Accountng Standards Board Accountng for Stock-based Compensaton. Statement of Fnancal Accountng Standards No Norwalk, CT: Fnancal Accountng Standards Board. Gay, C Hard to Lose: ÔReloadÕ optons promote stock ownershp among executves; but crtcs say theyõre a lot more costly than shareholders realze. Wall Street Journal, Aprl 8, 1999: R6. Heath, C., S. Huddart, and M. Lang Psychologcal factors and stock opton exercse. Quarterly Journal of Economcs (n press). Huddart, S Employee Stock Optons. Journal of Accountng and Economcs 18 (September): Huddart, S., and M. Lang Employee stock optons exercses: An emprcal analyss. Journal of Accountng and Economcs. 21 (February): Huddart, S Tax plannng and the exercse of employee stock optons. Contemporary Accountng Research 15 (Summer): Hemmer, T., S. Matsunaga, and T. Shevln Estmatng the "far value" of employee stock optons wth expected early exercse. Accountng Horzons 8 (December): Hemmer, T., S. Matsunaga, and T. Shevln An emprcal examnaton of reload employee stock optons. (October) Workng paper, Unversty of Washngton. 29

30 Hemmer, T., S. Matsunaga, and T. Shevln Optmal exercse and the cost of employee stock optons wth a reload provson. Journal of Accountng Research 36 (Fall): Kulatlaka, N., and A. Marcus Valung employee stock optons. Fnancal Analysts Journal (November-December):

31 Table 1 Prevalence of Reload Opton Grants a Year Number of Opton grants ,884 7,599 7,719 9,642 9,673 Number of Reload grants ,135 Reload grants as a Percent of opton grants a SourceÑStandard & PoorÕs Execucomp Database. The number of reload grants s the total number of second-generaton optons ssued as a result of a reload feature. Frms do not report conventonal optons separately from optons wth a reload feature (.e. frstgeneraton optons). Thus, the number of optons wth a reload feature s unknown. 31

32 Table 2 Prevalence of Reload Opton Grants by Industry a Industry Number of Frms wth No reload b Medan sales Number of Frms wth at Least 1 reload c Medan sales (mllons $) (mllons $) Prmary ndustry Transportaton Trade Food and Drug Servces Ol and gas Fnancal servces Manufacturng Computers Health care Utltes SIC code changes ,080 1, , , ,509 26,031 1,128 8,741 4,964 2,656 3,863 3, NA 6,887 2,691 All ndustres 1, ,926 a SourceÑStandard & PoorÕs Execucomp Database. b Includes all frms grantng at least one executve opton grant durng c Includes all frms that granted second-generaton optons when an opton wth a reload feature was exercsed. 32

33 Table 3 Reload Opton Values a Dvdend Yeld Volatlty Number 5-year Optons 10-year Optons of Reloads Allowed unrestrcted unrestrcted unrestrcted unrestrcted a Values are computed usng the Bnomal Opton Prcng Model for grants of optons wth varyng restrctons on reloads, volatltes of the underlyng stock, dvdend yelds, and tmes to expraton. All optons have strke prces equal to the market prce on the grant date. Assumes one new opton s granted on reload for each share of stock surrendered n payment of the exercse prce, and an annual nterest rate of 7%. Zero reloads correspond to a conventonal employee stock opton. Values are as at the grant date, and are expressed per dollar of stock prce on the grant date. The bnomal trees used to produce these estmates have one step per month.

34 Fgure 1 Opton Value as a Functon of the Number of Reloads Ths fgure plots grant-date value n dollars as a functon of the number of tmes the opton may be reloaded. Optons have a strke prce equal to the grant-date stock prce of $14.53 and a 10 year lfe. The stock parameters are those of Norwest (.e., a 3% dvdend yeld and volatlty of 27.3%). A 7% nterest rate s assumed. The reload factor, Z, corresponds to KovacevchÕs optons as descrbed n the text. The horzontal lne n the fgure s the value of optons that may be reloaded an arbtrary number of tmes. The bnomal trees used to produce these estmates have one step per month.

35 Fgure 2 Optmal exercse regon for an opton that may be reloaded once Assumng no dvdends, volatlty of 27.3%, an nterest rate of 7%, strke and grant date stock prces of $14.53, and the reload factor, Z, of KovacevchÕs optons, ths fgure plots the optmal exercse regon n a bnomal tree for an opton that may be reloaded once. Optmally, exercse occurs when the stock prce frst passes nto a node coded ÔxÕ. In regons coded Ô+Õ, t s optmal to hold the opton untl the next perod. Snce t s not optmal to exercse when the opton s out of the money, stock prces below the strke prce are not plotted. The bnomal trees used to produce these estmates have one step per month. 35

36 TABLE A1 Array of Exercse and Holdng Values of a Standard Opton =3 =2 =1 =0 Node G: S=$24.60 Node D: S=$18.23 V x =$14.60 Node B: S =$13.50 V x = $8.23 V h = $8.87 Node H: S=$13.50 Node A: S =$10.00 V x =$3.50 V h =$5.24 Node E: S =$10.00 V x =$3.50 V x = $0.00 V h =$3.03 Node C: S = $7.41 V x = $0.00 V h =$1.77 Node I: S=$7.41 V x = $0.00 V h =$ 0.89 Node F: S = $5.49 V x = $0.00 V x =$0.00 V h =$0.00 Node I: S=$4.07 V x = $0.00 The opton s ssued at perod =3 wth a strke prce equal to the current stock prce of $10.00 and a term to maturty of 3 perods. In each perod, the stock prce, S, can ncrease by a factor of u=1.35 or decrease by a factor of d=1/u=.741. At each node, V h s the value of the opton f held one more perod and (V x ) s the value of the opton f exercsed ths perod. The value of the opton, shown n bold, s the larger of the holdng and exercse values.

37 TABLE A2 Values for Reload Optons Issued at Nodes B and D Panel 1: Value of a reload opton ssued at node D wth an exercse prce of $18.23 and 1 perod to expraton: t = 2 t = 3 Node D: S = $18.23 V x = $6.37 Node G: S = $24.60 V x = $0.00 V h = $3.21 Node H: S = $13.50 V x = $0.00 Panel 2: Value of a reload opton ssued at node B wth an exercse prce of $13.50 and 2 perods to expraton: =2 = 1 = 0 Node D: S = $18.23 V x =$11.10 Node G: S = $24.60 Node B: S=$13.50 V x = $4.73 V h = $5.60 Node H: S = $13.50 V x = $0.00 V h = $2.83 Node E: S=$10.00 V x = $0.00 V x = $0.00 V h = $0.00 Node I: S=$7.41 V x = $0.00 S s the stock prce for that node. In each perod, the stock prce, S, can ncrease by a factor of u=1.35 or decrease by a factor of d=1/u=.741. V h s the value of the opton f held one more perod and (V x ) s the value f the opton s exercsed ths perod. The opton value, shown n bold, s the larger of the holdng value and the exercse value.

38 TABLE A3 Array of Values of a Reload Opton Issued at each node = 3 = 2 = 1 =0 Node G: S=$24.60 Node D: S=$18.23 V r = $0.00 Node B: S = $13.50 V r = $3.21 Node H: S=$13.50 Node A: S x =$10.00 V r = $2.83 Node E: S = $10.00 V r =$0.00 V r = $3.03 Node C: S = $7.41 V r = $1.77 Node I: S=$7.41 V r = $1.55 Node F: S = $5.49 V r = $0.00 V r = $0.97 Node I: S=$4.07 V r = $0.00 S s the stock prce. In each perod, the stock prce, S, can ncrease by a factor of u=1.35 or decrease by a factor of d=1/u=.741. V r s the value of a reload opton ssued at that node. The reload opton has an exercse prce equal to the market prce for that node and expres at =0.

39 TABLE A4 Array of Exercse (V x ) and Holdng (V h ) Values of an Opton wth a one-tme Reload Feature = 3 = 2 = 1 =0 Node G: S=$24.60 Node D: S=$18.23 V x =$14.60 Node B: S = $13.50 V x =$11.44 V h = $8.87 Node H: S=$13.50 Node A: S x =$10.00 V x =$6.33 V h =$6.53 Node E: S = $10.00 V x =$3.50 V x = $0.00 V h = $3.68 Node C: S = $7.41 V x = $1.77 V h =$1.77 Node I: S=$7.41 V x = $0.00 V h =$0.89 Node F: S = $5.49 V x = $0.00 V x = $0.00 V h =$0.00 Node J: S=$4.07 V x = $0.00 S s the stock prce for that node. In each perod, the stock prce, S, can ncrease by a factor of u=1.35 or decrease by a factor of d=1/u=.741. V h s the value of the opton f held one more perod. V x s the value f exercsed ths perod and ncludes both the gans from exercse and the value of a reload opton ssued at that node. The opton value, shown n bold, s the larger of the holdng value and the exercse value.

40 1 Other terms for reload optons nclude restoraton optons, replacement optons, contnuaton optons, replenshment optons, and accelerated ownershp optons. 2 Under some plans, the stock used to pay the exercse prce must have been held by the employee for a mnmum specfed tme. Frms may use ths feature to nduce executves to hold some of ther wealth n stock n perods before the executve exercses hs optons. Some reload plans also requre executves to hold shares acqured on exercse for some tme after exercse. See Hemmer et al. (1996). 3 The dfference between the market prce of the stock on the date of exercse and the exercse prce s ncome to the employee on the date of exercse. See Huddart (1998). 4 Reload features are more common n larger companes and fnancal servces busnesses. These frms tend to pay hgher-than-average dvdends. 5 Usng the same knd of analyss t can be shown that reload features are a hgher fracton of total opton value for dscount optons (.e., optons where the strke prce s below the grant-date stock prce) than for premum optons. 6 Nevertheless, these factors are lkely to be mportant. See Heath et al. (1999) for a dscusson of behavoral factors. 7 A typcal executve opton cannot be exercsed pror to vestng. Thus, the opton s forfeted f the executve leaves the frm pror to vestng. 8 Huddart and Lang (1996) document patterns of exercse and ther assocaton to volatlty, past stock prce movements, and the lapsng of vestng restrctons. 9 See Cox and Rubnsten (1985) for an excellent dscusson of the use of Bnomal Opton Prcng Model for valung other complex optons. 10 Negatve values of mean the stock prce has fallen snce the grant date. For example, f three perods after the grant date there have been two down moves and one up move, then the stock prce s S u S N 3 N =. 40