Journal of Economics

, Volume 110, Issue 1, pp 5–23 | Cite as

Irreversible exit decisions under mean-reverting uncertainty

Article

Abstract

Although many economic variables of interest exhibit a tendency to revert to long-run levels, mean reverting processes are rarely used in investment and disinvestment models in the literature. Previous work by Sarkar (J Econ Dyn Control 28(2):377–396, 2003), that focuses on irreversible entry decisions, showed that mean reversion has three effects on investment: (a) the “variance effect” (mean reversion reduces the long-run uncertainty and thus brings closer the critical investment level), (b) the “realized price effect” (the lower variance resulting from mean reversion makes it less likely to reach extreme high or low price levels, thereby reducing the likelihood of reaching the investment trigger) and (c) the “risk discounting effect” (mean reversion lowers the required rate of return, which affects both the project value and the value of the real option to invest). Metcalf and Hassett (J Econ Dyn Control 19(8):1471–1488, 1995) and Sarkar (J Econ Dyn Control 28(2):377–396, 2003) showed that (a) and (b) work in opposite directions, essentially canceling each other out, however the effect of (c) depends on parameter values, making the overall effect (a–c) of mean reversion on entry decisions ambiguous and parameter-dependent. In this paper, we show that as far as irreversible exit decisions are concerned, the effect of mean reversion is negative: Mean reversion unambiguously lowers the rate of irreversible disinvestment/exit for reasonable parameter values, since the mean reversion in this case only affects the value of the real option to exit and not the value resulting from (real) option exercise.

Keywords

Investment Uncertainty Real options Mean reversion 

JEL Classification

C61 E22 

1 Introduction

Virtually all investment and disinvestment decisions are made in an uncertain environment and some of them are impossible to reverse if economic conditions change unexpectedly. As early as the works of Robichek and Van Horne (1967, 1969), Dyl and Long (1969), Bonini (1977) and Myers and Majd (1990) researchers have shown that the option to completely abandon a project or irreversibly exit a market has significant value which substantially affects the desirability and the return on investment of the venture. Berger et al. (1996) were among the first that provided empirical evidence that investors actually value abandonment or exit options when assessing going—concern corporations.

Apart from the macabre example of suicide as the ultimate irreversible exit decision [mentioned in the seminal book of Dixit and Pindyck (1994)], there are several other important economic and societal decisions that are characterised by irreversibility. At the societal level, in the context of health economics, Mahul and Gohin (1999) analyse the irreversible decision made by French health authorities to slaughter and dispose infected animal herds at the outbreak of the highly contagious foot-and-mouth disease back in the 1990s.

At the microeconomic level, firms decide to close down producing plants (see the empirical investigations in Kovenock and Phillips 1997; Anderson et al. 1998), drop brands from their portfolio of consumer products in the face of demand uncertainty (Hitsch 2006) or leave oil extraction licences expire “unexercised” due to low fuel prices and high fixed costs (e.g. see Favero et al. (1994), for empirical evidence of irreversibilities involved in oil investments). At the international level, several researchers have provided empirical evidence that firms exit foreign markets in the face of fluctuating exchange rates and sunk-cost induced hysteresis (see Requena-Silvente 2005; Bernard and Wagner 2001; Harris and Li 2010, 2011, among others).1

A research stream that is known as the “real options” approach to capital investment (see Dixit and Pindyck 1994; Trigeorgis 1996, for an overview) has provided decision-makers with a plethora of optimal rules for irreversible (or partly reversible at a cost) investment and disinvestment decisions in the face of ongoing uncertainty, mainly by using the tractable geometric Brownian motion to model uncertain cash flow streams, prices, asset values or exchange rates (see McDonald and Siegel 1985, 1986; Brennan and Schwartz 1985; Pindyck 1988; Dixit 1989; Bentolila and Bertola 1990; Myers and Majd 1990, etc.). However, although the usually-employed geometric Brownian motion offers (dis)investment decision rules that are simple and intuitive, several researchers have noted that it is not a plausible price process under equilibrium conditions (see Bhattacharya 1978; Lund 1993; Bessembinder et al. 1995), and that mean-reverting processes should be preferred, for sound economic reasons.2

Despite this, very few studies have offered optimal investment and disinvestment decision rules for firms facing mean-reverting price uncertainty. Notable exceptions are the studies by Metcalf and Hassett (1995), Sarkar (2003) and Tsekrekos (2010). The first two studies concentrate on irreversible entry, while the last study focuses on costly (reversible) entry and exit decisions under mean-reverting uncertainty.

The papers by Metcalf and Hassett (1995) and Sarkar (2003), that focus on irreversible entry decisions, collectively establish that mean reversion has three effects on investment: (a) the “variance effect” (mean reversion reduces the long-run uncertainty and thus brings closer the critical investment level), (b) the “realized price effect” (the lower variance resulting from mean reversion makes it less likely to reach extreme high or low price levels, thereby reducing the likelihood of reaching the investment trigger) and (c) the “risk discounting effect” (mean reversion lowers the required rate of return, which affects both the project value and the value of the real option to invest). Metcalf and Hassett (1995) and Sarkar (2003) showed that (a) and (b) work in opposite directions, essentially canceling each other out, however the effect of (c) depends on parameter values, making the overall effect (a–c) of mean reversion on entry decisions ambiguous and parameter-dependent. This findings are confirmed by Tsekrekos (2010), that focuses on (costly) reversible entry/exit decisions. It is shown there that the overall effect of mean reversion is significant but ambiguous in direction (depending on parameter values) on reversible entry/exit, much like in Sarkar (2003) and irreversible entry.

This paper supplements this short list by focusing on the effect of mean reversion on irreversible exit decisions, which are clearly relevant in real-world applications and are empirically investigated by researchers. Irreversible exit/abandonment decisions, are also interesting and different from the above studies of Metcalf and Hassett (1995), Sarkar (2003) and Tsekrekos (2010) for the following reason: when there is (costly or not) reversibility or irreversible entry, the firm is affected by mean reversion both before and after “real options” exercise. Both the real option and the “underlying asset” that is earned after exercise are affected by the mean-reverting process. In contrast, when a firm is contemplating irreversible exit, by exercising its put (real) option to exit the firm completely and irreversibly abandons the market, and its position is no longer affected by mean reversion. This, as shown in this paper, makes a difference in the overall effect of mean reversion on firm decisions. We demonstrate that mean reversion unambiguously has a negative effect on irreversible exit decisions, when compared to the usually-employed geometric Brownian motion for reasonable parameter values. This is in contrast to the costly reversible exit case treated in Tsekrekos (2010) or the irreversible entry case treated in Sarkar (2003), where depending on parameter values the effect of mean reversion on (dis)investment can be positive or negative.

The rest of the paper is organized as follows: The next section presents the assumptions needed and describes the basic model setting. Section 3 derives and analyses the optimal disinvestment firm policy under mean reversion. Section 4 compares the findings under a mean-reverting price process with firm behavior under a geometric Brownian motion, and Sect. 5 concludes.

2 Basic setting and assumptions

In order to formulate the problem, assume that a firm operates in a market by having made in the past a discrete unit of investment: Namely, it has initiated a single project of a given fixed size that produces one unit flow of output per period (a simple normalization). The firm incurs a flow cost of \(c\) per period of time to produce the unit of output, and it can earn an equilibrium price \(P\) for it. The firm can at any point in time decide to abandon operations by incurring a lump sum exit cost \(K\). Such a decision to exit is completely irreversible, and takes place instantaneously.3 The magnitudes \(K, c\) and the risk-free interest rate \(r\) are constant and non-stochastic. The uncertainty arises from the output equilibrium price \(P\) which is exogenous to the firm (i.e. the firm is a price-taker).

In order to keep the exposition as close as possible to models that treat the case of entry or the costly reversible case (Dixit 1989; Sarkar 2003; Tsekrekos 2010), the firm is neither allowed to alter the scale of the investment at will, nor is its profit flow dependent on the installed capital stock.4

In the standard “real options” terminology, this setting corresponds to the firm having, while active in the market, a perpetual (American-style) put option to exit. If the output equilibrium price \(P\) is assumed to follow a geometric Brownian motion,
$$\begin{aligned} dP=\mu P dt+\sigma P dz \end{aligned}$$
(1)
the optimal exit policy of the firm is described by a time-independent trigger price, \(P_{L}\), with exit being optimal the first time that \(P_{L}\) is reached from above. It can be shown that in the presence of fixed exit costs
$$\begin{aligned} P_{L}<c-rK\equiv W_{L}, \end{aligned}$$
(2)
where \(W_{L}\) is the Marshallian (certainty case, \(\sigma \rightarrow 0\)) abandonment/exit trigger prices.5

As discussed in the introduction, our attention in this paper is on the effect of mean reversion on the optimal irreversible exit/abandonment decisions of firms. Previous work by Metcalf and Hassett (1995) and Sarkar (2003) that focused on irreversible entry has reached contrasting conclusions regarding the effect of mean reversion.

Following Sarkar (2003) and Tsekrekos (2010), I assume that the firm has to optimally decide its irreversible exit from the market in the face of the uncertain (and exogenous) mean-reverting equilibrium output price \(P\) that evolves according to the following process:
$$\begin{aligned} dP=\kappa \left( \theta -P\right) dt+\sigma P dz \end{aligned}$$
(3)
where \(\kappa \) is the speed of reversion, \(\theta \) the long-run mean price level, \(\sigma \) is the volatility of the process, and \(dz\) is the increment of a standard Brownian motion process (see also Bhattacharya 1978; Sick 1995, for models that employ this process).

Employing this mean-reverting process makes the findings of this paper regarding irreversible exit directly comparable to the decision rules for irreversible entry provided by Sarkar (2003). Moreover, as Sarkar (2003) and Tsekrekos (2010) establish, out of the alternative mean-reverting processes that have been used in the literature, the one in (3) is the most plausible as an equilibrium output price due to the homogeneity of degree one of its drift and diffusion terms in \(\left( P,\theta \right) \).6 Clearly, for \(\theta =0\) Eq. (3) becomes the geometric Brownian motion in (1) with \(\mu =-\kappa \), while for \(\kappa =0\) it becomes a geometric Brownian motion process with no drift. It is easy to verify that the price process in (3) reverts to its long-run mean in the limit, i.e. \(\mathbb E \left[ P_{\infty }\left. \right| P_{0}\right] =\lim _{t\rightarrow +\infty }\mathbb E \left[ P_{t}\left. \right| P_{0}\right] =\theta \) and has a finite conditional variance \(\mathbb V \left[ P_{\infty }\left. \right| P_{0}\right] =\lim _{t\rightarrow +\infty }\mathbb V \left[ P_{t}\left. \right| P_{0}\right] =\frac{\theta ^{2}\sigma ^{2}}{2 \kappa -\sigma ^{2}}\) that is decreasing in the speed of mean reversion \(\kappa \) and increasing in the long-run mean level \(\theta \).

3 Optimal irreversible exit under mean reversion

3.1 Deriving the optimal exit policy

Let the equilibrium output price follow the process in (3), and let \(\left( {\underline{P}},+\infty \right) \) denote the range of output prices over which the firm finds it optimal to operate in the market. Denote \(V\left( P\right) \) the expected net present value of the firm, starting with a price \(P\) in the operating state, and following the optimal exit strategy.

Over the range of prices \(\left( {\underline{P}},+\infty \right) \) where it is optimal for an active firm to continue in this state, the total return of the expected net present value of the firm, \(V\left( P\right) \), comprises of the expected capital gain \(\mathbb E [dV\left( P\right) ]/dt\), plus a cash inflow \(\left( P-c\right) \) per unit of time.

A simple application of Itô’s lemma yields
$$\begin{aligned} dV\left( P\right) +\left( P-c\right) dt&= V^{\prime }\left( P\right) dP+\frac{1}{2}V^{\prime \prime }\left( P\right) \left( dP\right) ^{2}+\left( P-c\right) dt\nonumber \\&= \left[ \frac{1}{2}\sigma ^{2}P^{2}V^{\prime \prime }\left( P\right) +\kappa \left( \theta -P\right) V^{\prime }\left( P\right) +P-c\right] dt \nonumber \\&+\sigma P V^{\prime }\left( P\right) dz \end{aligned}$$
(4)
From (4), the expected return and the standard deviation of the return of the active firm are
$$\begin{aligned} \mathbb{E }\left[ R\right]&= \frac{\mathbb{E }[dV\left( P\right) +\left( P-c\right) dt]/dt}{V\left( P\right) }\nonumber \\&= \frac{\frac{1}{2}\sigma ^{2}P^{2}V^{\prime \prime }\left( P\right) +\kappa \left( \theta -P\right) V^{\prime }\left( P\right) +P-c}{V\left( P\right) } \end{aligned}$$
(5)
and
$$\begin{aligned} \sigma \left( R\right) =\frac{\sigma P V^{\prime }\left( P\right) }{V\left( P\right) } \end{aligned}$$
(6)
respectively. From the intertemporal Capital Asset Pricing Model (CAPM) of Merton (1973), firm value must satisfy the following risk-return relationship
$$\begin{aligned} \mathbb E \left[ R\right] =r+\lambda \rho \sigma \left( R\right) =r+\lambda \rho \frac{\sigma P V^{\prime }\left( P\right) }{V\left( P\right) }, \end{aligned}$$
(7)
where \(\lambda =\frac{\mathbb{E }\left[ R_{m}\right] -r}{\sigma \left( R_{m}\right) }\) is the market price of risk (with \(\mathbb E \left[ R_{m}\right] \) and \(\sigma \left( R_{m}\right) \) the expected return and standard deviation respectively, of the return of the market portfolio, \(R_{m}\)), and \(\rho \) the correlation of changes in \(P\) with the market portfolio (i.e. \(dzdz_{m}=\rho dt\)). Both \(\lambda \) and \(\rho \) are assumed constant.
Substitute \(\mathbb E \left[ R\right] \) from (5) in Eq. (7), multiply both sides with \(V\left( P\right) \) and rearrange to get
$$\begin{aligned} \frac{1}{2}\sigma ^{2}P^{2}V^{\prime \prime }\left( P\right) +\left( \kappa \left( \theta -P\right) -\lambda \rho \sigma P\right) V^{\prime }\left( P\right) -rV\left( P\right) =c-P, \end{aligned}$$
(8)
i.e. the ordinary differential equation that the value of the operating firm must satisfy over the range of output prices \(P\) that it is optimal to remain active in the market.
The ordinary differential Eq. in (8) must be solved subject to the boundary condition
$$\begin{aligned} \underset{P\rightarrow +\infty }{\lim }V\left( P\right)&= \mathbb E \left[ \int \limits _{0}^{+\infty }\left( P\left( s\right) -c\right) e^{-rs}ds\right] \nonumber \\&= \frac{P}{r+\kappa +\lambda \rho \sigma }+\frac{\kappa \theta }{r\left( \kappa +\lambda \rho \sigma \right) }-\frac{\kappa \theta }{\left( r+\kappa +\lambda \rho \sigma \right) \left( \kappa +\lambda \rho \sigma \right) }-\frac{c}{r}\nonumber \\ \end{aligned}$$
(9)
that rules out explosive growth of firm value with high output prices. This “no-bubbles” condition essentially implies that for high equilibrium output prices, the option to exit becomes worthless and \(V\left( P\right) \) converges to the expected present value of operating in the market forever, starting from an initial price \(P\).7
As shown in Appendix, the solution of (8) subject to (9) is given by
$$\begin{aligned} V\left( P\right)&= BM\left( -\gamma ,b,\frac{2\kappa \theta }{\sigma ^{2} P}\right) P^{\gamma } +\frac{P}{r+\kappa +\lambda \rho \sigma }+\frac{\kappa \theta }{r\left( \kappa +\lambda \rho \sigma \right) }\nonumber \\&-\frac{\kappa \theta }{\left( r+\kappa +\lambda \rho \sigma \right) \left( \kappa +\lambda \rho \sigma \right) }-\frac{c}{r} \end{aligned}$$
(10)
where \(M\left( a,b,x\right) \) is Kummer’s confluent hypergeometric function (see Abramowitz and Stegun 1972), \(B\) is a constant to be determined by boundary conditions, \(\gamma \) is the negative root of the quadratic equation
$$\begin{aligned} \frac{1}{2}\sigma ^{2}\gamma \left( \gamma -1\right) -\left( \kappa +\lambda \rho \sigma \right) \gamma -r=0, \end{aligned}$$
(11)
given by
$$\begin{aligned} \gamma =\frac{2\left( \kappa +\lambda \rho \sigma \right) +\sigma ^{2}-\sqrt{8r\sigma ^{2}+\left( -2\kappa -2\lambda \rho \sigma -\sigma ^{2}\right) }}{2\sigma ^{2}} \end{aligned}$$
(12)
and \(b=2-2\gamma +\frac{2\left( \kappa +\lambda \rho \sigma \right) }{\sigma ^{2}}\).

The second line of (10) corresponds to the expected present value of operating in the market forever, starting from an initial price \(P\), while the first line corresponds to the firm’s option to exit the market if economic conditions become extremely unfavorable. It is easy to verify that while seemingly complex, the value function of the active firm under mean-reverting prices in Eq. (10) collapses to the geometric Brownian motion case, for \(\kappa =0\) (or \(\theta =0\)), since from the properties of the confluent hypergeometric function, we know that \(M\left( a,b,0\right) =1\).

To complete the solution, the optimal exit price threshold, \({\underline{P}}\), along with the constant \(B\) are determined by a standard value-matching
$$\begin{aligned} V\left( {\underline{P}}\right) +K=0 \end{aligned}$$
(13)
and a smooth-pasting condition
$$\begin{aligned} V^{\prime }\left( {\underline{P}}\right) =0 \end{aligned}$$
(14)
that apply at \({\underline{P}}\). At the optimal exit price level \({\underline{P}}\) the active firm exercises its exit option by paying \(K\) in order to abandon the project and exit the market.

The above boundary conditions highlight the difference between the effect of mean reversion on irreversible entry (or reversible exit, see Sarkar 2003; Tsekrekos 2010) and the irreversible exit case that we treat here. In boundary conditions (13) and (14), mean reversion only affects the value of the real option to exit and not the value resulting from (real) option exercise (i.e. the right-hand side of the boundary conditions). In contrast, when a firm is contemplating irreversible entry (Sarkar 2003) or reversible exit (Tsekrekos 2010), both sides of the relevant boundary conditions are affected by the mean reversion parameters \(\kappa \) and \(\theta \). As demonstrated in the sections that follow, this makes a significant difference, making the overall effect of mean reversion on irreversible firm exit decisions unambiguously negative for reasonable parameter values.

Equations (13)–(14) constitute a system that uniquely determines the optimal irreversible exit threshold, \({\underline{P}}\) and the constant \(B\). Substituting the value function (10) in the system of Eq. (13)–(14), it can be shown that the optimal solution \(\mathbf X =\left[ {\underline{P}},B\right] ^{\prime }\) is uniquely determined by solving the non-linear equation system
$$\begin{aligned} \mathbf F \left( \mathbf X \right) =\mathbf 0 \end{aligned}$$
(15)
where
$$\begin{aligned}&\displaystyle \mathbf F \left( \mathbf X \right) =\left[ \begin{array}{c} Q+M\left( {\underline{P}}\right) \underline{P}^{\gamma }B+\eta {\underline{P}}\\ W\left( {\underline{P}}\right) \underline{P}^{\gamma }B+\eta {\underline{P}} \end{array}\right] ,&\end{aligned}$$
(16)
$$\begin{aligned}&\displaystyle \xi \equiv 2\kappa \theta /\sigma ^{2}, \quad \eta \equiv -\left( r+\kappa +\lambda \rho \sigma \right) ^{-1},&\end{aligned}$$
(17)
$$\begin{aligned}&\displaystyle M\left( x\right) =M\left( -\gamma ,b,\frac{\xi }{x}\right)&\end{aligned}$$
(18)
$$\begin{aligned}&\displaystyle W\left( x\right) =\gamma \left( M\left( x\right) +\frac{\xi }{bx}M\left( 1-\gamma ,1+b,\frac{\xi }{x}\right) \right)&\end{aligned}$$
(19)
and
$$\begin{aligned} Q=\frac{\kappa \theta }{r\left( \kappa +\lambda \rho \sigma \right) }-\frac{\kappa \theta }{\left( r+\kappa +\lambda \rho \sigma \right) \left( \kappa +\lambda \rho \sigma \right) }-\frac{c}{r}+K \end{aligned}$$
(20)
For any given set of parameter values, the system equation in (15) can be easily inverted for \(\mathbf X =\left[ {\underline{P}},B\right] ^{\prime }\).

3.2 Analysis of solution

As a means of analysis of the optimal exit/abandonment policy under mean reversion, one can verify that as \(K\) tends to zero, the exit threshold \({\underline{P}}\) collapses to the level of the variable cost flow \(c\). In other words, in the absence of sunk costs, the firm will only exit whenever \(P<c\). Moreover, as \(\sigma \rightarrow 0\), \({\underline{P}}\rightarrow W_{L}\): the irreversible exit threshold under mean reversion converges to the Marshallian trigger in the absence of uncertainty. This can be seen in Fig. 1, where \(\frac{\underline{P}}{W_{L}}\), the optimal decision threshold under mean-reverting output prices, scaled by the corresponding Marshallian (certainty) trigger (Eq. (2)) is plotted as a function of volatility, for different levels of \(\kappa \), as well as for the case of a geometric Brownian motion with no drift, \(\kappa =0\).
Fig. 1

The ratio of the optimal irreversible exit threshold \({\underline{P}}\) under the mean-reverting price process in (3), over the corresponding Marshallian threshold, \(W_{L}\), as a function of volatility, \(\sigma \), for different speed of mean reversion levels \(\kappa \). The rest of the parameters are \(\theta =1\), \(r=0.04\), \(\rho =1\), \(\lambda =0.4\), \(K=2\) and \(c=2\)

Figure 2 plots \(V\left( P\right) \), the value of the firm, when the equilibrium output price reverts to two different levels in the long-run (a low one, \(\theta =1\), and a high one, \(\theta =3\)) at different speeds, \(\kappa \). It should be noted that the higher the level towards which the price process reverts, the less valuable the firm’s investment option to exit. Similarly, the stronger the mean reversion towards a high (low) long-run level, the less (more) valuable the firm’s abandonment option. When operating in the market is marginally profitable (\(\theta =1\), upper panel of Fig. 2), the higher the mean reversion speed is, the lower the value of the firm. In contrast, high mean reversion when the firm project is fairly profitable (high \(\theta \), lower panel of Fig. 2) increases firm value substantially.
Fig. 2

The value of the firm \(V\left( P\right) \) in Eq. (10), as a function of the output price \(P\) over the range \(\left( {\underline{P}},+\infty \right) \), for different levels of mean reversion speed \(\kappa \). For \(P\le {\underline{P}}\), \(V\left( P\right) =0\). The bold curve corresponds to the special case of a lognormally distributed output price (\(\kappa =0\)). The rest of the parameters are \(r=0.04,\rho =1,\lambda =0.4,c=2,\sigma =0.15\) and \(K=2\)

When active in the market, firm value consists of (a) the present value of operating cash flows and (b) a put option to exit the market. When mean reversion is introduced (i.e. as \(\kappa \) increases, say from 0 to 0.10 in terms of Fig. 2), the reduced variance of the price process decreases the put option to exit (the “variance effect”) and the reduced systematic risk increases the present value of operating in the market (the “risk discounting” effect). Which of the two effects dominates will determine whether the value of the firm \(V\left( P\right) \) will increase or decrease as \(\kappa \) increases. Evidently from Fig. 2, for higher long-run mean equilibrium prices it is the “risk discounting” effect that dominates.8

The effect of mean reversion on the optimal irreversible exit threshold is better demonstrated in Fig. 3 that plots \({\underline{P}}\) as a function of the speed of mean reversion \(\kappa \) for different long-run mean price levels \(\theta \). Recall that from the three effects of mean reversion identified in the literature and outlined in the introduction, only the “variance” and the “risk discounting” ones affect the level of the exit threshold; the “realized price” effect simply reduces the likelihood of reaching this threshold.
Fig. 3

The optimal irreversible exit threshold, \({\underline{P}}\), under the mean-reverting price process in (3) as a function of the speed of mean reversion, \(\kappa \), for different long-run mean price levels \(\theta \). The rest of the parameters are \(r=0.04\), \(\rho =1\), \(\lambda =0.4\), \(c=2\), \(\sigma =0.15\) and \(K=2\)

When the “risk discounting effect” (mean reversion reduces the systematic risk of the firm, thus firm cash flows are discounted at a lower rate) dominates the “variance effect” (mean reversion reduces long-run uncertainty, bringing closer abandonment price triggers), irreversible exit under mean reversion is postponed in comparison to the geometric Brownian motion case. This is seen in Fig. 3 as \(\kappa ,\theta \) increase: quickly reverting to a higher long-run equilibrium price makes irreversible exit more unlikely, decreasing the price threshold levels that must be reached before the firm optimally abandons the market.

Bearing in mind that the “realized price” effect reinforces this optimal postponement, by making it less likely that higher entry and lower exit thresholds are reached, one can conclude that mean reversion has an unambiguous negative effect on irreversible investment exit when the “risk discounting effect” dominates.

The “risk discounting effect” is relatively more important when interest rates are low and for long-lived investment projects with high intrinsic values (high \(\theta \), low \(K\)), one can summarize this section in the following:

Irreversible exit:Compared to the usually—employed geometric Brownian motion price process, mean reversion unambiguously decreases irreversible exit when the risk-free interest rate is low and when long-lived, high intrinsic value projects are considered.

The above confirms that the findings of Sarkar (2003) and Tsekrekos (2010) on the effect of mean reversion on irreversible or costly reversible entry are also valid for irreversible investment exit decisions.

4 The effect of mean reversion on irreversible exit decisions under uncertainty

In order to disentangle the net impact of the “variance”, the “realized price” and the “risk discounting” effects of mean reversion, one could examine whether mean reversion increases or decreases disinvestment and abandonment rates on an aggregate industry level, compared to the geometric Brownian motion case.

Of course answering such a question would require one to move to a general (rather than a partial) equilibrium model, where the price process should be endogenized as part of the resulting equilibrium. In spite of how interesting on its own such a research question is, it goes beyond the scope of this paper.

Here we will examine the effect of mean reversion in a simplistic manner by computing the ex-ante probability of exit/abandonment within a specified time horizon for an active firm. This approach has also been used in related papers by Sarkar (2003) and Tsekrekos (2010), thus adopting it here makes the findings of this paper directly comparable to previous research. As Tsekrekos (2010) points out, “[t]he rationale for using these probability measures is that although [...the optimal exit policy...] was derived in a single-firm setting, it happens to coincide with the optimal policy of firms under general equilibrium in a perfectly competitive industry/market, as shown byLeahy (1993)” (Tsekrekos 2010, p. 734).

An immediate implication is that if there is a large pool of active firms, individually contemplating exit from an industry/market that is characterized by the exogenous equilibrium mean-reverting price process in (3), then the ex-ante probability that a single firm will exit by time \(T\) will be a measure of the fraction of active firms, under competitive equilibrium, that will abandon the market by this time horizon. Lower (higher) probability of exit means a smaller (larger) fraction of active firms will disinvest and optimally abandon the market, i.e. mean reversion has a positive (negative) effect on investment. This way, the net impact of all three effects of mean reversion (the “variance”, the “realized price” and the “risk discounting” ones) can be examined and compared to the log-normally distributed price process (geometric Brownian motion).

The ex-ante probability that an active firm will exit the market by time \(T\) is equal to \(\Pr \left( {\underline{\tau }}\le T\right) \), with \({\underline{\tau }}=\inf \{t \ge 0 :P_{t} \le {\underline{P}}\}\) the first passage time of process (3) from level \({\underline{P}}\).

For the geometric Brownian motion process in (1) and the optimal exit threshold \(P_{L}\), these probabilities can be calculated in closed-form:
$$\begin{aligned} \Pr \left( \tau _{L} \le T\right)&= \Phi \left[ \frac{\ln \left( \frac{P_{L}}{P_{0}}\right) -\left( \mu -\frac{\sigma ^{2}}{2}\right) T}{\sigma \sqrt{T}}\right] \nonumber \\&+\left( \frac{P_{0}}{P_{L}}\right) ^{\frac{2 \mu }{\sigma ^{2}}-1} \Phi \left[ \frac{\ln \left( \frac{P_{0}}{P_{L}}\right) +\left( \mu -\frac{\sigma ^{2}}{2}\right) T}{\sigma \sqrt{T}}\right] \end{aligned}$$
(21)
where \(\Phi \left( .\right) \) is the standard normal cumulative distribution function, \(P_{0}\) the (known with certainty) current level of the price process in (1) and \(\tau _{L}=\inf \{t \ge 0 :P_{t} \le P_{L}\}\) (see Harrison 1985).
Unfortunately, for the mean-reverting price process in (3) the ex-ante probability of exit has to be computed numerically. We follow the procedure in ((Tsekrekos 2010, Appendix E)), that outlines how this probability can be computed via an implicit finite-difference approximation scheme or Monte–Carlo simulation. Table 1 reports estimates of the probabilities of exit, \(\Pr \left( {\underline{\tau }}\le T\right) \), for \(T=10\) years and a wide range of parameter values. In all reported estimates, the starting value \(P_{0}\) is set equal to \(P_{0}=\left( {\underline{P}}+1\right) \frac{\theta \sigma }{\sqrt{2\kappa -\sigma ^{2}}}\), i.e. one conditional standard deviation above the exit threshold \({\underline{P}}\).
Table 1

For a market with an equilibrium output price described by the mean-reverting process in Eq. 3, the table reports the probability that an active firm will optimally exit the market by time \(T\), calculated via Monte Carlo simulation

 

\(\theta = 0.40\)

\(\theta = \theta ^*\)

\(\theta = 1.40\)

 

\(\sigma = 0.10\)

\(\sigma = 0.15\)

\(\sigma = 0.25\)

\(\sigma = 0.10\)

\(\sigma = 0.15\)

\(\sigma = 0.25\)

\(\sigma = 0.10\)

\(\sigma = 0.15\)

\(\sigma = 0.25\)

Probability of exit, Panel (a):\(\rho = 1.0\)

\(\kappa = 0\)

   

1.0000

1.0000

1.0000

   

\(\kappa =0.05\)

0.7416

0.6608

0.5882

0.7088

0.6956

0.6996

0.0234

0.0720

0.1348

\(\kappa =0.10\)

0.9446

0.8590

0.7694

0.8914

0.8512

0.8198

0.0000

0.0144

0.1054

\(\kappa = 0.15\)

0.9908

0.9508

0.8586

0.9688

0.9444

0.9934

0.0000

0.0000

0.0344

Probability of exit, Panel (b):\(\rho = 0.0\)

\(\kappa = 0.05\)

0.7264

0.6366

0.5522

0.1500

0.2228

0.3586

0.0000

0.0000

0.0046

\(\kappa = 0.10\)

0.9394

0.8434

0.7314

0.0426

0.0838

0.2016

0.0000

0.0000

0.0000

\(\kappa = 0.15\)

0.9908

0.9462

0.8334

0.0048

0.0132

0.0702

0.0000

0.0000

0.0000

Probability of exit, Panel (c):\(\rho = -0.5\)

\(\kappa = 0\)

   

1.0000

1.0000

1.0000

   

\(\kappa = 0.05\)

0.7038

0.5798

0.4010

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

\(\kappa = 0.10\)

0.9348

0.8234

0.6450

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

\(\kappa = 0.15\)

0.9500

0.9392

0.7878

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

In each simulation, 100,000 paths and 100 time steps per year are used. \(\theta \) is the long-run output price level, \(\kappa \) is the speed of mean reversion, \(\sigma \) is the volatility of the output price process, \(\rho \) is the correlation between the equilibrium output price and the market portfolio and \(\theta ^*\) as explained in the text. In all cases, the initial price of the simulation is set to \(P_0=({\underline{P}}+1) \frac{\theta \sigma }{\sqrt{2\kappa -\sigma ^2}}\), i.e. one conditional standard deviation above the exit threshold \({\underline{P}}\). The time horizon is \(T = \)10 years, the market price of risk is \(\lambda =0.4\), the variable flow cost is \(c = 1\), and the exit sunk cost is \(\kappa = 2\) respectively. Under \(\kappa = 0\), the same probability under the geometric Brownian motion price process in (1), with \(\mu =0\) are reported. These are calculated via Eq. (21) and do not depend on \(\theta \)

The parameter values examined in this numerical investigation are in the same spirit as the ones in Tsekrekos (2010). Three \(\theta \) scenarios are selected: one (\(\theta =0.4\)) that makes the intrinsic value of operating in the market low, and in which we expect the “risk discounting effect” to be dominated by the “variance effect”, and one (\(\theta =1.4\)) where the exact opposite is true. Much like in Tsekrekos (2010), the third set of scenarios (under \(\theta =\theta ^{\star }\)) attempts to “neutralize” the opposing effects of “risk discounting” and “variance”, by setting \(\theta ^{\star }=\frac{rP_{0}}{r+\lambda \rho \sigma }\) for each different triplet \(\left( \kappa ,\rho ,\sigma \right) \) examined. It is not difficult to show that for a long-run mean equilibrium output price equal to \(\theta ^{\star }\), the firm finds it equally profitable to operate in the market for ever under both the mean-reverting price process in (3) and a geometric Brownian motion with no drift.

Turning to the results, Table 1 confirms numerically the negative effect of mean reversion on irreversible exit decisions that were identified in the previous section. This negative effect on disinvestment when compared to a logarithmic random walk with drift is more pronounced for high intrinsic value projects (\(\theta =1.4\)).

For the intermediate \(\theta =\theta ^{\star }\) scenarios, the probabilities to exit under mean reversion are lower than the corresponding geometric Brownian motion ones, for all correlation and volatility levels. The decrease in exit probabilities seems to be more pronounced the lower the correlation coefficient of the equilibrium output price \(P\) with the market portfolio. The numerical evidence seems to indicate that for reasonable parameter levels, mean reversion unambiguously has a negative effect on irreversible exit decisions, when compared to the usually—employed geometric Brownian motion. This is in contrast to the costly reversible exit case treated in Tsekrekos (2010), where depending on parameter values the effect of mean reversion on disinvestment can be positive.

The reason behind this difference is the following: If the firm can costly revert back to being active in the market (i.e. exit is not irreversible), when exercising its put option to exit, it acquires a call option to re-enter in the future. The “underlying asset” of this call option is the present value of future cash flows from operations (plus a put option to re-exit since the decision is reversible). This future stream of cash flows is affected by the “risk discounting effect”. In contrast, under irreversible exit, by exercising its put option to exit the firm completely and irreversibly abandons the market. Its position is then unaffected by mean reversion, and there is no “risk discounting effect” as in the reversible case. This leads firms to be less willing to make irreversible exit decisions from markets where equilibrium output prices are highly mean reverting, in comparison to cases where such decisions can be reversed, even at a cost.

In order to establish the robustness of the numerical results, in Table 2 the simulations are repeated for a different starting value \(P_{0}\). Unlike the estimates in Table 1, where the staring simulation value \(P_{0}\) is scenario-dependent (i.e. different for each \(\kappa ,\theta ,\sigma \) triplet), in Table 2 the starting value is set to a fixed level across all tabulated parameter scenarios. It is apparent that the numerical results are qualitatively unaffected by the choice of the starting simulation value. Similarly, in untabulated results, the main finding that mean reversion has an negative effect on irreversible exit decisions is found to be robust to increasing or decreasing the time horizon \(T=10\) in the simulations.
Table 2

For a market with an equilibrium output price described by the mean-reverting process in Eq. 3, the table reports the probability that an active firm will optimally exit the market by time \(T\), calculated via Monte Carlo simulation

 

\(\theta = 0.40\)

\(\theta = \theta ^*\)

\(\theta = 1.40\)

 

\(\sigma = 0.10\)

\(\sigma = 0.15\)

\(\sigma = 0.25\)

\(\sigma = 0.10\)

\(\sigma = 0.15\)

\(\sigma = 0.25\)

\(\sigma = 0.10\)

\(\sigma = 0.15\)

\(\sigma = 0.25\)

Probability of exit, Panel (a):\(\rho = 1.0\)

\(\kappa = 0\)

   

0.9910

0.9509

0.8785

   

\(\kappa = 0.05\)

0.0738

0.1828

0.3416

0.0072

0.0886

0.2998

0.0004

0.0134

0.1288

\(\kappa = 0.10\)

0.3964

0.4488

0.5178

0.0118

0.1660

0.4300

0.0000

0.0022

0.0640

\(\kappa = 0.15\)

0.7866

0.7146

0.6734

0.0162

0.2630

0.5476

0.0000

0.0000

0.0198

Probability of exit, Panel (b):\(\rho = 0.0\)

\(\kappa = 0\)

   

0.9904

0.9466

0.8605

   

\(\kappa = 0.05\)

0.0412

0.1078

0.2058

0.0000

0.0135

0.0851

0.0000

0.0000

0.0004

\(\kappa = 0.10\)

0.3514

0.3784

0.4096

0.0058

0.1004

0.2155

0.0000

0.0000

0.0000

\(\kappa = 0.15\)

0.7668

0.6746

0.6012

0.0138

0.1799

0.3314

0.0000

0.0000

0.0000

Probability of exit, Panel (c):\(\rho = -0.5\)

\(\kappa = 0.05\)

0.0176

0.0276

0.0252

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

\(\kappa = 0.10\)

0.3106

0.2960

0.2370

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

\(\kappa =0.15\)

0.7496

0.6358

0.5010

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

In each simulation, 100,000 paths and 100 time steps per year are used. \(\theta \) is the long-run output price level, \(\kappa \) is the speed of mean reversion, \(\sigma \) is the volatility of the output price process, \(p \) is the correlation between the equilibrium output price and the market portfolio and \(\theta ^*\) as explained in the text. In all cases, the initial price of the simulation is set to a level independent of the exit threshold. The time horizon is \(T = \) 10 years, the market price of risk is \(\lambda = 0.4\), the variable flow cost is \(c = 1\), and the exit sunk cost is \({\underline{K}} =2\) respectively. Under \(\kappa = 0\), the same probability under the geometric Brownian motion price process in (1), with \(\mu = 0\) are reported. These are calculated via Eq. (21) and do not depend on \(\theta \)

5 Conclusions

Although managers and investors truly apprehend and value the flexibility inherent in being able to abandon an investment project early, far fewer research papers are devoted to the analysis of irreversible abandonment/exit decisions than to irreversible investment/entry.

In this paper we examine the optimal irreversible exit decisions of firms under mean-reverting uncertainty. The aim is to investigate whether the findings of previous research that mean reversion has an ambiguous (i.e. parameter-dependent) effect on irreversible entry decisions are directly extendable to abandonment/exit decisions that cannot be reversed.

The results of the model in this paper indicate that they are actually not fully extendable. Mean reversion has a significant impact on firm optimal policies, as measured by the probability of exit for an active firm over a certain period of time. It is shown that as far as irreversible exit decisions are concerned, the effect of mean reversion is clearly negative: In this case mean reversion lowers the rate of irreversible disinvestment/exit, since the “variance”, “realized price” and “risk discounting” effects documented in the previous literature only affect the value of the real option to exit and not the value resulting from (real) option exercise. This is in contrast to the costly reversible exit case and the irreversible entry one.

Footnotes

  1. 1.

    Requena-Silvente (2005) reports that close to 10 % of UK small and medium-sized exporters irreversibly withdrew from the foreign markets they operated in the 1994–1998 period. This percentage does not include firms that only temporarily ceased exporting (which were dropped from his sample). Similarly, Bernard and Wagner (2001) report that between 1990–1997, about 8 % of German exporting plants left the foreign market every year on average.

  2. 2.

    The argument in Lund (1993) is that in equilibrium, a price processes should not be unbounded from above, as the logarithmic random walk with drift clearly is, since new entry or expanded production by incumbent suppliers will induce reversion to lower price levels.

  3. 3.

    This could easily be extended to the case where the irreversible exit decision takes time to enforce, using the treatment in Majd and Pindyck (1987).

  4. 4.

    See Abel and Eberly (1996) and Alvarez (2011) on how the optimal amount of installed capital is affected by varying degrees of costly reversibility.

  5. 5.

    The proof of (2) follows immediately from Dixit (1989), by setting the entry fixed cost to infinity (that is, \(k\rightarrow + \infty \) in his notation).

  6. 6.

    See also the discussion in ((Tsekrekos 2010, p.728 and footnote 4)).

  7. 7.

    See Bhattacharya (1978) for the present value of an infinite series of flows that evolve according to Eq. (3).

  8. 8.

    It should be stressed that, although not apparent in the upper panel of Fig. 2 due to the scale of the graph, the mean reversion parameters \(\theta \) and \(\kappa \) significantly affect the irreversible exit/abandonment price trigger, \({\underline{P}}\). In the upper panel, output equilibrium prices of \({\underline{P}}=\left[ 1.6398, 1.67204,1.70827,1.74946\right] \) lead to abandonment for values of mean reversion speed \(\kappa =\left[ 0, 0.02, 0.05, 0.10\right] \). Stronger reversion to low levels makes the firm optimally abandon the market at higher output equilibrium prices, which is the exact opposite of what is observed in the lower panel of the Figure.

Notes

Acknowledgments

This paper has greatly benefited from the valuable comments and suggestions made by the participants of the international workshop on “The Economics of Irreversible Choices” that was organized by the Lombardy Advanced School of Economic Research (LASER) and the DEFAP Graduate Business School in Public Economics and was hosted by the Università degli Studi di Brescia in Italy. Special thanks are due to David Schüller who acted as the discussant of the paper at the workshop, as well as to the organizers Giacomo Corneo, Luca Di Corato, Michele Moretto, Paolo Panteghini, Carlo Scarpa and last but not least Sergio Vergalli.

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Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Department of Accounting and FinanceAthens University of Economics and Business (AUEB)AthensGreece

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