Optimal timing of real estate development a real options and game theoretical framework

OPTIMAL TIMING OF REAL ESTATE DEVELOPMENT A REAL OPTIONS AND GAME THEORETICAL FRAMEWORK CHU YONGQIANG (B.S Beijing University) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE (ESTATE MANAGEMEMENT) DEPARTMENT OF REAL ESTATE NATIONAL UNIVERISITY OF SINGAPORE 2004 ACKNOWLEDGEMENT The author would like to express his gratitude to the following people, who made the completion of the thesis possible: Associate Professor Sing Tien Foo, My supervisor, for his invaluable guidance and advice throughout the whole process of my research in National University of Singapore His kindness and gentle personality also impressed me during the past two years It is my great fortune to be his student My wife, Ning Weiyi, for her love, support, encouragement and forgiveness Sorry, honey, I may not be able to accompany you for the next three years My parents, my sister, and my lovely niece, for their constant encouragement and love I owe a lot to my parents, especially my father, who passed away when I was writing this thesis Forgive me, father, and rest in peace Lastly to all my friends, who have helped me in one way or another during this research Chapter 1: Introduction 1.1Background 1 1.1 NPV vs Real Options .1 1.1.2 Market Structure: Competitive Monopoly or Oligopoly? 1.1.3 Symmetric or Asymmetric Investors 1.1.4 Complete Information vs Incomplete Information 1.2 Motivation 1.3 Research Questions .7 1.4 Research Methodology and Framework .7 1.4.1 Option Pricing Theory 1.4.2 Game Theory 1.5 Organization 1.5.1 Literature Review 1.5.2 Basic Real Estate Investment Model 10 1.5.3 Monopoly Real Estate Investment Model 10 1.5.4 Real Estate Investment Model in a Symmetric Duopoly Framework 11 1.5.5 Real Estate Investment Model in an Asymmetric Duopoly Framework 11 1.5.6 Duopoly Real Estate Investment under Incomplete Information 12 1.5.7 Conclusion .12 Chapter Literature Review .13 2.1 Option Pricing Theory .13 2.2 One Developer Real Options Theory 17 2.3 Game Theory 25 2.4 Real Options under Competition (Option Games) 28 Chapter One Firm Model Facing Exogenous Rental Flow 37 3.1 Model Assumptions 37 3.2 The Model 38 3.3 Neoclassical Cases 41 3.4 Numerical Example 41 3.4.1 The Optimal Timing .42 3.3.2 The Developer Value 44 3.5 Conclusion 45 Chapter Monopoly Real Estate Developer Model 46 4.1 Introduction .46 4.2 Problem Specification .49 4.3 The Model 51 4.4 Numerical Results and Comparative Static Analysis 57 4.4.1 The Volatility Effect 58 4.4.2 Demand Effect 61 4.4.3 Other Effects 63 4.5 Conclusion 64 Chapter Optimal Timing of Real Estate Development under Symmetric Duopoly .67 5.1 Introduction .67 5.2 Problem Specification and Model Assumptions .68 i 5.3 The Developers’ Value 70 5.3.1 The Follower’s Value .70 5.3.2 The Leader’s Value .72 5.4 The Equilibrium Exercise Strategies .73 5.4.1 Y0 < YL 73 5.4.2 YL < Y0 < YF .74 5.4.3 Y0 ≥ YF 74 5.5 Conclusion 76 Chapter Optimal Timing of Real Estate Development under Asymmetric Duopoly 78 6.1 Introduction .78 6.2 Model Specification 79 6.3 The Option Values of the Leader and The follower 82 6.4 The Equilibrium Exercise Strategies .91 6.4.1 Equilibrium Strategy when α1 ∈ [α1* , +∞) 91 6.4.2 Equilibrium strategy when α1 ∈ (α , α1* ) 95 Chapter Real Estate Development under Incomplete Information .99 7.1 Introduction .99 7.2 Model Specification 101 7.3 Equilibrium Strategies under Incomplete Information 103 7.4 Conclusion 117 Chapter Conclusion and Summaries 124 8.1 Contributions 124 8.2 Limitations 126 8.3 Further Directions 126 Bibliography 128 ii Summary This study is trying to apply the real options theory in real estate development, as real estate investment is irreversible and heterogeneous This study extends the existing literature in the following directions: (i) a monopoly real options model is proposed to examine the optimal timing and intensity simultaneously; (ii) the symmetric duopoly model is extended to asymmetric duopoly games; (iii) the incomplete information is incorporated in examine the asymmetric model… The Monopolistic model shows that the uncertain exogenous economic shock and the demand factors contribute to the option value of real estate development In the asymmetric model, the sub-game perfect Nash equilibrium is derived under different levels of comparative advantage for two different developers In the incomplete information model, a set of Bayesian Nash equilibrium is derived based on the results obtained in asymmetric duopoly model i List of Tables 3-1 The Basic Values of Relevant Variables…………………………………… 42 4-1 Input Assumptions for Numerical Analyses ……………………………… 58 4-2 Comparative Statics ………………………………………………………….64 7-1 Equilibrium Strategies ( ξ > ξ * , α1 = θξ , α = θξ )………………………… 119 7-2 Equilibrium Strategies ( ξ > ξ * , α1 = θξ , α = θ / ξ )……………………… 120 7-3 Equilibrium Strategies ( ξ > ξ * , α1 = θ / ξ , α = θ / ξ )…………………… 120 7-4 Equilibrium Strategies ( ξ < ξ ** , α1 = θξ , α = θ / ξ ) ………………….…121 7-5 Equilibrium Strategies ( ξ < ξ ** , α1 = θξ , α = θξ )……………………… 122 7-6 Equilibrium Strategies ( ξ < ξ ** , α1 = θ / ξ α = θ / ξ )…………………… 123 ii List of Figures 3-1 Trigger Value as a Function of Uncertainty……………………………….43 3-2 Developer Value as a Function of Rental Flow…………………………….44 4-1 Value Function of the Developer………………………………………… 54 4-2 Volatility Effect on Option Trigger Value………………………………… 58 4-3 Volatility Effect on Optimal Intensity……………………………………….59 4-4 Value of the Development Option ………………………………………… 60 4-5 Effects of Rental Sensitivity of Demand on Optimal Timing ……………….62 4-6: Optimal Intensity of Different Price Sensitivity of Demand…………… …63 4-7: Development Option Value………………………………………………….64 6-1 Developer 1’s Value………………………………………………………… 87 6-2 Developer 2’s Value When α1 > α1* > α ……………………………………….89 6-3 Developer 2’s Value When α * > α > α ……………………………… 90 iii Chapter 1: Introduction 1.1Background 1.1 NPV vs Real Options In the neoclassical economics, investment is an act of incurring an immediate cost in anticipation of future rewards Real estate investment is one of the most important categories of asset class of investment Investment is risky and no one can guarantee how much the rewards will be over a fixed holding period There is always uncertainty over the future market condition at the time when an investment decision is made How should an investor, facing uncertainty over future rewards, decide whether to invest or not The neoclassical economic theory offers a standard approach to evaluate the feasibility of an investment: First, investor should calculate the present value of the expected stream of profits that the investment project will generate Second, they should calculate the present value of the expected expenditure required for the investment Finally, they determine whether the difference between the two, which is known as Net Present Value (NPV), is greater than If the answer is yes, it is feasible to invest in the project Although the NPV rule has been used widely, some of the underlying assumptions appear to be unrealistic It is myopic to assume that an investment is reversible It implies that a wrong investment decision can be undone and the investment costs can be recovered should the market conditions turn out to be worse than expected The reversible investment decision is a now or never decision, that is, either the investor invests now or never invests These conditions may hold for some investment, they are, however, not satisfied in most investment decisions In real estate investment, irreversibility and the possibility of delaying an investment decision are important characteristics The recent development of the option pricing theory greatly challenges the propositions of neoclassical investment models An investment opportunity is regarded as an option an investor has a right but not an obligation to buy an asset (which is referred to, in this context, a finished project that will generate future cash flow) at some future time When an investor makes an irreversible investment, it kills the option of waiting to invest The option to invest, like a financial call option, does have value Thus the exercise of the option is equivalent to giving up the option to wait for possible increase in the value of an underlying asset, which can be viewed as opportunity cost foregone, which must be included as part of the investment cost Taking into consideration of this option value, the NPV rule must be modified as: invest when the present value of the expected stream of future income is at least as large as the present value of the expected expenditure plus the opportunity cost, that is, the value of option of waiting to invest In Trigeorgis (1996b), the new investment rule is defined based on the new concept of expanded net present value, that is: Expanded net present value=standard (static) net present value of expected cash flows +option premium Studies have shown that the opportunity cost of investing can be significant, and ignoring it can be erroneous Like the financial option, this opportunity cost is highly sensitive to uncertainty of the future cash flow To differentiate this investment from the standard options on financial asset, a new term, real option is used in the literature The study aims to use real options theory to analyze real estate investment, especially the timing problems in real estate development 1.1.2 Market Structure: Competitive Monopoly or Oligopoly? In traditional real options model, the market structure is not clearly defined Although most of the literature assumes that there exists only one firm in a market, in the literature the investment payoff, which always follows a geometric Brownian motion, is assumed to be exogenously determined The firm under this framework, is, therefore modeled as a price-taker, which is a key characteristic of competitive market But the market structure is not explicitly stated in most of the existing literatures, they model the only firm in the market, and assume it as a price taker, which may be misleading as if there is only firm in the market, and thus the market cannot be competitive In this study, I analyze real estate investment options that are modeled within clearly defined market structure, either in a monopoly market or in an oligopoly Table 7.6 Equilibrium Strategies ( ξ < ξ ** , α1 = θ / ξ α = θ / ξ ) Initial Economic Developer Shock Y0 ∈ (0, Y2aL2 ) Developer Transmission One developer T = inf{t ≥ : Y ≥ min{YL , ( at β1 r−µ )( )e( r − µ )τ I }} β1 − (θξ ) D (1) other One β1 r−µ )( )e( r − µ )τ I ) β1 − (θξ ) D(2) β1 r−µ )( )e( r − µ )τ I } β1 − (θ / ξ ) D (2) developer develops is revealed to the others Both developers’ until revealed to the One β1 r−µ )( )e( r − µ )τ I } β1 − (θ / ξ ) D (2) developer , at the other at is others Both developers’ information is revealed to the T = inf{t ≥ : Y (t ) ≥ ( β1 r−µ )( )e( r − µ )τ I , +∞ β1 − (θ / ξ ) D (2) parameters parameters T = inf{t ≥ : Y (t ) ≥ YN } Y0 ∈ (( Both developers’ immediately, while the other wait T = inf{t ≥ : Y (t ) ≥ ( Y0 ∈ (Y2bL1 ,( the at T = inf{t ≥ : Y (t ) ≥ ( Y0 ∈ (Y2aL2 , Y2bL2 ) Information β1 r−µ )( )e ( r − µ )τ I } β1 − (θ / ξ ) D (2) other If one developer develops first, No then is transmitted the other develops information immediately 123 Chapter Conclusion and Summaries 8.1 Contributions This study aims to developed a pure theoretical framework for analyzing optimal development timing decisions, that will be applicable in both monopoly and oligopoly market The followings are the finding in thesis that will contribute to better understand real options in real estate The first addresses explicitly the real options model in a monopolistic market structure The existing one firm real options model does not fit well into a monopolistic market framework or into a competitive market structure There is no competition in the one firm real options model, which is contradictory to the competitive market theory; at the same time the agent in the model is assumed to be price-taker, which is again inconsistent with the assumptions in the monopoly theory By explicitly assuming the market structure of monopoly, the optimal timing and optimal intensity problem can be solved simultaneously The optimal intensity is obtained endogenously when the market demand and supply are in equilibrium, compared to the model of deriving the optimal density using a Cobb-Douglas production function in a competitive market Technically this derivation is more complicated, nonetheless more realistic, especially for the case of real estate 124 development The results show that both the cash flow uncertainty and the demand function have significant impact on the optimal timing decision, and the demand elasticity also plays an important role in determining the development option value This thesis also examines the effect of asymmetric duopoly market, while most of the existing literature concentrates on only symmetric duopoly cases The asymmetric duopoly differs from the symmetric duopoly model in the equilibrium strategies Different level of asymmetry will significantly affect the equilibrium strategies, and the symmetric case can be modeled as an extreme case in this study The equilibrium strategies under different levels of asymmetry are theoretically evaluated Asymmetry gives the comparatively advantageous developer more power and flexibility in selecting the optimal timing of development that will maximize his profit In some cases the stronger developer can even have monopoly power, in the sense that he can ignore the preemptive competition from other developer This situation cannot be observed in symmetric cases The existing literature sparse on option game models under incomplete information Grenadier (2000) is one of the researches that have made significant contribution of research in this area In this study, I followed the idea in Grenadier (2000) that the information is only revealed only through option exercising actions While In Grenadier (2000) model, he assumes an exogenous payoff when exercising the option In our model, the payoff is 125 endogenously determined in an explicit duopoly model The equilibrium strategies are derived based on the complete information asymmetric duopoly model 8.2 Limitations However, being a master degree thesis, this study has also many limitations: The model presented in this study is not easily extended to the oligopoly case with more than players It will add great complexity to the model if the oligopoly model is analyzed in the same way There are no empirical tests conducted in this study The theory is not verified with real world real estate development cases due to the difficulty in obtaining relevant data There is, however, slight inconsistency with respect to the optimal density questions For brevity reason the optimal density that are modeled in the simple monopolistic cases is not extended to the duopoly games in the subsequent chapters In the duopoly models the development intensity is normalize to minimize the complexity of the mathematical derivation 8.3 Further Directions Limitations are not an obstacle Some further directions of research can be carried out along the following directions 126 A more generalized oligopoly model in the real options and game theoretical framework can be explored, like those proposed by Wang and Zhou (2004), who extend Grenadier (1996a) framework to a general oligopoly setting The present model could be extended to a general oligopoly market setting to allow more general applications The empirical testing of the theory obtained in this study can be done by collecting vast amount of data of real estate development There is already some empirical works done in testing the one firm real options model Therefore, empirical tests of the proposed duopoly models can be proposed in the future The study of the optimal intensity together with the optimal timing problem can be modeled by using more sophisticated mathematical tools and also numerical method, given the fact that analytical solutions are difficult to obtain 127 Bibliography Baldursson, F and Karatzas, I 1997 Irreversible Investment and Industry Equilibrium, Finance and Stochastics, 1, 68-89 Bertrand, J 1983, Théorié Mathématique de la richesse sociale, Journal des Savants Black, F and Scholes, M., the Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81, 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Players: The Case of Real Estate Markets, Working Paper, Cal State University Williams, J, 1991, Real estate Development as an Option, Journal of Real Estate Finance and Economics 4, 191-208 136 Williams, J, 1993, Equilibrium and Option on Real Assets, Review of Financial Studies, 6, 825-850 137 [...]... presence of other options, is generally less than its value in isolation, and it declines as more options are present In the field of real estate, there are also many literatures on optimal timing of 22 real estate investment Real estate investment is at least partially, if not all, irreversible and large amount of capital outlays are incurred when an investment option is exercised, therefore it is important... some assumptions to capture the special features of real estate development Another strand of real options research on real estate market is on lease valuation Grenadier (1995) uses real options approach to derive the entire term structure of lease rates, and thus provide an equilibrium framework for pricing a wide variety of leasing contracts using the same methodology Grenadier (1996b) provides a unified... has also gained attentions by many companies and interesting applications of real option models have been developed Real estate investment is very suitable to be analyzed using the real options 5 theory Real estate investment is obviously irreversible, or at least partially irreversible Real estate market is also full of uncertainties because of the irregularities in real estate cycles Compared to other... options analysis, especially when the focus is on timing games Thirdly, real estate markets are subject to real estate cycles, which may be different from the common business cycles Thus it is challenging to use real estate as a subject of this research and I hope real option theory can explain different investment behavior in the market and help better understanding of real estate markets, especially... types of investment, real estate investment has some special characteristics Firstly, real estate is heterogeneous due to its spatial characteristics The heterogeneity of real estate investment means that no two real estate projects are identical Secondly, real estate development process is complex and it takes relatively long time to complete The so called time-to-build feature cannot be ignored in real. .. Motivation The option pricing theory, since the seminal paper by Black and Scholes (1973), stimulated the growing literature on real options Real options has become a very important parts of the finance research, especially in the corporate finance research Researches have been expanded rapidly in various industries, such as natural resources, R&D and others The importance of real options theory has also... project payoff and investment cost is irreversible Capozza and Li (1994) propose a real option model of capital replacement, which is then applied to urban land market In the model the optimal capital intensity and optimal timing is determined simultaneously, and their results show that intensity interacts significantly with the timing, taxes and project value The ability to vary intensity raises hurdle...market In the oligopoly model, I put real estate investment problem in both real options and the game theoretical methodology As it is commonly accepted that real estate market are not a competitive market due to the special characteristics of real estate (i.e the product is heterogeneous); therefore real estate investment behavior will not be appropriate to be examined in a competitive market... monopoly real estate development market, what is optimal timing and density of development? (2) In an asymmetric duopoly real estate market, is there an equilibrium strategy for both developers in choosing their optimal timing? (3) If the comparative advantages of developers are private information, what is the equilibrium strategy and how the equilibrium strategies will be different compared with the case... classical game theoretical framework It is neither a standard form game nor extensive form games In the stochastic game, the sequence of movement itself is endogenous, rather than exogenous as in the extensive form games In the complete information game, the equilibrium strategy is the Markov Sub -game perfect Nash equilibrium, and the equilibrium in incomplete information game is the Markov sub-game ... (1987), and relaxes some assumptions to capture the special features of real estate development Another strand of real options research on real estate market is on lease valuation Grenadier (1995)... research and I hope real option theory can explain different investment behavior in the market and help better understanding of real estate markets, especially real estate cycles The real options. .. in both real options and the game theoretical methodology As it is commonly accepted that real estate market are not a competitive market due to the special characteristics of real estate (i.e