AN APPROXIMATE SOLUTION FOR PERPETUAL AMERICAN OPTION WITH TIME TO BUILD: THE VALUE OF ENVIRONMENTAL REMEDIATION INVESTMENT PROJECTS

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1 AN APPROXIMATE SOLUTION FOR PERPETUAL AMERICAN OPTION WITH TIME TO BUILD: THE ALUE OF ENIRONMENTAL REMEDIATION INESTMENT PROJECTS Espinoza, R.D. and Luccioni, L.X. ABSTRACT Most investos and eal estate developes conside bownfield edevelopment pojects isky and, to compensate fo the additional envionmental isk, demand highe etuns on the investment needed to cleanup and edevelop a contaminated popety. The peception of isk is aggavated by the fact that, depending upon the emediation technique adopted, emediation of a contaminated site takes time duing which, the eal estate maket conditions may change significantly. To estimate the value of bownfields consideing the envionmental and maket isks associated with popety clean up and edevelopment, both technical and maket isks ae integated. A closed-fom solution deived fo a pepetual Ameican call option was modified to calculate the optimal sequential investment that accounts fo the equied time to undetake the investment i.e., implement the poposed emediation fo a contaminated land. The poposed solution can be used to evaluate the value of a bownfield if the owne/develope has the option to delay emediation indefinitely waiting fo optimal conditions to stat. The esults of the poposed appoximation ae compaed to the esults obtained using numeical techniques to solve patial diffeential equations. INTRODUCTION Quantification of isk and economic cost associated with actual and/o potential envionmental pollution is cental to the success of bownfield edevelopment. The tem bownfield efes to abandoned, idled, o undeutilized envionmentally impaied popeties. A bownfield edevelopment includes cleaning the site up so it can be put back to a poductive use fo esidential, commecial, and/o industial puposes. As it is the case with any constuction poject, emediation of a contaminated site takes time to complete. Diffeent fom typical constuction pojects, eal estate tansactions that includes cleaning of a contaminated land is usually consideed isky by most investos and demand a highe etun on investment in ode to take the poject and the ate of etun demanded fo each poject depends upon each investos isk pefeence. GeoSyntec Consultants, Washington DC Office, , despinoza@geosyntec.com Cheokee Investment Patnes, Raleigh, Noth Caolina, , lluccioni@cheokeefund.com

2 Although emediation of contaminated sites in the United States ae enfoced by local o fedeal agencies, many sites ae cleaned voluntaily without a specific time table. This may be the case fo which the owne/developes, who ae not esponsible fo the site contamination and they have bought the popety with the intent to edevelop it as soon as the eal estate conditions ae appopiate. Fo these situations, voluntay clean up emediation of contaminated popeties can be viewed as a pepetual Ameican call option on the clean popety; that is the owne o develope has the ight, but not the obligation, to pay a total sunk emediation cost in etun fo a eal estate poject clean popety. PRELIMANARY FORMULATION Because emediation of a contaminated site takes time, as it is the case with any investment poject, a staight fowad application of the solution to the pepetual Ameican call option Samuelson, 965 cannot be applied. The solution to this poblem was povided by Majd and Pindyck 987. They analyzed the poblem of investing continuously and optimally until the poject is completed assuming that investment can be stopped and estated at no cost. The value of the cleaned land is assumed to follow a geometic Bownian motion pocess of the fom: d = α dt + σdz whee α is the expected ate of gowth; σ is the uncetainty associated to the value of the clean popety; dz is the diffeential of a standad Wiene pocess with mean 0 and vaiance dt. The execise pice i.e., the total emaining emediation cost, K is spend ove a peiod of time at a maximum ate of k. The total emaining emediation cost K and the ate of expenditue k ae elated as follows: kdt fo * K dk = 0 othewise whee * K epesents the optimal investment value of the cleaned popety as a function of the emaining emediation cost. Equation indicates that investment only takes place if the value of the cleaned popety is geate than the optimal value, * K, which needs to be obtained as pat of the solution. The investment poblem above descibed can be witten in a patial diffeential fom as:

3 3 fo < * K 0 = + C C C δ σ 3a fo * K 0 = + k K C k C C C δ σ 3b whee C is option value of the cleaned popety; is the isk-fee discount ate; δ = µ α µ >α is the oppotunity cost; µ is the maket isk-adjusted expected ate of etun fom owning the popety; K is the total emaining expenditue; k is the ate of expenditue. Equation 3a can be solved analytically wheeas Equation 3b must be solved numeically using finite diffeent techniques e.g., Majd and Pindyck, 987. APPROXIMATE SOLUTION As discussed above, voluntay clean up emediation of contaminated popeties can be viewed as a pepetual Ameican call option on the clean popety; that is the owne o develope has the ight, but not the obligation, to pay a total sunk emediation cost K spent a cetain ate in etun fo a eal estate poject clean popety whose cuent value is is stochastic and changes with time. If the emediation cost is assumed to be spent instantaneously, the closed-fom solution fo the pepetual Ameican call option on a dividend-paying stock deived by Samuelson 965 could be used to estimate the value of the investment: σ σ δ σ δ β + + = 4 K o * = β β 5 β * * K A o = 6 = othewise o fo o - o * β A K K C 7

4 In ode to use the closed fom solution deived fo the Ameican pepetual call option given by Equations 4 to 7, equivalents investment cost and value of the eal estate investment need to be used. The expessions that wee found to povide consistently equivalent esults of the optimal value of the eal estate investment and the option ae: i the pesent value of the total emediation cost; and ii the pesent value of the eal estate investment. The pesent value of the investment cost K o is obtained assuming that investment is made continuously ove the time peiod T=K/k such that: T K k k K t o = ke dt e / = 8 0 Similaly, The pesent value of the eal estate investment o is obtained assuming that investment cost is made continuously ove the time peiod T=K/k until the emediation is completed. The pesent value of the completed poject i.e., cleaned land that is expected to gow at a ate α can be estimated as using the discount ate µ as: α K / k e e µ K / k e δk / k o = = 9 Equations 8 and 9 can be used togethe with Equations 4 though 7 to obtain the value of an investment that takes time to build. ALIDATION Equations 4 though 9 can be used to evaluate the option value of an asset that takes time to build e.g., the emediation of a contaminated site. In this section, the poposed simplified equations ae validated against the esults of the moe igoous numeical solution povided by Majd and Pindyck 987. Table shows the esults of the value of the option to invest C assuming an annual isk-fee ate = %, annual ate of oppotunity cost δ = 6%, and annual standad deviation σ = 0%, a ate of expenditue k = million/yea, fo K =,, 3, 4, 5, and 6 million and selected values of i.e., values ae selected fom Majd and Pindyck, 987. Table also shows the esults fom Majd and Pindyck in italics obtained using numeical finite diffeence techniques. As shown in the table, the esults of the poposed appoximate solution agees well with the numeical esults. The aveage eo between the two sets of esults shown in Table is 0.79 pecent. Accoding to the esults povided by the finite diffeence solution and summaized in Table, fo a total emaining emediation cost of K=$3 million, investment should continue if the value of the eal estate is geate than * = $5. million i.e., the optimal option value C * = $.4 million. A value of C=5., K=3=$.44 million using the appoximate solution is shown in Table i.e., a.5% diffeence with the 4

5 numeical solution. This value, howeve, does not coespond top the optimal call option value as discussed below. The values of the optimal call option value, C* can be calculated using Equations 4 though 9. Fo instance, fo a total emaining emediation cost of K=$3 million, using Equations 5 and 7, the value of the optimal eal estate investment and its coesponding option value ae * = $5.0 million and C* = $.6 million, espectively. Fo this case, the diffeence between the option values calculated using the appoximate solution and the numeical scheme is $0.6 million i.e., that is.7% wheeas the diffeence between the optimal eal estate investment is $0. million i.e., that is 4.8%. A compaison of the optimal call option values C* fo each value of total emaining cost K obtained using the poposed appoximated solution and the esults fom the numeical solution by Majd and Pindyck 987 is pesented in Figue. Table Option Investment alue, C, alue of Completed Total Remaining Investment K, millions Poject, millions alues in italics wee obtained fom Majd and Pindyck 987. alues in bold and italics epesent the optimal investment value, C*, that coespond to each value of K. That is, investment takes place only if is geate than the coesponding optimal value. Although the calculated option value fo each pai,k in Table agees well with the ones calculated using the numeical finite diffeence scheme, as shown in Figue, the optimal option value may not be accuately calculated using the numeical algoithm scheme. 5

6 3.00 Optimal Option alue, C* millions Numeical Solution Appoximate Solution Remaining Total Cost, K millions Figue Compaison of Optimal Option alue C* The main eason fo the discepancy is the fact that the calculated values of the option C,K using the numeical algoithm ae obtained at the discete values of shown in Table may not necessaily include the optimal values *. Each of the values of selected fo the numeical analysis wee obtained as follows Majd and Pindyck, 987: = e 0.5 j 0 fo j = 0,,, 5. If follows fom Equation 0 that as j gows, the incement in gows exponentially. A bette ageement would likely be obtained if a fine mesh is used to fo the finite diffeent scheme. On the othe hand, as shown in Figue, the ageement between the calculated optimal value of completed eal estate poject using the appoximate solution and the numeical finite diffeent scheme is vey good. Theefoe, the optimal value fo stating the emediation poject can be easonable estimated using the appoximate solution pesented in this pape. 6

7 4 Optimal alue alue, * millions Numeical Solution Appoximate Solution Remaining Total Cost, K millions Figue Compaison of Optimal alue of Completed Poject * APPLICATION Fo the envionmental application discussed heein, the poposed modified Ameican pepetual option pesented in this pape can be used to answe the following questions:. What is the value of a eal estate investment call option value, C* if the land is contaminated and pospective selle o buye can emediate it at any time?. What is the optimal value of the poject * at which a develope should stat emediation of the site and pay the emediation cost K ove a peiod of time? 3. How these values ae affected if the emediation cost is uncetain? Assuming that the aveage cost to emediate the contaminated land is $5 million, the cuent value of the cleaned land is 6 million, the maket volatility of the eal estate poject is 0 pecent, the inteest ate is two pecent, the aveage time to edevelop is thee yeas, and the poject evenue steam, once it is completed, is six pecent of the cuent value, using Equations 4 to 9, the estimated value of the optimal investment * and the option to wait ae $6.96 and $0.7, espectively. Because the cuent value of the land is 6 million, it is optimal to wait. 7

8 Remediation poject ae uncetain egading cost and time to implement the emediation. In geneal, a pobability distibution can be estimated based upon technical infomation of the site and the emediation technique. These uncetainties can be integated to the uncetainty of the maket value of the cleaned popety using the model pesented in this pape along with Monte Calo simulations. To this end, the commecial softwae Cystal Ball TM is used to conduct the simulations. Fo this case, the emediation cost and the time to complete the emediation poject ae both consideed andom coelated vaiables log nomally distibuted with a mean and standad deviation values of $5 $.35 million and 3. yeas, espectively, and a coefficient of coelation of 0.8 Figues 3 and 4. The uncetainty associated with the eal estate maket is aleady accounted fo in Equations 4 though 9. Remediation Cost $ Millions Figue 3 Pobability Distibution fo Remediation Cost T Figue 4 Pobability Distibution fo Time to Completion The calculated optimal value to stat the emediation poject has the distibution shown in Figue 5 having a mean of $8.6 million and a standad deviation of $4.4 million. The optimal net pesent value of the contaminated popety i.e., the pepetual Ameican call option is $3. million Figue 6 having a standad deviation of $.7 million assuming that the develope can wait fo this optimal value. Fo this case, 8

9 because the uncetainty in the emediation cost and the time to complete is high, the optimal value to stat emediating the site is about 3 pecent highe than the case without uncetainty of these paametes. Thus, as expected, technical uncetainty favos delaying the poject. Foecast: Optimum Stat Poject alue 5,000 Tials Fequency Chat 9 Outlies $ Millions Figue 5 Pobability Distibution fo Optimum Poject alue * Foecast: Land alue 5,000 Tials Fequency Chat 3 Outlies $ m illions Figue 4 Pobability Distibution fo Option alue 9

10 CONCLUSIONS A simplified model to calculate the optimal option value fo pojects that equie time to implement has been pesented in this pape. The esults of the simplified model has been successfully compaed to the esults obtained using numeical techniques. Fo the envionmental application discussed heein, the poposed modified Ameican pepetual option pesented in this pape can be used to answe questions egading the value of bownfield pojects taking into consideation technical and maket uncetainty. ACKNOWLEDGEMENTS This wok was suppoted by the National Science Foundation NSF unde the Small Business Innovative Reseach SBIR Gant No , Phase I. REFERENCES Majd, S. and R. Pindyck [987]. "Time to build, option value, and investment decisions", Jounal of Financial Economics. 8. Samuelson, P.A. [965]. "Poof that popely anticipated pices fluctuate andomly", Ind. Management Review, 6[],