The Dividend Discount Model with Multiple Growth Rates of Any Order for Stock Evaluation. Abdulnasser Hatemi-J and Youssef El-Khatib

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1 he vdend scount Mode wth Mutpe Gowth Rates of ny Ode fo Stoc Evauaton bdunasse Hatem-J and Youssef E-Khatb UE Unvesty -n OBox 555 Unted ab Emates E-mas: and bstact In ths pape we povde a enea souton fo the dvdend dscount mode n ode to compute the ntnsc vaue of a common stoc that aows fo mutpe stae owth ates of any pedetemned numbe of peods mathematca poof s povded fo the suested enea souton numeca appcaton s aso pesented he souton ntoduced n ths pape s expected to mpove on the pecson of stoc vauaton whch mht be of fundamenta mpotance fo nvestos as we as fnanca nsttutons Runnn tte: Stoc Evauaton wth Mutpe stnct Gowth Rates JEL Cassfcaton: G C Keywods: Stoc Evauaton vdend scount Mode Mutpe Gowth Rates

2 Intoducton etemnn a measue that can epesent the ntnsc vaue of a stoc s an mpotant ssue fo nvestos and fnanca nsttutons seen poftabe nvestment pospects hs ssue mht be aso of nteest fo pocy maes n ode to desn appopate fnanca udance he dvdend dscount mode M whch has onay been deveoped by Godon and Shapo 956 and Godon can be used fo ths pupose Cuenty thee ae extensons of the mode n the teatue that aow fo the vauaton of a common stoc wth two dffeent owth ates acoss tme at the maxmum to ou best nowede In ths pape we suest a enea souton fo the vauaton of common stocs fo the M that aows fo mutpe owth ates of any pedetemned numbe he suested souton s poved mathematcay and an appcaton s povded he emann pat of ths pape s oanzed as foows he next secton pesents the mode and povdes a enea souton aon wth a mathematca poof Secton appes the suested souton to vauate a common stoc wth thee dffeent owth ates he ast secton offes the concudn emas he Mode wth Mutpe Gowth Rates fo Stoc Evauaton We consde and dffeent owth annuty peods wth owth we denote by : the end of peod to : wth n the foown: wth owth foowed by a pepetuty wth owth ate wth owth Fo he tme s to be on fom t : annutes and fom to wth a pepetuty as expaned un the owth ate s he dvdend at yea befoe s See aso Godon and Goud 978 as we as Fue and Hsa 98 It shoud be mentoned that thee ae aso atenatve modes fo stoc vauaton such as the capta asset pcn mode deveoped by Shape 96 and the abtae pcn theoy suested by Ross 976

3 he dvdend at the end of the peod s un the owth ate s he dvdend at yea between and s he dvdend at the end of the peod s Contnun ecusvey unt the foown: un the owth ate s he dvdend at yea between - and s he dvdend at the end of the peod s nd smay by contnun we have un the owth ate s he dvdend at yea between - and s he dvdend at the end of the peod s Moeove fo any and we can expesse the foown: he dvdend afte the m yeas s m m ow we can pesent ou man esut va the foown poposton oposton he vaue of a shae s ven by

4 whee : fo any : whee oof Fst notce that the vaue of a shae s ven -n enea tems- by the sees denotes the dvdend at yea wth espectve owth ate vaue of a shae can be wtten as 5 In the case of annutes peods the foowed by a pepetuty wth owth ate hen 6 he ast ne contans two sums of eometc sees that can be cacuated as foows

5 5 7 and 8 Substtutn equatons 7 and 8 n 6 we obtan whee fo ae ven by equatons and hs ends the undeyon poof In the pevous poposton we povde the vaue of a shae as a functon of dffeent owth ates and dvdends at end of peod fo the nteest ate We can deve a moe expct fomua as a functon of wth dffeent owth ates and the nteest ate as s shown n the foown cooay Cooay he vaue of a shae s ven by 9 whee fo ae ven by equatons and

6 oof he poof s stahtfowad by embeddn the vaue of nto the fomua fom Exampe In ths exampe we consde the case of owth ate and afte that a pepetuty wth owth ate at tme on fom to wth a owth ate fomua 9 fo ths patcua case can be wtten as and afte e when thee s one he tme s then wth owth ate he Exampe In ths exampe we consde the case of e we have annutes wth owth ates fo tme and afte that a pepetuty wth owth ate fo tme he tme s then on fom to wth a owth ate and equatons - we have whee and 6 wth a owth ate and afte wth the owth ate fom to Usn fomua hus the foown can be expessed: and

7 7 Exampe In ths exampe we consde the case of e we have annutes wth owth ates fo tme fo tme fo tme and afte that a pepetuty wth the owth ate an usn fomua and equatons - we have whee and hus we have n ppcaton Suppose a ban has ust pad a dvdend pe shae of $ Its dvdend s expected to ow at 5% dun the fothcomn yeas t s expected to ow by 7% n yeas to 7 and afte

8 that t s expected to ow at 6% pe yea n pepetuty he equed ate of etun s assumed to be 9% hs nfomaton means that the M wth thee owth ates can be used to fnd the vaue of a shae fo ths copoaton Based on fomua we can expess the foown: [ ] + [ + + ] + abe : he ppcaton Vaabe Vaue Vaabe 5 Vaue otes: s the dvdend n yea and s the owth ate at peod b means bea peod at tme b epesents the common stoc vaue based on the suested mode s t s evdent fom abe the pesent vaue of the undeyn common stoc s $7589 hs can be a benchma pce n ode to compae t wth the ea maet pce fo evauatn whethe the pce s coecty detemned o not If the stoc s not fay pced eadess f t s ovepced o undepced t can be used fo fndn a statey that esuts n a an by choosn appopate on o shot poston 8

9 Concusons Stoc vauaton s an mpotant ssue n fnanca maets One mode that s euay used fo ths pupose s the dvdend dscount mode M hs mode aows fo ony two dffeent owth ates to the best nowede In ths pape we suest a enea souton fo the M wth mutpe owth ates of any potenta numbe he suested souton s poved mathematcay and a numeca appcaton s aso povded he souton ntoduced n ths pape s expected to mpove on the pecson of stoc vauaton whch mht be of fundamenta mpotance fo nvestos fnanca nsttutons as we as pocy maes 9

10 Refeences Fue RJ and Hsa CC 98 Smpfed Common Stoc Vauaton Mode Fnanca naysts Jouna 9-56 Godon MJ 959 vdends Eanns and Stoc ces Revew of Economcs and Statstcs 99 5 Godon MJ 96 he Investment Fnancn and Vauaton of the Copoaton Homewood IL: R Iwn Godon MJ and Goud LI 978 he cost of equty capta: econsdeaton Jouna of Fnance Godon MJ and Shapo E 956 Capta Equpment nayss: he Requed Rate of oft Manaement Scence - Ross S 976 he btae heoy of Capta sset cn Jouna of Economc heoy 6 Shape WF 96 Capta sset ces: heoy of Maet Equbum unde Condtons of Rs Jouna of Fnance 9 5-