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Article Excerpt
This article derives a closed-form solution for an equilibrium real options exercise model with stochastic revenues and costs for monopoly, duopoly, oligopoly and competitive markets. Our model also allows one option holder to have a greater production capacity than others. Under a monopolistic environment we find that the optimal option exercise strategy in real estate markets is dramatically opposite to that in a financial (warrant) market, indicating the importance of paying attention to the institutional details of the underlying market when analyzing option exercise strategies. Our model can be generalized to the pricing of convertible securities and capital investment decisions involving both stochastic revenues and costs under different types of market structures.
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The option framework developed in the finance field has been used extensively for analyzing investment decisions related to nonfinancial assets. For example, Brennan and Schwartz (1985) and Paddock, Siegel and Smith (1988) use an option approach to evaluate natural resource investments and offshore petroleum leases. McDonald and Siegel (1986), Majd and Pindyck (1987) and Ingersoll and Ross (1992) explicitly analyze the impact of option value on capital investment decisions. McDonald and Siegel (1985) and Berger, Ofek and Swary (1996) address the termination option of an investment project. Childs, Ott and Triantis (1998) use the option valuation framework for analyzing the capital budgeting decisions of interrelated projects. Grenadier (1995) and Grenadier and Weiss (1997) apply the real options concept to value lease contracts and technological innovations. Schwartz and Zozaya-Gorostiza (2000) design a real options approach for evaluating information technology investments. However, it might be fair to say that, among all the areas embracing the application of the real options concept, real estate markets seem to draw the most attention from researchers. This is probably due to the large size of real estate markets and the availability of empirical data, which make it easier for researchers to empirically test or draw inferences from the real options theories developed in the field.
Titman (1985) and Williams (1991) are the first to apply the real options concept to value real estate developments. (1) Quigg (1993) and Holland, Ott and Riddiough (2000), among others, provide empirical evidence demonstrating that models based on the real options concept can indeed predict property values in real estate markets. Grenadier (1999) and Childs, Ott and Riddiough (2002) extend the literature by examining the impact asymmetric information and noise have on the exercise strategies of developers. On the other hand, recognizing that a development decision is not made in isolation and one developer's decision affects the decisions of others, researchers also advance the literature by analyzing option exercise strategies in an equilibrium setting. Williams (1993) first derives symmetric equilibrium exercise strategies for real estate developers. (2) However, because this equilibrium assumes that developers will exercise their options simultaneously, it might not be suitable to describe the exercise strategies of certain types of markets. For example, if there are three developers in an office property market and the minimum size of an office building is 20,000 square feet, under a symmetric equilibrium, all three developers will wait until the market demand reaches a level of around 60,000 square feet for each to build an office with a size of around 20,000 square feet. However, in reality, one developer will start construction when the demand level reaches about 20,000 square feet. Developers might build simultaneously (with a proportional share) if there is a large demand in a short time period that can be shared by them (say a demand for 200 single-family units in one month to be shared by 10 developers). Given this, it might be fair to say that the symmetric equilibrium characterized by Williams (1993) better describes the behavior of single-family property markets with a large number of developers (say 10 to 50), but may not be suitable for commercial property markets with fewer developers (say two to nine).
Grenadier (1996) extends Williams' model to allow for sequential exercise in a duopoly market. However, since most real estate markets (and most markets in which the real options concept has been applied) are not duopoly markets, (3) it is not clear if the result derived for a duopoly market can be generalized to oligopoly markets. Furthermore, there are many important questions that cannot be answered by examining duopoly markets only. For example, will an increase in the number of players in a market affect the exercise decision of the players in the market? Are developers' exercise strategies the same under different market structures (monopoly, duopoly, oligopoly and competitive)? More importantly, both Williams' symmetric equilibrium and Grenadier's duopoly market equilibrium assume that all developers have an identical production capacity. In reality, some developers might have a higher production capacity than others. For example, in a market with five developable lots, one developer might own three lots and the other two own one each. Will the exercise strategies with this market setting differ from a market where five developers own one lot each (or, at the other extreme case, one developer owns all the five lots)? These questions can be better answered by a model that allows for sequential exercises by multiple players.
In Williams' (1993) and Grenadier's (1996) strategic exercise models, the exercise price (the construction cost) is assumed to be fixed over time. While this assumption is a standard one for stock options with a fixed maturity date, it may be problematic for real estate markets. (4) This is true because the maturity of real options is infinite and the optimal exercise time of an option might be the main focus of the model. Simply put, if a developer can build a property with the same cost today or 20 years from today, there must be an incentive for the developer to delay the exercise of the option. Williams (1991) is the first to recognize the importance of construction costs in option exercise decisions and use the ratio between revenues and costs as the state variable for a developer's exercise decision. However, to the best of our knowledge, no attempt has been made to include a stochastic ratio (between revenues and costs) in an equilibrium model that allows for sequential exercises. Indeed, if the level of construction costs is an important factor that drives a developer's construction decisions, a real options model that examines only one side of a profit equation may result in counterintuitive explanations for the phenomena observed in the real world. (5)
We advance the literature on real options by developing an equilibrium model that incorporates both stochastic demand and construction costs and allows multiple developers in the market to exercise their development options sequentially or simultaneously. (6) We also allow one option holder to have a greater production capacity than others in order to examine the impact of market power on the exercise strategies of the holders. We solve the closed-form solution of the model and derive a unique Markovian subgame-perfect equilibrium for option exercise strategies in monopoly, duopoly, oligopoly and competitive markets. This model is developed using real estate markets as the laboratory because the large amount of empirical evidence available in the field allows us to compare our model's predictions with phenomena observed in the real world. It is also interesting to note that under a monopoly we find that the optimal real-option exercise strategy in real estate markets is opposite to the optimal warrant exercise strategy described by Constantindes (1984). This indicates a need to pay careful attention to the institutional details of a market when developing option exercise strategies. Having said this, we also believe that our model can be revised for the pricing of convertible securities and the analysis of capital investment decisions involving both stochastic revenues and costs under different types of market structures.
The article is organized as follows. The next section explains briefly the real estate market upon which our model is based. Following that, we devote three sections to introduce our model framework, solve the equilibrium option values and derive equilibrium exercise strategies for developers and then present the implications of our model. The next two sections examine the exercise strategies of a developer under a monopoly and report the exercise strategies of developers when they have different production capacities. The last section concludes the article.
A Description of Real Estate Markets
The development decision in a real estate market can be viewed as the exercise of a call option. The exercise price is the construction cost of a building at the time of exercise and the underlying asset is a newly developed property (a building with the underlying land). An owner will develop the property only if the rental level of the property is high enough and/or the construction cost of the property is low enough to justify the exercise of the option. However, because both the rental level and construction cost are a function of the demand and supply levels in the market, one developer's exercise decision will affect the decision of other developers. The worst scenario is that all developers start construction to capture the same level of demand. All developers will suffer under this circumstance. To avoid the situation where all developers exercise together, developers will have to find an equilibrium set of exercise points so that the profit from the exercise is the same among all developers. With this set of equilibrium points, developers will start to build in sequence whenever the demand level and the construction cost level justify the exercise of an option. To derive this set of equilibrium exercise points, developers will have to estimate the growth rates (and the variances of the growth rates) of future rental levels and construction costs.
The uncertainty about future rents and construction costs seems to be quite high in certain real estate markets. Legg Mason reports the national mean rental growth rates of four property types during the 1990 to 2002 period. (7) The mean growth rates (and the standard deviation) of apartment, office, retail and industrial properties are 2.07% (2.61%), 1.29% (7.14%), 1.35% (2.68%) and 0.95% (3.91%), respectively. The variance in the rental growth rate is highest for industrial and office properties, where the standard deviation is about 4 to 5.5 times that of the mean growth rate. The variance in the growth rates of the construction costs seems to be comparable to that of the income growth rate. The McGraw Hill Construction Dodge report provides a 20-city average construction cost index during the 1909 to 2002 period. (8) For the entire 94-year period, the mean construction cost growth rate is 4.89% and the standard deviation is 8.39%. (The construction cost growth rate is -19.52% during the 1920 to 1921 period.) (9) It should be noted that the rental rate and construction cost should be more volatile at the metropolitan level than at the national level (as reported here).
The construction cost consists of two components. The first is a lump-sum initial building cost and the second is the ongoing repair, replacement and renovation cost. After a building is completed, the owner has to incur continuous repair and replacement costs in order to maintain the physical condition of the building. Furthermore, as Grenadier (1996) pointed out, the use of new technologies in new buildings will affect the competitive level of existing buildings. Given this, in order to keep competitive as technologies progress, an owner will have to renovate the building even if its physical condition is well maintained. Consequently, the exercise decisions of developers at later periods affect the ongoing maintenance, repair and innovation costs of existing buildings.
The rental growth rate and construction costs data reported above also indicate that construction costs do not move in tandem with the rental level in the same market. During the 1990 to 2002 period, the correlation coefficients between the construction cost and the rental levels among apartment, office, retail and industrial proprieties are -7.45%, 0.28%, -8.78% and -13.40%, respectively. The literature also indicates that both the levels of housing starts and new construction are inversely related to construction costs (see, e.g., Blackley (1999) and Somerville (1999)) and that the level of construction costs negatively affects the investment decisions on commercial properties (see, e.g., Holland, Ott and Riddiough (2000)). Capozza et al. (2002) also indicate that the level of real construction costs plays an important role in dampening real estate cycles. It might be fair to say that both the literature and empirical evidence underscore the importance of having both rental rate and construction cost as decision variables in a real options model for real estate developments.
The production capacities among developers differ. Schwartz and Torous (2003) calculate the Herfindahl ratio using information of the top ten developers of office buildings in 34 metropolitan areas during the 1998 to 2002 period. The mean of the Herfindahl ratio for the 34 metropolitan areas is 0.11. However, the ratios seem to differ dramatically among big cities, with a low of 0.01 in New York City and a high of 0.85 in Indianapolis. Their evidence seems to confirm that developers in real estate markets might have uneven production capacities. Given this, it is also important to explore developers' exercise strategies when their production capacities differ within a given market.
Model Framework
We develop our real option model based on the market characteristics described in the previous section. We start our model by assuming a market with N identical parcels of vacant land owned by N different developers. Each developer has an option to develop the land at any given time in the future. This assumption will be relaxed in later sections where we allow one developer to hold all the N parcels of land or one developer to hold more parcels of land than other developers. We also assume that this N is not too large so that a developer's option value is not negligible (and her/his exercise strategies are relevant).
We assume that a new property will begin to produce rents at the time of completion. The rental level will be determined by a downward-sloping inverse demand function. This specification recognizes that an increase in the number of competing properties will reduce the rent levels in the market. The inverse demand function is subjected to continuous demand shocks and can be specified as
P(t) = X(t)D[Q(t)], (1)
where P(t) is the rent inflow at time t, Q(t) is the level of property supply at time t and D(*) is a differentiable function with D' < 0. (10) We define X(t) as a multiplicative demand shock that follows a geometric Brownian motion, or
[dx]/X = [[mu].sub.X] dt + [[sigma].sub.X] d[z.sub.1]. (2)
The constant [[mu].sub.X] is the instantaneous conditional expected percentage change in X per unit of time. The constant [[sigma].sub.X] is the instantaneous conditional standard deviation per unit of time with respect to [z.sub.1].
The specification of the construction cost (covering both the initial building and ongoing replacement costs) needs to reflect the fact that an increase in the number of competing properties will increase construction costs in the market. While we recognize that the initial building cost should be affected by the exercise decision of other developers at the time of exercise, in our continuous time model framework it is not feasible to specify a fixed time period within which the initial building cost increases with multiple developments. (11) For this reason, we specify the initial building cost as a series of cash outflows in the future so that the construction decisions of developers can affect one another in a continuous (and diminishing) manner. Under this specification, when two developers exercise their options within a short time period, the impact on construction costs is greater than if they exercise the options far apart. The ongoing replacement cost is also specified as a series of cash outflows in the future and a function of the number of competing properties in the market.
The construction cost outflow at each point will be determined by an upward-sloping inverse supply function. The inverse supply function is subjected to continuous supply shocks and can be specified as
C(t) = (1 + k)I(t)S[Q(t)], (3)
where C(t) is the cost outflow at time t. (1 + k) indicates that the cost outflow is the summation of the initial building cost outflow and the ongoing replacement cost outflow. With this specification, we implicitly assume that the ongoing replacement cost is proportional (k times) to the initial building cost. Q(t) is the level of property supply at time t, and S(*) is a differentiable function with S' > 0. We define I(t) as a multiplicative construction cost shock that also follows a geometric Brownian motion. We specify I(t) as
[dI]/I = [[mu].sub.I] dt + [[sigma].sub.I]d[z.sub.2]. (4)
The constant [[mu].sub.I] is the instantaneous conditional expected percentage change in I per unit of time. The constant [[sigma].sub.I] is the instantaneous conditional standard deviation per unit of time with respect to [z.sub.2]. Both the developers' initial building costs and ongoing replacement costs are stochastic and follow the same construction cost shock.
The instantaneous correlation coefficient [rho] between d[z.sub.1] and d[z.sub.2] is characterized by
d[z.sub.1]d[z.sub.2] = [rho] dt. (5)
We assume that construction will take [delta] years to complete. This assumption implies that, when a developer decides to exercise a development option (a decision to build) at time [tau], the property will not begin to receive rents until time...
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