Profitability and Risk Analysis for Investment Alternatives on C-R Domain

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1 roftablty ad sk alyss for Ivestmet lteratves o - Doma Hrokazu Koo ad Osamu Ichkzak Graduate School of usess dmstrato, Keo Uversty 4-- Hyosh, Kohoku-ku, Yokohama, , Japa Tel: , Emal: koo@kbs.keo.ac.p bstract. Ths paper vestgates a method for evaluatg proftablty ad rsk for multple vestmet alteratves, for both cases of cosstet retur over a plaed perod ad fluctuatg retur year by year. The paper frst exames a method for evaluatg a sgle alteratve from the vewpot of proftablty ad safety. The t proceeds to the evaluato of multple mutually exclusve alteratves, out of whch the best oe s selected. The paper proposes - doma whch comprses tal vestmet ad aual retur o each of horzotal ad vertcal axs. O ths doma, expected values of et preset proft ad aual mea proft are represeted. The the procedure for aalyzg ad evaluatg proftablty ad rsk s dscussed, ad the valdty of the proposed method s examed by usg umercal examples. Keywords: roftablty, sk, Safety, ultple vestmet alteratves, - doma. INTODUTION urret ucertates factors related to vestmet alteratves, such as tal vestmet ad aual retur, requre maufacturg compaes to pay careful atteto to methods for evaluatg proftablty ad rgorousess agast expected rsks. ethods for evaluatg ecoomc performace for a set of multple vestmet alteratves, for both cases of stable retur over the plag horzo, ad of fluctuatg retur year by year, are the ma area of focus of ths paper. Ths problem has bee vestgated the feld of egeerg ecoomy, ad basc procedures have bee clarfed ad modfed/exteded prevous research (Seu et al., 982, 986, 994; Nakamura, 98, 2002). Further, geeral ecoomc evaluato procedures wth cosderato of rsk have bee dscussed prevous research (Koo, 2003, 2009, 200, 20). The paper presets the basc model for aalyss the ext secto, ad the proceeds to the case of stable retur Secto 3, to be followed by the case of fluctuatg retur Secto 4. The smple umercal examples Sectos exame the effectveess of the methods proposed Secto 3 ad 4. The paper assumes the followg vestmet alteratve: Fgure 2.: Ivestmet alteratve wth stable retur Where each otato refers to : amout of tal vestmet : aual retur (crease of cash flow ad/or decrease cash outflow) : perod of vestmet : terest rate to be used as hurdle rate proft calculato The above fgure represets cosstet retur type. Fgure 2-2 represets aother case of fluctuatg retur year by year, where meas retur for the -th year. 2. ODEL FOULTION

2 2 3 4 Ths mples that the et preset value, defed by statemet (3,), ca be represeted as the horzotal legth o the - doma as Fgure 3.3. ual retur 4 Fgure 2.2: Ivestmet alteratve wth fluctuatg retur 3. THE SE OF ONSISTENT ETUN 3. epresetato of roft o the - Doma I ths case, the et preset value ad aual mea proft ca be obtaed by the followg equatos. Fgure 3.: epresetato of a vestmet alteratve o the - doma Ital vestmet, ad (3.) worth. (3.2) where s called uform seres preset factor obtaed by called captal recovery factor defed by ad, s. For the purpose of represetg such proft values as ad, ths paper proposes a doma whose horzotal axs correspods to the amout of tal vestmet, ad vertcal axs refers to the aual retur. Ths doma s hereafter referred to as - doma. The, a vestmet alteratve wth tal vestmet ad aual retur s represeted as a pot as show Fgure 3.. Depctg a le from (0,0) whose slope correspods to, aual proft of the vestmet alteratve, whch s obtaed by statemet (3,2), ca be represeted as Fgure3.. Here, t s clear that the values of ad are mutually verse ad holds the ext statemet of. (3.3) Therefore, o the - doma, the le wth the value ca be represeted as show Fgure 3.2. Slope of Slope of Fgure 3.2: The values of ad the - doma o Fgure 3.3: epresetato of o the - doma

3 3.2 epresetato of I ad ayback erod o the - Doma log wth the crease terest rate, the slope of the le of becomes steeper. Sce I (Iteral ate of etur) s defed to be the terest rate whch makes the value of et proft zero, the followg statemet s satsfed. r r 0. (3.4) It follows, r 0, (3.) Therefore, r r. (3.6) The, the value of I s depcted o the - doma as show s Fgure 3.4. Fgure 3.4: I ad payback perod N o the - doma O the other had, as the perod becomes smaller, the value of becomes larger. The value of payback perod N s gve by the perod where the et proft s zero, ad thus descrbed by the ext statemet. N 0, (3.7) Therefore, t follows, N I N N. (3.8) The the value of N ca be descrbed o the - doma as Fgure 3.4. ade Fgure 3.4 mples that, amog multple alteratves, oe wth hgher I always acheves shorter payback perod. Therefore f we evaluate alteratves based o I, aturally oes wth shorter payback perod are selected. It should be oted, however, that selecto by I s detcal wth selecto by payback perod, whch leads to selecto of alteratves of low rsk, ot ecessarly guarateeg hgh ecoomc proft. 3.3 Evaluato of sk The paper the aalyzes the rsk of vestmet alteratves. I ths paper, rsks ecompass those relatg to crease terest rate, crease tal vestmet, ad decrease aual retur. log wth the crease terest rate, the value of creases. Therefore, the slope of the le of o the - doma becomes steeper. I the same cotext, f the value of s creased to aual proft s decreased to r, the. (3.9) Ths mples that the decrease et aual proft ca be evaluated the same cotext as the case of crease terest rate. O the other had, the decrease aual proft from orgal value to ca be evaluated by the ext statemet.. (3.0) For all the proft o the le coectg (0,0) ad plot (, ), the decrease aual retur at the rato β decreases the proft equally. Therefore, t s clear that the rsk agast decrease aual retur ca be evaluated the same logc as the former two cases. From the above dscusso, rsk agast expected chages ca be evaluated smultaeously o the - doma, by the le coectg the pot (0,0) ad each plot (, ). The alteratve wth larger value of / (amely, steeper slope) s more rgorous terms of rsk. ut t should be oted that the alteratve wth hgher rsk averso level does ot guaratee ecoomc proftablty. 3.4 Evaluato of ultple lteratve The paper the exames the case of selectg the best oe out of mutually exclusve alteratves, whch ca be deleated Fgure 3..

4 : : Fgure 3.: utually exclusve alteratves The two alteratves ca be represeted o the - doma as show Fgure 3.6. Fgure 3.7: case of cocave le segmets N N r r Fgure 3.8: ovex le segmets o the - doma Fgure 3.6: ultple alteratves o the - doma Whe le segmets coectg (0,0), (, ) ad (, ) create covex, t should be oted that alteratve wth smaller tal vestmet always guaratees larger I ad shorter payback perod, although the value of et aual proft for may be larger tha. From the vewpot of rsk of crease terest rate, crease by the same rato tal vestmet, ad/or same rato of decrease aual retur, alteratve s more rgorous tha alteratve. What requres atteto s that, whe comparg more tha three alteratves, there mght be cases where le segmets coectg alteratves form cocave as Fgure 3.7. I such a case, eve f the terest rate fluctuates ad the slope of s chaged, the proft for a alteratve s always smaller tha alteratves or. Thus, the alteratve becomes dsqualfed terms of proftablty. It follows that the set of qualfed alteratves o the - doma creates covex le segmets as Fgure THE SE OF INONSISTENT ETUN 4. epresetato of roft o the - Doma Frst, the cash flow patter uder vestgato s descrbed as Fgure Fgure 4.: ash flow patter for the case of cosstet retur

5 I ths case, the et preset proft ca be calculated by the ext statemet, whereas et aual proft s depedet o the retur patter ad caot be obtaed drectly.. (4.) O the - doma, where the vertcal axs s coverted to the sum of aual retur, depctg the le wth slope from (0,0), the value of ca be represeted as show Fgure 4.2. The et preset proft for alteratves ad ca be calculated by the followg statemets, where subscrpts ad superscrpts refer to the ame of respectve alteratves.. (4.2). (4.3) The, these values ca be represeted o the - doma as Fgure 4.3. roftablty ca be evaluated by the legth of ad as Fgure 4.4. Fgure 4.2: epresetato of et preset proft o the - doma It should be oted that the values of I ad payback perod are depedet o the patter of aual retur, ad therefore caot be represeted o the - doma for the case of cosstet retur over the horzo. 4.2 Evaluato of ultple lteratves Ths secto aalyzes the comparso of multple alteratves, as llustrated Fgure 4.3. : : 2 3 Fgure 4.3: ultple alteratves 2 3 Fgure 4.4: ultple alteratves o the - doma For the purpose of evaluatg rgdty uder ucertates, ths secto cosders the case of crease the same rato tal vestmet, ad decrease the same rato aual retur,,,2,...,. I case where the value of tal vestmet creases from to α, as show Fgure 4., whe the plot reaches plot, the et proft becomes zero. Thus, the E for, deoted by, ca be gve by the ext statemet. O the other had, f the value of. (4,4) decreased, the plot o the - doma moves dowward. d f t reaches (refer to Fgure 4.6), the the et preset proft becomes zero. Therefore, E for aual retur decrease ca be gve by the ext statemet. s

6 Fgure 4.: E o the - doma maxmum value of, beg less tha ether or. I the same cotext, decrease or crease aual retur for the same rato over each year amog caddate alteratves, ca be evaluated by the le coectg (0,0) ad each plot. y plot o the same le gets the same mpact from the chage aual retur. Therefore, the decrease (or crease) aual retur ca be evaluated by the shft of the le wth slope startg (0,0) to upward (retur decrease) or dowward (retur crease). I ay case, alteratve Fgure 4.7 caot acheve the maxmum proft, ad ca be therefore dsqualfed selectg most proftable alteratve. Fgure 4.6: E o the - doma. (4.) Statemets (4.4) ad (4.) show that ad mutually verse, satsfyg the ext equato. are (4.6) 4.3 Elmato of Dsqualfed lteratves Ths secto exames the case of multple alteratves represeted o the - doma, where le segmets coectg adacet plots may be cocave. s regards the crease or decrease value of for the same rato amog caddate alteratves, ts mpact ca be a slope chage of the le startg from (0,0), upward wth vestmet crease ad dowward wth vestmet decrease. For both cases, alteratve caot atta Fgure 4.7: Dsqualfed alteratves o the - doma bove dscusso leads to a cocluso that a set of qualfed alteratves, ot from the vewpot of proft but from that of rsk averso uder ucertaty tal vestmet ad aual retur, ca create a set of covex le segmets as show Fgure 4.8. Fgure 4.8: ovex set or qualfed alteratves o the - doma

7 . NUEIL EXLES. The ase of osstet etur 700 Ths secto cosders the followg three alteratves, wth the terest rate =0%. ual mea proft for each alteratve ca be obtaed as follows: : : : Fgure.2: lots o the - doma It ca also be cofrmed that, whe tal vestmet s creased the rato for all alteratves, ad become equally proftable whe satsfes (.4) The, obtaed s.37. I the same cotext, f aual retur s decreased the same rato for all alteratves, proftable s gve by whch alteratves ad are equally Fgure.: Numercal example for the case of cosstet retur (.) (.2) (.3) Therefore, alteratve s most proftable. ut ths calculato caot aalyze robustess agast ucertates. The each alteratve should be plotted o the - doma, whch s show Fgure.2. It s clear from ths fgure that le segmets coectg plots,, ad are cocave, ad therefore, alteratve s dsqualfed. The fgure also shows that the slope coectg plots ad s 0.3. It follows that f the terstate rate s creased to satsfy 0.3, that s, >6%, alteratve becomes more proftable tha alteratve The ase of Icosstet etur Ths secto assumes the followg umercal example (=0%). : : Fgure.3: Numercal Example for fluctuatg aual retur 800

8 The et preset value of proft s gve by: Two alteratves ca be represeted o the - doma as Fgure.4. It ca be cofrmed that s larger tha ths fgure Fgure.4: epresetato of proft o the - doma The slope of the le segmet coectg plots ad s It follows that f tal vestmet s decreased for both alteratves to 93.8% from the curret estmato, proft for both alteratves becomes equal at the value I the same cotext, f the aual retur for each of both alteratves creases up to /0.938=.066 from the curret estmato, both alteratves become break-eve. If the expected chage rsk s lower, the we ca select alteratve after cosderato of rsk uder cosderato. Thus, the proposed aalyss procedure o the - doma helps ecoomc evaluato ad selecto of alteratves uder ucertates. 6. ONLUDING EKS Ths paper vestgated a problem of evaluatg proftablty ad rsk of vestmet alteratves, for both cases of cosstet retur ad fluctuatg retur over the plag horzo. aor outcome of ths paper s the procedure of vsually evaluatg proftablty ad rsk o the - doma. Especally, rsk evaluato o the - doma helps practcal decso makg uder ucerta stuatos. I ths cotext, ths paper has practcal purpose addto to theoretcal valdty of aalyss. The paper smply deoted aual retur by (or for the -th year). However, t actually comprses crease of come, such as sales crease, or decrease of producto cost, cludg materal cost ad processg cost. Therefore, retur ca be dvded to such factors as sales volume, producto volume, ut sales prce, ad ut varable cost. The, prevous research outcomes the feld of egeerg ecoomy applyg total-cost ut-cost doma, ad/or capacty surplus ad shortage dstcto, ca be combed to the aalyss o the - doma. Further research to combe these results of aalyss to help practcal decso makg s left as a topc for future research. EFEENES Koo, H. (2003) ethod for Evaluatg Ivestmet lteratves o the Total-ost Ut-ost Doma, roceedgs for the utum oferece of the Japa Idustral aagemet ssocato, Osaka, Koo, H. ad zumach, T. (2004) Ecoomc Evaluato for ultple Ivestmet lteratves uder Ucertaty, roceedgs of the th ual oferece of sa-acfc Idustral Egeerg ad aagemet Systems, Gold oast, ustrala, Koo, H. ad zumach, T. (2009) roft Sestvty alyss uder Ucertaty for ases of roducto apacty Surplus ad Shortage, Joural of Japa Idustral aagemet ssocato, 9 (6), Koo, H. (200) Ecoomc sk alyss for Ivestmet lteratves osderg Yeld ad apacty over ultple erods, Joural of Japa Idustral aagemet ssocato, 60 (6E). Koo, H. ad Ichkzak, O. (20) ethod ad rocedure for Ecoomc Evaluato of Improvemet ctvtes, Idustral Egeerg & aagemet Systems, 4 (2). Nakamura, Z. (98) Ecoomc Evaluato o the Varable-ost Fxed-ost Doma, roceedgs for the Sprg oferece of the Japa Idustral aagemet ssocato, Tokyo, Nakamura, Z. (2002) Safety Idces of roft uder Ucertates, roceedgs for the utum oferece of the Japa Idustral aagemet ssocato, Fukuoka, 4. Seu, S. ad Fushm, T. (982) Fudametals of Egeerg Ecoomy, Japa aagemet ssocato ress, Tokyo. Seu, S., Futa S., Fushm T., Yamaguch T. (986): Egeerg Ecoomc alyss, Nho Kkaku Kyoka, Tokyo. Seu, S., Nakamura, Z. ad Nwa,. (994) Exercses of Egeerg Ecoomy, Japa aagemet ssocato ress, Tokyo.