A Bayesian Approach to Real Options:

A Bayesian Approach to Real Options: The Case of Distinguishing between Temporary and Permanent Shocks Steven R. Grenadier and Andrei Malenko Stanford GSB BYU - Marriott School, Finance Seminar March 6, 2009 1 / 38

Traditional Real Options Models Important and in uential Based on a simple analogy Focus on uncertainty of future shocks Option to wait to achieve a higher cash ow level 2 / 38

Traditional Real Options Models Important and in uential Based on a simple analogy Focus on uncertainty of future shocks Option to wait to achieve a higher cash ow level Implications: Fixed upper trigger Record-setting news principle Maturity structure of project cash ows is irrelevant, for any given PV 2 / 38

This Paper: A Bayesian Approach Build a real options model that combines two sources of uncertainty: Uncertainty of future shocks Uncertainty regarding fundamental nature of past shocks 3 / 38

This Paper: A Bayesian Approach Build a real options model that combines two sources of uncertainty: Uncertainty of future shocks Uncertainty regarding fundamental nature of past shocks Consider a speci c application: Cash ow shocks can be fundamental (permanent) and non-fundamental (temporary) Firm is unable to distinguish the nature of past shocks As time passes, the rm updates its prior about past shocks: the longer a shock persists, the more likely it was permanent. 3 / 38

Results An additional real option: Option to learn about the nature of past shocks 4 / 38

Results An additional real option: Option to learn about the nature of past shocks Implications for investment dynamics: investment trigger depends on timing of past shocks investment not only in booms, but also at times of stable or decreasing cash ows sluggish response to cash ow shocks maturity structure of cash ows matters even for projects with the same PVs 4 / 38

Key Assumptions: Motivation 1. Both permanent and temporary shocks are important: Cash ows are more volatile than asset values Correlation between cash ows and asset values is far from being perfect Many shocks revert to mean 5 / 38

Key Assumptions: Motivation 2. Distinguishing between permanent and temporary shocks is often di cult: Most important, distinguishing between temporary and permanent shocks to commodity prices can be extraordinary di cult. The swings in commodity prices can be too large and uncertain to ascertain their causes and nature. The degree of uncertainty about duration of a price shock varies. For example, market participants could see that the sharp jump in co ee prices caused by the Brazilian frost of 1994 was likely to be reversed, assuming a return to more normal weather. By contrast, most analysts assumed that the high oil prices during the mid-1970s and early 1980s would last inde nitely. (World Bank s Global Economic Prospects annual report) 6 / 38

Commercial Real Estate (1980 s) O ce space in the oil-patch cities experienced incredible growth during the early 1980 s. For example, over the thirty-year period from 1960 through 1990, over half of all o ce construction in Denver and Houston was completed over the four-year interval: 1982 1985. Construction likely initiated over the period 1979-1983. During this period, o ce vacancies were at record levels (around 30%). What were developers thinking? 7 / 38

Domestic Crude Oil Prices (In ation-adjusted) 8 / 38

What were developers thinking? Quote from a Dallas o ce developer: We thought we were in the real estate business, but we were really in the oil business. The belief that high oil prices were the result of a permanent shock (OPEC) likely spurred on o ce development in oil-producing cities. Of course, this proved incorrect, but that is with hindsight. 9 / 38

Research Idea: Illustration 1.3 Cash flow X(t) 1.2 1.1 1 0.9 Shock X(t) 0.8 0 0.5 1 1.5 2 2.5 3 3.5 4 Time t 10 / 38

Research Idea: Illustration Traditional Models 1.3 Cash flow X(t) 1.2 1.1 1 0.9 Trigger Shock X(t) 0.8 0 0.5 1 1.5 2 2.5 3 3.5 4 Time t 11 / 38

Research Idea: Illustration Traditional Models 1.3 Trigger Cash flow X(t) 1.2 1.1 1 0.9 Shock X(t) 0.8 0 0.5 1 1.5 2 2.5 3 3.5 4 Time t 12 / 38

Research Idea: Illustration Bayesian Approach 1.3 Trigger Cash flow X(t) 1.2 1.1 1 0.9 Shock X(t) 0.8 0 0.5 1 1.5 2 2.5 3 3.5 4 Time t 13 / 38

Model: Overview A rm has an irreversible investment opportunity: At any time τ 0 pay an investment cost I > 0 and receive a perpetual cash ow X (t), t τ 14 / 38

Model: Overview A rm has an irreversible investment opportunity: At any time τ 0 pay an investment cost I > 0 and receive a perpetual cash ow X (t), t τ Cash ow process X (t) is subject to two types of shocks: A permanent shock occurs with intensity λ 1 and increases X (t)! (1 + ϕ) X (t) forever A temporary shock occurs with intensity λ 2 and increases X (t)! (1 + ϕ) X (t) temporarily until it reverts with intensity λ 3 14 / 38

Model: Versions 1 Simple Model only one shock which can be either permanent or temporary no drift or di usion 15 / 38

Model: Versions 1 Simple Model only one shock which can be either permanent or temporary no drift or di usion 2 Model with interaction between Bayesian and Brownian uncertainties only one shock presence of drift and di usion 3 Model with an unlimited number of shocks unlimited number of shocks presence of drift and di usion 15 / 38

After a Shock Reverses Let H (X ) be the investment option value after the shock reverses Before X (t) reaches the investment threshold X : rh (X ) = αxh 0 (X ) + σ2 2 X 2 H 00 (X ) This is solved subject to the appropriate boundary conditions: Value-matching: Smooth-pasting H (X ) = X r α H 0 (X ) = 1 r α I 16 / 38

After a Shock Reverses The solution is standard (e.g., Dixit and Pindyck (1994)) 17 / 38

After a Shock Reverses The solution is standard (e.g., Dixit and Pindyck (1994)) Investment trigger: 2 where β = 1 4 σ 2 α X = β β 1 I σ 2 s + α 2 3 σ 2 2 + 2rσ 2 2 5 > 1 17 / 38

After a Shock Reverses The solution is standard (e.g., Dixit and Pindyck (1994)) Investment trigger: 2 where β = 1 4 σ 2 α X = β β 1 I σ 2 s + α 2 3 σ 2 2 + 2rσ 2 2 5 > 1 Option value: H (X ) = ( X X β X r α I, if X < X X r α I, if X X 17 / 38

While the Shock Persists: Bayesian Learning Process p (t): conditional probability of the past shock being temporary Using Bayes rule: dp (t) = λ 3 p (t) (1 p (t)) dt with p (t 0 ) = λ 2 λ 1 + λ 2 Hence, p (t) = λ 2 λ 1 e λ 3(t t 0 ) + λ 2 18 / 38

While The Shock Persists: Option Value Let G (X, p) be the investment option value while the shock persists Before X (t) reaches the investment threshold X (p): rg (X, p) = αxg X (X, p) + σ2 2 X 2 G XX (X, p) X λ 3 p(1 p)g p (X, p) + pλ 3 H G (X, p), 1 + ϕ where H () is the option value after the shock reverts 19 / 38

While The Shock Persists: Option Value This is solved subject to the appropriate boundary conditions: Value-matching: 2 1 G ( X (p), p) = 4(1 p) r α + p 1 + r α + λ 3 Smooth-pasting with respect to X : λ 3 (r α)(1+ϕ) 1 G X ( X (p), p) = (1 p) r α + p 1 + r α + λ 3 Smooth-pasting with respect to p: 2 G p ( X (p), p) = 4 1 r α + 1 + r α + λ 3 λ 3 (r α)(1+ϕ) 3 λ 3 (r α)(1+ϕ) 3 5 X (p) I 5 X (p) 20 / 38

Investment Trigger Brownian e ect z } { σ 2 X (p) = ri + 2 X (p) 2 G XX ( X (p), p) X (p) X (p) +pλ 3 H I 1 + ϕ (1 + ϕ) (r α) {z } Bayesian e ect 21 / 38

Investment Trigger 1.4 1.35 1.3 Investment trigger X(p) 1.25 1.2 1.15 Brownian effect (σ =.10) Brownian effect (σ =.05) 1.1 1.05 Bayesian effect 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Firm's belief p 22 / 38

Before the Shock Arrives Let F (X ) be the investment option value before the shock arrives. Before X (t) reaches the investment threshold ˆX : rf (X ) = αxf 0 (X ) + σ2 2 X 2 F 00 (X ) + (λ 1 + λ 2 ) (G (X (1 + ϕ), p 0 ) F (X )). This is solved subject to the usual boundary conditions 23 / 38

Graphical Illustration: A Simulated Sample Path 1.4 1.35 1.3 1.25 Cash flow X 1.2 1.15 1.1 1.05 1 Shock 0.95 X(t) 0.9 0 0.5 1 1.5 2 2.5 3 Time t 24 / 38

Graphical Illustration: A Simulated Sample Path 1.4 1.35 Trigger 1.3 1.25 Cash flow X 1.2 1.15 1.1 1.05 1 Shock 0.95 X(t) 0.9 0 0.5 1 1.5 2 2.5 3 Time t 25 / 38

Graphical Illustration: A Simulated Sample Path 1.4 1.35 Trigger 1.3 1.25 Investment Cash flow X 1.2 1.15 1.1 1.05 1 Shock 0.95 X(t) 0.9 0 0.5 1 1.5 2 2.5 3 Time t 26 / 38

Model Implications Two forces that lead to later investment: the standard option to wait for realizations of future shocks option to learn more about past shocks 27 / 38

Model Implications Two forces that lead to later investment: the standard option to wait for realizations of future shocks option to learn more about past shocks Sluggish response of investment to past shocks: Consistent with slow response of aggregate investment and labor demand to shocks (e.g., Caballero and Engel (2004)) Investment in the face of stable or declining cash ows: A potential explanation for the period of rapid construction in Denver and Houston during 1982-1985 (Grenadier (1996)) 27 / 38

Importance of Timing of Project Cash Flows Consider the simple model (one shock, no drift or di usion) with one alteration Upon exercise at time τ, the rm gets a perpetual stream of payments 1 + k e k (t τ) X (t), t τ r k : measure of how "front-loaded" the project is 28 / 38

Importance of Timing of Project Cash Flows 1.2 1.18 Investment trigger X(t) 1.16 1.14 1.12 1.1 1.08 1.06 k = 0 k = 1 1.04 1.02 k = 5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Firm's belief p Investment in a more "front-loaded" project occurs earlier even when the projects are identical in other dimensions Example: immediate sale of the asset vs development of an oil well 29 / 38

Unlimited Number of Shocks Now, the belief process p (t) is a vector p (t) = (p 1 (t), p 2 (t),...) 0 : p k (t) is the probability at time t that there are k outstanding temporary shocks the dynamics of p k (t): dp k (t) dt = λ 3 p k (t) k! p i (t) i i =1 30 / 38

Unlimited Number of Shocks The investment threshold X (p) satis es h i X (p) = λ 3 i=1 X (p) p i i G 1+ϕ, p (p) + X (p) I S 1+ϕ, p (p) +ri + σ2 2 X (p) 2 G XX ( X (p), p). where p (p) is the updated vector of beliefs if the shock reverts immediately and S (X, p) is the present value of project cash ows. 31 / 38

Conclusion Introduce a novel kind of real options problem Bayesian uncertainty about past shocks leads to valuable option to learn and, hence, sluggish response to shocks investment at times of stable or decreasing cash ows importance of maturity structure of project cash ows Uncertainty about past shocks may be as important as uncertainty about future shocks 32 / 38

Real Options in a Signaling Equilibrium The paper presented studies learning of a decision-maker about the nature of the project. However, there is an alternative type of learning: The decision-maker and outsiders are asymmetrically informed about the nature ( type ) of the project. The exercise decision reveals information to the outsiders, because di erent types exercise the option at di erent thresholds. The payo of the decision-maker depends on the perception of his type by the outsiders. 33 / 38

Mathematical Formulation The option payo function from exercise at time τ is α (P (τ) θ) + G (P (τ)) W θ, θ, where W θ, θ is the function that captures the e ect of the outsiders incorrect beliefs. We normalize W (θ, θ) = 0. Then, for a given vector of inference function of the outsiders θ the informed agent s maximization problem is ˆP max ˆP n ˆP β α ˆP θ + G ˆP W θ ˆP, θ o. 34 / 38

Equilibrium Conditions In equilibrium, the inference function must be consistent with the agent s investment strategy: θ ( P (θ)) = θ 8θ 2 [θ, θ]. 35 / 38

Equilibrium Conditions In equilibrium, the inference function must be consistent with the agent s investment strategy: θ ( P (θ)) = θ 8θ 2 [θ, θ]. Hence, the rst-order condition yields the equilibrium di erential equation for P (θ): d P (θ) dθ = P (θ) G ( P (θ)) W θ (θ, θ) /α. (β 1) P (θ) βθ The di erential equation is solved subject to the appropriate boundary condition: If W θ θ, θ < 0, then P θ = P θ = θβ/ (β 1). θ, θ > 0, then P (θ) = P (θ) = θβ/ (β 1). If W θ 35 / 38

Equilibrium Properties The agent invests a suboptimal threshold in order to shift the beliefs of the outsiders. The direction of the e ect depends on whether the agent bene ts more when the outsiders believe that the project is more or less pro table: The agent bene ts from better beliefs ) exercise is earlier than in the symmetric information case. The agent bene ts from worse beliefs ) exercise is later than in the symmetric information case. Signal-jamming occurs: The outsiders infer the type correctly anticipating the suboptimal exercise by the agent. 36 / 38

Applications: Earlier Exercise Venture Capital Grandstanding: Perceived type of the general partner a ects the amount of new investment in his fund. Better projects are taken public earlier. Hence, an inexperienced venture capital rm has incentives to take the project public earlier in order to signal a better type. Managerial Myopia: Managerial compensation depends on the current stock price (e.g., because of incentives to exercise executive options). A manager is better informed about quality of the investment project than shareholders. Hence, a manager has incentives to invest at a lower threshold in order to signal higher quality. 37 / 38

Applications: Later Exercise Cash Flow Diversion: A portion of project value is observable only by the manager. The manager can divert it for his own consumption. Hence, a manager has incentives to invest later in order to signal worse quality of the project and be able to divert more. Sequential Investment in Product Markets: Two rms are asymmetrically informed about the value of a new product. Investment by the informed rm reveals information about the value of the product to the uninformed rm. Hence, the informed rm has incentives to invest later in order to understate pro tability of the project and thereby deter entry of the uninformed rm. 38 / 38