INTENSITY AND TIMING OPTIONS FOR INVESTMENTS IN A LESS THAN
PERFECTLY COMPETITIVE MARKET
CHU, Yongqiang
and
SING, Tien Foo
Department of Real Estate
National University of Singapore
Date: April 14, 2004
Abstract
The traditional real options models invariably assume that firms are price taker on one
hand, and they behave, on the other hand, like monopolists who face no competition
and competitive entry of new players in the future. The assumptions are relaxed in this
paper by firstly modeling the real estate development market in a monopolistic
framework, and substituting the exogenous rental or price variable with a more general
economic shock variable. The market rent in our model is endogenously defined in the
demand and the cost functions of firms with a simple linear monotonic production
technology. The capacity choice in our model is not a continuous and incremental
process. Instead, the developer will have to make the timing and the intensity decisions
simultaneously when the development option is exercised. In additional to demand
uncertainty, the numerical results show that the rental sensitivity to housing demand
will also have an inverse effect on the deferment option values of a real estate project.
Both the timing and intensity decisions are sensitive to the demand factors, which is
represented by the rental sensitivity variable.
Key Words:
Real Options, Optimal timing, Optimal Intensity, Real Estate Development
*Corresponding Author. Email rststf@nus.edu.sg. Address: Department of Real Estate, National
University of Singapore, 4 Architecture Drive, Singapore 117566. We wish to thank the comments by Prof.
Abdullah Yavas, Yuming Fu, and other participants at the research workshop at NUS on 12 April 2004.
INTENSITY AND TIMING OPTIONS FOR INVESTMENTS IN A LESS THAN
PERFECTLY COMPETITIVE MARKET
Introduction
The timing and intensity of investment are two important decisions faced by investors
in the process of selecting their production capacity. Real estate development is a
typical capital investment process, where a developer possessing a developable land has
to decide on when to start developing the land and how much to build on the land,
given the fact that zoning dictates the particular use type permitted on the land. The
timing question has been extensively studied in the real option and optimal stopping
time literature (McDonald and Siegel, 1986; Titman, 1985; Clarke and Reed, 1988;
Williams, 1991, et. al.). Pindyck (1988) first examines the capacity choice issue in an
irreversible investment framework. When investment is not perfectly reversible, he
found that firms hold less capacity than they would when the demand is uncertain. In
his model, the “lumpiness” of capital investment is ignored. The firms are, therefore,
assumed to have the flexibility of increasing their capacity as and when the needs arise.
This flexible capacity choice assumption may not be realistic in the real estate
development process. Firstly, there are statutory planning and technical constraints on
the maximum permissible density of development on a land parcel. Secondly, when the
foundation and floor plate of a building are set, it will be costly to reverse the intensity
decision on the land. The costs incurred are sunk. The investment is though not strictly
irreversible, any expansion of the land use capacity in the future would, however, come
at a high cost. Therefore, the intensity and timing decisions of a development are
important decisions, which are often made simultaneously in a real estate development
process (Williams, 1991; Capozza and Li, 1994). In other words, the development
intensity must be decided at the time the development option is exercised.
In most of the earlier real option models, prices or rental cash flows are used to proxy
investment payoffs. The price generating processes are assumed to follow specific
1
exogenous stochastic processes (Williams, 1991; Clarke and Reed, 1988; Capozza and
Helsley, 1989 and 1990; Capozza and Sick, 1994). Under this assumption, firms are
price takers who could not influence the price process in the future. The assumption is
only valid in a perfectly competitive market, where there exists multiple buyers and
sellers that will set the equilibrium price. However, in most of the earlier real option
models, the firm or the option holder is always assumed to be a monopolistic one, who
does not face any market competition. Given the monopolistic right, the firm’s
investment decision will always have significant influence on the price behavior of the
product. The firm is known to be a price-setter. The exogenous price function is,
therefore, rather restrictive in the real option model setting with only a single
monopolistic firm. The solution of the real option models offers at best a partial
equilibrium option estimate for a perfect competitive market.
The real estate market with the characteristics of product segmentation and high barrier
of entry for new players is truly not a competitive market. Few big players, instead,
dominate the market and establish significant monopolistic influence on the supply and
pricing of the products. The price setting behavior of the developers is not well
represented by the exogenous price assumption in most of the real option models on
development timing. To reflect the monopolistic power of the developer in a sub-sector
of real estate market, we model the development timing options using a more general
demand shift variable in place of the stochastic price function (Pindyck, 1988). In this
model setting, the developer is a monopolist, and yet his/her investment decision is not
restricted by the exogenous price function, but is driven by the demand shift variable.
The optimal development intensity of land parcels is another issue examined by
Williams (1991) and Capozza and Li (1994). In their models, they assume that the
developer can optimally pick the development density of a land at the time of
exercising the investment option. The optimal density is independent of the cost and
price functions, and it is fixed once determined. We relax the assumptions in Williams
(1991) and Capozza and Li (1991) models by modeling the density as an endogenous
2
variable in the price and cost functions.1 By imposing the microeconomic structure to
the price and cost functions with monotonic elasticity of scale, the capital choice and
development timing decisions are no longer separable. The optimal density decision
affects the boundary conditions, which in term affect the estimation of the development
timing option premium in the proposed model. Therefore, the optimal density and the
optimal timing of a development will have to be simultaneously and jointly solved in
the proposed real options models.
The paper is organized into six sections. Section II reviews the relevant real options
literature in optimal timing and intensity of investment. Section III explicitly specifies
the timing option and capacity choice problems with necessary assumptions. The
theoretical model and its respective analytical solutions are derived in section IV. Based
on a set of parameter inputs, the numerical analyses for the proposed optimal timing
and intensity of investment models are conducted, and the comparative statics are
summarized in section V. Section VI concludes the findings.
Literature Review
The paper of McDonald and Siegel (1986) was one of the earliest works that studies the
optimal timing of investment in an irreversible project. In their model, they assume that
the cost and value of the project follow exogenously defined continuous time stochastic
processes. They analyze the basic problem of timing of investment under the
uncertainty, and showed that the timing options do have value under uncertainty. The
investment rule derived from the traditional NPV method is inadequate when values
and costs of the project are not deterministic. The firm will invest only when the
benefits of undertaking the project exceed the costs by a positive amount that is large
enough to compensate the investors for giving up their timing options.
1
Unlike in Williams’s (1991) model, there is no statutory zoning restriction that imposes a
maximum permissible density of the land in our model and the model by Capozza and Li
(1994).
3
The typical optimal timing model of McDonald and Siegel (1986) has subsequently
been extended with different model assumptions and applied to evaluate investors’
behavior in different investments. Titman (1985) employs a simple binomial tree
method to explain why deferment of development of parking lots that are located in an
exclusive residential neighborhood is a feasible option when the market price is
uncertain. Williams (1991) then recast the urban property development problem by
modeling both the project cost and cash flows of the property as stochastic processes.
They show that developers can optimally decide on the density and timing of
development of the lands to maximize the market value of the land in a perfectly
competitive market framework. Clarke and Reed (1988) and Sing (2000) also apply the
real options framework to analyze the optimal timing problem in vacant developable
lands with different assumptions on the stochastic variables. They again confirm that
price uncertainty (Clarke and Reed, 1998), or rental uncertainty (Sing 2000) of the
developed property defer the land development decision, because the value of the
vacant land and its embedded timing options increase proportionately higher than the
developed property value.
In the urban land pricing model, Capozza and Helsley (1989) develop a simple dynamic
land price model that is composed of four additive components: the value of
agricultural land rent, the cost of conversion, the value of accessibility and the growth
premium. Capozza and Helsley (1990) then extend the urban land pricing model to a
stochastic city in which the household income and land rent follow stochastic processes.
Using the first hitting time approach to determine the optimal conversion of agricultural
land to urban use, the model shows that the rental uncertainty delays the conversion of
agricultural land to urban use, reduces the equilibrium city size and also imparts an
conversion option premium to agricultural lands. The growth premium also increases
the price of agricultural land at the boundary of urban land by a premium that is higher
than its opportunity cost. Capozza and Sick (1994) further improve the land pricing
model by adding the spatial and temporal risk structure to the model. By assuming that
the cash flows on land to follow additive diffusion process, they found that the price
4
agricultural land awaiting for conversion increases with the growth and unsystematic
risk of the urban rents, but decreases with risk aversion. They found that uncertainty
premium related to urban rents will have significant positive effects on both urban and
agricultural lands.
In the earlier land pricing model, capacity choice and intensity of investment is fixed
and constant. Pindyck (1988), on the assumption that firms can continuously and
incrementally expand the capital, shows that firms would hold less capacity in a market
with volatile and unpredictable demand. The opportunity cost of exercising the option
to increase the marginal unit of capital increases with uncertainty in demand. Williams
(1991) extend the capacity choice option to a model involving optimal development
density decision by including an exogenous density variable to the real estate
development option model. In his model, the optimal timing and optimal density of
development are driven by the uncertainties in both revenue of developed property and
the cost of development. Capozza and Li (1994) extend the density choice to modeling
the decision of converting vacant land to urban use. In an optimal stopping time
framework where the decisions to choose the timing and density of conversion of
vacant land are made simultaneously by developer, they found that stochastic density of
development increases the hurdle rate and delays development.
Compared to Williams (1991) and Capozza and Li (1994) models, where the stochastic
rents are exogenously defined, we follow Pindyck’s (1988) specification of a stochastic
market demand variable, which is a better reflective of capacity and timing of
investment decisions in a not so perfectly competitive real estate market. As a typical
real estate market is more akin to an oligopolistic or monopolistic market by nature, the
developer is unlikely to be a price taker as represented by an exogenous rent or price
function. Therefore, we assume a stochastic economic shock as the state variable that
drives the density and timing options in our model. In Pindyck’s (1988) model, capacity
expansion is a continuous and incremental process. Whereas, in our case, real estate
project is lumpy, the assumption of marginal increases in density from time to time is
5
not practical. We restrict our development intensity to a discrete process, which has to
be made simultaneously with the development timing decision.
There are other real option literature that examines issue relating to time to build
options (Majd and Pindyck 1986; Sing 2001), interest rate options (Ingersoll and Ross,
1992), entry and exit decision of firms (Dixit, 1989; Trigeorgis, 1991) and real estate
investment in oligopoly market (Grenadier 1996, Wang and Zhou 2002).
Problem Specification
Real estate market is generally characterized by its heterogeneous and immobile nature
of its products. Developers with sufficiently large capitals and strong track records
could exert monopoly power in a particular segment of markets. In our model with few
homogeneous developers, the market is not perfectly competitive. The monopolistic
influence of developers on the supply would indirectly determine the market prices and
rental cash flows of developed property. Therefore, the exogenously defined stochastic
price and rental functions are not feasible in our framework. The underlying driver of
the investment decision is represented in our model by a stochastic economic shock
variable that follows a multiplicative diffusion process.
To set up the framework of our model, we assume that the developer has already
acquired a vacant land. The decision faced by him is “when” and “how much” to
develop on this parcel of land. When dealing with the first problem of deciding on
when is the optimal time to develop, the decision is not a straight forward binary choice
of “now or never” as predicted by the traditional net present value (NPV) rule. When
the market is volatile, the simple rule of “NPV greater than or equal to zero” is no
longer applicable. Instead, many real options models (McDonald and Siegel, 1986;
Clarke and Reed, 1998; Sing 2000; et al) have suggested that the developer should only
initiate the development when the profit exceeds the cost by an amount that is
commensurate with giving the options of waiting for another period.
6
The question of intensity of development if ignored in the real options models will lead
to under-estimation of the urban land prices (Williams, 1991; Capozza and Li, 1994).
The flexibility given to the developer in adjusting the density of development will
interact in a positive and significant way to the conversion option premiums especially
when the market value for the urban land use is volatile. In a monopolistic market
framework, the intensity of development will have a direct influence on the price and
rental cash flows of the developed real estate. Therefore, unlike in Williams (1991) and
Capozza and Li (1994) models, the density variable in our model is defined as
endogenous functions of demand and supply of the undeveloped lands. The functional
forms of the demand and supply equations, which are dependent on the density variable,
will have significant effects on the timing options.
As real estate investment is lumpy and indivisible, the sequential expansion and
marginal increase in capacity investment processes like those advocated in the capacity
choice (Pindyck, 1988) and the time to build literature (Majd and Pindyck, 1986; Sing,
2001) may not be feasible. We, instead, follow the assumptions in Williams (1991) and
Capozza and Li (1994) models, which require that the intensity decision will have to be
made concurrently and simultaneously when the timing of development is decided.
The Model
Assuming that the developer faces the following inverse demand function:
R = YD(Q )
(1)
where R is the project cash flows, D(Q) is a concave market demand that increases at a
decreasing rate with respect to the density of new developed property built on the
vacant land, (Q), that is [ D ' (Q ) < 0 ], and Y is the exogenous economic shock that is
assumed to follow a geometric Brownian motion given below:
dY
= µdt + σdw
Y
(2)
In equation (2), µis the expected drift rate of Y, σ is the volatility of the incremental
change in Y, and dw is the incremental change of the standard Wiener process. The risk
7
variable, σ, in equation (2) is a constant firm specific idiosyncratic risk, which is not
related to the systematic market risk as in Capozza and Sick (1994).
The developer’s project cost is deterministic, i.e. it is independent of any stochastic
driver. It can be defined as a convex function of development density given below:
C = C (Q )
(3)
and is subject to a positive first order derivative constraint of [ C ' (Q ) > 0 ].
Like the earlier models by Williams (1991), the ownership of a vacant land give the
landowner a call option, when exercised, gives him a right to claim on the cash flows
generated from the developed property built thereon. Therefore, the vacant land value
can be modeled as a development option, [V], with an infinite maturity life, and it can
be defined as a function of the stochastic shocks variable,
V = V (Y )
(4)
Based on Ito’s Lemma, the incremental change in the option value over a short time
interval, [dt], can be expanded as follows:
1
∂ 2V
∂V
∂V
)dt + σ
dw
dV = ( σ 2Y 2
+ µY
2
2
∂Y
∂Y
∂Y
(5)
Since the firm specific idiosyncratic risk can be fully diversified away, we could
simplify equation (5) to the following ordinary differential equation (ODE) form:
1 2 2 ∂ 2V
∂V
σ Y
+ µY
− rY = 0
2
2
∂Y
∂Y
(6)
We assume that there exits a trigger value of [Y*], at which the development will
commence as long as the market shock exceeds Y*. The optimal time at which the
development occurs can be represented as [ T = inf[T ≥ 0, Y (T ) = Y * ], and at time T,
the value of the development option is simply the net discounted future cash flows of
8
the project, which is given as follows:
V (Y * , q) =
D(q )q − ( r − µ )τ *
e
Y − C (q)
r−µ
(7)
where τ is the time to build variable, which is assumed to be constant over time.
At time T when the development option is exercised, the developer will also have to
simultaneously decide on the optimal development density, q, which will maximize the
value of the vacant land as defined in equation (7). The optimal density could be
determined by taking the first order derivative of equation (7), and equating it to zero,
i.e. [
∂V
= 0 ].
∂q
To solve the optimal density of development, we need to explicitly define the functional
structure of demand and cost equations by generalizing the two the equations with the
following specifications:
D (Q ) = a − bQ
(8)
C (Q ) = c + dQ
(9)
Based on equations (6) to (9), the optimal density of development on the subject vacant
land, where [ q ≥ 0 ], can be defined as follows:
q* =
a
d
−
2b 2 KbY *
where K =
(10)
e − ( r − µ )τ
r−µ
The optimal value of the development option at time T in equation (9) can be redefined
as:
⎧
⎫
ad
a 2 KY *
d2
*
=
+
−c−
V
(
Y
)
; for (q * ≥ 0)⎪
⎪⎪
*
⎪
4b
2b
4bKY
⎨
⎬
⎪
⎪
⎪⎩V (Y * ) = 0;
otherwise ⎪⎭
(11)
9
The two parts of the value functions for different Y, and the optimal payoff value for the
land with embedded development option, V(Y*), corresponding to equation (11), are
shown in Figure 1.
After determining the optimal density of development, equation (11) will be used as the
traditional value matching conditions in solving the optimal timing of development.
The smooth pasting condition that will ensure the development will occur at time T is
given as follows:
V ' (Y * ) =
a2K
d2
−
4b
4bKY *2
(12)
A natural value-absorbing boundary condition when [Y=0]. At this point, the developer
will never exercise the option to start development and the option value is worthless,
which is defined as follows:
V ( 0) = 0
(13)
The ODE (6) is a homogenous second order differential equation, which has the
following general solution form:
V (Y ) = B1Y β1 + B2Y β 2
(14)
where β 1 and β 2 can be solved from the following quadratic equation:
1 2
σ β ( β − 1) + µβ − r = 0
2
(15)
Thus, we obtain:
β1 =
β2 =
− ( µ − 1 / 2σ 2 ) + ( µ − 1 / 2σ 2 ) 2 + 2rσ 2
σ2
− ( µ − 1 / 2σ 2 ) − ( µ − 1 / 2σ 2 ) 2 + 2rσ 2
σ2
>0
(16a)
<0
(16b)
Based on the value absorbing constrain, [ B2 = 0 ], and B1 of equation (14) could be
solved subject to the upper constraint in equation (11), where [q* ≥ 0]. The following
system of equations:
10
β
B1Y * 1 =
B1 β1Y *
a 2 KY *
d2
ad
+
−c−
*
4b
4bKY
2b
β1 −1
=
(17a)
a2K
d2
−
4b
4bKY *2
(17b)
The analytical solution for B1 and Y* can be given below:
a 2 KY *
B1 =
4b
(1− β1 )
+
ad
d2
− (c + )Y *( − β1 )
*(1+ β1 )
2b
4bKY
(18)
There are two groups of solutions for Y*, which correspond to value functions 1 and 2
in Figure 1 respectively:
(ad + 2bc ) β 1 + (ad + 2bc ) 2 β 1 − a 2 d 2 ( β 1 − 1)
2
Y =
*
(19)
a 2 K ( β 1 − 1)
(ad + 2bc ) β 1 − (ad + 2bc ) 2 β 1 − a 2 d 2 ( β 1 − 1)
2
Y* =
2
2
a 2 K ( β 1 − 1)
(20)
Subject to the minimum development density constraint, [ q * ≥ 0 ], the solution in
equation (20) will be truncated in the option payoff function. Therefore, the option
premium of the timing and intensity of development options for the vacant land can be
computed based on equations (18) and (19),
⎧V (Y ) = B1Y β1
⎪
⎨
a 2 KY
d2
ad
+
−c−
⎪V (Y ) =
4b
4bKY
2b
⎩
for Y < Y * ⎫
⎪
⎬
*
for Y ≥ Y ⎪
⎭
(21)
Numerical Analyses
Based on a base case scenario for a typical real estate development project that contains
the following input parameters as summarized in Table 1:
11
Table 1: Input Assumptions for Numerical Analyses
Input Parameter
Instantaneous drift of economic shock
Volatility of economic shock
Fixed cost
Variable cost (psm)
Demand parameter 1
Demand parameter 2 (psm)
Risk free interest rate
Time-to-build (year)
Base Value
0.06
σ = 0.2
c = $5x106
d = $200 0
a = $5000
b = $0.5
r = 10%
τ =3
We numerically analyze the effects of changes in volatility of economic shocks and
demand parameter on the trigger value, the optimal intensity and also the value of the
development options.
A. The Volatility Effect
Figure 2 shows the standard real options prediction (Dixit and Pindyck, 1994), which
suggests a positive relationship between the volatility and the trigger value of the
economic shocks, which can be represented by
∂Y *
> 0 . When the future market
∂σ
environment is uncertain, the developer will be less incline to commence the
development earlier. The higher trigger value for the development options is translated
into higher payoffs that are high enough to not just offset the development costs, but to
compensate the developer for giving up the waiting options. The results imply that
development activities will be relatively lower when economic uncertainty in a market
is high, and developer will likely to wait for clearer signals in the market before taking
any action to start development.
The volatility in the market increases the trigger value, which in term motivates the
∂q *
developer to increase the level of intensity of the development, i.e.
> 0 (Figure 3).
∂Y *
The result is consistent with that in Capozza and Li (1994), which predicts that
12
developer will delays the development when the economic is highly volatile, and when
the economic shock triggers the development option subsequently, the development
will likely to be more intensive and higher in its density. The intensity premium further
enhances the premium required for the developer to exercise the development options
earlier.
The higher intensity premium adds to the development option premiums in a highly
volatile environment. Figure 4 shows the development option value under different
volatility of economic shocks. By drawing a radius line from the origin, the payoffs that
trigger the development option can be determined as the tangency points that equate to
the optimal development timing option premium. Again, there is no surprise to the
results that show a positive volatility and value of development options.
B. Demand Effects
The demand effects, which are represented by the rental sensitivity to market demand,
[b],2 on the developer’s decision on the optimal timing and density of development are
numerically evaluated. In equation (8), the demand is an inverse function of density of
development. Thus, [b] measures the inverse multiplier effects of quantity of demand in
the market on the prices. Since the firms in the monopolistic framework in our model
are not a price taker, the demand is likely to respond to the changes in the rent and
quantity variables. The numerical analyses of the effects of [b] on the standard real
options indicators will likely to shed new light to the real options literature that
invariably assume a perfectly competitive market with exogenous price functions.
The positive relationship between the trigger value and b in Figure 5 indicates that
when the rental sensitivity to market demand is low, a higher trigger value is required
2
[b] in equation (8) can be considered as an inverse measure of the rental elasticity of demand,
which indicates a lower elasticity effects when the b is high for any given level of rent. This
rental sensitivity variable, which we prefer to term b in this paper, is a good proxy for our
analysis of the effects of monopolistic market power on the development options.
13
for the developer to exercise the development option. In other words, in a market when
the rental sensitivity is low, developers are less likely to start new projects and the
development activities in the market are likely to be depressed. The deferment options
become more valuable when the market rent is less sensitive to changing demand.
There is also a positive effect of [b] on the optimal intensity of development as shown
in Figure 6. When [b] increases, i.e. the rent sensitivity of market demand is low,
developers will likely to develop a project that is lower in density than it would
otherwise be in a normal market condition. This result is consistent in a monopolist
market, which predicts that low price elasticity will result in lower outputs.
The convex lines in Figure 7 show a negative relationship between [b] and the
development option values. The result shows that low rental sensitivity contributes in a
positive way to increasing the development option value. In a monopolist market,
developers can influence the demand and price of the product, and the sensitivity of the
product rent or price to demand will also add to the value of the development options.
When demand uncertainty is high and rental sensitivity is low, the waiting options
become highly valuable. The development activities will be significantly curtailed and
the city growth will also likely be slow.
C. Comparative Statics
The comparative statics in Table 2 summarize the relationships between various input
parameters with the trigger value, optimal intensity and also the development option
value.
14
Table 2: Comparative Statics
Trigger Value
(Y*)
Optimal Intensity
(q*)
Instantaneous Drift ( µ )
-
-
Value of the
Development
Option
+
Fixed Cost (c)
Variable Cost (psm) (d)
Demand Parameter 1(a)
Risk Free Interest Rate (r)
Time-to-Build (year) ( τ )
+
+
+
+
+
+
+
+
-
Input Parameter
Conclusion
The optimal timing problem has been extensively studied in the real options literature.
However, most of the models make two restrictive assumptions with respect to the price
function and the market structure. On one hand, the models assume that the investors
are monopolists who face no competition from future entry, and they are not price taker
for their products. On the other hand, the models invariable specify the price function
as an exogenous stochastic process, and the price changes are independent of the
demand shocks in the market.
The proposed real development option model in this paper reconciles the two
assumptions by providing a more general framework to analyze the optimal timing and
optimal intensity decisions faced by real estate developers. While maintaining the
monopolistic structure of the real option model, which is consistent with the
characteristic of many real estate sub-markets that are dominated by few large and
financially strong developers, we relax the rigid assumption of an exogenous defined
rental function. Instead, we allow the market rent to be endogenously and jointly
defined in an inverse demand function and a cost function. Like Williams (1991) and
Capozza and Li (1994), we also further extend the model by allowing the intensity of
development to be variable. Unlike in Pindyck’s (1988) marginal approach, where
capital investment is a continuous process embedded with a series of compound options,
the optimal timing and optimal intensity are joint decisions in our model. They will
15
have to be solved simultaneously when the development option is exercised.
Our numerical results of the volatility effects are not inconsistent with those observed
in the standard real option literature, which predicts a positive relationship between
volatility of demand shocks and the trigger value and the development option value.
However, the effects of market demand factor, which is represented by the rental
sensitivity variable, [b], shed new insight to the findings in optimal timing and intensity
options in the literature. The results show that when rental sensitivity to the market
demand is low, we would expect the development activities to slow down and also the
scale of development, if undertaken, will likely be smaller. The deferment option value
is not only dependent on the demand shock volatility; it is also negatively related to the
rental sensitivity to the market demand. For a market that is insensitive to the changes
in demand, we would expect the development activities to be depressed and the city
growth to be slower.
There are several extensions that can be made to the existing model framework
proposed in the paper. Firstly, the cost function, which is assumed to be deterministic,
can be assumed to follow a realistic stochastic process. Secondly, the monopolist
scenario can be further extended to cover oligopolist market using a duopoly model like
those proposed by Grenadier (1996 and 2001). In his duopoly real options papers, the
output is again exogenously determined. This assumption could be relaxed by allowing
the output to be endogenously determined in both the real option and Stackelberg
competition frameworks.
References:
Capozza, D.R. and R. Helsley, “The Fundamental of Land Prices and Urban Growth,”
Journal of Urban Economics, 1989, 26, 295-306.
Capozza, D.R. and R. Helsley, “The Stochastic City,” Journal of Urban Economics,
1990, 28, 187-203.
Capozza, D.R. and Y. Li, “The Intensity and Timing of Investment: The Case of Land,”
The American Economic Review, 1994, 84(4), 889-903.
16
Capozza, D.R. and G. Sick, “The Risk Structure of Land Markets” Journal of Urban
Economics, 1994, 35, 297-319.
Clarke, H.R. and W. Reed, “A Stochastic Analysis of Land Development Timing and
Property Valuation,” Regional Science and Urban Economics, 1988, 18, 357-381.
Dixit, A., “Entry and Exit Decision under Uncertainty, Journal of Political Economy,
1989, 97(3), 620-638.
Dixit, A. and R. Pindyck 1994, Investment under Uncertainty, Princeton University
Press, Princeton, NJ
Grenadier, S.R., “The Persistent of Real Estate Cycles”, Journal of Real Estate Finance
and Economic, 1995, 10, 95-119.
Grenadier, S.R. “The Strategic Exercise of Options: Development Cascades and
Overbuilding in Real Estate” Journal of Finance 1996, 51, 1653-1679.
Grenadier, S.R. 2000, Game Choices: The Intersection of Real Options and Game
Theory, Risk Books, London, UK.
Ingersoll, J.E. and S. Ross, “Waiting to Invest: Investment and Uncertainty,” The
Journal of Business 1992, 65(1), 1-29.
McDonald, R. and D. Siegel, “The Value of Waiting to Invest”. The Quarterly Journal
of Economics 1986,101 (4), 707-728.
Pindyck, R., “Irreversible Investment, Capacity Choice, and the Value of the Firm” The
American Economic Review, 1988, 78(5), 969-985.
Pindyck, R., “Irreversibility, Uncertainty, and Investment” Journal of Economic
Literature 1991, 29(3), 1110-1148.
Sing, T. F., “Irreversibility and Uncertainty in Property Investment” Journal of
Financial Management of Property and Construction 2002,7(1), 17-29
Sing, T. F., “Time to Build Options in Constructing Processes”, Construction
Management and Economics, 2002, 20(2), 119-130.
Sing, T. F., “Optimal Timing of a Real Estate Development under Uncertainty”, Journal
of Property Investment and Finance, 2000, 19(1), 35-52.
Titman, S., “Urban Land Prices under Uncertainty”, The American Economic Review,
1985, 75, 505-514.
Trigeorgis, L., “Anticipated Competitive Entry and Early Preemptive Investment in
Deferrable Projects”, Journal of Economics and Business 1991, 43, 143-156.
Williams, J.T., “Real Estate Development as an Option”, Journal of Real Estate
Finance and Economics, 1991, 4, 191-208.
17
Figure 1: Optimal Payoff Function for Development Options, V(Y*)
Figure 2: Volatility Effect on Option Trigger Value
Trigger Value (Y*
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
0.1
0.2
0.3
0.4
0.5
0.6
Volatility of Economic Shocks (σ)
18
Optimal Development Intesit
(q*)
Figure 3: Volatility Effect on Optimal Intensity
4800
4700
4600
4500
4400
4300
4200
4100
4000
0
0.1
0.2
0.3
0.4
0.5
0.6
Volatility of Economic Shocks (σ)
250
σ=0.05
σ=0.1
σ=0.2
σ=0.3
σ=0.4
200
5
($*10 )
Development Option Value, V(Y*
Figure 4: Value of the Development Option
150
100
50
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Economic Shocks (Y)
19
Figure 5: Effects of Rental Sensitivity of Demand on Optimal Timing
*
Trigger Value, (Y)
0.20
0.16
0.12
0.08
0.04
0.00
0
0.2
0.4
0.6
0.8
1
1.2
Demand Factor, (b)
Figure 6: Optimal Intensity of Different Price Sensitivity of Demand
Optimal Intensity of
Development (q*)
25000
20000
15000
10000
5000
0
0
0.2
0.4
0.6
0.8
1
1.2
Demand Factor, (b)
20
1200
b=0.2
b=0.4
b=0.6
b=0.8
1000
800
600
*
5
V(Y ) ($*10 )
Development Option Value,
Figure 7: Development Option Value
400
200
0
0
0.04
0.08
0.12
0.16
0.2
Economic Shocks (Y)
21