Real Options and Facilities Access Regulation

Transcription

1 Real Options and Facilities Access Regulation Godon Sick Haskayne School of Business Univesity of Calgay 22 Febuay 2008 Abstact Real options achieve thei value fom flexible management esponse to signals about uncetainty. Any constaint on flexibility will obviously impai the eal option value. Regulation imposes constaints on opeations, but also povides taiffs to the egulated entity. Thus, it is not obvious that the egulated entity necessaily suffes a value loss fom egulation, if the taiffs ae excessively geneous. The question of whethe the taiffs ae excessively geneous is a question of social optimality. We addess this question in the context of facilities access, which is a popula method of deegulating industies that had monopoly powe. The egulato detemines taiffs fo access to the poduction facilities of the oligopolist facility owne so that competitos can use the facilities and offe the consumption good in a competitive maket. This is the basis fo deegulation of powe, gas distibution and telecom industies. It has also been poposed in industies that wee not fomely egulated, such as the ail infastuctue fo integated mining industies. Pindyck has suggested that access taiffs in these cases should include compensation fo the capital costs plus the eal option pemium that was extinguished to establish the facility. But he offes no poof, no any analysis of the magnitude and diection of the distotion if compensation is only offeed fo the capital costs only. Also, he does not conside two-pat taiff stuctue with an up-font access taiff, plus an annual taiff fo capacity. In this pape, we investigate these issues numeically. 1 Intoduction In this pape, we investigate optimal taiffs fo facilities access when the facility Builde has a eal option to choose the optimal timing of constuction. Pindyck (2004, 2005) suggests that the cost base fo a egulated access taiff should include actual out-ofpocket costs of the facilities Builde plus the value of eal options that ae extinguished by committing capital and building. Pindyck suggests that not allowing fo a ecovey 1

2 of the eal option value will distot the ex ante incentives of the facilities Builde and not esult in a fist-best solution. Howeve, he focuses on the analysis of a one-pat taiff that would ecove the total cost (capital plus option oppotunity cost) without analyzing the natue (social cost, change in investment timing) of the distotions fom vaious taiff policies. Moeove, he does not conside the possibility of a two-pat taiff to mitigate those distotions. This pape analyzes those possibilities. We use a methodology fo analyzing enty and exit options fom Dixit et al. (1999); Sødal (2006) to analyze this model. 2 The Model We study a maket in which one o two poduces poduce a poduct that sells in a competitive maket. Demand is popotional to a stochastic vaiable X that follows the pocess dx = µ X dt + σ X dz. (1) We assume this isk is unsystematic, so the tue pobability pocess and the isk-neutal pobability pocess ae the same. The Builde has annual sales of q B = α B X, and the Seeke has sales of q S = α S X, whee 0 < α S α B ae constants.. Thus, the two face the same global shocks, but the Builde is a lage entity. A facilities povide o Builde, B, has eal options to build and to abandon a facility of capacity Q units of annual poduction fo a capital cost of K. Capacity is lumpy, and the Builde cannot build anything othe than exactly Q units of capacity. We suppose thee is facilities access legislation unde which anothe entity, S can seek access to a faction f (0, 1) of this poduction capacity to make its own poduct. The access Seeke must pay the Builde a two-pat taiff consisting of a fixed enty fee of k S f Q, whee k S is the taiff ate, and an annual take-o-pay capacity chage of T f Q, whee T is the annual toll ate. 1 This toll ate is in addition to any vaiable opeating costs fo the facility, which we take as an offset to selling pice. That is, we can embed any vaiable costs in the net selling pice P B, P S is net of these vaiable costs. The Builde and the access Seeke face the same poduct selling pice, and they ae pice takes, but the Seeke is assumed to have opeating costs that ae at least as lage as the Builde, so P S P B. To summaize the access policy, the access Seeke must nominate o eseve capacity f (and it must be excess to the then-cuent needs of the Builde) fo a fixed fee that is paid at the time it claims the capacity. Theeafte, the Seeke must pay an annual toll fee popotional to the capacity used, until it abandons the use of capacity. When 1 This includes the common single-pat taiff stuctue, when k S = 0. 2

3 it abandons the poject, it does not eceive a efund fom the Builde (thee is no way to equie the Builde to maintain liquidity as a efund to the Seeke). But, it does gain elief fom having to pay the annual capacity chage. If the Builde is using q B [0, Q] units of capacity at the time the Seeke nominates the capacity, then, since the Seeke can only use excess capacity, we must have f (0, 1 q B /Q). In paticula, if the Seeke waits until the Builde is using all capacity, it is not allowed to ente. 2 We fist analyze the access Seeke s poblem to detemine ational behaviou and optimal value. Given this behaviou, we study the Builde s poblem to detemine optimal behaviou and value. Behavious of the Builde and Seeke ae chaacteized by demand thesholds o tigges that they set optimally. The Builde entes the maket (builds) when the demand shock fist ises above an endogenous tigge theshold X BB. The Seeke entes the maket at a theshold X SE (which must be highe because the Seeke needs the facilities that the Builde constucts). At some highe theshold, X FB = Q(1 f ) α B, the Builde is using full capacity, and does not have stochastic poduction until demand falls below this theshold. At anothe point, which may be highe o lowe, X SF = f Q α S, the Seeke is using its full capacity and does not have stochastic poduction until demand falls below this theshold. On the othe hand, at some theshold X SA < X SE, the access Seeke will abandon the maket, with the only benefit being the elief fom having to pay the annual access taiff. At a lowe theshold S BA < min{x SA, X BB }, the builde also abandons the maket to eceive a salvage value S B [0, K). If S B = 0, the Builde will neve abandon, since it pays no fixed costs. If thee ae fixed costs of poduction, thei pesent value can be included in the capital cost K, salvage value S B and the taiff T. We assume the iskless inteest ate is. 3 The Access Seeke s Poblem The access Seeke has decision vaiables f, X SE, X SA and it has diffeent value functions when it is waiting fo development, W, when X < X SE, with value V S,W (X; f, X SE, X SA ) poducing at the ate α S X when poduction has been stated and X SA < X X SF f Q α S, with value V S,P (X; f, X SE, X SA ) poducing at the capped ate f Q when X X SF, with value V S,SF (X; f, X SE, X SA ) 2 If the Seeke entes at the same time as the Builde entes, it could, in pinciple, fom a joint ventue with the Builde, shaing the capital costs in the popotion f and foegoing the annual opeating taif, so that T = 0. 3

4 abandoned, A, and cannot e-open, with value V S,A (X; f, X SE, X SA ) = 0. The poblem is time invaiant, so the fundamental patial diffeential equation fo valuation is an odinay diffeential equation. Thee is an ODE fo each of these 3 egions, and they ae bound togethe by value-matching conditions at thei end-points, and they ae optimized by the smooth pasting conditions at the boundaies. This is the taditional solution to the poblem, as in Dixit (1989), fo example. Howeve, we can simplify the analysis using the discount facto appoach of Dixit et al. (1999); Sødal (2006). We will discuss both appoaches. 3.1 Access Seeke is Waiting fo Development σ 2 X 2 d 2 V S,W 2 dx 2 + µx dv S,W dx = V S,W (2) The value-matching condition is fom closed to open, whee it must pay the up-font taiff k S f Q: V S,W (X SE ; f, X SE, X SA ) = V S,P (X SE ; f, X SE, X SA ) k S f Q (3) It also must satisfy the feasibility constaint f 1 q Q (4) whee q = α B X SE is the poduction of the Builde at the time it entes. 3.2 Access Seeke is Opeating Below its Nominated Capacity f Q σ 2 X 2 2 d 2 V S,P dx 2 + µx dv S,P dx + α BXP T f Q = V S,P (5) One value-matching condition is fom the tansition of poducing feely to poducing at the nominated capacity: V S,P (X SF ; f, X SF, X SA ) = V S,SF (X SF ; f, X SE, X SA ) (6) The othe value-matching condition is fom the tansition of poducing feely to abandonment: V S,P (X SA ; f, X SF, X SA ) = 0. (7) 3.3 Access Seeke is Constained by its Nominated Capacity f Q σ 2 X 2 d 2 V S,SF 2 dx 2 + µx dv S,SF dx + QP T f Q = V S,SF (8) The value-matching condition is the same as equation (6). 4

5 4 The Discount Facto Solution Dixit et al. (1999); Sødal (2006) develop a nice pocedue fo solving these switching poblems. The pocedue extends the notion that we can deive the pesent value (PV) of an annuity by taking the pesent value of a pepetuity 1/ and subtacting the pesent value of a delayed pepetuity e t / epesenting the pesent value of the cash flows lost when the payments ae stopped at time t. The teminal point of the annuity is a bounday whee the cash flow steam tansitions to a new value. The discount facto appoach involves finding the expected PV facto fo the hitting time o fist tansition time to the bounday. We can take the value of pepetual cash flows, ignoing the bounday, and subtact the expected hitting-time PV of the change in value when the pocess hits the bounday. The change in value may equie knowing the value if the stochastic pocess etuns to the oiginal bounday, but this only esults in a linea system of equations fo the two bounday values. The expected PV facto is a value that is computed fom the fundamental PDE (ODE in ou case) fo valuation, coupled with a teminal value of 1 at the bounday. That is, conside the stochastic pocess (1) fo X. (We have assumed that this is both the tue and the isk-neutal pocess fo X, and what we need is the isk-neutal pocess.) Let D(X 1 ; X 2 ) be the expected PV facto fo the andom tansition time fom X = X 1 to a fixed value X 2. This is the value at a point whee X = X 1 of a secuity D(X; X ) that pays nothing pio to hitting the bounday, and pays 1 when the bounday is fist hit. Thus, B satisfies the fundamental ODE fo valuation: σ 2 D 2 2 d 2 D dd + µd dx2 dx = D (9) with the teminal condition D(X ; X ) = 1 (10) whee The geneal solution to the ODE (9) is D(X) = A 1 X γ 1 + A 2 X γ 2 (11) γ 1 = 1 2 µ σ 2 + (1 2 µ σ 2 ) σ 2 (12) γ 2 = 1 2 µ σ 2 (1 2 µ σ 2 ) σ 2 (13) Following the usual limiting aguments fo X 0 and X, we can sepaate the cases whee X has to incease to its bounday (A 2 = 0) and X has to decease to its 5

6 bunday (A 1 = 0), we can detemine the othe A i by the appopiate value matching condition, and conclude that the expected discount facto to the bounday is ( ) X γ1 D(X; X X ) = if X < X, ( ) X γ2 X if X X. (14) This solution woks diectly when thee is only one tansition bounday out of the egion. When the egion has two boundaies, (Sødal, 2006, Section 4) shows how to compute the conditional expected PV to the two tansition boundaies, and combine them to solve the the value fo the poblem with uppe and lowe boundaies, coesponding to enty and exit. To conside this two-bounday poblem, let D(X; X 1, X 2 ) be the expected discount facto to hitting the bounday X 1 fist befoe hitting X 2. That is, it is the value of a secuity that pays 1 at the fist passage time fo the bounday X 1, povided that the bounday X 2 has not yet been eached. We can have X 1 < X 2 o X 2 < X 1. The ODE fo the value of D is still equation (9), so the geneal solution is still equation (11), but we have new bounday conditions: D(X 1 ; X 1, X 2 ) = 1 D(X 2 ; X 1, X 2 ) = 0. The conditions says that the discount facto to the bounday is 1 if we stat at that bounday and 0 if we stat at the othe bounday. Substituting these into equation (11) gives two equations fo A 1, A 2 : A 1 X γ A 2X γ 2 1 = 1 A 1 X γ A 2X γ 2 2 = 0. Afte some elementay linea algeba, we find: 2 X γ 2 D(X; X 1, X 2 ) = Xγ 1 X (γ 1 γ 2 ) X γ 1 1 X(γ 1 γ 2 ) 2 X γ 2 1. (15) We can also veify that it povides sensible discount factos if the pocess X stats between the two boundaies: 0 D(X; X 1, X 2 ) 1 if and only if min{x 1, X 2 } X max{x 1, X 2 }. Also, when X 2, this appoaches the standad abandonment option with theshold X 1, and when X 2 0, it becomes the standad development option with theshold X 1. We now use these expected discount factos to solve the individual sub-poblems of Section 3. 6

7 4.1 Access Seeke is Waiting fo Development We use the expected discount facto to the development bounday and the change in value at the bounday. That is, pio to development, the Seeke has a pepetual steam of a 0 cash flow. But, at the time of development (X = X SE ), it exchanges this fo a pepetuity in sales net of taiffs, woth α S X SE P S /( µ) T FQ/. The pepetuity is offset by the expected PV of the value changes at the boundaies accessible afte development. One ( is the expected PV of the value lost if X hits the cap X SF, αs X which is D(S SE ; X SF, X SA ) SF P S ( µ) T f Q V S,SF (X SF ; f, X SE, X SA ). Note that we must use the conditional expected PV facto, since this is the case whee X hits the capacity bounday befoe it hits the abandoment bounday. The othe is the loss of pepetual evenue at abandonment, ( ) again with a conditional expected PV facto, which is αs X D(S SE ; X SA, X SF ) SE P S ( µ). Note that this implicitly means that the gain at abandonment is the elief fom having to make futue taiff expenditues. Thus: V S,W (X SE ; f, X SE, X SA ) = α SX SE P S ( µ) T f Q k S f Q ( αs X SF P S D(X SE ; X SF, X SA ) ( µ) T f Q ) V S,SF (X SF ; f, X SE, X SA ) ( ) αs X SE P S D(X SE ; X SA, X SF ) fo X X SE. (16) ( µ) Pio to development (X < X SE ), the value is the expected PV of the value at development: V S,W (X; f, X SE, X SA ) = D(X, X SE )V S,W (X SE ; f, X SE, X SA ). 4.2 Access Seeke is Opeating Below its Nominated Capacity f Q This case is the extension of subsection 4.1 to the situation whee X (X SA, X SF ). V S,W (X; f, X SE, X SA ) = α SXP S ( µ) T f Q ( αs XP S D(X; X SF, X SA ) ( µ) T f Q ) V S,SF (X SF ; f, X SE, X SA ) ( ) αs XP S D(X; X SA, X SF ) fo X (X SA, X SF ). (17) ( µ) 4.3 Access Seeke is Constained by its Nominated Capacity f Q This is a slightly poblematic condition, because the stochastic demand shock could, in theoy, hit this bounday and eflect back at the bounday. This only happens with 7

8 pobability zeo, and the elevant case is that it goes though the bounday befoe eflecting back. Thus, we will conside the case whee X > X SF and then conside the limit as X X SF, since the value function should be continuous at the bounday. When X > X SF, thee ae two possibilities of subsequent events: eithe X neve hits X SF again, o it does hit X SF again, at which time it evets to the unconstained opeation. The oveall value is the PV of pepetual opeation at the bounday minus the expected PV of the value change when it evets fom constained to unconstained opeation. Thus, fo X > X SF V S,SF (X; f, X SF, X SA ) = P Sf Q ( PS f Q D(X; X SF ) ) V S,SF (X SF ; f, X SE, X SA ) (18) **We need a bette chaacteization of the value in the capped egion, since D(X; X SF ) 1 as X 1, so that, in the limit, (18) does not succeed in identifying the value of V S,SF (X SF ; f, X SE, X SA ). The poblem is well-defined, but the solution is a little moe subtle than I have hee. One solution is to teat the egion [0, ) as a pepetual poduction egion whee the seeke has lost a set of upside call options on poduction. We could calculate Black-Scholes call option values fo each futue date and integate ove dates to get the total value of poduction lost to the cap. Then, this value would be adjusted by the pobability of going to the abandonment bounday. The solution would be numeical. 4.4 Access Seeke Optimization The valuations above ae conditional on the decision paametes f, X SE, X SA fo the access Seeke. We can numeically optimize the value pio to development ove these paametes, and we have the policy and value fo the access seeke. 5 The Facilities Builde Poblem The solution of the facilities Builde will poceed in a simila fashion, whee the Builde must detemine the theshold at which to build and to abandon. Given its decisions, the access Seeke will optimize its decisions, so the Builde will ationally anticipate this optimal esponse in detemining its own policy. We can numeically solve fo the Buildes decision policy and value. 8

9 6 Analysis Given the solution to the Seeke and Builde poblem, we can poceed to analyze how the solution vaies with the two-pat taiff policy. We can see how the Builde s decisions will change fo diffeent taiff policies, identifying how the taiff decisions cause it to poduce ealy o late. We can also sum the value fo the Builde and the Seeke to calculate a social value. This allows us to assess the social value losses fom specific taiff policies. Fo example, we can measue the social loss fom a single-pat taiff, and we can also measue the social loss when that taiff eimbuses out-off pocket constuction costs only. Refeences Dixit, A., Pindyck, R. S., and Sødal, S. (1999). A makup intepetation of optimal investment ules. The Economic Jounal, 109(455): Dixit, A. K. (1989). Enty and exit decisions unde uncetainty. Jounal of Political Economy, 97(3): Pindyck, R. S. (2004). Mandatoy unbundling and ievesible investment in telecom netwoks. NBER Woking Pape No. W Pindyck, R. S. (2005). Sunk costs and eal options in antitust. Keynote Addess, 9th Annual RO Confeence, Pais, June 2005, to appea in Issues and Competition Law and Policy, W.D. Collins Ed, ABA Pess. Sødal, S. (2006). Enty and exit decisions based on a discount facto appoach. Jounal of Economic Dynamics & Contol, 30(11):