Valuing Vacant Land

Vacant land has value because it represents an option to turn the vacant land into devel-

oped land. For example, a particular plot of land may be developed into condomini-

ums, an office building, or a shopping mall. In the future, the developer will have an

incentive to develop the property for the use that maximizes the difference between the

value of the project’s future revenues and its construction costs. However, the best pos-

sible future use for the land may not be known at the present time.

The real options approach can be used to determine the worth of an option to con-

struct one of a number of possible buildings with strike prices equal to the building’s

construction costs. One can value this option, and thus the land, by first computing the

risk-neutral probabilities associated with various outcomes. Example 12.3, adapted from

Titman (1985), uses the binomial approach to obtain the risk-neutral probabilities

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Markets and Corporate		Corporate Strategy	Companies, 2002
Strategy, Second Edition

Chapter 12

Allocating Capital and Corporate Strategy

433

necessary to value vacant land. One derives these probabilities from the observed mar-

ket prices of traded investments (for example, the price of existing condominiums and

the risk-free rate of interest).

Calculating these risk-neutral probabilities requires solving for probabilities that

generate expected cash flows for traded assets that equal their certainty equivalent cash

flows. In other words, with the correct risk-neutral probabilities, the expected cash

flows of traded assets, discounted at the risk-free rate, will equal the observed market

price of the traded asset. These same risk-neutral probabilities can then be applied to

the cash flows of the investment being valued to calculate the risk-neutral (or certainty

equivalent) cash flows, which are then discounted at the risk-free rate. Example 12.3

illustrates how this procedure can be followed to value vacant land.

Example 12.3:Valuing Vacant Land

An investor owns a lot that is suitable for either six or nine condominium units.The per unit

construction costs of the building with six units are $80,000 and with nine units $90,000.Con-

struction costs are the same whether construction takes place this year or next.The current

market price of existing comparable condominiums is $100,000 per unit.Their per year rental

rate is $8,000 per unit (net of expenses), and the risk-free rate of interest is 12 percent per

year.If market conditions are favorable next year, each condominium will sell for $120,000;if

conditions are unfavorable, each will sell for only $90,000.What is the value of the lot?

Answer:At the present time, building nine condominium units yields $90,000 profit

[ 9($100,000$90,000)] while building six units yields $120,000 profit [ 6($100,000

$80,000)].Hence, a six-unit building is best if building now.However, if the investor chooses

to wait one year to build, he will receive the payoffs illustrated in panel A of Exhibit 12.4,

which shows that, by waiting a year and constructing a nine-unit building if market condi-

tions are favorable, the investor will realize a total profit of $270,000.He will construct a six-

unit building and realize a total profit of $60,000 if unfavorable market conditions prevail.If

the present value of this pair of cash flows is larger than the $120,000 profit from building

a six-unit building now, waiting is the best alternative.Assuming that the investor waits, the

value of the lot is computed by valuing the two possible cash flow outcomes, $270,000 (favor-

able conditions) and $60,000 (unfavorable conditions).

To calculate the present value of this cash flow pair, first compute the risk-neutral prob-

abilities, and (1 ), associated with the two states.As the binomial tree in panel B of

Exhibit 12.4 shows, a $100,000 investment in a comparable condominium this year yields a

year-end value of either $120,000 plus $8,000 in rent or $90,000 plus $8,000 in rent, depend-

ing on market conditions.This implies that the risk-neutral probabilities must satisfy

$128,000(1 )$98,000

$100,000

1.12

7

which is solved by

15

Discounting next year’s expected cash flows at the risk-free rate of 12 percent, seen in

panel C, with expectations computed using the risk neutral “probabilities,”gives the current

value of the land under the assumption that it will remain vacant until next year.This current

value is

78

$270,000 $60,000

1515

$141,071

1.12

Since $141,071 is greater than the $120,000 profit that would be realized by building a six-

unit condominium immediately, it is better to keep the land vacant.The value of the vacant

land is $141,071.

Grinblatt882Titman: Financial	III. Valuing Real Assets	12. Allocating Capital and	© The McGraw882Hill
Markets and Corporate		Corporate Strategy	Companies, 2002
Strategy, Second Edition

434	Part IIIValuing Real Assets EXHIBIT12.4Binomial Trees forLand Valuation

Panel A

Cash from vacant land, developed next period

		Favorable (build 9-unit condominium)
		cash flow = $270,000
?
		= 9($120,000 – $90,000)
	1 – 
		Unfavorable (build 6-unit condominium)
		cash flow = $60,000
		= 6($90,000 – $80,000)

Panel B

Condominium values

$128,000 = $120,000 + $8,000

$100,000

1 – 
	$98,000 = $90,000 + $8,000

Panel C

Risk-free asset



$1.12

$1

1 – 

$1.12

Example 12.3 values vacant land as an option to build different kinds of structures,

depending on market conditions. How realistic is this? Chapter 8 indicated that option

values are increasing in the volatility of the underlying asset—in this case, developed

land. In a study of commercial properties in the Chicago area, Quigg (1993) found that

land was indeed more valuable with greater uncertainty. Result 12.3 summarizes this

view as follows:

Result 12.3

Vacant land can be viewed as an option to purchase developed land where the exercise priceis the cost of developing a building on the land. Like stock options, this more complicatedtype of option has a value that is increasing in the degree of uncertainty about the value(and type) of development.

Grinblatt884Titman: Financial	III. Valuing Real Assets	12. Allocating Capital and	© The McGraw884Hill
Markets and Corporate		Corporate Strategy	Companies, 2002
Strategy, Second Edition

Chapter 12

Allocating Capital and Corporate Strategy

435

Titman (1985) shows that development restrictions, such as ceilings on building

height or density, may reduce uncertainty, leading to both a lower value for vacant land

and a greater desire to exercise the option—that is, to develop the land. This curious

phenomenon—that development restrictions may lead to more development—arises

because the benefit of waiting is the greatest force keeping vacant landholders from

exercising the development option. The benefit of waiting is larger when the degree of

uncertainty about the option’s terminal value is greater, as Chapter 8 noted.

The valuation approach used here works because vacant land has payoffs like an

option and because the possibility of arbitrage keeps prices in line. Example 12.4 illus-

trates how to achieve arbitrage if the real estate market places a different price on the

value of the land than on the price derived from risk-neutral valuation.

Example 12.4:Arbitraging Mispriced Real Estate

If the land in Example 12.3 sells for $120,000, show how investors can earn arbitrage prof-

its by purchasing the land and hedging the risk by selling short the condominium units.

Answer:One achieves risk-free arbitrage by purchasing the land, selling short seven

comparable condominium units, and spending $626/1.12 on risk-free, zero-coupon bonds

maturing in one year.The present value of seven condominium units completely hedges the

risk from owning the vacant land, since the difference between the value of the units in the

favorable and unfavorable states, $210,000 ( 7 ($120,000 $90,000)), exactly offsets

the difference in land values in the two states ($270,000 $60,000).

The arbitrage opportunity is summarized as follows:

	Cash in	Cash in
Cash Inflow	Favorable State	Unfavorable
Today	Next Year	State Next Year
Investment(in $000s)	(in $000s)	(in $000s)

Short 7 condos	$700	$840$56	$630–$56
Buy vacant land	$120	$270	$ 60
Buy risk-free bonds	$626/1.12	$626	$626

Total	$21.071	$00000
		0$

The investment in Example 12.4 yields a risk-free gain of $21,071. Because this

kind of gain cannot exist in equilibrium, investors will bid up the price of the land from

$120,000 to its equilibrium value of $141.071.

It is tempting to argue that arbitrage is impossible in the situation described in

Example 12.4 because it is impossible to sell short seven condominium units. How-

ever, someone who already owns similar condominium units could sell seven of them,

and buy both the vacant land and the risk-free asset. At the margin, this looks like an

arbitrage opportunity because the change in cash flows associated with this decision is

riskless and yields positive cash today.