1

How does a development moratorium affect development timing choices and land values?

Jyh-Bang Jou

Department of Economics and Finance,

Massey University

National Taiwan University

Tan Lee

Department of Finance,

Auckland University of Technology

Yuan Ze University

2

Abstract

• This paper investigates how a development moratorium affects choices of development timing and land values in a framework where both the value of developed land evolves stochastically and the development costs are fully irreversible. We assume that a regulator initially announces that land is not allowed to develop during a finite period of time in the future.

3

• Developers thus must decide whether to develop land before the timing ordinance is imposed or after it expires. The development moratorium reduces the developer’s option value from waiting, and thus accelerates development. We also use simulation analysis to demonstrate how the other factors that relate to the demand and supply conditions of the real estate market affect this accelerating effect.

4

Introduction

5

• Local communities justify growth controls in terms of the benefits of improved environmental quality, lower municipal services costs and property taxes, and the preservation of small town character.

6

• One commonly employed growth control policy is a development moratorium. There are two forms of development moratoria (Turnbull, 2005b): (1) to specify the growth boundary for a specified date such as that provided by the Portland, Oregon, Metropolitan Council (Brueckner, 1990).

7

• (2) in the 2002 Supreme Court decision, Tahoe-Sierra Preservation Council v. Tahoe Regional Planning Agency. The Court decided that the planning agency can implement a six year moratorium on land development to ensure an essential tool of successful development. This paper will focus on (2).

8

• This paper is related to the literature on development regulation such as Brueckner (1990) and Mayo and Sheppard (2001).

• Brueckner investigates how the growth management timing ordinance affects land values and how to design an optimal policy of this kind in a framework where no uncertainty arises, there exists a negative population externality, and the policy is unanticipated. Brueckner also assumes that the city boundary shrinks as this policy is implemented.

9

• We abstract from the spatial factor, and assume that this policy restricts development during a finite period of time in the future. We focus on how this policy affects choices of development timing and land values in a framework where uncertainty arises and where the policy is preannounced.

10

• Mayo and Sheppard (2001) investigate how the uncertainty regarding the delay of development approval affects the current supply of developed properties. They build a two-period model in which a developer can develop either in period 0 at which he is uncertain about the delay of development approval, or in period 1 at which he realizes the exact delay of development approval. When deciding whether to develop in period 0, the developer must consider the net value from delaying development as an opportunity cost, a concern that is similar to our case.

11

• This paper is also related to the literature on the pricing of American Options. The developer in our model must consider the value of an American call option with a finite maturity with the expected present value of the other American perpetual call option. Barone-Adesi and Whalley (1987) and Carr (1995) provide the pricing formula for the former, while McDonald and Siegel (1986) provide the pricing formula for the latter.

12

Basic Assumptions

• A parcel of land is monitored by a developer, who must decide an optimal date to develop the parcel of land.

13

• The value of developing the parcel of vacant land, , evolves as

• is the expected rate of return, is the instantaneous volatility of the growth rate of , and is a standard Wiener process.

( )V t

( )( ). (1)

( )

dV tdt dZ t

V t

( )V t

( )Z t

14

• The developer is risk-neutral and faces a constant riskless rate .

• The development costs, denoted by , are assumed to be fully irreversible.

r

K

15

• We model the policy of development moratorium regulation as a development timing ordinance.

• At the current time the regulator announces that each parcel of vacant land will be subject to a development timing constraint such that from a certain date , the land is not allowed to develop until the date .

0t

1T

2T

16

• Each developer thus can develop either during time and or after .

• The developer must consider two American call options: one is a finite maturity option which can be exercised during and , and the other is a perpetual option which can be exercised at any time after .

0t 1T 2T

0t 1T

2T

17

• The developer will develop the vacant land before the moratorium is imposed if the value from development exceeds both option values.

• We will derive the level of land values that trigger development both after the development moratorium expires and before the moratorium is implemented.

18

Time Line of the Model

0t 1 1T 2T

1V

2

2V

19

The Development Triggers

• Denote for in the time interval and for in the time interval . After time , the developer holds a perpetual American call option.

1( )V t ( )V t 0 1[ , ]t T

2 ( )V t ( )V t

2[ , )T

2T

20

• Suppose that denotes this option value, which satisfies the following differential equation:

22 2 2 2

2 2 2222

( ( )) ( ( ))1( ) ( ) ( ( )) 0. (2)

2 ( )( )

F V t F V tV t V t rF V t

V tV t

.

2( ( ))F V t

21

• Equation (2) has an intuitive interpretation: if is an asset value, then the normal return equals the expected capital gain give by

2( ( ))F V t

2( ( ))rF V t

22 22 2 2

2 2 22 2

( ( ( ))) ( ( )) ( ( ))1( ) ( ) . (3)

( ) 2 ( )

E dF V t F V t F V tV t V t

dt V t V t

.

22

•The solution to Equation (2) is given by

where and are constants to be determined, and

1 22 1 2 2 2( ( )) ( ) ( ) , (4)F V t AV t A V t

1A 2A

21 2 2 2

1 1 2( ) , (5)

2 2

r

22 2 2 2

1 1 2( ) .

2 2

r

23

• Suppose that denotes the critical level of that triggers property development. This critical level and the two constants, and , are solved from the boundary conditions as follows:

*2V

2 ( )V t

1A

2A

22

( ) 0lim ( ( )) 0, (6)

V tF V t

* *2 2( ) , (7)F V V K

*2 2

2

2 ( )

( ( ))1. (8)

( )V t V

F V t

V t

24

• Solving Equations (6)-(8) simultaneously yields

• The net value from developing a unit of vacant land is thus equal to

* 12

1

. (9)( 1)

KV

*2

1

. (10)( 1)

KD V K

25

• Without any development moratoria, the developer will also exercise the development option when the value from developing land, , reaches . At that optimal exercise point, the developer will get a value equal to in Equation (10).

( )V t

*2V

D

26

• We can thus use in Equation (9) and in Equation (10) to compare their counterparts for the case where it is optimal for the developer to exercise the development option before the development moratorium is imposed.

*2V

D

27

• During any date before the moratorium is imposed, i.e., , the developer holds an American call option with a finite maturity. Suppose that denotes the value of this option at time . The value of this option net of the development cost is given by (Barone-Asedi and Whalley, 1987):

1

1 0 1[ , ]t T

*

1 1( )V

1

28

where

2* *1 1 1 2( ) ( ) ( )(1 ), (11 )hB V K V K e

2 1 1 1 1 *2

( )( ) 2 ( ) .K

h r T TV K

29

• At any , the developer also has an option to delay development until any future date where the moratorium expires, i.e., . Consequently, at time the developer must compare the net value from developing vacant land immediately with the expected discount value given by

1 0 1[ , ]t T

2

2 2[ , )T

1

2 1( ) *1 2 2 2 2 1 1 1

1

( , ) Prob ( ) | ( ) ( ) . (12)( 1)

r KC e V V V B K

30

• Equation (12) shows that the expected present value from delaying development is the product of three terms: the first one is the discount factor, the second one is the net value from developing vacant land immediately, and the last one is the probability that the value of developed land being higher than the trigger level (Harrison, 1985), which is given by

*2V

31

where is the cumulative distribution for a standard normal distribution. Given , let us denote as the that maximizes in Equation (12).

* 22

2 1* 1

2 2 2 1 1 1 1/ 22 1

ln( ) ( )( )( ) 2

Prob ( ) | ( ) ( ) ( ), (13)( )

V

B KV V V B K

( )

1

*2

2 1 2( , )C

32

• To derive the development rule, the developer must compare among the net value from developing land , in Equation (11), , , and in Equation (10). The development rule is stated in Proposition 1 below.

1 1( )V K 1( )B

*1 2( , )C 2 2( )V K D

33

• Proposition 1: (i) The developer will exercise the development option at if the following two conditions hold at the same time: and . Suppose that is the first that satisfies the above two constraints, then the developer will exercise the development option at .

1 0 1[ , ]t T

1 1 1( ) ( )V K B *1 1 1 2( ) ( , )V K C

*1

1

*1t

34

• (ii) If the conditions stated in (i) do not hold simultaneously for any , then the developer will exercise the development option at if . Suppose that is the first that satisfies the above constraint, then the developer will exercise the development option at .

1 0 1[ , ]t T

2 2[ , )T 2 2( )V K D **2

2

**2t

35

• Without any development moratoria, the developer will exercise the development option if . Equation (11) shows that , and Equation (12) shows that

( )V t K D 2

1( ) (1 )hB D e D

*2 1( )* *

1 2 2 2 2 1 1 1( , ) Prob{ ( ) | ( ) ( ) }rC e V V V B K D D

36

• That is, the net land value that triggers development without any moratorium, , is greater than both the net value that triggers development before the moratorium is implemented, , and the expected present value from delaying development after the moratorium expires, .

D

1( )B

*1 2( , )C

37

• Thus a development moratorium makes the developer more likely to develop before the development moratorium is implemented. Consequently, similar to a threatened development moratorium (Turnbull, 2004) and a threatened development prohibition as addressed in Riddiough (1997) and Turnbull (2002), the imposition of a development moratorium accelerates development.

38

• Our result stated in Proposition (ii) differs from that of Turnbull (2005b), which assumes that a developer chooses the date and the scale of development simultaneously in a non-stochastic framework.

39

• Turnbull (2005b) argues that if a development moratorium is imposed, the developer should develop vacant land as soon as it is allowed. That is, if the developer is at the date such that , then the developer should develop land once is reached.

3

3 1 2[ , ]T T

2T

40

Comparative Static Results

41

Proposition 2

• Suppose that the moratorium is effective. • (i) A developer will delay development if the

regulator delays the beginning date of the moratorium ( is increased), if the developer is farsighted (r is decreased), or if the developer incurs larger development costs (K is increased).

• (ii) A developer may delay development if the regulator puts forward the end date of the moratorium ( is decreased).

1T

2T

42

• (iii) The developer’s choice of development timing will be ambiguously affected as the developer expects to receive a higher expected rate of return from developing land ( is increased) or the developer faces a higher risk from developing land ( is increased).

• (iv) For all these three scenarios, however, the net value of developed property is ambiguously affected.

43

• will delay development because the two option values increase.

• may accelerate development because the perpetual option value decreases, and the finite-maturity option value remains unchanged.

1 1T T

2 2T T

1T 0t 1T 2T 2T

44

• , ?

Suppose that or is increased. The level of the net value of the developed land that triggers development will be increased, i.e., in Equation (10) and in Equation (11) will both be increased. This will lower the developer’s incentive to exercise the development option. However, this effect is offset by the following effect: As the value of developed land is expected to grow more rapidly or evolves more volatile, it is more likely to hit the trigger level of the value of the developed land.

D

1( )B

45

• accelerate development

An increase in reduces the net values of the two American options. Consequently, the developer will exercise the development option earlier. This is reasonable as a higher discount rate implies that the developer is shortsighted, and is thus reluctant to wait.

r

r

46

• delay development

If is increased, then the net values of the two American options will be increased. Consequently, the developer will exercise the development option later.

K

K

47

• The reason for the result outlined in Proposition 2(iv) is as follows. The net values of the two American-type options that a developer must sacrifice if developing immediately are decreased when the date is closer to the beginning date of the development moratorium, i.e., is decreased when evolves toward . Consequently, even if changes of an exogenous force induce the developer to delay development, the developer may receive a lower net value for the developed property, because his opportunity cost to develop property may be decreased instead.

1( )B

1 1T

48

Proposition 3: the moratorium is ineffective

• (i) If the developer is far-sighted ( is decreased) or incurs larger development costs ( is increased), the developer will develop vacant land later, and will thus receive a higher net value from development.

• (ii) Changes in the expected rate of return from developing land ( ) and the risk from developing land ( ) exhibit ambiguous effects on both the developer’s timing choice and the associated developed property value.

rK

49

Numerical Analysis

50

51

Figure 1: Simulation for the results of exercising the development option before

the development moratorium is implemented

This figure shows that a developer exercises the development option before the development moratorium is implemented. The jagged curve is the path of the value of developed property, , net of the development cost . The curve is defined in Equation (11), which is the level of the net value of developed property that triggers development. The curve is defined in Equation (12), which is the expected net present value of developed property (evaluated at the optimal exercise date ). The horizontal line is defined in Equation (10), which is the net value of developed property that triggers development in the absence of any regulation. When , the developer will exercise the development option immediately because the developer can receive a net value equal to 0.6027. The benchmark parameter values are , , 　 , , , and .

( )V tK 1( )B

*1 2( , )C

*2 2 D

1.16t

1 3T 2 5T 1K 0.01 0.2 (0) 1.1V

52

53

Figure 2: Simulation for the results of exercising the development option when T1 = 4

This figure shows the same keys in which we use the same benchmark parameter values as those in Figure 1 except for . When , the developer will exercise the development option immediately because the developer can receive a net value equal to 0.6251. The benchmark parameter values are , , , , , , and .

1T 1.94t

1 4T 2 5T 0.06r 1K 0.01 0.2

(0) 1.1V

54

Table 1: Optimal development timing and net developed land value

55

56

• Note: The terms , , , , , , , , and denote the beginning date of a moratorium, the ending date of the moratorium, the riskless rate, the development cost, the expected rate of return from development, the risk of development, the initial value of developed land, the optimal date of exercising the development option, and the net value from development.

1T 2T r K (0)V *1 * *

1 1( )V K

57

58

Figure 3: Simulation for the results of exercising the development option

after a development moratorium expires

This figure uses the same benchmark parameter values as those in Figure 1 except for the series of . The jagged curve is the path of the value of developed property, , net of the development cost . The curve is defined in Equation (11), which is the level of the net value of developed property that triggers development. The curve is defined in Equation (12), which is the expected net present value of developed property (evaluated at the optimal exercise date ). The horizontal line is defined in Equation (10), which is the net value of developed property that triggers development in the absence of any regulation. When , the developer will exercise the development option immediately and will receive a net value equal to 1. The developer will not exercise the development option before year 3 because the net value of developed property that triggers the developer to exercise the option before the moratorium is imposed, , has never reached. The benchmark parameter values are , , , , , , and .

( )V t K( )V t

K 1( )B

*1 2( , )C

*2 2 D

9.35t

1( )B

1 3T 2 5T 1K 0.01 0.2 (0) 1.1V 0.06r

59

Table 2: Optimal development timing and net developed land value

60

61

Conclusion

• Our results come from several simplified assumptions, which can be extended in the future study in several ways.

62

• We may assume that a developer needs to choose the date and the scale of development (e.g. Lee and Jou, 2007, Turnbull, 2005b) rather than choose the date of development only.

• Another example is to incorporate the spatial factor into analysis so as to investigate how the city boundary is affected by the development moratorium.