VALUING CORPORATE GROWTH USING REAL OPTIONS
Latypov Gennady
Faculty of economics, Kyoto University, e-mail: petrenko_spb@yahoo.com
Corporate DCF model has been often criticized for limitations in valuing investments with
significant multi-period growth opportunities. Current paper offers practical methodology that
employs recombining binomial trees and real option techniques for valuing companies pursuing
organic growth strategy in case of demand uncertainty. Specially constructed recombining tree
is adapted to valuations of multiple American type compound growth options. Computational
complexity is considerably reduced that makes these solutions feasible to implement in
business practice and empirical testing.
Keywords: real growth option, corporate valuation, organic growth.
JEL classification: G31, G11, D92.
I.
INTRODUCTION
Currently, traditional discounted cash flow (DCF) analysis based on shareholder value
maximization is being widely used in corporate budgeting practice as a criterion for
evaluating projects and companies. However, despite its popularity, corporate DCF
model has inherent limitations in valuing investments with significant flexibility options
(Mun [2004]) and is often criticized for undervaluation of investment opportunities that
lead to myopic decisions, underinvestment problem and potential loss of competitive
position. Instead, other alternative methods have been developed to effectively capture
management flexibility to respond to future uncertain developments, namely: revised
1
(dynamic) DCF model, decision analysis, and contingent claim analysis (real options).
All methods are concerned with valuing risky alternatives by backward recursion,
however, the probabilities and discount rate used in each type of analysis differ, and
hence, principally the results may substantially differ depending on the employed
approach.
To overcome the shortcoming of static DCF approach of assuming a predetermined
decision path, the dynamic DCF model employs decision trees to plot future
contingencies and expected future cash flows are estimated using real probabilities
extracted from similar projects and then discounted by the risk-adjusted opportunity
cost of capital. The dynamic DCF analysis offers a significant improvement over static
DCF in that it incorporates management reaction to uncertain future states; however,
this approach has main drawback in that it is highly sensitive to the subjective choice of
risk-adjusted discount rates. Decision analysis differs from dynamic DCF model in that
discounting of future cash flows is done by risk-free rate to capture investor’s time
preference, and adjustments for risk aversion appear in the characterization of the utility
function of a decision maker. Decision analysis typically focuses on the expected
utilities of an individual decision maker and, therefore, may not be a proper measure of
a project value from a shareholder’s viewpoint. Instead, option theory adopts the
perspectives of investors based on market equilibrium and determines the value of
investment opportunity using the values of other traded assets. The famous classic
option model is that of Black and Scholes [1973]. Later Cox, Ross and Rubinstein’s
[1979] developed a binomial option pricing method that converges in the limit to Black
and Scholes’s [1973] prominent model. Subsequently, based on Myers’ [1977] seminal
idea that one can view a firm’s discretionary investments as a call option on real assets,
2
the real option approach (McDonald and Siegel [1986], Dixit and Pindyck [1994],
Trigeorgis [1996]) extended the classic option theory to applications to partially or
completely irreversible investments with uncertain future payoffs and flexibility around
the investment timing. Unlike DCF and decision analysis, option valuation methods are
based on “no-arbitrage” assumption and use the portfolio composed of a traded
underlying asset (or traded twin asset perfectly correlated with the underlying if it is not
traded) and risk-free asset to replicate the payoffs of an option in any uncertain state.
Being independent of risk preferences of individual investors, such valuation enables
taking expectations of future payoffs using risk-neutral probabilities and discounting of
expected risk-neutral cash flows by risk free rate; therefore, actual probabilities of
future uncertain states are not necessary to estimate. Respectively, the option valuation
of investment opportunity is significantly simplified compare to dynamic DCF or
decision analysis, if the value of the underlying asset or the corresponding correlated
traded asset is observable and easy to model. However, if the value of underlying asset
is unobservable and there are no correlated traded twin-assets, which is frequently the
case of real options, special approaches should be used to estimate the value of
flexibility option. Copeland and Antikarov [2001], Mun [2002] have proposed an
appropriate solution algorithm based on the “Marketed Asset Disclaimer” assumption to
value options in case of incomplete markets. Under their assumption, to value such real
options, first, the stochastic process of the underlying non-traded asset without options
is simulated, and then various embedded options are treated as if they were assets whose
cash flows are derived from the value of the underlying asset without options. Hence,
the present value of the “as-if-traded” asset without options is used to construct a
replicating portfolio and flexibility options are quantitatively estimated using traditional
option pricing methods.
3
The main attention of the current paper is the application of real option theory to
sequential investment decision-making that maximizes a company’s value under
uncertainty. Trigeorgis [1993] points out that the presence of subsequent options can
alter the value of underlying risky asset, and hence, combined value of portfolio of real
options may differ from the sum of values of individual options. Addressing this issue,
the current research focuses on firm-level investment behaviour and explicitly considers
the effects of multiple corporate growth options on a firm’s market value. Using real
option framework it allows theoretical and quantitative capturing of the value of a
firm’s activities around building capabilities and exploring new opportunities in
dynamic environment. The paper is organised as follows. The following section
presents a brief review of the theoretical and empirical real option literature related to
corporate valuations. Section III provides detailed description of the model for arbitrage
free valuation of corporations with careful attention paid to the type of exogenous
uncertainty and the treatment of risk. The analysis will display the problem of choosing
the optimal reinvestment strategy in a discrete-time compound option framework to
obtain numerical arbitrage-free solutions of a corporate value with strategic flexibility.
Simple numerical examples that are discussed in section IV illustrate the results. Section
V concludes the discussion.
II.
THEORETICAL AND EMPIRICAL BACKGROUND
The way how corporations structure and govern their activities remains the central issue
to the theory of the firm. The real option thinking in corporate finance stresses the
4
additional value of flexibility that a firm’s governance structure has if it can efficiently
recognize, create and properly manage the various real options in different states of
uncertain environment.
The bulk of real option studies have showed that sequential decision-making better
describes organizational behaviour than the “now-or-never” commitment structure
implied by traditional valuation techniques like static DCF model. Earlier research has
been focusing on the analysis of the relationship between value of a single real option
and a firm’s value (McDonald and Siegel [1985], Pindyck [1988]). The recent
developments have extended the real option analysis in a broader context of corporate
strategy with focus on portfolio of options and studied the irreversible investments in
building strategic capabilities such as R&D, mergers & acquisitions, developments of
new technologies, etc. (See Copeland and Antikarov [2001], Mun [2002], Smit and
Trigeorgis [2004], Schwartz and Trigeorgis [2004] for a thorough review of real option
theory and practice). Real option analysis has been widely applied to analyze optimal
market entry (Folta and O’Brien [2004]), joint ventures (Kogut [1991]), and foreign
expansions (Campa [1994]). A number of recent papers in the strategic management
literature investigated the influence of uncertainty on the value of specific growth
options. For instance, Kogut [1991], Kogut and Kulatilaka [2001] explored the option
value associated with investing in platform capabilities based on which a firm may
better respond to uncertain changes in external environment. Similarly, McGrath and
Nerkar [2004] have found that pharmaceutical companies intend to create a growth
option by investing in patent in a new area of exploration if the surrounding uncertainty
and respective upside potential is large. Other authors (Hall [2000]) have showed the
presence of real option logic in the market valuations of innovations and R&D activities.
5
Levitas and Chi [2001] found that firms which signalled the holding of technological
competences through patenting enjoyed higher market values in more volatile
environment. Berger et al. [1996] analysed the market valuations of the option to
abandon a firm and showed that firms with greater exit value and less specialized assets
have a large value of flexibility to abandon the business in case of financial distress.
Several empirical studies have also examined how financial markets value real growth
options possessed by a firm. Kester [1984] measured the value of growth options as the
difference between a firm’s current market value and the value of its assets-in-place and
estimated that the value of the growth options is more than the half of the market value
of equity for a lot of firms and as much as 70 of 80% for volatile industries. Following
the work of Kester, other authors (Danbolt, Hall and Jones [2002], Tong and Reuer
[2006]) also suggested that the value of growth options can represent a sizable portion
of a firm’s market value, however, even adopting the real option framework, none of
these studies tested an analytic model of multiple growth options. Pindyck [1988]
employed numerical simulation and suggested that if demand volatility exceeds 0.2 the
value of growth options occupies more than half of a firm’s total value.
Most of the mentioned real option literature provides a compelling framework for
capturing management flexibility under uncertainty and implications for its quantitative
valuation in case of individual investment projects; however, it is important to recognize
several pitfalls in practical applications of the theory to valuing corporations (Philippe
H. [2005], Lander and Pinches [1998]). Proponents of real option theory (Kogut and
Kulatilaka [1994], etc.) stress its numerous advantages: no more biased discount rates,
ability to fully utilize the uncertain information, the quantitative measure of managers’
6
flexibility, etc. However, according to several studies the usage of real options is still
limited in business practice. Graham and Harvey [2001], Triantis [2001] found that real
option techniques are often being used only as a strategic qualitative tool rather than an
instrument for exact quantitative estimation. According to another investigation fulfilled
by Ministry of Economy Trade and Industry (METI [2003-04]), in 2003-04 years in
Japan there were no corporate finance officers actually using real option techniques in
investment appraisal. One more fact about real options was published in “Economist” in
April, 2000: 46% of American companies that tried real option techniques gave up
because those methods were too complicated to be used in practical evaluation of real
investments. As for the valuing of corporations, developing appropriate real option
methodology remains one of the hottest topics in research agenda (Triantis [2005],
Kellogg and Charnes [2000]). In the following, I briefly discuss the major conceptual
and practical issues concerning the applicability of the real option pricing techniques in
corporate valuations.
In general, the option pricing models may not be readily applicable to real corporations,
since corporate real options have several properties that are quite distinct from traded
financial options. The unique feature of real options embedded in corporate growth is
that they are usually considered as characteristics of a firm’s asset-in-place (Tong and
Reuer [2006]) and, hence, can not be separated and independently valued. One more
feature of corporate real options is their effect on the underlying company’s value. If
exercising of financial option has no impact on the value of underlying asset, the value
of real option may be severely influenced by managerial actions. If a company develops
or enters new markets or executes large-scale investments in R&D it may alter the
competitive structure of the firm’s industry and, hence, change the underlying asset
7
value (i.e. expected cash flows from market operations). Often, this impact on the
underlying asset’s value is not included in formal contracts but implicitly embedded in
strategic investments. The time of option expiration and exercising rules are also,
generally, not as clear as those for financial options.
Other problem lies in the way of formalizing uncertainty in real option models.
Uncertainty that has the strong impact on investment behavior is central to real option
theory of investment (Dixit and Pindyck [1994]); however, real option models
traditionally treat the value of a whole investment project itself as the exogenous
uncertainty variable and do not focus on the individual (usually multiple) sources of
uncertainty that drive the project’s value. This ambiguity creates difficulties for research
studies as to what are the most relevant and important sources of uncertainty. In
addition, current researches do not have a consensus on the empirical measure of
uncertainty (see, for example, Ghosal and Loungani [2000], Ogawa and Suzuki [2000]).
Regarding the application of quantitative models to growth reinvestments and corporate
valuations, the complexities of real-world investments create difficulties in specifying a
proper mathematical model for capturing the sequential interdependence among
multiple growth reinvestments over time. Traditionally, real option analysis of multiple
growth options focuses on multi-stage investment opportunities. In this framework,
first-stage investment is undertaken to create growth option, whereas second-stage
investment may be made to exercise this growth option (Copeland T. and Tufano P.
[2004], Panayi and Trigeorgis [1998], Luehrman [1998]). In case of multiple growth
opportunities, this model setting is limited to European type options. In contrast, the
practical real option problems usually involve American-type options with several state
8
variables and path-dependent pricing dynamic which is extremely difficult to solve
analytically. For multiple American-type growth options that give its owner the right to
exchange one asset for another at any time prior to expiration, complexity
accompanying real investments can even may make it difficult to solve the problem
computationally. Few attempts (Smit and Trigeorgis [2004], Copeland and Antikarov
[2005], Brandao, Dyer, Hahn [2005]) to model portfolio containing large number of
growth options have utilized non-recombining binomial trees and possess the problem
of exponential complexity that makes these solutions difficult to implement in practice.
Hence, testing propositions derived from applications of real option theory to research
in strategic management presents a challenge since the value of growth options
embedded in a company’s value could not be directly measured. As a result, past
researches haven’t relied on the precise measure of the value of real options and utilized,
instead, estimations based on reduced-form relationships between observed proxies that
in theory depend on the value of real options and strategic investments (Folta, Johnson,
O’Brien [2006]).
The main attention of this paper is to provide computationally efficient valuation
technique to directly measure the value of a firm with regard to the value of its portfolio
of multiple growth options. To overcome the gap between corporate finance theory and
practice, current paper utilises specially constructed lattice with recombining property
and applies Marketed Asset Disclaimer concept to properly value interacting multiple
options embedded in corporate growth. The results show that for high/moderate growth
firms operating in highly dynamic business environment the value of growth option
portfolio is significantly undervalued by traditional corporate Discounted Cash Flow
model. The paper offers several contributions to existing corporate finance literature and
9
practitioners. First, it builds a computationally effective model of a firm’s flexibility
that explicitly includes potentially very large number of compound American-type
growth options. Second, it validates an arbitrage-free valuation tool alternative to
traditional corporate DCF model that, the author believes, can find its useful practical
applications.
III.
THE MODEL
The strategic interest in any real option lies mainly in its ability to truncate the
distribution of unfavorable moves of the uncertain environment in the value of
investment. This section is to investigate the effects of introducing multiple growth
options in the value of corporations and the optimal operating rules for companies that
possess many American-type growth options.
Building on Myer’s [1977] original idea of treating the investment opportunities as
“growth options”, traditional corporate finance theory prescribes that a firm’s standalone value consists of two components: value of its assets in place and value of growth
opportunities (Damodaran [2002]; Copeland, Koller, Murrin [2000]). Corporate DCF
values both parts by dividing the infinite life of the firm into two periods:
10
•
period of abnormal growth when the company enjoys extra rents due to competitive
advantage (for example, economies of scale/scope, absolute cost advantage or
product differentiation), and/or high entry/exit barriers in an industry (like capital
intensity, minimum efficient scale, advertising intensity, etc.)
•
and period of stable growth when the state of competitive equilibrium has been
achieved and the firm can not earn excess return given current conditions of
technology, customer tastes, etc., thus, earning only market average return on its
capital.
DCF model further assumes that firm’s management execute growth reinvestments in
each period (Copeland T., Weston J. F., Shastri K. [2005], Damodaran [2001]). In
contrast, real option theory treats the pattern of reinvesting in a different way since each
reinvestment can be considered as an exercising of a “growth option” similar to
financial option with the underlying asset being “real” cash flows. Real option theory
maintains that the expansion investments made by corporations are often sunk in the
form of firm- or market-specific production/distribution infrastructure, brand
enhancement, knowledge base, etc. and executing such (partly) irreversible
reinvestment is not the obligation, but the right of the management (Dixit and Pindyck
[1994]); therefore, these rights should be exercised only if market turns out to be
favourable. Hence, before making a final investment decision a firm faces a critical
trade-off between flexibility and commitment.
On the one side, if a firm makes
investments to develop a market it will actually install additional expansion/growth
options that may set the path for future corporate development. On the other side,
despite earlier investment may bring immediate cash flows and provide platform for
11
future follow-on investments, it may “kill” the value of option to “wait and see”. As a
result, traditional cut-off rule like a positive NPV itself is not a sufficient criterion for a
viability of individual investment and instead of accepting all projects with positive
NPV it is necessary to choose the NPV-maximizing “wait-reinvest” strategy,
recognizing that future decisions and appropriate discount rates will be conditioned on
the realizations of stochastic variables and previously executed decisions. Thus, the
apparatus of real option theory enables a formal analysis of these sequential
discretionary reinvestment decisions.
There are several issues that should be considered in direct practical implementation of
real option theory to corporate growth valuation. Traditionally, real option models treat
the value of a project (or a company) itself as the exogenous stochastic variable. The
underlying assumption of modelling the stochastic process of the value of the whole
project lies in Samuelson’s [1965] proof that multiple sources of uncertainty driving the
project’s value can be eventually combined into a single random walk. Although
appealing from a theoretical perspective, the practical use of the company’s value in the
valuation process is limited due to the lack of intuition associated with the solution
techniques and the restrictive assumptions that are difficult to estimate empirically.
Despite greatly reducing dimensional complexity of valuation model to single
uncertainty, this approach has also the drawback of aggregating information about
individual risk factors into single number of dispersion of a company’s value. As the
result, based on the current models it is very difficult to provide clear threshold values
of individual uncertainty drivers that trigger investments. Moreover, Smith [2005]
showed that procedures for calculating integrated volatility may overstate the actual
uncertainty in the cash flows and, hence, overstate the value of many options. The
12
presence of multiple real options embedded in corporate growth also influences the
company’s value in nonlinear way. If a company has only one growth option, it is
possible to characterize the company’s value with a single growth option as following,
suppose, a lognormal process. However, if the company adds two or more subsequent
growth options, this inclusion results in the company’s value process being nonlinearly
transformed from the original and no longer following the same process. Hence, due to
truncation of the distribution of option pay-off, it is very difficult to consistently
estimate the volatility of underlying cash flows of the each individual growth option.
Moreover, it is not possible to separately identify the effects of individual uncertainty
drivers, company-specific factors (like tax shields, etc.) and compound growth options.
Based on these considerations, in the current model it is assumed that an individual
output factor, rather then a company’s value is exogenous stochastic variable. This
uncertainty is then further incorporated into the fundamental variable of a company’s
value: the company’s cash flows. These cash flows will therefore be the function of the
driving exogenous uncertainty and represent the underlying stochastic process that
drives the valuation model. In addition, the use of company’s cash flows provides a
greater level of detail in modeling a firm’s business processes and allows accounting for
non-linear effects of marginal income taxes, real tax shields, capital structure, etc.
First, consider no-growth case when the firm constantly produces and sells the same
output S0. Due to exogenous uncertainty, the future value of this output is also uncertain.
A common assumption regarding exogenous risk variable is that its current level
incorporates all public information available at this point in time and all future
fluctuations will be pure random and, thus, modeled as a random walk. Specifically, the
13
exogenous risk faced by a firm with a constant output is characterized as demand
uncertainty, Snomt, which follows a continuous-time diffusion process of the form:
dS nom
= μ ⋅ dt + σ ⋅ dW , where
S tnom
(1)
dW is a standard Wiener process, dW ∈ N (0,1) , while µ is inflation rate in the firm’s
product market and σ is volatility of demand changes. Next, for the general case when
the firm has the rights to expand capacity, assume that the life of distinctive competitive
advantage, during which the firm earns extra economic rents, is T periods, after which
shareholders get continuing value depending on a reinvestment policy during high
growth period 1 . To simplify the model, the share of variable cost in sales is assumed to
be state-independent (although, this assumption can be gradually relaxed to incorporate
cost uncertainty). Demand uncertainty is resolved and market opens at times t = 0, 1, …,
T. One restriction is placed on the form of demand volatility σ . At any future time t, σ
must be modeled as non-stochastic and known with certainty at the time of analysis. Let
Snom = {Snom1, Snom2, …, SnomT} denote the (finite) set of possible demand realizations.
At each time t, a firm possesses all the information ωt about past states of demand prior
to this time. The firm’s beliefs about future states based on information ωt available at
the current time t are captured by subjective probabilities p. Given the state of
information ωt the firm’s value is written E[ … ‫ ׀‬ωt ] or simply Et[…].
Following Tong and Reuer [2006] who found that firm-specific effects and
heterogeneity in the firm’s proprietary options always dominate over industry effects in
the value of a firm’s growth options, it is assumed that when considering a reinvestment
1
See Copeland, Weston, Shastri [2005] for deviation of the formula for continuing value.
14
strategy, a firm is engaged only in a game with nature (i.e., exogenous environmental
uncertainty) and agents formulate exercise strategies as a response to changes in
uncertain market demand in isolation. Reinvestment decision is made at the beginning
of the period and uncertainty surrounded demand is fully resolved at the end after the
investment installation.
Let t be the valuation date and T the growth option expiration date which is bounded by
the length of abnormal growth stage for any growth option. To fit the general model of
corporate flexibility, I characterize the firm that undertakes only replacement
investment to maintain sales and postpones net expansion reinvestment as being in
mode mt = 1 (“wait to reinvest”). Once a firm makes both replacement investment and
an additional net reinvestment to increase sales, the firm will be operating in mode mt =
2 (“reinvest”). By making net reinvestment of Inewt (ξ), a firm will expand the scale of
its operations by a factor ξ. Here, the exercise price of growth option which is
represented by net reinvestment costs of capital installation assumed to be nonstochastic and invariant to changes in the exogenous demand. At each state, a firm’s
management may execute either strategic decision (wait or reinvest) with each mode
having its own free cash flow function FCFmt for mt ∈ (1,2 ) . Operating in the mode 1
entails no switching costs, hence c1 = 0, whereas switching from mode “wait” to mode
“reinvest” has the cost in form of net reinvestment, hence, c2 = Inewt(ξ). The cash flow
in the interval (t, t+Δt) depends on the value of stochastic variable, Snom t, and on past
and current decision modes mt, hence FCFmt =FCF(Snom t, mt, mt-1, …, m0, t) 〜
FCF(Snom t, mt, t). After adjusting for inflation effect, the value of free cash flows from
operations (excluding net reinvestments) that are dedicated to mode mt can be expressed
as:
15
FCFt (S t , mt , t ) = EBITt (S t , mt , t ) ⋅ (1 − tax ) + At′(mt ) − I treplace (mt ) − dWCt′( mt )
(2)
where:
EBITt(St, mt, t) – real earnings before interest and taxes which are calculated as:
EBITt (S t , mt , t ) = S t (mt , t ) ⋅ (1 − VC ) − At′( mt )
VC is share of variable costs in real sales St,
tax is marginal corporate tax rate,
At′ is adjusted real depreciation2 ,
Itreplace is real replacement investment,
dWCt′ is adjusted change in working capital.
At the end of high-growth stage, future cash flows are represented by continuing value
of the firm CVt that can be calculated using simplified formula approach (see Gordon
growth model in Copeland, Weston, Shastri [2005]).
The valuation algorithm is fulfilled working backwards in a stochastic dynamicprogramming fashion (Dreyfus [1965]) and is similar to numerical methods for valuing
American call option on a stock. Unfortunately, closed form solution for a value of a
company with multiple American growth options is not available; therefore, I solve the
outlined optimization problem numerically by using recombining binomial-decision tree
that specifies cash flows associated with management decisions at each uncertain state.
The binomial model of Cox, Ross, Rubinstein [1979] gives accurate approximation of
continuous Geometric Brownian motion of demand uncertainty with the added
2
In order to match financial statements in nominal and real terms for the firms operating in high-inflation
environment, it is critical to adjust the levels of depreciation, interest expense, and changes in working
capital (see Copeland, Koller, Murrin [2000] for a detailed discussion of adjustment techniques).
16
advantage of providing necessary flexibility to adapt a model to real world situations
and allowing a solution for the value of embedded American options. However,
working through traditional recombining lattice becomes computationally cumbersome
and non-intuitive if a project involves several compound options and is not suitable for
dealing with multiple American growth options. Hence, tree was modified to
accommodate the dynamic of uncertain demand and, hence, cash flows both without
and with associated American-type growth options. In order for the modified tree to
recombine, I imposed following model restrictions:
•
Companies use straight-line amortization method both for financial reporting
and tax purposes.
•
There are no losses carried forward.
•
Firms do not have financial constraints and can raise additional debt or equity
capital to finance profitable investments.
One more condition is fulfilling all the calculations in real terms. Strictly speaking, this
is not the restriction because if done properly calculations in real terms match
estimations in nominal terms. This is rather technical condition that is necessary to
make the tree recombine and reduce the model’s dimension. For example, if the same
real volume of invested funds is calculated in nominal monetary terms, the amount of
nominal reinvestment will depend on the time of investing due to inflation effect. Hence,
level of capital calculated in nominal units at each node will be path dependent and the
tree will not recombine. However, if all calculations are done in real monetary terms,
path-dependence is eliminated. Applied for other cash inflows and outflows, this logic
serves as the basis for the whole methodology and secures the practical applicability of
Bellman algorithm.
17
Applying conventional binomial model for the case of a firm producing constant output
under demand uncertainty, the next period nominal sales can go up by factor u with
objective probability p or down by factor d with probability 1-p:
Insert figure 1 here
Factors u and d are the functions of dispersion in demand growth rate σ :
u = exp(σ ⋅ k −1 ) , d = 1 u , where
(3)
k is number of time steps in one period (used to increase a sample of the binomial
distribution in order to improve the approximation to GBM).
Transition probability among states p is the function of consumer inflation rate µ and
up/down factors:
p=
1+ μ k − d
u−d
(4)
Next, let St (j, q) indicate the volume of real sales at period t when demand jumped up jtimes, down t-j times and management executed growth reinvestment q-times before the
period t. Assume that management has the rights (growth options) to execute net
reinvestments in any period during abnormal growth phase; therefore, q may range from
zero (no growth case) to t. Mathematically, sales at each state (j, q) is equal initial level
of sales S0 multiplied by factor uj and dt-j due to market uncertainty and by expansion
factor ξ(q) due to reinvestments executed by the firm’s management (and adjusted
finally by inflation index 1+µ):
18
q
S t ( j, q ) =
So ⋅ u j ⋅ d t − j ⋅ ∏ ξ i
(1 + μ )
i =0
t
, where
(5)
0 ≤ j ≤ t , 0 ≤ q ≤ t , ξi – expansion factor of i-th growth option.
On figure 2 is shown the fragment of multinomial recombining tree containing
information about states of uncertain demand conditioned upon the past states and
decision policies and currently possible decision modes at each state. Due to the fact
that all cash flows in the model are estimated in real terms, the final tree of underlying
cash flows from operations possesses the recombining property similar to the tree
depicted in figure 2 that dramatically reduces the computational complexity of the
model from order 2n to order (n+1)2.
Insert figure 2 here
Given that a firm at each state is operating in a particular mode, the present value of
future
cash
(
flows
given
optimal
behaviour
henceforth
is
)
denoted Λ t FCF (S t + Δt , mt + Δt , t + Δt ) S t , mt , t . Optimal behaviour assumes that a firm
will maximize the value of current payoff plus the present value of expected future
payoffs net of switching costs. At each uncertain state the optimization algorithm
compares the conditional expected value from waiting with the conditional expected
value of immediate reinvesting. In my notation, the dynamic Bellman equation can be
written as (where the firm arrives at time t operating at mode mt with possible switching
to mode lt+△t=1,2):
19
(
)
Λ t FCF (S t + Δt , mt + Δt , t + Δt ) S t , mt , t =
{
((
= max FCF (S t , mt , t ) − cl + ρ ⋅ E t Λ FCF (S t + Δt , l ... m0 , t + Δt ) S t , mt , t
l
(6)
))}
where ρ denotes the risk-adjusted discount rate.
At terminal period T we have boundary condition that reflects payoffs of available
options:
(
)
Λ T FCF (S T + Δt , mT + Δt , T + Δt ) S T , mT , T =
{
((
= max FCF (S T , mT , T ) − cl + ρ ⋅ ET Λ CV (S T + Δt , l ... m0 , T + Δt ) S T , mT , T
l
))}
(7)
Where CV(l) (l=1,2) denotes continuing value of a firm at stable growth period in case
of “wait” and “reinvest” decisions.
At this step in order to value future cash flows that depend on uncertain state variable, S,
and can be modified by reinvestment decisions, q, it is necessary to derive the
appropriate discount rates in order to reflect properly the changing risk levels of these
cash flows. Traditional corporate DCF model discounts expected future cash flows by
risk-adjusted rate which is usually represented by weighted average cost of capital.
However, this approach does not provide a correct valuation for growth options. The
reason is that, if management decide to switch to “reinvest” mode, this decision will
change the expected future cash flows, and thereby alters the original risk profile of the
company. Thus, original WACC can no longer be considered as a suitable measure of
future corporate risks and it is necessary to apply different discount rate to adjust for
changed risk of cash flows. The best method is to make these cash flows riskless and
then discount them by risk free rate. Traditional procedure in option theory is to
20
construct a replicating portfolio with B units of risk-free asset and l units of a traded
underlying asset that generates cash flows which exactly match the option’s future
uncertain payoffs at all times and in all states. To justify the arbitrage-free valuation for
the general case of a company with growth options, I used the Marketed Asset
Disclaimer (MAD) approach as in Copeland and Antikarov [2001]. The fundamental
assumption underlying MAD approach is that the fair value of a non-traded underlying
asset without any flexibility options is “the price the asset would have if it were traded”
(Mason and Merton [1985]) and various embedded options are treated as if they were
assets whose cash flows are derived from the value of that underlying asset without
options. Hence, the present value of a company’s cash flows without any growth
options is characterized as a traded security to construct a portfolio that replicates the
payoffs of cash flows with growth options. The value of a company without growth
options is represented exactly by the current value of a firm’s “assets-in-place” which
may be consistently determined using traditional DCF analysis by applying original
WACC to discount expected cash flows. The dynamic value of a company with growth
opportunities is then replicated by the firm’s assets-in-place and risk-free asset and
estimated via traditional option pricing techniques.
It should be noted that using the framework of recombining tree it is easy and intuitively
clear to construct replicating portfolios for estimating the value of different managerial
alternatives. I provide an example of computational procedure with 2 growth options for
two-period tree that is enough to understand the algorithm which can be extended to any
number of periods and options. Let me again present a recombining tree similar to the
one described in figure 2, but now in each node will be Free Cash Flow (FCF) estimated
in formula (2) (see figure 3). At node j, slightly changed notation FCFt (uj, dt-j, ξq)
21
characterises cash flow in case market demand jumped up j times, down t-j times and
new reinvestments to bring this cash flow were executed q times (t is period of
estimation, j = t to 0). First, consider a node belonging to a path where management
were choosing “wait” mode in every state of nature (for example, node FCF1(u)). In
this case, the present value of the expected future cash flows is equal to the value of
cash flows generated by the company’s assets-in-place at time 0 which can be
determined using traditional corporate DCF model by taking the real expectations Et
over possible realizations of FCF and discounting them by a company’s WACC at
initial time t = 0:
PV wait (FCF1 (u )) =
FCF2 (u 2 ) ⋅ p + FCF2 (u, d ) ⋅ (1 − p )
1 + WACC
(8)
Further, under the MAD assumption, expected FCF without growth options calculated
in formula (8) are considered as if they were a traded asset, and used to replicate the
payoffs (cash flows) of the “reinvest” mode for the current node. Then algorithm moves
on the lower branch where management have already executed one growth option
(“reinvestment” decision) at a previous time point. Similarly, it is necessary to estimate
the arbitrage free expected value of FCF in case of „wait“ and “reinvest” modes for a
current node (FCF1(u,ξ)). However, due to the tree recombining property
„wait“ decision at the current node FCF1(u,ξ) exactly equals the value of
„reinvest“ mode of the node at the upper branch, FCF1(u). The latter value has been
already calculated using arbitrage free principle at the previous step and, hence, is used
to replicate the payoffs of the “reinvest” mode for the current node. The similar
procedures presented in the figure 3 are then applied sequentially to other nodes and the
present values of all other cash flows are estimated.
22
Insert figure 3 here
Moving backwards and repeating all the steps, one can solve both for the arbitrage free
value of the company at initial node and the optimal strategic mode at each uncertain
state. The calculated amount will represent the value of the company whose
management has flexibility to choose optimal reinvestment strategy in case of
uncertainty.
IV.
NUMERICAL EXAMPLE
I illustrate the valuation of a company with many growth options by applying it to a
hypothetical corporation XYZ. The abnormal growth phase was assumed to be 7
periods during which the company’s management has the right to execute 7 growth
options. To achieve the desired probability distribution, the number of sub-periods was
set to 40 per period. Numerical assumptions for three growth scenarios are summarised
in Appendix A. Corporate DCF model’s estimations of moderate growth scenario with
20% volatility in demand produced following valuations 3 :
Assets in place (no growth case):
41,148 million yen
Assets in place plus Net reinvestments (growth case):
84,254 million yen
The calculations based on the described model gave the company value of 86,339
million yen, which is 2.5% bigger than DCF “growth case”. If accounts only for the
value of growth opportunities, which in case of DCF model equals 43,107 million yen
and in case of proposed real option model 45,191 million yen, DCF results in 4.8 %
3
Continuing value was estimated simplistically assuming no growth in cash flows in stable period.
23
undervaluing of the company’s growth options because of inability to account for
management flexibility.
I now examine the impact of greater uncertainty on the value of a corporation.
Following the concept of mean-preserving spread as in Rothschild and Stiglitz [1970],
the increase of uncertainty over demand parameter S is defined in such a way that it
does not affect its mean. Both in the field of real option theory and strategic
management, it has been remarked that the value of real options is positively related to
the underlying uncertainty. In case of increasing market uncertainty, the value of
installed growth options should also increase since the possible loss is limited to the
initial investment and the potential gains from future growth opportunities has no upper
bound (Kulatilaka and Perotti [1998]). Hence, overall firm’s value should be positively
linked to the value and uncertainty of the underlying cash flows that can be acquired
through potential exercise of the growth options included in them.
Insert figure 4 here
Figure 4 summarizes the results of simulations of the mean-preserving increases in
volatility of demand changes and its effects on the values of the firm and its growth
options. The results suggest that the value of portfolio of compound growth options is
strictly increasing with the uncertainty over demand S. The figure 5 further clarifies that
the impact of volatility depends also on the degree of future potential growth gained by
exercising the strategic growth options. It shows that companies which expect high sales
growth with high volatility have much higher flexibility premium defined as:
24
Flexibility premium =
Value of the firm with flexibility
−1
DCF Value of the firm
(9)
Insert figure 5 here
Next, I investigate critical values of demand, S, that trigger option execution at different
growth scenarios. Figure 6 shows low-sales-growth scenario for different values of
demand volatility with appropriate critical boundary at which it is optimal to execute
growth reinvestment conditioned upon the previous waiting/reinvesting decisions (or in
other words, on the level of capital stock at time t).
Insert figure 6 here
The optimal investment policy given the capital level at each period t is “wait” when
actual level of sales lies lower critical boundary space and “reinvest” if actual level of
demand exceeds the boundary. Clearly, higher expected demand favours reinvestments
and reduces the value of waiting, whereas higher volatility lowers the critical threshold
and induces early reinvestment.
Next figure emphasizes the non-linear effect of growth and volatility on a firm’s
investment behavior. In figure 7 depicted the company that faces the same demand
uncertainty as the company XYZ, but expects much higher growth rates of demand in
future and, respectively, must invest more heavily in fixed capital to maintain growth.
Insert figure 7 here
25
Higher expected growth achieved through executing real options and, respectively,
higher levels of installed capital increase the absolute value of investment threshold;
however, the portfolio of growth options is more valuable under high uncertainty and
reinvestment will be committed even when the level of current demand is low.
Now I examine the effect of the heterogeneity in resources and capabilities on
investment patterns in relation to option creation and exercise. From a resource-based
view, one may argue that the optimal management of real options requires managerial
efforts that are enabled and constrained by firm-specific resources and capabilities.
Using numerical example, I show that different firms facing the same opportunity may
display different investment behaviour. Assume that there are 3 firms facing the same
demand uncertainty and the same growth prospects, but having differences in capital
productivity, say, high, average and low productivity of capital. The results of the
simulations are presented in Figure 8.
Insert figure 8 here
Figure 8 shows that a firm with high productivity of capital invest at much lower levels
of demand compare to its less productive competitors; hence, a capital-effective firm is
much more alert in dynamic environment and react quicker to changes in uncertain
market demand. Respectively, one of the conclusions of the current study is the
proposition that when managers of highly productive companies have the bundle of
26
successive growth options, they may be willing to accept lower levels of current
performance thresholds in order to gain access to future growth options.
Using the framework of the current model, it is also possible to observe the effects of
constraints on the exercise of growth options in the corporate portfolio. Consider the
situation of two identical companies that face the same demand uncertainty, but one
firm can choose m growth options whereas the other can execute only m – k options
(here, k is any positive number less then m) determined by the firms’ capital capacity
and the strength of competitive advantage. For example, the firm-2 can have production
capacity constraints limiting the number of available investments. Notwithstanding the
value of k variable, the model’s results presented in table 1 show the nonlinearity of the
differences between the values of companies with different level of flexibility provided
by growth options.
Insert table 1 here
Finally, I show the effect of corporate tax rate on the value of a firm’s growth options.
Taxes influence on cash flows in non-linear way, since they are levied in case there is
taxable income, and not if the firm incurs current loss. Figure 9 shows that although
absolute value of portfolio of growth option linearly decreases with marginal tax rate,
the value of management flexibility is much more profound at higher levels of
uncertainty and in high tax regime environments.
Insert figure 9 here
27
It can be seen from the analysis that the presence of firm-specific effects expands the
critical parameters in the real option analysis of determinants of a firm’s investment
behaviour from the focus on demand volatility to also inclusion of taxes, market
potential, which can be measured by attainable sales expansion ratio, and marginal
efficiency of capital and new investment. The current framework integrates all these
firm-specific factors and effectively captures the portfolio effects of a firm’s growth
options.
IV.1. DISCUSSION
The following discussion focuses on comparison between proposed algorithm and other
alternative numerical methods of valuing corporations and multiple American options
embedded in firm’s growth: namely, corporate DCF, decision binomial trees and
binomial lattices.
The key advantage of the current paper is that it integrates corporate valuation with the
real option framework and allows quantitative estimation of how capital allocation
made in any year can enhance the company’s value under uncertainty. Such “optionbased” thinking provides new focus for corporate CFOs by shifting the emphasis from
choosing projects with maximum static NPV to paying attention to NPV maximizing
flexible reinvesting policy. It was shown that the values and insights from the proposed
analysis may be significantly different from DCF model that assumes predetermined
reinvestment pattern and uses subjectively defined risk-adjusted discount rate to
calculate the present value of corporate cash flows. Concerning the real option
applications, the use of the MAD assumption (Copeland and Antikarov [2001]) has
28
been criticized in that the value of a non-traded asset without flexibility, which is used
to create a complete market, cannot be tested empirically and, hence, may lead to
significant errors in estimation of the value of option on that asset. However, the
described model recognizes the value of the company’s current assets-in-place as the
basis for arbitrage-free replicating for the company’s growth options. This value is
usually observable on the market or can be estimated by direct comparisons with peer
companies and, hence, the model’s results can be empirically tested.
The main drawback of traditional binomial lattices is in that that they do not deal with
multiple American growth options that are compound and path-dependent. As an
alternative, non-recombining binomial decision trees were proposed that can handle
potentially any number of American type growth options. The current model is able to
capture the same number of American type growth options; however, the computational
procedure is more efficient. The straight-forward computation at each node j of nonrecombining tree at each period t would require 2j calculations ( j ∈ [0, t ] ), whereas with
the proposed recombining tree, the number of endpoints is equal to (j+1)2. Model of this
dimension can be implemented in Excel by writing a VBA code or directly in a
spreadsheet. The difference in size become much more pronounced if somebody
considers more periods, either by increased length of abnormal growth period or finer
time steps. I further conducted a test of the computational performance of real option
evaluation using proposed recombining binomial tree. According to Hull [2003],
solving a lattice with number of periods 30 or more gives reasonable results; therefore, I
used it as an upper limit for the testing range. The results were obtained using notebook
with Intel(R) Core(TM)2 Duo 1.20 GHz processor and 0.99 Gb RAM. As indicated in
figure 10, recombining binomial tree is computationally very efficient especially for
29
cases with large number of options or when a high degree of accuracy is required. In
contrast, the binomial tree as in Brandao, Dyer, Hahn [2005] requires several hours to
evaluate using the powerful professional decision tree software.
Insert figure 10 here
Figure 11 depicts the visual layout of the model’s output example. Providing the same
clarity in visualising threshold values as a non-recombining decision tree, the
recombining tree is visually more compact and helps the management structure the
investment choice problem by mapping out all feasible alternatives contingent on the
uncertain “nature” moves in a clear-cut hierarchical manner. The author believes that
practical financial managers, even without technical background in real options, can
grasp the fundamental properties of the recombining decision tree.
Insert figure 11 here
Finally, real option theory with its origin in stochastic calculus is not a theory of a firm
that focuses on organizational and behavioural issues. Hence, no quantitative decisionmaking technique can guarantee the optimal policy and substitute for proper managerial
efforts and discretion. Therefore, for further research it is crucial to consider also
various organizational and managerial issues that can stimulate or, on the contrary, limit
the optimal execution of real growth options in business practice.
30
V.
CONCLUSION
Major source of value from growth reinvestments arises from their ability to enhance
the upside cash generating capacity of a corporation during favourable market
conditions by making follow-on expansion reinvestments. Myers [1987] observed that
“the standard discounted cash flow techniques will tend to understate the option value
attached to growing, profitable lines of business” and that “the most promising line of
research is to try to use option pricing theory to model the time-series interactions
between investments”. Trying to address this issue, current article contributes to
existing corporate finance literature in several directions:
z
It develops a general model of corporate flexibility where a company pursuing
organic growth uses capacity as a strategic variable to adjust discretely to a
stochastic demand. Each period a company may operate in two modes (wait or
reinvest) with costly switching that provides a set of multiple American-type
compound options.
z
It provides a modelling framework for valuing multiple growth options embedded
in corporate growth, which are essentially compound American options on
American options.
z
Finally, it validates a computationally effective tool for valuing corporations
providing an alternative to traditional corporate DCF model.
31
Described methodology shows that the combined value of multiple growth options can
be quite large and that corporate DCF criterion which does not account for these options
can be misleading. In general, generating the growth options should drive the corporate
value. At the same time, as growth reinvestments are costly and may be (partially)
irreversible, it is not certain that this generating process is always done efficiently and
creates value for shareholders. Applied dynamic-programming framework enables a
computationally feasible evaluation of the effects of many such growth options.
Proposed model explicitly accounts both for exogenous (market demand) and
endogenous (net reinvestments) determinants of the firm’s growth and shows that
binomial model combined with decision tree analysis allows quantitative estimation of
optimal reaction of a firm to uncertain nature moves. The primary advantage of this
approach is that it provides greater transparency in the modelling of real growth options
of a company by focusing on individual uncertainty factor that drives company’s future
cash flows.
The methodology also provides clear threshold information for
management (for example: based on currently employed capital, invest if and only if
sales reach the level of $X), because the model explicitly accounts for the values of each
individual cash flow in every period. Hence, applying Marketed Asset Disclaimer for a
firm’s current assets-in-place, the model offers arbitrage free algorithm for corporate
valuation that properly accounts for changing risk profiles in each period depending on
reinvestment decisions.
As was noted in the introductory sections, a big challenge to empirical research in
strategic management under the real option framework is the difficulty in measurement
of the value of a firm’s growth opportunities. The current study addresses this issue and
provides theoretically consistent and computationally effective algorithm for future
32
studies to improve the precision in measuring corporate growth options using public
accounting data. Recombining property of the final tree of FCF allows improving the
accuracy of the model output by increasing the number of subperiods without placing
excessive burden on computer memory. Moreover, the algorithm, although complicated,
does not use higher mathematics; therefore, it can be effectively presented to decisionmakers. Presented model may be also used to incorporate other risk factors such as cost
uncertainty for companies with volatile variable costs or foreign exchange rate
uncertainty for firms with high export/import exposure. The methodology has also
parallel application, because if slightly modified it can be used to value switching
options when the number of alternative assets is (infinitely) large.
This paper is not without limitations. It is based on several restrictive assumptions
concerning the real cash flows and does not account for the effects of competitive
interactions in the value of portfolio of growth real options and optimal investment
decisions. Future research should address the mentioned limitations, as well as
empirically test the arguments presented.
33
VI.
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37
VII. APPENDIX A
List of assumptions for numerical examples in the text:
Scenario of demand conditions
High sales growth
Moderate sales growth
Low sales growth
1
162.0%
130.0%
107.0%
Sales expansion factor of i-th growth option, i:
2
3
4
5
6
7
150.0% 145.0% 142.0% 129.0% 114.0% 109.0%
125.0% 120.0% 117.0% 114.0% 111.0% 108.0%
106.5% 106.0% 105.5% 105.0% 104.5% 104.0%
Capital efficiency
Assumption
4.73 yen of sales
per 1 yen of capital
5 yen of sales
per 1 yen of capital
Average productivity of initial assets-in-place
Marginal productivity of new investment
Income Statement
Initial level of sales
61,500 million yen
Length of high growth phase
7 periods
Sales expansion factor at stable growth phase
100.8%
Variable cost of sales
90.0%
Marginal income tax
40.7%
Balance Sheet
Working capital, % of sales
4.9%
Initial fixed capital
13,000 million yen
Remaining usage life
7 periods
Capital structure
Net debt ratio
19.8%
After-tax nominal cost of debt
0.6%
Nominal cost of equity
9.0%
Real WACC
6.9%
Market
Consumer inflation rate
0.5%
Inflation of capital assets
1.0%
Real risk free rate
1.0%
38
Figure 1. Demand uncertainty experienced by a firm with constant output S0.
S0*u
p
S0
1-p
S0*d
39
Figure 2. Fragment of recombining decision/binomial tree for reinvestment policy
in sales expansion *.
Now
Year 1
Year 2
u
wait
d
S1(1,0)=S0*u
S2(2,0)=S0*u2
S2 (2,1)=S0*u2* ξ
u
reinvest
d
u
S2 (2,2)=S0*u2* ξ2
u
d
wait
d
u
S1(1,1)=S0*u* ξ
S2 (1,0)=S0*u*d
reinvest
d
wait
S2(1,1)=S0*u*d* ξ
S0
u
reinvest
d
wait
u
S2 (1,2)=S0*u*d* ξ2
u
S1(0,0)=S0*d
reinvest
d
u
d
d
S2(0,0)=S0*d2
wait
u
S1(0,1)=S0*d*ξ
reinvest
S2(0,1)=S0*d2* ξ
d
S2(0,2)=S0*d2* ξ2
*
Inflation adjustment parameter is hided for preserving space.
“Wait” means execution of replacement investment to maintain sales and postponement of net
reinvestment.
“Reinvest” means execution both replacement investment and net reinvestment to increase sales.
For expositional simplicity and without loss of the generality, sales expansion factor ξ is represented by a
constant.
40
Figure 3. Applying MAD to properly value forecasted cash flows for 2-period
model with 2 American-type growth options.
STEP 1. Calculate the value of “wait” decision for a path with cash flows generated by the firm’s
original assets-in-place using objective probabilities and WACC discount rate.
STEP 2. Construct replicating portfolio to estimate arbitrage free value of “reinvest” decision.
STEP 3. Choose maximum value at decision node.
u
wait
d
FCF1(u)
FCF2(u2)
FCF 2 (u2, ξ)
u
reinvest
d
u
FCF 2 (u2, ξ2)
u
d
wait
d
u
FCF1(u, ξ)
FCF 2(u, d)
reinvest
d
wait
FCF 2 (u, d, ξ)
FCF0
u
reinvest
d
wait
u
FCF 2 (u, d, ξ2)
u
FCF1(d)
reinvest
d
u
d
d
FCF 2 (d2)
wait
u
FCF1(d, ξ)
reinvest
FCF 2 (d2, ξ)
d
FCF 2 (d2, ξ2)
Terminal period t=1
Period t=2
STEP 4. Go to lower branch decision node with already installed additional capital: here the value of
“wait” decision has been calculated at step 2.
STEP 5. Repeat steps 2 and 3.
STEP 6. Repeat steps 1 – 5 for decision nodes in all other states.
41
Figure 4. Value of the firm as a function of volatility of demand changes.
million yen
Flexibility
value
120,000
100,000
80,000
Value of growth opportunities
60,000
40,000
70.0%
67.5%
65.0%
62.5%
57.5%
60.0%
55.0%
52.5%
50.0%
47.5%
45.0%
42.5%
40.0%
37.5%
35.0%
32.5%
30.0%
27.5%
25%
22.5%
20.0%
17.5%
15.0%
12.5%
7.5%
10.0%
0
5.0%
20,000
Volatility of sales growth, %
Value of the firm with flexible growth strategy (real option value)
Value of the firm's current assets in place
Value of the firm with fixed growth strategy (DCF value)
Inputs: Moderate sales growth scenario.
Figure 5. Flexibility premium of different growth scenarios as a function of volatility of
demand changes.
Flexibility premium, %
70%
60%
50%
Low sales growth
Moderate sales growth
40%
High sales growth
30%
20%
10%
0%
5.0%
10.0%
15.0%
20.0%
25%
30.0%
35.0%
40.0%
45.0%
50.0%
55.0%
60.0%
65.0%
Volatility of sales growth, %
42
70.0%
Figure 6. Boundary space of critical values of sales that trigger reinvestments as a
function of capital level (previously installed options) and the time of option execution
(in parenthesis are shown the number of growth options executed in previous periods).
Volatility of sales
growth: 10%
Volatility of sales
growth: 30%
40,000
40,000
35,000
35,000
30,000
30,000
(5)
(6)
(Million yen)
(3)
Level of
(4)
capital
(Million yen)
(5)
(6)
1
2
3
4
5
6
Period
(1)
(2)
Level of
capital
(3)
(Million yen)
Inputs: Low sales growth scenario.
43
(4)
16,375
(2)
16,375
(1)
13,861
Period
20,000
14,716
3
20,000
13,000
1
2
6
20,997
(4)
16,375
(3)
20,997
Level of
capital
15,557
13,861
(2)
14,716
13,000
(1)
4
5
25,000
“Wait” zone
25,000
15,557
25,000
(5)
20,997
30,000
15,557
35,000
13,861
40,000
20,000
45,000
45,000
14,716
45,000
Sales
“Reinvest” zone
Sales
13,000
Sales
Volatility of sales
growth: 50%
(6)
1
2
3
4
5
6
Period
Figure 7. Boundary space of critical values of sales that trigger reinvestments as a
function of capital level (previously installed options) and the time of option execution
(in parenthesis are shown the number of growth options executed in previous periods).
Volatility of sales
growth: 10%
Volatility of sales
growth: 30%
.
Sales
Sales
250,000
250,000
200,000
200,000
150,000
150,000
100,000
100,000
50,000
(5)
(6)
(Million yen)
(2)
Level of
capital
(3)
(4)
62,241
(1)
30,589
Period
0
13,000
3
0
20,626
2
6
(5)
(Million yen)
80,088
(4)
1
4
5
44,039
(3 )
80,088
Level of
capital
44,039
(2)
62,241
(1)
30,589
20,626
13,000
50,000
(6)
1
2
3
4
5
6
Period
(1)
(2)
Level of
capital
(3)
(Million yen)
Inputs: High sales growth scenario.
44
(4)
(5)
80,088
50,000
44,039
100,000
62,241
150,000
30,589
200,000
13,000
250,000
20,626
Sales
0
Volatility of sales
growth: 50%
(6)
1
2
3
4
5
6
Period
Figure 8. The effect of heterogeneity in firm’s capabilities on demand threshold values.
Average productivity of capital at t = 0:
3.1 yen (sales) per yen (capital)
Sales
Sales
(4)
(5)
(6)
(2)
Level of
capital
(3)
(4)
(5)
(6)
1
3
4
Period
(1)
(2)
Level of
capital
21,041
2
(3)
(4)
29,488
Period
(1)
13,000
5
5
25,223
2
4
20,000
6
(5)
33,597
1
3
6
40,000
51,455
(3)
101,535
Level of
capital
76,516
(2)
89,258
(1)
64,023
51,531
40,000
70,000
38,666
120,000
60,000
45,180
170,000
80,000
32,280
220,000
100,000
20,000
160,000
140,000
120,000
100,000
80,000
60,000
40,000
20,000
270,000
25,895
Sales
20,000
High productivity of capital at t = 0:
4.8 yen (sales) per yen (capital)
16,860
Low productivity of capital at t = 0:
1.5 yen (sales) per yen (capital)
(6)
Inputs: Moderate sales growth scenario, Volatility of sales growth – 20 %, Number of available options in the whole high growth
period -7, Length of high growth phase – 7 periods, number of simulated time steps in one period – 40.
45
1
2
3
4
5
6
Period
Table 1. Difference between corporate values of firms with different supply of growth
options.
Firm-N,
Number of
growth
options N:
1
2
3
4
5
6
Value of
Firm-M
Firm-M, Number of growth options M:
1
2
3
4
5
6
Value
of
Firm-N
0
(7,848)
(15,148)
(22,084)
(28,294)
(33,440)
7,848
0
(7,300)
(14,236)
(20,446)
(25,592)
15,148
7,300
0
(6,936)
(13,145)
(18,292)
22,084
14,236
6,936
0
(6,209)
(11,355)
28,294
20,446
13,145
6,209
0
(5,146)
33,440
25,592
18,292
11,355
5,146
0
48,905
56,753
64,054
70,990
77,199
82,345
48,905
56,753
64,054
70,990
77,199
82,345
Inputs: Moderate sales growth scenario, Volatility of sales growth – 20 %, Number of available options in the whole high growth
period – variable.
46
Figure 9. Value of a firm and flexibility premium as a function of corporate tax rate and
expansion rates.
million yen
160,000
140,000
120,000
100,000
80,000
60,000
40,000
20,000
0
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
45.0%
50%
55.0%
60.0%
Marginal tax rate, %
Value of the firm with flexible growth strategy (real option value)
Value of the firm's current assets in place
Value of the firm with fixed growth strategy (DCF value)
Inputs: Moderate sales growth scenario, Volatility of sales growth – 40 %.
Volatility of sales
growth: 15%
Volatility of sales
growth: 40%
Flexibility premium, %
Flexibility premium, %
8%
60%
7%
50%
Low sales growth
6%
Moderate sales growth
5%
Moderate sales growth
40%
High sales growth
4%
Low sales growth
High sales growth
30%
3%
20%
2%
10%
1%
Marginal tax rate, %
60
.0
%
55
.0
%
50
%
45
.0
%
40
.0
%
35
.0
%
30
.0
%
25
.0
%
20
.0
%
15
.0
%
10
.0
%
60
.0
%
55
.0
%
50
%
45
.0
%
40
.0
%
35
.0
%
30
.0
%
25
.0
%
20
.0
%
0%
15
.0
%
10
.0
%
0%
Marginal tax rate, %
47
Figure 10. Computational efficiency of the proposed model*.
Total computational
time, minutes
13
10
8
5
3
0
1
2
3
4
5
6
7
8
9
10
(30) (60) (90) (120) (150) (180) (210) (240) (270) (300)
Number of growth options
(Total number of time steps in the model)
*
Number of simulated time steps in one period – 30, total number of time steps is shown in parenthesis.
48
Figure 11. Example of presentation of optimal reinvestment strategy in sales
growth dependent on the changes in state of nature.
Now
Year 1
Year 2
Year 3
u
wait
d
S1(1,0)=S0*u
S2(2,0)=S0*u2
S2 (2,1)=S0*u2* ξ
u
reinvest
d
u
reinvest
S2 (2,2)=S0*u2* ξ2
…
u
d
wait
d
u
S1(1,1)=S0*u* ξ
S2 (1,0)=S0*u*d
reinvest
d
wait
wait
S2(1,1)=S0*u*d* ξ
S0
u
reinvest
d
wait
u
wait
2
S2 (1,2)=S0*u*d* ξ
…
…
u
S1(0,0)=S0*d
reinvest
d
u
d
d
S2(0,0)=S0*d2
wait
wait
u
S1(0,1)=S0*d*ξ
S2(0,1)=S0*d2* ξ
d
reinvest
S2(0,2)=S0*d2* ξ2
49
…