Valuing Warrants: - PDF document

Valuing Warrants: Dilution and Down - Round Price Protection by Dwight Grant, PhD Managing Director PricewaterhouseCoopers LLP Reprinted with permission from Business Valuation Review www.pwc.com ValuingWarrants:DilutionandDown-RoundPrice Protection DwightGrant,PhD Thispaperaddressestwowarrantvaluationtopics:dilutionanddown-roundprice protection.Istartwithdilutionbecausethereislingeringconfusionamongappraisers onthissubject.Iaddressdown-roundpriceprotectionbecausetheSEChas highlightedthisissue,notingthatasimpleBlack-Scholes-Merton(BSM)equationmay notaccuratelyvaluethewarrantswhenthereisdown-roundprotection,thatis,where theexercisepriceonthewarrantswillbeloweredtomatchthepriceofanynew financingatalowerprice.Iillustratetwoalternativevaluationmethods:Onecombines alatticeandtheBSMequation,andtheotheremploysMonteCarlosimulation.With respecttodilution,Ishowthatitisnotanimportantconcerninvaluingwarrantsaslong asyouusecommonstockvolatilitywhenusingasimpleBSMequationandequity volatilitywhenusingtheBSMequationmodifiedforwarrants.Withrespecttodown- roundprotection,Iillustratethemagnitudeoftheeffectsontheprotection,whichare factsspecific. Introduction Thispaperaddressestwowarrantvaluationtopics: dilutionanddown-roundpric eprotection.Istartwith dilutionbecausethereislingeringconfusionamong appraisersonthissubject.Iaddressdown-roundprice protectionbecausetheSEChashighlightedthisissue, notingthatasimpleBlack-Scholes-Merton 1 (BSM)equa- tionmaynotaccuratelyvaluethewarrantswhenthereis down-roundprotection,thatis,wheretheexercisepriceon thewarrantswillbeloweredtomatchthepriceofanynew financingatalowerprice.Iillustratetwoalternative valuationmethods:OnecombinesalatticeandtheBSM equation,andtheotheremploysMonteCarlosimulation. 2 Withrespecttodilution,Ishowthatitisnotan importantconcerninvaluingwarrantsaslongasyouuse commonstockvolatilitywhenusingasimpleBSM equationandequityvolatilitywhenusingtheBSM equationmodifiedforwarrants.Withrespecttodown- roundprotection,Iillustratethemagnitudeoftheeffects ontheprotection,whicharefactsspecific. PlainVanillaWarrantsandDilution Althoughthevaluationofwarrantsiswelldocumented, thereissomeconfusionabouttheroleofdilutioninthat valuation.Therefore,thefirststepistoidentifytwo differentdilutioneffectsconnectedtowarrants.Ireferto onetypeasparticipationdilutionandtheotherasnon- fair-valuedilution. Theholderofawarrantbuyscommonstockfromthe issuingfirmwhenthestockisworthmorethanthe warrantholderpays. 3 Thistransactionreducesordilutes theproportionateownershipoftheexistingcommon stockinthefirm’sequityappreciation.Irefertothisas participationdilution.Firmsmostfrequentlycreate warrantsaspartofanissuanceofbondsorpreferred stockwithwarrantsattached.Itisreasonabletothinkof thosetransactionsasbeingatfairvalue,withthevalueof thewarrantbeingembeddedintheoverallexchangeof cashforsecurities.Becausetheyareatfairvalue,the valuesofothersecurities,inparticular,commonstock,do notchange.Therefore,participationdilutiondoesnot typicallygiverisetoanychangeinthevalueofthe commonstockatthetimethewarrantsareissued. Participationdilutioncanbeincorporatedinavaluation modelbymodifyingthestandardBSMequation. 3 Employeestockoptionsarelikewarrantsinthisregard.Incontrast, market-tradecalloptionsdonotdirectlyaffectthecommonstockofthe firm. 1 F.BlackandM.S.Scholes,‘‘Thepricingofoptionsandcorporate liabilities,’’ JournalofPoliticalEconomy 81(May–June1973):637–654. DwightGrantisaManagingDirectorinPwC 9 sValue AnalyticsandDerivativesPractice.HeisbasedinSan Francisco. 2 P.Boyle,‘‘Options:AMonteCarloapproach,’’ JournalofFinancial Economics 4(1978):323–338. BusinessValuationReview Volume33 N Number1–2  2014,AmericanSocietyofAppraisers BusinessValuationReview—Summer2014Page21 Warrantscanalsobevaluedquiteaccuratelyusingthe BSMequationwithoutadjustingforparticipationdilu- tion.Thetwomodelsdifferintheirdefinitionsofthe underlyingassetandtheassociatedvolatility. Textbooksoftenintroduceasecondsourceofdilution byconsideringexamplesinwhichwarrantsaredistrib- utedforfree.Becausethefirmgivesawaythewarrants, thestockpricemustdeclinesothatthetotalvalueof equity,commonstockpluswarrants,isunchanged.Irefer tothisasnon-fair-valuedilution. 4 Ibelievethatnon-fair- valuedilutionisthesourceoftwomisimpressions:(a) Issuingwarrantsdilutesthecommonstockand always reducesitsprice,and(b)oncethecommonstockprice respondstotheissuanceofwarrants,thereisnofurther dilutioneffect.Thefollowingtwoexamplesaddressthose misimpressions. Example1 Consideranall-equityfirmwith1.00million( N ) sharesofcommonstockpricedat$10.00( S )pershare. Thefirmissues0.50millionwarrants( M )tobuyone shareeachatapriceof$10.00( X )pershare.Thewarrants haveafive-yearterm( T ),thefirmpaysnodividends,the volatility( s )ofthefirmis40.00%,andtherisk-freerate ofinterestis3.00%peryearcompoundedcontinuously. FollowingHull’sexample,Iassumethewarrantsare distributedforfree(seefootnote4).Inthatcase,Hull,and others,haveshownthatthevalueofthewarrantwillbe N /( N + M ),[1.00/(1.00 + 0.50)]inthisexample, multipliedbythevalueofaBSMcalloptionwith S 5 $10.00, X 5 $10.00, r 5 3.00%, T 5 5.00,and s 5 40.00%.Inthisexample,thevalueofthewarrant( W )is: W ~ 1 : 00 = 1 : 50 ðÞ $ 3 : 9508 ðÞ ~ $ 2 : 6339 : Thewarrantsinaggregateareworth$1.3169million. Therefore,thecommonstockmustbeworth$8.6831 millionor$8.6831pershare.Thevalueofthecommon stockfallsfrom$10.00to$8.6831whenthewarrant issuanceisannounced.Again,forclarity,Irefertothisas thenon-fair-valuedilutioneffect. Havingtakenintoconsiderationthenon-fair-value effect,canweignoreanyotherdilutioneffectinvaluing warrants?Ingeneral,wecannot.Toseewhy,considerthe valuationofthiswarrantafterthestocktradesat$8.6831. Ifwevaluethewarrantasacalloptionwiththe parameters S 5 $8.6831, X 5 $10.00, r 5 3.00%, T 5 5.00,and s 5 40.00%,itsvalueis$3.0231.Thisis 14.78%higherthanthevaluealreadycalculated.This errorhastwosources.Byfarthemoreimportantofthe twosourcesoferrorisouruseofanincorrectvolatility. Wepreviouslycalculatedthevalueofacalloptiononthe totalfirm,commonstockpluswarrants;wearenow calculatingthevalueofacalloptiononthecommon stockonly.Becausethewarrantsaremorevolatilethan thecommonstock,theirvolatilitymustbegreaterthan 40.00%,andthevolatilityofthecommonstockmustbe lessthan40.00%.Byusing40.00%volatilityinour valuationofthewarrantasacalloptiononthecommon stock,weoverestimatedthevalueofthewarrant.I elaborateonthisrelationshipinexample2. Example2 Theparametersofthisexamplearethesameasin example1,butinsteadofgivingawaythewarrants,we sellthemforfairvalue.Ourchallengeistodetermine thevalueofthewarrants.Issuingwarrantsforcash increasesthevalueofthefirm.Thevalueofthefirmper shareofcommonstockis$10.00 + WM / N .Withthat change,thewarrantcanbevaluedas N /( N + M )fraction ofacalloptionwith S 5 $10.00 + W (0.5/1.00), X 5 $10.00, r 5 3.00%, T 5 5.00,and s 5 40.00%.Note thatwehaveasmallchallengeinthatthevalueofthe warrant, W ,isdefinedintermsofastockpricethat includes W .Weneedtouseasearchprocesstosolve oneequationwrittenintermsofoneunknown, W .The solutionisavalueperwarrantof$3.5280. 5 Inthis example,weeliminatedthenon-fair-valuedilutioneffect butcontinuetoincludetheparticipation-dilutioneffect.I nowexploreinmoredetailthesizeoftheparticipation- dilutioneffect. Inexample1,therewasasubstantialerrorwhenwe ignoredthedilutioneffectandvaluedthewarrantasacall optiononthecommonstock.Thaterrorhadtwosources. Onewastheuseofaninappropriatevolatility,andthe otherwastheomissionoftheparticipation-dilutioneffect. Iextendthisexampletopartitionthosetwoeffectsand documenttheirrelativemagnitudes.Todothat,the volatilityofthecommonstockneedstobeidentifiedina capitalstructurecomposedofcommonstockandwarrants onthecommonstock. Whaleydetailstherelationshipsthataddressthe valuationofwarrantsandothersecurities. 6 Thevolatility ofthecommonstock, s S ,isrelatedto(a)thevolatilityof totalequity, s E ,(b)thedeltaofthecommonstock, N ( d 1 ), 4 See,forexample,JohnHull, Options,FuturesandOtherDerivatives ,6th ed.(UpperSaddleRiver,NewJersey:PrenticeHall,2006),298–300. 5 Notethatifweapplythissameapproachtoexample1afterthewarrants havebeendistributed,wecorrectlyvaluethewarrants. S 5 $8.6381 + W (0.5/1.50), X 5 $10.00, r 5 3.00%, T 5 5.00,and s 5 40.00%providesa valueforthewarrantof$2.6339,andthevalueof S 5 $8.6381 + W (0.5/ 1.00) 5 $10.00isnotsurprising. 6 RobertE.Whaley, DerivativesMarkets,ValuationandRiskManagement (Hoboken,NewJersey:JohnWiley&SonsInc.,2006),chapter12,439– 444. BusinessValuationReview Page22  2014,AmericanSocietyofAppraisers and(c)theratioofthevalueoftheequity( E )tothevalue ofthecommonstock( S ): s S ~ N d 1 ðÞ E = S ðÞ s E : Thedeltaofthecommonstockisequaltooneminus thedeltaofthewarrant,whichisequalto M /( N + M ) multipliedbythevalueof N ( d 1 )calculatedinacalloption formulawiththeunderlyingassetequaltototalequity, E 5 $10.00 + W (0.5/1.0), X 5 $10.00, r 5 4.00%, T 5 5.00,and s 5 40.00%.Specifically, s S ~ 1 { Nd 1 ðÞ M = N z M ðÞ ½ E = S ðÞ s E ~ 1 { N 0 : 7966 ðÞ 0 : 5 = 1 : 5 ðÞ ½ 11 : 7640 = 10 : 00 ðÞ 40 : 00 % ~ 0 : 737611 : 7640 = 10 : 00 ðÞ 40 : 00 % ~ 34 : 71 % : Ifwecalculatethevalueofthewarrantwiththis volatilityandignoretheparticipation-dilutioneffect,we have S 5 $10.00, X 5 $10.00, r 5 3.00%, T 5 5.00, s 5 34.71%,and W 5 $3.5560.Thisisonly0.80%larger thantheearlierestimateof$3.5280.Ifweuseavolatility of40.00%,thewarrantvalueis$3.9508,adifferenceof 11.99%.Therefore,thevastmajorityofpricingerroris attributabletofailingtomatchtheunderlyingassetand thevolatility,asopposedtofailingtomodelthe participation-dilutioneffect.Table1expandsthisresult forexample2usingarangeofdilutionfactors, M /( N + M ),from5%to50%.Theparticipation-dilutionerroris thedifferencebetween(a)thewarrantvaluewhenyou usetotalequityastheunderlyingsecurity,totalequity volatility,andyouconsiderdilution,and(b)thewarrant valuewhenyouusecommonstockastheunderlying asset,commonstockvolatility,andyouignoredilution. Thiserrorisrelativelysmall,rangingfrom0.09%to 1.41%. 7 Theerrorwhenyouusecommonstockasthe underlyingasset,totalequityvolatility,andignore dilutionisrelativelylarge,rangingfrom1.33%to 21.89%. Insummary,theonlyimportantdilutionthatchanges thesharepriceofacompany’sstockoccurswhen warrantsareissuedandthecompanyreceiveslessthan theirfairvalue.Ibelievethisrarelyoccurs.Both outstandingwarrantsandto-be-issuedwarrantscanbe valuedaccuratelyusingeitherthestandardBSMformula ortheBSMformulamodifiedtoincorporatetheeffectof upsideparticipationbythewarrants.ThestandardBSM formularequiresanestimateofthevolatilityofthe commonstock,whichismorereadilyavailablefrom tradingdatawhenthewarrantsarealreadyoutstanding. ThemodifiedBSMformularequiresanestimateofthe volatilityofthecommonstockandwarrantscombined, whichismorereadilyavailablefromtradingbefore warrantshavebeenissued. WarrantswithDown-RoundProtection Whenwarrantsarecreatedaspartofapackageof securitiesthatfinanceearlystageprivatecompanies,they sometimesincludedown-roundprotection.Atypicalreset provisionwouldlowerthestrikepriceofthewarrantinthe eventthatthefirmissuessecuritiesatalowerpriceinthe future.Tolimitthecomplexityofthediscussion,Iwill considerthevaluationofw arrantsoncommonstock. Moreover,Iwillusethepe rspectiveaboutdilution developedintheprevioussection;namely,Iwillvaluethe warrantsascalloptionsonthecommonstock,keepingin mindthatthevolatilityIuseisthecommonstockvolatility. Apopularapproachtovaluingwarrantswithreset provisionsistoignoretheresetfeatureandapplythe BSMformula.Onejustificationforignoringthereset provisionistheclaimthattheprobabilityofissuing securitiesatalowerpriceiszero.Thisviewfliesinthe faceofthedecisionofthecontractingpartiestoincludea resetprovision:Ifitcanneverhappen,whyisitpartof thecontract?Amodifiedviewisthattheprobabilityis 7 ThisisconsistentwithresearchbyJacobSidenius,‘‘Warrantpricing— Isdilutionadelusion,’’ FinancialAnalystsJournal (September/October 1996):77–80. Table1 ComparisonofWarrantValuations E 5 $10Million, N 5 1.0Million, X 5 $10, r 5 3%, T 5 5.0Years,and s E 5 40% TotalEquityasthe UnderlyingSecurity CommonStockastheUnderlying SecuritywithCommonVolatility CommonStockastheUnderlying SecuritywithEquityVolatility DilutionFactorWarrantValueStockVolatilityWarrantValueErrorWarrantValueError 4.8%$3.899039.3%$3.90250.09%$3.95081.33% 9.1%$3.849838.7%$3.85670.18%$3.95082.62% 20.0%$3.715837.0%$3.73160.43%$3.95086.33% 33.3%$3.528034.7%$3.55600.80%$3.950811.99% 50.0%$3.241431.2%$3.28701.41%$3.950821.89% ValuingWarrants:DilutionandDown-roundPriceProtection BusinessValuationReview—Summer2014Page23 sufficientlylowthattheeffectoftheresetprovisionon thevalueofthewarrantisimmateriallysmall.Thatmay oftenbethecase.Itdependsonboththeprobabilityofa securityissuanceandthesecuritycharacteristicsthat determinethevalueofaresetwhenitoccurs.My objectiveistoprovideevidencerelevanttothisassertion. Iconsidertwoapproachestovaluingresetprovisions, onerelativelystraightforwardandonemorecomplex. Theformerallowsforareseteventatonlyonedate.It requiresthecreationofalatticeandtheapplicationofthe BSMformula.Thelatterallowsresetsatmultipledates andincorporatesaprobabilityofresetateachdate. Forasingleresetdate,thewarrantanditsreset provisioncanbevaluedusingalatticeandtheBSM formulaasshownintheTable2A.Thispresentation assumesastandardCox-Ross-Rubinstein 8 latticecover- ingtheperiodfromthevaluationdatetotheforecastreset date.Inthisillustration,Iassumethatthewarrant’s exercisepriceresetsbasedonanewissuanceofcommon stockafter a T years,where T istheoriginaltimeto expirationofthewarrant.Tomaketheformulasgeneral,I assumeanevennumberofstepsinthelattice.The expressionBSM( S , X , r , s ,T )representstheBSMvalue ofacalloptionfortheassociatedvariables.Thepayoffs attheendnodesofthelatticearetheBSMoptionvalues ofcalloptionswitharemaininglifeof(1 2 a ) T years. Thecalloptionvaluesarediscountedbackthroughthe latticetodeterminethevalueofthewarrant.Wherethe stockpricesoftheendnodesinthelatticeareabove X , theexercisepriceinthecalloptioncalculationisthe originalexercisepriceof X .Wheretheend-nodestock pricesarelessthan X ,theexercisepricesintheoption calculationsarethoselowerstockprices. Table2BprovidesvaluesthatmatchTable2Aforan exampleinwhichafirmwithacommonstockpriceof 8 J.Cox,S.Ross,andM.Rubinstein,‘‘Optionpricing:Asimplified approach,’’ JournalofFinancialEconomics 3(September1979):229–263. Table2A AnEven n -StepValuationLatticeforaWarrantwithaSingleResetafter a T Years D t ~ a T = n , u ~ e s  D t p , d ~ 1 = u , and p ~ ( r D t { d ) = ( u { d ) 0 D t 2 D t 3 D t a T PayoffatDate a T Su n BSM(S u n ,min( X , Su n ), r ,(1 2 a ) T , s ) … Su n 2 1 d BSM(S u n 2 1 d ,min( X ,S u n 2 1 d ), r ,(1 2 a ) T , s ) Su 2 Su 2 ……… SuSud SSud … Su n/2 d n/2 BSM( Su n/2 d n/2 ,min( X , Su n/2 d n/2 ), r ,(1 2 a ) T , s ) SdSud 2 Sd 2 … Su n/2 2 1 d n/2 + 1 BSM( Su n/2 2 1 d n/2 2 1 ,min( X , Su n/2 2 1 d n/2 + 1 ), r ,(1 2 a ) T , s ) Sd 3 ……… Sd n BSM( Sd n ,min( X , Sd n ), r ,(1 2 a ) T , s ) Table2B ValuationLatticeforaWarrantwithaSingleResetafter a T Years S 5 $10.00, X 5 $10.00, r 5 2.0%, s 5 50%, T 5 8.0, a 5 0.5, n 5 100 0.000.040.080.124.00PayoffatDate4.00 $220,265$220,255 …$180,337$180,328 $13.50 $12.21……… $11.05$11.05 $10.00$10.00…$10.00$4.08 $9.05$9.05 $8.19…$8.19$3.34 $7.41 ……… $0.00045$0.00019 BusinessValuationReview Page24  2014,AmericanSocietyofAppraisers $10.00hasissuedeight-yearwarrantswithanexercise priceof$10.00.Iassumeacommonstockissuanceoccurs after4.00years,andtheexercisepriceofthewarrantsis loweredifthestockpricehasdeclined.Therisk-freerateof interestis2%compoundedcontinuously,andthevolatility ofthecommonstockis50%.Thenumberofstepsinthe latticeis100.Thetwotopnodesinthelatticerepresent extremelysmallprobabilityoutcomeswithextremelyhigh warrantvaluesandnoreset.Thethirdnodeshown representsthecaseofanequalnumberofupanddown movesinthelatticeandastockpriceafter4.00yearsequal tothestartingvalue,$10.00.Inthatcase,theexerciseprice isnotreset,andthevalueofafour-yearwarrantis$4.08. Thenextnoderepresentsthecaseofforty-nineupand fifty-onedownmoveswitharesultingstockpriceafter 4.00yearsof$8.19.Iftheexercisepriceonthewarrant werenotreset,thevalueofthewarrantwouldbe$2.84. Withtheexercisepriceresetto$8.19,thevalueis$3.34. Eachofthelatticeoutcomesinthelowerhalfofthelattice producesahigherwarrantvaluethanwouldbethecase withouttheresetfeature.Itisworthnotingthatforallofthe prices$10.00andlower,thepayoffinthelatticeisthatof anat-the-moneywarrant,whichinthisexampleisworth 40.8%ofthecommonstockprice.Takingintoaccountthe resetfeature,thevalueofthewarrantwhenissuedis$6.04, whichis8.1%higherthanthevalueofasimilarwarrant withoutaresetfeature,$5.59. Table3reportsthevaluesofwarrantswiththesame basicfeaturesandarangeofcommonstockvolatilities andtimestoreset.Onapercentagebasis,theresetfeature hasamaximumeffectforvolatilitiesintherangeof20% to30%butissmallerforbothhigherandlower volatilities.Themaximumpercentageeffectoccurswhen theresetisapproximatelyhalfwaythroughthelifeofthe warrant.Iftheresetoccursveryearly,thepotentialfor downwardmovementintheexercisepriceissmaller;if theresetoccursverylate,youhavealowertimevalueof theoptionassociatedwiththelowerreset.Inthis example,thevalueoftheresetfeatureislessthan11% ofthevalueofthewarrant.Supposetheprobabilityofa securityissuanceis20%.Inthatcase,thevalueofthe resetprovisionrelativetothevalueofthewarrantislikely tobelessthan2.2%[(11%)(20%)].Whileeachsetof circumstanceswilldiffer,thisanalysissuggeststhat ignoringtheeffectonvalueofwarrantresetprovisions maybereasonableinmanyinstances. IfweuseaMonteCarlosimulationanalysis,wecan introducemorecomplex,andperhapsmorerealistic, financingandresetscenarios.ToillustratetheMonte Carloapproachandthepotentialeffectsofamore complexsetoffinancingpossibilities,Iconsidernew issuanceattheendofeachofthefirstsevenyears.Itreat thedecisiontoissuenewsecuritiesasarandomvariable withaconstantprobabilityofoccurrenceeachyear.The Table3 ValueofaWarrantwithaSingleResetafter4.00Years S 5 $10.00, X 5 $10.00, r 5 2.0%, s 5 50%, T 5 8.0 FractionofTimetoReset 0.10.20.30.40.50.60.70.80.9 Volatility WarrantValue 10%$2.08$2.11$2.12$2.12$2.11$2.10$2.08$2.06$2.02 20%$3.11$3.16$3.19$3.19$3.19$3.18$3.15$3.12$3.06 30%$4.12$4.18$4.22$4.23$4.23$4.22$4.19$4.15$4.08 40%$5.05$5.13$5.17$5.18$5.18$5.17$5.15$5.10$5.02 50%$5.90$5.98$6.02$6.04$6.04$6.03$6.00$5.96$5.88 60%$6.66$6.73$6.77$6.78$6.79$6.78$6.75$6.71$6.63 70%$7.31$7.38$7.42$7.43$7.43$7.43$7.40$7.36$7.29 80%$7.88$7.93$7.96$7.98$7.98$7.97$7.95$7.91$7.84 90%$8.35$8.40$8.42$8.43$8.43$8.42$8.40$8.36$8.29 IncreaseinValueCreatedbytheResetFeature 10%7.2%8.6%9.0%8.9%8.6%8.0%7.1%5.9%4.3% 20%7.9%9.7%10.6%10.9%10.8%10.3%9.5%8.2%6.2% 30%7.4%9.1%10.0%10.4%10.4%10.0%9.4%8.2%6.3% 40%6.5%8.1%8.9%9.3%9.3%9.1%8.5%7.6%5.9% 50%5.6%7.0%7.7%8.1%8.1%7.9%7.5%6.6%5.2% 60%4.8%6.0%6.6%6.8%6.9%6.7%6.4%5.7%4.5% 70%4.0%5.0%5.5%5.7%5.7%5.6%5.3%4.7%3.7% 80%3.3%4.1%4.5%4.7%4.7%4.6%4.3%3.8%2.9% 90%2.7%3.3%3.6%3.8%3.7%3.6%3.3%2.9%2.0% ValuingWarrants:DilutionandDown-roundPriceProtection BusinessValuationReview—Summer2014Page25 issuanceisindependentofthestockpriceandofany previousfinancing.Inareal-lifesituation,itmaybe appropriatetochangebothoftheseassumptions. Table4reportsthevaluesofthewarrantswithareset featureforrangesofprobabilitiesofnewissuancesand volatilities.Thetablealso reportsthepercentage increaseinvalueattributabletotheresetfeature.I selectedtheprobabilities0.14,0.29,…,1.0(1/7,2/7, …,7/7)suchthattheexpectednumberoftimesanew financingoccursis1,2,…,7.Itseemsreasonableto focusontheresultsfornewfinancingonetothree times.Foronefinancing,thevalueincreasesrange from6.3%to9.6%.Thesevaluesaresimilartothose showninTable3,wheretheaveragevalueacrossall combinationsofasinglefinancingatdifferentpointsin theeight-yearperiodis6.6%.Whentheexpected numberoffinancingsistwo,thevalueofthereset provisionrangesfrom9.3%to13.5%;whenitisthree, therangeis11.4%to17.9%.Theseindicativevalues maybeusefulindeterminingwhenaresetfeature shouldbevaluedandwhenitcanreasonablybeignored. Summary Therearetwotypesofdilutionthatplayarolein warrantvaluation.Non-fair-valuedilutionoccurswhen warrantsaredistributedforlessthantheirfairvalue.In thiscase,thevalueofthecommonstockdecreases,andit isimportanttotakethatintoconsiderationwhenvaluing warrants.Itisalsoreasonabletobelievethatthiscaseis primarilyanartifactoftextbookexamples.Participation dilutionoccursbecausewarrantssharetheappreciationof thefirm’svalue.Thisdilutioncanbemodeled,ornot, becauseitseffectisrelativelysmall.However,itis importanttoappreciatethatwhenmodeled,thevolatility usedshouldbethevolatilityofthecommonstockand warrants,andwhenthisdilutionisnotmodeled,the volatilityshouldbethatofthecommonstockonly. Iillustratedtwomethodstovaluewarrantswithareset feature.Onemethodassumesthatthereisasinglefuture financingataknowndateandrequiresthecombinationof alatticeandtheBSMformula.Foraneight-yearwarrant, theresetfeatureincreasedthewarrantvalueby2%to11%, dependingonthestockvolatilityandthetimingofthenew financings.Iftheprobabilityofanewfinancingissmall, forexample20%,thentheoverallinfluenceonvalueis likelytobesmallenoughtoignore,i.e.,0.4%to2.2%of thevalueofthewarrant,whichisitselftypicallyasmall percentageofthecapitalstructure.Thesecondmethod employsMonteCarlosimulationandallowsformultiple financingsatrandomtimesoverthelifeofthewarrant.As theexpectedvalueofthenumberoftimesthefirmissues commonstockincreases,thevalueoftheresetfeature increases.Whentheexpectednumberoffutureissuancesis threeoveraneight-yearperiod,theresetfeatureadds between11%and18%tothevalueofthewarrant. Whetherthisislargeenoughtorequireinclusionina valuationwilldependonthesizeofthewarrantposition. Table4 ValueofaWarrantwithaProbabilityofResetEachYear S 5 $10.00, X 5 $10.00, r 5 2.0%, s 5 50%, T 5 8.0 ProbabilityofAnnualReset Volatility0.140.290.430.570.710.861.00 10%$2.06$2.16$2.23$2.28$2.32$2.35$2.38 20%$3.10$3.27$3.40$3.49$3.56$3.62$3.67 30%$4.12$4.33$4.49$4.60$4.70$4.77$4.83 40%$5.07$5.30$5.47$5.60$5.70$5.78$5.84 50%$5.94$6.18$6.35$6.48$6.57$6.65$6.71 60%$6.74$6.96$7.13$7.25$7.34$7.41$7.47 70%$7.48$7.69$7.83$7.94$8.02$8.09$8.14 80%$8.19$8.37$8.50$8.59$8.66$8.72$8.77 90%$8.91$9.06$9.17$9.25$9.31$9.36$9.40 IncreaseinValueCreatedbytheResetFeature 10%6.3%11.0%14.6%17.3%19.5%21.3%22.7% 20%7.8%13.5%17.9%21.1%23.6%25.7%27.4% 30%7.5%13.0%17.1%20.1%22.5%24.4%26.0% 40%6.9%11.8%15.4%18.1%20.1%21.8%23.2% 50%6.4%10.6%13.7%15.9%17.7%19.1%20.2% 60%6.2%9.7%12.2%14.1%15.6%16.7%17.7% 70%6.4%9.3%11.4%12.9%14.1%15.1%15.8% 80%7.4%9.8%11.5%12.7%13.7%14.4%15.0% 90%9.6%11.5%12.8%13.8%14.5%15.1%15.6% BusinessValuationReview Page26  2014,AmericanSocietyofAppraisers Thispublicationhasbeenpreparedforgeneralinformationonmattersofinterestonly,anddoesnotconstituteprofessionaladviceonfactsand circumstancesspecifictoanypersonorentity.Youshouldnotactupontheinformationcontainedinthispublicationwithoutobtainingspecific professionaladvice.Norepresentationorwarranty(expressorimplied)isgivenastotheaccuracyorcompletenessoftheinformationcontained inthispublication.PricewaterhouseCoopersLLP,itsmembers,employees,andagentsshallnotberesponsibleforanylosssustainedbyany personorentitywhoreliesonthispublication. PwCreferstotheUnitedStatesmemberfirm,andmaysometimesrefertothePwCnetwork.Eachmemberfirmisaseparatelegalentity. Pleaseseewww.pwc.com/structureforfurtherdetails.ThisproposalisprotectedunderthecopyrightlawsoftheUnitedStatesandother countries. Abouttheauthor DwightGrant isamanagingdirectorinPwC’sValueConsultinggroup,basedintheSanFranciscooffice.He advisesclientsoncomplexfinancialinstrumentsandoverseesprofessionalstandardsandtechnicaltrainingforthe practice. PwC’sValueConsultinggroupadvisesclientsontheanalysisandvaluationofcomplexfinancialinstruments,such asderivatives,hybridsecurities,hedginginstrumentsandcontingentpaymentagreements.Servicesprimarilyfocus onvaluationsandmeasuringthefairvalueofassetsforfinancialreporting,taxplanning,transactionsanddecision- makingpurposes. Dwighthasover30yearsofexperienceprovidingvaluationservicestoclients,Hepreviouslytaughtat Thunderbird,TheGarvinSchoolofInternationalManagementandwasalsoavisitingprofessoratFundação ArmandoAlvaresPenteadoinSãoPaulo,Brazil.HeholdsaBAineconomicsfromUniversityofWesternOntario, anMBAinfinancefromWhartonatUniversityofPennsylvaniaandaPh.D.infinancefromtheUniversityof Pennsylvania. DwightGrant,PhD (415)498-8116 dwight.grant@us.pwc.com