Dept of Real Estate and Construction Management
Div of Building and Real Estate Economics
Master of Science Thesis no. 487
A Practical Application of Real Option Valuation to Large-Scale
Commercial Real Estate Development Projects
– A Case Study Example Utilizing Binomial Trees
Author:
Supervisor:
Geoffrey Gerring
Åke Gunnelin
Stockholm 2009
Master of Science Thesis
Title:
A Practical Application of Real Option Valuation to LargeScale Commercial Real Estate Development Projects: A
Case Study Example utilizing Binomial Trees
Author:
Geoffrey Gerring
Department:
Department of Real Estate and Construction Management
Division of Building and Real Estate Economics
Master Thesis number:
487
Supervisor:
Åke Gunnelin
Keywords:
Real Option Valuation, Real Option Analysis, Commercial
Real Estate Development, Large-Scale Development,
Binomial Trees
Abstract
This thesis seeks to reduce the complexity of the option valuation mathematics, known as
partial differential equations, which underlay the theory of real option valuation by utilizing
binomial trees. The binomial tree methodology is then used via a case study approach to value
a large-scale commercial real estate development project in Denver, Colorado. The valuation
results are consistent with the theory – that high volatility and managerial flexibility add
significant value to a project. In this case study the additional value added by the sequential
option to delay construction of the second phase and/or the third phase of the development
represents between 13 and 16 percent (depending on source of volatility) of the total present
value of the project.
2
Acknowledgement
This Master of Science Thesis has been conducted at the Division of Building and Real Estate
Economics at the Royal Institute of Technology in Stockholm, Sweden, during the spring of
2009.
I would like to express my gratitude to my thesis advisor, Åke Gunnelin, Associate professor
at the Division of Building and Real Estate Economics, for his knowledge, time and energy
assisting me to understand and to write on a very complicated and interesting topic.
I am grateful to the real estate practitioners who took the time to meet with me to discuss this
thesis. These deliberations broadened my understanding of how real estate professionals
evaluate and make decisions about development projects in the “real world” and how real
option analysis would be utilized.
Finally, I would like to thank my family for their love and support, especially my wife, Elin
and my two boys, Erik and David. These past two years have been a fantastic adventure!
Uppsala, Sweden. 2009
Geoffrey Gerring
3
Table of Contents
List of Figures ........................................................................................................................... 5
List of Tables............................................................................................................................. 5
Chapter 1: Introduction ........................................................................................................... 6
1.1
1.2
1.3
Background ............................................................................................................. 6
Objective ................................................................................................................. 7
Methodology ........................................................................................................... 7
Chapter 2: Real Options .......................................................................................................... 9
2.1
2.2
2.3
Financial versus Real Options................................................................................. 9
Real Option Types and Terminology .................................................................... 10
Option Money States ............................................................................................. 11
Chapter 3: Real Option Valuation ........................................................................................ 12
3.1
The Binomial Tree ................................................................................................ 12
A Simple Example ...................................................................................................... 12
Moving Beyond The Simplified – Getting To The Numbers...................................... 14
Calculating Risk-Neutral and Actual Probabilities ..................................................... 16
3.2
3.3
3.4
Dividend Yield ...................................................................................................... 17
Assumptions .......................................................................................................... 18
Proxies ................................................................................................................... 18
Chapter 4: Case Study ........................................................................................................... 20
4.1
Office Development, Denver, Colorado, USA ..................................................... 20
Project Description ...................................................................................................... 20
Methodology ............................................................................................................... 21
Step 1: Compute Base Case Present Value without Flexibility/Options ..................... 21
Step 2: Identify Volatility and Model Impact on Present Value ................................. 23
Step 3: Use Volatility to Build Present Value Binomial Tree ..................................... 28
Step 4: Conduct Real Options Valuation (ROV) ........................................................ 29
Case Study Observations ............................................................................................. 30
Chapter 5: Conclusion............................................................................................................ 36
Reference List ......................................................................................................................... 37
Appendix ............................................................................................................................... 39
4
List of Figures
Figure 1.1: The Four-Step Process (Source: Adopted from Copeland and Antikarov, 2003) ... 8
Figure 2.1: Mapping a Development Opportunity onto a Financial Call Option .................... 10
Figure 2.2: Call Option Payoffs (Source: Adopted from Kodkula and Papudesu, 2006) ........ 11
Figure 3.1: One Period Binomial Tree of Project Value .......................................................... 12
Figure 3.2: One Period Binomial Tree of Call Option ............................................................. 12
Figure 3.3: Replicating Portfolio of Stocks and Bonds ............................................................ 13
Figure 3.4: Lognormal Distribution ......................................................................................... 14
Figure 3.5: Binomial Tree - Multiplicative Stochastic Process (Source: Adopted from Cox
et al., 1979 and Copeland and Antikarov, 2003)................................................... 15
Figure 3.6: American Call Option Tree ................................................................................... 15
Figure 3.7: Net Present Value .................................................................................................. 16
Figure 3.8: Present Values ex Dividends ................................................................................. 17
Figure 3.9: Net Present Value of Dividend Paying Asset ........................................................ 18
Figure 3.10: Expected Return Distribution of Investment ....................................................... 19
Figure 4.1: Illustrative sketch of development concept (Source: Developer) .......................... 20
Figure 4.2: Site Plan of Proposed Development (Source: Developer) ..................................... 21
Figure 4.3: Historical Denver Office CBD Quarterly Rental Index ........................................ 24
Figure 4.4: Mean Reverting Rental Growth ............................................................................. 24
Figure 4.5: An Example of Projected Future Mean Reversion Rents ...................................... 25
Figure 4.6 - Metropolitan Denver New Office Construction Absorption ................................ 26
Figure 4.7: Metropolitan Denver New Office Construction Absorption ................................. 27
List of Tables
Table 3.1: The Probability That the Project Value Will Exceed the Strike Price .................... 16
Table 4.1: Property Characteristics and Market Assumptions ................................................. 22
Table 4.2: Sample of Project Future Project Rents .................................................................. 25
Table 4.3: Hurdle Conditions for the Construction of Buildings 2 & 3 ................................... 27
Table 4.4: Base Case Present Value without Flexibility/Options ............................................ 32
Table 4.5: Binomial Tree Mapping the Multiplicative Stochastic Process of the Project’s
Present Value ........................................................................................................ 33
Table 4.6: Project Dividend Payout (% of yearly NPV) .......................................................... 33
Table 4.7: Binomial Tree Mapping the Project’s Real Option Values .................................... 34
Table 4.8: Optimal Decision based on the Real Option Valuation .......................................... 35
5
Introduction
Chapter 1: Introduction
1.1 Background
While the deterministic discounted cash flow (DCF) / net present value (NPV) analysis is the
most widely utilized investment valuation tool it suffers from several deficiencies making it a
less than ideal choice for analyzing commercial real estate development investments (Dixit &
Pindyck, 1994; Hoesli, Jani, & Bender, 2006; Copeland & Antikarov, 2003). Dixit and
Pindyck (1994) note that there are three conditions that make valuation of a potential
investment via the DCF / NPV approach less accurate; (1) when the cost of the investment is
partially or completely irreversible, (2) when there is uncertainty in the future expected cash
flows, (3) when there is flexibility in the timing of the investment.
Each of the above mentioned conditions is readily prevalent in most all development projects.
For example, (1) large irreversible investments are made to construct a project, (2) exogenous
macroeconomic factors impact real estate supply and demand thus varying the size of future
potential cash flows, and (3) sequentially-staged approvals provide for flexibility in the timing
of investment payments through continue/delay/abandon decisions as each stage is completed.
Additionally, the gap between a project’s expected NPV and its “true expected NPV” grows
wider as each of the above conditions increase in magnitude. This is because the DCF / NPV
analysis assumes that investment expenditures can be reversed or completely recovered if the
full potential of an investment cannot be realized; or if the investment is assumed to be
irreversible than it becomes a now of never decision, if not taken now than it will never be
taken in the future (Dixit & Pindyck, 1994).
Dixit and Pindyck (1994) argue that when investment decisions take into account
irreversibility, uncertainty and timing it makes those decisions more option-like thus changing
the investment rules of the “standard neoclassical investment model.” They elaborate that the
opportunity cost of killing an option, by taking the investment decision, and not waiting for
new information must be included in the NPV rule: “invest when the net present value of a
project is greater than zero.” Instead the present value of future cash flows must be greater
than the present value of investment cost plus the present value of keeping the investment
option alive.
How does this “investment under uncertainty” approach apply to large-scale1 commercial real
estate development?
To illustrate, let’s talk a quick look at typical development project – in this case, an incomegenerating office building. The typical project generally has three sequential phases; (1)
planning and entitlement, (2) design and engineering, and (3) construction and lease-up. As a
project moves through each stage new risks are encountered and/or new market information is
revealed. This future information allows the development firm to adjust the course of a project
1
In this thesis, the term large-scale is defined as a project that includes a property which must be re-entitled for
the proposed use and/or a building which has the ability to be constructed in two or more distinct successive
phases. The term large-scale would also apply to a property with several distinct building envelopes which can
be developed in several successive phases.
6
Introduction
to stem losses or to capitalize on gains. The current financial crisis provides many instances
where project courses have been altered based on new market information/risk. For example,
projects in the planning phase have been put on hold and projects currently under construction
are evaluating the option to abandon or phase the project. Clearly the choice to change the
course of a project has value; value which cannot be accounted for in the DCF / NPV
analysis.
This flexibility does add significant value to potential projects and, interestingly, this value
grows as the volatility of potential project cash flows grows; the opposite of what will be
projected via the NPV calculation (Copeland & Antikarov, 2003). The value of flexibility
comes from the fact that the real estate developer has the right but not the obligation to make
a decision, therefore, the best way to value a development project is through option valuation
theory.
1.2 Objective
The objective of this thesis is twofold: (1) to contribute to the practical application of real
option valuation in the field of real estate development by utilizing a transparent,
computationally efficient model (a binomial tree model) and (2) to illustrate by way of case
study the use of the real option valuation model to determine the option value and
consequently the net present value of a potential development project.
1.3 Methodology
The research methodology that will be employed in this thesis will be quantitative in nature.
A discrete-time, real options valuation model will be developed. This model will utilize a
binomial tree structure to graphically illustrate the stochastic process that forms the value of a
large-scale real estate development project; the underlying risky asset. One of the first
examples of a binomial option valuation approach was developed by Cox et al. (1979).
Additionally, the valuation model used in this thesis will avoid the use of complex stochastic
differential equations, as used in the renowned Black-Scholes-Merton model, and will instead
utilize the binomial tree because its algebraic solutions limit the calculation complexities that
are associated with the valuation of real options. The model’s methodology is based on the
approached outlined in Copeland and Antikarov’s (2003) text titled Real Options: A
Practitioner’s Guide. In their text they advocate a four-step process for valuing real options:
1. Step one starts with the creation of a traditional Discounted Cash Flow analysis of a
project to determine its best, unbiased estimate of expected project value without
flexibility (options). The present value of the expected cash flows represents the “t
equals 0” value of the underlying risky asset.
2. Step two begins with identifying sources of uncertainties (for example: rental rates,
absorption (lease-up) rates, sales price per unit, cost of construction, and/or interest
rates, etc.). The behavior of each source of uncertainty is analyzed and modeled. For
instance, if the historical per annum rental rate increase of office space exhibits a mean
reversion tendency, that behavioral type is modeled. For simplicity and ease of
evaluating / interpreting the final option values this thesis will assume that each source
of uncertainty is unrelated; in other words, all sources of uncertainty are uncorrelated.
7
Introduction
Once each source of uncertainty is appropriately modeled, a Monte Carlo simulation is
performed to convert the multiple sources of uncertainty (volatility) into one.
3. Step three takes the volatility derived from the Monte Carlo simulation in step two to
determine the up and down movements within the binomial tree which models the
stochastic value of the underlying risky asset as a normal random walk. The nodes of
the binomial decision tree are determined by the point at which a managerial decision
can be made. Again, for simplicity, these nodes occur at regular time intervals (i.e.
one-year increments). Each node indicates points where a real option occurs.
4. Step four, the final step, is to value the payoffs (options) of the binomial decision tree
by working backward in time, from node to node using the risk-neutral probabilities
method.
Step 1
Compute base case PV
without flexibility
Objectives:
To compute base case
net present value
without flexibility
(options) at t=0
Comments:
Net present value
derived from
traditional DCF
valuation analysis
Step 2
Step 3
Identify
uncertainty and
model impact on
present value
To understand how
uncertainty varies
the growth or
reduction of
present value over
time
Multiple sources
may exist but best
to select fewer
variables which
have the greatest
impact.
Estimate and
model uncertainty
using either
historical data or
management
estimates
Use standard
deviation to build
binomial tree
To illustrate up and
down value
movements that
occur when real
options respond to
new information
Decision points
determine the
node locations in
the binomial
decision tree.
The flexibility has
altered the risk
characteristics;
therefore, the cost
of capital has
changed.
Step 4
Conduct Real
Options Valuation
(ROV)
To value the total
project using a simple
algebraic methodology
and an Excel
spreadsheet
ROV will include the
base case NPV without
options plus the option
value.
Under high uncertainty
and managerial
flexibility, option value
will be substantial.
Figure 1.1: The Four-Step Process (Source: Adopted from Copeland and Antikarov, 2003)
8
Real Options
Chapter 2: Real Options
In the early 1970’s Fischer Black and Myron Scholes, and Robert Merton published their
seminal articles on the pricing of financial options. Since then, their Nobel Prize winning
works have opened the floodgates from which hundreds of theoretical papers have poured.
The early focus of these articles were on securities option pricing given that historical data
were plentiful for measuring past price variance and given that the current market price of the
underlying risky asset could be directly observed in the market (Brealey, Myers, & Allen,
2006). However, others quickly began to see the utility of applying option valuation theory to
different types of risky assets.
The term “real option” was first coined by Professor Stewart Myers at the Massachusetts
Institute of Technology (MIT) Sloan School of Management in 1977 (Borison, 2005). The
term real option originates from the fact that the option’s underlying asset is a tangible asset,
such as property, a natural resource or even a pharmaceutical, as opposed to a financial asset,
such as stocks or bonds. While the underlying asset may differ from the financial asset, the
definition of a real option essentially remains the same to that of a financial option. “A real
option is the right, but not the obligation, to take an action (e.g., deferring, expanding,
contracting, or abandoning) at a predetermined cost called the exercise price, for a
predetermined period of time – the life of the option” (Copeland & Antikarov, 2003, p. 5).
But while the financial markets have been quick to embrace the option valuation theories, the
real assets markets have not been as quick. Many reasons have been cited to explain this fact;
(1) knowledge of stochastic differential equations, a graduate level of mathematics, is needed
to solve the Black, Scholes and Merton option valuation model, (2) the market for real assets
are not as efficient as that of financial assets, and (3) historical data on real assts is usually
inconsistent or nonexistence (Borison, 2005; Triantis & Borison, 2005; Barman & Nash,
2007; Copeland & Antikarov, 2003). However, binomial trees (in place of stochastic
differential equations), assumptions (to counter market inefficiencies), and proxies (to
compensate for lack of historical data) and can all be utilized to overcome the challenges thus
releasing the potential of real option valuation.
In order to better understand real options it is useful to examine how financial and real options
are alike. The basic terminology of the various option types and their associated money states
is also covered.
2.1 Financial versus Real Options
Similar to financial options, real options have five main variables. Figure 2.1: Mapping a
Development Opportunity onto a Financial Call Option illustrates the relationship between
financial options and real options.
1. The value of the underlying risky asset (V). This is the value of the development to be
built. This value corresponds to the price of the underlying stock.
2. The exercise/strike price (X). The cost to construct the project or the cost to invest in
the next phase of development.
3. The time to expiration (t). The length of time the option is available; i.e. the length of
time the development firm can defer the construction.
9
Real Options
4. The risk-free rate of interest over the life of the option (rf). The time value of money is
the risk free rate of return. What the firm can expect to earn in a completely riskless
investment.
5. The variance of the value of the underlying risky asset (σ²). The uncertainty about the
future value of the project’s cash flows measured in the project’s standard deviation;
i.e. the project’s risk.
Development Opportunity
Variable
Financial Call Option
Base case present value
without flexibility
V
Stock price of
underlying asset
Cost to construct
or invest
X
Exercise/strike price
Length of time the
decision may be deferred
t
Time to expiration
Time value of money
rf
Risk-free rate of return
Riskiness of the project
assets
σ²
Variance of returns on
stock
Figure 2.1: Mapping a Development Opportunity onto a Financial Call Option
(Source: Adopted from Luehrman, 1998)
A sixth variable should be included when or if the underlying asset has cash payouts (i.e. free
cash flows, after tax cash flows, etc.) or noncapital gains returns as these payouts will reduce
the value of the option which in turn maybe impact the timing of exercise. This thesis will
recognize the net operating income payments as free cash flow payouts from the underlying
asset therefore this sixth variable will be included in the ROV process.
2.2 Real Option Types and Terminology
Looking more closely at the above Chapter One definition of a real option there are two
distinct aspects worthy of further explanation; (1) a real option is the right, but not the
obligation, to take an action at a predetermined cost, (2) for a predetermined period of time.
The first aspect pertains to an option’s right, which there are two; the right to buy and the
right to sell. A call option is the right to buy the underlying asset by paying the exercise price.
A put option is the right to sell the underlying asset by accepting the exercise price.
The second aspect pertains to an option’s predetermined time period, which can also be
generally classified in two categories; European and American. A European option is an
option that can be exercised only on the specified date, the date of expiration. An American
option is an option that can be exercised at any time during its life. American options most
similarly represent the options available to development projects.
10
Real Options
The following is a list of potential real options in real estate development projects; adopted
from Copeland and Antikarov (2003):
o Option to construct –Call Option: The right to execute the project
o Option to abandon –Put Option: The right to terminate or sell project
o Option to defer / delay / wait-to-invest –Call Option: The right to delay a project until
better market information is available
o Option to expand / grow –Call Option: The right to increase the scale of a project
o Compound Option: An option that is contingent on the execution of another option.
For example, the option to construct a project is contingent on the execution of the
option to invest in construction documents.
o Switching option: An option to switch from one product type to another. For example,
the option to change from a residential use to an office use.
o Rainbow option: An option which has more than one source of uncertainty.
2.3 Option Money States
There are three money-states used to describe option value; (1) out-of-the-money, OTM, (2)
at-the-money, ATM, and (3) in-the-money, ITM. A call option is in-the-money if its strike
price is less than the current market price of the underlying asset. A call option is at-themoney if its strike price is equal to the current market price of the underlying asset. A call
option is out-of-the-money if its strike price is greater than the current market price of the
underlying asset.
Figure 2.2 illustrates a call option which costs $2 and has a strike price of $19. The gray line
is the option’s payoff and the dashed black line is the option’s profit. Notice that theoretically
this option would be exercised when it is at-the-money even though its payoff is not
technically profitable. This is because the cost to acquire the option is a sunk cost; a cost
which cannot be recovered once it has occurred. Thus according to economic theory, if an
investor can derive more value by exercising the option than not (in this case, when
),
then the investor should exercise the option regardless of what he paid to acquire that option
(Brealey, Myers, & Allen, 2006).
Option Payoff
6
Out-of-the-Money (S<X)
5
4
In-the-Money (S>X)
Strike Price (S=X),
At-the –Money
3
2
1
0
-1
Option Price
-2
Underlying Asset Value
-3
14
15
16
17
18
19
20
21
22
23
24
Figure 2.2: Call Option Payoffs (Source: Adopted from Kodkula and Papudesu, 2006)
11
Real Option Valuation
Chapter 3: Real Option Valuation
The purpose of this chapter is to describe in more detail steps three and four in the four-step
process.
3.1 The Binomial Tree
A Simple Example
There are two distinct advantages of using the binomial tree to value real options. The first is
the ability to explicitly illustrate the evolution of the underlying asset’s value, therefore
illustrating the option value’s evolution. This makes the valuing an American call option
possible since this option type can be exercised at any point in time, much like a developer’s
decision to construct or abandon a project. The second is that the computations are relatively
simple, so they can be easily understood and explained.
The binomial tree maps out the evolution of the project’s value by utilizing discrete time
intervals. Thus, each node in the tree represents a possible value of the underlying asset at a
particular point in time. The discrete time intervals should be continued into the future to
match the expiration date of the longest option. For example, if the longest option of a
development is five years, then the up and down evolution of the value of the project should
be carried forward from
to
.
The following simplified example, which follows closely to the example presented by Cox,
Ross, and Rubinstein (1979), illustrates a potential development project that has an option to
construct which expires at the end of one period. As the term binomial indicates, there are two
possible future values at the end of the first period. Let
represent the value of the project,
the underlying risky asset in the up state at the end of the first period with probability and
represent the value of the project in the down state with probability . The up and down
movements are illustrated in the one period binomial tree below:
V
uV, with probability p
dV, with probability q
Figure 3.1: One Period Binomial Tree of Project Value
While the value of the project is determined from today expanding into the future, the present
value of the value of the option is determine by working from the future back to today.
Therefore, in this example, the present value of an option to construct, , is determined by
applying the rational exercise rule (the maximum of either zero or the future value of the
underlying asset, , minus the construction cost to build the asset, , also called the strike
price) to end of the first period project values,
and . To illustrate, let
be the value of
the call option at the end of the first period for the up state value of the project,
and
be the value of the call option for the down state,
.
C
Cu=max[0,uV-X], with probability p
Cd=max[0,dV-X], with probability q
Figure 3.2: One Period Binomial Tree of Call Option
12
Real Option Valuation
Once the values for and
are determined via the rational exercise rule, there are two
methods for determining the final present value of the option, ; (1) this value can be
determined through the replicating portfolio approach, or (2) this value can be determined
through the certainty-equivalence, risk-neutral probability approach. This thesis utilizes the
certainty-equivalence, risk-neutral probability approach as introduced by Cox, Ross, and
Rubinstein in 1976. Their approach is based on the same arguments that underlie the
replicating portfolio approach; that a hedging position, a combination of shares in the
underlying asset and riskless bonds, can be utilized together with the law of one price to
create an identical portfolio of securities to value the call option. The following is a brief
overview of the replicating portfolio approach which then leads into the risk-neutral approach.
If we begin by assuming that represent a constant risk-free rate of interest plus one
, and assuming that individuals can borrow or lend without restriction at this rate, and
assuming there are no taxes or transaction costs, then it should be possible to create a
replicating portfolio which will produce identical cash flows to one option on the underlying
asset; a portfolio of Δ shares and bonds borrowed at the risk-free rate (Copeland &
Antikarov, 2003). Figure 3.3 illustrates a replicating portfolio of stocks and bonds with
identical payouts as the above option.
ΔV+B
ΔuV+rB, with probability p
ΔdV+rB, with probability q
Figure 3.3: Replicating Portfolio of Stocks and Bonds
Since the above portfolio is constructed to have identical cash flows as the option to be priced,
we can set the end of period values of the portfolio equal to the possible outcomes of the call
option:
Solving the above equations, we find:
Then solving for the present value of the call option we find:
3.1
Plugging Δ and B into equation 3.1 we arrive with equation 3.2:
3.2
Simplify the equation 3.2 with
13
Real Option Valuation
Then equation 3.2 can be rewritten to equation 3.3, the certainty-equivalence, risk-neutral
probability approach to solving option values:
3.3
It should be noted that a binomial tree can be constructed to model either an additive
stochastic process or a multiplicative stochastic process. The easiest way to determine which
process to model begins by determining whether the future values of a project follow a normal
distribution, a distribution from
, or a lognormal distribution, a distribution
from
. Since it is highly unlikely that the present value, not to be confused with net
present value, of a commercial development project will go below zero, this thesis will utilize
the multiplicative binomial tree.
Figure 3.4: Lognormal Distribution
Moving Beyond The Simplified – Getting To The Numbers
What is the real option value of the ability to delay construction (from today up to three
periods) of a real estate development project that’s static DCF estimates that the present value
of that project be 100? If the present value cost to construct the project is 110, then one can
determine that the NPV of this project is -10, and thus the project would be rejected.
However, given a present value standard deviation of 25 percent and the risk-free rate of five
percent we can map out the evolution of the present value and then solve for the delay option
value.
To answer the above question we follow an example presented by Cox et al. (1979) and an
example presented by Copeland and Antikarov (2003). As always the first step is to map out
the evolution of the present value of the proposed project. The movement of the present value
at each node over each period can have two possible movements, an up-movement,
or a
down-movement,
. This thesis assumes that the down-movement is the inverse of the
up movement;
. These up and down movements are related to the volatility of the
14
Real Option Valuation
potential present value cash flows of the underlying project. Therefore, as illustrated in Figure
3.5, if the volatility of the underlying is
and the time period is one year
, then
the up and down movements at each node will be
and
, respectively. Thus, if the current present value of the project is
, the project value at the end of the first period will be either
with
probability or
with probability .
Figure 3.5: Binomial Tree - Multiplicative Stochastic Process (Source: Adopted from Cox et
al., 1979 and Copeland and Antikarov, 2003)
Figure 3.6 illustrates the fold back formulas used to determine the present value of the call
option to delay the construction of the project for up to three periods. The final period’s endof-period payouts, which depend on the value of the risky asset contingent on its state of
nature minus the exercise price (the cost to construct), are determined with the rational
exercise rule. Then those terminal values (i.e.
and
) would fold back into the
subsequent nodes where they are multiplied by their associated risk-neutral probabilities and
then discounted at the risk-free. This process is repeated until a final value is achieved at
.
Figure 3.6: American Call Option Tree
The final values are presented in Figure 3.7.
15
Real Option Valuation
Figure 3.7: Net Present Value
In this example, the actual NPV of a project that has an option to delay construction up to
three years is 20.1, therefore the option value of this project is
and
represent 30.1 percent or close to a third of the present value of the project.
Looking more closely at Figure 3.7 one will notice that there are three instances where the
option value is zero. These zero values indicate instances where the option to construct would
not be exercised because the underlying asset value does not exceed the exercise price. The
option values within the binomial tree represent net present values of the project / investment
at various states of nature and time. For instance, if the option to construct where exercised at
then the net present value of the investment would be
.
Calculating Risk-Neutral and Actual Probabilities
What is the chance that the value of the underlying will be greater than the cost to exercise the
option (
) at the end of the exercise period? To determine the answer, a distinction
must be made between the risk-neutral probabilities that are used in the option valuation in the
binomial tree and the actual (observable) probabilities (Amram & Kulatilaka, 1999). The
actual probability of the upward, q, is based on the on the risk-adjusted discount rate, :
Using the observable probabilities one can calculate the percent chance that the value of the
underlying asset will be greater than the option exercise (strike) price at any period .
Continuing with the above example, if we were to use a risk-adjusted discount rate of
, the actual probability of an upward movement is
. Adding together the ending value’s observable probabilities which
are greater than the strike price of 110, we find that there is a 77 percent probability that the
underlying asset’s value will exceed the exercise price.
Table 3.1: The Probability That the Project Value Will Exceed the Strike Price
Value of Development
at Time = T
Observable
Probability
Risk-Neutral
Probability
Prob.(up) 69%
Prob.(up) 54%
Prob.(down) 31% Prob.(down) 46%
Observable
Probability (%)
Risk-Neutral
Probability (%)
211.00
32.85%
129.00
44.28%
77.88
19.89%
34.55%
46.73
2.98%
9.94%
16
}
= 77%
15.47%
40.04%
Real Option Valuation
3.2 Dividend Yield
The above example has thus far assumed that no dividends have been paid out by the
underlying asset. While at times this might be the case, where no income is generated by
the asset or all income generated is reinvested back into the asset, at other times it
might be more appropriate to view the income streams generated by the asset as
dividend payments.
The cash flow analyses within this thesis stop at the Net Operating Income (NOI) level
because capital expenditures and taxes are generally unique to individual properties and
their owners. However, if we assume that the NOI generated from the underlying asset is
channeled from the asset to the owner (viewed as yearly streams of income from the
project; i.e. dividends) then we apply the dividend yield analogy to adjust the underlying
value accordingly. This is because the project paying a periodic dividend (NOI) is
analogous to an option written on a dividend paying commodity (Cox, Ross, &
Rubinstein, 1979). Furthermore, dividend payouts affect option prices through their
effect on the underlying asset value because the asset value is expected to drop by the
amount of the dividend on the ex-dividend date (Copeland & Antikarov, 2003).
Therefore, as Dixit and Pindyck (1994) note there is an opportunity cost of holding the
option instead of the underlying asset. That opportunity cost is the value of the foregone
income stream provided by the dividend payment and as the dividend rate growths so
grows the opportunity cost of the options. “At some high enough price, the opportunity
cost of foregone dividends becomes great enough to make it worthwhile to [early]
exercise the option” (Dixit & Pindyck, 1994, p. 149)
Returning to our three-period example above, if the underlying asset was to provide a periodic
three percent dividend yield,
, how would this impact the value of the American call
option to delay construction of the project? The impact of the dividend is to reduce the present
value of the underlying asset on the day that the dividend is paid. Equation 3.4 illustrates the
impact of the dividend (the NOI paid out from the project) to the present value of the
underlying in its period one up-state, reducing it from 128 to 124.16. Since the dividend is
paid periodically equation 3.4 would be applied to all states of nature.
3.4
Figure 3.8 displays the impact of dividend yield to asset’s present value.
Figure 3.8: Present Values ex Dividends
17
Real Option Valuation
Figure 3.9 illustrates the impact of dividend yield to the NPV of a flexible investment;
reducing the value from 20.1 to 17.89. Therefore the actual real option value has been reduced
7.3 percent, from
to
.
Figure 3.9: Net Present Value of Dividend Paying Asset
3.3 Assumptions
Copeland and Antikarov (2003) suggest the use of two main assumptions in order to apply
real options methodology to real-world settings. The first assumption is a concept they call the
marketed asset disclaimer (MAD), which states the value of a project without flexibility
(without real options), determined via the traditional NPV approach, “is the best unbiased
estimate of the market value of the project were it a traded asset” (2003, p. 94). Their
argument is what identical asset matches the risk and cash flow characteristics better than the
project itself. Further they note that “MAD makes assumptions no stronger than those used to
estimate the project NPV in the first place” (Copeland & Antikarov, 2003, p. 95) and if an
investment manager is basing decisions off the NPV approach there exists no new set of
assumptions to use for real option analysis. One should recall that the NPV base assumption is
that future cash flows and discount rate are based on comparable (identical) projects to
determine the projects value if it were to be trade in the marketplace.
The second assumption employed by Copeland and Antikarov is that properly anticipated
prices and cash flows fluctuate randomly; following a geometric Brownian motion or GBM.
They point to Paul Samuelson’s proof published in 1965 in Industrial Management Review
titled “Proof That Properly Anticipated Prices Fluctuate Randomly”. In his proof, Samuelson
sets the foundation of the efficient-market hypothesis by demonstrating that the “rate of return
on any security will be a random walk regardless of the pattern of cash flows that it is
expected to generate in the future as long as investors have complete information about those
cash flows” (Copeland & Antikarov, 2003, p. 222). This second assumption becomes the
rationale for utilizing a binomial lattice for calculating values.
3.4 Proxies
A very important variable of the ROV methodology is the measure of volatility of the total
value of the underlying asset over its lifetime. Volatility is measured as the standard deviation
of the variance in the rate of return of the underlying asset (Kodukula & Papudesu, 2006). A
higher standard deviation results in a larger distribution of returns thus leading to more
uncertainty in the likelihood that the expected return will occur.
18
Probability of r
Real Option Valuation
Figure 3.10 illustrates the volatility in the expected returns, , of the underlying asset. The
dashed line represents a project with a lower volatility of returns compared to the solid line.
r
Figure 3.10: Expected Return Distribution of Investment
Here it is interesting to highlight that higher volatility is interpreted as higher risk; therefore,
to compensate for this risk, the traditional DCF uses a larger discount rate; this, in turn, drives
down the NPV of the investment. The use of a larger discount rate to compensate for risk
might be appropriate if the investment possesses absolutely no managerial flexibility.
However, as can be observed in Figure 3.10, there is just as much probability that the realized
return will exceed the expected return. Additionally an investment that possesses managerial
flexibility can take appropriate actions to avoid losses thus impacting the distribution of
returns. The traditional DCF does not account for this but ROV does.
Not only is volatility an important input variable, but also it is one of the most difficult
variables to ascertain. This is because there exist no market of traded identical projects from
which to observe historical returns volatilities. Therefore volatility estimations can be derived
a couple of different ways. Kodukula & Papudesu (2006) suggest five different ways to
estimate volatility; (1) Logarithmic cash flow returns method, (2) Monte Carlo simulation, (3)
Project proxy approach (4) Market proxy approach, and (5) Management assumption
approach. Copeland & Antikarov (2003) suggest two ways; (1) Monte Carlo simulation, and
(2) Management assumption approach.
This thesis will utilize both the Monte Carlo simulation and market proxy approach. For the
purpose of simplicity this thesis will also assume that there is no correlation between the input
varibles used in the Monte Carlo simulation approach.
Each approach is described in more detail within Chapter 4: Case Study below.
19
Case Study
Chapter 4: Case Study
The objective of this case study is to apply the concepts in Real Option Valuation (ROV) to a
current office development project in Denver, Colorado. At the request of the developer both
the company and the project will remain anonymous. Therefore the actual project specifics
have been adjusted to comply, however all market information obtained by the author remains
in its original form.
4.1 Office Development, Denver, Colorado, USA
Figure 4.1: Illustrative sketch of development concept (Source: Developer)
Project Description
This Office Development is located less than three miles (5 km) from Denver’s central
business district and at the convergence of several modes of transportation within the city.
The goal of this redevelopment “is to create and implement a world class urban village that
maximizes city-wide assets, takes responsibility for integrating with existing neighborhoods
and captures the benefits of light rail transit”. The 18.5 acre (7.5 hectare) transit-oriented site
is located directly adjacent to a future transit station that will serve as a major multi-modal
transit facility for the City of Denver. The site is zoned transit mixed-use, which is the City’s
highest density zone category, allowing for a 5:1 floor area ratio. However, a view plane
ordinance limits the heights of the proposed office buildings to no more than 100 feet (30
meters) or approximately eight stories.
20
Case Study
Parcel 1
Station
Parcel 2
Parcel 3
The Site is also easily accessible by car as it is
located at the intersection of two of the City’s
major thoroughfares; an interstate which not only
bounds the site to the north but also provides future
tenants with unprecedented visibility. The other
thoroughfare is one of Denver’s historic
commercial boulevards which bounds the site to the
east. The adjacent multi-modal transit station is at
the confluence of three light rail lines that not only
connects the site to three important destinations in
the near-term, the CBD, the southeast suburban
office centers and many large residential
communities to the southwest, but also links the
site to the entire city in the long term; as the Denver
Metro area has initiated an ambitious mass transit
program which will provide light rail transit
throughout the City.
The site is envisioned to provide a high degree of
pedestrian activity and energy as people transfer
Figure 4.2: Site Plan of Proposed
between light rail lines and exchange modes of
Development (Source: Developer)
transportation, from foot to car to bus to rail.
Connections to adjacent uses and neighborhoods will be promoted through an integrated
network of public spaces and streets.
The preliminary concept of land uses consists of ground-floor retail with multiple floors of
office space above. In addition to the transportation connectivity, the site offers some of the
best uninterrupted views of the Rocky Mountains.
Methodology
As mentioned above, the application of ROV to case projects will follow the four-step
approach as outlined by Copeland and Antikarov (2003). Refer to Chapter 1.3 above.
Step 1: Compute Base Case Present Value without Flexibility/Options
The base case present value represents the value, , of the underlying risky asset at
.A
standard discount cash flow has been created to determine this present value. The following
case study project information was gathered and used in the DCF:
Property Characteristics and Assumptions:
Table 4.1 summarizes the property characteristics and market assumptions made to value the
proposed project from both the static DCF/NPV and ROV perspective. Many of the
characteristics and assumptions have been slightly simplified to provide for a more straightforward modeling approach. For example, all the buildings are modeled as being 100 percent
office even though it is anticipated that each will have some amount of ground floor retail
space, the temporal durations occur in one year increments, rental payments are made in
arrears, and building numbers indicate phase chronology. In the base case DCF construction
phasing is staggered; i.e. building one construction start is 2008, building two’s start is 2009,
21
Case Study
Table 4.1: Property Characteristics and Market Assumptions
Parcel One
Floor Plate
Floors
Gross Building Area (GBA)
Efficiency
Gross Leasable Area (GLA)
20,250
7
141,750
90%
127,575
1
191
2
$ 155.25
Floor Plate
Floors
Gross Building Area (GBA)
Efficiency
Gross Leasable Area (GLA)
Required Parking
56,350
6
338,100
90%
304,290
456
2
$ 168.75
Required Parking
Parcel One Total Costs
SF
Flrs
SF
Parking Requiments
Office - 1 space per 500 SF
Parking Reduction
SF per structure parking space
SF
Construction Costs
Space
Office Building
Below Grade Parking Cost
Above Grade Parking Cost
$
500
25%
350
SF
135
SF
$25,000
$15,000
SF
Space
Space
Parcel Two
Parcel Two Total Costs
SF
Flrs
SF
SF
Space
Construction Cost Growth
3.2%
Operating Expenses
Operating Expenses Growth
$8.00
3.5%
Interest Rates
10 Year US Treasury - 2008 Average
3.66%
3
Parcel Three
Risk Free Rate
Floor Plate
59,750
Floors
Gross Building Area (GBA)
Efficiency
Gross Leasable Area (GLA)
Required Parking
7
418,250
90%
376,425
565
2
$ 168.75
Parcel Three Total Costs
Project Summary
Total Gross Area
Total Rentable Area
Parcel
898,100
808,290
18.5
SF
Stabilized Class A Risk Premium
Flrs
SF
2.66%
4
Stabilized Discount Rate (r = rf + RP)
Development Discount Rate
Optioned Discount Rate
SF
Space
3.61%
5
6.27%
12.00%
12.00%
Exit Cap Rate - 2008 Ave. Denver CBD
7.60%
Inflation - Average CPI 1987-2008
Long-Term Building Vacancy
3.0%
8.00%
SF
SF
AC
SF
Notes:
1. Parcel One parking provided in adjacent transit parcel above grade parking structure
2. Blended development cost includes cost of parking
3. Risk Free Rate = 10 Yr Treasury Less 100 bps (Geltner et. al. 2007 p. 251)
4. Geltner et. al. (2007) p. 252
5. Discount rate for stabilized 'Class A' office
and building three’s start is 2010. The construction duration lasts two years for both the base
case and dynamic DCF. The base case DCF assumes a 70 percent building occupancy once
the building is completed with an addition 10 percent of the remaining space is leased the
subsequent years until the building is fully leased (three years after completion of
construction), at which time the indicated long-term building vacancy rate is subtracted from
the potential gross income.
An 18-year cash flow has been generated to value the property. The indicated exit cap rate is
applied to the 19th year to determine the building’s terminal value. The future cash flows are
discounted with two different rates, a stabilized discount rate, which is applied to the a
buildings cash flows once that asset has achieved a 90 percent lease rate (10 percent vacancy)
and a development discount rate, to account for the higher risk of asset
22
Case Study
construction/stabilization. For example, cash flows that occur on and after building
stabilization are discounted at the indicated stabilized discount rate back to the year in which
building stabilization occurs. This value and any cash flows that occur prior to stabilization
are discounted with a higher development discount rate. Therefore, the NPV of the DCF
represents both the risk of a development project and the risk of an existing stabilized asset.
Based on the developer’s assumption, the following is an example of building one to illustrate
the base case cash flow concept. Construction starts in the beginning of 2008 and is
completed at the end of 2009. The building is occupied at 70 percent in beginning of year
2010 and rent is collected at the end of the year. In 2011 the building is occupied at 80
percent, and then in 2012 the building occupancy increases an additional 10 percent to 90
percent. At this time the building is considered to be stabilized. All future cash flows
occurring after 2012 are discounted back to 2012 using the stabilized discount rate. The 2012
total is then discounted back to the present with the risk adjusted development discount rate.
The same process is repeated for the other two buildings to complete the step 1 process –
Computing the base case present value of the underlying asset without flexibility.
Table 4.4 on page 32 contains the projects Base Case Present Value without
Flexibility/Options.
Step 2: Identify Volatility and Model Impact on Present Value
The second step begins by identifying the uncertainties which cause the value of the
underlying to change over time. While there may exist multiple sources of uncertainty it is
best to select the fewest number which have the greatest impact (Copeland & Antikarov,
2003). For this case project two main causal uncertainties have been identified; (1) Annual
rental income on a per square foot basis within Denver’s central business district office space
market and (2) the absorption of newly constructed office space in metropolitan Denver. As
mentioned above, in 3.4 Proxies, these inputs will not be cross-correlated.
The following approaches for projected future rents, project future project adsorption and
hurdle conditions within the Monte Carlo simulation are adapted from Anthony C. Guma’s
(2008) thesis titled “A Real Options Analysis of a Vertically Expandable Real Estate
Development”.
Annual Rental Income – a mean-reversion stochastic process
The concept is that the annual rental growth rate follows a stochastic process around a longterm mean; a mean-reverting process. In this case that long-term mean is the expected annual
rental increase. To model the future potential office rental rates for the project, historical
rental rates were analyzed. illustrates that the historical data of quarterly rate changes in
Denver office rents in the central business district exhibit a strong mean reverting behavior.
23
Case Study
Rental Index
Denver Office CBD Quartly Rental Index
170
160
150
140
130
120
110
100
90
80
70
60
Qtrly Index
Trend
0
8 16 24 32 40 48 56 64 72 80 88 96
Quarterly Measure - 1985-2008, 1985=100
Figure 4.3: Historical Denver Office CBD Quarterly Rental Index
The following discrete-time equation is used to model the mean reverting behavior of yearly
changes in the rental rate growth (Dixit & Pindyck, 1994, p. 76):
4.1
Where
is the rate of change from one year to the next, is the speed at which
the rate change returns to the long-term mean after every deviation, is the long-term mean of
the rate of yearly rental change,
is the rate of change for the last period, and is a white
noise process with zero mean and variance of :
Annual Change in Rent
Mean Reverting Rental Rate Growth
15.00%
12.50%
10.00%
7.50%
5.00%
2.50%
0.00%
-2.50% 0
-5.00%
-7.50%
-10.00%
-12.50%
-15.00%
Δrt
2
4
6
8
10 12 14 16 18 20
Mean(r ̅)
Years
Figure 4.4: Mean Reverting Rental Growth
Confident that Denver CBD’s ex post rental rates are a good indicator of future, the future
achieved rental rates were calculate with the following variables for 4.1:
,
24
Case Study
,
= previous years
Achieved Rent,
year,
. Using the preceding
Year
Achieved
E(Rent)
Δrt
Rent
input variables, Table 4.1 and Figure
0
24.25
0.057
5.75%
$25.64
4.5 illustrate one possible scenario of
1
25.02
-0.081
-8.70%
$22.84
projected future project rental rates.
2
25.81
-0.168 -14.20%
$22.14
Since the white noise process
3
26.62
0.050
8.78%
$28.96
generates random outputs each time
4
27.46
-0.029
-4.11%
$26.34
the excel model is run different mean
5
28.33
-0.011
0.49%
$28.47
reverting results occur in the
column. The current years change is then multiplied to the previous year’s Achieved Rent to
determine the current year’s rent:
.
Table 4.2: Sample of Project Future Project Rents
Annual Rent per Squarefoot
Projected Future Project Rents
46.00
44.00
42.00
40.00
38.00
36.00
34.00
32.00
30.00
28.00
26.00
24.00
E(Rent)
Achieved Rent
0
2
4
6
8
10 12 14 16 18 20
Years
Figure 4.5: An Example of Projected Future Mean Reversion Rents
Newly Constructed Office Space Absorption
The second casual uncertainty is the rate at which newly constructed office space is absorbed
into the existing market. Historically, as Figure 4.6 illustrates, Denver’s newly constructed
office absorption has been highly volatile; in one instance ranging from a positive absorption
of 3.6 million square feet in 2000 to a negative 3.7 million in 2001.
25
Case Study
Metro Denver Office Absorption
4,000
3,000
2,000
1,000
SF in 1,000
*2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
-1,000
1997
0
Trendline
-2,000
-3,000
-4,000
Figure 4.6 - Metropolitan Denver New Office Construction Absorption
Generally office demand is related to employment growth and the replacement of outmoded
space. In 2000, the City and County of Denver’s comprehensive land use and transportation
plan, known as Blueprint Denver (2002), predicted that Denver’s employment would increase
by 109,200 jobs, from 411,000 jobs to 520,200 jobs, by 2020; a yearly increase of 1.33
percent. Replacement of outmoded space is expected to be approximately 2 percent of
existing supply; however, several large office development proposals have been waiting for
several years for office employment growth and absorption to bring vacancies to more
reasonable levels (Denver, 2002). Given the real possibility of newly constructed office space
over-saturation it is likely that historical office space absorption volatility is a fair
representation of future volatility.
To simulate the projected future absorption of the project based on the past volatility of the
Denver market, the absorption rates of the project were divided into three time zones. Time
zone one, representing the initial absorption of the project, year’s one through three, is
modeled to be a cumulative random draw uniformly distributed between 0.0% and 15.0% of
the entire project area, 0 SF to 136,458 SF respectively. Time zone one seeks to capture the
initial uncertainty the market regards to new projects. Here it is important to recognize that
the developer is speculatively constructing the first building, which is approximately 15
percent of the total gross leasable area. Additionally it is important to recognize that the
cumulative nature of uniformly distributed random draws equate to an 88 percent leased
building in three years, which is in line with the developers base assumption of a 90 percent
leased (stabilized) building within three years after completion of construction. See Step 1:
Compute Base Case Present Value without Flexibility/Options above.
Time zone two, representing the acceptance of the project in the market place occurring year
four through nine, is modeled to be a random draw uniformly distributed between 0.0% and
80.0% of the entire project area, 0 SF to 727,776 SF respectively. Time zone three which
occurs from year ten to nineteen is modeled to be a random draw uniformly distributed
between 0% and 60% of the entire project area, 0 SF to 545,832 SF respectively. Figure 4.7
represents one possible absorption scenario.
26
Case Study
Projected Future Project Absorption
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Lease-Up Per Year
Project Vacancy
Cumulative Absorption
1
2
3
4
5
6
7
8
9
10
Years
Figure 4.7: Metropolitan Denver New Office Construction Absorption
Model Impact on Present Value
Monte Carlo simulation
To complete the final objective of step two, model impact on present value, the two sources of
uncertainty are linked into a dynamic DCF. The dynamic cash flow model incorporates the
uncertainties within a set of hurdle conditions for the development of buildings two and three.
Once the hurdle conditions are met the developer initiates the construction of the buildings.
As illustrated in Table 4.3, The construction of building two would commence in 2010
because each of the two hurdle conditions are satisfied; (1) cumulative average growth of rent
is greater than 3.25% and (2) pre-leasing of building two is greater than 80.0%. Similarly, the
construction of building 3 would commence in 2012.
Table 4.3: Hurdle Conditions for the Construction of Buildings 2 & 3
2008
2009
2010
2011
2012
Year
1
2
3
4
5
Rent Growth Hurdle:
3.25%
6.3%
9.0%
6.6%
2.4%
4.1%
B2 Pre-Leasing Hurdle:
80.0%
0.0%
58.6%
100.0%
100.0%
100.0%
B2 Exercise Year
3
NO
0
NO
0
YES
3
NO
0
YES
0
Rent Growth Hurdle:
3.00%
4.1%
9.0%
6.6%
2.4%
4.1%
B3 Pre-Leasing Hurdle:
70.0%
0.0%
0.0%
10.2%
100.0%
100.0%
B3 Exercise Year
5
NO
0
NO
0
NO
0
NO
0
YES
5
Monte Carlo simulation is utilized to ascertain an estimate for the volatility of the project’s
values given the impact of the uncertainties and hurdle rates. Two thousand scenarios of the
proposed development are simulated. The value of the range of present values is represented
in the annual standard deviation of the project
.
27
Case Study
Market proxy approach
The comparative market proxy approach measure that will be utilized for benchmarking
purposes in this thesis is the Transactions-Based Index (TBI) of Institutional based Index of
Institutional developed by Massachusetts Institute of Technology Center for Real Estate. The
TBI is based on NCREIF database transactions and provides a quarterly, total return index on
commercial real estate within the United States at the property-type level. The TBI also
allows for a separate tracking index called the Demand Side index (“constant liquidity”)
which illustrates what buyers are willing to pay and is considered to be a true indicator of
market volatility. Fisher et al. (2003) best describe the value of the Demand Side index:
“The volatility observed in securities markets indices reflects an ability to sell
investments quickly in all market conditions (“constant liquidity”).The volatility
reflected in private market [i.e. the commercial real estate market] transaction
prices reflects an “apples-vs-oranges” distinction between transaction prices
observed in up-markets and in down-markets. The up-market prices reflect an
ability to sell more assets, more quickly and easily, than the down-market prices.
The constant-liquidity price index adjusts for this difference, which suggests a
related motivation for the development of such an index.”
Based on the TBI Demand Side index from 1994 to 2008, the quarterly volatility (standard
deviation) is
(
annualized).
Volatility Conclusion
Given that there does not exist an index which tracks the volatility of real estate development
projects, it would appear that the simulated volatility would be a fair representation of the
volatility of the project since the Demand Side index essentially represents the risk of built,
in-place assets (office building) where as the project volatility includes both the risk of
constructing a new building and the risk of a built, in-place asset.
Step 3: Use Volatility to Build Present Value Binomial Tree
The third step is to use the project’s base case DCF present value together with the volatility
of the investment determined via the Monte Carlo simulation performed in the previous step
to build a present value binomial tree. The nodes in the binomial tree are determined from and
correspond to the identified points where real options occur (and thus can be exercised) in the
project. The duration of the binomial tree corresponds with the life of the options.
For this case study analysis, a sequential, compound option is identified. This option is known
as a defer option (wait-to-invest/wait-to-construct option). It is a sequential compound option
because in this case building one must be construct before or at the same time as building two
and three; and building two must be constructed before or at the same time as building three.
The total duration of this option is ten years, however, the choice to exercise (kill) this option
occurs at the end of each year. Therefore, this option can be thought of as an American call
option on an American call option.
In this case study, the developer has a sequentially-occurring, yearly-repeating option for next
ten years to-construct or not-to-construct building two and/or three. The reasons why these
options are sequential is because: (1) the developer would like to limit his exposure by
28
Case Study
requiring that buildings two and three be pre-leased to a certain percentage (however, it is
feasible that each building meets pre-leasing requirements at the same time and thus be
constructed concurrently); (2) building one is the smallest of the three and is required to be
constructed as a part of the land acquisition “deal”; building two is the largest of the three and
would likely (according to the developer’s intuition) be easier to pre-lease once building one’s
construction has proved to the market the developer’s commitment to the project and because
there might be a “market-entry advantage” since only a fraction of the planned transit-oriented
developments would be past the planning stages; and finally building three might experience
more market competition but would have the advantages of not being a substantial leasing
burden (given it’s available space) and it would be a part of an established development
project.
The reason why the developer’s option duration is ten years is because the sources of
investment financing are a part of a 15-year strategic investment fund. Therefore, the
developer would like to ensure that if the final building is not constructed until the final date
of expiration there would be enough time to construct, stabilize and sell the asset in time to
close the fund and issue returns to the investor group.
Table 4.5 below illustrates the evolution of the present value of the project over the next ten
years. The NOI generated from the base case DCF represent streams of revenue which are
channeled back to the developer; i.e. dividend yields. Table 4.6 presents the project dividend
payouts as a percentage of yearly NPV. Essential the dividend yield is the project’s capital
rate:
Step 4: Conduct Real Options Valuation (ROV)
To value this sequential compound option we employ a methodology called backward
induction. We first start at the final nodes of the last (third) option (the option to delay
construction of the third building) and work our way to the beginning of the first option (the
option to delay the construction of the first building); refer to Table 4.7 on page 34. At each of
the eleven ending nodes of third option we apply the rational exercise rule; the maximum of
either zero or the expected value of exercising the option; the future present value of the
underlying asset, , minus the construction cost2 (the strike price) to build the third building,
denoted as
in Table 4.5. For example, in the state-of-nature node 0,10 (i.e. zero down
states, ten up states) the rational exercise rule would be as follows:
. Here the zero value represents the fact that if this option is not exercised it
expires and thus would have no value.
Once the rational exercise rule has been applied to all the year ten ending nodes, we move
backward to the previous year (year nine nodes). Here the rational exercise rule is slightly
different. It is the maximum of either the expected asset value of keeping the option alive
(delay construction) or the expected value of exercising the option:
. The value of holding the option is the discounted (at the risk-free rate) weighted
average (using the risk-neutral probabilities) of the potential future option values and the
2
The costs of constructing the buildings, shown in 2008 dollars, have been inflated yearly at the Construction
Cost Growth as indicated in Table 4.1: Property Characteristics and Market Assumptions on page 22.
29
Case Study
value of exercising the option is future state of nature minus the exercise price:
This intermediate option value rule is applied to the option nodes, folding back upon one
another until
is reached for option three.
Once option three has been valued, we move to valuing option two. Calculate the option
values for this predecessor option (constructing building two) for its ten-year life using the
option values from the successor option (constructing building three) as the underlying asset
values (Kodukula & Papudesu, 2006; Copeland & Antikarov, 2003). Again we begin at the
end of the life of option two; whether to construct building two or not at the end of the tenth
year. Here the rational exercise rule for option two is the maximum of either zero or the future
value of call option three,
, minus the cost to build the second building, :
. The option values are folded back upon on another, as
above, until
.
Finally, option one is valued on option two. The rational exercise rule for option one is the
maximum of either zero or the future value of call option two,
, minus the cost to build the
second building, .Since option one expires at the end of the first period the rational exercise
rule is applied to the end of period one option value of option two:
.
The beginning option value of the first option at
can be interpreted as the net present
value of the project with flexibility (options). Therefore, the $17.1 million is the NPV of a
development project that has present value of $152.0 million today, whose expected present
value deviation has a standard deviation of 21.93% per year, and has three sequential options
to defer construction; one for one year and two for ten years.
Table 4.7 illustrates the option values at each node for all three sequential options. Table 4.8
illustrates the optimal decision at each node; exercise, hold or abandon.
Case Study Observations
The lattice in Table 4.8 shows the optimal decision if you were to make the decision to delay,
construct, or abandon the project at the end of a particular period t considering the
expectations of what the value of the project will be in the future. For example, at
for
option one, the best decision is to delay the construction of building one for one period
because you increase the ENPV by $26.2 million by doing so:
–
.
If for option one at
, you observed the value of the project going down in period one,
than you would abandon the project all together because this option expires after one period,.
However, if at
, you observe the value of the project going up in period one, than you
would exercise the option to construct building one, as shown by “Construct”, thus bringing
to life the second option (sequential options).
30
Case Study
Notice that for the second option at the middle state of
, the optimal decision is “Delay”
while at the same state for the third option the optimal decision is “Construct”. The reason for
this is because the project option is a compound sequential option; therefore, the third option
is not “alive” until the execution of the second option occurs. Hence, even though the optimal
decision for the third option is to construct, this will not occur since the optimal decision of
the second option is to delay. On the other hand, if at
the second option was in the up
state (Construct), then the second option would be executed thus bringing to life the third
option, would also be immediately executed because the optimal decision in this state is to
construct building three.
31
Case Study
Table 4.4: Base Case Present Value without Flexibility/Options
($ in Thousands)
Year
2008
1
24.25
0%
0%
0%
2009
2
25.02
0%
0%
0%
2010
3
25.81
70%
0%
0%
2011
4
26.62
80%
70%
0%
2012
5
27.46
90%
80%
70%
2013
6
28.33
100%
90%
80%
2014
7
29.23
100%
100%
90%
2015
8
30.15
100%
100%
100%
2016
9
31.10
100%
100%
100%
2017
10
32.09
100%
100%
100%
2018
11
33.10
100%
100%
100%
2019
12
34.15
100%
100%
100%
2020
13
35.22
100%
100%
100%
2021
14
36.34
100%
100%
100%
2022
15
37.49
100%
100%
100%
2023
16
38.67
100%
100%
100%
2024
17
39.89
100%
100%
100%
2025
18
41.15
100%
100%
100%
2026
19
42.45
100%
100%
100%
(22,007)
3,094
(3,094)
0
0
0
3,191
(3,191)
0
0
0
3,292
(988)
2,305
(792)
1,513
3,396
(679)
2,717
(937)
1,780
3,504
(350)
3,153
(1,091)
2,062
3,614
(289)
3,325
(1,255)
2,071
3,729
(298)
3,430
(1,298)
2,132
3,846
(308)
3,539
(1,344)
2,195
3,968
(317)
3,651
(1,391)
2,260
4,093
(327)
3,766
(1,440)
2,326
4,223
(338)
3,885
(1,490)
2,395
4,356
(348)
4,008
(1,542)
2,465
4,494
(360)
4,134
(1,596)
2,538
4,636
(371)
4,265
(1,652)
2,613
4,782
(383)
4,400
(1,710)
2,690
4,933
(395)
4,539
(1,770)
2,769
5,089
(407)
4,682
(1,832)
2,851
5,250
(420)
4,830
(1,896)
42,680
39,745
5,416
(433)
4,983
(1,962)
3,021
11,495
(1,149)
10,345
(3,591)
6,754
11,858
(949)
10,909
(4,130)
6,780
12,233
(979)
11,254
(4,274)
6,980
12,619
(1,010)
11,610
(4,424)
7,186
13,018
(1,041)
11,976
(4,578)
7,398
13,429
(1,074)
12,355
(4,739)
7,616
13,854
(1,108)
12,745
(4,905)
7,841
14,291
(1,143)
13,148
(5,076)
8,072
14,743
(1,179)
13,564
(5,254)
8,310
15,209
(1,217)
13,992
(5,438)
8,554
15,689
(1,255)
14,434
(5,628)
8,806
16,185
(1,295)
14,890
(5,825)
9,065
16,697
(1,336)
15,361
(6,029)
135,732
126,400
17,224
(1,378)
15,846
(6,240)
9,606
11,002
(1,100)
9,901
(3,448)
6,453
11,349
(908)
10,441
(3,965)
6,476
11,708
(937)
10,771
(4,104)
6,667
12,078
(966)
11,112
(4,248)
6,864
12,460
(997)
11,463
(4,397)
7,066
12,853
(1,028)
11,825
(4,550)
7,275
13,259
(1,061)
12,199
(4,710)
7,489
13,678
(1,094)
12,584
(4,875)
7,710
14,111
(1,129)
12,982
(5,045)
7,937
14,557
(1,165)
13,392
(5,222)
8,170
15,017
(1,201)
13,815
(5,404)
8,411
15,491
(1,239)
14,252
(5,594)
125,931
117,273
15,981
(1,278)
14,702
(5,789)
8,913
E(Rent)
% of P1 Absorbed
% of P2 Absorbed
% of P3 Absorbed
Parcel One
Development Cost
Potential Gross Inc.
Less Vacancy
Effective Gross Inc.
Operating Expenses
Net Operating Income
Reversion Value
PV Stabilized
PV Development
Parcel Two
Development Cost
Potential Gross Inc.
Less Vacancy
Effective Gross Inc.
Operating Expenses
Net Operating Income
Reversion Value
PV Stabilized
PV Development
Parcel Three
Development Cost
Potential Gross Inc.
Less Vacancy
Effective Gross Inc.
Operating Expenses
Net Operating Income
Reversion Value
PV Stabilized
PV Development
Total Develop. Cost
PV of the Project
NPV
152,044
(9,143)
38,747
24,194
(66,342)
9,839
(9,839)
0
0
0
(68,465)
10,150
(10,150)
0
0
0
10,470
(10,470)
0
0
0
0
69,558
0
0
(72,838)
9,128
(9,128)
0
0
0
0
9,417
(9,417)
0
0
0
(77,574)
9,714
(9,714)
0
0
0
10,021
(10,021)
0
0
0
0
58,293
(161,187)
0
0
0
10,801
(3,240)
7,561
(2,607)
4,954
11,143
(2,229)
8,914
(3,084)
5,830
124,551
10,338
(3,101)
7,237
(2,504)
4,733
10,665
(2,133)
8,532
(2,961)
5,570
116,690
32
Case Study
Table 4.5: Binomial Tree Mapping the Multiplicative Stochastic Process of the Project’s Present Value
($ in Thousands)
Input Parameters
Calculated Parameters
Exercise Prices
Annual Risk-free Rate (rf)
2.66%
Up movement per step
1.2452
Option 1, X1
$22,007
Current value of underlying, V0
$152,044
Down movement per step
0.8031
Option 2, X2
$76,073
Annual standard deviation of PV
Periods per year
21.93%
1
Annual Risk-free Rate plus 1
Risk neutral prob. (up)
Risk neutral prob. (down)
3.28%
5.68%
1.0266
0.505558
0.494442
5.77%
Option 3, X3
$70,580
5.83%
5.96%
6.11%
6.27%
6
452,665
291,941
188,284
121,432
78,316
50,509
32,575
7
530,799
342,333
220,783
142,392
91,834
59,227
38,198
24,635
8
621,531
400,849
258,523
166,732
107,532
69,351
44,727
28,846
18,604
9
726,649
468,644
302,247
194,931
125,718
81,081
52,292
33,725
21,751
14,028
10
848,120
546,986
352,772
227,516
146,734
94,635
61,033
39,363
25,387
16,373
10,559
Dividend
0.00%
0.00%
1.35%
Present Value Event Tree for the underlying (ex dividend)
0
1
2
0
152,044
189,326
232,556
1
122,104
149,984
2
96,731
3
4
5
6
7
8
9
10
3
280,085
180,637
116,500
75,135
4
328,959
212,158
136,829
88,246
56,913
5.82%
5
385,774
248,801
160,461
103,488
66,743
43,045
Table 4.6: Project Dividend Payout (% of yearly NPV)
($ in Thousands)
Year
0
1
2
3
4
5
6
7
8
9
10
Total Present Value
Capital Invest. (Construct Costs)
24,194
(22,007)
105,002
(68,465)
190,724
(77,574)
212,099
234,968
261,664
281,792
284,096
286,258
288,094
289,570
NPV
Dividend Payout ( as % of NPV)
2,187
0.00%
36,537
0.00%
111,637
1.35%
205,365
3.28%
222,343
5.68%
247,268
5.82%
266,428
5.77%
268,445
5.83%
270,146
5.96%
271,506
6.11%
272,492
6.27%
33
Case Study
Table 4.7: Binomial Tree Mapping the Project’s Real Option Values
($ in Thousands)
Option 1
0
1
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
17,067
1
36,579
0
0
33,327
1
59,290
11,162
0
81,785
1
127,552
43,888
$17 million is the NPV of a project valued with Real Options. The option value is approximately $26
million which is approximately 17% of the present value of the project.
2
104,342
20,900
2,779
2
185,361
74,815
17,697
Option 2 [Expires in 10 years]
3
4
5
6
170,927
246,517
340,794
459,298
38,798
71,335
123,351
189,241
5,513
10,868
21,259
41,204
407
873
1,870
4,009
0
0
0
0
0
0
Option 3 [Expires in 10 years]
3
4
5
6
254,538
332,804
429,842
551,196
113,619
157,622
212,399
281,139
32,493
56,772
87,220
125,671
4,998
9,766
18,904
36,169
870
1,865
3,997
0
0
0
34
7
607,101
271,924
78,967
8,592
0
0
0
0
8
790,330
374,925
135,780
18,415
0
0
0
0
0
9
1,016,758
502,714
206,786
39,470
0
0
0
0
0
0
10
1,295,677
660,641
295,058
84,597
0
0
0
0
0
0
0
7
701,939
366,763
173,805
62,722
8,566
0
0
0
8
888,204
472,798
233,654
95,982
18,361
0
0
0
0
9
1,117,763
603,720
307,791
137,428
39,353
0
0
0
0
0
10
1,399,915
764,879
399,296
188,834
67,674
0
0
0
0
0
0
Case Study
Table 4.8: Optimal Decision based on the Real Option Valuation
Option 1
0
1
0
Delay
1
Construct
Abandon
Option 2 [Expires in 10 years]
0
1
2
3
4
5
6
7
8
9
10
0
Delay
1
2
Construct
Delay
Delay
3
Construct
Delay
Delay
Delay
4
Construct
Construct
Delay
Delay
Abandon
5
Construct
Construct
Delay
Delay
Abandon
Abandon
6
Construct
Construct
Delay
Delay
Abandon
Abandon
Abandon
7
Construct
Construct
Construct
Delay
Abandon
Abandon
Abandon
Abandon
8
Construct
Construct
Construct
Delay
Abandon
Abandon
Abandon
Abandon
Abandon
9
Construct
Construct
Construct
Delay
Abandon
Abandon
Abandon
Abandon
Abandon
Abandon
10
Construct
Construct
Construct
Construct
Abandon
Abandon
Abandon
Abandon
Abandon
Abandon
Abandon
1
2
Construct
Construct
Delay
3
Construct
Construct
Construct
Delay
4
Construct
Construct
Construct
Delay
Delay
5
Construct
Construct
Construct
Delay
Delay
Abandon
6
Construct
Construct
Construct
Construct
Delay
Abandon
Abandon
7
Construct
Construct
Construct
Construct
Delay
Abandon
Abandon
Abandon
8
Construct
Construct
Construct
Construct
Delay
Abandon
Abandon
Abandon
Abandon
9
Construct
Construct
Construct
Construct
Construct
Abandon
Abandon
Abandon
Abandon
Abandon
10
Construct
Construct
Construct
Construct
Construct
Abandon
Abandon
Abandon
Abandon
Abandon
Abandon
Delay
Delay
Option 3 [Expires in 10 years]
0
1
2
3
4
5
6
7
8
9
10
0
Delay
Construct
Delay
35
Conclusion
Chapter 5: Conclusion
Clearly, there is value in the ability to wait to make a decision until more and/or better
information is known. This delay value becomes even more significant when making an
irreversible, uncertain investment decision that could be made at a different time. This holds
true for most everything, whether it’s going to the movies (waiting to find out if it’s worth
making an investment which has a cost equal to the price of a movie ticket to receive a payout
which is the ability to see the film) or it’s investing millions of dollars to construct a building.
This thesis looked closer at the value of the option to delay. Its goal was to illustrate, by way
of real world example, the embedded delay value that exists in large-scale, commercial real
estate development projects which cannot be demonstrated through the neoclassical
discounted cash flow / net present value valuation approach. In fact, the only way to
accurately quantify the value of the ability to make future decisions based on better
information is through the Real Option Valuation approach.
Though the theoretical mechanics behind option pricing theory are indeed complex, especially
when viewed through the lens of partial differential equations, this thesis has shown, with the
help of many contributors (Cox, Ross, & Rubinstein, 1979; Dixit & Pindyck, 1994; Copeland
& Antikarov, 2003; Kodukula & Papudesu, 2006), that this complexity can be greatly reduced
via the use of binomial trees and the undergraduate-level algebra knowledge needed to solve
them. Furthermore, the market inefficiencies which exist in the realm of traded real assets
thus making difficult to value real options can be overcome with assumptions and proxies that
are no stronger than the current assumptions and proxies used to value investments by way of
the neoclassical DCF / NPV.
This thesis has also shown that the results of the binomial tree methodology used in the case
study approach to value a large-scale commercial real estate development project in Denver,
Colorado are consistent with real option theory – that high volatility and managerial flexibility
add significant value to a project. In this case study the additional value added by the
sequential option to delay construction of the second phase and the third phase of the
development represented between 13 and 16 percent (depending on source of volatility) of the
total present value of the project.
However, in the course of presenting the concept of this thesis to several real estate
investment firms, the author has come to realize several somewhat limiting conditions; (1)
real options valuation is based on valuing future flexibility; this implies that there must be
some flexibility in the project at some future date. While this exists for most projects it does
not exist for all. In fact, when utilizing construction and/or permanent financing (as is the case
with most real estate investments) underwriting specifications usually preclude a project from
making certain changes. (2) Real option value is derived from making the optimal decision at
the optimal time. Human behavior and marketplace perceptions may entice an actor to take a
sub-optimal action thus realizing less than expected net present value.
Ultimately, the success of an investment comes down to making the right decision at the right
time and this is the true value of ROV.
36
Reference List
Reference List
Amram, M., & Kulatilaka, N. (1999). Real Options: Managing Strategic Investment in an
Uncertain World. Boston, Massachusetts: Harvard Business School Press.
Barman, B., & Nash, K. E. (2007, September). A streamlined real options model for Real
Estate Development. MS Thesis. Retrieved June 29, 2008, from Massachusetts Institute of
Technology: http://hdl.handle.net/1721.1/42010
Borison, A. (2005). Real Options Analysis: Where Are the Emperor's Clothes? Journal of
Applied Corporate Finance , 17 (2), 17-31.
Brealey, R. A., Myers, S. C., & Allen, F. (2006). Principals of Corporate Finance. New
York: McGraw-Hill/Irwin.
Copeland, T., & Antikarov, V. (2003). Real Options: A Practitioner's Guide. New York:
Cengage Learning.
Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option Pricing: A Simplified Approach.
Journal of Financial Economics , 222-263.
Denver, C. a. (2002). Blue Print Denver. Retrieved March 11, 2009, from DenverGov.org:
http://www.denvergov.org/portals/145/documents/BlueprintDenver.pdf
Dixit, A. K., & Pindyck, R. S. (1994). Investment under Uncertainty. Princeton, New Jersey:
Princeton University Press.
Fisher, J., Gatzlaff, D., Geltner, D., & Haurin, D. (2003). Controlling for the Impact of
Variable Liquidity in Commercial Real Estate Price Indices. Real Estate Econmics , 269-303.
Geltner, D. M., Miller, N. G., Clayton, J., & Eicholtz, P. (2007). Commercial Real Estate:
Analysis and Investment. Mason: Thomson Higher Education.
Guma, A. C. (2008, September). A Real Options Analysis of a Vertically Expandable Real
Estate Development. Retrieved January 15, 2009, from Massachusetts Institute of
Technology: http://web.mit.edu/cre/alumni/pdf/msred-thesis-08_aacre-award_anthonyguma_real-options.pdf
Hoesli, M., Jani, E., & Bender, A. (2006). Monte Carlo Simulations for Real Estate Valuation.
Journal of Property Investment and Finance , 24 (2), 102-122.
Kodukula, P., & Papudesu, C. (2006). Project Valuation Using Real Options: A Practitioner's
Guide. Fort Lauderdale: J. Ross Publishing, Inc.
Lister, M. J. (2007, September). Towards a New Real Estate: Innovative Financing for a
Better Built Environment. MS Thesis. Retrieved September 9, 2008, from Massachusetts
Institute of Technology: http://hdl.handle.net/1721.1/42031
Luehrman, T. A. (1998). Investment Opportunities as Real Options: Getting Started on the
Numbers. Harvard Business Review , 51-66.
37
Reference List
Teach, E. (2003). Will real options take root? Why companies have been slow to adopt the
valuation technique. CFO , 19, 73.
Triantis, A., & Borison, A. (2005). Real Options: State of the Practice. Journal of Applied
Corporate Finance , 8-24.
38
Appendix
Appendix
39
Appendix
Appendix 1: Transactions-Based Index (TBI)
http://web.mit.edu/cre/research/credl/tbi.html
Year
Qtr
Price
Index
Demand
Index
Supply
Index
Total Return
Index
1994
1994
1994
1994
1995
1995
1995
1995
1996
1996
1996
1996
1997
1997
1997
1997
1998
1998
1998
1998
1999
1999
1999
1999
2000
2000
2000
2000
2001
2001
2001
2001
2002
2002
2002
2002
2003
2003
2003
2003
2004
2004
2004
2004
2005
2005
2005
2005
2006
2006
2006
2006
2007
2007
2007
2007
2008
2008
2008
2008
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
97.19
95.27
98.54
92.12
94.65
97.14
102.53
102.32
102.8
108.42
104.58
112.89
118.57
124.62
133.66
143.42
151.78
152.7
149.55
150.16
150.79
151.87
149.56
147.68
151.22
155.07
158.71
159.5
158.22
154.43
148.84
144.49
151.03
145.82
155.83
153.1
154.8
152.57
157.34
157.96
158.12
164.35
166.02
172.47
177.27
194.27
207.54
227.71
237.73
237.79
253.71
268.11
277
295.45
297.71
296.1
288.82
266.17
261.03
241.84
90.41
88.61
94.72
88.54
93.32
95.75
101.14
100.88
105.08
110.76
111.51
120.19
119.09
125.05
140.34
150.27
157.44
158.06
156.39
156.79
151.39
152.37
149.95
147.95
151.48
155.22
161.3
161.95
153.35
149.64
136.09
132.17
148.16
143.12
152.61
150.02
147.97
145.86
158.24
158.84
155.65
161.71
171.27
177.71
181.24
198.3
220.8
241.61
245.7
245.26
264.43
278.88
288.06
306.54
300.2
298.13
270.27
249.05
235.42
218.62
104.29
102.22
102.32
95.67
95.81
98.35
103.73
103.57
100.37
105.93
97.9
105.82
117.82
123.95
127.04
136.62
146.04
147.23
142.74
143.52
149.89
151.09
148.89
147.12
150.66
154.62
155.86
156.78
162.92
159.05
162.46
157.64
153.65
148.29
158.8
155.95
161.62
159.28
156.14
156.78
160.31
166.7
160.62
167.05
173.05
189.95
194.7
214.19
229.58
230.09
242.95
257.26
265.85
284.2
294.66
293.52
308.03
283.92
288.86
267.01
98.6
97.93
102.82
97.46
101.86
106.26
113.67
114.42
116.51
124.76
122.09
133.09
141.91
151.49
164.84
178.89
191.7
195.26
193.5
196.2
199.82
204.6
204.29
204.01
212.04
220.35
228.8
233
235.19
233.18
228.71
226.01
240.81
236.75
257.24
256.28
262.45
262.07
273.71
277.53
281.69
295.92
302.04
316.39
328.22
362.72
389.78
430.54
453.13
456.62
490.35
520.44
542.11
582
590.15
589.6
579.49
537.92
531.74
495.6
Quarterly
Annualized
40
Price
Demand
Supply
Total
Return
-2.02%
3.32%
-6.97%
2.67%
2.56%
5.26%
-0.21%
0.47%
5.18%
-3.67%
7.36%
4.79%
4.85%
6.76%
6.81%
5.51%
0.60%
-2.11%
0.41%
0.42%
0.71%
-1.54%
-1.27%
2.34%
2.48%
2.29%
0.50%
-0.81%
-2.45%
-3.76%
-3.01%
4.33%
-3.57%
6.42%
-1.78%
1.10%
-1.46%
3.03%
0.39%
0.10%
3.79%
1.01%
3.74%
2.71%
8.75%
6.39%
8.86%
4.21%
0.03%
6.27%
5.37%
3.21%
6.24%
0.76%
-0.54%
-2.52%
-8.51%
-1.97%
-7.93%
-2.03%
6.45%
-6.98%
5.12%
2.54%
5.33%
-0.26%
4.00%
5.13%
0.67%
7.22%
-0.92%
4.77%
10.89%
6.61%
4.55%
0.39%
-1.07%
0.26%
-3.57%
0.64%
-1.61%
-1.35%
2.33%
2.41%
3.77%
0.40%
-5.61%
-2.48%
-9.96%
-2.97%
10.79%
-3.52%
6.22%
-1.73%
-1.39%
-1.45%
7.82%
0.38%
-2.05%
3.75%
5.58%
3.62%
1.95%
8.60%
10.19%
8.61%
1.66%
-0.18%
7.25%
5.18%
3.19%
6.03%
-2.11%
-0.69%
-10.31%
-8.52%
-5.79%
-7.68%
-2.03%
0.10%
-6.95%
0.15%
2.58%
5.19%
-0.15%
-3.19%
5.25%
-8.20%
7.48%
10.19%
4.95%
2.43%
7.01%
6.45%
0.81%
-3.15%
0.54%
4.25%
0.79%
-1.48%
-1.20%
2.35%
2.56%
0.80%
0.59%
3.77%
-2.43%
2.10%
-3.06%
-2.60%
-3.61%
6.62%
-1.83%
3.51%
-1.47%
-2.01%
0.41%
2.20%
3.83%
-3.79%
3.85%
3.47%
8.90%
2.44%
9.10%
6.70%
0.22%
5.29%
5.56%
3.23%
6.46%
3.55%
-0.39%
4.71%
-8.49%
1.71%
-8.18%
-0.68%
4.76%
-5.50%
4.32%
4.14%
6.52%
0.66%
1.79%
6.61%
-2.19%
8.27%
6.22%
6.32%
8.10%
7.85%
6.68%
1.82%
-0.91%
1.38%
1.81%
2.34%
-0.15%
-0.14%
3.79%
3.77%
3.69%
1.80%
0.93%
-0.86%
-1.95%
-1.19%
6.15%
-1.71%
7.97%
-0.37%
2.35%
-0.14%
4.25%
1.38%
1.48%
4.81%
2.03%
4.54%
3.60%
9.51%
6.94%
9.47%
4.99%
0.76%
6.88%
5.78%
4.00%
6.85%
1.38%
-0.09%
-1.74%
-7.73%
-1.16%
-7.29%
4.29%
18.30%
3.88%
16.44%
3.95%
16.74%
5.08%
21.93%
Appendix
Appendix 2: Binomial Tree Mapping the Project’s Present Value Based on the TBI Demand Side Index Volatility
Input Parameters
Annual Risk-free Rate (rf)
Current value of underlying, V0
Annual standard deviation of PV
Periods per year
2.66%
$152,044
21.93%
1
Dividend
0.00%
0.00%
1.35%
Present Value Event Tree for the underlying (ex dividend)
0
1
2
0
152,044
189,326
232,556
1
122,104
149,984
2
96,731
3
4
5
6
7
8
9
10
Calculated Parameters
Up movement per step
Down movement per step
Annual Risk-free Rate plus 1
Risk neutral prob. (up)
Risk neutral prob. (down)
3.28%
5.68%
5.82%
3
280,085
180,637
116,500
75,135
4
328,959
212,158
136,829
88,246
56,913
5
385,774
248,801
160,461
103,488
66,743
43,045
41
1.2452
0.8031
1.0266
0.505558
0.494442
5.77%
6
452,665
291,941
188,284
121,432
78,316
50,509
32,575
Exercise Prices
Option 1, X1
Option 2, X2
Option 3, X3
$22,007
$76,073
$70,580
5.83%
5.96%
6.11%
6.27%
7
530,799
342,333
220,783
142,392
91,834
59,227
38,198
24,635
8
621,531
400,849
258,523
166,732
107,532
69,351
44,727
28,846
18,604
9
726,649
468,644
302,247
194,931
125,718
81,081
52,292
33,725
21,751
14,028
10
848,120
546,986
352,772
227,516
146,734
94,635
61,033
39,363
25,387
16,373
10,559
Appendix
Appendix 3: Binomial Tree Mapping the Project’s Real Option Values Based on the TBI Demand Side Index Volatility
Option 1
0
0
10,627
1
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
25,462
0
81,465
$11 million is the NPV of a project valued with Real Options. The option value is approximately $20 million
which is approximately 13% of the present value of the project.
1
21,580
0
1
44,291
7,579
1
116,488
49,266
2
76,368
13,876
1,549
2
157,387
74,815
22,063
Option 2 [Expires in 10 years]
3
4
5
6
118,898
162,615
214,107
275,504
25,280
45,814
77,133
114,781
2,962
5,645
10,710
20,219
187
379
769
1,563
0
0
0
0
0
0
Option 3 [Expires in 10 years]
3
4
5
6
202,510
248,902
303,155
367,402
103,063
132,101
166,182
206,679
38,926
56,772
77,842
103,022
6,009
11,239
20,869
36,169
984
1,997
4,056
0
0
0
42
7
347,969
159,503
37,954
3,173
0
0
0
0
8
432,851
212,169
69,843
6,443
0
0
0
0
0
9
531,931
273,926
107,529
13,083
0
0
0
0
0
0
10
647,172
346,037
151,823
26,568
0
0
0
0
0
0
0
7
442,808
254,342
132,793
54,401
8,236
0
0
0
8
530,724
310,043
167,717
75,925
16,725
0
0
0
0
9
632,936
374,932
208,534
101,218
32,006
0
0
0
0
0
10
751,409
450,275
256,061
130,805
50,023
0
0
0
0
0
0
Appendix
Appendix 4: Optimal Decision based on the Real Option Valuation Based on the TBI Demand Side Index Volatility
0
1
Hold
Option 1
0
Exercise
Abandon
0
0
1
2
3
4
5
6
7
8
9
10
Hold
1
Hold
Hold
0
0
1
2
3
4
5
6
7
8
9
10
Exercise
1
2
Exercise
Hold
Hold
1
Exercise
Exercise
2
Exercise
Exercise
Hold
Option
3
Exercise
Hold
Hold
Hold
2 [Expires
4
Exercise
Exercise
Hold
Hold
Abandon
in 10 years]
5
6
Exercise
Exercise
Exercise
Exercise
Hold
Hold
Hold
Hold
Abandon Abandon
Abandon Abandon
Abandon
Option
3
Exercise
Exercise
Exercise
Hold
3 [Expires
4
Exercise
Exercise
Exercise
Hold
Hold
in 10 years]
5
6
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Hold
Hold
Abandon Abandon
Abandon
43
7
Exercise
Exercise
Exercise
Hold
Abandon
Abandon
Abandon
Abandon
7
Exercise
Exercise
Exercise
Exercise
Hold
Abandon
Abandon
Abandon
8
Exercise
Exercise
Exercise
Hold
Abandon
Abandon
Abandon
Abandon
Abandon
8
Exercise
Exercise
Exercise
Exercise
Exercise
Abandon
Abandon
Abandon
Abandon
9
Exercise
Exercise
Exercise
Hold
Abandon
Abandon
Abandon
Abandon
Abandon
Abandon
Exercise
Exercise
Exercise
Exercise
Abandon
Abandon
Abandon
Abandon
Abandon
Abandon
Abandon
9
Exercise
Exercise
Exercise
Exercise
Exercise
Abandon
Abandon
Abandon
Abandon
Abandon
10
10
Exercise
Exercise
Exercise
Exercise
Exercise
Abandon
Abandon
Abandon
Abandon
Abandon
Abandon