Using Binomial Decision Trees and Real Options Theory to
Evaluate System Dynamics Models of Risky Projects
Burcu Tan
University of Texas, McCombs Business School
1 University STA B6000,
Austin, TX 78712-0201, US
512-471-1671
Burcu.Tan @ phd.mccombs.utexas.edu
Edward Anderson
University of Texas, McCombs Business School
1 University STA B6000,
Austin, TX 78712-0201, US
512-471-6394
Edward.Anderson@mccombs.utexas.edu
James Dyer
University of Texas, McCombs Business School
1 University STA B6000,
Austin, TX 78712-0201, US
512-471-5278
Jim.Dyer@mccombs.utexas.edu
Geoffrey Parker
Tulane University, Entergy Tulane Energy Institute
A. B. Freeman School of Business
9 McAlister Dr.,
New Orleans, LA 70118, US
504-865-5472
gparker @tulane.edu
Many important risky projects are characterized by stochastic processes embedded in
non-linear, feedback structures with delays. System dynamics models may be used to
estimate the cash flow resulting from these projects. If these projects include managerial
flexibility (real options), a correct financial evaluation of these cash flow requires the use
of real options methodology. We adapt prior work on real options valuation in the
decision analysis literature to develop a methodology that avoids the need to estimate a
risk-adjusted discount rate for the project with options. We illustrate this approach with a
model drawn from the wind power industry, which is characterized by numerous
uncertainties and high managerial flexibility. We conclude with a discussion comparing
this methodology to the previous methods and describe under what conditions each one
might be a more appropriate choice.
Keywords: System dynamics, real options, decision analysis
1. Introduction
It has been well recognized that the traditional discounted cash flow (DCF)
methods of valuation fail to account for the value of managerial flexibility inherent in
many types of projects. For instance, a simple net present value analysis does not capture
the values of the options to delay, expand or abandon the project. The real options
valuation approach is the state-of-the-art method to value capital investment projects that
involve managerial flexibilities. It applies financial options theory to value options
derived from managerial flexibility, which are called “real options” to reflect their
ssociation with real assets rather than with financial assets (Myers, 1987; Trigeorgis and
Mason, 1987; Trigeorgis, 1988; Dixit and Pindyck 1994).
Traditional real options solution approaches typically rely on models that are
highly stylized closed-form formulations based on the assumption that the value of the
real asset over time can be modeled as a stochastic process (e.g. McDonald and Siegel,
1986; Paddock et al. 1988; Capozza and Li, 1994) or they are based on the use of a
discrete dynamic programming approximation of a stochastic process (e.g. Trigeorgis,
1991; Trigeorgis, 1993; Kogut and Kulatilaka, 1994). Neither of these approaches can
generally handle complex projects that include rework, learning curves or other stochastic
processes embedded within nonlinear feedback structures characterized by delays
(Forrester, 1961; Forrester, 1975).
In this paper, we adapt prior work on real options valuation in the decision
analysis literature to support the use of Systems Dynamics (SD) methodology to evaluate
real options. This work builds on a burgeoning stream of research extending the
capabilities of SD methodology in modeling complex systems. Ford and Sobek (2005)
built a product development project model that uses real options concepts to manage
product design risk. Barghav and Ford (2006) examined the relationship between project
management quality and the value of flexible strategies. Johnson et al. (2006) used an SD
model to value flexibility in a large petrochemical project. These papers provide
examples of how real options logic can be incorporated into SD models of projects; yet
they analyze projects with a single option and they do not focus on formalizing an
algorithm to estimate the market value of projects with complex option structures using
SD models.
Closest to our paper, Tan et al. (2009) proposed a formal, decision-tree based
algorithm to evaluate SD models of projects that account for managerial flexibility. That
paper develops a methodology to transform the data generated by Monte Carlo
simulations of the SD model into a decision tree representation. The decision tree is then
evaluated using a risk-adjusted discount rate for the project. This is commonly called the
“naive approach” to modeling real options in the finance literature (Copeland and
Antikarov, 2001) as it does not capture the changing risk characteristics of the expected
future cash flows by adjusting the discount rate. Adding the options changes the expected
future cash flows and thereby alters the risk characteristics of the project so that the risk-
adjusted discount rate for the project without options may not be appropriate after the real
options have been included in the model.
In this paper we propose an approach (which we shall refer to as the diffusion
approximation approach) which modifies the traditional decision tree approach by using
concepts from Copeland and Antikarov (2001) and Brandao et al. (2005a) to overcome
this flaw. The method avoids the problem of selecting an appropriate risk-adjusted
discount rate for the analysis by using a “risk neutral”! valuation and provides a more
accurate estimate of the market value of the project. Similar to the approach in Tan et al.
(2009), the diffusion approximation approach is a decision-tree based method that relies
on system dynamics simulations; hence, it takes advantage of the complementary
strengths of system dynamics and decision anal in representing stochastic models and
decision processes (Tan et al. 2009). However, unlike Tan et al. 2009, this paper
espouses an approach that more accurately reflects the market value of the project.
The remainder of the paper is as follows. Section 2 provides an overview of the
traditional decision tree approach and discusses its limitations. Section 3 introduces the
diffusion approximation algorithm. Section 4 briefly presents the motivating example
that was analyzed in Tan et al. (2009) and illustrates the steps of valuing this project
using the diffusion approximation algorithm. A comparison of the two valuation
approaches as well as a discussion of some limitations is provided in Section 5, followed
by a short conclusion in Section 6.
2. Overview of the Traditional Decision Tree Approach
The traditional decision tree approach developed by Tan et al (2009) for linking to
an SD model is based on transforming data generated by Monte Carlo simulations of the
SD model of the project into a decision tree. The key virtue of the method is taking
advantage of the system dynamics methodology in modeling the underlying uncertainty.
Real options valuations traditionally model uncertainty by assuming an analytically
tractable stochastic process. Yet, the simple stochastic processes that are often used to
value financial options may not capture the complex real-world behavior of uncertainties
associated with real options. System dynamics, on the other hand, is a methodology
developed to analyze and manage complex feedback systems. Consequently, it is
powerful in handling nonlinearity and path-dependence. Modeling the structure that
produces the complex dynamics associated with many risky projects using SD may
improve the accuracy of the ultimate valuation. Yet, to incorporate the real options in the
valuation of a project, one needs to model and optimize a sequential decision process,
which is not an inherent capability of the SD simulation environment. To model the
decision process, Tan et al. (2009) resort to decision tree analysis, which provides an
intuitive approach in modeling managerial flexibilities and discrete approximations of the
project uncertainty. Hence, the traditional decision tree approach aims to benefit from the
' Risk Neutral Measure is an important concept in the context of mathematical finance and risk neutral
valuation is an important general principle in option pricing (Hull 2006). A risk-neutral measure is a
probability measure in which today’s arbitrage-free price of a derivative security is equal to the discounted
expected value (under the measure) of the future payoff of the derivative. The measure is in general
different than the “physical” measure of probability and is employed to determine the worth of derivative
securities. Please refer to Dixit and Pyndick (1994) for further details.
complementary strengths of system dynamics and decision tree analysis. The steps of the
algorithm are as follows:
I. Identify the managerial flexibilities and decision sequences: The first step is
identifying the “real options” in the project. At each period, the manager needs to
decide whether or not to exercise the available options; hence there is a sequence
of decisions to be made throughout the horizon of the project. Each possible
sequence of these decisions is called a decision sequence.
Il. Build the deterministic SD model that captures the project dynamics
Il. Model the uncertainty by specifying the random variables and their distributions:
In general, there are multiple sources of uncertainty to be specified in a project.
IV. Run Monte Carlo simulations of the SD model for each decision sequence: A
Monte Carlo run for a specific decision sequence gives a cash flow distribution
for each period. Hence, Step 4 results in M xT cash flow distributions, where T
is the time horizon of the project and M is the number of possible decision
sequences.
V. Obtain the discrete distribution approximations for the first period cash flow
distribution for each decision sequence: The bracket median approximation
technique (Clemen 1997) is used to obtain a k-point discrete distribution
approximation of the first period cash flow distributions.
VI. Obtain the conditional discrete approximations for the remaining periods for
each decision sequence: The cash flow distributions for period ¢ are conditional
on the cash flow distributions of period t-/, as well as on the decision sequence
that is chosen. These conditional continuous distributions are discretized applying
the bracket median method recursively.
VII. Solve the decision tree by backwards induction using the risk adjusted discount
rate: In practice, this step is handled easily by using decision analysis software
such as DPL™.
The major limitation of the traditional decision tree approach just described is that
the risk-adjusted discount rate for the project without options is used as the discount
factor for the entire decision tree (Teisberg, 1995). Essentially, the risk-adjusted discount
rate that a financial analyst should choose to value the project with options may be
different from the one he should choose to value the project without options because of
the alternatives’ different risk levels.
The classical approach to incorporating market information is based upon
identifying a replicating portfolio for the project under consideration and using the
volatility information of this replicating portfolio to obtain an appropriate discount rate
for the project. However, the replicating portfolio assumption is difficult to use in
practice when evaluating individual corporate investments because it is hard to find a
single replicating asset or even a portfolio of publicly traded assets with returns that are
perfectly correlated with those from the project (Borison 2005), and the appropriate
replicating portfolio may change if the risk of the project is changed by the addition of
options.
This criticism can be overcome by using a decision tree based real options
valuation method developed by Copeland and Antikarov (2002) and modified by
Brandao, Dyer and Hahn (2005a). We will make some modifications to this method and
call it the diffusion approximation approach. Copeland and Antikarov (2002) suggest the
use of a Monte Carlo simulation of a pro forma spreadsheet model to obtain an estimate
of the risk associated with the project without options, which is then used to construct the
required decision tree. Instead we substitute simulation runs from an SD model and
obtain a reliable, and theoretically correct (from the point of view of the finance
literature) valuation of the investment projects.
3. The Diffusion Approximation Approach
In its simplest form, the diffusion approximation method assumes that the changes
in the project’s value over time approximately follows a geometric Brownian motion
(GBM) diffusion process, which is a standard assumption in the finance literature (e.g.
Copeland and Antikarov, 2001). The method relies on the market asset disclaimer
(MAD) assumption, which assumes that the value of the project without options is the
best unbiased estimator of the market value of the project. Hence, the expected NPV of
the project without options is taken as the market price of the project as if it were traded
(Copeland and Antikarov, 2001).? Then, the value of the project without options is
assumed to change over time according to a GBM process, which is the same process
used to model the changes in the price of a stock when the Black and Scholes (1973)
option pricing model is used. The assumptions behind the GBM model may not hold for
all projects, in which case other models of stochastic processes (e.g. mean reverting) may
be used (Brandao et al 2005a, Hahn and Dyer 2008).
We use the GBM model because the use of a binomial lattice approximation to a
GBM process is well established in the literature (Hull, 2003) and it is straightforward to
build a corresponding decision tree to value the project once the parameters of the
process are provided. In the decision tree representation, the project values are
discounted with the risk-free rate since the risk-neutral probabilities are used, which
eliminates the need to estimate different risk-adjusted discount rates as options are added
to the project.
The first three steps of the diffusion approximation approach are exactly the same
as the traditional decision tree approach: 1) Identify the decision variables and decision
sequences 2) Build the deterministic SD model 3) Specify the distributions for the
uncertain variables. The remaining steps that distinguish the diffusion approximation
algorithm are as follows.
> The MAD assumption is used in order to create a complete market for an asset that is not traded in the
market. It is a strong modeling assumption made to justify the use of risk-neutral valuation. Nevertheless, it
eliminates the reliance on the existence of a replicating portfolio. Instead, it uses the project itself as the
twin security and is claimed to “make assumptions no stronger than those used to estimate the project NPV
in the first place” (Copeland and Antikarov, 2001, p. 67). For further discussion of the MAD assumption,
see also Borison 200Sand Smith 2005.
The fourth step requires calculating the expected NPV of the project without
options at time t=0 using a DCF analysis. To do that, we run Monte Carlo simulations of
the project without options, obtain the NPV for each iteration using the WACC? and use
the mean value of these iterations as an estimate of the expected NPV.
Step 5 of the diffusion approximation algorithm is to obtain the cash flow payout
rate 6, in each period, which is defined as the ratio of the expected cash flow in period t
to the remaining value of the project in period ¢. Let V, and C be random variables
representing the uncertain project values and cash flows in period t, and V, and C, be
their corresponding means. The project value at time ¢ is simply the present value of the
remaining project cash flows. Hence, the cash flow payout rate in period t is defined as
6, =C,/V,. The cash flow payout rate is used to calculate the cash flows that are paid out
at the end of each time period as a function of the project value. This implies that the
cash flows will vary over time reflecting the uncertainty in the project value, but that they
will remain a constant fraction of the residual value of the project in each time period
(Copeland and Antikarov 2001).
Step 6 is to estimate the volatility (o) of the project returns. Smith (2005)
suggests an approach to estimating this volatility that can be used in the SD simulation
environment. First, we model the GBM approximation of the project cash flows using the
present value computed in Step 4 and the cash flow payout rates computed in Step 5.
Then, we search for a volatility that best mimics the uncertainty in the original SD model;
ie., that minimizes the difference between the cash flow distributions generated by the
original SD simulation model and the cash flow distributions obtained by the GBM
approximation.
Step 7 is calculating the parameters of the binomial approximation of this GBM
diffusion process. A binomial lattice is a probability tree with binary chance branches that
go up (u) or down (d) with the unique feature that the outcome resulting from moving up
and then down is the same as the outcome from moving down and then up. In particular,
the binomial lattice model assumes that with probability p the value of the project V will
go up to Vu, and with probability 1—p it will go down to Vd at the end of one period.
The parameter wu is greater than | (reflecting a proportional increase), whereas d =1/u
(reflecting a proportional decrease).
3 See Brandao et al (2005) for a discussion on the choice of the discount rate for this step.
Vu 3
Vu 2
Vu
Vu ?d
Vv
Vu d?
Vd
Vd?2
vds
Figure 1: A Binomial Lattice
In order to calculate these three parameters u, d and p, it is sufficient to know the
volatility o of the GBM process, which is estimated in Step 6, and the risk-free discount
; im 1+rAt—d ‘ : ' ‘
rate r since u=e*, d=1/u, and p -—— where Aris the time period used in
the binomial lattice. The probabilities p and 1— p are the probabilities that a risk-neutral
investor would assign to the two outcomes; therefore they are often called “risk-neutral”
probabilities. Finally, we need the initial value V of the project to build the binomial
lattice, which is approximated by the expected NPV of the project without options
determined in Step 4.
The lattice may also be “unfolded” and represented as an equivalent binomial
tree, which increases the number of endpoints in the model decreasing computational
efficiency, but allows these problems to be solved using “off the shelf’ decision tree
software with an intuitively appealing visual representation. Brandao et al (2005a, b) and
Smith (2005) discuss the pros and cons of this transformation of the problem.
Once the project without options is modeled with a binomial decision tree
approximation (Step 8) of the GBM process, options can be added to the decision tree by
using decision nodes. For example, an abandon option can be modeled by adding a
decision node without any subsequent chance nodes (i.e. no further cash flows), whereas
simple expansion and contraction options can be modeled as percentage changes in the
cash flows (for details, see Brandao et al. 2005a).
These eight steps are summarized in Table 1.
Steps of the Algorithm
Step 1 Identify the managerial flexibilities and decision sequences
Step 2 Build the deterministic SD model that captures the project dynamics
Model the uncertainty by specifying the random variables and their
Step 3 distributions
Run Monte Carlo simulations of the SD model for the project without options
Step 4 and calculate the expected NPV
Step 5 Obtain the cash flow payout rates for each period
Step 6 Estimate the volatility of the project returns
Calculate the parameters of the binomial approximation to the GBM process
Step 7 (i.e. u, d and p)
Step 8 Build the binomial lattice
Step 9 Add the options and_solve the decision tree by using the risk free rate
Table 1: Summary of the Algorithm
4. Illustration of the Solution Approach with a Motivating Example
We will use the same hypothetical wind power project described in Tan et al.
(2009) to illustrate the diffusion approximation algorithm. For convenience, we directly
quote their description of the project: A hypothetical firm needs to evaluate an investment
opportunity to build a 40-MW wind farm with the option to add 50 MW within the first 4
years. The firm can also delay the beginning of the project by up to 2 years. The
expansion option may be considered after the 40-MW wind farm comes online, which
takes a year. The option can be distributed over the remaining 2 years, but due to
economies of scale, the firm does not want to have less than 25 MW built at a time. So,
the firm can either expand high (build all 50 MW at once) or expand low (build 25 MW
one year) with the option to build another 25 MW in the successive year or suspend
investment, i.e. continue operating the wind farm at its current capacity. Given these
managerial flexibilities, the list of decision sequences can be constructed as follows:
Strategy ID_| Decision in 2008 | Decision in 2009 | Decision in 2010 | Decision in 2011
1 I N/A s Ss
2 | N/A Ss L
3 | N/A Ss H
4 l N/A H Ss
5 l N/A L Ss
6 | N/A L L
i D | N/A L.
8 D | N/A H
9 D | N/A Ss
10 D D | N/A
Table 2: Strategies for the Example Problem
The next step is building the SD model that captures the project dynamics. Figure
2 is a sector diagram of the SD model for evaluating the wind-power project. The most
important sectors of the model are the natural gas price sector and the supplier sector in
which the learning curve is modeled. A more detailed description of the model is
provided in Tan et al. (2009).
Step 3 is adding the uncertainty by specifying the random variables and their
distributions. Three major uncertainties are captured by the model: natural gas price, the
learning curve and the expiration date of the production tax credit’. Each uncertainty has
several components, making the project harder to evaluate with most traditional methods.
REGULATION Global capacity acquisition SUPPLIER
+} -Leaming Curve
Firm's capacty aoquistion
pre
“PTC expiration date
CASH FLow | Descente INVESTMENT
“Generation Revenue Pen DECISION
“Operating and -Discount factor Lavaltabilty
Maintenance Cost <aavaias Lconstructon time
t
Fims capacty
Revenue accusiton
Generation by unt capacity
GAS PRICE ELECTRICITY FRM
Unit Exploration Cost PRICE “Total Capacity
“FeeservesiProducton Ratio -Capaciy factor
ReservesiProduction Ratio | 6) Cost Etlct Generation
-Supply/Demand Ratio
Figure 2: Sector Diagram of the SD Model
At Step 4, the expected NPV of the project without options at time t=0 is
calculated using a DCF analysis. To do that, we use the Monte Carlo simulation of the
strategy “invest-suspend-suspend”, which reflects the project without options. Then, we
calculate the NPV of each simulation iteration using the WACC and estimate the mean
value of these iterations to obtain an estimate of the expected NPV. In the example
project, the expected NPV of the project without options is found to be $7.79 million.
Then, as the next step, cash flow payout rates are to be estimated for each time period. To
do that, the present value of the remaining cash flows is calculated for each sample path
generated by the Monte Carlo simulation, which yields the random project values v, at
* Renewable energy producers currently receive a 1.9 cent benefit for each kilowatt-hour of generation,
known as the production tax credit (PTC). The uncertain expiration date of the PTC has been a major
consideration in wind capacity investment decisions
5 See Brandao et al (2005) for a discussion on the choice of the discount rate for this step.
period t. Then, the average cash flow at period t, C,, is divided by the of average project
value V, to obtain the following cash payout rates:
Period 6, Period | ©
1 | 0.027 1 0.183
2 | 0.107 12 0.162
3 | 0.128 13 0.180
4 | 0.132 14 0.195
5 | 0.139 15 0.218
6 | 0.136 16 0.246
7 | 0.148 17 0.302
8 | 0.152 18 0.369
9 | 0.157 19 0.530
10 | 0.169 20 1.000
Table 3: Cash Flow Payout Rates
The PV and the cash flow payout rates are used to estimate the volatility of the
GBM approximation. First, using this PV and the cash flow payout rates, and assigning
an arbitrary value for volatility, a GBM approximation is modeled (Hull, 2003). Then, we
search for the volatility that best mimics the uncertainty in the original SD model. One
way of doing this is comparing the 10", 50 and 90" percentiles of the cash flow
distributions obtained through the GBM approximation for a given volatility to the
corresponding values given by the original SD model. Note that this results in 3x7 pairs
to be compared, where T is the number of periods. We take the sum of squared errors
between these pairs to obtain a measure of fit for the GBM approximation under the
given volatility. We repeat this procedure for a predetermined set of candidate values for
volatility. Then, we choose the value that minimizes the sum of squared errors. We find
that when o = 0.06, the GBM approximation mimics the cash flow distributions given by
the original SD model quite closely (Figure 3). In this case, we limited the search to a
predetermined set of candidate values for volatility; however, when more precision is
required one can solve a stochastic optimization problem to determine the volatility that
minimizes the difference between the original and the approximated cash flows.
As the next step (Step 7), we calculated the parameters of the binomial
approximation of this GBM diffusion process. We set At=0.5 years to increase the
accuracy of the binomial approximation, and we obtained u=1.04, d=0.96 and
p=0.78. We also need the initial value V of the project to build the binomial lattice,
which is approximated by the expected NPV of the project without options.
10
GBM Approximation vs. Original Model Cash Flows
8 if -. . 90th percentiles
= GBM App -10th
Original Modet 10th
| —»—"GBM App 50th"
4 i 10th percentiles ~~ Original Modeb50th
-2-GBM App. 20th
= Original Mode!30th
Cash Flows ($M)
50th percentiles
1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Year
Figure 3: GBM Approximation ys. the Original Model Cash Flows
The final step is adding the options to the decision tree by using decision nodes
(for details, see Brandao et al. 2005a). For the example project, the cash flows change
proportionally with the changes in the capacity. For example, the option to expand by 25
MW (“Expand Low”) increases the capacity by 62.5%. The revenues from electricity
generation are proportional to the capacity; hence, the increase in revenue when the
option is exercised can be modeled by simply increasing the cash flows by 62.5%. Yet,
this scheme does not directly allow for incorporating the uncertainty in the cost of
capacity (the learning curve uncertainty) because the cost of capacity is a one time
payment at the exercise time of the option, which does not affect the cash flow stream
afterwards. To handle this issue, we modeled the learning curve uncertainty as a private
risk.
In many projects, there are project-specific risks that cannot be hedged by trading
securities, such as technological risks. In view of that, real options studies make the
distinction between public (market-priced) risks and private (project-specific) risks
(Smith and Nau. 1995). When the investment under concern is dominated by private
risks, dynamic programming based approaches (such as decision tree analysis) should be
preferred rather than the traditional option pricing techniques that were developed mainly
for market-priced risks (Dixit and Pindyck, 1994). Many projects have both kinds of
risks, though; in which case the recommended strategy is to separate the public and
private risks and use the appropriate risk adjustment for each one (Borison, 2005,
Brandao et al. 2005a, Smith and Nau, 1995).
Fortunately, treating different types of risks separately is straightforward once a
decision tree is created. Public and private risks may be represented with separate chance
nodes. Risk-neutral probabilities are used for the former and subjective probabilities are
used for the latter. For the example project, the cost of capacity is discretized so that at
11
any period f, it is high, nominal or low. The values for each of these branches were
obtained by discretizing the Monte Carlo simulation data for the cost of capacity. This is
done the same way as the cash flow distributions are discretized: A three-point bracket
median method is used while preserving the path-dependence of the learning curve
uncertainty by carefully computing the conditional probabilities of the branches. For
example, chance nodes HighC, NominalC and LowC in Figure 4 discretize the learning
curve uncertainty for the first period. Note that the uncertainty in the cost of capacity
does not affect the volatility of the subsequent cash flows.
The decision tree was built and solved using DPL™ (Figure 4). The expected PV
of the project is estimated to be $57.48 million. The optimal policy suggests that the
investment should be undertaken and it is optimal to expand high afterwards. Note that
the traditional decision tree approach in Tan et al. (2009) results in an expected PV of
$55.386 million and suggests investing immediately and expanding afterwards, the
amount depending on the cash flow realization (Tan et al. 2009).
ut
High [20.48]
"
Expand_H [2.787]
OPTION1
High [9.787]
High
OPTION1
Nominalc [12]
333
Invest
Expand_L [10.64]
8 ©
vW
Suspend [7.941] 2
al
Expand_H [1482]
INVESTMENT_DEC1
[16.88]
OPTION1
Lowc [14.82]
Suspend [10.97]
INVESTMENT_DEC2
No [5.704]
0.03
Figure 4: The Optimal Policy Obtained Using the BDH method
5. Benefits and Limitations
For the example project, the traditional decision tree approach and the diffusion
approximation approach yielded similar results. However, this will not always be the
case. The diffusion approximation algorithm overcomes the major flaw of the traditional
12
decision tree approach (Tan et al. 2009), which is the use of the same risk-adjusted
discount rate for the project with and without options regardless of the changing risk
character. In general, for projects with long lives or for projects whose risk-profiles
change significantly, the errors caused by using the wrong discount rate are magnified,
and the differences between the two approaches become larger (Smith and McCardle
1998, Teisberg 1995).
Nonetheless, diffusion approximation algorithm has some limitations of its own.
The algorithm makes two strong assumptions. One is that the cash flow of the project
without options can be represented by a GBM (or other similarly tractable) stochastic
process. This implies that the algorithm will only yield a reasonably accurate valuation if
each period’s cash flow from the project without options has an approximately lognormal
distribution. Fortunately, because of the numerous influences upon the individual cash
flows that one can represent in a system dynamics model, this is often the case.
A second restriction is that the diffusion approximation approach assumes the
cash flows of the project with an option are proportionate to the cash flows of the project
without the options. For example, if the firm exercises an expansion option to increase its
capacity by x %, the revenues should change by a linear function of x. If this is not the
case, then the diffusion approximation approach may not be an appropriate modeling
approach. In such cases, the traditional decision tree approach might be a better
alternative since it provides a greater fidelity to the details of the modeled project and is
not subject to the assumptions associated with the use of the GBM _ diffusion
approximation. Still, for a reliable valuation the analyst needs to make sure that the
valuation results are not highly sensitive to the choice of discount rate.
6. Summary and Conclusion
Smith (1999) makes the observation that when evaluating risky projects, there has existed
a fundamental trade-off between what he terms “detail complexity” and “dynamic
complexity.” He suggests that financial theory has tended to sacrifice detail complexity,
the fidelity of a model at a detailed level, to better capture market information to
appropriately discount risky cash flows. On the other hand, he suggests that decision
analysis has often focused on detail complexity at the expense of keeping some model
dynamics unrealistically simple, for example by using a single risk-adjusted discount rate
to value future cash flows even if the risks of these cash flows change when different
project options are selected. In this paper, we have attempted to show that by using a
system dynamics model as an input to evaluating managerial flexibility it is possible to
improve this trade-off between dynamic and detail complexity.
We proposed a method that relies on an SD model of the project to model the
project uncertainty and a binomial tree approximation of the uncertainty to employ risk
neutral valuation. The former brings high fidelity to model details due to the unique
capabilities of SD in modeling complex feedback systems, whereas the latter avoids the
need to estimate a risk-adjusted discount rate for the project with options ensuring better
fidelity to dynamic complexity compared to traditional decision tree methods. Hence, our
13
method potentially improves the trade-off between dynamic and detail complexity
especially for evaluating projects whose viability is determined by the interaction of
stochastic processes within a complex nonlinear feedback structure.
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