A Decision Model for Invesmen Timing Using Real Opions Approach Jae-Han Lee, Jae-Hyeon Ahn Graduae School of Managemen, KAIST 207-43, Cheongrangri-Dong, Dongdaemun-Ku, Seoul, Korea ABSTRACT Real opions approach has recenly received growing aenions in he research on business and echnology managemen. Despie he applicabiliy of he approach, i gained much momenum in pracice. This paper addresses a problem of deciding he opimal invesmen ime as he uncerainy resolves over ime. The opimal iming for invesmen decisions was formulaed, and he characerisics of he opimal decision were idenified. KEYWORDS Decision-Making; Opimal invesmen iming; Real opions approach; Telecommunicaions. Inroducion Real opions approach is based on he idea of evaluaing financial opions. In he financial opions evaluaion, purchase or sales decisions would be based on he volailiy of he fuure sock prices [, 2]. In he real world, he uncerain fuure is considered as a risk o he business decisions. However, as he volailiy in he financial opion is considerable valuable, he uncerainy in he real decision problems can be a good source of opporuniy raher han a risk if i is appropriaely assessed and managed [3,4,5]. Rapid echnological developmen and ever-changing business environmen caused a lo of uncerainies o deal wih. Because of he uncerainy, gahering informaion before major commimen and making coningen decisions based on he undersanding is criical. In ha sense, delaying commimen of resources or delaying opion could be a good sraegy o ake [4,5,6,7]. In his paper, real opions approach was used o esimae he value of delaying opion in he inves decision-making. Using he valuaion approach, a model for deciding opimal invesmen iming was developed. Undersanding delaying opion and making coningen decisions based on he undersanding would help business managers o raionally evaluae he business siuaions and make inelligen invesmen decisions. Empirical sudy is also done for a mobile elecommunicaions company in Korea. Since he Personal Communicaions Service (PCS) was inroduced in 997, five mobile elecommunicaions companies, 2 cellular and 3 PCS companies, have been compeing seriously. Wih heir heavy subsidy for mobile phones and markeing effors, he number of subscribers have increased dramaically over o 20M and even surpassed he number of wired elephone users in 999. In 998, he Korean governmen allowed o provide he mobile elecommunicaions service hrough he resale of he exising companies channel capaciies. A Korean company had o decide wheher hey wan o ener he marke or no. In his paper, he company s sraegy for marke enry was evaluaed while marke is mauring. Addiionally, he value of waiing for he informaion, especially poenial growh of he marke was analyzed.
2. A Decision Model for Invesmen Timing If an invesor is absoluely cerain how he oucome of an invesmen would urn ou in he near fuure, hen s/he could easily decide wha o do. I would be he case of deposiing money on he bank or invesing on he risk-free CD. However, wih lo of uncerainies involved in he invesmen decisions, i maybe worhwhile o consider delaying decision and see how he immediae uncerainies are resolved. Wih he informaion, decisions can be make more inelligenly. However, i is likely o incur waiing cos. Therefore, he decision o delay he commimen would be based on he radeoff beween he value of waiing vs. cos of waiing. Le R() be he value of waiing unil ime. I is assumed o be a non-decreasing and wice differeniable funcion of. Le C() be he cos of waiing unil ime. I is also assumed o be a non-decreasing and wice differeniable funcion of. Wih he above assumpions, he problem of finding he opimal ime * can be represened as follows [ ] Max P () = Max R () C(), where P() is he ne profi if decision is delayed unil ime. Le MVW and MCW be a marginal value of waiing and marginal cos of waiing, respecively. Marginal value of waiing (MVW) would come from he fac ha decisions can be made depending on he uncerainy resoluions over ime. Marginal cos of waiing (MCW) happens when cos is incurred when decision is delayed. For example, i could happen when poenial cusomers are los o he compeiors. An invesor would inves now if MVW were less han MCW. Oherwis e he would raher wai and see wha happens. The opimal decision ime for profi maximizaion can be found when he firs derivaive of P() equals zero or he firs derivaives of R() and C() are equal. Tha is, P () R () R () = = MVW MCW = 0 Furhermore, for P() o have a single maximum poin over [ 0, T], he second-order condiion would be required. Because i is a consrained opimizaion over [ 0, T], he opimal ime * can be can be characerized in he following wo cases. Case : * is inerior poin of [0, T], where P () U () D () * = Arg = = 0. 0 < < T Figure : Inerior Poin Case
Case 2: * is a boundary poin If P () > 0 hen * = T. If P () < 0 hen * = 0. Figure 2: Boundary Poin Case When MVW and MCW are linear or consan, he opimal invesmen ime can be easily be found by comparing he derivaives of hem. For example, if he derivaive of MVW is zero and ha of MCW is posiive, hen * equals o 0. On he oher hand, if he derivaive of MCW is zero, hen, he * equals o T, he expiraion dae of he invesmen opporuniy. However, more ineresing case is when he derivaive of MVW is negaive and he derivaive of MCW is posiive. Tha represens he real marke siuaion well and shown in Figure 3. The only ime he opimal invesmen ime can be an inerior poin is when MVW and MCW show he relaionship shown in Figure 3-. Bu, if he MVW and MCW increase as he Figure 3-2 and 3-3, hen he opimal invesmen ime becomes zero or T. Figure 3: Characerisics of he opimal ime 3. An Empirical Sudy In he empirical sudy, a Korean mobile elecommunicaions company was chosen and heir daa were used o esimae he opimum marke enry ime. The company was rying o resale he channels of oher Korean mobile elecommunicaions company. Their business plan was ha hey buy he channels from oher company and bundle i wih heir exising elecommunicaions services. Their plan may make sense because cusomers wan a bundled service wih a single bill and single cusomer conac.
Forecas of Subscribers in Korean marke In order o esimae fuure subscribers for wireless elecommunicaions service, several assumpions are made. One of he assumpions was ha oal number of subscribers is o follow an S-shape curve which approaches o he ceiling value over ime. Based on he assumpions, number of subscribers unil he end of year 2002 was esimaed using non-linear regression mehod. The Resul is as follows. Table : Esimaed number of subscribers by he regression model (Uni: Thousands) 99. Q3 99. Q4 00. Q 00. Q2 00. Q3 00. Q4 0. Q Toal subscriber 7,753 9,222 20,469 2,496 22,320 22,969 23,472 0. Q2 0. Q3 0. Q4 02. Q 02. Q2 02. Q3 02. Q4 Toal subscriber 23,856 24,47 24,366 25,5 25,638 25,732 25,802 Expeced marke size for he company A survey was done for,500 people o undersand heir inenions o subscribe. The reference probabiliy ha would decide wheher a non-subscriber will subscribe for a mobile elephone service was esimaed o be p* = 0.3549. If P i p* = 0.3549, hen a subscripion is expeced, else no subscripion is expeced. Based on he probabiliy, he company is expeced o acquire he following subscribers, if he company sars a wireless resale service in Oc. 999. Table 2: Esimaion of he subscribers (Churn rae = 0.03/monh) (Uni: acual) 99.Q3 99.Q4 00.Q 00.Q2 00.Q3 00.Q4 0.Q Accumulaed 34,557 8,533 236,768 297,9 358,29 45,353 464,820 New 34,557 46,976 55,235 60,423 6,00 57,062 49,467 Average 3,866 76,087 229,665 288,275 347,542 402,892 450,875 0.Q2 0.Q3 0.Q4 02.Q 02.Q2 02.Q3 02.Q4 Accumulaed 505,00 535,977 558,878 575,303 586,828 594,790 600,23 New 40,90 30,967 22,90 6,425,525 7,962 5,44 Average 489,860 59,898 542,2 558,044 569,223 576,946 582,224 Esimaion of he number of subscribers caused by invesmen delay The subscripion probabiliy for a non-user is assumed o be consan over ime. Tha is, f() = q F(), where f() is he probabiliy of a cusomer would subscribe a ime, and F() is he probabiliy of a cusomer o have subscribed by he ime. Then, he company s poenial subscriber will be
K The size of laen marke a = K Toal laen marke size D 0 = D MaximumPoenial D 0 K 0 0 Max Subscriber : Ceiling of he oal number of subscriber D() : Toal number of subscriber a K 0 : Poenial subscriber, if invesed in Oc. 999 (No delaying of decision) Table 3: Change of subscribers caused by decision delay (Uni: housands, Marke Ceiling=30,467) 99.Q3 99.Q4 00.Q 00.Q2 00.Q3 00.Q4 0.Q K 582 354 278 25 64 24 94 0.Q2 0.Q3 0.Q4 02.Q 02.Q2 02.Q3 02.Q4 K 70 52 38 28 7 3 0 Esimaion of Volailiy One way of esimaing volailiy or uncerainy of he invesmen projec would be from he pas sock prices. However, he company in sudy has no pas hisory of providing any wireless business. Therefore, pas 2 years sock price for he oher elecommunicaions company daa was used insead. Le n : Number of daa S i : Sock price a he end of i-h period (i = 0,, Κ, n) τ : The number of period in a quarer u : The average value of s = n n i= ( u i u) u 2 i 2 S, T ƒð ln ƒó (ƒê ) T, ƒðt S 2 According o his, he sandard deviaion of u is ƒð T. Therefore, s is he esimae of ƒð T and s* can be used as he esimae of σ iself. s* can be found as s s * = (S.E = s 2n ƒñ Average of s = n u i = 0.049 n ( u i u ) i= * ) 2 =0.088 i Wih ha, τ and s* were esimaed o be τ=/3 and s * = s =0.294864. 4. Evaluaion of MVW & MCW Black-Scholes model assumes he marke condiions are consan unil he opion s expiraion dae. Therefore, his sudy uses he cos of mainaining he subscriber level a =0 (Oc. 999), as he cos of delaying invesmen decision.
Here e was used as a cusomer reenion weigh o keep he laen subscriber from leaving o oher service providers. The value of he real opion can be esimaed by using he Black-Scholes model. The resuling value is shown in Figure 4. As shown in he figure, he cos increases far more rapidly han he value of he real opion. This difference can be more easily recognized from Figure 5, he ne profi. This figure shows ha he company loses heir value if invesmen decision is delayed. Figure 4: B-S Opion value & Cos Figure 5: Ne profi Considering he fac ha he Korean mobile elecommuncaions marke has already grown close o is sauraion sage by he ime he company was considering o sar he wireless resale business, i migh be useless o argue ha i is beer o inves now han wai. Too see he difference in marke condiions, i is proposed o esimae invesmen iming assuming ha he marke is a is early sage. Figure 6 represens he marke growh and Poin B represens curren siuaion (998.8). Poin A in Figure 6 is choosen as a suiable poin for he earlier sage of he marke growh, from where he value and he cos of waiing is calculaed. The resuls are as follows. Figure 6: Marke peneraion
Figure 7: The MVW & he MCW(997.0) Figure 8: Ne profi (997.0) From Figure 8, he opimal invesmen iming is o wai 5 quarers o observe he change in marke condiions before making a decision o ener he marke. In he early ime, he cos of waiing is no subsanial enough o make a hecic sar for he service. 5. CONCLUSIONS This paper addressed a problem of deciding he opimal invesmen ime as he uncerainy resolves over ime. The opimal iming for invesmen decisions was formulaed, and he characerisics of he opimal invesmen ime were idenified. The invesmen ime model was applied o a Korean company looking for he enry o he resale marke for he personal mobile communicaions service. The analysis showed ha i is beer o wai abou before commiing resources. Because here is no much o lose for already sauraing marke, i seems o be beer o wai and see how he grea uncerainy regarding he poenial marke size of he marke is laer resolved. However, here may be some opion value o ener he marke early ha was no considered in his paper. The cusomer base esablished by enering he marke early may prove o be a good invesmen serving as a sepping-sone for he nex generaion service. Tha would be anoher research area o pursue. REFERENCES. Black, F. and Scholes, M(973). The pricing of opions and corporae liabiliies. Journal of Poliical Economy, 3, 637-654 2. Hull, J. C.(997) Opions, Fuures, and oher Derivaives, Prenice Hall Inernaional Inc., New Jersey USA. 3. Copeland, T. E. and Keenan, P. T.(998). Making Real Opions Real. The McKinsey Quarerly, No. 3, 28-4 4. Dixi, A. K. and Pindyck, R. S.(995). The Opions Approach o Capial Invesmen. Harvard Business Review May-June, 05-5 5. Luehrman, T. A.(998a). Invesmen Opporuniies as Real Opions: Geing Sared on he Numbers. Harvard Business Review, July-Augus, 3-5 6. Lander, D. M. and Pinches, G. E.(998). Challenges o he Pracical Implemenaion of Modeling and Valuing Real Opions. The Quarerly Review of Economics and Finance, 38, 537-567 7. Meron, R. C.(998). Applicaions of Opion-Pricing Theory: Tweny-Five Years Laer. The American Economic Review, June, 323-349