Barriers and Optimal Investment 1

Transcription

1 Barriers and Optima Investment 1 Jean-Danie Saphores 2 bstract This paper anayzes the impact of different types of barriers on the decision to invest using a simpe framework based on stochastic discount factors. Our intuitive approach proposes an aternative to the rea options methodoogy that does not rey on the smooth-pasting condition. n appication to MacDonad and Siege s canonica investment probem (1986) shows that the standard investment threshod over-estimates the optima threshod when the ower barrier is absorbing and under-estimates it when the ower barrier is refecting. Key words: investment; uncertainty; irreversibiity; barriers; rea options. JEL cassification: D92, D81, E22. 1 The hepfu comments of participants at the 2003 Econometrics Society Summer Meetings at Northwestern University, at the 7 th nnua Internationa Conference on ea Options, and at the 2003 Meetings of the European Economic ssociation are gratefuy acknowedged. I am, of course, responsibe for a remaining errors. 2 ssistant Professor, Schoo of Socia Ecoogy and Economics Department, University of Caifornia, Irvine Phone: (949) Fa: (949) E-mai: saphores@uci.edu.

2 I. Introduction Barriers are often assumed away in the stochastic investment iterature, yet intuitivey they shoud matter. This paper fis this gap for simpe investment probems by making two contributions. First, it etends the canonica investment mode of MacDonad and Siege (1986) when the cost of the investment is fied. It shows that their investment threshod overestimates the optima investment threshod with a ower absorbing barrier and underestimates this threshod with a ower refecting barrier. Our numerica resuts show that the nature of the ower barrier is important for investment decisions at higher eves of uncertainty. Second, this paper generaizes stochastic discount factors to the case where the autonomous stochastic variabe of interest is constrained by a barrier. This provides an intuitive aternative to the conventiona rea options methodoogy that can be readiy etended to more compe investment probems. When to pay a constant (sunk) amount I for a payoff X that foows an autonomous diffusion process is probaby the most basic investment probem. s such, it has aready received a ot of attention (e.g., see McDonad and Siege 1986, Diit and Pindyck 1994, or Diit, Pindyck, and Søda 1999, and the references herein). Surprisingy, however, with the eception of Brock, othschid, and Stigitz (1982), who anayze a basic probem of stochastic capita theory, the potentia impact of a ower barrier on the decision to invest has not been anayzed in this canonica framework. 1 Instead, a ower unattainabe barrier (unreachabe in finite time) is usuay assumed in order to derive a cosed-form soution. n eampe is 0 with the geometric Brownian motion (GBM) for the perpetua ca option. Intuitivey, however, we epect to invest more conservativey in the presence of a ower absorbing barrier, which, if reached, makes investing permanenty unattractive, than if a ower 1

3 refecting barrier aows the investment payoff to rebound and grow arger with voatiity. 2 n absorbing barrier coud resut, for eampe, from demand shifts foowing innovations by competitors (in eectronics, pharmaceuticas ), from gradua changes in tastes, from bankruptcy if the investment opportunity is a ca option to purchase another firm, or from the disappearance of a natura resource (overfishing may permanenty depress a fish stock, for eampe). Conversey, a refecting barrier may arise from government imposed price foors (as for some agricutura commodities), or when a resource has residua vaue from aternate uses. For eampe, the owner of a vacant urban pot of and can erect a commercia buiding if the economy is booming, or buid a surface parking ot if the rea estate market is depressed. In addition, the rea options approach (Diit and Pindyck 1994), which is now the standard methodoogy for soving simpe stochastic investment probems, may deter non speciaists because of its reiance on a technica condition (often caed smooth-pasting ) for which the underying theory is hardy accessibe to most economists (Søda 1998). By contrast, the stochastic discount factors approach presented herein reies essentiay on concepts from deterministic optimization probems, so it shoud be appeaing to economists with itte background in finance. This paper is organized as foows. Section 2 introduces the stochastic discount factor approach and shows that it is equivaent to the rea options methodoogy. Section 3 anayzes the decision to invest in the presence of a refecting or an absorbing barrier in a canonica framework and presents resuts from a numerica iustration. Section 4 concudes. n appendi outines most of the proofs presented herein. 2

4 II. II.1 Barriers and Optima Stopping ues ssumptions and Definitions Consider again the simpe framework where a firm can invest a fied amount I for an investment whose epected net present vaue foows the time homogeneous diffusion process dx = α( X ) dt + σ ( X ) dw, (1) where dw is an increment of a standard Wiener process (Karin and Tayor 1981) and X is defined on an interva Γ of the form [L,], (L,], [L,), or (L,) with - L< +. 3 Once undertaken, this investment is irreversibe. discussed in MacDonad and Siege (1986), this framework refects some of the key characteristics of the probem of a monopoist or of the probem of a firm in a competitive industry enjoying temporary rents (provided X ehibits a decreasing trend). For simpicity, the vaue of this investment opportunity is discounted using a constant discount rate. 4 To simpify our anaysis, we suppose that: ssumption 1: X is reguar on Γ, i.e., there is a non-zero probabiity that X can reach any point of Γ in finite time starting from any other point in Γ. This is usefu to epress the investor s decision in terms of vaues of X instead of time. Moreover, the infinitesima trend of X, α(.), and the infinitesima standard deviation, σ(.), are continuous on Γ; in addition, σ(.)>0 on the interior of Γ. ssumption 2: X admits a finite barrier at (L,). We focus mosty on two cases: either is refecting, so X simpy rebounds upon reaching, or it is absorbing, so the investment opportunity disappears as soon as X hits <I. 3

5 Let us now reca a coupe of usefu definitions about barriers. though we focus on ower barriers, these concepts can easiy be etended to upper barriers. Definition 1. ower barrier [ L, ) is said to be attracting if there is a non-zero probabiity that X reaches before any interior point. Let p ; ( y) denote the probabiity that X reaches before starting from y. Conversey, if is non-attracting, then X is certain to reach any interior point before, and thus p ; = 1 y. It is important to note that the attracting property of a barrier hods for a interior points as a resut of the requirement that the function σ(.) be stricty positive on (L,) and the definition of p ; ( y). Indeed, Karin and Tayor (1981) show that p ; S( y, ) ( y) =, S(, ) (2) where, for L< 1 < 2 <, the scae function S( 1, 2 ) is defined by 2 ξ 2 αζ ( ) S ( 1, 2) = ep dζ dξ. 2 ( ) 1 ξ σ ζ 0 (3) In Equation (3), ξ 0 is an arbitrary constant that has no bearing on the vaue of p ; y ; changing ξ 0 is akin to mutipying the numerator and the denominator of p ; y by the same constant. From (2) and (3), we see that is attracting if and ony if im S( ξ, z) ξ + is finite for z (, ). 4

6 Definition 2. Let (, ) and y (, ). ower barrier is said to be attainabe if and ony if the epected time it takes X to reach either or starting from y is finite. If is not attainabe, it is unattainabe. Unattainabe barriers may or may not be attracting, but a attainabe barriers are attracting (Karin and Tayor 1981, Chapter 15). Barriers may then be cassified as foows: ttainabe and attracting, which incude refecting and absorbing barriers; Unattainabe but attracting, such as + for a Brownian motion with a positive trend; and Unattainabe and unattracting. Let us now eamine specific absorbing, refecting, and unattainabe barriers in the contet of our simpe investment probem. II.2 Objective Function Let us first assume that a ower barrier is absorbing, so as soon as X hits, X stays stuck at <I, and the investment opportunity disappears. This eementary probem embodies two key differences compared to its deterministic counterpart. First, because of uncertainty, we don t know how ong it may take for X to reach the investment threshod. Second, and most importanty here, X never reaches if it hits first. With this in mind, if y=x(0), to find the investment threshod we need to sove 5

7 Ma Dy ;[ I], (4) T ; where D y y ; E e ρ is the epected discount factor for T, time 0 (now) and the first time X hits conditiona on not hitting first. y, the eapsed time between From Karin and Tayor (1981), we know that for y (, ), W( y) D y ; verifies the inear, second-order, ordinary differentia equation 2 2 σ ( y) d W( y) dw( y) + α( y) ρw( y) = 0. (5) 2 2 dy dy ; ( T, y By definition, D = 0 =+ because starting from, X never reaches ) and D ; = 1 so the two boundary conditions needed to fuy define W(y) are simpy W( ) = 0 and W( ) = 1. (6), Then, if W 1 (y) and W 2 (y) are two independent soutions of (5) defined over [, ], D W () W ( y) W () W ( y). () ( ) () ( ) = y; W2 W1 W1 W2 (7) Let us now suppose instead that ( L, ) is refecting. Because X is reguar, it wi aways reach the investment threshod denoted here by, but we don t know how ong it wi take. This probem can be written: Ma Dy ;( I), (8) T ; where D y y ; E e ρ is the epected vaue of the discount factor for T y;, the random 6

8 duration between time 0 when y=x(0) and the moment where X first hits. W( y) D y ; aso verifies (5) and a boundary condition for finding W(y) is simpy W( ) = 1. (9) We aso need to write a boundary condition at to epress that it is refecting. We have: Lemma 1: If is refecting then dw dy y= = 0. (10) Proof. Let us therefore suppose that, at time 0, X=. In the neighborhood of, X behaves as a Brownian motion with infinitesima mean α() and variance σ 2 (). Now consider a discrete approimation of the Brownian motion, as in Diit (1993). Since X cannot take a vaue ower than, after a sma time increment t, X moves up by a sma deterministic increment >0 (i.e., X( t)= + ), where t t. Then, for (,), T W( ) = E {ep( ρ dτ)} = E {ep( ρ dτ)ep( ρ dτ)} 0 0 t 0 0 T t 0 T + = [1-ρ t+o( t)] E {ep( ρ dτ)} = [1-ρ t+o( t)]w( + ) 0 dw = [1-ρ t+o( t)][w( ) + y= + o( )]. dy The transition from the first to the second ine above reies on the aw of tota probabiity and the Markov property (Karin and Tayor 1981). The transition from the second to the third ine is a 7

9 Tayor etension of W( + ). Simpifying, dividing by, and taking to 0, gives (10). If W 1 (y) and W 2 (y) again denote two independent soutions of (5), when we combine the two boundary conditions (9) and (10), we get D ' ' W2() W1() y W1() W2( y) = y; ' ' W2 W1 W1 W2. () ( ) () ( ) (11) In economics, a ower barrier is typicay assumed to be unattainabe to simpify derivations. This seems to impy, however, that ower barriers have a negigibe impact on the investment decision. If is unattainabe, et us denote reevant discount for our simpe investment probem by D y. Mathematicay, one of the two independent soutions of (5) and its first derivative typicay goes to infinity when approaches (think of zero for the geometric Brownian motion); suppose it is the case for W 2 (y). Then it is easy to see from (7) and (11) that viewing as the imit of either an absorbing or a refecting barrier eads to the same discount factor, and therefore to the same investment threshod. II.3 First Order Necessary Condition s time eapses, y, the current vaue of X changes randomy. It woud thus seem that the firstorder necessary condition depends on a random variabe, as discussed in Diit, Pindyck and Søda (1999). 5 The key to this probem is to note that the first order condition is verified at the optimum, so this condition needs to be written at =y=. This eads to: 8

10 Lemma 2. Whether is absorbing, refecting, or unattainabe, the optimum investment threshod verifies the first order necessary condition D y ; = y= ( I) + 1= 0. (12) where we omit the subscript or for simpicity. Proof. See Proposition 1 beow. From Lemma 2, the sum of two margina changes at equas zero: first, waiting a bit onger impacts the present vaue of the project through the epected discount factor; and second, it affects the net payoff from reaizing the investment opportunity (its margina vaue is 1 here). II.4 Link with the ea Options pproach Let us now eamine how the stochastic discount factor approach described above reates to the standard rea options approach. s above, W 1 (y) and W 2 (y) denote two independent soutions of Equation (5) defined over (, ). Proposition 1. With either a refecting or an absorbing barrier at, the standard rea options approach and our approach are equivaent. Proof. Consider first the absorbing case. From Diit and Pindyck (1994), the vaue of the option to invest I to get, denoted by ϕ(), verifies the Beman equation (5), so et us write it 9

11 ϕ ( ) = W ( ) + W ( ), where 1 and 2 are two unknown constants to be determined simutaneousy with the investment threshod; denoted here by a to distinguish it from, which soves (12). Since is absorbing, the option to invest at is 0 so that W () + W () = 0. (13) When the option is eercised, at a, it is echanged for the net vaue of the investment (the a a continuity condition ). ϕ ( ) = I impies 1 1 a 2 2 a a W ( ) + W ( ) = I. (14) Since there are three unknowns ( 1, 2, and a ), another condition (the smooth-pasting condition ) is needed (Diit and Pindyck 1994). Here, it equas ' ' 1 1 a 2 2 a W ( ) + W ( ) = 1. (15) Combining (13) and (15) gives 1 and 2 ; inserting these epressions in Equation (14) gives ' ' 2() 1( a) 1() 2( a) ( a I ) () 1( a) 1() 2( a) W W W W + = W W W W (16) This is aso Equation (12) so a =. We proceed simiary for the refecting case. The vaue of the option to invest, denoted here by ψ(), again verifies the Beman Equation (5), so ψ ( ) = BW 1 1( ) + B2W2( ), where B 1 and B 2 are two unknown constants, and we denote by r the investment threshod. To find the boundary condition at, the ogic foowed to derive Equation (10) eads to ψ ' () = 0(see aso Diit 1993), so that 10

12 ' ' 1 1 B2W2 BW () + () = 0. (17) The continuity and smooth-pasting conditions (Equations (14) and (15)) are simiar, so the equation foowed by the investment threshod is ' ' ' ' 2() 1( r) 1() 2( r) ' ' ( r I ) 1 0, 2() 1( r) 1() 2( r) W W W W + = W W W W (18) which again is equivaent to (12), so r =. In this simpe framework, the vaue of the option to invest is simpy the net present vaue of the investment at y (, ). s epected, the investor seeks the investment threshod that maimizes the vaue of the option to invest at eercise. More importanty, the stochastic discount factor approach provides an intuitive aternative to the rea options approach that does not rey on the smooth-pasting condition. This approach can readiy be etended using functiona forms defined in Karin and Tayor (1981; Chapter 15) to more compe probems invoving mutipe payoffs as we as fows of costs or benefits. III. ppication to a Simpe Investment Probem Let us now consider the case where X foows the geometric Brownian motion (GBM) dx = µ Xdt + σ Xdz, (19) where µ and σ >0 are respectivey the infinitesima trend and voatiity parameters. It is we known that the GBM is reguar and that =0 is unattainabe. 6 The case =0 is discussed in detais 11

13 in Diit and Pindyck (Chapter 5). For convenience, we define the dimensioness parameters 2α ρ κ 1, λ =. 2 σ α κ provides an inde of variabiity for X: for α>0, the more negative κ is, the ess voatie X is; conversey, a vaue of κ between 0 and 1 indicates high voatiity for X. λ on the other hand, scaes the discount factor with the epected rate of growth of X; as show beow, λ must be ess than one in order for our investment probem to have a finite soution even with an absorbing ower barrier. Two independent soutions of (5) here are W θ 1 W2 1 2 ( ) and ( ), θ ξ = ξ ξ = ξ (20) where θ 1 and θ 2 verify 2 2 σ 2 σ θ + α θ ρ = 0, 2 2 (21) so that 2 2 κ κ κ κ 1 2 θ = + + λ(1 κ) >0, θ = + λ(1 κ) < 0. (22) itte bit of agebra shows that ( θ ) ( ρ α ) 1 > 1 >. (23) Now suppose that there is a barrier on X at >0 with <I, otherwise the investor is guaranteed to make money, and denote I / by J; hence, J>1. 12

14 Let us first suppose that >0 is absorbing. Inserting (20) into (7) eads to D θ2 θ1 θ1 θ2 y y y ; =. θ θ θ θ (24) The resuting first order necessary condition, based on (12), can be written F ( / ) = 0, where θ θ + 1 θ θ F ( z) (1 θ ) z + θ Jz + ( θ 1) z θ J. (25) Likewise, if >0 is refecting, inserting (20) into (11) gives D θ2 1 θ1 θ1 1 θ2 θ2 y θ1 y y ; = θ2 1 θ1 θ1 1 θ θ θ 2 2 1, (26) and (12) becomes F ( / ) = 0, with θ θ + 1 θ θ θ θ2 F z z + Jz + z J (27) ( ) (1 θ ) θ θ θ. where again J I /. F (.) and F (.) are dimensioness. We have: Proposition 2. F ( / ) = 0 ( absorbing) and F ( / ) = 0 ( refecting) admit unique soutions, denoted respectivey by and if and ony if λ>1 (i.e., ρ>α). If λ 1, it is optima to wait forever since the discounted epected vaue of the investment keeps on growing. Proof. Let us first suppose that is absorbing. We note that θ1 θ2 F ( J) = J( J 1) > 0, ' θ θ F J = θ2 J > ( ) (1 )( 1 2 1) 0, (28) since J I / >1, θ 1 > 0, and θ 2 < 0. Differentiating F (z) twice gives 13

15 ' θ θ θ θ 1 θ2 1 F z = z Jz + 1 θ2 " θ1 θ2 2 F ( z) = ( θ1 θ2) z {(1 θ1)( θ1 θ2 + 1) z+ θ1( θ1 θ2 1) J}, ( ) (1 θ )( θ θ 1) θ ( θ θ ) θ, (29) so F " ( z ) has the sign of the inear function f ( z) = (1 θ1)( θ1 θ2 + 1) z+ θ1( θ1 θ2 1) J and f( J) ( θ θ )(1 θ θ ) J θ θ = has the sign of α 1 θ θ =, i.e. the sign of α. σ Knowing the sign of F " ( z ) aows us to make inferences about F ' ( z ) and F ( z ). The refecting case is handed simiary after noting that " " F ( z) = F ( z). Tabe 1 summarizes the variations of F (z), F (z) and their first two derivatives on (J,+ ). Detais are provided in the appendi. Let us now compare and. We have: Proposition 3. Suppose that ρ>α so the investment probem admits a unique soution. ssuming an unattainabe barrier (i.e., assuming =0) overestimates the optima investment threshod if >0 is absorbing, and it underestimates the optima investment threshod if >0 is refecting: θ I 1 < < θ1 1, (30) θ1 where θ1 1 I is the investment threshod if =0 (McDonad and Siege 1986). Proof. See the appendi. 14

16 This resut impies that the wedge between the critica vaue and I is infuenced by the presence and the nature of a ower barrier. This wedge is smaer when is absorbing and arger when is refecting. Since we ony have impicit epressions for and, we need to compare them numericay. Before iustrating our resuts on a numerica eampe, et us see how and vary with uncertainty (σ), the cost of the project (I), and the ower barrier ( ) when ρ is fied. 7 Proposition 4. the project (I). However, Proof. See the appendi. increases with uncertainty (σ), and both decreases as increases whie and increases with the cost of increases with. Proposition 4 is compatibe with the findings of Brock, othschid, and Stigitz (1982). The investment threshod increases with the cost of the project (I) as the investor needs to wait onger to secure higher gains. Moreover, the investment threshod increases with uncertainty when the ower barrier is refecting because more uncertainty increases epected net gains. Likewise, a higher ower refecting barrier truncates the ow vaues of X from beow, thus increasing the epected net present vaue of the project; it is therefore optima to invest ater. However, a higher ower absorbing barrier increases the ikeihood that the project wi oose its vaue so the investor needs to act more swifty. Unfortunatey, it is not possibe to concude 15

17 anayticay how increases with σ, so we conduct a numerica investigation. Suppose here that I=$1, ρ=5% per year, and α=2% per year, so λ>1. We vary σ between 0 and 1.0 (the unit of σ is ( year) 0.5 ), and between $0.0 and $0.9 to see how and vary with these parameters. esuts are presented on Figures 1 to 2B., θ 1 = θ1 1 From Figure 1, we see that for reativey ow vaues of the voatiity (more generay, I, for negative vaues of κ when α>0), there is itte difference between,, and. But for higher vaues of σ (for κ (0,1) if α>0), the difference between these optima investment threshods can be substantia: when σ=0.4 for eampe, = $4.46, = $5.00, and = $6.09. These differences matter as they are captured in the present vaue of the investment opportunity (Figure 1B). For X(0)=$1.5, when κ=0, the present vaue of the investment opportunity for the absorbing case is ony 2.6% beow that of the unattainabe case (=0), and 5.4% above present vaue of the investment opportunity for the refecting case. However, when σ=0.4, these differences jump to 12.9% and +51.2% respectivey. The importance of the ocation of the ower barrier, in addition to its nature, is highighted in Figures 2 and 2B. When is reativey far from the optima threshod (which depends aso on the voatiity of X), the ower barrier has reativey itte impact on the investment decision ( $0.25 on Figure 2). s gets coser to I, however, its impact starts to be fet: for =$0.50 for eampe, = $3.50, and = $4.03 (for =0 here, = $3.72 ); these vaues change to $3.02 and $4.03 respectivey for =0.80. gain, these differences matter: when 16

18 =0.80 for eampe, the present vaue of the investment opportunity is 22.0% ower for absorbing and 74.7% higher for refecting compared to the case =0 (Figure 2B). In addition, a comprehensive numerica investigation around the parameter vaues seected did not revea a parameter combination that decreases when σ increases. This may not be the case if X foowed another process, as mentioned in Brock, othschid, and Stigitz (1982, page 42) who find that, in genera, the effect of an increase in the variance of X is ambiguous in the presence of an absorbing barrier. IV. Concusions Whie barriers are often assumed away in stochastic investment probems, this paper shows that barriers matter when uncertainty is high enough. We provide an intuitive methodoogy based on stochastic discount factors to derive simpe investments rues for autonomous diffusion process in the presence of common types of barriers. Using functionas anayzed in Karin and Tayor (1981, Chapter 15), this approach can easiy be etended to many other investment probems, incuding for eampe barriers with more compe payoffs or investments that modify a monetary fow. n iustration based on a canonica investment probem (a particuar case of MacDonad and Siege 1986) shows that investment rues based on the perpetua ca option may overestimate the investment threshod in the presence of a ower absorbing barrier and may underestimate the investment threshod with a ower refecting barrier. These resuts have impications for testing empiricay the theory of investment under 17

19 uncertainty. For eampe, empirica rea options modes appied to dataset that incude investment opportunities with refecting and absorbing barriers may yied inconcusive or biased resuts if the nature and ocation of different barriers is not accounted for. More generay, barriers may pay an important roe in the soution of stochastic investment probems when voatiity is high enough. Future work coud consider the impact of barriers on investment opportunities with time imits, anayze the interpay between barriers and discount rates, and revisit the pricing formuas of financia options when the underying is imited by a barrier. 18

20 ppendiequation Section (Net) Proof of Proposition 2. Let us first suppose α<0, so f( J ) < 0. From (23), θ 1 > 1, so im f( z) z + = ; since f ( z ) is inear, these 2 inequaities impy that f( z ) < 0 on (J,+ ); the same hods for F " ( z ), so ' F ( z ) is stricty decreasing on (J,+ ). From (28), ' F ( J ) > 0, and since ' im F ( z) = (as θ 1 > 1), F ( z ) first increases and then decreases on (J,+ ). From z + F ( J ) > 0 (see (28)) and im F ( z) = (again, θ 1 > 1; see (25)), we infer that F ( z ) has a unique zero on (J,+ ). z + Let us now suppose 0 α<ρ. From (23), θ 1 > 1, im f( z) z + =, but now f( J) 0, F " ( z ) starts positive on (J,+ ), and then becomes stricty negative. s a resut, ' F ( z ) first increases and then decreases towards - (from θ 1 > 1); as ' F ( J ) > 0 (see (28)), F ( z ) first increases and then decreases. From F ( J ) > 0 and im F ( z) z + Finay, suppose α ρ. gain f( J) 0 but now θ 1 1 so im f( z) =, we concude as above. z + =+, making both f ( z ) and F " ( z ) positive on (J,+ ). Hence, ' F ( z ) increases on (J,+ ) and it is stricty positive because ' F ( J ) > 0 (see (28)). Therefore, F ( z ) stricty increases on (J,+ ). F ( J ) > 0 (from (28)) then impies that F ( z ) has no zero on (J,+ ). Low discounting in this case just does not prevent the epected net present vaue of the project to keep on increasing. The same resuts hod for the refecting case, using the same ogic and " " F ( z) = F ( z). 19

21 Proof of Proposition 3. From Tabe 1, the sign of F aows us to compare and since F ( z ) > 0 for z ( J, ) and F ( z ) < 0 for z >. fter using that F = 0 to repace the higher powers of with a inear epression in, a itte agebra shows that F θ2 θ 1 θ = (1 θ2) + θ2j. Here 2 θ 1 θ2 θ2 > 0 ( θ 2 θ 1 < 0 and θ 2 < 0 ), and as > J > 1, (1 θ ) + θ J > (1 θ ) J + θ J = J > 0 so F > 0, and <. But from F ( / ) = 0, we have θ1 θ2 2 2J 1 1 (1 θ ) + θ = (1 θ ) + θ J so 2 θ2 (1 θ ) + J > 0 impies that 1 θ1 (1 θ ) + J > 0 and therefore θ 1 < θ1 1 I, which gives haf of (30). eca indeed that ρ>α impies that θ 1 >1. Finay, use F = 0 to repace the inear terms in with higher powers of in F to get F θ θ θ1 θ2 = (1 θ1) + θ1 θ1 1 2 J. The inequaity < 20

22 impies that F < 0, so 1 θ1 (1 θ ) + J < 0 and therefore θ θ I <. Let us now outine the comparative statics anaysis for the investment threshods with different types of barriers. We assume that λ >1 in order to have a soution. Proof of Proposition 4. To get started, et us eamine how θ 1 and θ 2 vary with v=σ 2 and ρ. Differentiating (21) gives 2 dθ θ ( θ 1) =, so that dv 2 vθ + 2ρ dθ2 dθ1 < 0, if ρ > α, > 0, dv dv 0, if ρ α. (.1) Likewise, dθ 2θ = and d ρ 2 vθ + 2ρ dθ1 dθ2 > 0, and < 0. (.2) dρ dρ To anayze how the investment threshod changes with one of the mode parameters, which we designate genericay by ω, we appy the impicit function theorem using either (25) or (27) to get a reationship that can be written Gzω (, ) = 0(with the appropriate subscript). Then d G G = dω ω 1. (.3) 21

23 is refecting. Let z / =. Here, G ' = F < 0 z from Tabe 1. Now et 2 v σ. We have G dθ1 dθ2 θ1 θ2 = n( z) z {[ 1 θ1] z + Jθ1} + v dv dv dθ θ 1 dθ θ [ ] dv 1 θ1 θ z z J + z J + z 2 θ2 dv θ2 θ2 1 1 θ1 θ2+ 1 θ1 θ2 From (30), (1 θ1) z + Jθ1 < 0, and since z J = 1 z Jz, θ2 θ1 z θ1 θ [ ] 1 z J θ z J z θ θ + + = < 0. With (.1), (.2), and (.3), we concude θ2 θ1 d 0. dv > (.4). Let us now consider the impact of. In this case, G G z = < 0 z since G ' = F < 0 z and z > 0. Moreover, after using that F ( z ) = 0, G 1 θ = 1 > 0. 1 [ θ2 θ1] {[ θ2 ] z θ2j} θ2 Hence, d d > 0. (.5) 22

24 Finay, et us consider dj. s above, G d < 0 G θ1 θ2 and now = θ 1 z 1 > 0, J so d 0. dj > (.6) is absorbing. Let z / = and 2 v σ. s above, G ' = F < 0 z from Tabe 1. We now have G dθ1 dθ2 θ1 θ2 = n( z) z {[ 1 θ1] z + Jθ1} + v dv dv dθ1 θ d 1 θ θ z + [ z J]. dv dv This time, however, [ θ ] 1 z + Jθ > 0 (see (30)), so the right hand side terms in factor of 1 1 dθ1 dθ2 are negative, whie the terms in factor of z J are positive. It is difficut to dv dv compare the two because of the term in n( z ). Let us now consider the impact of. Using the same arguments as before, G G z = < 0 z since again G ' = F < 0 z and z > 0. Moreover, G 1 θ1 θ2 = [ θ2 θ1] z {[ 1 θ1] z + θ1j} < 0, which eads to, 23

25 d d < 0. (.7) Finay, et us consider dj. s above, G d < 0 G θ1 θ2 and now = θ z θ > J 1 2 0, so d 0. dj > (.8) 24

26 eferences Brock, W.., M. othschid, and J. Stigitz, Stochastic Capita Theory. I. Comparative Statics, NBE Technica Paper No. 23, Diit,.K. (1993). The rt of Smooth Pasting (Chur: Switzerand: Harwood cademic Pubishers) Vo. 55 in Fundamentas of Pure and ppied Economics, eds. Jacques Lesourne and Hugo Sonnenschein. Diit,.K., Pindyck,.S. (1994). Investment Under Uncertainty (Princeton University Press). Diit,.K., Pindyck,.S., Søda, S. (1999). markup interpretation of optima investment rues. The Economic Journa, 109 (pri), Dumas, B. (1991). Super contact and reated optimaity conditions. Journa of Economic Dynamics and Contro 15, Karin, S., Tayor, H. M. (1981). Second Course in Stochastic Processes (San Diego, C: cademic Press). McDonad,., Siege, D. (1986). The vaue of waiting to invest. Quartery Journa of Economics 101, Søda, S. (1998). simpified epression of smooth pasting. Economic Letters 58,

27 Tabe 1: Variations for F (z) and F (z). α<0 (so θ 1 >1) 0 α<ρ (so θ 1 >1) Z J + Z J + F (z) - aways - - F (z) + + then - - F (z) + - F (z) + - F(z) + - F(z) + - α ρ (so θ 1 1) Z J + F (z) + aways F (z) + - F(z) + Note: these resuts appy to both F (z) and F (z) so the reevant subscript is omitted. 26

28 $8.00 refecting Optima Investment Threshod $6.00 $4.00 $2.00 =0 absorbing $ σ Figure 1. Optima Investment Threshod as a function of σ. 27

29 $2.00 Vaue of Investment Opportunity $1.50 $1.00 $0.50 refecting absorbing =0 $ σ Figure 1B. Vaue of Investment Opportunity as a function of σ. Notes for Figures 1 and 1B: these resuts were generated using I=$1.0 (the cost of the investment), α=2% per year (the epected rate of growth of the investment), =$0.5 (for the absorbing and refecting cases), and ρ=5% per year (the discount rate). The voatiity coefficient, σ, is in ( year) 0.5. The vaue of the investment opportunity is cacuated at y=x(0)=$

30 $6.00 refecting Optima Investment Threshod $4.00 $2.00 =0 absorbing $- $- $0.25 $0.50 $0.75 $1.00 Figure 2. Optima Investment Threshod as a function of. 29

31 $1.50 refecting Vaue of Investment Option $1.00 $0.50 =0 absorbing $- $- $0.25 $0.50 $0.75 $1.00 Figure 2B. Vaue of Investment Opportunity as a function of. Notes for Figures 2 and 2B: these resuts were generated using I=$1.0 (the cost of the investment), α=2% per year (the epected growth rate of the investment), σ=0.3 (in ( ) 0.5 year ), and ρ=5% per year (the discount rate). The vaue of the investment opportunity is cacuated at y=x(0)=$

32 1 Brock, othschid, and Stigitz (1982) anayze what the standard tree cutting probem. Their comprehensive anaysis appies to genera stochastic processes but it reies on advanced mathematics and on what has become the rea options methodoogy; in addition, they assume the eistence of a singe stopping vaue and they do not anayze the GBM case. By contrast, the anaysis herein presents an aternate approach that reies ony on eementary mathematica toos, and it proves the eistence of a singe stopping vaue for the GBM with barriers. 2 For other types of barriers, see Dumas 1991 or Diit parenthesis means that an interva is open at that end, whie a square bracket means that it is cosed. Thus (a, b] incudes b but not a. 4 This is ceary a strong assumption but the presence of an attainabe barrier makes it difficut to use the Capita sset Pricing Mode to find the nondiversifiabe risk of the investment opportunity. Technica difficuties may detract us from our goa, i.e. anayzing the impact of a barrier on the decision to invest in a simpe framework. 5 Diit, Pindyck and Søda (1999) dea ony with an unattainabe ower barrier. 6 It corresponds to - for n(x), which foows a Brownian motion, and we know that diffusions have ony finite variations in finite time. 7 s discussed in Diit and Pindyck (1984, page 150), assuming that σ varies independenty of other parameters (such as the discount rate ρ or the epected rate of growth α of X) is often not very satisfactory. For simpicity, we adopt this assumption here because we are interested in possibe impacts of barriers on the decision to invest. Note, however, that the discount rate is typicay mandated eogenousy for pubic projects. 31