Stickiness of Rental Rates and
Developers' Option Exercise Strategies
Rose Neng Lai, Ko Wang and Jing Yang*
June 2005
Submitted to
Cambridge-Maastricht 2005 Symposium
Abstract: In this study we incorporate sticky rents into a real options model to rationalize the widely documented overbuilding puzzle in real estate markets. Given the fact that developers’ objective function is to maximize revenue by selecting an optimal occupancy level, our model provides a better explanation on the phenomena we observed in the real world than the traditional market-clearance based real options models. We also show that developers’ exercise strategies can be affected by market conditions (such as the size of market) and the type of property markets. In other words, developers’ exercise strategies could differ among various markets and under different conditions.
*Rose Neng Lai is Assistant Professor, Faculty of Business Administration, University of Macau, Taipa, Macao, China. e-mail: . Ko Wang is Professor, Department of Finance, California State University-Fullerton, e-mail: . Jing Yang is Assistant Professor, Department of Finance, California State University-Fullerton, e-mail: .
1. Introduction
Overbuilding exists in basically all kinds of property markets. For instance, vacancy rates of the rental market and the office market in the United States are exceptionally high in the 1980s. In particular, the former exceeded 10% in 1986 (Belsky and Goodman, Jr., 1996) while the latter reached as high as 18% in 1990 (Grenadier, 1995a). The National Association of Realtors predicts that the vacancy rates are 14.5%-15.5% for offices, 11-11.5% for industrial spaces, 8%-8.5% for retail spaces, and 5.5%-6% for multi-family rental units in 2005 (Lereah, 2004). Furthermore, it is not uncommon for the hotel industry to experience 40% or higher vacancy rate (see for example McAneny et. al., 2001). All these vacancy levels are too high to be merely caused by costly searching and matching processes between households and landlords.
While oversupply in property markets has been widely studied from different aspects,[1] only Grenadier (1996) and Wang and Zhou (2000) offer a complete theoretical model for the justification. Grenadier (1996) studies the timing of real estate developments using a strategic option exercise approach[2]. He provides an interesting rationale for the commonly observed phenomenon: increasing development in the “cold” real estate markets. He believes that panic in the market can induce preemptive competition in constructions during the market downturn. (He terms this phenomenon “irrational construction”.) Wang and Zhou (2000) are among the first to investigate vacancy rate and existing supply with a game theoretical approach. Their model demonstrates that developers under competition will start construction whenever there is an opportunity and that sticky rents led by an implicit collusion can be optimal for developers.[3] In other words, under their model framework, excess vacancies could be an equilibrium solution to real estate markets.
However, it is fair to say that both models, while interesting, have certain weaknesses. Upon describing developers’ behavior explicitly in a dynamic setting, Grenadier’s (1996) model is basically a market clearance model, and is therefore not designed for the purpose of resolving the excess vacancy puzzle. This is because any excess space will be absorbed when the rental rates are appropriately adjusted no matter how panic developers are; and the occupancy rate should always be adjusted to the full occupancy level. While Wang and Zhou (2000) focus on the supply side of the markets, and model the vacancy behavior explicitly, their model does not describe developers’ behavior in a dynamic setting, and fails to explicitly characterize the sequential exercise decisions for development. In other words, the model does not address the timing issues of developments, nor detailed conditions affecting the timing of developments.
Our paper attempts to build a more complete model by incorporating sticky rent argument developed by Wang and Zhou (2000) to the dynamic options exercise model setting of Grenadier (1996). To the best of our knowledge, this model is the first in the literature that has incorporated real estate sticky rents into real options framework. We will then demonstrate that, with the sticky rents setting, our model performs better than traditional market clearance based real options models in terms of interpreting the essence of overbuilding and vacancies.
Our model starts with an assumption that developers will seek a revenue-maximizing occupancy level and will allow an optimal excess vacancy rate in the market. This assumption differs from that used by Grenadier (1996), for which developers will adjust rent levels to eliminate all excess supplies. Our model also differs from Grenadier (1996) in that we use the expected average time to estimate development intensity as opposed to the expected median time used by Grenadier. The use of expected averaged time allows us to obtain closed form solutions (as opposed to simulations in Grenadier, 1996) for the timing of developments and the time lag between developers’ constructions decisions. With closed form solution, it is easier to analyze how the various factors might affect a developer’s supply decision in terms of magnitude and direction.
This paper is organized as follows. The next section presents the preliminary construction strategy model framework. In Section 3, we will form duopolistic market equilibrium where rents are adjusted to clear the market so that all the excess capacity is eliminated. In Section 4, we will deduce the optimal rigid rent that developers should adopt and the conditions when overbuilding is preferred. We also generalize the explanation of overbuilding when rigid rate exists. Section 5 compares the overbuilding model with the market clearance model. Finally, Section 6 concludes.
2. Model Framework
In a dynamic market, equilibrium rental rates with two suppliers should generally be lower than that with only one supplier. As such, sequential versus simultaneous constructions generate different payoffs to developers. Furthermore, a rational developer will build the unit only if the rental rate is high enough to generate attractive payoff and/or the construction cost is low enough to justify building while giving up the opportunity to wait for more information. However, too high a rental rate resulting in a less than full occupancy may not be as desirable as a lower rate that attracts enough demand to exhaust the new supply. In section 3, we will derive market equilibrium development strategies that maximize allowable profits to both developers and at which demand meets supply. In section 4, we will deviate from market equilibrium and obtain sticky rental rates that can maximize the total profit while simultaneously permit some level of vacancy under certain circumstances.
We begin by considering the preliminary assumptions. First, we need the standard assumptions for financial options such as a complete market, investors prefer more to less, the price process follows random walk with known growth rate and variance, and the risk-free rate is constant and known. Second, we need an incremental demand function which depends on the random incremental demand level, and is inversely proportional to rental rate, that is
(1)
where a and b are constants, X(t) is the multiplicative demand shock at time t, which may represent changes in the number of households or in standard of living due to changes in market conditions, consumers’ taste and other factors affecting demand. R(t) is the rental rate at any time t that meets total demand with total supply.
The multiplicative demand shock X(t) in equation (1) follows geometric Brownian motion
(2)
where w is a Wiener process, which is a random variable drawn from a normal distribution with E(dw) = 0, and Var(dw) = dt. m is the constant instantaneous growth rate of the demand shock per unit time, and s is the constant instantaneous standard deviation per unit time with respect to w. Equation (2) states that the instantaneous change in demand shocks governing the rental rates is its growth rate plus the standard deviation times the instantaneous change in the Wiener process governing its randomness.
Next, consider a simple duopoly market in which two developers own identical parcel of land for construction. Both are said to own construction options that can be exercised whenever they see a fit. Unlike financial options that generate perpetual dividend yield immediately after exercise, there may be a few years of construction period before rental income can be realized. Hence, assuming the new rental unit will begin to produce rental income soon after construction is completed and rented, a factor representing the construction time lag is still required. Suppose it takes d years to complete the construction. A developer who decides today to build the rental unit at time t will receive rental income only after time t + d. The developer who chooses to build first is called the Leader, while the latter will be the Follower.
The value of the building option for each unit a developer supplied at the time of decision-making is the discounted perpetual rent series received after the rental unit is built, that is
, (3)
where r is the risk-free rate, and t is the date when construction commences. The first term within the inner parentheses in equation (3) represents the perpetual rent flow discounted to time t; while the second term represents any costs incurred during construction and assumed paid in one lump sum at time t.
Applying Ito’s Lemma, the option value before exercising possesses an instantaneous rate of change of
(4)
If the time of valuation is assumed as today (t = 0), the payoff function will be invariant with time, and becomes
. (5)
As there is no interim payoff from the bare land before construction, the expected rate of return from holding the development option should be equal to the risk-free rate according to the risk neutrality argument (see Cox and Ross, 1976, or Merton, 1975, for details). Hence, equating the expectation on equation (4) to the risk-free rate, we obtain an equation governing the rental flow of the Follower as
(6)
In the following section, we will deduce the optimal construction conditions in sub-game Nash equilibrium, where the rental rate considered is the ordinary market equilibrium rate that clears the market. Sticky rates will be imposed later in Section 4.
3. Developers' Game with A Market Clearance Condition
When a developer begins construction, she/he will pay the construction cost and receive rent flow when the building is completed. At equilibrium with market clearance, the rent will always be adjusted until all units are occupied. This means that all the existing space, M, is already fully occupied. Therefore, the incremental demand level in equation (1) will be equal to newly developed property Q(t), and hence the rental rate determined by demand takes the form
(7)
where . The parameter, b, provides a measure of price elasticity.
Specifically, when the market experiences only the Leader’s supply QL, the Leader can enjoy a rental rate of
(8)
When the Follower decides to enter the market by building QF units of rental space, the increased supply will drive down the rental rate to
(9)
after d years, where Q = QL + QF. The Follower will experience only one demand function (9), while the Leader experiences both demand functions, firstly (8) and subsequently (9). Thus, the Leader's strategy embeds that of the Follower. This chain of interactive decisions constitutes a game between both developers in equilibrium.
We first solve for the Follower’s market equilibrium construction strategy and then work backward to determine that of the Leader (see for example, Dutta and Rustichini, 1993, and Grenadier, 1996, for reasons supporting such sequence). We redefine the payoff V(X) in equation (5) as F(X) for the Follower, and as L(X) for the Leader.
To solve equation (5) for the Follower, F(X) must satisfy the following boundary conditions (see Dixit and Pindyck, 1994)
(10a)
(10b)
(10c)
where XF, the ‘trigger point’, is the level of demand shock with which the Follower should start construction. The first boundary condition is the absorbing barrier specifying that the development option will remain zero forever once the rental rate hits zero. The value-matching condition (10b) simply states the payoff at the time of exercising the option. The last is the smooth-pasting condition that ensures an optimal value for F(X) at XF (also known as the ‘high-contact’ condition by Merton, 1973).
Proposition 1. In the equilibrium with market clearance, the optimal demand shocks that trigger constructions are:
, (11)
, (12)
where .
Proof. See Appendix ■
It is easy to see that the trigger shock for the Leader is smaller than that for the Follower, (that is, XLe < XFe), because is always true. A developer should wait for the demand shock to hit at least as high as her/his trigger point (as equation (11) or (12)) before starting the construction of the rental units, rather than exhausting her/his right of waiting for optimal development. Developers receive equal benefits either when developers construct sequentially (the former taking place when X = XLe, and the latter at X = XFe), or simultaneously whenever the demand shock bypasses XLe and reaches XFe. Thus, market equilibrium is attained not only at a unique circumstance, but an unlimited set of conditions soon after some critical thresholds are reached.