Proceedngs of the 2001 Wnter Smulaton Conference B. A. Peters, J. S. Smth, D. J. Mederos, and M. W. Rohrer, eds. A REAL OPTIONS DESIGN FOR PRODUCT OUTSOURCING Harret Black Nembhard Leyuan Sh Department of Industral Engneerng Unversty of Wsconsn-Madson 1513 Unversty Avenue Madson, WI 53706-1572, U.S.A. Mehmet Aktan Department of Industral Engneerng Atatürk Unversty College of Engneerng Erzurum, 25240, TURKEY ABSTRACT We develop a fnancal model to assess the opton value of outsourcng. We value the real optons assocated wth outsourcng an tem usng Monte Carlo smulaton. Ths valuaton gves decson makers a way to choose the approprate outsourcng strategy based on an ntegrated vew of market dynamcs. A smulaton example s used to demonstrate the applcaton of real optons to value outsourcng. The smulaton program code was wrtten n JavaScrpt so that the valuaton task would be accessble to other users because of ts web enabled feature. 1 INTRODUCTION Increased competton n the global market has caused organzatons to realze that the most compettve way of survval s hgh value. Ths can often be acheved through ncreased flexblty. Then the queston becomes: Precsely how valuable s flexblty? The fnancal arena was the orgnal ground for the applcaton of the optons-based framework to the valuaton of flexblty. More recently, manageral operatng flexblty has been lkened to fnancal optons. The goal of our research s to vew the flexblty surroundng manufacturng operatons usng fnancal optons. Nembhard, Sh, and Park (2000) develop a framework for the broad scope of ths research actvty. In Nembhard, Sh, and Aktan (2001), we consder the decson to ntroduce statstcal process control (SPC) charts to montor qualty. In ths paper, we specfcally consder the manufacturng decson to ncrease flexblty through outsourcng, whch s recognzed as a source of great compettve advantage (Gupta and Zhender 1994). The classcal net present value approach often falls short n the analyss of such decsons, due to ts nablty to address market dynamcs wth respect to the key determnng varables. We use the optons approach to fnd the value of outsourcng durng a specfed length of tme, consderng future uncertanty. The problem s analyzed usng Monte Carlo smulaton. Results of the smulaton are provded wth numercal examples. Usng the proposed desgn, a company wll be able to answer questons about the long-term value of product outsourcng. Ths paper s organzed as follows. Approaches for multvarate opton valuaton are dscussed n Secton 2. The fnancal model that wll be used to fnd the opton value of outsourcng s defned n Secton 3. Secton 4 dscusses the Monte Carlo smulaton. An example and numercal results are gven n Secton 5. We make some concludng remarks n Secton 6. 2 OPTION MODELS An opton s the rght, but not the oblgaton, to take an acton n the future (Amram and Kulatlaka 1999). Some optons are assocated wth nvestment opportuntes that are not fnancal nstruments. These operatonal optons are often termed real optons to emphasze that they nvolve real actvtes or real commodtes, as opposed to purely fnancal commodtes, as n the case, for nstance, of stock optons (Luenberger 1998). A European opton gves the rght to exercse the opton on the expraton date. Here, we formulate the outsourcng problem as a seres of n European optons, where all optons start at tme zero, and each opton expres at one of the n equally spaced tme ntervals. In our context, ths means that outsourcng may or may not be appled (whch s the opton) n any tme nterval. Black and Scholes (1973) developed a closed form soluton for valung a European opton wth one varable. In the case of one varable, the bnomal lattce approach of Cox, Ross, and Rubnsten (CRR) (1979) s a powerful numercal procedure for valung optons. Boyle (1988) developed an extenson of the CRR procedure for opton valuaton n the case of two state varables. In our outsourcng model (whch wll be defned n the next secton), there are multple sources of uncertanty. 548
Nembhard, Sh, and Aktan Valung real optons for such a model wll requre an analyss of multple varables. Boyle, Evnne and Gbbs (1989) developed an n-dmensonal extenson of the CRR procedure usng multnomal lattces. Kamrad and Rtchken (1991) developed a smlar multnomal lattce technque for valung projects for one or more state varables. However, multnomal lattce approaches are not practcal for opton valuaton f we must deal wth a large number of varables, because these approaches generate many nodes and requre extensve calculatons. Monte Carlo smulaton provdes a good alternatve for valung European optons wth more than one varable. 3 A FINANCIAL MODEL FOR OUTSOURCING In ths secton, we provde a framework for the fnancal model for product ourscourcng. It parallels Kouvels (1999) whch develops a general framework for evaluatng the total expected cost of outsourcng from a global network of supplers when the purchasng frm faces uncertan exchange rates. Consder an tem that s a part of a fnal product. The total cost for the tems sold has three man sources of uncertanty: unt producton cost of the tem durng the tme nterval begnnng at tme t, S 1 (; unt outsourcng prce of the tem durng the tme nterval begnnng at tme t, S 2 (; and unt delvery cost of the outsourced tem durng the tme nterval begnnng at tme t, S 3 (. We assume that the demand D s constant. Assumng there s no swtchng cost, and the manufacturer can change the decson about outsourcng at each tme nterval. If the unt outsourcng cost of the tem s less than the unt producton cost of the tem, then total cost reducton R( from outsourcng the tem durng the tme nterval that begns at tme t s [ S ( S ( S ( t ] D R = ) (. 1 2 3 If K denotes the fxed cost of outsourcng per tme nterval (contractng cost, and other possble costs to make the contrac, then net cost reducton F( due to the outsourcng opton s { 0, [ S ( S ( S ( t ] D K} F( = max 1 2 3 ). (1) Fgure 1 shows the relatonshp between the varables, total producton and outsourcng costs, the opton value, and ts effect on decsons. The three varables affect the total costs. Whether we outsource or not also nfluences the total costs. If the outsourcng cost of the tem s less than the producton cost of the tem, then we may decde to outsource the tem nstead of producng t. The opton value s found by evaluatng the cost reducton that may be possble by outsourcng n future. Dependng on the opton value, we decde whether to outsource now, possbly later, or never. Unt Producton Cost Unt Outsourcng Prce Do Not Outsource Producton Cost Outsource Cost Compare Opton Value Decson Unt Delvery Cost Outsource Fgure 1: Relaton between varables and the opton value 4 MONTE CARLO SIMULATION To value the three-varable outsourcng opton, we wll use Monte Carlo smulaton. Snce there are no swtchng costs, a decson gven n a tme nterval does not affect the decsons n other tme ntervals. Ths feature makes t relatvely easy to use Monte Carlo smulaton n ths problem. Smulaton models may be used to gve numerous possble paths of evoluton for underlyng state varables from the present to the fnal date n the opton. In the commonly used Monte Carlo smulaton method, the optmal strategy on each path s determned and the payoff s calculated (Amram and Kulatlaka 1999). Suppose that the process followed by the underlyng varable S n a rsk-neutral world s ds = µ Sdt + σ Sdz (2) where z s a Wener process, µ s the expected return n a rsk-neutral world (µ = r), and σ s the volatlty. To smulate the path followed by S, we dvde the lfe of the underlyng varable nto n short ntervals of length Dt and approxmate Equaton (2) as S( t + S( = µ S( t + σs( ε t (3) where S( denotes the value of S at tme t, and ε s a random sample from a normal dstrbuton wth a mean zero and unt standard devaton. Ths enables the value of S at tme Dt to be calculated from the ntal value of S, the value at tme 2Dt to be calculated from the value at tme Dt, and so on. One smulaton tral nvolves constructng a complete path for S usng n random samples from a normal dstrbuton (Hull 1997). 549
From Ito s lemma (see Hull (1997) for a dscusson of Ito (1951)), the process followed by lns s so that σ 2 d ln S = µ dt + σ dz 2 Nembhard, Sh, and Aktan σ 2 S( t + = S( exp µ t + σ ε t 2 (4) Ths equaton s used to construct a path for S n a smlar way to Equaton (3). Whereas Equaton (3) s true only n the lmt as Dt tends to zero, Equaton (4) s exactly true for all Dt (Hull 1997). In our model, we use Equaton (4) to generate values for the three varables S 1 (, S 2 (, and S 3 (. For each t, we calculate the average of max{0, [S 1 ( S 2 ( S 3 (] D K} values (see Equaton (1)) obtaned from the smulaton runs, and dscount that average to the present tme. In order to fnd the estmated opton value, we add the dscounted averages for all t untl T, where T s the expraton tme of the opton. Snce the three varables may be correlated, we need correlated samples ε, (=1,2,3) from normal dstrbutons (see Equaton (4)) where the coeffcent of correlaton between sample and sample j s ρ,j. We frst sample three ndependent varables x (=1,2,3), from unvarate standardzed normal dstrbutons. The requred samples ε are ε = α x. k = 1 For ε to have the correct varance and the correct correlaton wth the ε j (1 j < ), we must have and, for all j <, j k= 1 k = 1 α 2 k k = 1 α α = ρ. k jk The frst sample, ε 1, s set equal to x 1. These equatons for the α s can be solved so that ε 2 s calculated from x 1 and x 2 ; and ε 3 s calculated from x 1, x 2 and x 3 (Hull 1997). k, j Fgure 2: Input wndow for the smulaton 5 A SIMULATION EXAMPLE The program code for the Monte Carlo smulaton was wrtten n JavaScrpt, so that t can be executed wth Mcrosoft Internet Explorer 3.0 or Netscape Navgator 2.0, and later versons. Frst, the user enters all parameter values nto the nput boxes and these values are checked to ensure that they are vald. In order to store the generated values, the program generates an array for each of the three varables. The frst array contans the S 1 ( values for all tme ntervals untl the expraton date of the opton. Smlarly, the second array contans the S 2 ( values, and the thrd array contans the S 3 ( values. Then, the value of Equaton (1) at tme t s calculated for all smulaton runs, and the average of those values s found. Then, ths average s dscounted to the present tme. A dscounted average s calculated for each tme pont,.e., t,2t T. The opton value estmate s the sum of these dscounted averages. Fgure 2 gves the nput wndow for the smulaton program. For ths smulaton example, the tme untl opton expraton s one year, and length of each tme nterval s one month snce there are 12 ntervals. The rsk-free nter- 550
est rate r s 6% per year. The number of smulaton runs s 5,000. The other parameters for modellng varables, volttly, and correlaton can be seen n the nput felds. Mcrosoft Internet Explorer verson 5.5 was used to run the smulaton code. Fgure 3 gves the output wndow where the estmated opton value, standard devaton of the opton value estmate, and mnmum and maxmum opton values obtaned from the 5,000 smulaton runs are presented. Also, for each tme nterval, the fracton of smulaton runs where outsourcng resulted n a cost reducton s gven as a rate n the output wndow. The estmated opton value s $247,610. Ths means that consderng outsourcng as an opton for one year has an estmated value of $247,610. Standard devaton of the estmated opton value s $4,437. Then, a 95% confdence nterval for the opton value s $247,610±1.96*$4,437 whch gves ($238,912, $256,308). Nembhard, Sh, and Aktan We see that outsourcng rate s zero at t=0. Ths means that outsourcng does not reduce the cost now. From t=1 to t=12, outsourcng rate s between 46.92% and 49.78%. Therefore, our chance of outsourcng from the frst month to the twelfth month s nearly between 47% and 50%. In other words, wth 47 to 50 percent probablty, outsourcng wll cost less than producng. We also see that the maxmum opton value obtaned from the 5,000 smulaton runs s $2,072,012. Ths means that cost reductons as much as $2,072,012 may be possble by outsourcng. Mnmum opton value for the 5,000 smulaton runs s $0. Ths means that market condtons were never favorable for outsourcng n any of the twelve months. 6 SUMMARY In ths paper, we have shown how the value of outsourcng can be determned usng a real optons framework. The need for ths approach s due to the nablty of classcal net present value methods to address dynamcs n the market condton wth respect to unt producton cost, unt outsourcng prce and unt delvery cost for an tem. By connectng the dynamc aspects wth the manufacturng operatonal aspects, we now have a way to address a key ssue: the bottom-lne cost assocated wth the outsourcng decson. Monte Carlo smulaton was key n ths study because for three sources of uncertanty, t was more practcal to use than the other methods. Monte Carlo smulaton also provded the maxmum and mnmum opton values obtaned durng the smulaton, whch represent the best and the worst scenaro for the opton. Fgure 3: Output wndow of the smulaton REFERENCES Amram, M. and Kulatlaka, N. 1999. Real Optons: Managng Strategc Investment n an Uncertan World. Harvard Busness School Press. Black, F. and Scholes, M. 1973. The Prcng of Optons and Corporate Labltes. Journal of Poltcal Economy 81:637-659. Boyle, P. P. 1988. A Lattce Framework for Opton Prcng wth Two State Varables. Journal of Fnancal and Quanttatve Analyss 23(1):1-12. Boyle, P. P., Evnne, J., and Gbbs, S. 1989. Numercal Evaluaton of Multvarate Contngent Clams. The Revew of Fnancal Studes 2(2):241-250. Cox, J. C., Ross, S. A., and Rubnsten, M. 1979. An Opton Prcng: A Smplfed Approach. Journal of Fnancal Economcs 7:229-263. Gupta, M. and Zhender, D. 1994. Outsourcng and ts Impact on Operatons Strategy. Producton and Inventory Management Journal. 35(3):70-76. 551
Hull, J. 1997. Optons, Futures, and Other Dervatves Securtes. Thrd Edton, Prentce Hall Englewood Clffs, N.J. Ito, K. 1951. On Stochastc Dfferental Equatons. Memors, Amercan Mathematcal Socety 4:1-51. Kamrad, B., and Rtchken, P. 1991. Multnomal Approxmatng Models for Optons wth k State Varables. Management Scence, 37(12):1640-1653. Kouvels, P. 1999. Global Sourcng Strateges Under Exchange Rate Uncertanty. In Quanttatve Models for Supply Chan Management, ed. Tayur, S., Ganeshan, R., and Magazne, M. 625-667. Boston: Kluwer Academc Publshers. Luenberger, D. G. 1998. Investment Scence. Oxford: Oxford Unversty Press. Nembhard, H. B., Sh, L., and Aktan, M. (2001). A Real Optons Desgn for Qualty Control Charts. Workng paper, n revew wth The Engneerng Economst. Nembhard, H. B., Sh, L., and Park, C. S. (2000). Manufacturng Transtons as Real Optons n the New Economy. The Engneerng Economst, 45, 3, 232-258. AUTHOR BIOGRAPHIES HARRIET BLACK NEMBHARD s an Assstant Professor n Industral Engneerng at the Unversty of Wsconsn-Madson. She receved her Ph.D. from the Unversty of Mchgan. She s a member of INFORMS, IIE, and ASQ. She served as Proceedngs Co-Edtor for 1999 WSC. Her nterests nclude qualty engneerng, manufacturng systems, and dscrete-event smulaton. Her emal and web addresses are <hbnem@engr.wsc.edu> and <http://www.engr.wsc.edu/e/faculty/ nembhard_harret.html>. LEYUAN SHI s an Assocate Professor n Industral Engneerng at the Unversty of Wsconsn-Madson. She receved her Ph.D. from Harvard Unversty. She s a member of INFORMS, IIE, IEEE, and IAFE. Her research nterests nclude modelng and analyss of dscrete dynamc systems, dscrete-event smulaton, and large-scale optmzaton. Her emal and web addresses are <leyuan@engr.wsc.edu> and <http:// www.engr.wsc.edu/e/faculty/sh_leyuan.html>. MEHMET AKTAN s a Ph.D. Student n the Department of Industral Engneerng at the Unversty of Wsconsn- Madson. Hs emal address s <aktan@ cae.wsc.edu>. Nembhard, Sh, and Aktan 552