ENTRY AND EXIT DECISIONS PROBLEM: A SURVEY

In this survey, the entry-exit decisions problem is studied. In its simplest version, the problem concerns to the investment and disinvestment decisions process faced by a firm, when its output price is considered being stochastic. This is essentially the problem faced by a firm exploiting natural resources (i.e. oil, copper, etc.) whose output prices are daily traded in the commodities markets.

ENTRY AND EXIT DECISIONS PROBLEM: A SURVEY Andrea Girometti∗ Master in Advanced Studies in Finance University of Ză urich - ETH Ză urich andreagirometti@yahoo.it January 2004 Abstract In this survey, the entry-exit decisions problem is studied The problem concerns to the investment and disinvestment decisions process of a firm, when its output price is stochastic Typically, this is the problem faced by an oil company or by a firm involved in the commodities markets The first approach, here analyzed in detail, was proposed by Dixit in 1989 and it is based on the contingent claim theory In particular, the values of a firm in both activity and inactivity states are determined by using the theory of real options pricing Thanks to the so-called value-matching and smooth-pasting conditions, it is possible to interlink these two firm’s values and, therefore, to determine a pair of trigger prices, giving an optimal decision policy The optimal policy fixes the levels of the output prices at which it is economically convenient either to start the production or to abandon the market In addition, some interesting extensions of the basic model are presented and, finally, the two most recent approaches are explained The first one is based on the ”mark-up” concept, the second one is based on the optimal stopping time theory ∗ The author is grateful to Ente Luigi Einaudi for the support He is also grateful to Paolo Casini, Paolo Verzella and Antonio Sciarretta for useful discussions 1 Introduction In this survey, the entry-exit decisions problem is studied In its simplest version, the problem concerns to the investment and disinvestment decisions process faced by a firm, when its output price is considered being stochastic This is essentially the problem faced by a firm exploiting natural resources (i.e oil, copper, etc.) whose output prices are daily traded in the commodities markets As a general framework, a single firm having access to a single good production opportunity is considered If {Zt }t≥0 is a stochastic process taking discrete values 0, and indicating the current firm’s state, at time t such a monopolistic firm is supposed to be either inactive Zt = or already active Zt = In the inactive state there is no production at all, and the firm is waiting the best conditions to enter in the market and to produce If the firm is already active, it has full capacity utilization, the resource is infinite and it can abandon the market if its profitability is not satisfying Suppose that the firm, changing its state Zt , switches production on or off instantaneously Switching production on, the firm can invest a lump-sum cost KI in order to start the production and, in the opposite case, the firm has to pay a lumpsum cost KE in order to exit to the market It is assumed that KI + KE > 0, otherwise a firm could have an infinite profit switching continuously The firm’s activity implies also a variable production cost wo per each unit of output flow This is the operating cost Moreover, it is supposed that the firm has to pay again the lump-sum entry cost KI if it exits and wants to re-entry later All these costs are supposed constant over time The riskless cost of capital, at which all values will be discounted, is r and it is constant The uncertainty comes uniquely from the market price of the output This price is represented by the stochastic process {Pt }t≥0 , and it is assumed that it is driven by a geometric Brownian motion1 : dPt = αPt dt + σPt dBt , (1) where α and σ are constant and strictly positive, and {Bt }t≥0 is a standard Brownian motion The output price represents the profit flow per unit of production, and its expected value grows at the rate α2 We will identify these hypotheses as standard assumptions Later, some of them will be relaxed The case of a geometric Brownian motion is the simplest case because of the existence and uniqueness of a solution for the stochastic differential equation (1) However, the prices of commodities not seem to be well described by such a process In fact, it is known that, for example, the oil price seems to be better represented by a mean-reverting process Obviously, this complicates much more the analysis and, for the moment, nobody should have presented any interesting result As we can see in Dixit and Pindyck [DP], an interesting interpretation of the drift can The aim of a firm is to determine, jointly, optimal bounds (PL , PH ) This pair represents the trigger prices for entry and exit into the market At a price level PH it begins to be profitable for a firm starting the production If the output price is lower than PL , producing is not profitable anymore and the firm leaves the market Various approaches to the problem and several possible extensions are presented in this survey All these methods are alternative to the standard capital budgeting process based on the NPV rule The inadequacy of this approach is widely acknowledged, because of total neglect of the stochastic nature of the output prices In general, the real option valuation theory seems to be more suitable and more useful The first and simplest model was pioneered by Mossin [Mo] in 1968, but the first formal discussion on the entry decisions problem was proposed by McDonald and Siegel [MS] in 1985 The combined problem concerning the entry and the exit was faced by Brennan and Schwartz [BS] in 1985 They applied, for the fist time, the well known options pricing theory developed by Black, Merton and Scholes in 1973 in order to evaluate active and inactive firms, and they defined the concepts of option to enter and option to abandon as part of the firm’s value A formal and complete discussion was presented by Dixit [D2] in 1989 In particular, he focused on entry and exit trigger prices as fundamental indicators for firm’s decision policies Moreover, he made comparisons between the standard Marshallian theory and the new option theory-based approach, introducing the hysteresis concept, and he proposed a numerical analysis with a comparative statics analysis of the results he found3 During the ’90s, many authors developed the Dixit’s basic model considering various possible extensions like adding taxes or an investment’s lag, considering a restricted number of switches or the possibility of laying-up or scrapping the production Other authors discussed the possibility and the effects of an interest rate risk or a currency rate risk, or focused on studying be done According to the CAPM model, the appropriate risk-adjusted discount rate for the firm’s cash-flows should be: µ = r + βσρpm , where β is the market price of the risk and ρpm is the correlation between the output price and the market portfolio µ is the total risk-adjusted expected rate of return that investors would require if they are to own the project α is viewed as the expected capital gain Assuming α < µ, the difference δ = µ − α represents a kind of dividend In case of a storable commodity like oil, copper etc., δ is called net marginal convenience yield from storage and represents the benefit coming from last stored unit This benefit is given, for example, by the possibility of smoothing production These economic results will not be presented in this survey The interest reader can refer to the Dixit’s paper [D2] the economic equilibrium between firms using the Dixit decision’s rules (see respectively, for example, Ingersoll and Ross [IR], Kogut and Kulatilaka [KK], Leahy [Le]) Gauthier [GL] applied the theory of exotic options (in particular parisienne options) in order to study the case of the delay (always) existing during the capital budgeting process However, a definitive and rigorous treatment of the problem was proposed by Brekke and Øksendal in 1994 [BØ] They analyzed the entry-exit decisions problem applying both the option pricing theory and the dynamic programming theory, gave a formal proof of the existence of a solution, and extended the classical approach considering the case of a finite resource In 2001 Sødal [Sø] proposed a totally new approach, based on the ”mark-up” concept Finally, in 2003, Chesney and Hamza [CH] proposed a probabilistic approach The survey is organized as follows In section the basic model proposed by Dixit will be examined In sections 3, 4, the most interesting extensions of the basic model are presented They concern respectively the presence of an investment’s lag, of a restricted number of possible switches, of the possibility of either laying-up or scrapping the project Section will be devoted to the problem viewed as a sequential optimal stopping problem, as Brekke and Øksendal did, and the case of finite resource will be introduced Finally, sections and will present the new approaches based on the mark-up concept developed by Sødal and on probabilistic tools proposed by Chesney and Hamza The basic model Through this paragraph we will refer to the fundamental Dixit’s paper [D2] Consider the standard assumptions At level price Pt , the firm could be either active Zt = or inactive Zt = Then, the first step is determining the firm’s value in both states In state (Pt , 0), suppose that the inactive firm has an expected net present value V0 (Pt ) Such a firm can observe the current output price and then, following optimal policies4 , decides whether to continue being inactive or to enter in to the market Assume, for the moment, that this decision is irreversible The firm has simply an option to invest and its value is completely represented by the value of this option If the option is exercised, the firm starts the production Analogously, in state (Pt , 1), the active firm has an expected net present value V1 (Pt ) In this case, the firm decides whether to continue being active or to exit Suppose that this decision is again irreversible In this case, In general, the firm maximizes its expected net present value the firm’s value is given by the value of the current profit and the value of an option to abandon If this option is exercised, the firm goes back to the inactive state Viewed as options to enter and to abandon, the two expected net present values V1 (Pt ) and V0 (Pt ) can be determined by using the contingent claim analysis Therefore, the market must supposed to be complete, i.e every traded asset is supposed to be spanned by other assets in the economy In particular, both the values of the active and inactive firms are supposed to be positively correlated with either a traded asset or a basket of traded assets, in order to make possible the replication the firms values Once the two values V1 (Pt ) and V0 (Pt ) are determined, it is possible to determine the two trigger prices (PL , PH ) considering the so-called valuematching conditions and the smooth-pasting conditions, as we will see in the following sub-section 2.3 2.1 Value of an inactive firm Consider an inactive firm having only one possibility to enter in its market Intuition suggests that such firm finds optimal to remain inactive as long as the output price is lower a certain threshold PH and it will invest as soon as the price reaches PH Therefore, over the range (0, PH ) the inactive firm’s investment opportunity, i.e the value of the inactive firm, is equivalent to a perpetual call option, where the strike is the entry cost KI The decision to invest corresponds to the decision of exercising the option Therefore, in order to obtain the inactive firm’s value, it can be applied the contingent claim theory, as McDonald and Siegel [MS] have done Applying the Itˆo’s Lemma to dV0 (Pt ) and substituting the price dynamic (1), we have: dV0 (Pt ) = V0 (Pt )dPt + V0 (dPt )2 = 
  2 = V0 (Pt )αPt + V0 (Pt )σ Pt dt + σPt dBt The term in square brackets is the expected value of dV0 (Pt ), and there being no operating profit, it is the only expected capital gain In a risk-neutral world5 it has to be equal to rV0 (P )dt, the riskless return Thus, we obtain Here the fundamental assumption regarding the completeness of the market plays a central role Thanks to this assumption, in fact, it is possible to replicate the expected value of dV0 (Pt ) the following differential equation: 2  σ Pt V0 (Pt ) + αPt V0 (Pt ) − rV0 (Pt ) = 0, (2) with boundary condition limPt →0 V0 (Pt ) = Equation (2) is a second order differential equation and is homogeneous and linear It is known that its general solution is given by substituting P ξ One can obtain: σ ξ(ξ − 1) + αξ − r = φ(ξ) = ξ − (1 − m)ξ − ρ = 0, or and ρ = σ2r2 The convergence condition is r > m, and where m = 2α σ2 φ(0) = −ρ < and φ(1) − (ρ − m) < Since φ (ξ) = > 0, one root must be greater than and the other must be less than 0:  (1 − m) + (1 − m)2 + 4ρ > 1, β1 = β2 = (1 − m) −  (1 − m)2 + 4ρ < Finally, the general solution of (2) is given by: V0 (P ) = A1 P β1 + A2 P β2 , (3) where coefficients A1 and A2 have to be determined As we said, this solution is valid for P ∈ [0, PH ] 2.2 Value of an active firm Analogous considerations can be made in order to determine the net present value of the active firm Consider, thus, for the moment, an active firm having only one possibility to exit to its market An active firm persists in this state as long as the output price is higher than a certain threshold PL and it will abandon the market as soon as the price falls down PL Then, in the interval [PL , ∞) an active firm holds its option to abandon, and the firm’s value is given by both the operating profit and the option to abandon Considering the value of an active firm V1 (P ) and applying the Itˆo’s Lemma, we obtain: dV1 (Pt ) = V1 (Pt )dPt + V1 (dPt )2 =
   2 = V1 (Pt )αPt + V1 (Pt )σ Pt dt + σPt dBt The term in square brackets is the expected value of dV1 (P ) and, as before, in a risk-neutral world it has to be equal to the riskless portfolio value rV1 (P )dt Note that, this time, there exists a dividend, namely the operating flow of operating profit (Pt − wo ) in addition to the expected capital gain coming from the option to abandon Therefore, under the fundamental assumption regarding the completeness of the market, we have the following differential equation:  V (Pt )σ Pt2 + V1 (Pt )αPt − rV1 (Pt ) + (Pt − wo) = (4) P − wro EquaThe boundary condition is now limP →+∞ V1 (P ) = limP →+∞ r−α tion (4) is not homogeneous anymore For the homogeneous part, it is possible to proceed as before For the non-homogeneous part one should try a linear form and solving for the coefficients6 Finally, we get the following general solution:   wo P β1 β2 − , (5) V1 (P ) = B1 P + B2 P + r−α r for P ∈ [PL , ∞), where coefficients B1 and B2 have to be determined Because the homogeneous part is identical in equations (2) and (4), coefficients β1 and β2 are the same as before.7 2.3 The trigger prices Our problem is, now, to determine the constants A1 , A2 , B1 , B2 in equations (3) and (5) and the two thresholds (PL , PH ) It is possible to simplify the situation, considering the endpoints conditions When the output price is very small, the probability of raising to PH is small for any fixed timehorizon Therefore, the option to invest becomes out-of-the-money and thus, For further explanations, see Dixit and Pindyck [DP], p.187 As we said, the value of the active firm is given by the sum of the current profit and the option to abandon We can see that the terms in parentheses can be written as:
 ∞  wo Pt −ru − =E e (Pu − wo )du r−α r This is just the expected present value of a project that lives forever, starting from an initial price P Therefore, the remaining part, B1 Ptβ1 + B2 Ptβ2 , can be considered as the value of the option to abandon the coefficient A2 must to be zero Similarly, as P → ∞ the probability that the output price raises to PL is small for any fixed time-horizon Therefore, the option to abandon becomes out-of-the-money and, thus, the coefficient B1 must to be zero Finally, the problem simplifies considering the following two equations: V0 (P ) = AP β1 ,  V1 (P ) = BP β2 + P wo − r−α r (6)  (7) Moreover, economically meaningful solution for V0 (P ) and V1 (P ) requires their non-negativity and, therefore, both coefficients A and B have to be non-negative Those two solutions have to be linked by the so-called value-matching condition (or high-contact condition) and its relative smooth-pasting condition At the threshold PH , the firm pays KI to exercise the option to invest, abandoning an asset of value V0 (P ) and getting another one of value V1 (P ) Therefore, at level PH it must be satisfied, as feasibility conditions, the following condition: V0 (PH ) = V1 (PH ) − KI , =⇒ V0 (PH ) = V1 (PH ) Similarly, at the threshold PL the firm pays KE to exercise the option to disinvest, abandoning an asset of value V1 (P ) and getting another one of value V0 (P ) Therefore, at level PL the feasibility conditions are: V1 (PL ) = V0 (PL ) − KE , =⇒ V1 (PL ) = V0 (PL ) Substituting equations (6) and (7) in the feasibility constraints we obtain the following system:   BPLβ2 +         β2   BPH + PL r−α − wo r = APLβ1 − KE PH r−α − wo r = APHβ1 + KI   β2 −1  +  β2 BPL       β BP β2 −1 + H (8) β1 APLβ1−1 r−α = r−α = β1 APHβ1−1 In the following, we can call these conditions also feasibility conditions These four equations determine the four unknowns of the entry-exit problem A, B, PL and PH They are non-linear in PL and PH , so that an analytic solution in a closed form is impossible However, it can be proved that a solution exists9 The thresholds satisfy < PL < PH < ∞, and the coefficients of the option value terms, A and B, are non-negative Further results require a numerical solution With this model it can be possible to make a comparison with the Marshallian approach, and making a comparative statics analysis The interested reader should see the Dixit and Pindyck book [DP] In the book is also presented a useful example regarding the application of the model in the copper industry Investment’s Lag So far, the model assumed that the project is brought on line immediately after the decision to invest is made However, many investments take time and the lag between the decision to invest and the start of the production can be quite long For example, McRee noted that, on average, the lag is years in case of building a power generating plant10 In this paragraph, we intend to extend the basic model considering a so-called time-to-build dt ≥ This problem was faced for the first time by Bar-Ilan and Strange [BaS] in 1992 Starting from the Dixit’s basic model and considering the standard assumptions, they supposed that there exists an investment lag with timeto-build equal to dt ≥ This implies that a project started at time t will begin generating revenues and incurring marginal costs at time t + dt The entry cost KI is supposed to be paid at the end of construction, but the commitment is irreversible once the decision is made This is equivalent to say that at time t the entry cost is given by e−rdt KI Therefore, instead of two, we have three state of nature In fact, the firm can be inactive, Zt = 0, active Zt = and ”under construction” Zt = c, where c = 0, is an arbitrary constant 3.1 The value of the firm in the three states Proceeding as before, we have to determine the value of a firm in the three states of nature If a firm is inactive, it has the opportunity of exercising the A formal proof was given by Brekke and Øksendal in 1994 See [BØ] See McRee K.M., Critical issues in electric power planning in 1990s, Canadian Energy Research Institute, 1989 10 option to invest It is sure that in the range (0, PH ) it will remain inactive and its value is given by V0 (Pt ) This value can be obviously written as: V0 (Pt ) = e−rdt V0 (Pt+dt ) =⇒ dV0 (Pt ) = e−rdt dV0 (Pt+dt ) The value of the inactive firm at time t is equal to the value of the inactive firm at time t + dt discounted for the lag dt It is possible to apply to dV0 (Pt+dt ) the same analysis of sub-section 2.1 Moreover, because the discount factor is not affected by the output price, taking the limit as dt → 0+ , we obtain the same differential equation (2) with the same boundary condition, and therefore the same solution: V0 (P ) = AP β1 , (9) where the constant A has to be determined Analogously, in the range (PL , ∞) an active firm will not switch to the inactive state and its value is given by:
 dt  −rdt −ru V1 (Pt ) = e V1 (Pt+dt ) + E e (Pu − wo )du =⇒ −rdt =⇒ dV1 (Pt ) = e  dV1 (Pt+dt ) + dt e−ru (Pu − wo )du This means that V1 (Pt ) is given by the sum of the discounted future value V1 (Pt+dt ) of the firm and the expected cash flow coming from the operating gain in the interval [t, t + dt] Applying the Itˆo’s Lemma to dV1 (Pt+dt ), substituting the geometric Brownian motion to Pt+dt and taking the limit as dt → 0+ , we obtain the same differential equation (4) with the same boundary condition, and therefore the solution is always the following:   wo Pt β2 − V1 (P ) = BP + , (10) r−α r where the constant B has to be determined To complete the solution we must determine the value function Vc (Pt , θ), giving the value of a project under construction, where θ ∈ [0, dt] is the remaining time until completion of the investment This value is given by: Vc (Pt , θ) = e−rdt Vc (Pt+dt , θ − dt) =⇒ 10 dVc (Pt , θ) = e−rdt dVc (Pt+dt , θ − dt) ... enter and option to abandon as part of the firm’s value A formal and complete discussion was presented by Dixit [D2] in 1989 In particular, he focused on entry and exit trigger prices as fundamental... interval [PL , ∞) an active firm holds its option to abandon, and the firm’s value is given by both the operating profit and the option to abandon Considering the value of an active firm V1 (P ) and applying... Marshallian approach, and making a comparative statics analysis The interested reader should see the Dixit and Pindyck book [DP] In the book is also presented a useful example regarding the application