Noise, Real Estate Markets, and Options on Real Assets: Applications*
Paul D. Childs Carol Martin Gatton College of Business and Economics University of Kentucky, Lexington, KY 40506-0034 (859) 257-2490 /
[email protected] Steven H. Ott Belk College of Business and Administration University of North Carolina at Charlotte, Charlotte, NC 28223-0001 (704) 547-2744 /
[email protected] Timothy J. Riddiough Department of Real Estate University of Wisconsin-Madison 53706 (608) 262-3531 /
[email protected] April 2001 Abstract This paper explores the implications of noise in real estate markets by examining two applications that focus on real option valuation and optimal exercise policy. The first application examines an imperfectly competitive market for real estate development in which agents compete over the timing of lead investment. Information spillover and freerider incentives are shown to cause significant delay in lead investment. Delay together with competitive response once pioneering development has occurred explains observed patterns of development in blighted urban land markets and for large-scale multi-stage development projects. The second application examines valuation and default exercise policy of risky coupon debt that is secured by a lease-encumbered noisy real asset. Our main result is that, relative to the noiseless case, the borrower will generally delay default exercise until the noisy signal of asset value is far into-the-money. This finding provides an information-based explanation for the apparent under-exercise of the mortgage default option that has been observed in the literature.
We are grateful to Jerome Detemple, David Geltner, Steve Grenadier, David Mauer, Alex Triantis and Bill Wheaton for helpful discussions, and to workshop participants at the University of Cincinnati, University of Connecticut, University of Miami, MIT, University of North Carolina, University of North Carolina at Charlotte, University of South Carolina, Southern Methodist University and the 1997 Conference on Real Options.
*
Abstract This paper applies theory developed in Childs, Ott and Riddiough (2001a) to explore the implications of noise in real estate markets. We consider two applications that specifically focus on real option valuation and optimal exercise policy. The first application examines an imperfectly competitive market for real estate development in which agents compete over the timing of lead investment. Information spillover and freerider incentives are shown to cause significant delay in lead investment. Delay together with competitive response once pioneering development has occurred explains observed patterns of development in blighted urban land markets and for large-scale multi-stage development projects such as DisneyWorld. The second application examines valuation and default exercise policy of risky coupon debt that is secured by a lease-encumbered noisy real asset. Our main result is that, relative to the noiseless case, the borrower will generally delay default exercise until the noisy signal of asset value is far into-themoney. This finding provides an information-based explanation for the apparent underexercise of the mortgage default option that has been observed in the marketplace.
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Noise, Real Estate Markets, and Options on Real Assets: Applications
1 Introduction This paper applies theory developed in Childs, Ott and Riddiough (2001a) (hereafter COR (2001a)) to more thoroughly explore the implications of noise in real estate markets. We are particularly interested in the valuation and optimal exercise policy of options that are contingent on noisy real asset values. This is an important topic—options on noisy real assets are ubiquitous and have substantial real economic effects—but has received relatively little attention in the literature. For example, there is a substantial literature on the implications of fundamental asset value noise in real and financial markets (e.g., Case and Shiller (1989), Goetzmann (1993), Black (1986)) as well as on portfolio allocation decisions (e.g. Gennotte (1986), Merton (1987)). There is also a voluminous market microstructure literature that has focused on asymmetric information and noise trading effects on financial asset values (see, e.g., O’Hara (1997) for a summary). However, with the exception of Williams (1995), Zhang (1997), Merton (1998) and Grenadier (1999), the direct effects of noise on options whose values are derived from underlying real asset values have largely been ignored. To explore the implications of noise on real option values and exercise policy we consider two applications. The first application examines an imperfectly competitive market for real estate development in which agents compete over the timing of lead investment. In this setting the precise location of the pre-investment rental-rate demand curve is observable only with noise. However, once pioneering development has occurred, its relative success or failure is revealed to the market vis-à-vis realized demand. The information externality associated with lead development creates free-rider problems and hence a tendency to delay until the expected returns to initial investment are (super) super-normal. A tendency towards delay in lead development together with (non-) competitive response by follow-on investors explain supply boom (and bust) patterns observed in many localized real estate markets. Relevant examples that reflect the described market structure include well located but blighted urban land, urban-fringe residential and retail development, and certain types of largescale multi-stage development projects such as DisneyWorld in Florida. Redevelopment of
1
blighted urban land is, in particular, plagued with information related free-rider problems. Pioneering developers worry that, on the one hand, profits associated with successful development will be eroded by rapid entry in a setting where land use regulators are eager to facilitate additional investment. On the other hand, losses can occur and are exacerbated if financial distress and other costs follow from an unexpectedly low realized demand outcome. Concern over information spillover and a desire to internalize benefits from lead investment explain why Disney (as well as numerous other large scale developers) assembled large tracts of contiguous land prior to undertaking their multi-stage development projects. We complement and extend earlier research by modeling the strategic development problem as a continuous time Cournot competition game of incomplete information, in which agents compete over the timing (as opposed to the absolute quantity) of new supply.1 A conditional expected perfect Markov equilibrium is shown to obtain when agents are indifferent between leading and following at the point at which initial investment occurs. An important outcome of the analysis is an extreme “no investment” proposition that follows when product demand is infinitely elastic. The intuition for this result is that free-rider incentives completely dominate any business-stealing effects when there are no monopoly rents associated with moving first. Delay is less extreme, but prominent nonetheless, when temporary monopoly rents can be realized from assuming the role of a pioneer investor. We analyze the characteristics of the lead investment threshold as a function of several important explanatory variables, including the elasticity of product demand, parameters that describe the dynamics of asset value noise, and time from initial asset value observation. The probability of immediate follow-on development is also quantified, and is shown to be significant for realistic parameter values. High lead investment thresholds coupled with high likelihoods of immediate follow-on investment explain why localized development is often delayed for long periods of time, but that, once lead investment occurs and information is
1
Our model directly relates to at least three strands of literature. One has to do with the strategic exercise of options that are not in zero net supply (e.g., Emanuel (1983), Spatt and Sterbenz (1988), Williams (1993), and Grenadier (1996b)). A second relevant strand of literature is in the industrial organization area, which considers preemption under uncertainty (e.g., Spatt and Sterbenz (1985) and Fudenberg and Tirole (1985)). A third strand of literature considers herding behavior and information cascades as explanations for the rational ‘clumping' of investments. There is a long list of papers that fall into this latter category; see in particular, Rob (1991), Banerjee (1992), Bikhchandani, Hirshleifer and Welch (1992), Caplin and Leahy (1995) and Grenadier (1999). Our application is perhaps most similar to Caplin and Leahy (1995) in its emphasis on information externalities as a rationale for long waiting times in new product markets, and for potentially rapid market development once initial investment is made.
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revealed as to the market’s economic viability, it is often the case that a development boom quickly follows. Our second application addresses valuation and default exercise policy associated with risky coupon debt that is secured by a lease-encumbered noisy real asset. Low risk, fixed-term lease flows are attractive to capital constrained property owners, who often require significant amounts of debt to finance their investments. These lease flows are attractive because lenders, who are concerned about collateral value risk in historically volatile and often highly illiquid property markets, will typically request that fixed-term leases are in place in order to secure debt financing. Fixed term space-lease contracting and debt financing are therefore complementary in terms of facilitating investment and providing liquidity in real asset markets. This analysis merges two rather distinct literatures that have developed on leasing and risky debt valuation, with the additional innovation that we account for noise in the underlying real asset value. Grenadier (1995, 1996a) analyzes fixed-term leases and the term structure of lease rates in a full information (FI) option pricing framework. His analysis assumes all-equity financing, however, and therefore does not consider the interaction of debt finance and lease contracting. Titman and Torous (1989), Kau, Keenan, Muller and Epperson (1990) and others have considered the FI valuation of mortgages secured by income-producing real estate in a contingent-claims setting. They model asset income as constantly proportional to lognormally distributed property value, suggesting that leases are continuously resigned in a spot market. In reality, lease contracting is costly due to frictions associated with search and negotiation in decentralized space markets. These frictions effectively prevent agents from recontracting on a continuous basis and hence obscure the precise observation of spot lease prices. Once a lease contract is in place, real asset value has two components: the full information value of the fixed-term lease payments (the leased-fee interest) plus the noisy residual asset value. Just after the lease contracting date the value of the leased-fee component will be relatively prominent. However, as the lease term-to-maturity shortens and there are fewer contracted lease payments remaining, the noisy residual asset value becomes an increasingly important component of capitalized asset value. This suggests that the borrower’s ability to make accurate default decisions will typically deteriorate subsequent to the lease signing date. We analyze default option exercise policy at and subsequent to the lease re-signing date to gauge the combined effects of lease contracting and residual asset value noise.
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Default option exercise policy is shown to differ from exercise policy that would be implemented if asset value information were complete. Our main result is that, relative to the noiseless case, the borrower will generally delay exercise until the noisy signal of asset value is far into-the-money. An implication of this finding is that, if the claimholder recognizes that noise exists, but the empiricist—who is trying to compare observed exercise policy with that predicted by a noiseless model of asset prices—does not, a “sub-optimal” exercise policy may be inferred when in fact the policy is rational given the information available. This explanation is consistent with evidence from mortgage default studies as to why the observed default exercise boundary is found to be significantly lower than that predicted by standard theoretical option-based models (see, e.g., Vandell (1992), Quigley and Van Order (1995)). The main body of the paper is organized as follows. Section 2 briefly reviews the modeling framework developed in COR (2001a) as it applies to topics of interest in this paper. Section 3 addresses the issue of strategic real estate development with information spillover. Section 4 considers default exercise policy in the context of secured debt and when the underlying real asset can only be observed with noise. Section 5 concludes the paper.
2 Modeling Framework and Applicability to Real Estate Markets Theory developed in COR (2001a) underlies the applications examined in this paper. Specifically, we adopt a reduced form modeling approach in which the full information (FI) value is a lognormal variable as expressed in equation (1) of that paper. The FI value is obscured by noise for reasons described in sections 1 and 2.1 of the COR paper. Noise is mean reverting and evolves according to equation (2). The observed value, defined as the product of the FI value and noise, is as defined in equation (3) of that paper. The conditional expected value, which provides an efficient unbiased estimate of the FI value, is as described in equation (19) of COR. Additional measures that are of particular importance to issues discussed in this paper include the residual variance, as defined in equation (17), the conditional dynamics of the expected value, as defined in equation (29), and the revealed variance, as defined in equation (31a). All of these formulas are summarized in Table I. To simplify the analysis, we will assume that agents are risk neutral. See COR (2001a, 2001b) for additional discussion and supporting rationale as to the risk neutrality assumption.
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Table I Key Formulas from COR (2001a) Eqn # Definition
Formula
COR (2001a)
FI or True Value
dX (t ) = µ X X (t )dt + σ X X (t )dWX
(1)
Noise
dY (t ) = (− κ ln Y (t ) )Y (t )dt + σ Y Y (t )dWY
(2)
Observed
Z (t ) Z (t )dt + σ X Z (t )dWX + σ Y Z (t )dWY dZ (t ) = µ X − κ ln X ( t )
(3)
Value
Conditional Expected Value
Residual Variance
1 (γ ( t )−σ Y20 ) µ X ∫ t (1−ξ ( s ))ds e 0 M ( t ) = Z ( 0 ) e 2 • t ξ ( s ) dz ( s )+κ ln Z ( s ) + 1 γ ( s ) ds 1 t M( s ) 2 κ γ ( s )ds ∫0 2 ∫0 e e
γ (t ) =
e
2 ρκt
(2σ
(
2σ X2 σ Y20 − c 2 X
(
))
)
(
+ ρκ σ − c − ρκ σ Y20 − c 2 Y0
)+c
(19)
(17)
Conditional Dynamics
dM (t ) = µ X M (t )dt + ξ (t )σ Z M (t )dWZ
Revealed Variance
ν (t | I ( s )) = ∫ ξ 2 ( h )σ Z2 dh = σ x2 (t − s) − (γ (t ) − γ ( s) )
t
(29)
(31a)
s
The perspective in COR (2001a) was primarily that of an asset valuation specialist (AVS), who “is an impartial, fully rational agent who has free access to publicly available information, including continuous knowledge of all comparable asset sales. The AVS does not trade for its own account in this market nor does it facilitate market transactions as a market maker.” The 5
perspective in this paper differs in that we focus on valuation by the real optionholder. Real optionholders are typically principals as opposed to agents, in the sense that they have a direct ownership interest in the underlying asset upon which option value is contingent. Because of its “insider” position, the real optionholder may have information regarding the underlying asset value that a third party, such as an AVS, might not have. However, as noted in COR (2001a), because of transaction costs and unique asset characteristics, the real optionholder will be unable to execute trades (real or synthetic) that identify the asset’s FI value. Indeed, in some cases (which we analyze in this paper), the asset does not even exist until option exercise occurs. Consequently, even though it may be possible that the real optionholder’s estimate of noisy asset value is somewhat more precise than certain outsiders, its information set will nevertheless be incomplete and it will be forced to rely on information that is quite similar to that utilized by the AVS. Market realities imply, therefore, that the real optionholder can only observe underlying asset value with noise. Necessary conditions for persistent noise are that i) markets are incomplete in an Arrow-Debreu sense, ii) any realized pre-option exercise cash flows cannot be used to perfectly infer asset value, and iii) sale of any portion of the underlying asset is restricted over the option exercise period. Finally, to focus ideas, we will further rule out the purposeful— and costly—acquisition of private information that might be used to increase the precision of the value estimate. See COR (2001b) for analysis of endogenous information acquisition incentives for options on real assets.
3 Strategic Development and Information Externalities in Real Estate Markets 3.1 The Basic Setting
Consider development strategy in an imperfectly competitive real estate market. Prior to any land development, agents are unable to infer the exact location of the built property rentalrate demand curve. However, once initial development occurs and actual rental demand is realized, the relative success or failure of the lead developer can be accurately assessed. The option to undertake lead development within this market is therefore written on a demand function that is observable only with noise. In a dynamic setting a potential lead developer will consider the tradeoff between capturing temporary monopoly rents through early investment (a 6
“business-stealing” effect) with the possibility of miscalculating product demand. An additional cost to moving first is that investment provides information regarding the position of the demand curve to competitors. This latter effect suggests that incentives for delay will exist in order to free-ride on information generated by a first-mover. When free-riding incentives dominate business-stealing effects, long lags in lead development may be realized. However, once initial investment does occur and product demand is revealed, rapid follow-on investment will often be realized due to information spillover and the likelihood of high realized demand.2 We model the strategic development problem as a continuous time Cournot competition game in which suppliers have the option to invest capital to create one unit of production capacity (space). A Cournot (as opposed to a Bertrand) modeling approach is appropriate in a real estate development setting since the demand for space is derived from surrounding economic activity, implying that developers will not compete directly on price when formulating their business plans. Instead, competition over quantity is the crucial strategic variable in development markets. Quantity competition assumes a rather different form than that characterized in the standard Cournot model, however. In a real estate development setting, land use regulations typically constrain choices along the capacity dimension by imposing maximum density restrictions. As a result, developers often take capacity as given and instead compete over the timing of product delivery in markets in which space is strategic substitute. Once developed, a built asset will produce a stream of currently observable but dynamically uncertain cash flows for an indefinite time period. Prior to lead investment, however, product price is observable only with noise. Consequently, from an ex ante, predevelopment perspective, determining a best estimate of FI demand is crucial in the formulation of competitive investment policy. An efficient unbiased estimate of inverse product demand exists and is described by the following relation: p(q, t ) = M (t ) D(q ) ,
2
(1)
We abstract from the particulars of pre-leasing strategies in this paper in order to focus on the generic first-mover problem. Mechanisms such as pre-leasing are sometimes used in an attempt to resolve demand uncertainty in development markets. In many development situations, however, space consumers will themselves recognize uncertainty in demand and be reluctant to sign a lease until development has actually happened. Moreover, certain real estate markets—such as those for rental housing, hotel and entertainment—resemble a continuous cash flow spot market for space usage, thus rendering a pre-leasing approach ineffective. It is these types of markets that best fit the setting we describe in this application.
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where p(q,t) is the time t estimated product price per unit (net of ongoing operating costs) as a function of realized production capacity, q = 1, 2 is the total production capacity (units of space) developed, M(t) is the value of multiplicative demand shock term, and D(q) is a non-increasing function. The pre-development estimated product price depends on the best estimate of the unobservable FI demand shock term, X(t). We will assume that dynamics associated with X(t) are as described in equation (1) of COR (2001a) (see Table I). We will also assume that noise follows mean reverting dynamics as described in equation (2) of COR (2001a), which in turn produces an observed value, Z(t), as stated in equation (3) of that paper. In this case the best estimate of the current X(t) is the conditional expected value, M(t), as defined in equation (19) in COR (2001a). Furthermore, conditional on all of the information available as of time t, proposition 4 of COR (2001a) shows that the dynamics for M(t) evolve according to equation (29) in that paper. The dynamics of M(t) are identical to the dynamics associated with the FI shock term, X(t), with the exception that the variance rate in the conditional dynamics is modified to reflect the expected arrival rate of information in a setting in which observed values cointegrate with the unobserved FI value. It is also important to recognize that we now characterize M(t) as determining a per unit revenue flow, whereas we previously specified M(t) as an asset price. Results derived in COR (2001a) are easily modified to address this case, since the functional relation between revenue flow and price is homogeneous of degree one. To isolate the economic consequences of information spillover on investment policy, we rule out dominant complementarities in product demand. Weak non-complementarity in demand (i.e., supplied space is a weak strategic substitute since D(2)≤D(1)) implies that, although positive payoff externalities may exist to affect inverse demand, they are (weakly) dominated by standard substitution effects that occur in local space markets and negative common resource externalities that are often priced when undertaking additional development (e.g., congestion effects, pollution). In the sub-sections below we value the option to develop for the follower, and then determine the optimal investment policy for the leader. Although our primary focus is on the effects of noise on competitive lead investment decisions, we will also provide the leader's FI
8
optimal exercise policy in order to build intuition and to gauge the relative effects of noise on investment decisions. 3.2 Investment by the Follower
Dynamic programming requires that we first determine the follower's contingent investment value, and then use the results to derive the investment policy of the leader. The follower’s optimal exercise policy and investment value are determined as a function of the FI demand term, X(t). This follows because previous investment by the leader fully reveals the demand curve. Full information with respect to demand implies that X(t) is substituted in for M(t) in equation (1). Consider the follower’s optimal exercise policy.3 Assume that the unit cost of investment is K for both competitors. Further assume that once the investment decision is made, development is instantaneous and irreversible. Irreversibility suggests that, once lead investment has occurred, the follower holds a proprietary option on the development of the second project. Given all this and the fact that demand is fully observable, the critical value, p f * = X
f*
D(2) , at
which follow-on investment is triggered is
X
f*
D( 2 ) =
β δX K, β −1
(2)
where
2
1 2 1 σ X − µ X + σ X2 − µ X + 2rσ X2 2 2 β= 2 σX
3
Valuation and exercise policy in the case of full information and non-strategic interaction are well known; consequently, we will state the analytic results with only a sparse narrative. A more detailed presentation of the noiseless version of the model can be found in Dixit and Pindyck (1994, pp. 309-314).
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(3)
and where parameter values are as defined previously and in COR (2001a). Note that X
f*
β D( 2 ) = K , is the capitalized value of expected cash flows (the built asset value) at the δX β −1
point of follow-on investment. For properly defined parameter values, it is easily shown that 11 (as seen in the e1/2β(β-1)γ(t) term in equation (8)), which is due to the existence of noise and the fact that payoff functions to the leader and follower are non-linear in the range [0, X
f*
]. Finally observe
that the equilibrium relation stated in equation (8) is Markov in the sense that solution depends
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only on current time t. However, M(t) is itself a specific composite of historical values as defined in equation (19) of COR (2001a). Properties of the lead investment boundary can be illustrated by the special cases of κ→∞ (no noise vis-à-vis a high rate of cointegration between the observed value and the FI value),
κ=0 (steadily accumulating noise with no cointegration between the observed value and the FI value) and σ Y20 = c (a constant rate of noise over time). As κ→∞, the observed value, Z(t), reverts to the FI value, X(t), immediately. This implies that γ(t)→0 for all t>0 and the lead investment threshold relation expressed in equation (8) reduces to the certainty relation expressed in equation (6). For κ=0, noise steadily accumulates at a rate of ρ 2σ Y2 , with terms as defined in equations (2) and (7) in COR (2001a). In this case, as well as other cases in which γ(t) is increasing in t, it can be shown that the lead investment threshold is increasing in time. This follows because, as residual variance grows over time, the probability of making an exercise error also increases. Increasing error probabilities, in turn, create an incentive for the leader to further delay investment. Interestingly, given κ=0, the lead investment threshold in this strategic investment case increases over time even though the investment threshold in the proprietary option case is time homogeneous (see COR (2001b)). This is due to the effect of information spillover that is strictly to the advantage of follow-on investors in a strategic investment setting. Lastly, when σ Y20 = c , residual variance is constant over time to result in a time homogeneous lead investment threshold. For realistic parameter value constellations it will often be the case that the lead investment boundary, M l * , will exceed the follow-on investment threshold of X
f*
. For
example, as D(1) approaches D(2) from above and for increasing σ Y2 , M l * increases and eventually exceeds X
f*
. This follows because information generated by lead investment is a
public good, thereby creating powerful incentives for delay. In direct relation to our claim that M l * (t ) can exceed X
f*
, we state the following result for the special case of constant returns to
industry scale: Corollary 1 (No Investment Outcome): If D(1) = D(2) and demand is observable only with noise, investment never occurs.
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Proof: See Appendix
The intuition for this no-investment result is that, if marginal returns to (pure) capital investment are constant in scale, then there is no first-mover advantage with respect to capturing temporary monopoly rents. Because there is a positive probability that the leader will make a mistake—i.e., investment might provide ex post sub-normal returns given irreversible investment cost, K—and because the information externality generated by lead investment is strictly a benefit to the follower, free-rider incentives dominate all other effects and neither competitor will be willing to move first. Long delays of this type would appear to be costly from a social perspective, and suggest a potential role for central planning in the development of new product markets. We will now consider the characteristics of the lead investment threshold, M l * (t ) , in greater detail. In Figures 2 through 4 we vary certain key parameter values as they effect the relation between the lead threshold value (seen on the y-axis) and the temporal noise volatility parameter, σ Y (seen on the x-axis). These figures directly illustrate that the lead investment threshold increases as the residual variance increases, since it is true that
∂γ (t ) > 0 for any given ∂σ Y
time t. Incentives to delay are derived from two complementary sources of uncertainty: waiting for the arrival of information in order to resolve future price uncertainty as well as waiting to resolve current price uncertainty. A higher level of current price uncertainty, as summarized by
σ Y , interacts with future price uncertainty to induce competitors to set a higher threshold value in order to compensate for information spillover effects that follow from lead investment. Figure 2 shows that the lead investment threshold is increasing in the price elasticity of demand. As D(2) approaches D(1) from below a flatter-sloped demand “curve” obtains to provide relatively lower expected monopoly rents, and thus induces the leader to invest at a higher threshold value. Figure 3 illustrates that, as the observed value is more tightly cointegrated with the FI value vis-à-vis a higher κ parameter, the lead investment threshold decreases. This follows because an increase in κ results in a higher relative rate of information arrival, which reduces errors in exercise policy to mitigate incentives for delay.
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Figure 2 Lead Investment Threshold as a Function of Temporal Noise Volatility: Alternative Demand Elasticities, D(2)
Leader's Investment Threshold
3
2 D(2)=.95
D(2)=.90 Follower Boundary D(2)=.80
1 0
0.05
0.1
0.15
0.2 σY
Parameter Values: r=.06, δX=.08, σX=.15, σY0=.10, κ=.15, D(1)=1.0, K=10, t=5.
Figure 4 considers the effects of time on the lead investment threshold value. Interestingly, for low values of σ Y the lead investment threshold is higher for t=2 than for t=10. The relation flips, however, when σ Y is sufficiently large. This outcome follows from the complicating effects of initial estimation error, as summarized by σ Y0 , which in this case equals .10. The value of σ Y for which residual variance equals initial variance, and therefore results in a constant residual variance through time, is σ Y =.056. For σ Y < (>).056, residual variance is decreasing (increasing) over time. This implies a relatively high (low) rate of information arrival for small t, and therefore a stronger (weaker) tendency for delay in order to realize the benefits of the new information. The point at which the lines cross in Figure 4 is σ Y =.056, which is where residual variance is constant and therefore the lead investment threshold is time homogeneous. 18
Figure 3 Lead Investment Threshold as a Function of Temporal Noise Volatility: Alternative Rates of Mean Reversion in Noise, κ
Leader's Investment Threshold
2
κ =.05
1.5 κ =.15
Follower Boundary κ =.40
1 0
0.05
0.1
0.15
0.2 σY
Parameter Values: r=.06, δX=.08, σX=.15, σY0=.10, D(1)=1.0, D(2)=.90, K=10, t=5.
In Figure 5 we graph the lead investment threshold against the level of initial noise, σ Y0 . In this case the value toward which the residual variance asymptotes (which is the parameter c as defined in equation (18) of COR (2001a)) is constant regardless of σ Y0 . As can be seen, increasing the level of initial noise increases the lead investment threshold. This follows because ∂γ (t ) > 0 ; i.e., all else equal, higher initial noise increases the residual variance, which increases ∂σ Y0 the likelihood of an error in exercise policy. This in turn leads to further delay in lead investment.
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Figure 4 Lead Investment Threshold as a Function of Temporal Noise Volatility: Alternative Times from Initial Observation, t
Leader's Investment Threshold
2
t =10
1.5
t =5
Follower Boundary
t =2
1 0
0.05
0.1
0.15
0.2 σY
Parameter Values: r=.06, δX=.08, σX=.15, σY0=.10, κ=.15, D(1)=1.0, D(2)=.90, K=10.
Rational development cascades can be characterized by calculating the probability that the revealed X(t) exceeds the follower's investment threshold, X
f*
, given that the leader invests
( (
at M l * (t ) . The probability of immediate follow-on investment is given by Φ − d X
f*
))
,0, γ (t * ) ,
where t* corresponds to the time at which lead investment occurs. If revealed demand is sufficiently strong, there will be a rush to develop in order to harvest high realized market rents. Conversely, if revealed demand is sufficiently weak, the market leader may have to “go it alone” for a period of time until the market can support further entry. Probabilities of immediate followon investment are shown in Figures 6 and 7. Observe that the probability of a development cascade is increasing in σ Y and D(2), and is decreasing in κ. The intuition for these comparative static results follow directly from comparative statics associated with the lead investment 20
threshold. These comparative statics, as well as others that can be obtained from a further examination of equation (8), provide empirically testable relations regarding local development outcomes when irreversibility, value uncertainty, an imperfectly competitive market structure, and information spillover interact to affect investment incentives.
Figure 5 Lead Investment Threshold as a Function of Initial Noise Volatility, σ Y
0
Leader's Investment Threshold
1.5
Follower Boundary
1.25
1 0
0.05
0.1
0.15
0.2 σY0
Parameter Values: r=.06, δX=.08, σX=.15, κ=.15, σY=.10, D(1)=1.0, D(2)=.90, K=10 t=5.
Given their similarities in the context of markets for real estate development, it is relevant to compare our model with Grenadier’s (1999) model of option-based investment cascades. In Grenadier’s model developers are endowed with private signals of the true built asset value, which vary by their precision. In the absence of agent-generated information, noise remains constant over time and value signals are independent of one another. Heterogeneous value signals result in a precision-based ordering of development. In our model value signals are commonly observed by developers, and hence are perfectly correlated. Noise variance increases 21
or decreases over time depending on initial levels of noise relative to the steady state. Because of the symmetry of information, developers are equally likely to assume the role of lead developer in our model.
Figure 6 Probability of Immediate Follow-On Development as a Function of Temporal Noise Volatility: Alternative Demand Elasticities, D(2)
Probability of Investment Herding
1
0.75
D(2) =.95
0.5 D(2) = .90
0.25 D(2) = .80
0 0
0.05
0.1
0.15
σY
0.2
Parameter Values: r=.06, δX=.08, σX=.15, σY0=.10, κ=.15, D(1)=1.0, K=10, t=5.
Information is revealed based on the actions of developers in both models. Information revelation may be partial or complete in Grenadier’s model, in which developers with less precise value signals learn by observing exercise or delay decisions by those with more precise value signals. Developers do not learn by observing revenue flows from built projects, however. Learning by observing may lead developers to throw away their own information and jump on the “investment bandwagon” as relatively better informed agents commit to development. Significant underinvestment is not a crucial issue in this model, since the information structure is such that inaction gradually reveals the true asset value.
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Figure 7 Probability of Immediate Follow-On Development as a Function of Temporal Noise Volatility: Alternative Rates of Mean Reversion in Noise, κ
Probability of Investment Herding
1
0.75
κ =.05
0.5
κ =.15
0.25 κ = .40
0 0
0.05
0.1
0.15
0.2 σY
Parameter Values: r=.06, δX=.08, σX=.15, σY0=.10, D(1)=1.0, D(2)=.90, K=10, t=5.
In contrast, information revelation is complete in our model due to observation of realized revenue flows from lead development. However, prior to lead development, no information is revealed by a lack of action among agents. Symmetric ex ante value signals and complete ex post value revelation are responsible for the long delays in lead development in our model. Thus, useful information that is tossed away after the fact may be responsible for overinvestment and herding in Grenadier’s model, whereas no information is lost as a result of investment in our model. It is the incentive to delay to avoid mistakes and the inability to infer information prior to lead investment that results in initial under-investment and then potential herding in our model. That is, information is potentially under-utilized after development occurs in the Grenadier model, whereas under-utilization problems occur in the pre-investment stage in our model. 23
4 Coupon Debt Financing Secured by Renewable Lease Income 4.1 Set-Up
This application considers the affects of asset value noise on debt valuation and the default exercise behavior of a borrower when real asset value depends on income derived from a finite-term fixed payment lease. Analysis in this application covers the period from an arbitrary lease recontracting date, t r ≥ 0 (sometime after the debt issuance date), through the debt maturity date, Td, under the assumption that the new lease term, Tl, exceeds the remaining debt term, Td tr (see Figure 8). The lease we analyze is riskless from a tenant default perspective.6 The debt is risky, however, since the lease term is finite and the residual asset value is stochastic.
Figure 8 Timing of Lease Contract and Debt Maturity Td: Debt Term
0
Debt Issuance Date
tr
Td
Lease Recontracting Date
Debt Maturity Date
tr + T l
Lease Termination Date
Tl: New Lease Term
The risky debt is a compound contingent-claim, in the sense that the fixed coupon payment is the payoff required to keep the default option alive for another period. Default exercise policy on and after the lease recontracting date will depend on lease flows relative to scheduled debt service as well as the current asset value estimate relative to debt value. If, at the time of lease recontracting, the rate of lease payment equals or exceeds the required rate of debt service, the default option is in effect European, since killing the default option early means
6
For an analysis of the impact of lessee default risk on equilibrium lease terms, see Grenadier (1996a).
24
foregoing a positive stream of income.7 At the other extreme, if lease income and estimated asset value are sufficiently low, the borrower will simply default at the recontracting date. These extreme cases bound a third possible outcome in which the lease flow by itself is insufficient to cover debt service, but asset value is high enough to warrant funding the cash flow deficit with outside resources in order to wait to resolve additional asset value uncertainty. Exercise policy is dynamic in this middle region, and is the primary focus of our analysis. 4.2 Model
Given this setup, a first step in the formal analysis is to establish the continuous rate of fixed lease payment, L, as of the lease recontracting date, tr.8 Assume that the time tr estimate of asset value equals Z(tr)=M(tr), where the variance in the initial FI asset value estimate is σ Y2t ≥ 0 r
and parameters are as defined in COR (2001a) (see in particular section 2.1 of COR (2001a) for additional background). Subsequent to time tr, and conditional on M(tr), the asset will be expected to produce a stochastic flow of income at a rate of δXΜ(t) over the time interval t ∈ [t r , t r + Tl ] . The dynamics of the conditional expected spot value, M(t), over this time interval
are as described in equation (29) of COR (2001a). By further applying forward valuation results from Corollary 4 of COR (2001a), the lease flows from time tr to tr+Tl are valued at
(
)
M (tr ) 1 − e −δ X Tl . This in turn suggests that the continuous rate of fixed lease payment is
determined by the following relation: Tl
∫0
(
Le − rs ds = M (tr ) 1 − e −δ X Tl
)
which implies that L=
(
rM (tr ) 1 − e −δ X Tl 1 − e − rTl
)
(10)
7 This is analogous to Merton's (1973) result with respect to early exercise of a non-dividend paying call option, and is true even if the default option is deep in-the-money at any point prior to the debt termination date. 8 We will assume that leases are net, which means that lessees pay all expenses associated with operating the asset. For simplicity of presentation, we will ignore transaction costs to lease recontracting as they impact the determination of lease payments.
25
The upper and lower bounds of the dynamic default region can now be determined. Let the continuous rate of debt payment equal P. This payment, which was determined prior to tr, depends on the initial loan amount as well as other exogenous contractual elements. As previously discussed, if the rate of lease income flow, L, exceeds the rate of debt payment, P, default will not occur prior to the debt maturity date. Thus, L=P establishes an upper bound of the region for which default exercise policy is dynamic. In terms of time tr asset value estimate, M(tr), we can use Equation (10) to determine this upper bound as
M (tr ) =
(
P 1 − e − rTl r 1 − e −δ X Tl
(
) )
(11)
Thus, for M (tr ) ≥ M (tr ) , default exercise policy simplifies to analyzing a European option in which the underlying asset value is noisy. See COR (2001b) for further analysis of valuation and exercise policy associated with this type of contingent claim. Determination of the lower bound of the time tr exercise continuation region, M (tr ) , is a dynamic programming problem that depends on potential realizations of debt value, D( Mˆ (t ), t ) , as a function of the lease-encumbered asset value estimate, Mˆ (t ) . At the lease renewal date, M (tr ) = Mˆ (tr ) . However, for t>tr, these two value estimates—one of which is based on valuing
stochastic spot revenue flows ( M (t ) ) and the other of which is based on a combination of deterministic fixed-term lease flows and stochastic residual revenue flows ( Mˆ (t ) )—will almost surely differ from one another. The composite time t, t≥tr lease-encumbered asset value estimate can be expressed as
(
)
L 1 − e − r (t r +Tl −t ) ˆ + M (t )e −δ X (t r +Tl − t ) M (t ) = r
(12)
where M(t) is the expected asset price conditioned on initial conditions and information available at time t, as summarized in equation (19) of COR (2001a).
26
In order to understand the dynamics for coupon debt value, D( Mˆ (t ), t ) , and associated default exercise policy, the dynamics for Mˆ (t ) must first be developed. Determination of the dynamics for Mˆ (t ) is non-trivial, and is summarized in the following proposition:
Proposition 2 (Lease-Encumbered Asset Value Dynamics):
(
)
(
)
dMˆ (t ) = rMˆ (t ) − L dt + ξ (t )σ Z Ω Mˆ (t ), t dW ,
where
(
(13)
)
L 1 − e − r (t r +Tl − t ) Ω Mˆ (t ), t = Mˆ (t ) − r and other parameters are as defined herein and in COR (2001a).
(
)
Proof: See Appendix.
The dynamics for the lease encumbered asset value, Mˆ (t ) , are intuitive and descibe a natural adjustment to the spot pricing process, M(t). The drift term in Equation (13) is modified from the usual constant proportional rate of payout to reflect that the rate of dividend payout (lease revenue flow) is fixed rather than variable over the remaining lease term. The volatility associated with the lease encumbered asset is seen to increase in t. This follows because more weight is placed on the residual risky asset value, and less weight is placed on the deterministic lease flows, as lease maturity shortens. The time-to-maturity volatility effect can also be seen by
(
)
recognizing that Ω Mˆ (t ), t equals M (t )e −δ X (t r +Tl − t ) , as shown in Equation (12). This expression explicitly highlights the fact that lease-encumbered asset volatility increases over time in direct proportion with the present value of the residual noisy asset value component. Given the asset value dynamics expressed in Equation (13), the dynamics for continuoustime coupon debt value can finally be stated. Again, by applying Ito's lemma and recalling that P is the continuous rate of debt repayment, the dynamics for equilibrium debt value are
(
)
(
)
1 2 ξ (t )σ Z2 Ω 2 Mˆ (t ), t DMˆ (t ), Mˆ (t ) + rMˆ (t ) − L DMˆ (t ) + Dt + P = rD, 2
where D = D( Mˆ (t ), t ) is debt value and subscripts on D indicate partial derivatives.
27
(14)
Recall that we have described this process in order to determine the dynamic exercise policy region at (as well as subsequent to) the lease renewal date, tr. The lower bound of this region is the point at which the leased real asset value, Mˆ (tr ) , equals the time tr value of the risky coupon debt, D( Mˆ (t r ), t r ) .9 A similar logic applies within this region to determine the time t exercise boundary, t r < t ≤ Td . To obtain the mortgage and endogenous default boundary values we discretize Equation (14) and solve it numerically, subject to the appropriate boundary conditions.10 4.3 Numerical Results
Figure 9 shows how the lower bound of the time tr default exercise region changes depending on the noise volatility parameter, σY. This lower bound is seen to be increasing in the temporal noise volatility. A positive relation follows because revealed variance, ν(t), as defined in equation (31a) in COR (2001a), decreases as noise volatility increases. Lower revealed variance implies a slower rate of information arrival. This in turn induces the equityholder to act as if FI asset value were less volatile than it actually is, and therefore to exhibit less patience (and what might be perceived as greater confidence) in her exercise decisions. Consider now an initial value, M (t r ) , that lies in the middle of the dynamic default policy region. Figure 10 displays three different default boundaries from the lease resigning date, tr, to the debt maturity date, Td: i) the boundary resulting from the full information case, σ Yt = σY r
= 0; ii) the boundary resulting from use of the noisy conditional expected asset value, M * (t ) , when σY>0, and iii) the boundary resulting from use of the noisy observed value, Z * (t ) , when
σY>0, and where Z(t) is as defined in equation (3) of COR (2001a).11 In all three cases the
Although not the focus of our analysis, determination of the debt default boundaries suggest that lease contracting at time tr might assume a strategic dimension. For example, the property owner might shorten or lengthen the lease term to effect lease flows relative to the debt flows. This type of strategic behavior may explain why vacancy persists in space markets, since a property owner may hold out to sign a tenant at a higher long-term lease rate when current short-term lease rates are too low to support debt payments. 10 See Titman and Torous (1989) for an analogous set of boundary conditions corresponding to the noiseless asset case (where we make a constant interest rate assumption). For valuation purposes, we apply the dynamic programming technique of Hull and White (1990). 11 Recall that the default boundary based on the conditional expected value, M * (t ) , results from using revealed as opposed to actual asset variance. In this example, use of revealed variance increases the boundary relative to the 9
28
default boundary first declines and then increases as the time to debt maturity decreases. The initial decline in the default boundary is non-standard in a full information setting, and is due to the fact that value estimation precision deteriorates at a slower rate than the variance in the asset value increases as a result of a shortening lease term. This increases the effective rate of information arrival to decrease the default boundaries.
Figure 9 Default Bounds at the Lease Contracting Date, tr, as a Function of Temporal Noise Volatility 115
No Default Prior to Maturity
Asset Value Bounds
110
105
100
Dynamic Default Policy
95 Immediate Default
90
85 0
0.05
0.1
0.15
0.2
0.25
0.3 σY
Parameter Values: r=.06, δX=.08, σX=.15, σY0=.10, κ=.15, D(1)=1.0, D(2)=.90, K=10, tr=0, Tl=10, Td=5, Loan Amount: 100 (interest-only), P=8.
Note that the zero noise default boundary is bracketed by the two boundaries that obtain when noise is present. The boundary based on the conditional expected value, M*(t), is seen to exceed the noiseless boundary. Total noise in this example increases (linearly) over time to slow noiseless case. Given the parameter values used to generate this figure, once M * (t ) is known it is straightforward to determine Z * ( t ) .
29
the arrival rate of information relative to the noiseless benchmark. This reduces the incentive to wait for the arrival of additional information and therefore increases the default boundary. Alternatively, the boundary that is based on the observed value, Z*(t), is lower than the other two boundaries. This is due to the weight given to the asset's forward value which, for parameter values used in this example, remains well above the conditional default boundary for all time t, t r ≤ t ≤ Td . Hence, because weight is given to the forward value in the determination of M(t), exercise policy is such that the observed value, Z(t), must be relatively low for the equityholder to be sufficiently confident that default is the correct policy decision.
Figure 10 Alternative Default Boundaries as a Function of Time 100
Default Boundary
95
M*(t) when σY = σY0 = .10
90
X∗(t) when σY = σY0 = 0
85
Z* when σY = σY0 = .10
80 0
1
2
3
4
5
t-tr
Parameter Values: r=.06, δX=.08, σX=.15, σY0=.10, κ=0, σY=.10, D(1)=1.0, D(2)=.90, K=10, tr=0, Tl=10, Td=5, Loan Amount: 100 (interest-only), P=8.
30
Figure 11 M*(t) as a Function of Time: Alternative Mean Reversion Parameters, κ
Default Boundary M* (t)
100
95
κ =.05
90 κ = .40
κ = .15
85 0
1
2
3
4
5 t-tr
Parameter Values: r=.06, δX=.08, σX=.15, σY0=.10, σY=.10, D(1)=1.0, D(2)=.90, K=10, tr=0, Tl=10, Td=5, Loan Amount: 100 (interest-only), P=8.
Exercise policy based on a noisy observed value may shed some light on research as to the causes and consequences of borrower default decisions. For example, Vandell (1992) and Quigley and Van Order (1995) provide empirical evidence suggesting that borrowers underexercise their default option relative to predictions made by the standard (noiseless) contingent-claim model of coupon debt value. They propose borrower transaction costs (e.g., tax consequences, reputation costs) or liquidity constraints as possible explanations for observed underexercise of the default option. We provide an alternative explanation of this behavior. When there is uncertainty as to the current FI value of the asset securing the debt, the default exercise boundary of the observed value Z(t)—which is information that is most readily available to the empiricist—will typically be lower than the FI exercise boundary. Therefore, when noise is present, but when the econometrician assumes full information, the exercise policy may appear sub-optimal when it is rational given the information available.
31
Figure 12 M*(t) as a Function of Time: Alternative Temporal Noise Volatilities, σY
Default Boundary M*(t)
100
95
σY = .20
90 σY = .10 σY=.0
85 0
1
2
3
4
5 t-tr
Parameter Values: r=.06, δX=.08, σX=.15, σY0=.10, κ=.15, D(1)=1.0, D(2)=.90, K=10, tr=0, Tl=10, Td=5, Loan Amount: 100 (interest-only), P=8.
Figures 11 through 13 display comparative static effects on the optimal default exercise boundary, M*(t), as functions of the speed of mean reversion of noise, κ, the temporal noise volatility rate, σY, and the initial noise, σ Yt . Increasing the speed of mean reversion increases the r
rate of information arrival to increase the value of waiting to default. In contrast, increases in the temporal rate of noise volatility decreases the rate of information arrival to decrease the value of waiting to default. Finally, an increase in initial noise results in a somewhat unintuitive decrease in the default boundary. This follows because, all else equal, higher initial levels of noise increase the relative rate of information arrival, which in turn increases the value of waiting to default.
32
Figure 13 M*(t) as a Function of Time: Alternative Initial Noise Volatilities, σ Y
tr
Default Boundary M*(t)
100
95
σY0(tr) = .10
90
σY0(tr) = 0 σY0(tr) =.20
85 0
1
2
3
4
5 t-tr
Parameter Values: r=.06, δX=.08, σX=.15, κ=.15, σY=.10, D(1)=1.0, D(2)=.90, K=10, tr=0, Tl=10, Td=5, Loan Amount: 100 (interest-only), P=8.
5 Conclusion We have examined the effects of real asset value noise on option prices and exercise policy by analyzing two real estate market applications. The first application considers competition over lead investment in the market for real estate development. In this setting the inverse demand for space is uncertain prior to lead investment, but is fully revealed ex post. Information spillover combined with irreversibility and uncertainty in future rental rates create strong incentives for delay in order to free ride on information generated by the first mover. In the special case of infinitely elastic product demand, we show that investment never occurs. We also show that, conditional on the lead investment threshold value, the likelihood of a development cascade is significant. We therefore provide an information-based explanation for boom (as well as bust) in localized markets for real estate development. 33
The second application examines default decisions on risky coupon debt that is secured by an asset whose value depends on long-term lease income and a noisy residual spot price. We show that, when measured by the conditional expected value, a borrower will exhibit less patience and default sooner than she would in a full information setting. This follows from the effects of temporal noise, which slows the arrival rate of information. However, when measured by the observed value—which is information that is most readily available to market analysts and empiricists—a borrower may appear to delay default beyond the point predicted in an FI option model setting. Thus, what might otherwise appear as sub-optimal delay is instead a rational response by a borrower to incomplete asset value information.
34
Appendix
Proof of Proposition 1: Proving this proposition amounts to showing that b
X
∫0
f*
1 (b −1)γ (t ) Φ −d X X (t ) g ( X (t ) I (t ) ) dX (t ) = M (t )e 2
b
( (
f*
, b, γ (t )
))
(A1)
1 Begin by recognizing that g(ln(X(t)|I(t)) is normally distributed with mean ln (M (t ) ) − γ (t ) and 2 variance γ (t ) . See equations (19) and (17) of COR (2001a) for further detail regarding the
X (t ) 1 + γ (t ) ln M (t ) 2 properties of the mean and variance. Now define the transformations U (t ) = γ (t )
and W (t ) = U (t ) − b γ (t ) . Use the defined relation between U(t) and X(t) and complete the square, and then rewrite the left-hand side of equation (A1) as follows: b
1 1 f* (b −1)γ ( t ) − (U ( t ) − b γ ( t ) ) M (t )e 2 1 − d ( X , 0,γ ( t ) ) e 2 dU (t ) . Substitute W(t) in for U(t) and the 2π ∫− ∞ right-hand side of equation (A1) is verified. Finally, equation (A1) can be used to obtain equation (8) from equation (7).
Proof of Corollary 1: If D(1)=D(2), then product demand is insensitive to supply. In this case it is a simple exercise to show the leader's expected value conditional on investment is M(t)D(2)/δX – K. Once the leader enters the market, the true demand becomes known. If demand is revealed to be above X f * , the follower enters and product value for both suppliers is XtD(2)/δX - K. Alternatively, if the true demand is below X f * , the follower defers investment. In this case the follower's product value is (X(t)/Xf*)β(Xf*D(2)/δX - K), which is greater than the product value of the leader since X (t ) < X f * and β>1. Now, for any value M(t) for which γ(t)>0, there is some
positive probability that the FI value, X(t), is below X f * . Therefore, given any M(t), the expected product value of the follower exceeds the expected product value of the leader. Since investment only occurs if the expected product values of the leader and the follower are equal, neither competitor wishes to lead and investment never occurs. Proof of Proposition 2: Apply Ito’s Lemma to equation (12) and recall that the dynamics of M(t) can be expressed as seen in equation (29) of COR (2001a). The result follows by performing the appropriate substitutions and simplifying.
35
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