Valuing the Conversion Option Afforded by Form-Based Zoning Ordinances in Different Economic Environments
By: W. Keener Hughen, Ph.D. Assistant Professor of Finance University of North Carolina at Charlotte 9201 University City Blvd. Charlotte, NC 28223 [email protected] (704) 687-7638
& Dustin C. Read, PhD/JD Assistant Professor of Property Management and Real Estate College of Liberal Arts and Human Sciences Virginia Tech 295 W. Campus Drive Blacksburg, VA 24061 [email protected] (540) 231-0773
Abstract: Form-based zoning ordinances often provide real estate developers with the ability to alter the mix of residential and commercial space included in their projects so long as they comply with heightened urban design requirements governing the exterior of buildings and the relationship of these structures to the public realm. The real option pricing model presented in this paper seeks to value this flexibility and assess the impact on both development decisions and developer profits in different economic environments. Focusing on the incremental value afforded by a form-based code in a single-period setting allows for the derivation of closed-form solutions in some scenarios, thereby contributing to the existing real options literature. Results generated by the model also suggest form-based codes can serve as a useful urban revitalization tool. These regulations are found to have a greater relative impact on developer profits in weaker markets and those with more volatile demand, which bodes well for their use to stimulate development in markets where it might not otherwise occur. Nonetheless, form-based codes may not encourage a greater amount of mixed-use development in comparison to other regulatory alternatives allowing for the integration of residential and commercial space at the project level. Policymakers must therefore be mindful of their objectives when evaluating the merits of this type of land use regulation.
Keywords: Form-based zoning, real estate development, real options, and urban revitalization
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Introduction
Form-based zoning ordinances provide real estate developers with the ability to include a market-driven mix of residential and commercial space within their projects so long as they satisfy a series of urban design requirements put in place to ensure the external features of buildings support a community’s articulated vision (Langdon, 2006; Barry, 2008 Talen, 2013). To the extent the option to convert space from one use to another over time is valued by the private sector, these regulations are anticipated to encourage development activity in markets where it might not otherwise occur (Walters and Read, 2014). The real option pricing model presented in this paper examines this hypothesis to determine if form-based codes are a useful urban revitalization tool.
Factors such as conversion costs, diminishing marginal revenues and economies of scale in construction are taken into account to estimate the impact of a stylized form-based code on development decisions and developer profits when market demand is uncertain and evolving stochastically over time. Unlike many real option models set in continuous time that focus on the impact of regulatory flexibility on the timing of real estate development, this model examines the value afforded by a form-based code in a single-period setting. This simpler setting allows for the derivation of closed-form solutions in some scenarios and greatly facilitates the sensitivity analyses. The results generated by the model suggest this type of zoning can act as a catalyst for development in markets with relatively weak and/or volatile demand. This bodes well for municipalities interested in using form-based codes to foster urban renewal. However, the flexibility afforded by form-based codes may not produce a greater amount of mixed-use development in comparison to other regulatory tools permitting the integration of residential and commercial space at the project level. Policymakers representing municipal governments should
2 be mindful of these outcomes when evaluating the potential advantages and disadvantages of implementing form-based zoning ordinances in the communities they serve.
The study begins by reviewing the structure of form-based codes and discussing the appropriateness of examining them through the lens of real option theory. Key features of the model presented in this paper are outlined next, along with the numerical results generated by a set of base case parameters. Sensitivity analysis is then performed to estimate the potential impact of form-based zoning on development decisions and developer profits in the presence of different market conditions. The paper concludes with an assessment of the model’s limitations and a summary of issues in need of further investigation. Steps are taken throughout to emphasize the relevance of the research to the study of economic development, real estate and urban planning.
Literature
Form-based zoning represents a dramatic departure from the land use regulations that have shaped the built environment of cities throughout the United States over the course of the last century. Rather than imposing somewhat arbitrary restrictions on allowable land uses within defined geographic districts, as is typically the case with Euclidean zoning, form-based codes authorize a much more liberal mix of residential and commercial space in areas that are urban in character (Walters, 2007; Gellar, 2010). Stakeholder interests are protected throughout the process by requiring developers to act in accordance with heightened urban design standards influencing how privately-owned buildings interact with the public realm. Common features of modern form- based codes include requiring buildings to meet the street in a uniform manner to frame public areas; eliminating blank walls along pedestrian thoroughfares through the inclusion of transparent
3 windows and doors; screening parking and moving it to the rear of structures; and establishing height and width requirements for buildings and roads to improve walkability (Talen, 2009).
Municipalities interested in adopting form-based codes begin by soliciting input from stakeholder groups regarding the desired physical character of a place or space (Sitkowski, 2006).
Diagrams, drawings and pictures illustrating the vision for future development are then created through an iterative process, along with text describing the relationship between buildings, streets and public spaces (Rangwala, 2005; Cable, 2009; Morphis, 2010). The resulting code includes all of these elements and is prescriptive, as opposed to proscriptive, in the sense that it tells developers what they can build on a given site as opposed to what they cannot (Inniss, 2007). Controlling development activity in this manner is anticipated to combat urban sprawl and improve the quality of life enjoyed by local residents through the elimination of regulatory barriers that have historically served as obstacles to mixed-use development (Talen, 2013).
Depending upon the policy objectives, form-based codes can impose mandatory design standards or voluntary ones that a real estate developer can opt into as an alternative to the existing zoning requirements. These regulations can also be structured to cover a variety of geographic scales ranging from targeted revitalization districts to a municipality as a whole (Langdon, 2006).
Irrespective of these differences, most form-based codes include five common components including a regulating plan illustrating the desired location for different types of buildings and public areas; urban regulations governing size, massing and in-building use standards; street regulations controlling the width and location of thoroughfares and sidewalks; and in some instances landscaping and architectural controls imposing significant restrictions on the design of both structures and open spaces (Inniss, 2007). Codes including these components have withstood a host of legal challenges, especially when architectural requirements are implemented in a style-
4 neutral manner to influence interactions between buildings and the public realm (Sitkowski, 2006).
Resilience to legal scrutiny has encouraged a growing number of municipalities to adopt form- based codes to govern real estate development in green-field locations and on infill sites.
Those in favor of using form-based zoning as an urban revitalization tool contend that this type of regulation can produce a wide array of benefits for economically disadvantaged neighborhoods when it encourages the incorporation of housing into mixed-use development projects (Gonzalez & Lejona, 2009). Some notable examples include improving access to employment opportunities for the working poor; deterring crime through the integration of housing into active commercial districts; enhancing pedestrian mobility as a result of greater spatial connectivity between complimentary land uses; attracting retail amenities to low-income communities through the elimination of barriers to entry; and promoting healthier lifestyles through the provision of functional public spaces (Gellar, 2010). Municipalities can additionally benefit from an expanded tax base and a lower marginal cost of providing infrastructure when form-based codes serve as a catalyst for development in economically fragile areas (Barry, 2008).
The benefits associated with form-based zoning have created substantial support among planners and economic developers working on behalf of local governments (Talen, 2005). Real estate developers, on the other hand, have expressed more tenuous enthusiasm in many cases as a result of fears such regulations will complicate the entitlement process or impose costly aesthetic requirements on future projects (Walters & Read, 2014). Nonetheless, form-based codes may offer the private sector several advantages worthy of consideration. These include greater regulatory predictability and flexibility to respond to market conditions (Langdon, 2006; Cable, 2009).
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Since form-based codes are created through extensive collaboration between the public and private sectors, they should provide a clear picture of a community’s expectation for future development and expedite the review process for conforming projects (Berg & BenDor, 2010).
This is often preferable to the regulatory environment created by Euclidean zoning ordinances where discretionary modifications to existing codes are nearly always required to accommodate development of any significant scale (Woodward, 2013). Form-based codes may also reduce the risk of negative externalities through the creation of a more predictable urban environment. This can encourage developers to undertake projects in previously untested markets if they are confident nearby construction will be of equal quality and reflect a cohesive plan (Barry, 2008). A final advantage of form-based codes relates to the flexibility they provide the private sector to respond to the market. By focusing on the physical character of development, as opposed to the uses allowed within, these regulations typically permit developers to modify their proposed mix of residential and commercial space without the need for a costly rezoning (Walters & Read, 2014).
All of the aforementioned features differentiate form-based codes from other regulations encouraging compact, mixed-use development because they expand the opportunities available to the private sector instead of constricting the supply of developable land (De la Cruz, 2009).
Projects in compliance with time-tested principles of urban design can therefore be undertaken, which include a market-driven mix of residential and commercial space that would be very difficult to achieve under traditional Euclidean zoning (Walters & Read, 2014). By taking this approach, form-based codes are capable of aligning the interests of profit-driven developers, design professionals and municipal governments seeking to address urban sprawl (Carmona, 2009). This contention naturally leads to questions regarding the efficacy of form-based zoning in practice and its ability to simultaneously achieve a wide range of policy objectives. One such question is
6 whether allowing developers to alter the mix of residential and commercial space within projects over time is a sufficient incentive to encourage development in otherwise challenging markets.
Real option theory offers an appropriate means of addressing this issue by framing real estate ownership as an option to develop or redevelop a site at some point in the future. The option has value in volatile markets where future demand for space is unclear at the time semi-irreversible investment decisions must be made (Titman, 1985). The value associated with the right, but not the obligation, to postpone development decisions until additional information is available can therefore be modeled using traditional option pricing methodologies (Lucius, 2001; Ott, 2002).
Numerous studies have used this approach to examine the timing and intensity of development
(Sing, 2001; Capozza & Li, 2002; Plantinga, Lubowski & Stavin, 2002; Cunningham, 2006); the effects of competition on developer behavior (Wang & Zhou, 2006; Chu & Sing, 2007; Bulan,
Mayer & Somerville, 2009); factors encouraging construction booms and prolonged real estate cycles (Cauley & Pavlov, 2002; Grenadier, 1995; Grenadier, 1996; Ott, Hughen & Read, 2012); and the impact of government interventions on development decisions (Riddiough, 1997;
Cunningham, 2007; Hughen & Read, 2014). These studies illustrate the usefulness of real option theory as a means of analyzing topics related to real estate development and urban planning.
Many of the research endeavors examining “redevelopment” options focus implicitly or explicitly on situations where there is an opportunity to increase the intensity of development on a site, transform an asset that has become obsolete, or remediate a property contaminated by hazardous waste (Williams, 1997; Lentz & Tse, 2005; Wang, Hipel & Kilgour, 2009; Clapp &
Salavei, 2010; Wang, Hipel & Kilgour, 2011; Clapp, Jou & Lee, 2012; Clapp, Bardos, & Wong,
2012; McMillen & O’Sullivan, 2013; Chen & Lai, 2013). Less attention has been devoted to the flexibility afforded by progressive land use regulations allowing real estate developers to quickly
7 alter the mix of residential and commercial space within projects in response to evolving market conditions. Notable exceptions include the work of Childs, Riddiough & Triantis (1996), as well as Gunnelin (2001), who broach the subject directly. These studies serve as a foundation for an extension of real option theory to the study of form-based zoning as an urban revitalization tool.
However, both are set in continuous time to focus on the impact of regulatory flexibility on the timing of real estate development. The model presented in the following section offers a simplified single-period setting allowing for the derivation of closed-form solutions in some scenarios. This methodological approach additionally provides a means of determining whether form-based zoning can encourage investment in relatively weak or volatile markets, as well as whether it is more likely than other regulatory tools to encourage mixed-use development.
The Model
Consider a scenario in which a developer controls a tract of land that can be developed into a total of N units. A form-based code is in place allowing for a mix of two possible uses within the project, which can be thought of generically as commercial and residential space. Construction begins today (time 0) and is completed at the end of the period (time T). The starting point of the project is fixed in order to focus the analysis on the incremental value afforded by the ability to shift between alternative land uses, as opposed to the impact of form-based zoning on the timing of development. Before construction begins, the developer must decide on the percentage of units
Xi that will be dedicated to each property type 푖 (푖 = 1,2).
The revenue per unit generated by type 푖 property is not revealed until the end of the period when the initial construction is complete. Constant or declining marginal revenues are allowed for as a function of total developed units. Additional supply of a particular product type may have
8 little impact on rent levels in undersupplied markets, and revenues per unit may therefore be constant. On the other hand, the revenue per unit may be decreasing in quantity in overbuilt markets. The revenue from selling 푥 units of type 푖 property is therefore modeled generically as
휃푖푅푖(휆푖, 푥), where the elasticity parameter 휆 determines the degree of decline in marginal revenue and the parameter 휃 determines the level of demand. The revenue functions 푅푖(휆푖, 푥) are increasing, concave-down functions of units sold and satisfy the following conditions: i)
2 2 푅푖(0, 푥) = 푥, ii) 푅푖(휆푖, 0) = 0, iii) 휕푅푖(휆푖, 푥)⁄휕푥 > 0, iv) 휕 푅푖(휆푖, 푥)⁄휕푥 < 0, and v)
2 휕 푅푖(휆푖, 푥)⁄휕푥휕휆푖 < 0. For each property type, the demand parameter 휃 is assumed to evolve stochastically according to geometric Brownian motion, 푑휃푖 = 휃푖(µ푖푑푡 + 𝜎푖푑푧푖), where the two
Brownian motions are allowed to be correlated: 퐸[푑푧1푑푧2] = 𝜌푑푡.
Construction costs are paid upfront before construction begins. The cost for each property type is modeled generically as 훾푖퐶푖(휀푖, 푥) , where 훾 determines the cost level and 휀 determines the economies of scale in construction. The cost functions 퐶푖(휀푖, 푥) are increasing, concave-down functions of units built and satisfy the same conditions as the revenue functions 푅푖(휆푖, 푥) above.
The developer observes current demand levels 휃10 and 휃20 for the two property types and decides on the percentage of units Xi to construct of each type so as to maximize the discounted expected revenue less the construction costs. The rate at which the cash flows should be discounted is generally difficult to determine and depends on risk preferences. However, the profit may be viewed as a contingent claim on the stochastic demand state variables, and standard arguments imply profit may be computed by setting the discount rate equal to the risk-free rate r and risk- adjusting the drifts 휇푖. Because the demand state variables are not traded assets, the risk-adjusted
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∗ drifts 휇푖 are found by subtracting a risk premium from the true drifts and need not be equal to r.
Thus, the developer simply chooses X1 and X2 to maximize the profit
rT e E[1T R1 (1, X1 ) 2T R2 (2 , X 2 )] 1C1 (1, X1 ) 2C2 ( 2 , X 2 )
over the triangular region 0 ≤ X1 + X2 ≤ 1, 0 ≤ X1, 0 ≤ X2, where the expectation is in the risk neutral measure. The solution is straightforward. Since X1 and X2 are determined today, they do not depend on the state next period and the expectation term is simply
1T 2T E[1T ]R1 (1, X1 ) E[2T ]R2 (2 , X 2 ) e 10R1 (1, X1 ) e 20R2 (2 , X 2 ) . A standard two variable optimization problem emerges to determine the optimal profit :
maxA1R1(1, X1) A2R2 (2 , X 2 ) 1C1(1, X1) 2C2 ( 2 , X 2 )
over the triangular region 0 ≤ X1 + X2 ≤ 1, 0 ≤ X1, 0 ≤ X2 , where Ai i0 exp((i r)T) .
Because the objective function is separable in X1 and X2, necessary conditions for an
2 2 2 2 interior solution (X1 , X 2 ) are Ai Ri x i Ci x at X i for i 1,2. In particular, with constant marginal revenues it is not optimal to mix development—either it is not profitable to build any units or it is optimal to develop all of one property type or the other. In contrast, if construction costs are sufficiently low and revenue sufficiently sensitive to supply, it is always optimal to mix.
Now suppose the form-based code permits the conversion of type 1 property to type 2 after observing the demand outcome. Allowing the conversion of space in only one direction limits the flexibility afforded to the developer, but also serves to take into account practical factors such as parking requirements, mechanical system capacities or differences in bay depths anticipated to impinge upon the unfettered conversion of newly constructed commercial space to residential use.
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The cost to convert property is paid at time T and is assumed to be proportional to the amount of property converted. The developer now faces an additional decision. For each choice of X1 and X2 and for each demand outcome (휃1푇, 휃2푇) , the developer must decide at time T how much type 1 property y to convert to type 2. Given the demand outcome, the developer solves
, where is the per unit conversion cost. max1T R1(1, X1 y) 2T R2 (2 , X 2 y) c0 y 푐0 y X1
When marginal revenues are constant, the objective function is linear in 푦 and the solution
∗ ∗ is 푦 = 0 when 2T 1T c0 and is 푦 = 푋1 when 2T 1T c0 . With declining marginal revenues, the objective function is concave-down and therefore the solution is
cr 푐푟 y* minX1,max(0, y ), where 푦 is the critical point determined by the first order condition.
This value is then substituted into the profit function, and the optimal profit opt with the option to convert is determined by the optimization problem:
opt rT * * maxe E [1T R1(1, X1 y ) 2T R2 (2 , X 2 y ) c0 y ]1C1(1, X1) 2C2 ( 2 , X 2 )
∗ over the triangular region 0 ≤ X1 + X2 ≤ 1, 0 ≤ X1, 0 ≤ X2. Because the optimal 푦 depends on the demand outcome, the expectation term cannot be simplified as before and the solution typically must be found by numerical dynamic optimization. The flexibility to convert type 1 property to type 2 clearly adds value by allowing the developer to trade poor type 1 demand outcomes for higher type 2 demand outcomes. The value added by the conversion option is the difference opt
in project values with and without the option to convert.
When marginal revenues are constant, the profit function with the conversion option is separable and an analytic solution for is available. This case is investigated further in the
11 subsequent discussion, where it is shown that the conversion option can be interpreted as a spread option on the two demand variables. And although an analytic solution for the value added by the conversion option is not generally available when marginal revenues are not constant, it is possible to determine how this value is affected by changes in the underlying parameters. For example, because the value of the project without the conversion option depends only on expected
demand, an increase in either volatility parameter i has no effect on the value of the project without the conversion option. On the other hand, higher volatility obviously increases the value of the conversion option because the developer can trade poor demand outcomes for type 1 property for higher demand outcomes for type 2 property. Conversely, an increase in conversion costs has the contrasting effect of reducing the value of the conversion option. In fact, if conversion costs are too high the option to convert property is worthless.
An increase in construction costs clearly reduces the project value with and without the
conversion option. However, an increase in the linear cost component for type 2 property, 2 , increases the value of the conversion option (i.e., the value of the project without the conversion
option decreases more than the value of the project with the option) and an increase in 1 decreases the option value. In both cases, the option to convert becomes less valuable as it becomes relatively more expensive to develop type 1 property. To see this mathematically, first note that it can never be optimal to build less type 1 property in the presence of a conversion option than the amount of type 1 property optimally built without the option. Similarly, it is never optimal to build more of type 2 property with the conversion option. That is, the optimal amounts built with the conversion
opt opt opt opt option (X1 , X 2 ) and without the option (X1, X 2 ) satisfy X1 X1, X 2 X 2 . Because the cost functions are increasing in their second arguments, it follows that
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opt opt C1 (1, X 1 ) C1 (1, X 1 ) 1 1
opt opt C2 ( 2 , X 2 ) C2 ( 2 , X 2 ) 2 2 as claimed.
In a similar fashion, an increase in the economies of scale in construction costs for type 1
property, 1 , increases the value of the conversion option as the marginal cost of building more
type 1 property decreases, and an increase in 2 decreases the option value:
opt opt C1 (1, X 1 ) C1 (1, X 1 ) 1 1 1 1 1 1
opt opt C2 ( 2 , X 2 ) C2 ( 2 , X 2 ) 2 2 2 2 2 2
2 Here, marginal cost decreases with the degree of economies of scale: 휕 퐶푖(휀푖, 푥)⁄휕푥휕휀푖 < 0.
Intuitively, as the relative cost of building type 1 property decreases, again the option to convert becomes more valuable.
Because the demand parameters 10,20,1, and 2 appear inside the expectation in the profit function whether or not the conversion option is included, the effect of these parameters on the conversion option is more subtle. For example, the sensitivities of the profits with and without
the conversion option to the demand parameter 10 are given by
opt d rT 1T opt e E R1 (1 , X 1 y ) d10 10 and
d rT 1T e E R1 (1 , X 1 ) . d10 10
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Profits increase as 10 increases, but it is not clear which increases more. In some states it is optimal to convert all type 1 property and in some states it is optimal to convert nothing; thus in some
opt states R1 (1, X1 y ) R1 (1, X1 ) , while in other states it is the other way around.
Intuitively, the conversion option depends positively on when it is optimal to convert
opt fewer than X1 X1 units on average. This is the case because the profit with the conversion option is more sensitive to increases in . On the other hand, if it is optimal to convert more than units on average, then an increase in has a smaller effect on the profit with the conversion option and the conversion option depends negatively on . Similarly, the conversion
opt option depends positively on 20 when it is optimal to convert more than X 2 X 2 and it
opt depends negatively on when it is optimal to convert less than X 2 X 2 on average. This intuition is verified in the numerical example presented below.
The sensitivities of the profits with and without the conversion option to type 1 property
elasticity, 1 , are given by
opt d rT R1 opt e E1T (1, X1 y ) d1 and
d rT R1 e E[1T ] (1, X1 ) . d1
opt If X1 E[y ] X1 , the conversion option will increase with due to the condition that
opt the marginal revenue declines with the elasticity parameter. Conversely, if X1 E[y ] X1 ,
the conversion option will decrease with . Suppose that 2 0 and 1 0 , but otherwise the two property types have the same parameters. If construction costs are low it is optimal to build
14 all type 1 property, and if there is uncertainty in future demand, then with the option to convert it is optimal to convert some of the type 1 property to type 2 property in the appropriate states. For
small increases in 1 , it is again optimal to build all type 1 property, and it will be optimal to convert even more space in the marginal states. Therefore, the loss in revenue from the increased elasticity is partly mitigated by the additional conversion, thereby increasing the value of the
conversion option. When 1 2 , by symmetry it is optimal to build 50% type 1 property and 50% type 2 property without the conversion option, while it may again be optimal to build 100% type
1 property with the conversion option if there are sufficient economies of scale in construction.
However, it is optimal to convert less than 50% of type 1 property on average when there are
positive conversion costs. Therefore, the conversion option is less valuable. In fact, as 1 gets
very large relative to 2 it will no longer be optimal to build any type 1 property and the conversion option will have no value.
Thus, the value of the conversion option has a hump shape with respect to each of the demand level and elasticity parameters. As any one of these parameters gets very small or very large, the conversion option will have lower value. In fact, if construction costs are relatively low and the same for the two property types, then the value is near its maximum when the demand
parameters for the two property types are also the same: 10 20 and 1 2.
When marginal revenues are constant, the profit optimization problems simplify and an analytic solution for the valued added opt is available. The following discussion investigates this special case further. First note that with constant marginal revenues, the revenue without the conversion option is linear in X and the cost function is linear or concave down; thus, the profit
15 function is necessarily optimized at a corner: (0,0) , (1,0) , or (0,1). It follows that the optimal profit
is max0, A1 1C1 (1,1), A2 2C2 ( 2 ,1), where Ai i0 exp((i r)T) .
When the option to convert is available, the revenue is linear in the amount converted and
∗ therefore the developer optimally converts all or nothing: 푦 = 0 when 2T 1T c0 and
∗ otherwise 푦 = 푋1. It follows that the revenue is equal to 1T X1 2T X 2 when ,
and is equal to 2T (X 2 X1 ) c0 X1 when 2T 1T c0 . The revenue may be written more
succinctly as 1T X1 2T X 2 X1 max0,2T 1T c0 , and thus the profit function with the conversion option is equal to the sum of the profit function without the option and the product of
X1 and the value of the spread option on the two demand variables that pays out the maximum
of zero and 2T 1T c0 . The profit function is again concave up in X and is therefore also optimized at one of the corners, and so the optimal profit in this case is given by
opt max0, A1 1C1 (1,1), A2 2C2 ( 2 ,1).
The conversion option adds no value when demand is sufficiently low so that it’s not optimal to develop any property even with the option to convert. It also adds no value whenever
20 is much larger than 10 so that it is optimal to develop all type 2 property with or without the
conversion option. On the other hand, when 10 is large relative to 20 so that it is optimal to
opt develop all type 1 property, then A1 1C1 (1,1) and A1 1C1(1,1). In this case, the conversion option again adds little value because it is equivalent to an out-of-the-money spread option. Alternatively, the conversion option adds the most value when the developer is indifferent between developing all type 1 property or all type 2 in the absence of the option. This state is
reflected when A1 1C1 (1,1) A2 2C2 ( 2 ,1) ), in which case the value added with the conversion option is . To see this, note the conversion option adds value only when
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opt A1 1C1 (1,1) . If it is suboptimal to develop anything without the option, so that
A1 1C1 (1,1) 0, then the value added is A1 1C1(1,1) which is less than . Similarly,
the value added is A1 1C1(1,1) A2 2C2 ( 2 ,1), which is also less than , when it is optimal to develop all type 2 property. The value added is exactly when it’s optimal to develop
all type 1 property. Because the spread option value depends negatively on 10 and positively
on 20 , the value added is largest when A1 1C1 (1,1) A2 2C2 ( 2 ,1) .
Figure 1 summarizes the aforementioned constant marginal revenue results. In Region 0, the costs outweigh the revenues and it is not optimal to develop any property even with the conversion option. In Region 1, the demand for type 2 property is much larger than the demand for type 1 property and it is optimal to develop all type 2 property (and none of 1 property) even with the conversion option. The curved line separating Regions 1 and 2 indicates the values along which the developer with the conversion option is indifferent between developing all type 1
property and all type 2 property (i.e., along this curve, A1 1C1(1,1) A2 2C2 ( 2 ,1)). The diagonal line separating Regions 2 and 3 indicates the values along which the developer without the conversion option is indifferent between developing all type 1 property and all type 2 property
(i.e., along this line, A1 1C1(1,1) A2 2C2 ( 2 ,1)). Similarly, the curved line separating
Regions 0 and 4 indicates the values along which the developer with the conversion option breaks even by developing all type 1 property.
The conversion option adds value in Regions 2, 3, and 4. In Region 2, the demand for type
1 property is sufficiently large to encourage type 1 development with the option but not without, whereas in Region 3, the demand is so large that all type 1 property is developed without the option. In Region 4, the option is necessary for development to occur. The value added is largest
17 along the diagonal line separating Regions 2 and 3; however, the relative value added, defined as the ratio of the value of the conversion option to the profit without the conversion option, is necessarily largest in Region 4 where the option accounts for the entire value of the project.
Though the value of the conversion option may increase along the diagonal line as demand grows, the relative value is necessarily largest when demand is relatively weak.
INSERT FIGURE 1 HERE
The preceding analyses show that the conversion option can stimulate real estate development in relatively weak markets when it would not otherwise be profitable to do so. It can also stimulate mixed-use development in certain scenarios. For example, in the absence of the conversion option it is not optimal to mix the two property types when marginal revenues are constant or nearly constant. However, if the spread option is sufficiently valuable, offering the conversion option always results in mixed-use development. This is the case when there is sufficient demand for type 1 property to encourage the developer to move forward with the project and the demand for type 2 property is not so overwhelming as to encourage the production of only that type of space, as represented in Regions 2, 3, and 4 in Figure 1.
Numerical Results
To help verify the intuition that the relative value of the conversion option is greater in relatively weak markets when 0, but may not always produce mixed-use development, numerical solutions are presented in this section including selected functional forms for revenues and costs. Tables 1 and 2 present the base case parameters and numerical results respectively. It should be noted that the cost of constructing type 1 property is 10% higher than the cost of constructing type 2 property to create tension in the model and reflect a premium associated with constructing space with attributes making it suitable for future conversion.
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INSERT TABLE 1 HERE
In Table 2, each 4x2 block of cells shows a unique set of outcomes with defined initial demand levels ϴ10 and ϴ20, both ranging from 100 to 130. The first row of each of these blocks shows developer profits without the conversion option on the left and with the conversion option on the right. The amount of type 1 and type 2 property initially constructed in each state are presented in a similar manner in rows two and three respectively. The fourth row shows the average amount of type 1 property converted to type 2 at time T in the presence of a conversion option.
INSERT TABLE 2 HERE
The 4x2 block of cells located in upper left-hand corner of Table 2 shows no development occurs at all, irrespective of the conversion option, when demand levels for both types of property are sufficiently low. While unsurprising, these results illustrate the limitations of form-based zoning as an urban revitalization tool. These regulations are reliant on economic forces and are only capable of encouraging development when there is ample market demand. There are also scenarios where demand is strong enough to encourage development activity, but the conversion option has no value because the cost of constructing convertible space is simply too high relative to the demand. One such example, among many, can be observed when ϴ10 = 110 and ϴ20 = 110.
Demand levels warrant only the development of non-convertible space, which results in the same profits with or without the conversion option because it effectively has no value.
Nonetheless, there are scenarios where the option to convert type 1 property to type 2 encourages development when it would not otherwise occur. This can be seen when initial demand levels are ϴ10 = 110 and ϴ20 = 100. Development only occurs in the presence of the conversion
19 option, with 100% of the developable units initially constructed as type 1 property and approximately 25% of the developable units later converted to type 2. The profits achieved by the developer in this setting are solely a product of the conversion option. In related scenarios, development is feasible with or without the conversion option, but the ability to convert space increases the developer’s profits significantly. For example, when ϴ10 = 120 and ϴ20 = 100 the ability to convert space increases the developer’s profits by approximately 32% in comparison to the outcome in the same setting without the conversion option.
Although total profits increase with demand levels in all scenarios, the relative value of the conversion option tends to be greater in weaker markets so long as the economic benefits provided by the option to convert are not completely washed out by the associated costs. This can be observed by comparing the ϴ10 = 120 and ϴ20 = 110 scenario to the ϴ10 = 130 and ϴ20 = 120 scenario. The percentage increase in profits generated by the conversion option is 51% in the first of these scenarios and only 28% in the second holding all other factors constant. This finding is generally consistent with real option theory because the value of the conversion option comprises a greater percentage of the total project value in weak markets. Alternatively, “intrinsic value” or the value of a real estate development project in the absence of demand uncertainty, comprises a greater percentage of the total project value in strong markets. Hence, the relative benefits of the conversion option afforded by a form-based code are often greater when market demand is weaker.
It should be noted that there are many scenarios in Table 2 where it is optimal to develop all type 1 property or all type 2 property. This is the product of economies of scale in construction, relatively low levels of demand elasticity for both property types, and cost premiums associated with the construction and subsequent conversion of convertible space. In fact, there are only five scenarios where the conversion option encourages more mixing of type 1 and type 2 property in
20 comparison to the development patterns that occur in the absence of the option. There are also two scenarios where the conversion option reduces the extent to which type 1 and type 2 property are integrated. In both of these cases, the projects are comprised of 50% type 1 property and 50% type 2 property in the absence of a conversion option, only to become less integrated when the conversion option is introduced. Projected demand for type 2 property is just high enough to encourage the development of a substantial amount of convertible type 1 property in these scenarios, without being so high as to simply encourage the development of nonconvertible type
2 property at a lower construction cost from the onset. These nuanced relationships and interrelated variables highlight the potential limitations of form-based codes if the primary objective is to encourage the integration of residential and commercial space at the project level.
Table 3 presents the results generated by the model after increasing the elasticity parameters 휆1and 휆2 from 10% to 20%. As would be expected, this creates a greater incentive to mix type 1 and type 2 property in order to reduce the impact of downward sloping demand. The benefits derived from economies of scale in construction are mitigated and developer profits decline in comparison to the base case. It also becomes optimal to build-out less than 100% of the developable units in more of the scenarios where market demand for one property type, or both, is weak to moderate in strength. The conversion option does, however, tend to encourage the construction of a greater percentage of the total developable units than would occur in the absence of the ability to convert type 1 property to type 2 property.
One exception to the aforementioned generalization exists when ϴ10 = 120 and ϴ20 = 110.
This represents a market setting where a conversion option could conceivably constrain the scale of real estate development in comparison to a regulatory environment allowing mixed-use projects, but not the unilateral right to convert one type of property to another as might be possible in the
21 presence of a form-based code. First note that the incentive to convert type 1 property to type 2 property diminishes in outcome in which ϴ1T is relatively large and ϴ2T relatively small. The developer is better off constructing more type 1 property and less type 2 property in these states, subject to an understanding that high levels of demand elasticity may impinge upon such a strategy.
In contrast, conversion occurs more frequently when ϴ1T is relatively small and ϴ2T relatively large. The developer is better off constructing more type 1 and type 2 property in these states, while converting as necessary to accommodate evolving market demand. The amount of property converted from type 1 property to type 2 property tends to decrease as the total scale of development increases and only a portion of the type 1 property is converted when demand levels are moderate. Thus, a conversion option may increase or decrease the total amount of development depending upon demand levels and correlation, the relative size of economies of scale in construction in comparison to the demand elasticity parameters, and the costs associated with conversion. This observation is important because it suggests the flexibility afforded by the implicit conversion option in a form-based code may not increase the total scale of development in select market environments.
The results presented in Table 3 also provide further evidence that conversion options have greater relative value in markets with weaker demand. This can be observed as both ϴ10 and ϴ20 increase from 110 to 120 to 130. The conversion option increases projected profits in each of these cases by 6.25%, 3.78% and 2.74% respectively in comparison to the same settings without the conversion option. Once again, total profits generated by real estate development are greater when demand levels are higher, but the relative value of conversion options is greater in relatively weak markets. This may help form-based codes serve as an effective urban revitalization tool.
INSERT TABLE 3 HERE
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Table 4 is included to illustrate a fundamental premise of real option theory; the impact of demand volatility on option values. By increasing the volatility parameter σ to 30% the conversion option afforded by a form-based code becomes significantly more valuable. Since a greater portion of the project’s total value is associated with the option value, as opposed to the intrinsic value, the option to convert type 1 property to type 2 property encourages development in even the weakest demand scenario presented in the upper left-hand corner of Table 4 where both ϴ10 and
ϴ20 are set at 100. From the perspective of those interested in using form-based codes to stimulate urban renewal, this finding is noteworthy because it suggest conversion options are not only valuable when market demand is relatively weak, but also when demand levels are uncertain.
Conditions such as these are likely to exist in many urban areas in need of revitalization.
INSERT TABLE 4 HERE
Conclusions
The model presented in this paper examines the potential impact of form-based zoning on developer profits in a single-period setting. This approach allows for the derivation of closed-form solutions in some scenarios where real estate developers have the flexibility to alter the mix of residential and commercial space included in their projects as market conditions evolve. Providing developers with the ability to shift from one land use to another within their projects, so long as they conform to defined urban design requirements, appears capable of spurring development activity in markets where it would not otherwise occur. It may also increase the scale of development projects in a number of economic environments. The greatest relative impact on developer profits is likely to occur in weaker markets, as well as those with higher demand
23 volatility. Taken as a whole, these findings are consistent with real option theory and suggest form-based zoning can serve as one component of a market-driven urban revitalization strategy.
Form-based zoning ordinances may, however, fail to act as a catalyst for mixed-use development in some instances. Despite regulatory flexibility, real estate developers may still find it advantageous to engage in single-use development if doing so allows them to avoid conversion costs or benefit from economies of scale in construction. The profitability of such an approach is amplified when marginal revenues remain constant irrespective of the amount of residential or commercial space delivered to the market. All of these factors highlight the limitations of form- based zoning and the need to consider other regulatory alternatives in specific market settings if stimulating mixed-use development is a municipality’s primary policy objective.
Despite the importance of the aforementioned findings, it should be noted that the model presented in this paper examines only the value of the conversion option afforded by form-based zoning and not several of the other benefits discussed in the extant literature. The model does not consider the regulatory predictability offered by this type of zoning, nor does it take into account its ability to encourage the development of cohesive urban neighborhoods reflecting a community’s vision. These issues require further exploration using both qualitative and quantitative research methods. Furthermore, the numerical solutions presented in this paper effectively compare the outcomes of a form-based code to those of an alternative land use regulation also permitting mixed-use development, but not providing the developer with the ability to convert one type of space to another over time as market conditions evolve. The flexibility afforded by form-based codes is therefore the primary focus of the research, as opposed to the value associated with the right to engage in mixed-use development from the onset of a project.
This distinction must be kept in mind when evaluating the results generated by the model.
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The paper also stops short of considering several pragmatic concerns that may limit form- based zoning’s effectiveness as an urban revitalization tool. First and foremost, many communities may simply lack the financial resources and human capital needed to implement this type of land use regulation (Garvin and Jourdan, 2008). Institutional biases serve as another potential obstacle to mixed-use development, irrespective of the regulatory environment, when capital providers and real estate practitioners favor more traditionally types of development that perpetuate low-density land use patterns (Bohl and Plater-Zyberk, 2006). Finally, political forces may limit the benefits economically disadvantages members of a community derive from a form-based code if urban elites and real estate development interests have a disproportionate amount of power in the planning and implementation process (Inniss, 2007). These questions must be addressed in the future to better understand the advantages and disadvantages of form-based zoning ordinances.
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Figures
Figure 1. Optimal Development Regions*
*The regions of no development, all type 1 development and all type 2 development with and (휇∗−푟)푇 without the conversion option for different initial demand level combinations (퐴푖 = 휃푖0푒 푖 ). In Region 0, the optimal amount developed without and with the option is 푋 = 푋표푝푡 = (0,0); in Region 1, 푋 = 푋표푝푡 = (0,1); in Region 2, 푋 = (0,1), 푋표푝푡 = (1,0); in Region 3, 푋 = 푋표푝푡 = (1,0); and in Region 4, 푋 = (0,0), 푋표푝푡 = (1,0).
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Tables
Table 1. Base Case Parameters
Parameters for the stochastic demand functions = 100 to 130 휃푖0 is the stochastic demand parameter for property type i at time zero 휃0
휆푖 is the elasticity parameter determining declining marginal revenue for property type i = 20% 𝜎푖 is the demand volatility rate for type i space 𝜎푖 ∗ 휇∗ = 5% 휇푖 is the risk-neutral drift in demand for property type i 푖 Parameters for the construction cost functions = 110; = 100 훾푖 is the cost level per unit for property type i 훾1 훾2 휀푖 determines the economies of scale in construction for property type i 휀푖 = 10% 푐0 is the per unit conversion cost 푐0 = 10 Other parameters N is the total number of units that can be developed N = 100 T is the length of time required to complete construction T = 1 r is the risk free rate r = 5%
Table 2. Numerical Solutions Derived from the Base Case Parameters
ϴ20=100 110 120 130
ϴ10=100 0.00 0.00 9.00 9.00 18.00 18.00 27.00 27.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100.00 100.00 100.00 100.00 100.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 110 0.00 4.45 9.00 9.00 18.00 18.00 27.00 27.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100.00 100.00 100.00 100.00 100.00 100.00 0.00 25.08 0.00 0.00 0.00 0.00 0.00 0.00 120 9.00 11.91 9.50 14.36 18.00 18.00 27.00 27.00 100.00 100.00 50.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 50.00 0.00 100.00 100.00 100.00 100.00 0.00 18.09 0.00 27.28 0.00 0.00 0.00 0.00 130 18.00 19.83 18.00 21.67 19.00 24.31 27.00 27.93 100.00 100.00 100 100.00 50.00 100.00 0.00 100.00 0.00 0.00 0.00 0.00 50.00 0.00 100.00 0.00 0.00 12.52 0.00 19.87 0.00 29.40 0.00 38.54
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Table 3. Numerical Solutions After Increasing the Elasticity Parameter (λ = 20%)
ϴ20=100 110 120 130
ϴ10=100 0.00 0.00 2.08 2.08 7.14 7.14 14.06 14.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 42.00 42.00 71.00 71.00 94.00 94.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 110 0.00 0.39 2.08 2.21 7.14 7.18 14.06 14.07 0.00 20.00 0.00 11.00 0.00 6.00 0.00 3.00 0.00 0.00 42.00 38.00 71.00 70.00 94.00 93.00 0.00 5.05 0.00 1.59 0.00 0.84 0.00 0.15 120 1.92 3.23 4.00 4.84 9.00 9.34 15.24 15.45 38.00 55.00 38.00 74.00 33.00 39.00 21.00 25.00 0.00 0.00 42.00 0.00 67.00 61.00 79.00 75.00 0.00 9.99 0.00 21.57 0.00 3.27 0.00 2.17 130 6.66 8.55 8.70 10.73 12.75 13.69 17.90 18.39 67.00 89.00 63.00 100.00 48.00 100.00 35.00 44.00 0.00 0.00 37.00 0.00 52.00 0.00 65.00 56.00 0.00 14.95 0.00 25.57 0.00 33.11 0.00 4.84
Table 4. Numerical Solutions Increasing the Demand Volatility Parameter (σ = 30%)
ϴ20=100 100 120 130
ϴ10=100 0.00 2.16 8.98 8.98 17.98 17.98 26.98 26.98 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100.00 100.00 100.00 100.00 100.00 100.00 0.00 40.12 0.00 0.00 0.00 0.00 0.00 0.00 110 0.00 8.65 8.98 12.70 17.98 17.98 26.98 26.98 0.00 100.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 100.00 0.00 100.00 100.00 100.00 100.00 0.00 30.71 0.00 41.16 0.00 0.00 0.00 0.00 120 8.98 15.76 9.48 19.07 17.98 23.25 26.98 28.21 100.00 100.00 50.00 100.00 0.00 100.00 0.00 100.00 0.00 0.00 50.00 0.00 100.00 0.00 100.00 0.00 0.00 25.55 0.00 32.35 0.00 42.24 0.00 48.75 130 17.98 23.18 17.98 26.06 18.98 29.51 26.98 33.81 100.00 100.00 100.00 100.00 50.00 100.00 0.00 100.00 0.00 0.00 0.00 0.00 50.00 0.00 100.00 0.00 0.00 21.76 0.00 26.75 0.00 33.94 0.00 43.10
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