Using discounted flexibility values to solve for the decision costs of sequential investment policies.

Transcription

1 Using discounted flexibility values to solve for the decision costs of sequential investment policies. Steinar Ekern NHH - Norwegian School of Economics, Helleveien 30, 5045 Bergen, Norway Mark B. Shackleton (corresponding author, m.shackleton@lancaster.ac.uk) Lancaster University Management School, Bailrigg, Lancaster, LA1 4YX, UK Sigbjørn Sødal (sigbjorn.sodal@uia.no) University of Agder, Kristiansand, Postboks 422, 4604 Kristiansand, Norway May 2016

2 Abstract Optimal stopping investment problems typically locate the policy triggers that correspond to associated decision costs numerically. Whilst these search procedures can in principle tackle problems of any scale, in practice investment systems have studied few decision points because no structure for option variables and their conditions has been available. By assuming policy trigger values, we present a framework that uses discounts to solve for option values and decision costs explicitly. This is possible for investment networks of increasing scale and complexity and affords greater intuition than numerical work alone. We extend classical hysteresis including a third interim mode; firstly with three thresholds we assume this mode is negotiable only one way, secondly with four thresholds we assume it can be traversed up or down using two way discounting. The framework presented supports systems of even greater complexity and is suitable for extending numerical work to greater problems. Keywords: (S) Decision support systems; optimal stopping, smooth pasting, discount functions and investment graphs. C61, G31. Highlights Provides a framework for multi level, multi stage decision making under uncertainty Places discounts from a decision network into a valuation matrix Given policy thresholds, provides explicit solutions for option values anddecisioncosts Collates variables in suitable form to search numerically for policy thresholds that match specified costs. 0

3 1 Introduction This article follows the literature on managing real assets with flexibility to switch cashflow modes. Many articles have described operational flexibility using options, where the present values of revenues and running costs (plus capital investment) are viewed as the underlying and strike of a call or put option. 1 These techniques have been employed to evaluate capacity, investment and other financial decisions. 2 Although less has been written on the creation and transformation of options, sequential and modular values with repeated changes have been treated. 3 Our procedure tracks sequential flexibility, potentially indefinite, through investment networks by accounting for the creation and use of options whose value depends on cashflow changes. By proposing a policy framework of points where options are used or created, we collate values at investment or divestment times, determining their value via expectations with respect of astatevariable. For an option s value to be maximised by choice of its decision time, so called smooth pasting conditions must hold where two value functions join with equal first derivatives (with respect to this state variable). We extend the intuition embedded in this first order condition as a rate of return. Options have a beta matching their relative growth to the underlying investment and this can be used as a metric in optimal exercise timing. In addition to equations that track the value before and after transitions, we represent smooth pasting by including the beta of options weighted by their value. A discount factor 4 representation of options provides both the means to determine betas and to link option values that are separated by threshold and time. 1 We follow Black and Scholes [5], Merton [19] and Cox, Ross and Rubinstein [8] in pursuing risk neutral valuation of claims including options. Myers [20], Brennan and Schwartz [6], McDonald and Siegel [18], Alvarez [4] and the texts of Dixit and Pindyck [12], Trigeorgis [27] all detail real options. 2 These include capacity (Pindyck [21] and Dangl [9]), land (Capozza and Li [7]), costly reversibility (Abel and Eberly [2] and Eberly and van Mieghem [14]), marginal cost of capital (Abel, Dixit, Eberly and Pindyck [1]), information (Lambrecht and Perraudin [17]), capital structure, debt valuation (Sarkar and Zapatero [22]), competition (Siddiqui and Takashima [24]) and leasing (Trigeorgis [26]). 3 Gamba and Fusari [16] motivate and value project design using six modularity principles and Ekern [15] modelled hysteresis with repeated, but limited, switches. 4 Dixit, Pindyck and Sødal [13] and Sødal [25] developed the discount factor approach. 1

4 Whilst small systems have been solved in the literature numerically, this typically occurs with less intuition; our framework s contribution is to make a solution clear and visible. This allows examination of systems with more modes of flexibility (in terms of thresholds and cashflows). In principle this has been feasible to date but in practice in the literature it has not been demonstrated due to the limited ability to track increasing numbers of variables. Section two applies this method to canonical hysteresis and under illustrative geometric Brownian motion extends this to incorporate a third threshold and cashflow mode. Section three introduces a fourth threshold and solves a more advanced problem with two way options. Section four concludes. 2 Investment and divestment sequences 2.1 Canonical hysteresis Economic hysteresis (Dixit [11]) occurs when a business entity can switch between two operational modes; e.g. a firm can either be idle or operating. In this example, when fully on, the firm earns a continuous stochastic revenue e viewed as a constant dividend yield ( ) on a present value e i.e. e = e We label the firm s constant operating cost rate ; with interest rate this is equivalent to a perpetual present value =. Without switching costs, the stochastic profit ratee or naive net present value e might drive decisions but potential irreversibility must be taken into account. In addition to the present value cost, we include an extra, non recoverable, switching friction 0; in value terms switching from idle to full incurs + and switching off to idle spares (the roundtrip frictions are 2 ). Labelling idle to full transitions and full to idle a hysteresis switching policy consists of two non-stochastic thresholds in the present value revenue 5 or revenue. Total cost changes are given by = ± where so that. 5 The stochastic notation for revenues and flows is suppressed to but the subscripted times distinguish stochastic or dynamic variables from non stochastic thresholds such as We also use a sequence of stopping times = 1 2 at which reaches thresholds such as ; is the next to be encountered. 2

5 In the region the operating status of the firm is determined by the last policy threshold encountered by ;ifthiswas the firm is full on but if was last, the firm is idle. That is to say, whilst the stochastic present value exceeds the full to idle policy point i.e. the firm may be operating but it will switch to idle the instant the present value revenue falls to =. Whilst the firm may not be operating but it will switch from idle to full immediate a rise to = occurs. Before a switch is made, the firm possess one of two timing options; a call or a put on exercised at different times These options are potentially long lived and do not depend directly on time (other than through )so we denote their dynamic values ( ). Whilst idle, the value of the option to call (i.e. switch to fully capture when it reaches )islabelled with subscript ( ); this option has a strike price of Whilst fully on, the value of the option to put (i.e. switch to idle giving up when it reaches ) is labelled with, ( ) and this option has strike The call ( ) is created when = (firm goes idle) and is used to switch to full later at = At creation the call has low value and at its use it has maximum value. The put ( ) is created when the call is used at = and the put is used later when =. The put s value at creation is also less than at its time of exercise or use. 2.2 Matching value at times of optimal stopping We track these complementary options ( ) ( ) over time and in particular their value when they convert. Equation (1) compares the dynamic conversion of the call value (on the left) to the put and operational proceeds (on the right) and equates them at two different points of (and times ). When = and going from idle to full with + (in the forward direction) and when going full to idle at = with (in the reverse direction). In Equation (1), transition and equality at a threshold is achieved with or signs; where these separate two states, they can be combined denoting equality but under different frictional sign ±. ( ) ( )+ ( ± ) at = = (1) The call function ( ) depends on the choice of process for but its values ( ) ( ) where it is created and used are non-stochastic. 3

6 The put is created when = andutilisedwhen = and its nonstochastic values are ( ) and ( ) at these two policy points. 2.3 Scaled smooth pasting at stopping times Optimal decision making at both thresholds requires a first order or smooth pasting condition which can be framed as rate of return equivalence (see Shackleton and Sødal [23]). The value equations in (1) are dynamic in until the policy points are reached, smooth pasting requires that the first derivative of each with respect to matches across the transition. Rather than just differentiate by we also rescale by before evaluating the result at each threshold = The first line of Equation (2) shows this applied to Equation (1), for its dynamic quantities; the total rate of capital return (or decline) 6 with respect to must match at the two policy points. Due to the beta definition ( ) = ( ) ( ) the second line of Equation (2) is useful for the options and cashflows because it identifies the value weighted beta of components (note that fixed costs ± have no sensitivity to and zero beta). Equation (2) must be evaluated at times where = and =. ( ) ( ) + (2) ( ) ( )+1 when = = at = Geometric Brownian motion example With stochastic flow and value following a Brownian motion geometrically, we now tackle the call and put functions. Under risk neutral diffusion (Equation3,withriskfree, volatility and yield ) revenue flow and value = followthesamediffusion (any power of the cashflow ( ) 6 Applying rates of return to the timing values ( ) ( ) and flows is equivalent to examining their value weighted betas. Using a risk return relationship = + (where is a risk premium), a total rate of return equation can be simplified to Equation (2) by subtracting Equation (1), then by dividing by ( + ) ( ) + ( )+( + 1 ) +( + 0 )( ± ) 4

7 also follows a GBM but with yield b = ( ) ( 1)). 7 =( ) + = (3) Option claims ( ) that are contingent on reaching a certain condition satisfy a Bellman equation. The firstlineofequation(4)isarestrictionderived from the risk neutral expectation of a local change giving a risk free return (expectations of an Ito expansion generate [ ( )] = ( ) ). The pricing equation in the first line of Equation (4) generates option solutions ( ) and ( ) For time homogeneous problems, beta constants (for the call) or (put) are given by a quadratic in the third line of Equation (4). 0 = ( ) 2 = ± ( ) +( ) ( ) (4) s µ The elasticities or betas of ( ) ( ) are constant at 1 and 0 Restricting = = 2 (one degree of freedom) gives =2 1 which are used for the results in Appendix A, B and C. 2.5 Using discounts for option values Given that we introduced the value of a new option as a fraction of its value when used, it is natural to represent the call and put option values as discounted fractions of their final values ( ) ( ) Equation (5) shows each option as a dynamic fraction of these final values, but where the dynamic variable is carried in discount functions ( ) ( ) 7 The value of a perpetuity of the power of the cashflow arises from a constant yield ( implies 0).Thepresentvalueofapowerflow reverts to for =1or to a constant 1 for =0(note that the beta of [ ] ( ) = ( ) = is = ). 5

8 for the call or put. ( ) = ( ) ( ) (5) ( ) = ( ) ( ) Presenting the GBM solutions ( ) in the form of Equation (5) gives discount functions unit value at the random stopping times that determine their maturity ( is the next time at which reaches its goal) in Equation (6). 8 With the final option values ( ) ( ) acting as scaling constants, the discount factor carries the beta of each option (from Equation (5), ( ( )) ( ) = where the omitted boundary depends on the type ). ( ) = ( ) = i h ( ) = = h ( ) i = = µ µ (6) The general optimality conditions (Equations 1, 2 and 5) for the simple hysteresis case are collected within the left panels of Table 1 in Appendix A. Substituting the discount factors from Equation (6), the right panels show the solutions, with assumed optimal thresholds =1 4. For a small number of options, ad hoc methods can be used to find these but as their number grows a framework for solving larger systems is necessary. To illustrate this, we next add a third cashflow mode with its own cost rate and option involving two additional critical values. 2.6 A third mode with different power Before the full state with cashflow,value is launched, suppose that there is operational flexibility to engage with ( ) i.e. the power of underlying revenue (labelled and with 0 1). This might arise when a test market with scale economies different to those with full cashflow is possible. We leave the present value of operations in this mode open (and 8 Mean reverting and arithmetic Brownian flows (e.g. used in Sarkar and Zapatero [22] and Alexander, Mo and Stent [3]) also have discount solutions; these are documented in Dixit, Pindyck and Sødal [13]. Whilst these processes have constants solving a similar fundamental quadratic, unlike GBM their betas are not iso elastic. 6

9 not equal to ). For GBM, footnote 7 shows that the expectation of ( ) is proportional to the power of the current flow and the perpetuity value has fixed yield, b therefore we set its value to Before launching,we consider calling a preliminary cashflow we label this state gamma (subscript ) with a call option ( ) The beta of the present value is but the beta of the call (or put) option within this mode ( ) is not ; under GBM it remains the same (or ). 9 We amend the simple hysteresis transition from idle to full (in Section 2.2) into two calls; one from idle to gamma and one from gamma to full. Threshold is replaced by two; when switching from idle to gamma and from gamma to full. In the idle region the call is labelled ( ) (as before but with threshold now ) but the second call which is used in the gamma mode at is labelled ( ). Once in full mode, we assume that the option remains as in Section 2.2; a put ( ) that is created when = andisusedwhen = (note the assumed policy inputs follow ). Since we are solving for costs combined with frictions, we no longer track separately but the total present value cost difference at each threshold. At when the put is used to relinquish we model a present value of cost savings (including frictions) of At when the idle state ends and the gamma state starts, a present value cost of is incurred but is gained. Finally when the gamma state ends and the full state commences an incremental is incurred to relinquish and gain. 2.7 Vector and matrix representation Now we put all option values at transition thresholds along with the associated cashflows into vectors U W for the option values at their creation and use respectively. Vectors Y Z carry the cashflows at option creation, use while X carries the present value cost changes (including switching frictions). The options are dynamic in between thresholds but Equation (7) 9 Claims ( ) can be functions of or = Under GBM, ( ) and ( ( )) generate asset pricing equations with betas that satisfy =. Solutions ( ) or are equivalent because = = (for calls or puts ). Similarly, exchange options between and (elasticity ) also have the same power. 7

10 shows them evaluated at the three thresholds; in rows one. W + Z = U + Y X ( ) ( ) ( ) + 0 = ( ) + ( ) ( ) 0 (7) The last row of Equation (7) shows the put ( ) along with the present value of full operations being used to convert into zero revenue (idle) plusthepresentvalueofcostsavings plus the call ( ). The second row of Equation (7) shows the call ( ) at in the idle state being converted into a gamma operational flow less costs plus the call in the gamma state ( ) The top row (top threshold) of Equation (7) shows the call ( ) in the gamma state being used along with the power value to convert into a full flow less costs, plus the put ( ) Appendix B shows the network of transitions. The three options come in a strict sequence, ( )+ ( ) ( )+ ( )+ each conversion occurring at a different (stopping) time = 1 2 etc. associated with arrival of at thresholds The expectation of the discounted stopping times that separate these events is captured by matrix Equation (8). U = D W ( ) 0 0 ( ) ( ) ( ) = ( ) 0 0 ( ) ( ) 0 ( ) 0 ( ) (8) The top line of Equation (8) says that the value of the put when it is created at the top threshold ( ) is ( ) times its value at the bottom thresholdwhenitisused ( ). The second and third lines say that upon creation each call ( ) ( ) is worth a fraction of its value at use ( ) ( ) The discounts ( ) ( ) ( ) are given by Equation (6) but with arguments reflecting new boundaries. The third set of Equations (9) ensures that each of the value matching Equations (7) smooth pastes across the transition. Taking cash flow proceeds into account, we require (scaled) differentiation in to give the same growth rate of total value with respect to. Equation (9) implements this by 8

11 multiplying each element of (7) by its beta ( for both calls, for the put and 1 for with zero beta, all fixed costs (and frictions) disappear). Although only the vector β Y β Z is required, diagonal matrices for Y Z are given by β β = {1 1} { 1 1}. β W = β U + β Y β Z 0 0 ( ) 0 0 ( ) 0 0 ( ) = 0 0 ( ) ( ) 0 0 ( ) (9) 2.8 Matrix solution In the top two lines of Equation (10), Equations (9 and 8) solve for U W then in the last line Equation (7) solves for X the vector of costs. The elements within D Y Z that determine the flexibility structure depend explicitly on the policy points Appendix B details the solution for this extended system with =1 2 4 (solution (10) was also used for simple hysteresis results in Appendix A, but using simpler two row or column vectors and matrices that are not detailed). W = [β W β U D] 1 (β Y β Z) (10) U = D [β W β U D] 1 (β Y β Z) X = U W + Y Z In the next Section the system is adapted to incorporate more advanced switching features. Due to the extension of the discount definition in the next Section, the β β matrices are not obtained as directly as Section 2 (Appendix B presents Equation (19), a dynamic version of Equation (7), to show why beta matrices β β are easy for one way discount factors). Although the contents of D Y Z and method for β change, Equation (10) continues to hold. 3 Two way discounting As well as progressing from the gamma mode to the full, in this Section we accommodate the possibility that progression could occur downward to idle 9

12 from gamma, i.e. not only upward. We do this by splitting the put from full to idle into two. The first put is assumed to return the firm to the gamma mode, the second is included in the gamma mode which now holds a call on full (as described in Section 2) and also a put to idle simultaneously. Thus we have four policy trigger points and value matching equations summarised in Equation (11). Prior threshold (and its transition) is replaced by two where the firm goes full to gamma and gamma to idle (resp.) and four policy thresholds are used. With extra flexibility, vectors U W accommodate two more option points (a total of eight) in the four rows that correspond to evaluation of dynamic quantities (in )at and (summarised in Appendix C and Equation 21). Again vectors Z Y contain the present values of cashflows lost or gained (left or right) and vector X as a present value of incremental costs (on the right) ensures that values matching at the four policy points. W + Z = U + Y X ( ) ( ) ( ) ( ) + 0 = ( ) ( ) + ( ) ( ) 0 (11) The other vector or matrix equations (that do not contain X) are required to solve for the new contents of U W These must (scale) smooth paste Equation (11) and capture the extended discount relationships between the elements of U W if the gamma option has two possibilities. 3.1 Two options in the gamma mode When is between and and operations are in the gamma mode (with cashflow present value ), one of two mutually exclusive outcomes is possible. If over time diffuses to the level then the value of the dynamic option ( ) will reach ( ) (not depending on ( )) but if diffuses to then ( ) attains ( ) (not depending on ( )). Since the firstisanupmovementandthesecondadown, ( ) is a linear combination of a call on ( ) and a put on ( ) This can be seen in Equation (12) which shows ( ) as a discounted combination of ( ) and ( ) However unlike Equations (5 and 6), in Equation (12) 10

13 discount factors have three arguments; the first is the current level (dynamic, if labelled ), the second is the level at which the discount factor achieves unity and the third the level at which attains a zero value. This condition of achieving zero worth ensures that at the end of its life, ( ) depends fully on one value and not on the other at all. ( )= ( ) ( )+ ( ) ( ) (12) These amended calls and puts are given by stopping time expectations conditioned on reaching the target threshold whilst not reaching the complementary threshold. 10 They can be simplified in Equation (13) to linear combinations of the simple call put discounts (from Equation (6) with two arguments that cannot achieve a zero value). ( ) = ( ) ( ) (13) ( ) = ( ) ( ) Equation (13) includes normalisation constants that do not depend on the initial state ; = ( ) and = ( ) where =(1 ( ) ( )) 1. These ensure the complementary boundary conditions at both thresholds are met. 3.2 Dynamic discount and growth matrices The two way discount factors in Equation (13) that depend on the current state can be put into discount matrix D ( ) for use in the dynamic relationship U ( )=D(P t ) W in Equation (14). Compared to (8), Equation 10 See Darling and Siegert [10] for the treatment of the conditional stopping time. ( ) = ) = min ( ) ( ) = = max ( ) They encompass one way factors as a special case (left panel with boundaries at 0) and two way bounds (right panels). ( 0) = ( ) ( ) = ( ) ( )=0 ( )=1 ( )=1 ( )=0 11

14 (14) has contents based on a dynamic set of time option values e.g. ( ) U( ) = D( ) W ( ) 0 ( )0 0 ( ) ( ) ( ) = ( ) 0 0 ( ) ( ) ( ) 0 0 ( ) ( ) ( ) 0 0 ( ) 0 ( ) (14) To separate dependency on the initial from the final states further, D ( ) can be represented as a product of matrices with one way factors only in the first (dynamic) matrix and (static, between outer gamma thresholds ) normalisation factors in the second matrix. D ( )= 0 ( ) 0 0 ( ) 0 0 ( ) ( ) 0 0 ( ) 0 0 ( ) An inverse or growth mapping of dynamic W( ) from static U via W( )= G ( ) U is also possible. Equation (15) shows G ( ) and its decomposition into dynamic and static matrices (the latter depends on normalisation factors on the inner separation; b = b ( ) and b = b ( ) where b =(1 ( ) ( )) 1 ). This tracks the dynamic growth of option values from past static values in U at fixed thresholds (i.e. prior stopping time 1 ). 0 ( ) ( ) 0 G ( ) = ( ) ( ) (15) 0 ( ) ( ) 0 0 ( ) ( ) = ( ) b b ( ) 0 b b 0 0 ( ) ( )

15 When matrices D ( ) G ( ) are restricted line by line to stopping times by setting = they are annotated D G (as before). 11 The inverse discount or growth matrix G contains factors greater than one such ³ as ( )= 1 (because ). The three argument, two direction growth factors also have magnitude greater than unity. 3.3 Scaled differentiated for beta matrices These specifications and separations are useful for determining the betas of the option values. Equations (14) and (15) for D ( ) and G ( ) can be differentiated and scaled with respect to (Appendix B also completes this task for the simpler three by three D in Section 2.6). In Equation (16), the result of this beta operation labelled by ( ) 0 is to premultiply the one way discount factors by their betas, leaving normalisation weights unchanged i.e. D 0 ( )= 0 ( ) ( ) 0 0 ( ) ( ) 0 0 ( ) ( ) (16) and G 0 ( )= 0 ( ) ( ) ( ) b b ( ) 0 b b 0 0 ( ) ( ) (17) Applying this operation and Equations (16 and 17) to U( )=D( ) W retrieves β ( ) U the weighted beta of U in Equation (18). Similarly expressing W( ) from static U via W( )=G( ) U recovers β ( ) W (U ( )) 0 = D ( ) 0 W = D ( ) 0 GU = β ( ) U (18) (W( )) 0 = G ( ) 0 U = G ( ) 0 DW = β ( ) W 11 Multiplying GD gives I, some elements give unity straight away (one way options e.g. the put growth times the put discount ( ) ( )=1)butthetwoway options take more algebraic expansion beforereducingto1orcancellingto0. 13

16 When Equation (18) is restricted to the vector of starting thresholds and times (i.e. by setting U ( )=U or W( )=W) it gives expressions for the beta matrices β = D 0 G and β = G 0 D required in solution (10). With = Appendix C shows the results of two way discounting in the gamma mode including a figure plotting dynamic values ( ) etc. Other more complex and nested structures are possible by including additional gamma modes e.g. b. 4 Conclusion The option value to time the launch flexibility or suspension of different cashflow modes can be represented using discounts. The discount function captures the dynamics of a stochastic process and performs two key functions. Firstly it quantifies an expectation of time and value separation between option values at the (static) policy thresholds when created and used. Secondly it quantifies the (dynamic) elasticity or option beta; this is key to the smooth pasting first order conditions for threshold optimality. Discount factors with functional forms and betas dependent on assumed diffusion dynamics have been used before but with limited interaction. We extend their use so that more options can interact within an investment network. This is done by separating the beginning and end of life network value of each option in two vectors and solving with discount matrices whose size and composition reflect the scale and form of flexibility present. The discount formulation is quite general, it holds for many processes and a network of policy transition thresholds can be formulated before a diffusion assumption is made. The advantage of choosing the policy points first is that explicit solutions for option values can be derived from discounting and smooth pasting conditions alone. Since the fixed operational costs only appear in one set of conditions (value matching) it is necessary to solve for them last. It is unlikely that the initially chosen set of policy points will generate fixed costs and frictions that conform to the actual cost structure of the firm, so a numerical search for the roots of the system is then necessary. The method presented here facilitates this search by providing a framework to structure and solve complex problems on trial policy points first which allows results to be interpreted. 14

17 VM DI SP Conditions ( )= + ( ) ( )= + ( ) ( ) ( ) ( ) q q ( ) ( ) ( ) ( )=1 + ( ) ( )=1 + ( ) Solutions (bold), inputs = = q q = = Table 1: Given = , these six equations are used to solve for ( ) ( ) ( ) ( ). VM stands for value matching, DI for discounting and SP for smooth pasting. The first and last apply at the two thresholds and but the middle panel operates between the two policy thresholds. 5 Appendix A: Simple hysteresis results Normally the operating characteristics (i.e. )ofthefirm are knownandthepolicypoints mustbedetermined. Wehaveshown that it is possible to solve for given explicitly. Although costs and values X U W can be inferred from discounts, betas and thresholds D β etc., because these items all depend on the policy points, Equation (10) cannot isolate thresholds from inputted costs; this remains a root finding problem. However the approach we have taken shows using linear algebra the system of policy points whose root must be found. If Section 2 concluded with classical hysteresis, i.e. two thresholds and modes, then this set of option values and costs solves for thresholds =1 4. Once =1 643 =2 286 are calculated, values of =1 965 and =0 322 can be inferred. If policy points consistent with say values of =2 0 and =0 3 are required (instead of =1 965 and =0 322 derived from =1 4), this method facilitates a numerical search for the root finding of ( ) that generates ( ).Aswellasthefirst order (smooth pasting) conditions second order conditions for maximal values in U W need checking. The examples in Appendices A, B and C were checked numerically for maxima. 15

18 full flow ( )+ ( )+ power flow ( )+ ( )+ + idle no flow ( ) ( ) stopping time policy threshold =1 =2 =4 option used ( ) ( ) ( ) option gained ( ) ( ) ( ) Table 2: Investment network graph at three thresholds (horizontal; put, call, call) for three state system (vertical; idle, power, full). Value matching at investment occurs vertically, diffusion and discounting horizontally. Type Condition Solutions (bold) inputs ( )= ( ) = VM ( )= ( ) = ( )= ( ) = DI ( )= ( ) ( ) ( )= ( ) ( ) ( )= ( ) ( ) = = = SP ( )= ( )+ ( )= ( )+ ( )= ( ) = = = Table 3: ( ) ( ) ( ) ( ) ( ) ( ) and 9 equations 6 Appendix B: Three threshold results With increasing along the horizontal axis and value increasing vertically, Table 2 shows the network of transitions in Section 2.6; each vertical transition corresponds to a line of Equation (7). Table 3 shows the solutions for = Note that the total roundtrip costs + = are negative Although the betas were straightforward to put into β β it can also be done via extending Equation (1) to Equation (19). ( ) ( )+ at 2 = ( ) ( )+ at 1 = (19) ( ) ( ) + at 3 = 16

19 The elements on the left of Equation (19) can be depicted as a dynamic U( )=D( )W in the first line of Equation (20). U( ) = D( ) W ( ) 0 0 ( ) ( ) ( ) = ( ) 0 0 ( ) ( ) 0 ( ) 0 ( ) (U( )) 0 = (D( )) 0 W ( ) 0 0 ( ) ( ) ( ) = ( ) 0 0 ( ) ( ) 0 ( ) 0 ( ) Scaled differentiation ( ) 0 in the last line of Equation (20) allows repetition of Equation (18) but for the 3 3 case. Using W = GU leads to recovery of β = D 0 G (the other 3 3 diagonal beta matrix from Equation (9), β = G 0 D runs similarly). (U( )) 0 = (D( )) 0 W =(D( )) 0 GU = β U where 0 ( ) 0 G = 0 0 ( ) ( ) Appendix C: Four threshold results ( ) ( )+ at 2 = ( ) ( ) + + at 3 = ( ) ( )+ at 1 = ( ) ( ) + at 4 = (21) Equation (21) captures the dynamic representation and Table 4 the investment network (a bipartite and directed graph 12 ) for Section 3. Working down (21) and from the top threshold (on the right of Table 4) is where the two way gamma option is used (as a call) to go from gamma flow to full at 2 Policy point (next down or left) is where the put from full is used to return to the gamma state at 3 Point is where the idle call is used to move to the gamma state at 1 and (bottom and far left) is where the 12 Bipartite nodes are either beginning (U) orend(w), directed means moving from one to the other type without immediate reversal (see Wilson [28]). (20) 17

20 ( )+ ( )+ + ( )+ ( )+ ( )+ ( )+ + ( ) ( ) time 4 (= 70) 1 (= 10) 3 (= 50) 2 (= 30) policy =1 =2 =3 =4 used ( ) ( ) ( ) ( ) gained ( ) ( ) ( ) ( ) Table 4: Investment graph with four thresholds (horizontal) and three states (vertical); switching times of 10,30,50,70 match those shown in the time series plot of Figure 1. Value matching at investment occurs vertically, diffusion and discounting occurs horizontally. Note two way discounting in the gamma state. adapted gamma option is used (as a put) to go to idle at 4. Matrices were used to solve for the extended hysteresis including an interim gamma state with thresholds = The results along with a summary of the conditions are shown in Table 5. These provided the basis for Figure 1, which shows plots in two panels either side of the conditions that link them. The first panel of Figure 1 shows a stylized sample path for against time; a (predictable) saw tooth that traverses the four thresholds in turn at times = (down the long edge of the figure). Holding the page in portrait mode, the horizontal axis in the top panel represents and the vertical (following the longest edge down) ; however turning the page counter clockwise ninety degrees (to landscape) allows the time series tobeshowninthispanelregularlywiththe dimension (up) and the time dimension across (to the right) as usual. The cashflow modes of the firm are captured by being shown in blue for idle, purple for gamma and red for full (see below). The second panel shows the cashlow and option values against that result from the sample path in the first panel. Here (in portrait mode only), the option values ( ) and + ( ) are plotted on the vertical against on the horizontal. The advantage of this portrayal is that the time series values of in the top panel are aligned down the page to their corresponding option values in the bottom. The fixed transition points that 18

21 correspond to four smooth pasted policy transition points and (set to 1,2,3,4) run down the length of the page and are common across both panels. In between the policy points, the values for options and the firm are dynamic. With the call function ( ) represented as a discount of its final payoff e.g. ( )= ( ) ( ) etc. interim values for ( ) are plotted in the second panel for the sample path of in the first. The values of the firm including operational cashflowsineachmode(zero, and ) are shown; idle crosses (blue call), gamma circles (purple cashflow and green inc. gamma option) and full plusses (red cashflow and black inc. put). The fixed switching costs ( and ) separate the transitions and complete the value matching conditions. Smooth pasting is apparent from a matching of the slopes of total value functions at and Discounts separating option values at different thresholds are represented in the middle panel across the page (see Table 4). When breaches (from below) at 1 =10the firm uses ( ) to switch from idle to gamma and gains a call/put plus cashflow less costs, less friction, i.e. ( )+ When breaches (from below) at 2 =30the firm uses ( ) to switch from gamma to full and gains a put plus net change of cashflow, less costs, less friction, i.e. ( )+ On the way down, when breaches (from above) at 3 =50the firm uses ( ) to switch from full to gamma and gains a put/call plus net change of cashflow, less costs, less friction, i.e. ( ) + + When breaches (from above) at 4 =70the firm uses ( ) to switch from full to gamma and gains a call plus net change of cashflow, less costs, less friction, i.e. ( ) + (note that this order holds only for the sample path real random paths will generate different switching sequences). The round trip costs are negative + + ( = 0 044); if follows a GBM with parameters e.g. = 4% 4% 20% (from Dixit and Pindyck [12]) they are consistent with threshold separation = being optimal. 19

22 Type VM VM DI DI SP SP Condition: Values (sols in bold) ( )= ( )+ ( )= ( )+ + ( )= ( )+ ( )= ( ) = = = = ( )= ( ) ( ) ( )= ( ) ( )+ ( ) ( ) ( )= ( ) ( )+ ( ) ( ) ( )= ( ) ( ) = = = = β W = β U + β Y β Z = = = = Table 5: Eight option values, four frictions and 12 equations. 20

23 References [1] Andrew B. Abel, Avinash K. Dixit, Janice C. Eberly, and Robert S. Pindyck. Options, the value of capital, and investment. The Quarterly Journal of Economics, 111(3): , [2] Andrew B. Abel and Janice C. Eberly. Optimal investment with costly reversibility. Review of Economic Studies, 63: , [3] David R. Alexander, Mengjia Mo, and Alan F. Stent. Arithmetic Brownian motion and real options. European Journal of Operational Research, 219: , [4] Luis H.R. Alvarez. Optimal exit and valuation under demand uncertainty: A real options approach. European Journal of Operational Research, 114: , [5] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81(May June): , [6] Michael John Brennan and Eduardo S. Schwartz. Evaluating natural resource investments. Journal of Business, 58(2): , [7] Dennis Capozza and Yuming Li. The intensity and timing of investment: the case of land. American Economic Review, 84(4): , [8] John C. Cox, Stephen A. Ross, and Mark Rubinstein. Option pricing: a simplified approach. Journal of Financial Economics, 7: , [9] Thomas Dangl. Investment and capacity choice under uncertain demand. European Journal of Operational Research, 117: , [10] D. A. Darling and A. J. F. Siegert. The first passage problem for a continuous markov process. Annals of Mathematical Statistics, 24: , [11] Avinash K. Dixit. Entry and exit decisions under uncertainty. Journal of Political Economy, 97: , [12] Avinash K. Dixit and Robert S. Pindyck. Investment under Uncertainty. Princeton University Press,

24 [13] Avinash K. Dixit, Robert S. Pindyck, and Sigbjørn Sødal. A markup interpretation of optimal investment rules. The Economic Journal, 109(455): , [14] Janice C. Eberly and Jan A. Van Mieghem. Multi-factor dynamic investment under uncertainty. Journal of Economic Theory, 75: , [15] Steinar Ekern. Entry and exit decisions with restricted reversibility. NHH Working Paper 93-08, [16] Andrea Gamba and Nicola Fusari. Valuing modularity as a Real Option. Management Science, 55(11): , [17] Bart Lambrecht and William R.M. Perraudin. Real options and preemption under incomplete information. Journal of Economic Dynamics and Control, 27: , [18] Robert L. McDonald and Daniel R. Siegel. Investment and the valuation of firms when there is an option to shut down. International Economic Review, 26(2): , [19] Robert C. Merton. The theory of rational option pricing. Bell Journal of Economics, 4(1): , [20] Stewart C. Myers. Determinants of corporate borrowing. Journal of Financial Economics, 5: , [21] Robert S. Pindyck. Irreversible investment, capacity choice and the value of the firm. American Economic Review, 78: , [22] Sudipto Sarkar and Fernando Zapatero. The trade-off model with mean reverting earnings: Theory and empirical tests. The Economic Journal, 113(490): , [23] Mark B. Shackleton and Sigbjørn Sødal. Smooth pasting as rate of return equalization. Economics Letters, 89(2 November): , [24] Afzal Siddiqui and Ryuta Takashima. Capacity switching options under rivalry and uncertainty. European Journal of Operational Research, 222: ,

25 [25] Sigbjørn Sødal. Entry and exit decisions based on a discount factor approach. Journal of Economic Dynamics and Control, 30(11): , [26] Lenos Trigeorgis. Evaluating leases with complex operating options. European Journal of Operational Research, 91: , [27] Lenos Trigeorgis. Real options: Managerial flexibility and strategy in resource allocation. MIT Press, [28] Robin J. Wilson. Introduction to Graph Theory. Longman, third edition,

26 Figure 1: Time series of sample path Pt (down page), VM and DI equations and matching value plots V(Pt) Pt whilst idle Pt whilst gamma Pt whilst full Time t Value of revenue process Pt (0 5) against time t (0 90) PV process Pt Vp(Pfg) < Vp(Pgf) +Pfg +Pgf Vg(Pgi) < Vg(Pig) < > =Vg(Pfg) > =Vg(Pgf) Value matchings +Pgi +Pig +Pfg +Pgf Xgi Xig +Xfg +Xgf =Vc(Pgi) > =Vc(Pig) Vp(Pfg) * Dp(Pt,Pfg) = Vp(Pt) Discountings Vg(Pgi) * Dp(Pt,Pgi,Pgf) + Dc(Pt,Pgf,Pgi) * Vg(Pgf) = Vg(Pt) Vc(Pt) = Dc(Pt,Pig) * Vc(Pig) Pt levels Pgi=1 Pig=2 Pfg=3 Pgf=4 6 Pt + Vp(Pt) 5 Pt 4 (Pt)^g + Vg(Pt) Total value inc, option (Pt)^g Vc(Pt) Total value by mode for values of Pt in sample path PV process Pt