Growth Options in General Equilibrium: Some Asset Pricing Implications1

811 Uris Hall Graduate School of Business 3022 Broadway Columbia University New York City New York 10027

First draft: May 2004 Please do not quote or circulate

1 Abstract I develop a general equilibrium model of a production economy which has a risky production technology as well as a growth option that can expand the scale of the productive sector of the economy. I show that the optimal exercise of the growth option differs qualitatively from the predictions of models of real options that ignore feedback effects of potential exercise on equilibrium quantities [such as consumption] and wealth-dependent risk aversion. An important result is that the number of options endowed to the consumer influences the optimal exercise policy due to trade-offs between consumption and early exercise. Equity price in the economy can be characterized as the sum of the stock of the good in the economy plus the value of the [perpetual] call option whose value depends on the endogenous consumption policy which plays the role of dividends and changes dramatically when the growth option is near the money. This leads to time-varying equity risk premium: the conditional volatility of equity returns depend on the delta of the growth options and the conditional expected returns depend on both delta and gamma of the growth options. As a result, significant fluctuations in moments of equity returns are predicted when the growth options are near the money. The moneyness of the growth option is the key factor which determines the extent to which the book to market ratios will influence the conditional moments of equity returns.

2 1Introduction

In this paper I examine how real options will be exercised in general equilibrium settings and how optimal exercise strategies differ from those derived in options pricing literature. I construct a single good, general equilibrium production model of an economy with a risky production technology which exhibits linear, stochastic constant returns to scale. This technology may be thought of as assets that are already in place. The infinitely lived representative consumer can expand the scale of the production technology by investing an amount of the only good in the economy at any time. This is naturally interpreted as a growth opportunity. Since the consumer must pay for the investment, it stands to reason that the consumer will wait until the stock of good reaches a sufficiently high level before he decides to exercise the option to expand the economy. This waiting period is optimally chosen by the consumer by controlling his rate of consumption at each instant. The critical level of the stock of good at which exercise occurs is endogenous to the equilibrium that I study in this paper. The problem of optimal exercise of options has received extensive attention in the academic literature. The seminal paper by Merton (1973) laid out the economic circumstances under which options may be optimally exercised. In the real options literature, McDonald and Siegel (1986) and Dixit and Pindyck (1995) have characterized the economic circumstances under which real options will be optimally exercised. Our approach can be contrasted with the approaches in the literature along the following dimen- sions:

1. The investor in our framework owns the “tree” or the “assets in place” that produces the good and derives utility at each instant from a ﬂow rate consumption from the tree. The investor has, in addition, an

3 option to expand the tree at any time2. The investor therefore faces atrade-off between his current rate of consumption and future rate of consumption that will reflect the expanded size of the tree after the option is exercised. This is in contrast to the previous literature where the decision maker does not own the tree but has an option to acquire the value of the tree. It is reasonable to think that this difference matters when the consumer is able to decide on the dividend sequence before and after exercise.

2. The objective of the investor in our framework is to maximize the discounted expected life-time utility of consumption. The objective in much of the options pricing literature is to maximize the discounted expected payoﬀ of the option taking the stream of dividends [consumption] as given. In our framework the stream of dividends or consumption is endogenously determined. The investor must optimally select the sequence of consumption to implement an exercise policy that maximizes his life-time discounted expected utility. In this important sense the real options that I study in this paper are distinct from the ones studied in the options pricing literature. In options pricing theory, the dividend [consumption] policy is either exogenous or often implicitly assumed away. In a closed economy, optimal consumption rate must be managed to optimally ﬁnance the exercise of growth options.

A number of authors have laid the groundwork to link asset pricing and the presence of growth options in the economy. Notable contributions include Berk, Green and Naik (1999), and Kogan (2001, 2004)3.Berk, Green and Naik (1999) consider a partial equilibrium model to establish a link between stock returns and ﬁrm characteristics. Kogan (2001, 2004)

4 constructs a general equilibrium framework with convex adjustment costs and irreversibility to link equity returns and firm characteristics in a two- sector growth model. My focus as well as the modeling approach differ from these earlier contributions. I study a general equilibrium model like Kogan but examine a much simpler investment problem than Kogan (2001, 2004) and stay in the realm of single-sector growth models. The modeling approach is one in which the economy has a simple (endogenous) structural break: the economy begins with a live growth option, but once it is exercised it reverts to a “Robinson Crusoe” economy similar to Lucas (1978) and Cox, Ingersoll and Ross (1985). While this assumption is admittedly strong, it allows one to examine optimal exercise strategies in much greater detail and study how the feedback effects on equilibrium quantities influence optimal exercise. In addition, it allows me to explicitly link the role played by consumption in the exercise of growth options and hence on asset pricing. At a basic level, the introduction of growth options is natural in production- based models of asset pricing. Economic intuition suggests that the consumer when confronted with growth options may sharply alter consumption rates to optimally exercise the growth option. This implies that the sen- sitivity of the consumption rate to wealth may be qualitatively different in economies with growth options than conventional models might suggest. This property will have a corresponding consequence for asset prices and equilibrium interest rates which I characterize in this paper. In addition, the framework proposed in the paper can form the basis for modeling more intricate ways of modeling economy’s production opportunity set. For example, one natural extension is to let the growth option alter the riskiness of the economy upon its exercise. This will allow one to study the correlation between the new technology arising from the exercise of growth options with that of the existing old technology. Another extension is to endow the econ-

5 omy with more than one [possibly interdependent] growth options. These are topics for further research. The paper is organized as follows: the next section reviews the a baseline model of growth options. I also motivate its construction by relating the model to previous contributions in the literature. This model is then extended to a situation where the investor derives beneﬁts from consuming dividends from the ownership of a tree [production process] on which he has an option to expand. In section 3, I characterize the endogenous level of the stock of good at which the option is exercised by the representative consumer in a general equilibrium economy. This trigger policy and its in- timate connection to relative risk aversion, and optimal consumption policy are addressed in section 4. The exercise policy in the general equilibrium framework is compared with the policies implied by the baseline models. I develop in section 5 a complete characterization of equity prices in the economy. Section 6 concludes.

2 Baseline Model of Real Options

Papers by Merton (1973), McDonald and Siegel (1986) and Dixit and Pindyck (1995) provide the reference points for this paper. I brieﬂy reformulate their results in the context of my model. We ﬁrst consider the baseline case in which the consumer does not derive any dividend income but has an option to invest in a new technology. In other words there are no assets in place. The stock of good W follows a Geometric Brownian motion process with a drift parameter µ and has a volatility parameter of σ. Assume that the subjective discount rate of the consumer is ρ. In order to exercise the growth option, the consumer will have to make an investment expenditure of I.

The objective of the consumer is to maximize the discounted expected

6 payoﬀs from exercising the growth option as stated below:4

ρT V (W )=MaxE e− (αWT I) (1) − An important implication of the speci£ fication can¤ be better understood α by rewriting the option payoffsasfollows:letusset I =1, and note that the payoffs upon exercise of option at a critical level of stock W ∗ can be written as follows:

α Payoffsuponexercise=αW ∗ I = I W ∗ 1 = I [W ∗ 1] (2) − I − − h i This means that the growth option that we have modelled is equivalent to a number #I of options each with an investment outlay or a strike price α of unity. With the constraint I =1firmly in place, increasing I corresponds to an increase in the number of growth options that the consumer has. Note that the time T at which the growth option is exercised is determined by an endogenous level of the stock of good denoted by W ∗. The dynamics of the stock of good is governed by the stochastic differential equation given next.5

dW =(µ δ)dt + σdB (3) W −

Following the work of Merton (1973), McDonald and Siegel (1986) and

Dixit and Pindyck (1995) the optimal exercise boundary W ∗ for a consumer to invest in a growth option will then be given by the equation below:

I γ γ W ∗ = . (4) α γ 1 ≡ γ 1 − − where,

7 2 σ2 σ2 2 µ δ 2 + µ δ 2 +2ρσ γ = − − − r − − > 1. (5) ³ ´ σ³2 ´ and the value of the growth option can be represented as follows.

I W γ W γ V (W )= I (W ∗ 1) (6) γ 1 W ≡ − W − µ ∗ ¶ µ ∗ ¶ In this formulation, we have not enforced any restrictions that may be implied by equilibrium considerations. For example, if we let µ = ρ = r, then the formula for the valuation of growth option is that of the perpetual call derived by Merton (1973). In this case, absence of dividends will lead to the implication that the call option will never be exercised6.Notethat the optimal exercise boundary depends on the parameters of the production technology, subjective discount rate [degree of impatience], the dividend payout factor and the parameters relating to the option itself [I ]. Dividend payouts accelerate the trigger and this implies earlier exercise and a reduction in the value of the call. A couple of observations are worthy of note. First, there is no trade off between consuming now and consuming later in this baseline model: there is no utility attached to flow rate of consumption or dividends. Second, any dividend payout is exogenous to the model. As a consequence, W ∗ and V (W ) depend on dividend payout parameter only by exogenous payout specification parametrized by δ. In the context of this framework, several important results have been established. Irreversibility of growth options and the ability to delay leads to a range of inaction [i.e. no investment] even when αW>I.In Hubbard (1994) this wedge of inaction is defined as follows:

αW γ ∗ = > 1. (7) I γ 1 −

8 γ Note that the region of inaction is dependent only on the ratio γ 1 − : whether the growth option doubles or triples the wealth, the region of inaction will still be the same. This homogeneity property or the scale inde- pendence is well known in the literature. This region of inaction increases as σ increases, holding other factors ﬁxed. Likewise, the region decreases as the subjective discount rate increases. By specializing the model, we can get the same values and the exercise boundary obtained by McDonald and Siegel (1986)7.

2.0.1 Modiﬁcation to Baseline Model

In order to compare the baseline model with the implications of equilibrium models of growth options, it is useful to make one modification to the baseline model of growth options that we sketched out earlier to include some benefits from receiving dividends. We permit the investor to own the basic technology governed by the stochastic process specified in equation (3). We therefore allow the investor to receive an exogenously specified dividends from the tree. The investor must decide when to expand the tree by a α fraction α > 1. We will continue to impose the restriction that I =1. The decision problem facing the investor may be formulated as follows:

T ρs ρT αδWT V (W )=MaxE e− δWsds + e− I (8) ρ µ − ·Zt µ − ¶¸ Or,

T ρs ρT δWT V (W )=MaxE e− δWsds + e− I 1 (9) ρ µ − ·Zt µ − ¶¸ The optimal exercise boundary W ∗ for a consumer to invest in a growth option will then be given by the equation below:

9 I (ρ µ) γ W ∗ = − (10) (α 1) δ γ 1 · − ¸ − This can be written as follows to gain better economic intuition:

(α 1) δW γ − ∗ = > 1 (11) (ρ µ) I γ 1 − − The net present value of the payoﬀs from incremental expansion of the tree must still be above the option value identiﬁed by McDonald and Siegel (1986). The sum of the value of the dividends and the value of the growth option can be represented as follows.

δ I W γ V (W )= W + (12) ρ µ γ 1 W − − µ ∗ ¶ Note that the optimal exercise boundary depends on the parameters of the production technology, subjective discount rate [degree of impatience], the dividend payout factor and the parameters relating to the option itself [I]. When the subjective discount rate is high enough, i.e., ρ >µ, the consumer will prefer a positive dividend payout: dividends increase the value derived from current consumption, but reduces the value of the growth option This tension in an equilibrium model will determines the optimal [consumption] payout policy endogenously. We have so far sketched out the trade-offs between current consumption and the exercise of growth options without enforcing any equilibrium requirements formally. In the rest of the paper, we focus on a general equilibrium setting in which a representative consumer is confronted with the choice as to when growth options should be exercised when he derives utility from a flow rate of consumption at each instant. The focus is then on the manner in which such a decision may influence equilibrium quantities [such as consumption and risk aversion] and prices in the economy.

10 3 The Model

Equilibrium asset pricing in exchange economies with a representative consumer can be traced back to the papers by Lucas (1978) and Rubinstein (1976). Models asset pricing with production economies have been developed by a number of authors including Cox, Ingersoll and Ross (1985), Kyd- land and Prescott (1982) and Jermann (1998). As has been pointed out, one of the difficulties with production models is that the consumer is able to use the production technologies to smooth his consumption to a point that these models have difficulty in generating sufficient movements in the pricing ker- nel. Adjustment costs in production, time to build and other enhancements have been explored by authors to overcome this difficulty in models of asset pricing in which the production side of the economy is explicitly accounted for. One of the natural ways in which lumpy investments in production arises is through the growth options that economies often have. Recent ex- amples of such growth options would include the internet boom, fiber optic networks and communication technologies and bio tech investments. Such growth options may not only significantly expand the scale of the economy but also the risk-return character of the economy. They require lump sum investments. This is the avenue that I pursue in my paper. In the real options literature initiated by the early work of McDonald and Siegel (1986) and Dixit and Pindyck (1995) such lump sum investments in irreversible production technologies have been investigated. But such papers consider situations in which the exercise of real options do not have any consequences for equilibrium quantities such as consumption or for equilibrium prices of assets and interest rates. I make that connection in this paper. I consider a single good production economy setting with a representative consumer. The consumer is assumed to have a constant relative risk

11 aversion utility function. The economic environment is fully described by the speciﬁcation of the production technology, option to expand the scale of production, and the preferences of the consumer.

3.0.2 Production Technology

There is a single good in the economy and it serves as the numeraire. The production sector has a risky technology. Once an amount qt of the good is invested in the technology at time t, the output will evolve as shown below:

dqt = µdt + σdzt (13) qt where the instantaneous expected rate of return µ and the diﬀusion 8 coeﬃcient σ are exogenous positive constants. The process zt is a standard Brownian Motion on the underlying probability space (Ω, , ). In addition F P to this technology, the consumer has the option of investing I units of the good to expand the scale of the technology by a factor α > 1. It is assumed that the initial endowment is such that the expansion cannot be undertaken at time 0.

3.0.3 Preferences and Endowments

The risk-averse consumer is endowed with an initial wealth of x0 and has an exclusive access to the risky production technology. He maximizes his life- ρt time discounted expected utility of consumption: E0[ 0∞ e− u(ct)dt]where u is his von Neumann-Morgenstern utility function andR ρ is his time prefer- ence rate. In this paper we examine a special class of utility functions whose relative risk aversion in consumption is a positive constant:9

1 1 A u(c)= c − , A>0, A = 1 (14) 1 A 6 − This speciﬁcation has been widely used in the theory of intertemporal

12 consumption-portfolio selection problems, default-free term structure theory andinassetpricing.

3.0.4 A Brief Recap of CIR (1985) Results

CIR (1985) developed an asset pricing framework and considered a simple case of their framework for studying the term structure of interest rates. In their economy, CIR (1985) did not consider explicitly growth options. In the simplest version of the Cox, Ingersoll and Ross (1985) paper with the assumptions about the preferences and technology that we have made earlier, the optimal consumption policy, the value function and the equilibrium default-free term structure can be characterized in closed form as follows.

ct = kWt (15)

where, k is the marginal propensity to consume [MPC] from wealth. This is given by the expression below.

2 ρ (1 A)(µ Aσ ) k = MPC = − − − 2 > 0. A The value function of the representative consumer in this baseline case is provided next.

W 1 A J (W )=k A t − (16) B t − 1 A − and

2 rCIR = µ Aσ − When there are many risky technologies, the CIR (1985) model gives qualitatively very similar conclusions: the risk-free rate is still a constant

13 and the functional forms of the optimal consumption and the value function are still the same with the exception that the drift and the volatility parameters will now reﬂect the weighted drifts and weighted covariances of the outputs from the diﬀerent technologies. This baseline case forms the boundary condition to the economy that we study in this paper: after the growth option is exercised, our economy will approach the CIR (1985) equilibrium.

3.1 Optimization Problem and Equilibrium

We now describe the equilibrium in this economy. Given the production opportunity set α,I , the controls of the consumer will be the amount qt { } invested in the risky technology, the consumption rate ct and the optimal investment boundary level W .Deﬁne the t -stopping time: τ =inf t ∗ {F } { ≥ 0 Wt >W . The wealth dynamics facing the consumer can be formally | ∗} represented as:

dWt =[rt(Wt qt) ct] dt + µqtdt + σqtdzt for 0 t<τ − − ≤

where rt is the endogenously determined default-free interest rate. Let us denote the set of admissible controls by (W0). The objective function fac- A ing the consumer is the expected life-time discounted utility maximization. Formally, the consumer maximizes the value function J which is deﬁned as the supremum of the expected utility over the set of admissible controls:

∞ ρt J(W0)= sup E0 e− u(ct)dt (18) (W0) 0 A ·Z ¸

Let us denote the optimal policy be (c∗,q∗,W∗). The equilibrium in this economy is deﬁned as following:

Deﬁnition 1 An equilibrium r, c ,q is a set of stochastic processes { ∗ ∗} (r; c∗,q∗) which satisfy market clearing condition: qt∗ = Wt.

14 The market clearing condition implies that there is no risk-free lending or borrowing at equilibrium, as in CIR (1985). Critical to the characterization of our equilibrium with default is the optimal boundary W ∗ at which the growth option will be exercised. It can be shown that the value function satisﬁes the following properties: (i) J(.) is strictly increasing and strictly concave. (ii) J(.) is continuous on [W , )withJ(W )=JB(αW I), ∗ ∞ ∗ ∗ − where JB(.) is the consumer’s valuation function after the growth option is

∂JB(αW ∗ I) exercised. (iii) (smooth pasting condition) limW W + J0(W )= ∂W − . → ∗ ∗ (iv) (dynamic programming principle)

τ ρt ρτ J(W0)= sup E0 e− u(ct)dt + e− JB(αW ∗ I) (19) (W0) 0 − A ·Z ¸ It can be shown that for any t<τ, the value function J (.) is the unique C2(W , + ) solution of the Bellman equation: ∗ ∞

1 2 2 ρJ = σ W JWW + µW JW +max[u(c) cJW ](W

1 c∗(W )=(u0)− (JW (W )) (21)

The approach to solve this problem is by backward induction. We ﬁrst solve the general equilibrium of the economy after thegrowthoptionisexer- cised. This follows directly from the contributions of Cox, Ingersoll and Ross (1985). We then use this value function of the consumer as the boundary condition to search for the optimal exercise boundary of the representative consumer.10 We now directly proceed to illustrate our numerical results in the next few sections.

15 4 Optimal Exercise of Growth Options

We examine a baseline case where we set the consumer’s subjective discount factor µ =0.15 and the risk aversion parameter A =2.0. We characterize the exercise strategy for growth options as follows: we keep the ratio α I ﬁxed at 1 but change I within this constraint to model varying degrees of scale expansion.

In table 1 below, the results are presented for the optimal exercise of the growth option. The long-run marginal propensity to consume out of wealth is shown in parentheses. They are computed using the basic CIR (1985) equilibrium that prevails in our economy after the growth option is exercised. The parameters are chosen at α = I =2.0 in this illustration. This means that the economy can essentially double its wealth by exercising the growth option. One would therefore expect the feedback eﬀects on equilibrium quantities and prices to play a big role in the manner in which the option is exercised.

Table 1 Wedge of inaction α 1 −I W ∗ µ = 15%, and A =2.0. α = I =2 100% increase in scale. MPC in parentheses. −→ σ =0.15 σ =0.20 σ =0.25 r =10.5% r =7% r =2.5% ρ = 10% 1.282 (11.375%) 1.309 (10.50%) 1.346 (9.375%) ρ = 12% 1.244 (12.375%) 1.275 (11.50%) 1.303 (10.375%) ρ = 14% 1.213 (13.375%) 1.2475 (12.50%) 1.2775 (11.375%) ρ = 16% 1.1885 (14.375%) 1.224 (13.50%) 1.2555 (12.375%)

16 Note that as the subjective rate of discount increases, the growth option is exercised sooner. This is due to the fact as ρ increases, the marginal propensity to consume increases, increasing the incentive to exercise early. In Table 2, we present the optimal exercise boundary when there is only a 10% increase in wealth due to exercise of growth options. Note the increase in the optimal exercise boundary in Table 2 relative to the ones we found in Table 2. This emphasizes a crucial result in our set up: scale of the option matters in equilibrium model.

Table 2 Wedge of inaction α 1 −I W ∗ µ = 15%, and A =2.0. α = I =1.10 10% increase in scale. MPC in parentheses. −→ σ =0.15 σ =0.20 σ =0.25 r =10.5% r =7% r =2.5% ρ = 10% 1.3204 (11.375%) 1.3625 (10.5%) 1.4145 (9.375%) ρ = 12% 1.2695 (12.375%) 1.3162 (11.50%) 1.3575 (10.375%) ρ = 14% 1.2301 (13.375%) 1.2793 (12.50%) 1.3224 (11.375%) ρ = 16% 1.1991 (14.375%) 1.2494 (13.50%) 1.2935 (12.375%)

In order to compare these results with the baseline model of real options, it is essential to pin down a) dividend yields that are speciﬁed exogenously and, b) adjust for the presence of risk premium. I will set the exogenous dividend yield at the long-run MPC implied by the CIR (1985) model given the parameters that we use in the baseline model. The issue of risk premium is fully addressed in the general equilibrium model developed in the paper. I

17 provide a comparison of the optimal exercise policies with the baseline case below in Table 3.

Table 3 Wedge of inaction γ γ 1 − µ = 15%, Baseline Model calibrated with long-run MPC Long-run MPC in parentheses.

σ =0.15 σ =0.20 σ =0.25 ρ = 10% 1.9341 (11.375%) 2.432 (10.5%) 3.3093 (9.375%) ρ = 12% 1.6000 (12.375%) 1.9066 (11.50%) 2.3619 (10.375%) ρ = 14% 1.4324 (13.375%) 1.6563 (12.50%) 1.9633 (11.375%) ρ = 16% 1.3333 (14.375%) 1.5112 (13.50%) 1.7439 (12.375%)

Note that the exercise policy implied by the baseline model leads to delays in exercise of the growth option. For example, at ρ = 12%, and σ =0.20, we find that the exercise occurs in the baseline model at 1.9066 as opposed to a wealth level of 1.2793 in Table 2. The difference represents a percentage increase in wealth of (1.9066 1.2793)/1.9066 = 32.9 02%. − The economic rationale for this follows from the differences in the objective functions that correspond to the baseline models and the ones proposed here. First note from (9) that in the baseline model, the investor has no choice over the consumption stream over his life. No matter how attractive the growth option is the investor still consumes at the same rate δ prior to the exercise of growth option. In addition, the investor’s payoff upon exercise of the growth option is linear in wealth. On the contrary, the equilibrium model with growth options allows the consumer to choose the consumption sequence simultaneously with the optimal exercise strategy. This is evident

18 from equation (20). When the growth option is very attractive, the consumer can reduce his rate of consumption to a level that is less than the long-run equilibrium value to essentially ﬁnance the growth option and thus accelerate the exercise. The concavity of the value function contributes to the fact that the scale of the growth option is a relevant factor in the early exercise decision.

4.1 Relative Risk Aversion in Wealth

I characterize the behavior of the indirect value function. The relative risk aversion in wealth (RRA) of the representative consumer is measured using his value function in our economy prior to the exercise of the growth options in the economy. Note that the relative risk aversion in wealth for the general equilibrium economy of CIR (1985) with no growth options under our hypothesized assumptions is simply a constant given by A. In ﬁgure 1, we plot the wealth along the X-axis and the relative risk aversion in wealth along the Y-axis.

19 A=2, Mu=10%,Rho=12%, & Sigma=20% 2

1.8

1.6

1.4

1.2

Risk Aversion in Wealth in Aversion Risk 1 I=3, W*=1.815 I=1.5, W*=3.749 I=1.25, W*=6.295 0.8

0.6 0 1 2 3 4 5 6 7 Wealth

Figure 1: RRA in wealth vs. Wealth. We find that the relative risk aversion decreases in a significant manner as the wealth level increases from 0.to W ∗. This decrease in RRA is due to the fact that the growth option goes more and more in the money as the wealth level increases. A consequent action for the consumer to curtail the flow rate of consumption relative to the long-run mean rate of consumption [as we will show later] to try to improve future wealth level. We obtain time varying risk aversion in our model by virtue of the presence of growth option — this comes from the production side of the economy as opposed to models in which habit formation produces time varying risk aversion to help explain aggregate stock market behavior11. Note that the rate at which the RRA drops is a function of the scale of the growth option. When the economy is on the verge of exercising a major growth option, RRA drops much more rapidly.

20 4.2 Optimal Consumption

In the absence of growth options, the CIR general equilibrium model implies that the optimal consumption is given by kW. Moreover, the elasticity of optimal consumption with respect to wealth is a constant. In ﬁgure 2, we C(W ) plot the we plot the normalized optimal consumption kW .

A = 2, Mu = 10%, Rho = 12% & Sigma = 20% 1

I = 1.25, W* = 6.295. 0.9 I = 1.5, W* = 3.749

0.8 I = 3, W* = 1.815

0.7

0.6 Normalized Consumption

0.5

0.4 0 1 2 3 4 5 6 7 Wealth

Figure 2: Normalized Consumption vs. Wealth The consumption rate relative to the long-run value dramatically declines as the growth option goes into the money. For I =3, the effect is rather pronounced: the consumption rate falls to less than 80% of the long-run level as the exercise boundary is reached. It is this flexibility that is missing in models of real options that ignore feedback effects of growth options on the decisions made by the investor. As the scale of the project goes down the effect on consumption rates becomes predictably much less pronounced. It is

21 important to stress that consumption changes in this model reflect dynamic financing of the growth options by the representative consumer. We have restricted the consumer from borrowing or other financing opportunities. Then, the only way for the consumer to finance the growth option is by dynamic adjustment of consumption flow rates which depends on the extent to which the growth option is in the money.

5 Characterization of Equity Prices

Equity prices in our economy can be characterized as the sum of the current wealth plus the value of a perpetual call option to expand the economy by a factor (α 1) at a strike price I which represents the fixed invest- − ment expenditure. To see this, firstnotethatequityprice,E,willsolvethe following partial differential equation12.

1 2 2 c∗(W ) r(W )E +(r(W )W c∗(W ))EW + σ W EWW = 0 (22) − − 2 Implicitly, we assume that the equity holders receive dividends equal to the optimal consumption rate. In the absence of any growth options, note that E = W trivially satisﬁes the valuation equation above. This is the same as saying that Tobin’s q is always equal to one in a general equilibrium model without growth options or other enhancements. When growth options are present, the above equation must satisfy the boundary condition shown below.

E(W ∗)=αW ∗ I (23) − Let us write the equity price as the sum of wealth W plus an option

22 C so that E = W + C. Plugging this into the valuation equation and the boundary condition, we get the following result.

1 2 2 r(W )C +(r(W )W c∗(W ))CW + σ W CWW = 0 (24) − − 2 The boundary condition becomes13:

C(W ∗)=(α 1) W ∗ I (25) − − Equity in our model is a portfolio of a) a long position in the wealth of the economy, plus b) a long position in a perpetual real option to expand the wealth of the economy by a factor of (α 1) at a strike price of I. The − value of this real option depends on the optimal consumption policy which plays the role of dividends and on the risk-free rate which is no longer a constant. An immediate consequence is that the model will produce interesting interdependency between Tobin’s q, consumption rates and interest rates as the growth option moves in and out of the money.

23 A = 2, Mu = 10%, Rho = 12%, & Sigma =20%. 0.7 I = 3 0.6

0.5

0.4 I = 1.5 I = 1.25 0.3 Call Option Value Option Call

0.2

0.1

0 0 1 2 3 4 5 6 7 Wealth

Figure 3: Value of Growth Option vs. Wealth.

Let ∆ denote the delta of the growth option and Γ denote the gamma of the growth option.

24 A = 2, Sigma = 20%, Mu = 10%, Rho = 12% 1.4

1.2

1 I = 3.0

0.8

0.6 Call Option Delta

0.4 I = 1.5

0.2 I = 1.25

0 0 1 2 3 4 5 6 7 Wealth

Figure 4: Growth Option Delta vs. Wealth.

25 A = 2, Mu = 10%, SIgma = 20%, Rho =12%. 1.4

1.2

0.8 I = 3

0.6 Call Option Gamma Option Call 0.4

I = 1.5

0.2 I = 1.25

0 0 1 2 3 4 5 6 7 Wealth

Figure 5: Gamma of growth option vs. Wealth.

We can now characterize the equity prices as follows: E = W + C(W ). This implies the following dynamics [gross of consumption/dividends] for equity prices:

1 dE = dW + C dW + C dW 2. W 2 WW This can be written out explicitly as:

dE = µEdt + σEdB

where, the expected change in equity prices are denoted by µE and the volatility of the changes is denoted by σE.

1 µ = (1 + ∆) µW + Γσ2W 2 E 2 · ¸

26 σE =(1+∆) σW

The conditional moments of equity price changes depend on the wealth levels as well as the extent to which the growth options in the economy are near the money. The expected equity price change is inﬂuenced both by the delta and the gamma of the growth option whereas the volatility of the equity price change is driven only by the delta of the growth option. Equity prices are expected to increase as the option goes deeper in the money as does the volatility of equity price changes. Once the option is exercised, the drift of the equity price changes drops and the volatility drops as well. This is due to the assumption in our model that there are no additional growth options in our economy. Note that the equity returns will now depend on E the ratio of market to book q = W at each instant. In fact, the equity returns can be written as follows:

dE = µ0 dt + σ0 dB for W

(1 + ∆) µ 1 Γσ2W µ0 = + (27) E q 2 q · ¸

(1 + ∆) σ σ0 = (28) E q · ¸

27 A = 2, Mu = 10% Sigma = 20%, Rho = 12% 0.22

0.2 I = 3

0.18

0.16

0.14 I = 1.5

Conditional Expected Return on Equity I = 1.25 0.12

0.1 0 1 2 3 4 5 6 7 Wealth

Figure 6: Conditional Expected Return vs. Wealth

28 A = 2, Sigma = 20%, Mu = 10% and Rho = 12% 0.32

0.3 I = 3

0.28

0.26

I = 1.5 0.24

I = 1.25 Conditional Volatilityof Equity Returns Conditional 0.22

0.2 0 1 2 3 4 5 6 7 Wealth

Figure 7: Conditional Volatility vs. Wealth. Sharpe ratios can be computed for various wealth levels. In our model theseratioswillbeinﬂuenced by the feedback eﬀects on risk-free rates as well. For the base case CIR model the Sharpe ratio is 4, given the parameters chosen in the illustrations.

29 A =2, Sigma = 20%, Mu = 10%, Rho = 12%. 0.52

0.5

0.48 I = 3 0.46

0.44 I = 1.5 0.42 I = 1.25

Sharpe Ratio 0.4

0.38

0.36

0.34

0.32 0 1 2 3 4 5 6 7 Wealth

Figure 8: Sharpe Ratio vs. Wealth Sharpe ratio peaks at about 0.51 when the economy’s option to expand three-fold is approaching in-the-money status and then begins to drop rapidly as the option is about to be exercised. We conclude by reporting Tobin’s q for the three illustrative case below.

30 A = 2, Sigma = 20% Mu = 10% and Rho = 12% 1.4

1.35

I = 3 1.3

1.25

1.2

1.15 Market to Book (q)

1.1 I = 1.5 1.05 I = 1.25

1 0 1 2 3 4 5 6 7 Wealth

Figure 9: Tobin’s q vs. Wealth.

6 Conclusion

We have presented a general equilibrium production model with growth options. The model allows us to determine endogenously the optimal consumption, and equity prices. A key implication of our model is that the risk aversion in wealth of the consumer displays time variation through endogenous wealth dependency which arises from the presence of growth options in the economy. In particular, it declines with increases in wealth as the growth option gets more and more in the money. This reduction is accompanied by a fall in normalized consumption and an acceleration of the exercise boundary relative to the baseline models of real options. The equity prices can be characterized as a portfolio of wealth of the economy plus the option to expand the economy. The delta and the gamma of the growth option play

31 acriticalroleininﬂuencing the properties of the conditional equity returns and the impact of book to market ratios on equity returns.

7Appendix

7.1 Boundary properties of consumption and interest rates

The following limiting results can be obtained for our model. A k− 1 A (i) limW J(W )= W − . →∞ 1 A − A k− A (ii) limW JW (W )=(1 A) 1 A W − . →∞ − − A k− A 1 Proposition 1 (iii) limW JWW(W )=(1 A)( A) 1 A W − − . →∞ − − −

The convergence of J, JW , JWW implies that not only the asymptotic behavior of value function J converges to that of J0, but also all the interesting variables which depend up to second derivative of J will converge to those variables in the standard CIR economy. For example the shadow default-free (instantaneous) interest rate r(W ) and the optimal consumption policy will satisfy:

lim r(W )=µ Aσ2 W − →∞ c(W ) lim = k (29) W W →∞ We also show that in the limit as the economy approaches the exercise boundary, the consumption policy and the default-free interest rates can be solved in closed form.

From the boundary condition J(W )=JB(αW I)andthe“smooth ∗ ∗ − ∂JB (αW ∗ I) pasting” condition limW W + J0(W )= ∂W − , we can characterize the → ∗ ∗ limiting behavior of r(W )and c(W ) when wealth level is close to default:

2ρ W I/α 2A 1 1 W I/α lim r(W )= − µ α − A k − W W + 1 A W − − 1 A W → ∗ − − 1 1 I lim c(W )=α − A k(W ) (30) W W + − α → ∗ 32 7.2 References

33 References

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34 [10] Hubbard, Glenn, “Investment Under Uncertainty: Keeping One’s Op- tions Open,” Journal of Economic Literature, (December 1994), pages 1816-1831.

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35 [21] Mark Rubinstein, “The Valuation of Uncertain Income Streams and the Pricing of Options,” The Bell Journal of Economics, Vol. 7, No. 2. (Autumn, 1976), pp. 407-425.

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36 Notes

1Preliminary draft. Comments invited. I thank Ganlin Chang, Urban Jermann and Tano Santos for discussions on the topic. Hayong Yun provided expert computational assistance.

2We can modify the set up so that the option can also change the risk characteristics of the tree.

3See also the contributions by Dow and Ohlson (1992) which studies the eﬀect of irreversibility in stochastic growth economies.

4The expectation is with respect to the ﬁrstpassgetimedensityofthe process Wt to an endogenous boundary level W such that T =inf t> { } ∗ { 0,Wt >W . ∗} 5 The process Bt is a standard Brownian motion process on the underlying probability space (Ω, z,P) .

6McDonald and Siegel (1986) impose equilibrium restrictions on the subjective rate of discount as well: they require that the subjective rate of discount should be the weighted average of the equilibrium expected returns on assets with the same risk as that of the forcing variables in their model.

7In their model McDonald and Siegel (1986) consider more general options with stochcastic strike prices. Our special case leads exacttly to their numerical results under the following parameters. Setting µ = δ, σ =0.20, and ρ = 10%, we get the value of investment opportunity to be 16% higher than the one implied by the net present value rule. The exercise boundary is 56% higher than the zero NPV rule.

37 8We restrict attention to a constant opportunity set to get tractable results.

9We would like to point out that all of our major results still hold for a general von Neumann-Morgenstern utility function u: which is a strictly 3 increasing, strictly concave C (0, + ) function with limc 0 u0(c)=+ ∞ → ∞ γ and limc + u0(c)=0andsatisﬁes the condition u(c) M(1 + c) for → ∞ | | ≤ some positive constants M and γ.

10See Chang and Sundaresan (2004) for an implementation of this tech- nique.

11See Sundaresan (1989), Constantinides (1990) and Cochrane and Camp- bell (1999). The instantaneous default-free instantaneous interest rate in our 2 model is given by r(W )=µ WJWWσ . The presence of growth option in the − JW economy has important effects on the equilibrium risk-free rate. As wealth approaches infinity,weknowthattherisk-freeinterestratemustconverge to the CIR (1985) r = µ Aσ2. At wealth levels below W , the equilibrium − ∗ interest rate is well below the level implied in an economy with no growth options. This variation in risk-free rate will turn to have important implications for equity prices in our economy. We can characterize the default-free term structure in this economy. Let us denote P (t, T )asthepriceattime t for a zero coupon bond which pays one unit consumption good at time T . P (t, T ) will satisfy the following partial differential equation (PDE) with R(T τ) boundary conditions P (W ∗, τ)=e− − , P (W, T )=1:

1 2 2 r(W )P +(r(W )W c∗(W ))PW + σ W PWW + Pt = 0 (31) − − 2 In standard equilibrium setting, the term structure will be ﬂat: as the yield to mature for a zero coupon bond R(t, T )= (ln P (t, T ))/(T t)issimply − − a constant equal to the instantaneous interest rate µ Aσ2.Howeverinour −

38 model, the shape of the term structure is wealth dependent.

12See Cox, Ingersoll and Ross (1985) for a derivation of this result under fairly general circumstances.

13 In addition, we impose the “high contact” condition that C0 (W ∗)= α 1, and the condition that C(0) = 0. −