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Helsinki University of Technology Systems Analysis Laboratory Research Reports A92, August 2005 PORTFOLIO OPTIMIZATION MODELS FOR PROJECT VALUATION Janne Gustafsson Dissertation for the degree of Doctor of Technology to be presented with due permission for public examination and debate in Auditorium E at Helsinki University of Technology, Espoo, Finland, on the 26th of August, at 12 o'clock noon. Helsinki University of Technology Department of Engineering Physics and Mathematics Systems Analysis Laboratory
Distribution: Systems Analysis Laboratory Helsinki University of Technology P.O. Box 1100 FIN-02015 HUT, FINLAND Tel. +358-9-451 3056 Fax +358-9-451 3096 [email protected] This report is downloadable at www.sal.hut.fi/Publications/r-index.html ISBN 951-22-7790-5 ISSN 0782-2030 Otamedia Oy Espoo 2005
Title: Portfolio Optimization Models for Project Valuation Author: Janne Gustafsson Cheyne Capital Management Stornoway House 13 Cleveland Row London SW1A 1DH [email protected] Publication: Systems Analysis Laboratory Research Reports A92, August 2005 Abstract: This dissertation presents (i) a framework for selecting and managing a
portfolio of risky multi-period projects, called Contingent Portfolio Programming (CPP), and (ii) an inverse optimization procedure that uses this framework to compute the value of a single project. The dissertation specifically examines a setting where the investor can invest both in private projects and securities in financial markets, but where the replication of project cash flows with securities is not necessarily possible. This setting is called a mixed asset portfolio selection (MAPS) setting. The valuation procedure is based on the concepts of breakeven selling and buying prices, which are obtained by first solving an optimization problem and then an inverse optimization problem.
In the theoretical part of the dissertation, it is shown that breakeven prices are consistent valuation measures, exhibiting sequential consistency, consistency with contingent claims analysis (CCA), and sequential additivity. Due to consistency with CCA, the present approach can be regarded as a generalization of CCA to incomplete markets. It is also shown that, in some special cases, it is possible to derive simple calculation formulas for breakeven prices which do not require the use of inverse optimization. Further, it is proven that breakeven prices for a mean-variance investor converge towards the prices given by the Capital Asset Pricing Model (CAPM) as the investor’s risk tolerance goes to infinity. The numerical experiments show that CPP is computationally feasible for relatively large portfolios both in terms of projects and states, and illustrate the basic phenomena that can be observed in a MAPS setting.
Keywords: Project valuation, Project portfolio selection, Mixed asset portfolio selection,
Multi-period projects, Ambiguity
Academic dissertation Systems Analysis Laboratory Helsinki University of Technology Portfolio Optimization Models for Project Valuation Author: Janne Gustafsson Supervising professor: Professor Ahti Salo, Helsinki University of Technology Preliminary examiners: Professor Don N. Kleinmuntz, University of Illinois, USA
Professor Harry M. Markowitz, Harry Markowitz Company, USA
Official opponent: Professor James S. Dyer, University of Texas at Austin, USA Publications The dissertation consists of the present summary article and the following papers: [I] Gustafsson, J., A. Salo (2005): Contingent Portfolio Programming for the
Management of Risky Projects. Operations Research (to appear). [II] Gustafsson, J., B. De Reyck, Z. Degraeve, A. Salo (2005): Project Valuation in
Mixed Asset Portfolio Selection. Systems Analysis Laboratory Research Report E16. Helsinki University of Technology. Helsinki, Finland.
[III] Gustafsson, J., A. Salo (2005): Project Valuation under Ambiguity. Systems Analysis Laboratory Research Report E17. Helsinki University of Technology. Helsinki, Finland.
[IV] Gustafsson J., A. Salo (2005): Valuing Risky Projects with Contingent Portfolio Programming. Systems Analysis Laboratory Research Report E18. Helsinki University of Technology. Helsinki, Finland.
[V] Gustafsson, J., A. Salo, T. Gustafsson (2001): PRIME Decisions - An Interactive Tool for Value Tree Analysis, Proceedings of the Fifteenth International Conference on Multiple Criteria Decision Making (MCDM) Ankara, Turkey, July 10-14, 2000, Lecture Notes in Economics and Mathematical Systems, Murat Köksalan and Stanley Zionts (eds.), 507 165–176.
Contributions of the author Janne Gustafsson has been the primary author of all of the papers. He initiated Papers [I], [II], and [IV], whereas the topics for Papers [III] and [V] were suggested by Prof. Salo.
PREFACE This dissertation is a compilation of two of my central works on project portfolio
selection and project valuation, Papers [I] and [II], together with three more or less
related papers. My aim has been to improve my own understanding, as well as that
of practitioners and the general academic, about how to value and select projects
and other non-tradable investments, especially when it is possible to invest in
securities as well. This topic is fundamental to many academic fields, including
corporate finance and management science.
First and foremost, my thanks go to my supervisor Professor Ahti Salo for providing
invaluable support and for suggesting me to work on this topic as a part of my
Master’s Thesis. Since that time, the ideas have evolved significantly further and
become what you see in this dissertation. I would especially like to thank Professors
Bert De Reyck and Zeger Degraeve of London Business School for their support with
one of the papers. London Business School was indeed a truly outstanding research
environment.
I would also like to thank Professor Jim Dyer of University of Texas at Austin whose
review comments on Paper [I] led to a substantial improvement in the quality of the
paper. It is unlikely that I will ever receive such quality feedback on any of my present
or possible future papers. My sincere thanks also go to Professor Harry M. Markowitz
for the time he had for reading and commenting on my manuscripts. The central
ideas of his pioneering model and his exemplary writing style have had much
influence on my research and the way in which I try to write articles.
Last, I am grateful to Dr. Hannu Kahra and Mr. Lauri Pietarinen for creating an
exciting work environment at Monte Paschi Asset Management, where I was
employed during the writing of the first version of the summary article. Likewise, I
would like to thank my present colleagues Petteri Barman and Mikko Syrjänen for
providing an opportunity to work in an outstanding team in London.
London, U.K., June 27, 2005,
Janne Gustafsson
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1 INTRODUCTION 1.1 Context and Background The evaluation and selection of risky projects, such as research and development
(R&D) projects, has attracted substantial interest among both academicians and
practitioners (see, e.g., Martino 1995, and Henriksen and Treynor 1999). Especially
in high-technology firms, the selection of the R&D portfolio can be a major
determinant of the future performance of the company.
The task of selecting risky projects has been widely studied within several scientific
disciplines, most prominently in corporate finance, operations research, and
management science (see, e.g., Brealey and Myers 2000, Ghasemzadeh et al. 1999,
Heidenberger 1996, Smith and Nau 1995, Gear and Lockett 1973). Many of the
developed approaches aim at placing a value on the project; if this value is positive
the project is started; otherwise it is not. For example, in the literature on corporate
finance, the value of a risky project, like that of any other risky investment, is
calculated as the net present value (NPV) of its cash flows, discounted at a discount
rate that reflects the riskiness of the project (Brealey and Myers 2000). It is typically
suggested that this rate should be the expected rate of return of a security that is
“equal in risk” to the project. Under the Capital Asset Pricing Model (CAPM; Sharpe
1964, Lintner 1965), two assets are regarded equally risky if they have the same
beta, and therefore the finance literature suggests that one should use the beta of the
project, or equivalently, the covariance between the project and the financial market
portfolio, to determine the discount rate for the project (Brealey and Myers 2000).
However, the use of the CAPM in project valuation relies on the assumption that the
firm is a public company maximizing its share price; yet, many companies make
decisions about accepting and rejecting projects before their initial public offering.
Several methods have been proposed to value projects of a private firm. These
include decision analysis (French 1986, Clemen 1996) and options pricing analysis,
which is also referred to as contingent claims analysis (CCA; Merton 1973b, Hull
1999, Dixit and Pindyck 1994, Trigeorgis 1996). Still, conventional methods based on
decision analysis, such as decision trees, may lead to biased estimates of project
value, because they do not take into account the opportunity costs imposed by
alternative investment opportunities. Options pricing analysis accounts for the effect
that financial instruments have on project value, but its applicability is limited,
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because in practice it may be difficult to replicate project cash flows using financial
instruments.
When used to select a project portfolio, most single-project valuation methods are
problematic in that they neglect the effect of project interactions, such as synergies
and diversification, on the overall performance of the portfolio. Also, these methods
do not take into account the firm’s resource constraints, which may strongly limit the
projects that can simultaneously be included into the portfolio. Therefore, such
methods may lead to a suboptimal project portfolio both in terms of expected cash
flows and the risk of the portfolio. For this reason, it is advisable to employ a project
portfolio selection method instead. Such a method can potentially determine the most
valuable portfolio where project synergies and the effect of diversification on the risk
of the portfolio are taken into account, although it may not directly put a value on any
single project. Still, as shown in this dissertation, these methods can also be applied
to value single assets in the firm’s portfolio through a specific inverse optimization
procedure.
Earlier project portfolio methods have, however, suffered from various shortcomings
that have hindered the use of the methods in practice. For example, many of the
currently available methods, such as the method by Gear and Lockett (1973) and
Heidenberger (1996), make restrictive assumptions about the nature of the investor’s
risk aversion, failing to imply diversification, for instance. Further, some methods do
not consider uncertainty, while others fail to properly account for the multi-period
nature of projects.
1.2 Aims and Practical Relevance This dissertation has two primary aims. First, it aims at developing a framework for
selecting a portfolio of risky multi-period projects which is (i) well-founded in the
theories of finance, management science, and operations research, and also (ii)
practically applicable in the sense that it (a) captures most of the phenomena that are
relevant to R&D portfolio selection and (b) is computationally feasible for portfolio
selection problems of realistic size. Second, the dissertation aims at developing a
procedure for project valuation in a setting where the firm can invest both in private
projects and publicly-traded securities, but where replication of project cash flows
with securities is not necessarily possible. This setting is called the mixed asset
portfolio selection (MAPS) setting.
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While a framework for selecting a portfolio of risky projects enjoys general interest of
practitioners of corporate finance, also MAPS-based project valuation is important in
practice. On the one hand, MAPS-based project valuation is, in principle, called for
when a corporation or an individual makes investments both in securities and
projects, or other lumpy investment opportunities. For example, many investment
banks invest in a portfolio of publicly traded securities and undertake uncertain one-
time endeavors, such as venture capital investments. On the other hand, a
fundamental problem in the literature on corporate finance is the valuation of a single
project while taking into account the opportunity costs imposed by securities (see,
e.g., Brealey and Myers 2000). This is a MAPS setting that includes one project and
several securities.
1.3 Structure of Dissertation The dissertation includes five papers. Papers [I] and [II] form the core of the
dissertation. Paper [I] presents a modeling framework, called Contingent Portfolio
Programming (CPP), for selecting and managing a portfolio of risky multi-period
projects. The framework is taken into use and extended in Paper [II], which examines
the valuation of risky projects in a MAPS setting. The valuation procedure is based
on the concepts of breakeven selling and buying prices (Luenberger 1998, Smith and
Nau 1995, Raiffa 1968), which rely on the comparison of MAPS problems with and
without the project being valued. Paper [II] shows that breakeven buying and selling
prices exhibit several important properties and that they are therefore consistent
valuation measures. Paper [I] constitutes the primary modeling contribution of the
dissertation, whereas Paper [II] contains the dissertation’s main methodological and
theoretical contribution.
The three other papers provide additional contributions. Papers [III] examines a
single-period MAPS setting where the investor is either unable to give probability
estimates or where the estimates are ambiguous. In particular, we concentrate in this
paper on the Choquet-Expected Utility (CEU) model, which is able to capture
ambiguity, and develop two models to solve MAPS models when the investor is a
CEU maximizer. Paper [IV] discusses multi-period project valuation, compares the
present approach to other multi-period approaches in the literature, and produces
further computational results. Paper [V] describes a decision support system based
on an interval value tree method called Preference Ratios In Multi-attribute
Evaluation (PRIME; Salo and Hämäläinen 2001), and presents a case study where it
is used to develop scenarios for the market share of Sonera SmartTrust, a Finnish
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high-technology company. A similar approach can be used for scenario generation in
project portfolio selection.
2 EARLIER APPROACHES
2.1 R&D Project Selection Models Several methods for the selection of R&D projects have been developed over the
past few decades (see, e.g., Martino 1995, and Henriksen and Traynor 1999). Many
of these methods are based on mathematical optimization. Such methods are
typically focused on capturing some specific characteristics of R&D portfolio selection
such as synergies or follow-up projects, as described in Table 1. An ideal project
selection framework would implement all of the characteristics in Table 1 in a
theoretically rigorous way. However, few methods aim at capturing all of these
features, and many of those that do resort to theoretically questionable approaches
in modeling some of the features.
Optimization-based R&D project selection methods, such as the ones in Table 1, can
be viewed as extensions of standard capital budgeting models (see, e.g., Luenberger
1998). These models capture rather complex problems with project
interdependencies and resources constraints, but they do not usually address
uncertainties associated with the projects’ outcomes, which makes it impossible to
Table 1. Overview of Approaches to R&D Project Selection.
Features Model RN FP VA CO PV RC RD SY Ghasemzadeh et al. 1999 X X X Heidenberger 1996 X X X X Santhanam & Kyparisis 1996 X X X Czajkowski & Jones 1986 X C Fox et al. 1984 X X Mehrez & Sinuary-Stern 1983 X Aaker & Tyebjee 1978 X X X Gear & Lockett 1973 X X X X Gear et al. 1971 Bell et al. 1967 X X Watters 1967 X C Brandenburg & Stedry 1966 X X
Key: CO = correlation or other probabilistic interaction between project outcomes, FP = follow-up projects, PV = project versions, RC = resource constraints, RD = resource dynamics, RN = reaction to new information, SY = synergies (cross terms for project outcomes), VA = variability aversion, X= feature present in basic model, C = chance-constrained model
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attach risk measures to project portfolios. Also, these models do not usually offer
possibilities for reacting to new information. In Table 1, the scarcity of X’s in columns
“VA”, “CO”, and “RN” highlights these shortcomings.
Even though some methods do deal with project uncertainties and the investor’s risk
aversion, they often do so by resorting to unrealistically restrictive assumptions or
theoretically unfounded approaches. For example, the method by Mehrez and
Sinuany-Stern (1983) relies on restrictive assumptions about the investor’s utility
function while Czajkowski and Jones (1986) employ chance-constraints that may
lead to preference models that are inconsistent with expected utility theory and other
well-founded preference frameworks. Also, the models of Gear and Lockett (1973)
and Heidenberger (1996) do not account for the variability of portfolio returns, even
though they allow the investor to react to new information.
Another limitation in some optimization models (e.g., Gear and Lockett 1973,
Czajkowski and Jones 1986, and Ghasemzadeh et al. 1999) is that project inputs are
separated from outputs, wherefore projects cannot produce inputs for other projects,
for instance. These models typically also assume that there exists a predefined,
static supply of resources in each time period (see, e.g., Ghasemzadeh et al. 1999
and Gear and Lockett 1973) which makes it impossible to invest profits for later or
immediate use. There is consequently a need for dynamic modeling of resources;
early attempts into this direction have been presented by Brandenburg and Stedry
(see Gear et al. 1971).
In summary, there appears to be need for an optimization method that rigorously
captures (i) project uncertainties, (ii) the investor’s risk preferences, and (iii) dynamic
production and consumption of resources.
2.2 Stochastic Programming
Stochastic programming models analogous to R&D portfolio selection models have
appeared in investment planning as well as in asset-liability management (e.g.,
Bradley and Crane 1972, Kusy and Ziemba 1986, Mulvey and Vladimirou 1989, Birge
and Louveaux 1997, Mulvey et al. 2000). These two problem contexts share
similarities with the selection of R&D projects in that (i) the investor seeks to
maximize the value of a portfolio of risky assets in a multi-periodic setting and (ii)
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there are several asset categories which parallel the multiple resource types
consumed and produced by R&D projects.
A key difference between R&D portfolio selection models and the financial stochastic
programming models is that in financial optimization, the (dis)investment decisions
are unconstrained quantities that do not restrict the investor’s future decision
opportunities (e.g., security trading). In contrast, R&D project selection involves
“go/no go”-style decisions (Cooper 1993) where the “go”-decision leads to later
project management decisions while the “no go”-decision terminates the project
without offering further decision opportunities. Table 2 contrasts the key
characteristics of selected financial models of stochastic programming to CPP, which
is developed in Paper [I]. Among these, CPP has close parallels with the dynamic
model of Bradley and Crane (1972), as well as the models of Birge and Louveaux
(1997) and Mulvey et al. (2000) which employ state (scenario) trees.
2.3 Project Valuation Methods The literature on corporate finance contains a large number of apparently rivaling
methods for the valuation of risky projects. The most popular approaches include (i)
decision trees (Hespos and Strassman 1965, Raiffa 1968), (ii) expected utility theory
(von Neumann and Morgenstern 1947, Raiffa 1968), (iii) the risk-adjusted NPV
Table 2. Comparison of Some Stochastic Programming Approaches to Portfolio Selection
Bradley and Crane (1972)
Kusy and Ziemba (1986)
Mulvey and Vladimirou (1989)
Birge and Louveaux (1997, §1.2) CPP (Paper [I])
Model type Linear Linear Quadratic or non-linear, network
Linear Linear, mixed integer
Multiple time periods Yes Yes but only two stages
Yes but only two stages Yes Yes
Model of uncertainty State (event) tree
States for second stage; for external cash flows only
States for second stage State tree State tree
Objective Expected value of terminal wealth
Expected discounted net revenues
Mean-variance model or Expected utility of terminal wealth
Expected utility of terminal wealth
Utility of terminal wealth; mean-risk model
Model of risk aversion / risk measure
Loss constraints
Penalty from constraint violations
Variance or power utility function
Piecewise linear utility function
LSAD or EDR
Type of decisions Quantitative (asset trading)
Quantitative (asset trading)
Quantitative (asset trading)
Quantitative (asset trading)
Choices between actions
Decisions influence future decision possibilities
No No No No Yes
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method (see, e.g., Brealey and Myers 2000), (iv) real options (Dixit and Pindyck
1994, Trigeorgis 1996), (v) Robichek and Myers’ (1966) certainty equivalent method
(see also Brealey and Myers 2000, Chapter 9), (vi) Hillier’s (1963) method, and (vii)
Smith and Nau’s (1995) method. These methods are summarized in Table 3. The
three columns under the heading “Purpose” indicate the purpose to which the
method is intended. As Table 3 describes, decision trees and expected utility theory
are complementary techniques to the other methods, which aim at calculating the
present value of an investment. In a complete project valuation framework, there is a
specific method addressing each of the three purposes.
Table 3. Methods for the valuation of risky multi-period investments.
Purpose Method CE PV ST Formula / explanation
Risk-adjusted NPV X [ ]
1 (1 )
Tt
tt adj
E cNPV I
r=
= − ++∑
Decision tree X A chart with decision and chance nodes
Expected utility theory X [ ] [ ]( )1 ( )CE X u E u X−=
Contingent claims analysis X 0
n
i ii
NPV I S x ∗
=
= − + ∑
Robichek and Myers (1966) X [ ]
1 (1 )
Tt
tt f
CE cNPV I
r=
= − ++∑
Hillier (1963) X 1 (1 )
Tt
tt f
cNPV I CEr=
⎡ ⎤= − + ⎢ ⎥
+⎢ ⎥⎣ ⎦∑
Smith and Nau (1995) X
NPV = breakeven selling or buying price Preference model for cash flow streams: ( ) ( )1 2 1 2[ , ,..., ] , ,...,T TU c c c E u c c c∗= ⎡ ⎤⎣ ⎦
MAPS (Paper [II]) X NPV = breakeven selling or buying price Preference model for terminal wealth levels
Key: CE = Certainty equivalent for a risky alternative, PV = Present value of a risky cash flow stream, ST = Structuring of decision opportunities and uncertainties, I = investment cost, ct = risky cash flow at time t, radj = risk-adjusted discount rate, u = utility function, Si = price of security i, ix ∗ = amount of security i in the replicating portfolio, rf = risk-free interest rate, u* = intertemporal (multi-attribute) utility function.
The MAPS valuation approach developed in Paper [II] is most closely related to the
method by Smith and Nau (1995), the main difference being in the employed
preference model. It is also consistent with contingent claims analysis, and in some
special cases, with Hillier’s (1963) method, as discussed in Papers [II] and [IV]. On
the other hand, the CPP framework, which is developed in Paper [I] and
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consequently used in Papers [II] and [IV], implements a decision-uncertainty
structure similar to decision trees, and also allows the use of a wide range of
preference models, including the expected utility model. Paper [III] employs a
Choquet-expected utility model, which is an alternative to expected utility theory.
In the following, the methods in Table 3 and their limitations and possible uses are
discussed in more detail.
2.3.1 Decision Trees and Related Approaches A decision tree describes the points at which decisions can be made and the way in
which these points are related to unfolding uncertainties. Conventionally, decision
trees have been utilized together with expected utility theory (EUT; von Neumann
and Morgenstern 1947) so that each end node of the decision tree is associated with
the utility implied by the earlier actions and the uncertainties that have resolved
earlier. This decision tree formulation does not explicitly include the time axis or
provide guidelines for accounting for the time value of money.
In corporate finance, decision trees are used to describe how project management
decisions influence the cash flows of the project (see, e.g., Brealey and Myers 2000,
Chapter 10). Here, decision trees are typically applied together with the risk-adjusted
NPV method, whereby an explicitly defined time axis is also constructed. However,
the selection of an appropriate discount rate for NPV is often problematic, mainly
because the rate is influenced by three confounding factors, (i) the risk of the project,
which depends on the project’s correlation with other investments, (ii) the opportunity
costs imposed by alternative investment opportunities, and (iii) the investor’s risk
preferences.
Several methods for determining the discount rate have been proposed in the
literature. However, most of them have problematic limitations. For example, the
weighted average cost of capital (WACC) is appropriate only for average-risk
investments in a firm, whereas discount rates based on expected utility theory do not
account for the opportunity costs imposed by securities in financial markets. The real
options literature suggests the use of contingent claims analysis (CCA) to derive the
appropriate discount rate by constructing replicating portfolios using market-traded
securities. Still, it may be difficult to construct replicating portfolios for private projects
in practice. Last, the use of a CAPM discount rate is appropriate only for public
companies whose all shares are traded in markets that satisfy the CAPM
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assumptions. Even when these assumptions are satisfied, a CAPM discount rate is
applicable for single projects only when there are no synergies between projects.
2.3.2 Robichek and Myers’ and Hillier’s Methods Robichek and Myers’ (1966) and Hillier’s (1963) methods are two alternative ways of
determining a risk-adjustment to a discount rate in a multi-period setting. These
methods have been widely discussed in the literature on corporate finance (see, e.g.,
Keeley and Westerfield 1972, Chen and Moore 1982, Ariel 1998, and Brealey and
Myers 2000). They both employ expected utility theory or a similar preference model
to derive a certainty equivalent (CE) for a risky prospect.
In Robichek and Myers’ method, the investor first determines a CE for the cash flow
of each period, and then discounts it back to its present value at the risk-free interest
rate. Yet, because CEs are taken separately for each cash flow, the method does not
account for the effect of cash flows’ temporal correlation on the cash flow stream’s
aggregate risk; hence, it may lead to an unnecessarily large risk-adjustment. In
contrast, in Hillier’s (1963) method, we first determine the cash flow streams that can
be acquired with the project in different scenarios and then calculate the NPVs of
these streams using the risk-free interest rate. The result is a probability distribution
for risk-free-discounted NPV, for which a CE is then determined. However, the use of
the risk-free interest rate essentially means that any money received before the end
of the planning horizon is invested in the risk-free asset. Yet, it might be more
advantageous to invest the funds in risky securities instead. Therefore, Hillier’s
method is, strictly speaking, applicable only in settings, where the investor cannot
invest in risky securities.
2.3.3 Smith and Nau’s Method The idea behind Smith and Nau’s (1995) method, which Smith and Nau call “full
decision tree analysis,” is to explicitly account for security trading in each decision
node of a decision tree. The main advantage of the approach is that it appropriately
accounts for the effect that the possibility to invest in securities has on the discount
rate of a risky project. Incorporating decision trees, a preference model, and security
trading, Smith and Nau’s method is one of the most complete project valuation
methods to date. However, it does not consider alternative projects, which impose an
opportunity cost on the project being valued, wherefore it is applicable only in a
setting where the investor can invest in a single project and several securities. In a
multi-project setting, the method is subject to the usual shortcomings of single-project
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valuation methods; in particular, it fails to account for the effect of diversification and
project synergies.
Also, the practically appealing form of the method, the integrated rollback procedure,
relies on several restrictive assumptions: (i) additive independence (Keeney and
Raiffa 1976), (ii) constant absolute risk aversion (CARA), and (iii) partial
completeness of markets. Yet, as pointed out by Keeney and Raiffa (1976), additive
independence entails possibly unrealistic preferential restrictions. The CARA
assumption may also be questionable, because it leads to an exponential utility
function with utility bounded from above. This is known to result in an unrealistic
degree of risk aversion at high levels of outcomes (see, e.g., Rabin 2000). In
practice, it may also be difficult to create a replicating portfolio for market-related
cash flows of a project, as it is assumed in partially complete markets.
In view of the limitations of Smith and Nau’s (1995) method, it appears that there is a
need for project selection framework that (i) considers all projects in the portfolio,
implements (ii) decision trees and (iii) security trading, and allows for (iv) a realistic
array of risk preferences without resorting to overly restrictive assumptions.
3 PROJECT PORTFOLIO SELECTION MODEL The Contingent Portfolio Programming (CPP) framework presented in Paper [I] is the
underlying modeling framework that is used throughout this dissertation, except in
Paper [V]. CPP allows risks to be managed both through diversification (Markowitz
1952, 1959) and staged decision making (Cooper 1993), and accounts for the firm’s
resource constraints. The framework has also the advantage that, when the firm’s
risk measure satisfies a linearity property, it leads to linear programming models,
which can readily solved for portfolio selection models of realistic size.
3.1 Framework In CPP, projects are regarded as risky investment opportunities that consume and
produce several resources over multiple time periods. Analogously to Gear and
Lockett (1973) and the Stage-Gate process of Cooper (1993), the staged nature of
R&D projects is captured through project-specific decision trees, which support
managerial flexibility by allowing the investor to take stepwise decisions on each
project in view of most recent information (Dixit and Pindyck 1994, Trigeorgis 1996,
Brandao and Dyer 2004, Brandao, Dyer, and Hahn 2004). Uncertainties are modeled
using a state tree, representing the structure of future states of nature, as shown in
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the leftmost chart in Figure 1. In general, the state tree is a multinomial tree that can
have different probability distributions in its branches.
0 1 2
Stat
es
50%
30%
70%
40%
60%
50%
15%
35%
20%
30%
ω0
ω1
ω2
ω12
ω21
ω22
ω11
Start? Continue?Yes
No
Yes
No
0 1 2Time
0
Start?Yes
No
2
Continue?Yes
No
Yes
No
1Time
ω11
ω12ω11
ω12ω21
ω22ω21
ω22
ω1
ω2
Time
Figure 1. A state tree, a decision sequence, and a decision tree for a project.
Projects are modeled using decision trees that span over the state tree. The two
rightmost charts in Figure 1 describe how project decisions, when combined with the
state tree, lead to project-specific decision trees. The specific feature of these
decision trees is that the chance nodes are shared by all projects, since they are
generated using the common state tree. This allows for taking into account the
correlations between projects.
Also securities can be straightforwardly incorporated into the CPP framework in order
to develop a MAPS setting. While Paper [I] briefly mentions this possibility, a proper
development of the resulting CPP MAPS model is given in Papers [II] and [IV]. In
such a model, security trading is implemented through state-specific trading
variables, which are similar to the ones used in financial models of stochastic
programming (e.g. Mulvey et al. 2000) and in Smith and Nau’s (1995) method. In
addition to introducing security trading, Paper [II] presents a risk-constrained mean-
risk version of the CPP model, which is not discussed in Paper [I]. Paper [III] employs
a one-period version of this model, formerly called “general deviation-based mean-
risk model” in a previous version of Paper [II], which appears in my Licentiate Thesis
(Gustafsson 2004).
Four types of constraints are imposed on a CPP model: (i) resource (budget)
constraints, (ii) decision consistency constraints, (iii) risk constraints, which apply to
risk-constrained models only, and (iv) deviation constraints. Resource constraints
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ensure that there is a nonnegative amount of cash in each state. Decision
consistency constraints implement the projects’ decision trees. These constraints
require that (i) at each decision point reached, only one action is selected, and that
(ii) at each decision point that is not reached, no action is taken. Deviation constraints
are needed in the formulation of many deviation-based risk measures.
3.2 Objective Function The preference models in CPP can be further classified into two classes: (1)
preference functional models, such as the expected utility model, and (2) bi-criteria
optimization models. In general, we can refer by a “mean-risk model” either to a
preference functional model (like in Paper [I]) or to a bi-criteria optimization model
that uses optimization constraints (like in Papers [II] – [IV]). I adopt here the latter
terminology.
In a preference functional model, the investor seeks to maximize the utility of the
terminal resource position
max [ ]U X ,
where U is the investor’s preference functional and X is the random value of the
resource position in period T. Under expected utility theory, the preference functional
is given by [ ] [ ]( )=U X E u X , where u is the investor’s von Neumann-Morgenstern
utility function. A risk-constrained mean-risk model can be formulated as follows:
max [ ]E X ,
subject to
[ ]X Rρ ≤ ,
where ρ denotes the risk constraint and R the investor’s risk tolerance.
One of the main differences between Papers [I]–[IV] is the objective function used
within the CPP framework. Paper [I] concentrates on linear preference functionals,
mean-lower semi-absolute deviation (mean-LSAD) model and mean-expected
downside risk (mean-EDR) model, which lead to linear CPP models. Such models
can typically be solved when there is a reasonably large number of states and
projects (e.g., 100 projects and 200 states), as shown by the numerical experiments
in Paper [I]. Lower semi-absolute deviation (LSAD; Ogryczak and Ruszczynski 1999,
Konno and Yamazaki 1991) and expected downside risk (EDR; Eppen et al. 1989)
are defined as
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LSAD: ( ) ( ) ( )X X
X X X X Xx dF x x dF xµ µ
δ µ µ−∞ −∞
= − = −∫ ∫ and
EDR: ( ) ( ) ( )X X XEDR x dF x x dF xτ τ
τ τ−∞ −∞
= − = −∫ ∫ ,
where Xµ is the mean of random variable X, τ is some constant target value, and FX
is the cumulative density function of X. The mean-LSAD model exhibits linear pricing
and consistency with stochastic dominance (Levy 1992, Ogryczak and Ruszczynski
1999), whereas the mean-EDR model is consistent with expected utility theory (von
Neumann and Morgenstern 1947, Fishburn 1977) and hence also dynamically
consistent (Machina 1989). Also, as long as the preference model and the underlying
random variables satisfy the assumptions of Dyer and Jia’s (1997) relative risk-value
models, in particular the relative risk independence condition and non-negativity of
outcomes, the preference model exhibits also the properties observed with relative
risk-value models, such as the decomposition to the relative risk-value form.
In contrast, Paper [II] focuses on the risk-constrained mean-variance model, because
hereby it is possible to contrast the results with the CAPM, which is based on the
Markowitz (1952) mean-variance model. Similarly, Paper [IV] employs the risk-
constrained mean-variance model in its numerical experiments, although it is not
otherwise limited to this model. The numerical experiments in Paper [IV] involve also
expected utility maximizers exhibiting constant absolute risk aversion (CARA;
Keeney and Raiffa 1976).
Paper [III] uses the Choquet-Expected Utility (CEU) model (Choquet 1953, Gilboa
1987, Schmeidler 1989, Wakker 1990, Camerer and Weber 1992), which under
stochastic dominance reduces to the Rank Dependent Expected Utility (RDEU;
Quiggin 1982, 1993) model. This model is given by
[ ] [ ] [ ] ϕ∞
−∞
′= = = −∫( ) ( ) (1 ( )) ( )c X XU X CEU X E u X u x F x dF x ,
where ϕ is the transformation function describing the investor’s ambiguity aversion.
Paper [III] focuses on two special cases of the transformation function, quadratic and
exponential functions, and presents two alternative formulations of a CEU MAPS
model, the binary variable model and the rank-constrained model. Also, Paper [III]
presents a MAPS model using Wald’s maximin criterion (Wald 1950), which is a
limiting special case of the CEU model.
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4 PROJECT VALUATION While Paper [I] provides a framework for determining the value of a project portfolio,
the methodology for calculating the value of a single project within a portfolio is
developed in Paper [II]. With a single exception in Section 5.2 of Paper [II], examined
project valuation settings in Papers [II] – [IV] are MAPS settings with at least the risk-
free asset available. That is, the investor is able to invest in both (i) securities, which
can be bought and sold in any quantities, and (ii) projects, which are lumpy all-or-
nothing type investments.
4.1 Breakeven Buying and Selling Prices Since a project is a non-tradable investment opportunity, the value of a project is
defined as the amount of money at present that is equally desirable to the project,
which corresponds to the conceptual definition of NPV in the literature on corporate
finance. Still, the procedure obtained in Paper [II] is quite different from the
conventional NPV formula found in course books on corporate finance (e.g., Brealey
and Myers 2000). Nevertheless, as shown in Paper [IV], the approach coincides with
many of the conventional project valuation approaches when the investor can invest
only in a single project and the risk-free asset.
In a portfolio context, the above definition for project value can be interpreted so that
the investor is indifferent between the following two portfolios: (A1) a portfolio with the
project and (B1) a portfolio without the project and cash equal to the value of the
project. We may alternatively define the value of a project as the indifference
between the following two portfolios: (A2) a portfolio without the project and (B2) a
portfolio with the project and a debt equal to the value of the project. The project
values obtained in these two ways will not, in general, be the same. The first type of
value is called the “breakeven selling price” (BSP), as the portfolio comparison can
be understood as a selling process, and the second type of value the “breakeven
buying price” (BBP).
As discussed in Paper [II], finding a BSP and BBP is an inverse optimization problem
(see, e.g., Ahuja and Orlin 2001): one has to find a change in the budget so that the
optimal value of the second portfolio optimization problem matches the optimal value
of the first problem. Such problems can be classified into two groups: (i) finding an
optimal value for the objective function, and (ii) finding a solution vector. The problem
of finding a BSP or BBP falls within the first class; inverse optimization problems of
this class can be solved by finding a root to a strictly increasing function. To solve
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such root-finding problems, we can use usual root-finding algorithms, such as the
bisection method and the secant method.
4.2 Theoretical Results 4.2.1 Consistency of Breakeven Prices As shown in Paper [II], breakeven prices exhibit sequential consistency, consistency
with CCA, and sequential additivity. Due to sequential consistency, the investor will
behave rationally in sequential decision problems. Technically, this means that when
an investor first buys a project and then sells it, or vice versa, his/her (sequential)
buying and selling prices will be equal to each other.
Consistency with CCA refers to the property of the breakeven prices that, whenever
CCA is applicable, i.e. whenever there exists a replicating portfolio for the project, the
breakeven buying and selling prices are equal to each other and yield the same
result as CCA (see also Smith and Nau 1995). Due to this property, the breakeven
prices can be regarded as a generalization of CCA.
Finally, sequential additivity states that the (sequential) BSPs / BBPs of two or more
projects will always add up to the BSP / BBP of the portfolio composed of the same
projects. This is a result of the fact that BSPs and BBPs are added values; if valued
non-sequentially, projects’ breakeven prices are non-additive in general.
4.2.2 Equality of Prices and Valuation Formulas Paper [II] also shows that breakeven prices exhibit two important properties for a
broad class of risk-constrained mean-risk investors: (i) the breakeven prices are
equal to each other, and (ii) they can be solved through a pair of optimization
problems without resorting to possibly laborious inverse optimization.
Papers [III] and [IV] produce analogous valuation formulas for other settings. Each of
these formulas implies that, under the specified circumstances, the breakeven prices
will be equal to each other and that they can be solved through a pair of optimization
problems without using inverse optimization. Paper [III] develops the formula for
expected utility maximizers exhibiting CARA and investors using Wald’s maximin
criterion. Both of these preference models have been widely used in the literature.
On the other hand, Paper [IV] develops a formula for BSP when the investor can
invest only in a single project and in the risk-free asset. A similar formula holds for
both BSP and BBP when the investor exhibits CARA; further, it is now possible to
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16
use the certainty equivalent operator directly on the project’s future value distribution.
The paper also shows that when the investor exhibits linear pricing the breakeven
prices coincide with the Hillier’s (1963) method. In contrast, the prices will almost
never give the same result as Robichek and Myers’ (1966) method, because this
method typically overestimates the risk of the project.
4.2.3 Relationship to Capital Asset Pricing Model Paper [II] also derives analytical formulas for BSP and BBP in the case where (a) the
investor is a mean-variance optimizer, (b) the optimal mixed asset portfolio at present
is known, and (c) projects are uncorrelated with securities. Using first these formulas
and then generalizing the result, the paper proves that the breakeven prices of a
mean-variance investor will converge, as risk tolerance goes to infinity, towards the
price that the CAPM would place on the project. This result is valid regardless of the
correlation of the project with market securities or other projects, as long as the
optimal project portfolio in the limit is the same with and without the project. If the
portfolios differ, the value of the project will converge towards the CAPM price of the
difference of portfolios with and without the project.
4.3 Valuation of Real Options An interesting extension to the breakeven price methodology is the valuation of
opportunities, especially that of real options. The term “real option” originates from
the fact that management’s flexibility to adapt later decisions to unexpected future
developments shares similarities with financial options (Dixit and Pindyck 1994,
Trigeorgis 1996, Copeland and Antikarov 2001, Black and Scholes 1973, Merton
1973, Hull 1999). For example, possibilities to expand production when the markets
are up, to abandon a project under bad market conditions, and to switch operations
to alternative production facilities can be seen as options embedded in a project.
Typically, the real options literature employs CCA to value real options, which
requires that project cash flows can be replicated using financial securities. When
replication is possible for all assets, markets are said to be complete. However, it can
be difficult to construct replicating portfolios for private projects in practice, especially
when the projects are developing innovative new products that do not resemble
existing market-traded assets. Therefore, it is relevant to examine how real options
could be priced in incomplete markets. Since real options of a project have
conventionally been valued in the presence of securities, which is a MAPS setting, it
is natural to consider the application of the breakeven price methodology to real
option valuation.
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Because a real option gives the investor an opportunity but not the obligation to take
an action, we need for the valuation of real options concepts that rely on comparing
the situations where the investor can and cannot take an action instead of does and
does not, as breakeven prices do. Such prices are called opportunity selling and
buying prices. Opportunity prices are always non-negative, because an opportunity
cannot lower the value of the investor’s portfolio. It is also straightforward to show
that the opportunity prices can be obtained by taking a maximum of zero and the
respective breakeven price. Since breakeven prices are consistent with CCA, also
opportunity prices have this property, and hence they can be regarded as a
generalization of the standard CCA real option valuation procedure to incomplete
markets.
5 RESULTS FROM NUMERICAL EXPERIMENTS Numerical experiments are conducted in all of the papers. Paper [I] carries out an
extensive numerical study on the computational performance of CPP models. These
experiments indicate that CPP models of realistic size can be solved in a reasonable
time. When solved as linear programming models, where integer variables are left
continuous, CPP models of about a hundred five-staged projects and several
hundreds of states can be solved in a reasonable time. Mixed integer programming
models with a couple of tens of three-staged projects and less than a hundred states
have usually an acceptable solution time. A similar computational experiment is
conducted in Paper [V], indicating that PRIME models can be solved in a relatively
short time using the PRIME Decisions software.
Also Papers [II] – [IV] conduct numerical experiments. These aim at demonstrating
the properties of breakeven prices in different contexts. Because it is not immediately
obvious how generalizable the results are, the conclusions are necessarily limited to
a rather general level. Overall, two main points can be highlighted.
5.1 Effect of Alternative Investment Opportunities The results from the numerical experiments indicate that alternative investment
opportunities, both projects and securities, do have a major impact on the value of a
project. Beginning from the setting where it is possible to invest only in the project
being valued, the experiments in Paper [II] show that introduction of new projects to
the set of available investment opportunities typically lowers project values. This is
understandable in view of the definition of breakeven prices: the value of the optimal
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18
portfolio without the project will increase, which lowers the value of the project. Only
in the case where the value of the optimal portfolio with the project increases more
than the portfolio without the project will the value of the project grow. This might be
the case when the project is negatively correlated with other projects.
Similarly, securities can also lower project values by increasing opportunity costs (i.e.
by increasing the value of the portfolio without the project), and they can also raise
them by providing better diversification of risk (i.e. by raising the value of the portfolio
with the project). The total effect depends on the relative magnitude of these two
phenomena. We also see in the numerical experiments that, as predicted by the
related proposition in Paper [II], when a project has a replicating portfolio, its value is
consistent with the project’s CCA value at all risk levels.
5.2 Pricing Behavior as a Function of Risk Tolerance Another set of insights from numerical experiments is related to the behavior of
breakeven prices when the investor’s risk tolerance increases. First, the experiments
show that, when securities are not available, project values can rise non-
monotonically as the risk tolerance is increased. This is because, without securities,
opportunity costs are imposed in a lumpy manner, and the price of a project is
determined by the projects that fit into the portfolio (limited by the risk constraint) with
and without the project. This may result in a lumpy up and down movement in
breakeven prices.
In contrast, when securities are available, project values change monotonically by
risk tolerance, because securities can be bought in a continuous manner. However,
the project values can either rise or decrease by risk tolerance, depending on the
correlation of the project with the rest of the mixed asset portfolio. Indeed, it is the
correlation with the rest of the mixed asset portfolio that determines the limit
behavior, not that with the security portfolio only. Therefore, one cannot use the
project’s beta to make estimates about the sign of change in the project value when
the investor’s risk tolerance increases.
The limit behavior of the breakeven prices is also examined in the numerical
experiments. As predicted by the related propositions in Paper [II], it is observed that
the breakeven prices for a mean-variance investor converge towards the CAPM price
of the project as the investor’s risk tolerance goes to infinity. Similarly, we see in
Paper [III] that when an expected utility maximizer becomes less risk averse, project
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values approach values that are close to the projects’ CAPM prices. Indeed, in
neither case do the prices converge towards the values given by a risk-neutral
investor. Finally, it is also observed, as expected, that when the investor becomes
increasingly averse to risk or ambiguity, project values approach the values given by
a maximin investor, the investor using Wald’s (1950) maximin criterion.
6 IMPLICATIONS AND FUTURE RESEARCH DIRECTIONS The methodology developed in this dissertation provides a way to calculate
theoretically justifiable values for risky projects and portfolios of risky projects so that
the investor’s risk preferences, the opportunity costs of alternative investment
opportunities, and project interactions are properly accounted for. In particular, it is
now possible to determine the theoretically appropriate discount rate for a risky
project, even when the project is owned by a private company or when there are
synergies between projects. In addition, the methodology makes it possible to solve
project portfolio selection problems where risks are managed using both
diversification and staged decision making. Uncertainties can also be modeled with
relative accuracy, because the models can be formulated using linear programming,
which permits the use of a large number of states.
This dissertation opens up avenues for further work both in practical and theoretical
areas. On the practical side, applications of the methodology are called for. For
example, the analysis of oil field investments appears to be a promising area for the
present methodology, because many oil companies possess portfolios of oil fields
and the associated uncertainties are mostly related to external sources. Another
interesting application area is the valuation of collateralized debt obligations (CDOs),
which are portfolios of assets with a possibly complex tranche structure. In addition,
in order to ease the use of the methodology in practice, the development of a
dedicated software application to solve CPP models is called for.
On the theoretical side, several extensions of the methodology can be made. It is, for
example, possible to include transactions costs and capital gains tax into a CPP
MAPS model by using several resource types, each indicating the number of specific
assets held, and state-specific decision trees that implement the trading decisions for
each asset bought in the associated state. Further work is also needed in the
modeling of synergies and follow-up project structure of R&D projects. For example,
some synergies can be modeled by describing each project as a sequence of tasks,
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20
each modeled as an individual project in CPP’s sense, that are interconnected so
that a task cannot be started before the task preceding it is finished. The synergy
then arises from that a single task is used in the task sequence of several projects.
In general terms, the dissertation provides several new research topics for the related
academic disciples. In corporate finance, where the limitations of CAPM discount
rates have hopefully become more apparent than what they were previously, we
need more research to develop procedures to value projects of public companies
when all shareholders are not CAPM investors. This is relevant, for instance, with
partly state-owned companies, because in order to maintain the necessary voting
power the state has to possess a large amount of shares in the company, and
therefore it cannot efficiently diversify its investment portfolio, as the CAPM requires.
A similar situation may arise also in other cases where the investor is interested both
in the financial return and the voting right provided by the share. For researchers of
operations research and management science, the present work shows that decision
trees, real options, and project portfolio selection models can be profitably combined
together into a unified framework, where project values can be determined using
inverse optimization. I am confident that these three project selection methodologies
and inverse optimization have much to offer to each other and in terms of future
research possibilities.
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Forthcoming in Operations Research
i
!"$#%& %' () * +-,.0/ 213 456,
798;:<:>=?A@CBEDF8HGIBJBJK<:• LNM D9OQP8RSK
Systems Analysis Laboratory, Helsinki University of Technology,
Otakaari 1M, P.O. Box 1100, 02015 HUT, Finland
[email protected], [email protected]
Abstract: Methods for selecting a research and development (R&D) project
portfolio have attracted considerable interest among practitioners and aca-
demics. This notwithstanding, the industrial uptake of these methods has
remained limited, partly due to the difficulties of capturing relevant concerns
in R&D portfolio management. Motivated by these difficulties, we develop
Contingent Portfolio Programming (CPP) which extends earlier approaches in
that it (i) uses states of nature to capture exogenous uncertainties, (ii) mod-
els resources through dynamic state variables, and (iii) provides guidance for
the selection of an optimal project portfolio that is compatible with the deci-
sion maker’s risk attitude. Although CPP is presented here in the context of
R&D project portfolios, it is applicable to a variety of investment problems
where the dynamics and interactions of investment opportunities must be
accounted for.
Keywords: Research and development, project selection; Decision analysis,
theory; Programming, linear, applications
27
1
T UQV3WQXZY\[$]_^WQ`aYbV
The selection of research and development (R&D) projects has attracted considerable in-
terest in the literatures on technology management and operations research (OR). These
projects involve many characteristics – such as uncertainties and interdependent resource
constraints – that are potentially amenable to analysis by OR techniques. Indeed, there
exists a variety of related methods, ranging from scoring methods such as value trees
(Keeney and Raiffa 1976, French 1986) to optimization models (see, e.g., Gear and Lockett
1973, Heidenberger 1996, Ghasemzadeh et al. 1999) and dynamic programming methods
such as decision trees and real options (French 1986, Dixit and Pindyck 1994, Smith and
Nau 1995, Trigeorgis 1996). Yet, despite the plethora of methodological approaches, these
methods have not enjoyed widespread industrial use, possibly due to the difficulties of
capturing the full range of phenomena that are relevant to the problem of selecting and
managing R&D projects.
Building on the literatures on decision analysis, R&D management, and portfolio optimi-
zation, we develop Contingent Portfolio Programming (CPP) as a modeling framework which
accommodates most of the characteristics that are relevant to the selection of risky pro-
jects. In CPP, projects are regarded as risky investment opportunities that consume and
produce several resources over multiple time periods. The staged nature of R&D projects
is captured through project-specific decision trees (cf. Gear and Lockett 1973) which sup-
port managerial flexibility by allowing the decision maker (DM) to take stepwise decisions
on each project in view of most recent information (Trigeorgis 1996). Uncertainties, on the
other hand, are modeled through a state tree in the spirit of stochastic programming (see,
e.g., Birge and Louveaux 1997).
While CPP permits a wide range of risk attitudes, we focus on a class of objective func-
tions that are a combination of a mean-risk model (Markowitz 1959, 1987) and a multi-
attribute value function (Keeney and Raiffa 1976). In particular, we consider two objective
functions which lead to linear programming models, permitting the solution of relatively
large-scale project portfolios. The first one is a mean-lower semi-absolute deviation model,
28
2
(mean-LSAD model; Ogryczak and Ruszsynski 1999) which is a special case of generalized
disappointment models with a standard measure of risk (Jia and Dyer’s 1996, Jia et al.
2001). The second is a mean-expected downside risk model (mean-EDR model; Eppen et
al. 1989) which is consistent with expected utility theory (Fishburn 1977).
The rest of the paper is structured as follows. §2 provides a brief overview of earlier ap-
proaches and §3 presents an introductory example. A formal development of CPP is given
in §4, followed by an analysis of computational complexity in §5. §6 concludes the paper
with suggestions for future research directions.
c defXhgi`aj$Xlk)m\m_XZY\e^Anojp
Several methods for the selection of R&D projects have been developed over the past few
decades (for a review, see Martino 1995 and Henriksen and Traynor 1999). These methods
can be categorized into three aggregate groups: (1) scoring models, (2) optimization mod-
els, and (3) dynamic programming models. Among these, the two latter groups are more
relevant to CPP, although some ideas of scoring models (e.g., consideration of multiple at-
tributes) are also included in CPP.
Optimization models for project selection can be viewed as extensions of standard capital
budgeting models (see, e.g., Luenberger 1998). These models capture project interdepend-
encies and resources constraints, but they do not usually address uncertainties associ-
ated with the projects’ outcomes, which makes it impossible to attach risk measures to
project portfolios. There are some approaches based on utility functions, fuzzy set theory,
and chance-constraints, but the resulting models are problematic as they make restrictive
assumptions about the nature of uncertainty or the DM’s risk preferences.
Stochastic optimization models analogous to R&D portfolio selection models have ap-
peared in investment planning as well as asset and liability management (e.g., Birge and
Louveaux 1997, pp. 20–28, and Mulvey et al. 2000). These two problem contexts share
similarities with the selection of R&D projects in that (i) the DM seeks to maximize the
value of a portfolio of risky assets in a multi-periodic setting and (ii) there are several asset
categories which parallel the multiple resource types consumed and produced by R&D
29
3
projects. A key difference, however, is that in financial optimization the (dis)investment
decisions are unconstrained quantities that do not restrict the DM’s future decision op-
portunities (e.g., security trading). In contrast, R&D project selection involves “go /no go”-
style decisions where the “go”-decision leads to later project management decisions while
the “no go”-decision terminates the project without offering further decision opportunities.
The staged nature of R&D projects has motivated the development of dynamic program-
ming approaches, most notably (1) decision trees based on decision analysis (see French
1986) and (2) real options (see, e.g., Dixit and Pindyck 1994, Trigeorgis 1996); for a com-
parative analysis of these two approaches, we refer to Smith and Nau (1995). Dynamic
programming methods capture the structure of consecutive decisions and uncertainties of
an R&D project, but they do not explicitly address projects interactions or resource con-
straints. In consequence, researchers have sought to link decision trees with portfolio
models (e.g., Heidenberger 1996 and Gear and Lockett 1973); this notwithstanding, the
resulting models have failed to capture many relevant phenomena in R&D portfolio selec-
tion, such as risk aversion and resource dynamics.
The ability to yield theoretically defensible discount rates is often stated as a major advan-
tage of the real options approach over decision trees (Trigeorgis 1996). However, this ap-
proach assumes that the cash flows from the project can be replicated with financial in-
struments for all states of nature, which may be unrealistic if the project results in inno-
vative products that are not similar to market-traded assets. Furthermore, much of the
real options literature employs continuous stochastic processes in the modeling of uncer-
tainties, whereas the uncertainties of R&D projects often relate to discrete events. These
features may make it difficult to use the real options approach in practical project selec-
tion problems.
30
4
q roXJj$gi`tsu`iV\e$XJvwdAx\e$s"m\gaj
In CPP, the DM makes decisions about which projects are started, when they are started,
what resources are allocated to them, and in what situations they are terminated or ex-
panded, among others. The decisions are subjected to relevant constraints (e.g., availabil-
ity of resources), and they influence the resource flows that are acquired from the project
portfolio.
For each project, the resource flows depend on the future states of nature. The resource
flows associated with a given portfolio management strategy are consequently uncertain,
which means that the final resource position at the end of the planning horizon is risky. It
is assumed that the DM seeks to maximize the utility (or equivalently, certainty equiva-
lent) of her final resource position.
For illustrative purposes, let us assume that the DM can invest in projects A and B in two
phases (see Figure 1). In period zero, she can start either one or both of the projects. If a
project is started, she can make an additional investment in period one, in which case the
project generates a positive cash flow in period two; otherwise, the project is terminated in
which case it yields no further cash flows. Any surplus that is not invested can be depos-
ited at an 8% risk-free interest rate. The initial budget is $9 million.
Uncertainties are captured through a state tree with seven states, of which two are asso-
ciated with period one and four with period two (see Figure 2). Taken together, Figures 1
and 2 correspond to the decision trees in Figures 3 and 4 where the projects’ cash flows
are shown as a function of the indexed action variables X (X = 1, if action is selected, X=0
if it is not). In Figures 5 and 6, these cash flows are shown as a function of action vari-
ables, while Figure 7 shows the cash flows from the entire portfolio. The two projects are
negatively correlated so that that if project A performs poorly, project B yields a high re-
turn, and vice versa.
31
5
Proj
ects
Start? Continue?Yes
No
Yes
No
Start? Continue?Yes
No
Yes
No
B
A
0 1 2 Time
s1
s2
s11
s12
s21
s22
0 1 2
Stat
es
Time
50%
50%
30%
70%
60%
40%
s0
Figure 1 Decisions for projects A and B Figure 2 States of nature
Start?
YesXASY
NoXASN
s1
Continue?YesXACY1
NoXACN1
YesXACY2
NoXACN2
s2
s11
s12
s21
s22
$20m
$0
$0
$0
$0
$0
$5m
$10m
$0XASY = “Start project A Yes”
XASN = “Start project A No”XACY = “Continue project A Yes”XACN = “Continue project A No”
s11
s12
s21
s22
0 1 2 Time
-$1m
-$3m
-$3m
Start?
YesXBSY
NoXBSN
s1
Continue?YesXBCY1
NoXBCN1
YesXBCY2
NoXBCN2
s2
s11
s12
s21
s22
$2.5m
$0
$0
$0
$0
$0
$25m
$1m
$10XBSY = “ Start project B Yes”
XBSN = “ Start project B No”
XBCY = “ Continue project B Yes”XBCN = “ Continue project B No”
s11
s12
s21
s22
0 1 2 Time
-$2m
-$2m
-$2m
Figure 3 Decision tree of project A Figure 4 Decision tree of project B
Based on Figure 7, resource constraints can now be written as
0921 0 =−+⋅−⋅− sBSYASY RSXX
008.123 1011 =−⋅+⋅−⋅− ssBCYACY RSRSXX
008.123 2022 =−⋅+⋅−⋅− ssBCYACY RSRSXX
008.15.220 11111 =−⋅+⋅+⋅ ssBCYACY RSRSXX
008.1110 12111 =−⋅+⋅+⋅ ssBCYACY RSRSXX
008.1255 21222 =−⋅+⋅+⋅ ssBCYACY RSRSXX
008.110 2222 =−⋅+⋅ ssBCY RSRSX ,
where RS’s are nonnegative real-valued variables that denote the resource surplus in each
state (i.e., resource position).
32
6
-1·XASY
-3·XACY1
-3·XACY2
5·XACY2
0
20·XACY1
10·XACY1s1
s2
s11
s12
s21
s22
0 1 2
Stat
es
Time
s0 -2·XBSY
-2·XBCY1
-2·XBCY2
25·XBCY2
2.5·XBCY1
1·XBCY1s1
s2
s11
s12
s21
s22
0 1 2
Stat
es
Time
s0
10·XBCY2
Figure 5 Cash flows of project A Figure 6 Cash flows of project B
-1·XASY +-2·XBSY
-3·XACY1 +-2·XBCY1
-3·XACY2 +-2·XBCY2
5·XACY2 +25·XBCY2
20·XACY1 +2.5·XBCY1
10·XACY1 +1·XBCY1
s1
s2
s11
s12
s21
s22
0 1 2
Stat
es
Time
s0
0 +10·XBCY2
Figure 7 Cash flows of the project portfolio
For the measurement of risk, a second set of constraints is developed by introducing de-
viation variables sV −∆ ( sV +∆ ) which indicate by how much the value of the DM’s resource
position in a specific terminal state falls short of (or exceeds, respectively) the expected
value of the resource position at the end of the planning horizon. These constraints are
0=∆+∆−− −+sss VVEVV , where sV denotes the value of the resource position in state s,
EV is the expected value of the resource position over all terminal states, and the sum
s sV V+ −−∆ + ∆ measures by how much sV differs from EV . Since there are no other re-
source types except money, EV is given by
33
7
11 12 21 2250% 30% 50% 70% 50% 40% 50% 60% ,s s s sEV RS RS RS RS= ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅
while the deviation constraint for state s11 is 0111111 =∆+∆−− −+sss VVEVRS . The con-
straints for other terminal states can be expressed similarly. Because positive realizations
of −∆ sV will be penalized in the objective function by a negative coefficient, only one of the
terms +∆ sV and −∆ sV can be positive.
Because continued investments in period 1 are possible only if the project was initially
started, the following consistency constraints apply (cf. Figures 3 and 4):
1
1
=+=+
BSNBSY
ASNASY
XX
XX
BSYBCNBCY
ASYACNACY
XXX
XXX
=+=+
11
11 BSYBCNBCY
ASYACNACY
XXX
XXX
=+=+
22
22
The action variables X’s are nonnegative integers; in fact, they are binary variables due to
the two leftmost consistency constraints.
The DM seeks to maximize the certainty equivalent of her terminal resource position. It is
assumed that this can be approximated in the mean-risk form by deducting a risk term
based on lower semi-absolute deviation (LSAD) from the expected resource position in pe-
riod 2 (see, e.g., Ogryczak and Ruszsynski 1999). For example, if the risk aversion coeffi-
cient for LSAD has been estimated at y = 0.5, the objective function is
Maximize CE = EV – 0.5 z LSAD =
11 12 21 2250% 30% 50% 70% 50% 40% 50% 60% 0.5s s s sRS RS RS RS⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ − ⋅
11 12 21 2250% 30% 50% 70% 50% 40% 50% 60%s s s sV V V V− − − − |
⋅ ⋅ ∆ + ⋅ ⋅∆ + ⋅ ⋅ ∆ + ⋅ ⋅∆ ~.
With this objective function, the optimal strategy is to start both projects, but to terminate
project A in period 1 if state s2 occurs and project B if state s1 occurs (i.e., decision vari-
ables XASY, XACY1, XACN2, XBSY, XBCN1, and XBCY2 are one while all other X’s are zero). In pe-
riod 2, the corresponding expected resource position is EV = $18.80 million and the LSAD
term is $2.95 million, which leads to the optimal CE value $18.80 – 0.5 × $ 2.95 = $17.33
million. In period 2, the resource position attains its lowest level in state s12 at $13.76 mil-
lion, well above the 1.082 × $9 ≈ $10.50 million obtained by depositing the initial budget at
the risk-free interest rate. Thus, even though both projects entail the risk of losing most of
the initial investment when evaluated in isolation, the optimal strategy yields a return that
surely exceeds the risk-free interest rate.
34
8
Assuming that the same risk-free interest rate is applied to all states, the present value of
the optimal project portfolio can be readily calculated. That is, by discounting the CE of
the final resource position at the 8 % risk-free interest rate and by deducting the initial
budget of $9 million from this value, the net present value of the portfolio is found to be
$5.85 million. The risk-adjusted discount rate ρ that accounts for both time and risk
preferences can be computed from 22 )1/()1/( frCEEV +=+ ρ , which gives
=−+= 1/)1( CEEVr fρ 1.08 18.80 /17.33 1 12.5%⋅ − ≈ .
;Af;a$Abb$A
The constraints and the objective function of a CPP model are defined by resource types,
the state tree, and project-specific decision trees. We first define these three concepts and
then discuss the constraints and the objective function in CPP.
$ ACN ¡¢
Resources are inputs and outputs that are either consumed or produced by projects. They
can be production factors (e.g., money, equipment), intangibles (e.g., intellectual property
rights), or other relevant assets that the DM may be interested in. A resource type is de-
noted by r and the set of all resource types by R .
$i£ ¤A¥¦¥-)§©¨ª¦¥NF
The time-state model of CPP is a state tree which represents the structure of future states
of nature. Each state prevails during one period within the planning horizon 0,...,T . The
set of states in period t is denoted by tS , and the set of all states is « Tt
tSS0=
= . The time
period of state Ss ∈ is denoted by t(s).
The state tree starts with a single base state 0s in period 0. Each state 1' ,0ts S t T−∈ < ≤ is
followed by at least one state ts S∈ . This relationship is modeled by the function
SSB →: which returns the unique (immediate) predecessor 1' −∈ tSs of state , 0,ts S t∈ >
(by convention, 0 0( )B s s= ). The n-th predecessor of ( )ts S t n∈ ≥ is defined recursively by
1( ) ( ( ))n nB s B B s−= , where 0 ( )B s s= . This function can be used to obtain the states on a
35
9
path from the base state 0s to state s . These states, together with state s , are contained
in ( ) ' | 0 such that ( ) 'B kS s s S k B s s= ∈ ∃ ≥ = .
States result through uncertain events (e.g., “markets went up”). The probability that state
( 0)ts S t∈ > obtains, subject to the assumption that its predecessor ( )B s has occurred, is
given by the conditional probability ( )( )B sp s . The base state 0s occurs with probability
one, i.e., 0( ) 1p s = . Unconditional probabilities for the other states ( 0)ts S t∈ > are com-
puted recursively from the equation ))(()()( )( sBpspsp sB ⋅= .
¬$i® ¯°²±>³Z´µ>¶E·
¸¹aº!¹t» ¼½>¾H¿ÁÀ¿ÄÃ!ÅÆÇÃÈ¿ÉÅHÊÀ
The DM takes decisions with regard to projects z ∈ Z . Following Gear and Lockett (1973),
the decision opportunities for each project are structured as a decision tree which consists
of decision points: that is, for each project z ∈ Z , there is a set of decision points zD such
that at decision point zDd ∈ , the DM chooses one of the actions in da A∈ . The decision
point at which action a can be taken is ( )d a . The first decision point of project z is the
base decision point 0zd .
At each decision point d , the DM has information about (i) what state Sds ∈)( prevails at
this point, and (ii) what actions she has taken earlier on with regard to project z , if any;
this information is conveyed by the action that immediately preceeds d. For all decisions
points zd D∈ other than the base decision point, this action, called the parent action of
d , is given by the function ( )ap d . It is assumed that the decision points form a consis-
tent tree so that each decision point has a unique parent action.
For each action a there is an action variable Xa that is equal to the number of times that
this action is selected at decision point ( )d a (e.g., 1 if the action is selected once; 0 if the
action is not selected). Apart from binary choices, action variables can be used to model
decisions that correspond to nonnegative integers or continuous real numbers.
A project management strategy zX is defined by the action variables that are associated
36
10
with the decision points zd D∈ of project z . A portfolio management strategy X is the
DM’s complete plan of action for all projects z ∈ Z and all states s S∈ .
ËÌaÍ!ÌaÎ Ï<ÐCÑJÒÈÓ<ÔtÕJÐÖH×SÒCØÑ
The project management strategy zX induces a resource flow ( , )rzRF szX of resource type
r in state s . Letting )(sc ra denote the flow of resource type r in state s due to action a ,
this flow is given by
:( ) ( )
( , ) ( )z dB
r rz a a
d D a As d S s
RF s c s X∈ ∈∈
= ⋅Ù Ù
zX ,
where the restriction in the summation of decision points ensures that actions influence
resource flows only in the current state and relevant future states. The aggregate resource
flow ( , )rRF sX in state s is obtained by adding the flows for all projects, i.e.,
:( ) ( )
( , ) ( , ) ( )z dB
r r rz a a
z Z z Z d D a As d S s
RF s RF s c s X∈ ∈ ∈ ∈
∈
= = ⋅Ú Ú Ú Ú
zX X .
For the time being, we assume that resource flows are linear in the action variables, which
means that interactions among actions (e.g., synergies) are not accounted for. In principle,
such interactions can be captured through cross-terms for pairs of action variables. In
this case, the aggregate resource flow becomes
,: :
( ) ( ) ( ) ( )
( , ) ( )z d z dB B
r ra a a a
z Z d D a A z Z d D a As d S s s d S s
RF s c s X X′
′ ′′ ′ ′∈ ∈ ∈ ∈ ∈ ∈
′∈ ∈
= ⋅ ⋅Û Û Û Û Û Û
X .
Ü$ÝaÜ Þ\ßNà!á¢âZã²äNåaàâ-á
The four main constraint types in CPP are (i) decision consistency constraints, (ii) resource
constraints, (iii) optional constraints, and (iv) deviation constraints.
æçèæçté êë>ìHíÁîÂíÄï!ð2ñï<ðCîÇíòîEóFë¢ð>ì-ôñï<ðCîóõtö<íèðó²î
The structure of decision points influences how many actions can be selected at a given
decision point d . For instance, if the parent action of d was selected once, the DM ar-
rives at d and chooses one of the actions in dA ; but if the parent action was not selected,
the DM does not arrive at d so that none of the actions at d can be selected. Thus, at
each decision point other than the base decision point, the number of selected actions is
the same as the number of times that its parent action was selected. At the base decision
point 0zd , the DM usually chooses one of the alternative actions; however, multiple project
37
11
instantiations (i.e., exact copies) can be modeled by allowing the number of selected ac-
tions to be greater than one, say Lz.
The above requirements imply the following decision consistency constraints
0dz
a za A
X L z∈
= ∀ ∈÷
Z (1)
0( ) \
d
a ap d z za A
X X d D d z∈
= ∀ ∈ ∀ ∈ø
Z . (2)
Unconstrained quantitative decisions, which do not form a decision tree, can be modeled
by omitting the constraints (1) and (2). For example, decisions in securities trading can be
modeled by associating with each state a continuous unconstrained action variable which
is equal to the amount of transactions in that state.
ùúèùúaû ü<ýCþJÿJýÿCþ þ
Resource constraints can be employed to ensure that there is a nonnegative stock of re-
sources in each state. They are modeled through resource surpluses that would remain in
state Ss ∈ , if the DM were to choose the portfolio strategy X . The surplus of resource
type r in state Ss ∈ is
( ) ( )
( ) ( , )
( ) ( , )
r rrs r r r r
B s s B s
b s RF sRS
b s RF s RSα →
+
=
+ + ⋅ X
X
0
0
if
if
ss
ss
≠=
,
where )(sb r is the initial endowment of resource r in state Ss ∈ and rssB →)(α is the rate at
which the surplus in state ( )B s is transferred to s . Initial endowments cannot be influ-
enced by the DM and they do not entail any costs. The transfer rate may depend on the
resource type and the state: it may be equal to (1 + risk-free interest rate) for money, while
the rate for perishable goods may be zero.
The resource surplus variables rsRS are continuous, and for a given portfolio strategy X ,
they can be solved using the following resource constraints
00 0( , ) ( )r r rsRF s RS b s r R− = − ∀ ∈X
( ) ( ) 0( , ) ( ) \ r r r r rB s s B s sRF s RS RS b s s S s r Rα →+ ⋅ − = − ∀ ∈ ∀ ∈X .
Resource surplus variables are usually constrained to non-negative values. However,
negative values may also be permitted in order to allow for the possibility to borrow funds,
for instance.
38
12
!#"%$'&%%!)(*"+!,(
Optional constraints include any other constraints that may apply. The two most com-
monly discussed optional constraints in the literature are prerequisite constraints, which
define relations between follow-up and prerequisite projects, and project version con-
straints which model alternative versions of a project. Additional examples on optional
constraints are given by Ghasemzadeh et al. (1999). Note that in CPP project versions can
be modeled with a single project in which the choice among the project versions is made
at the base decision point so that no dedicated constraints for this purpose are needed.
-/.10 2436587:9#;=<?>7A@CBEDC9F;=<GED
The DM seeks to maximize the utility of the terminal resource position, viz.
max [ ]U X ,
where U is the DM’s preference functional and X is the value of the resource position in
period T. Under expected utility theory, the preference functional is given by
[ ] [ ]( )=U X E u X , where u is the DM’s von Neumann-Morgenstern utility function.
HIJCIK LMN#O#PQRSQOSTOQO)N#UVOXWZY[)O\^]
Because nonlinear utility functions can entail computational challenges in the context of
large-scale portfolios, we focus on two special cases of the objective function that (i) im-
plement a reasonable model of risk aversion and (ii) lead to a linear programming model.
In both cases, we assume that the DM’s preference functional can be approximated as a
mean-risk model; such models have been widely used in the field of portfolio selection (see
Markowitz 1952, 1959).
We also assume that the value of the final resource position is (i) additive with regard to
resource types and (ii) linear with respect to the amount of surplus of each resource (see,
e.g., Keeney and Raiffa 1976). These two latter assumptions are not overly restrictive as
linear pricing is widely employed in financial modeling. For a given state, the total value of
resource surpluses can be obtained by associating state-dependent weights rsw with each
resource type. These weights can be interpreted as unit prices so that the monetary value
of resource surpluses in state s is given by
39
13
( ) ,r rs s s
r R
V w RS∈
= ⋅_
sRS
where rsw is the unit price of resource type r in state s ∈ ST. The expected (monetary) value
of the terminal resource position is thus given by
( ) ( ) ( ) ( )T T
r rT s s s
s S s S r R
EV p s V p s w RS∈ ∈ ∈
= ⋅ = ⋅ ⋅` ` `
T sRS RS , (3)
where TRS is a vector of all rsRS ’s for which r ∈ R and s ∈ ST.
abcCbd eCf^gh4ikjmlngpoqj)g
Several dispersion statistics have been proposed in the literature on portfolio selection.
Markowitz (1952, 1959) suggests the use of variance and semivariance for the selection of
securities. Expected downside risk (EDR) has been employed in capacity planning (Eppen
et al. 1989), while absolute deviation has been applied in real-time stock market analysis
(Konno and Yamazaki 1991). These last two measures are linear, which make them at-
tractive for large-scale portfolio selection problems. Also, these measures lead to mean-
risk models that are consistent with the first and second degrees of stochastic dominance
(FSD and SSD; Levy 1992, Ogryczak and Ruszczynski 1999, Fishburn 1977), which sug-
gests that the resulting models are theoretically reasonable.
In particular, we employ expected downside risk (EDR; Eppen et al. 1989, Fishburn 1977)
and lower semi-absolute deviation (LSAD) as measures of risk. Specifically, EDR is given
by
all : all :
[ ] ( ) | | ( )( )x x
x t x t
EDR X p x x t p x t x
< <
= − = −r r
,
where ( )p x is the probability mass function of X and t is the target value from which de-
viations are computed. When the target value is equal to the expression t = µX = E[X], we
have
all : all :
[ ] ( ) | | ( )( )
X X
X Xx x
x x
LSAD X p x x p x x
µ µ
µ µ< <
= − = −s s
.
Both measures can be calculated from deviation constraints as follows. Let +∆ sV and −∆ sV
be nonnegative deviation variables which measure how much the total value of the re-
source surpluses in state s ∈ ST (i.e., sV ) differs from the risk measure’s target value t.
For EDR, these variables satisfy the equations
( ) 0 ,+ −− − ∆ + ∆ = ∀ ∈s s s TV t V V s SsRS (4)
40
14
where only one of the variables +∆ sV and −∆ sV can be positive, because −∆ sV has a nega-
tive coefficient in the objective function. The EDR of the value of the final resource posi-
tion is given by the sum
( )T
ss S
p s V −
∈⋅ ∆
t. (5)
The LSAD measure can be computed by using t = ( )T TEV RS in (4) instead of a fixed target
value t. This leads to
( ) ( ) 0+ −− − ∆ + ∆ = ∀ ∈s T T s s TV EV V V s SsRS RS , (6)
whereafter the LSAD can be obtained from (5).
uvwCvx yz#+|~C^8yZ)mz
The objective function can now be stated in the mean-risk form
max ( ) ( )T TEV RP −−T TRS V , (7)
where EVT is defined by Equation (3), and ( )T TRP −∆V is given by
( ) ( )T
T ss S
RP p s Vλ− −
∈
= ⋅ ⋅ ∆
T
V , (8)
where deviation variables −∆ sV terms are obtained either from (4) (EDR) or (6) (LSAD).
When the mean-risk model gives a certainty equivalent for a random variable, like the
mean-LSAD model does, RPT can be interpreted as the DM’s risk premium. By substituting
Equations (3) and (8) into (7), the objective function becomes
max ( )T
r rs s s
s S r R
p s w RS Vλ −
∈ ∈
⋅ − ⋅ ∆
.
Importantly, the mean-EDR model falls within the scope of expected utility theory1 (Fish-
burn 1977), and can therefore be regarded as an acceptable model of risk aversion; in par-
ticular, it does not suffer from dynamic inconsistencies (Machina 1989), which may occur
with the mean-LSAD model and other non-expected utility models. On the other hand, the
mean-LSAD model can be motivated by its link to disappointment models and standard
measures of risk (Jia and Dyer 1996, Jia et al. 2001) and the properties of constant abso-
1 Since [ ] [ ] ( )all : all :
( ) ( ) ( )x x
x t x t
E X EDR X p x x t x p x xλ λ< ≥
− ⋅ = − − +
, the mean-EDR model is
equivalent to the utility function (1 ) ,
( ),
x t x tu x
x x tλ λ+ − <
=
≥ .
41
15
lute risk aversion and constant relative risk aversion (see French 1986): if the value of the
final resource position is subjected to a positive affine transformation, the certainty
equivalent undergoes a similar transformation so that CE[a⋅X + b] = a⋅CE[X] + b for con-
stants a > 0 and b. In expected utility theory, no utility function implies both risk aversion
and such a linear pricing property.
¡ £¢¤¦¥¨§ª©«¬¤ªE
®¯,° ±³²:´µE¶:·¸¹)µ
Although CPP is based on linear programming, the required computational effort may be-
come prohibitive if the model is very large. It is therefore instructive to examine the num-
ber of decision variables and constraints in a CPP model (see Tables 1 and 2). Here, ºZz
zDD∈
= is the set of all decision points and » »Zz Dd
dz
AA∈ ∈
= is the set of all actions.
Because deviation variables and resource surplus variables are continuous, there are at
most A integer variables in a CPP model. The number of integer variables can be reduced
by not constraining one of the action variables at each decision point to integer values.
Still, this variable can assume integer values only, because it is related to the other inte-
ger-valued actions at the same decision point through Equations (1) and (2). The upper
bound for the number of integer variables can thus be reduced to A D− .
Table 1 Number of Decision Variables Decision variable Number Type Action variables (X’s) |A| Typically integer Resource surplus variables (RS’s) |S|⋅|R| Continuous Deviation variables (∆V’s) 2⋅|ST| Continuous TOTAL |A| + |S|⋅|R| + 2⋅|ST|
Table 2 Number of Constraints Constraint Number Decision consistency constraints |D| Resource constraints |S|⋅|R| Deviation constraints |ST| Optional constraints O TOTAL |D| + |S|⋅|R| + |ST| + O
42
16
For example, a CPP model with three resource types, five time periods, a binary state tree
(where each state is split into two further states in the next period) and thirty projects
with four consecutive “go/no go” decisions leads to 15·30 = 450 integer action variables
and equally many continuous action variables, as well as 31·3 = 93 continuous resource
surplus variables and 2·16 = 32 continuous deviation variables (cf. Table 1). This leads to
a total of 450 integer variables and 575 continuous variables. The number of constraints
is 450 + 31·3 + 16 + 0 = 559. This model can be readily solved using standard techniques
of mixed integer programming (MIP).
In comparison, a conventional decision tree for the same portfolio selection problem would
contain an enormous number of decision and chance nodes. The first decision node of the
tree would entail 230 alternative decisions, one for each possible project portfolio, and each
of these decisions would lead to a binary chance node that resolves at the end of the first
period. At the start of the second period, there would be 230·2 decision nodes, each pre-
ceded by the earlier portfolio decision and the chance outcome. Assuming that these and
ensuing decision nodes entail at most 230 decision alternatives each, we obtain an upper
limit of 1 + 230·2 + (230·2)2 + (230·2)3 ¼ 1028 for the number of decision nodes. Apart from
being intractable, the resulting tree would still require that the sufficiency of resources is
verified at each decision node. On the other hand, building a separate conventional deci-
sion tree for each project would not support the consideration of project interactions or
the variability of portfolio returns. Nevertheless, in this case, each decision tree would
contain 15 decision nodes and 15 chance nodes, resulting in a total of 450 decision nodes
and equally many chance nodes.
A challenge with all state tree based approaches, including CPP, is that the size of the
state tree can become excessively large if the outcome of the project portfolio depends on a
large number of risk factors. This is often the case when the outcome of each project de-
pends on a project-specific risk factor, because the number of states then increases expo-
nentially with the number of projects. For example, in a setting where each project either
fails or succeeds and the success of each project is independent of that of other projects,
the number of corresponding terminal states becomes 2n, where n is the number of pro-
43
17
jects. With a sufficiently large n, this necessarily leads to a CPP model that cannot be
solved in a reasonable time.
The above situation does not arise if CPP is used to capture external uncertainties, such
as market risks and regulatory changes that are not influenced by the projects. This focus
on external uncertainties is natural in CPP, because the state tree is shared by all projects
and project decisions do not influence state probabilities. Hence, CPP may be particularly
suitable for scenario analysis where portfolio management strategies are optimized with
regard to a relatively small number of states.
½¾À¿ ÁÂZÃÅÄÆÈÇÉÈÇ=ÊÂEË:ÉÍÌÏÎÈÐÑÄCÒEÓÊÔÃÕÒEËÈÇÖ
The computational performance of CPP models was tested with a dedicated C++-
application that runs under Windows XP operation system using LP Solve 4.0.1.9 software
package for LP and MIP models (available at ftp://ftp.es.ele.tue.nl/pub/lp_solve/). The
optimizations were run on a laptop computer with 512 MB of memory and 1.06 GHz Pen-
tium III processor.
×CØÙCØÚ ÛÜÈÝSÞßáàâãÞ)äSåçæè'é%Þ#å,êÝ
In our numerical experiments, the number of projects, stages per project, time periods,
and resources varied as described in Tables 3 and 4. For each model type, 30 models with
randomized resource flows and state probabilities were generated. The timeout for the so-
lution algorithm was set to 20 minutes to ensure that the total computation time re-
mained reasonable even in cases where the median solution time was several minutes and
the worst case solution time could have been several hours.
___________________________________________________________________________________
INSERT TABLES 3 AND 4 AROUND HERE
___________________________________________________________________________________
In most cases, we used two resource types, (i) money with the risk-free interest rate of 5%
and (ii) a perishable capacity resource with a zero transfer rate. Any additional resource
types, if present, were perishable capacity resources. The weight (i.e., unit price) of capac-
ity resources in the objective function was 0 and that of money was 1. In the first period,
the initial budget for money was $2 million multiplied by the number of projects, while
44
18
initial monetary endowments in other periods were zero. In each period, initial endow-
ments for other resources were equal to one unit of resources multiplied by the number of
projects.
The state tree had a binary structure such that each non-terminal state was split into two
states in the next period. The probabilities for the terminal states were computed by gen-
erating real numbers from the uniform distribution over the unit interval and by normaliz-
ing the resulting numbers. The probabilities of the other states were aggregated from
those of the terminal states.
Project decision trees consisted of binary “go/no go”-decisions in two or more stages. The
first decision was taken in period 0, followed by the next decision in every consecutive pe-
riod up to the total number of stages. Each “go”-decision entailed an immediate cost. Each
cost was obtained by (i) deciding on the most likely value of the cost and (ii) by multiplying
this value with a random number from the (0,1)-lognormal distribution, implying a log-
normal distribution for all costs. The most likely values for costs were assumed to rise
linearly with time so that later stages were more expensive to carry out than earlier ones.
We used the value of $1 million for the first stage, implying a cost of $2 million of the sec-
ond stage, $3 million for the third stage, and so on. Costs for other resources were mod-
eled similarly, using a most likely value of 1 resource unit for the first stage and a linear
growth model.
The revenues began one period after the last “go”-decision. Similarly to costs, revenues
were randomized by first selecting a most likely value for the revenue and then multiply-
ing this number by a random number drawn from the (0,1)-lognormal distribution. For
each project, revenues were distributed evenly over time, in the sense that the most likely
revenue was the same for each period. The cumulative sum of the most likely values for
revenues over the time horizon was assumed to be 1.15 times the sum of most likely val-
ues for costs over the time horizon. For example, for a two-staged project the sum of the
most likely values for costs was $3 million and hence the sum of the most likely values for
revenues was $3.45 million. In a model with four periods, this would imply that the most
likely revenue in the last period and in the second last period was $1.725 million. Projects
45
19
did not yield inflows for other resources.
In most cases, a mean-LSAD model with the risk aversion coefficient 0.5λ = was em-
ployed. We also conducted experiments with the mean-EDR model with 0.5λ = and the
target value of 1.05Tb ⋅ , where b is the initial budget and T denotes the last time period.
These are indicated by the symbol ¤ in Tables 3 and 4.
×CØÙCØÙ ëÞ)ìêèíå,ì
Some CPP models were solved as MIP models (Table 4) and some as LP models where inte-
ger variables were left continuous (Table 3). The results for LP models suggest that CPP
models for realistic portfolio selection problems can be solved in a reasonable time, and
that relatively few projects involve action variables with non-integer values. For example,
CPP models with 250 3-staged projects, 2 resources, and 5 time periods (16 terminal
states) were solved in a median time of 46.7 s. Models with 50 5-staged projects, 2 re-
sources, and 9 time periods (256 terminal states) had a median solution time of 6 min 59
s. Typically, LP models with less than 6,000 variables and 3,000 constraints could be
solved within the 20-minute time-out limit. In most cases, the number of non-integer ac-
tion variables was small, about 3%–10% of all action variables.
In LP models, the possibility to borrow additional resources and the DM’s risk neutrality
led to (i) a lower number of fractional integer variables and (ii) a higher probability of at-
taining an integer solution (Table 3). In the presence of both assumptions, action variables
always assumed integer values, suggesting that fractional project management decisions
were either due to limited resources or the DM’s risk aversion.
Typically, MIP models were much more time-consuming to solve than LP models (Table 4):
for example, an MIP model with 30 3-staged projects, 2 resources, and 5 time periods
could be solved in a median time of 49.4 s, while the LP model took only 0.32 s to solve.
The mean and standard deviation of the solution time seemed to grow exponentially with
the number of integer variables, wherefore models with more than 350 integer variables
could not usually be solved within the 20-minute time-out limit. However, the possibility
to borrow resulted in significantly shorter solution times, as models with 100 3-staged
46
20
projects, 1 resource, and 5 time periods were solved in a median time of 3.335 s which is
not much more than the median solution time of 2.835 s for LP models. When both the
possibility to borrow and the assumption of risk neutrality were introduced, there was no
significant difference in the solution times of MIP and LP models.
î ïñðóòôòöõ/÷pøùõúkûýüÿþ ú ð þú
The CPP modeling framework presented in this paper is applicable to the portfolio man-
agement of correlated R&D projects and, more generally, to the analysis of investment
problems where the dynamics and interdependencies of risky investment opportunities
must be accounted for. This framework has several appealing characteristics, such as the
explicit consideration of resource dynamics and managerial flexibility. It also accommo-
dates a wide range of risk attitudes, including two risk averse preference models that lead
to linear CPP models.
Our simulations indicate that LP and MIP formulations for CPP models of realistic size can
be solved in a reasonable time. LP models of about a hundred five-staged projects and
several hundreds of states can be solved in a reasonable time by using a standard per-
sonal computer and a public domain C++ LP package. MIP models, on the other hand,
usually take much more time solve; MIP models with a couple of tens of three-staged pro-
jects and less than a hundred states have usually an acceptable solution time.
There are several avenues for further research. On the theoretical side, CPP needs to be
extended to settings where more complex resource dynamics must be accounted for (e.g.,
cost of storage, proactive management of multi-purpose resources) or where the DM’s ac-
tions influence the structure of the state tree. In terms of future applications, CPP seems
particularly useful in settings where separate decision trees for each project can be devel-
oped, but where the optimal decisions are interlinked by resource constraints and the
need for a portfolio management strategy which accounts for the DM’s risk attitude.
÷ /ú
Birge, J. R., F. Louveaux. 1997. Introduction to Stochastic Programming. Springer, New York, NY.
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21
Dixit, A. K., R. S. Pindyck. 1994. Investment Under Uncertainty. Princeton University Press, Prince-
ton, NJ.
Eppen, G. D., R. K. Martin, L. Schrage. 1989. A scenario approach to capacity planning. Opera-
tions Research 37(4) 517–527.
Fishburn, P. C. 1977. Mean-Risk Analysis with Risk Associated with Below-Target Returns. Amer.
Econ. Rev. 67(2) 116–126.
French, S. 1986. Decision Theory – An Introduction to the Mathematics of Rationality. Ellis Horwood
Limited, Chichester.
Gear, A. E., A. G. Lockett. 1973. A Dynamic Model of Some Multistage Aspects of Research and
Development Portfolios, IEEE Trans. Eng. Management EM20(1) 22–29.
Ghasemzadeh, F., N. Archer, P. Iyogun. 1999. A zero-one model for project portfolio selection and
scheduling. J. Oper. Res. Soc. 50(7) 745–755.
Heidenberger, K. 1996. Dynamic project selection and funding under risk: A decision tree based
MILP approach. European J. Oper. Res. 95(2) 284–298.
Henriksen, A., A. Traynor. 1999. A Practical R&D Project-Selection Scoring Tool. IEEE Trans. Eng.
Management 46(2) 158–170.
Jia, J. J. S. Dyer. 1996. A standard measure of risk and risk-value models. Management Science
42(12) 1691–1705.
Jia, J., J. S. Dyer, J. C. Butler. 2001. Generalized Disappointment Models. Journal of Risk and Un-
certainty 22(1) 59–78.
Keeney, R. L., H. Raiffa. 1976. Decisions with Multiple Objectives: Preferences and Value Tradeoffs.
John Wiley & Sons, New York, NY.
Konno, H., H. Yamazaki. 1991. Mean-Absolute Deviation Portfolio Optimization and Its Applica-
tions to the Tokyo Stock Market. Management Sci. 37(5) 519–531.
Levy, H. 1992. Stochastic Dominance and Expected Utility: Survey and Analysis. Management Sci-
ence 38(4) 555–593.
Luenberger, D. G. 1998. Investment Science. Oxford University Press, New York, NY.
Machina, M. 1989. Dynamic Consistency and Non-Expected Utility Models of Choice Under Risk.
Journal of Economic Literature 27(4) 1622–1668.
Markowitz, H. M. 1952. Portfolio Selection. J. Finance 7(1) 77–91.
–––––. 1959. Portfolio Selection: Efficient Diversification of Investments. Cowles Foundation, Yale.
Martino, J. P. 1995. Research and Development Project Selection. John Wiley & Sons, New York,
NY.
Mulvey, J. M., G. Gould, C. Morgan. 2000. An Asset and Liability Management Model for Towers
Perrin-Tillinghast. Interfaces 30(1) 96–114.
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Ogryczak, W., A. Ruszczynski. 1999. From stochastic dominance to mean-risk models: Semidevia-
tions as risk measures. European J. Oper. Res. 116(1) 33–50.
Smith, J. E., R. F. Nau. 1995. Valuing Risky Projects: Option Pricing Theory and Decision Analy-
sis. Management Sci. 41(5) 795–816.
Trigeorgis, L. 1996. Real Options: Managerial Flexibility and Strategy in Resource Allocation. MIT
Press, Massachussets.
úkþ û ò ú
This research has been supported by the Academy of Finland. We would like to thank area
editor James S. Dyer, an anonymous associate editor, and three anonymous referees for
their helpful comments on earlier versions of this paper. This work has also benefited from
discussions with Victor De Miguel, Jaakko Dietrich, Tommi Gustafsson, Toni Jarimo,
Juha Koljonen, Pertti Laininen, Harry M. Markowitz, and Juha Ojala. We would also like
to thank Kjell Eikland for his invaluable help with LP Solve software package. The finan-
cial support of the Finnish Cultural Foundation, and the Foundation of Jenny and Antti
Wihuri is gratefully acknowledged.
49
23
Table 3 Solution times for LP CPP models. 30 iterations were performed per setting. Solution time (s) No. of fractional integer variables
Projects
Stages
Period
s
Resou
rces
No. of
variab
les
No. of
constrain
ts
No. of in
teger va
riables
Exp
ectation
Stan
dard
devia
tion
25%
qu
artile
Med
ian
75%
qu
artile
Exp
ectation
Stan
dard
devia
tion
25%
qu
artile
Med
ian
75%
qu
artile
% in
teger solu
tions
20 3 5 2 374 218 140 0.184 0.028 0.161 0.18 0.201 13.67 7.24 8 13 18 3 % 30 3 5 2 514 288 210 0.332 0.064 0.3 0.32 0.35 14.07 6.91 10 14 16 0 % 60 3 5 2 934 498 420 1.208 0.232 0.992 1.152 1.312 15.03 6.57 10 13 19 0 % 100 3 5 2 1494 778 700 4.11 0.671 3.645 4.016 4.516 19.03 7.53 12 19 23 0 % 250 3 5 2 3594 1828 1750 47.541 7.042 43.032 46.738 50.923 23.53 5,75 19 23 26 0 % 100 4 5 2 3094 1578 1500 19.353 5.178 15.723 18.015 19.348 47.23 11.92 36 46 58 0 % 100 3 6 2 1590 858 700 7.272 1.352 6.299 6.9 8.322 24.6 7.89 19 24 29 0 % 100 4 6 2 3190 1658 1500 35.672 5.917 30.884 35.451 39.036 58.97 15.51 46 57 65 0 % 100 5 6 2 6390 3258 3100 285.30 77.21 214.92 278.89 335.79 126.9 34.99 100 120 140 0 % 100 4 9 2 4534 2778 1500 481.58 111.30 374.06 470.59 550.49 65.1 16.8 53 64 69 0 % 25 5 9 2 3084 2053 775 145.95 38.66 117.64 141.68 166.22 123.27 31.25 94 117 145 0 % 50 5 9 2 4634 2828 1550 413.54 100.53 318.94 418.52 452.75 152.53 38.19 115 146 181 0 % 30 4 6 5 1279 797 450 4.821 0.791 4.226 4.667 5.198 60.33 17.34 45 57 72 0 % 100 4 6 5 3379 1847 1500 73.554 25.105 53.196 64.153 77.932 65.8 18.72 49 61 81 0% 20 3 5 1 343 187 140 0.146 0.031 0.13 0.14 0.15 12.93 5.35 10 11 15 0 % 100 3 5 1 1463 747 700 3.386 0.598 2.884 3.195 3.745 19.6 6.16 14 19 21 0 % 100* 3 5 1 1463 747 700 3.167 1.017 2.674 2.835 3.004 6.47 4.64 2 7 10 20 % 100# 3 5 1 1463 747 700 2.869 0.402 2.574 2.804 3.145 11.6 3.46 9 11 13 0 % 100*# 3 5 1 1463 747 700 2.45 0.493 2.263 2.323 2.423 0 0 0 0 0 100 % 1000*# 3 5 1 14063 7047 7000 618.18 268.27 323.91 605.19 827.79 0 0 0 0 0 100 % 100¤ 3 5 1 1463 747 700 3.047 0.401 2.704 3.004 3.194 11.87 3.9 9 12 13 0 % 100¤ 3 5 2 1494 778 700 3.581 0.437 3.255 3.545 3.936 12.7 3.45 10 12 14 0 % *: borrowing is allowed, #: risk neutrality, ¤: mean-EDR model.
50
24
Table 4 Solution times for MIP CPP models. 30 iterations were performed per setting. Solution time (s)
Projects
Stages
Period
s
Resou
rces
No. of
variab
les
No. of
constrain
ts
No. of in
teger va
riables
Exp
ectation
Stan
dard
de-
viation
25%
qu
artile
Med
ian
75%
qu
artile
Tim
eouts
(1200 s)
10 3 5 2 234 148 70 1.626 1.861 0.471 0.952 1.552 0 15 3 5 2 304 183 105 7.807 22.855 0.971 1.622 4.076 0 20 3 5 2 374 218 140 10.371 12.612 2.484 7.05 12.147 0 25 3 5 2 444 253 175 41.841 58.301 6.35 22.442 43.943 0 30 3 5 2 514 288 210 128.988 196.7 7.28 49.391 130.498 0 35 3 5 2 584 324 245 242.688 319.226 30.263 99.162 277.799 1 40 3 5 2 654 358 280 406.089 389.605 90.05 285.33 484.627 4 20 3 5 1 343 187 140 7.394 11.98 0.912 2.203 9.383 0 20* 3 5 1 343 187 140 0.247 0.073 0.21 0.23 0.24 0 60* 3 5 1 903 467 420 1.573 0.648 1.141 1.272 1.773 0 100* 3 5 1 1463 747 700 4.31 2.402 2.904 3.355 4.727 0 200* 3 5 1 2863 1447 1400 26.062 16.252 14.661 18.917 28.772 0 100# 3 5 1 1463 747 700 > 1200 - > 1200 > 1200 > 1200 30 10¤ 3 5 2 234 148 70 1.626 3.15 0.32 0.691 1.412 0 *: borrowing is allowed, #: risk neutrality, ¤: mean-EDR model.
51
52
Helsinki University of Technology Systems Analysis Laboratory Research Reports E16, June 2005
PROJECT VALUATION IN MIXED ASSET PORTFOLIO
SELECTION
Janne Gustafsson Bert De Reyck Zeger Degraeve Ahti Salo
AB TEKNILLINEN KORKEAKOULUTEKNISKA HÖGSKOLANHELSINKI UNIVERSITY OF TECHNOLOGYTECHNISCHE UNIVERSITÄT HELSINKIUNIVERSITE DE TECHNOLOGIE D’HELSINKI
53
Title: Project Valuation in Mixed Asset Portfolio Selection Authors: Janne Gustafsson Cheyne Capital Management Limited Stornoway House, 13 Cleveland Row, London SW1A 1DH, U.K. [email protected] Bert De Reyck London Business School Regent's Park, London NW1 4SA, U.K. [email protected] Zeger Degraeve London Business School Regent's Park, London NW1 4SA, U.K. [email protected] Ahti Salo Systems Analysis Laboratory Helsinki University of Technology P.O. Box 1100, 02015 HUT, FINLAND [email protected] www.sal.hut.fi/Personnel/Homepages/AhtiS.html Date: June, 2005 Status: Systems Analysis Laboratory Research Reports E16 June 2005 Abstract: We examine the valuation of projects in a setting where an investor, such
as a private firm, can invest in a portfolio of projects as well as in securities in financial markets. In project valuation, it is important to consider all alternative investments opportunities, such as other projects and financial instruments, because each imposes an opportunity cost on the project under consideration. Therefore, conventional methods based on decision analysis may lead to biased estimates of project values, because they typically consider projects in isolation from other investment opportunities. On the other hand, options pricing analysis, which does consider financial investment opportunities, assumes that a project’s cash flows can be replicated using financial assets, which may be difficult in practice. The main contribution of this paper is the development of a procedure for project valuation in a setting where the firm can invest in a portfolio of projects and securities, but where exact replication of project cash flows is not necessarily possible. We consider both single-period and multi-period models, and show that the valuation procedure exhibits several important analytical properties. We also investigate the pricing behavior of mean-variance investors through a set of numerical experiments.
Keywords: finance, capital rationing, portfolio selection, decision analysis, decision
analysis, mixed integer programming.
54
1
1 Introduction In the literature on corporate finance (e.g. Brealey and Myers 2000, Chapter 9), it is proposed that
a project’s cash flows should be discounted at the rate of return of a security that is “equivalent in
risk” to the project. Under the Capital Asset Pricing Model (CAPM; Sharpe 1964, Lintner 1965),
two assets are equivalent in risk if they have the same beta. Therefore, an appropriate discount
rate for a project is [ ]( )f M fr r E r rβ= + − ,
with rf the risk-free interest rate, [ ]cov[ , ] / varp M Mr r rβ = the project’s beta, Mr the random
rate of return of the market portfolio, and pr the random rate of return of the project. The use of
the CAPM relies on the assumption that the firm is a public company maximizing its share price.
When the markets abide by the CAPM assumptions, the share price is maximized when all
projects with a rate of return higher than what is given by the CAPM are started (Rubinstein
1973). Yet, many companies make decisions about accepting and rejecting projects before their
initial public offering (IPO). At this stage, the CAPM assumptions are not typically satisfied,
because shareholders are often entrepreneurs and venture capitalists. Also, since there is no share
price to maximize, such firms are typically assumed to maximize the value of their project
portfolio instead. In such a setting, the firm is regarded as an individual investor with well-
defined preferences over risky assets.
Several methods have been proposed to value projects in a setting where the firm is an individual
investor, including decision analysis (French 1986, Clemen 1996) and options pricing analysis,
also referred to as contingent claims analysis and real options analysis (Dixit and Pindyck 1994,
Trigeorgis 1996). However, conventional methods based on decision analysis may lead to biased
estimates of project values, because they typically consider projects in isolation from other
investment opportunities, neglecting the opportunity costs that these investment opportunities
impose. Options pricing analysis does account for the effect that financial instruments have on
project values, but its applicability is limited, because replicating the project’s cash flows using
financial instruments may be difficult in practice. Smith and Nau (1995) have extended decision
analytic methods to include the possibility of trading securities. They do not, however, consider
other projects in the portfolio.
55
2
The main contribution of this paper is the development of a procedure for project valuation in a
setting, where the firm can invest in a portfolio of projects and securities, but where replication of
project cash flows with securities is not necessarily possible. We call this setting a mixed asset
portfolio selection (MAPS) setting. The valuation procedure is based on the concepts of
breakeven selling and buying prices (Luenberger 1998, Smith and Nau 1995, Raiffa 1968), which
are the investor’s own selling and buying prices for a project. We formulate the procedure for
both expected utility maximizers and mean-risk optimizers. Using the Contingent Portfolio
Programming (CPP) framework (Gustafsson and Salo 2005), we develop a general multi-period
MAPS model, which allows us to study a broad variety of risk-averse preference models in a
multi-period setting. This model accommodates Markowitz’s (1952) mean-variance (MV) model,
Konno and Yamazaki’s (1991) mean-absolute deviation (MAD) model, the mean-lower semi-
absolute deviation (MLSAD) model (Ogryczak and Ruszczynski 1999), and Fishburn’s (1977)
mean-risk models where risk is associated with deviations below a fixed target value. We also
prove several analytical properties for breakeven selling and buying prices.
MAPS-based project valuation is important in practice as well as in theory. On the one hand,
MAPS-based project valuation is required when a corporation or an individual makes investments
both in securities and projects, or other lumpy investment opportunities. For example, in June
2004, Microsoft had invested nearly $44.6 billion in short-term investments in financial markets
in addition to carrying out a large number of projects at the same time (Microsoft 2004). Also,
many investment banks invest in a portfolio of publicly traded securities and undertake uncertain
one-time endeavors, such as venture capital investments and organization of IPOs. On the other
hand, one of the fundamental problems in the literature on corporate finance is the valuation of a
single project while taking into account the opportunity costs imposed by securities (see e.g.
Brealey and Myers 2000). This is a MAPS setting that includes one project and several securities.
This paper is structured as follows. Section 2 introduces single-period MAPS models and
discusses different formulations of the portfolio selection problem. Section 3 presents the multi-
period MAPS model. In Section 4, we introduce the valuation concepts, and produce theoretical
results on the valuation properties of different types of investors. Section 5 demonstrates the
pricing properties of mean-variance investors through a series of numerical experiments. In
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3
Section 6, we summarize our findings and discuss the managerial implications of our results.
2 Mixed Asset Portfolio Selection In a MAPS problem, available investment opportunities are divided into two categories: (1)
securities, which can be bought and sold in any quantities, and (2) projects, lumpy all-or-nothing
type investments. From a technical point of view, the main difference between these two types of
investments is that the projects’ decision variables are binary, while those of the securities are
continuous. Another difference is that the cost, or price, of securities is determined by a market
equilibrium model, such as the CAPM, while a project’s investment cost is an endogenous
property of the project.
Portfolio selection models can be formulated either in terms of rates of return and portfolio
weights, like in Markowitz-type formulations, or by using a budget constraint, expressing the
initial wealth level, and maximizing the investor’s terminal wealth level. When properly applied,
both approaches yield identical results. We use the second approach with MAPS, because it is
more suitable to project portfolio selection. We first formulate single-period MAPS models,
where the investments are made at time 0 and the objective at time 1 is optimized. These models
will allow us to generate several insights and show how MAPS is related to Markowitz (1952)
and the CAPM. We then develop the multi-period MAPS model based on Contingent Portfolio
Programming (CPP, Gustafsson and Salo 2005).
Early portfolio selection formulations (Markowitz 1952) were bi-criteria decision problems
minimizing risk while setting a target for expectation. Later, the mean-variance model was
formulated in terms of expected utility theory (EUT) using a quadratic utility function. However,
there are no similar utility functions for most other risk measures, including the widely used
absolute deviation (Konno and Yamazaki 1991). Therefore, we distinguish between two classes
of portfolio selection models: (1) preference functional models, such as the expected utility
model, and (2) bi-criteria optimization models or mean-risk models.
A single-period MAPS model using a preference functional can be formulated as follows. Let
there be n risky securities, a risk-free asset (labeled as the 0th security), and m projects. Let the
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4
price of asset i at time 0 be 0iS and the corresponding (random) price at time 1 is 1
iS . The price
of the risk-free asset at time 0 is 1 and 1 fr+ at period 1, where rf is the risk-free interest rate. The
amounts of securities in the portfolio are denoted by , 0,...,ix i n= . The investment cost of
project k in time 0 is 0kC and the (random) cash flow at time 1 is 1
kC . The binary variable zk
indicates whether project k is started or not. The investor’s budget is b. We can then formulate a
MAPS model using a preference functional U as follows:
(i) maximize utility at time 1: 1 1
, 0 1max
n m
i i k ki k
U S x C z= =
⎡ ⎤+⎢ ⎥⎣ ⎦∑ ∑x z
subject to
(ii) budget constraint at time 0: 0 0
0 1
n m
i i k ki k
S x C z b= =
+ =∑ ∑
(iii) binary variables for projects: 0,1 1,...,kz k m∈ =
(iv) continuous variables for securities: free 0,...,ix i n= .
The budget constraint is formulated as equality, because in the presence of a risk-free asset all of
the budget will be expended at the optimum. In this model and throughout the paper, it is
assumed that there are no transaction costs or capital gains tax, and that the investor is able to
borrow and lend at the risk-free interest rate without limit. These assumptions can be relaxed
without introducing prohibitive complexities. For expected utility theory, the preference functional is [ ] [ ]( )U X E u X= , where u is the investor’s von Neumann-Morgenstern utility
function. When the investor is able to determine a certainty equivalent for any random variable X,
U can be expressed as a strictly increasing transformation of the investor’s certainty equivalent
operator CE. Hence, the objective can also be written as max [ ]CE X , which gives the total value
of the investor’s portfolio.
In addition to preference functional models, mean-risk models have been widely used in the
literature. We concentrate on these models, because much of the modern portfolio theory,
including the CAPM, is based on a mean-risk model, namely the Markowitz (1952) mean-
variance model. Table 1 describes three possible formulations for mean-risk models: risk
minimization, where risk is minimized for a given level of expectation (Luenberger 1998),
expected value maximization, where expectation is maximized for a given level of risk (Eppen et
al. 1989), and the additive formulation, where the weighted sum of mean and risk is maximized
(Yu 1985). In Table 1, ρ is the investor’s risk measure, µ is the minimum level for expectation,
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and R is the maximum level for risk. The parameters λ are tradeoff coefficients. The Markowitz
(1952) model can be understood as a special case of a mean-variance MAPS model where the
number of projects is zero.
Table 1. Formulations of the mean-risk optimization problem.
Objective Constraints Risk minimization 1 1
, 0 1min
n m
i i k ki k
S x C zρ= =
⎡ ⎤+⎢ ⎥
⎣ ⎦∑ ∑x z
1 1
0 1
n m
i i k ki k
E S x C z µ= =
⎡ ⎤+ ≥⎢ ⎥
⎣ ⎦∑ ∑
0 0
0 1
n m
i i k ki k
S x C z b= =
+ =∑ ∑
Expected value maximization
1 1
, 0 1max
n m
i i k ki k
E S x C z= =
⎡ ⎤+⎢ ⎥⎣ ⎦∑ ∑x z
1 1
0 1
n m
i i k ki k
S x C z Rρ= =
⎡ ⎤+ ≤⎢ ⎥⎣ ⎦∑ ∑
0 0
0 1
n m
i i k ki k
S x C z b= =
+ =∑ ∑
General additive 1 1 1 1
1 2, 0 1 0 1max
n m n m
i i k k i i k ki k i k
E S x C z S x C zλ λ ρ= = = =
⎡ ⎤ ⎡ ⎤⋅ + − ⋅ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑ ∑ ∑ ∑x z
0 0
0 1
n m
i i k ki k
S x C z b= =
+ =∑ ∑
Sharpe (1970) 1 1 1 1
, 0 1 0 1max
n m n m
i i k k i i k ki k i k
E S x C z S x C zλ ρ= = = =
⎡ ⎤ ⎡ ⎤⋅ + − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑ ∑ ∑ ∑x z
0 0
0 1
n m
i i k ki k
S x C z b= =
+ =∑ ∑
Ogryczak and Ruszczynski (1999)
1 1 1 1
, 0 1 0 1max
n m n m
i i k k i i k ki k i k
E S x C z S x C zλ ρ= = = =
⎡ ⎤ ⎡ ⎤+ − ⋅ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑ ∑ ∑ ∑x z
0 0
0 1
n m
i i k ki k
S x C z b= =
+ =∑ ∑
The general additive form can be turned into the model employed by Sharpe (1970) by dividing the additive form by 2λ , provided it is nonzero, and into the form used by Ogryczak and
Ruszczynski (1999) by dividing it by 1λ . Apart from expectation and risk constraints, the Karush-
Kuhn-Tucker (KKT) conditions of all of the formulations are identical and therefore will yield the
same efficient frontiers, as long as optimal solutions in the additive formulation remain bounded.
However, if limitless borrowing and shorting are allowed, the additive formulation can give
unbounded solutions unless a risk constraint is introduced. Yet, in this case the formulation
essentially coincides with expected value maximization in terms of KKT conditions.
Both risk minimization and expected value maximization models have advantages and
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6
disadvantages. Risk minimization requires the investor to set a minimum level for expectation, a
readily understandable quantity, while the interpretation of a maximum risk level in expected
value maximization may not always be clear. However, expected value maximization allows us to
include multiple risk constraints, e.g. one for variance, one for expected downside risk, and one
for the probability of getting an outcome below a particular level, so that the risk profile of the
portfolio can be matched to the risk preferences of the investor as closely as possible. Due to this
flexibility we focus on this formulation in the following.
3 A Multi-Period MAPS Model
3.1 Framework We develop a multi-period MAPS model using the Contingent Portfolio Programming (CPP)
framework developed by Gustafsson and Salo (2005). In this framework, uncertainties are
modeled using a state tree, representing the structure of future states of nature, as depicted in the
leftmost chart in Figure 1. The state tree need not be binomial or symmetric; it may also take the
form of a multinomial tree with different probability distributions in its branches. In each non-
terminal state, securities can be bought and sold in any, possibly fractional quantities.
0 1 2
Stat
es
50%
30%
70%
40%
60%
50%
15%
35%
20%
30%
ω0
ω1
ω2
ω12
ω21
ω22
ω11
Start? Continue?Yes
No
Yes
No
0 1 2Time
0
Start?Yes
No
2
Continue?Yes
No
Yes
No
1Time
ω11
ω12ω11
ω12ω21
ω22ω21
ω22
ω1
ω2
Time Figure 1. A state tree, decision sequence, and a decision tree for a project.
Projects are modeled using decision trees that span over the state tree. The two right-most charts
in Figure 1 describe how project decisions, when combined with the state tree, lead to project-
specific decision trees. The specific feature of these decision trees is that the chance nodes are
shared by all projects, since they are generated using the common state tree. Security trading is
implemented through state-specific trading variables, which are similar to the ones used in
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7
financial models of stochastic programming (e.g. Mulvey et al. 2000) and in Smith and Nau’s
(1995) method. Similar to a single-period MAPS model, the investor seeks either to maximize the
utility of the terminal wealth level, or the expectation of the terminal wealth level subject to a risk
constraint.
3.2 Model Components The two main components of the model are (i) states and (ii) the investor’s investment decisions,
which imply the cash flow structure of the model.
3.2.1 States
Let the planning horizon be 0,...,T . The set of states in period t is denoted by tΩ , and the set
of all states is 0
T
tt=
Ω = Ω∪ . The state tree starts with base state 0ω in period 0. Each non-terminal
state is followed by at least one state. This relationship is modeled by the function :B Ω→Ω
which returns the immediate predecessor of each state, except for the base state, for which the function gives 0 0( )B ω ω= . The probability of state ω , when ( )B ω has occurred, is given by
( ) ( )Bp ω ω . Unconditional probabilities for each state, except for the base state, can be computed
recursively from the equation ( )( ) ( ) ( ( ))Bp p p Bωω ω ω= ⋅ . The probability of the base state is
0( ) 1p ω = .
3.2.2 Investment Decisions
Let there be n securities available in financial markets. The amount of security i bought in state ω is indicated by trading variable ,ix ω , 1,...,i n= , ω∈Ω , and the price of security i in state ω
is denoted by ( )iS ω . Under the assumption that all securities are sold in the next period, the cash
flow implied by security i in state 0ω ω≠ is ( ), ( ) ,( )i i B iS x xω ωω ⋅ − . In base state 0ω , the cash
flow is 00 ,( )i iS x ωω− ⋅ .
The investor can invest privately in m projects. The decision opportunities for each project, 1,...,k m= , are structured as a decision tree, where we have a set of decision points kD and
function ( )ap d that gives the action leading to decision point 0\k kd D d∈ , where 0kd is the
first decision point of project k. Let dA be the set of actions that can be taken in decision point
kd D∈ . For each action a in dA , a binary action variable az indicates whether the action is
selected or not. Action variables at each decision point d are bound by the restriction that only
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one az , da A∈ , can be equal to one. The action in decision point d is chosen in state ( )dω .
For a project k, the vector of all action variables az relating to the project, denoted by kz , is
called the management strategy of project k. The vector of all action variables of all projects,
denoted by z, is the project portfolio management strategy. We call the pair ( , )x z , composed of
all trading and action variables, the mixed asset portfolio management strategy.
3.2.3 Cash Flows and Cash Surpluses
Let ( , )pkCF ωkz be the cash flow of project k in state ω with project management strategy kz .
When ( )aC ω is the cash flow in state ω implied by action a , this cash flow is given by
:( ) ( )
( , ) ( )k dB
pk a a
d D a Ad
CF C z
ω ω
ω ω∈ ∈∈Ω
= ⋅∑ ∑kz
where the restriction in the summation of the decision points guarantees that actions yield cash
flows only in the prevailing state and in the future states that can be reached from the prevailing state. The set ( )B ωΩ is defined as ( ) | 0 such that ( )B kk Bω ω ω ω′ ′Ω = ∈Ω ∃ ≥ = , where
1( ) ( ( ))n nB B Bω ω−= is the nth predecessor of ω ( 0 ( )B ω ω= ).
The cash flows from security i in state ω∈Ω are given by
( ),
, ( ) ,
( )( , )
( )i is
ii i B i
S xCF
S x xω
ω ω
ωω
ω
⎧− ⋅⎪= ⎨⋅ −⎪⎩
ix 0
0
if if ω ωω ω=
≠
Thus, the aggregate cash flow ( , , )CF ωx z in state ω∈Ω , obtained by summing up the cash
flows for all projects and securities, is
( )
1 1
, 01 :
( ) ( )
, ( ) , 01 :
( ) ( )
( , , ) ( , ) ( , )
( ) ( ) , if
( ) ( ) , if
k dB
k dB
n ms p
i ki k
n
i i a ai d D a A
d
n
i i B i a ai d D a A
d
CF CF CF
S x C z
S x x C z
ω
ω ω
ω ω
ω ω
ω ω ω
ω ω ω ω
ω ω ω ω
= =
= ∈ ∈∈Ω
= ∈ ∈∈Ω
= +
⎧ − ⋅ + ⋅ =⎪⎪⎪= ⎨⎪ ⋅ − + ⋅ ≠⎪⎪⎩
∑ ∑
∑ ∑ ∑
∑ ∑ ∑
i kx z x z
Together with the initial budget of each state, cash flows define cash surpluses that would result
in state ω∈Ω if the investor chose portfolio management strategy ( , )x z . Assuming that excess
cash is invested in the risk-free asset, the cash surplus in state ω∈Ω is given by
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9
( ) ( )
( ) ( , , )( ) ( , , ) (1 )B B
b CFCS
b CF r CSωω ω ω
ω ω
ω ω →
+⎧⎪= ⎨ + + + ⋅⎪⎩
x zx z
0
0
if if ω ωω ω=≠
,
where ( )b ω is the initial budget in state ω∈Ω and ( )Br ω ω→ is the short rate at which cash
accrues interest from state ( )B ω to ω . The cash surplus in a terminal state is the investor’s
terminal wealth level in that state.
3.3 Optimization Model When using a preference functional U, the objective function for the MAPS model can be written
as a function of cash surplus variables in the last time period, i.e. ( )
, ,maxU Tx z CS
CS
where CST denotes the vector of cash surplus variables related to period T. Under the risk-
constrained mean-risk model, the objective is to maximize the expectation of the investor’s
terminal wealth level:
, ,max ( )
T
p CSωω
ω∈Ω
⋅∑x z CS
Table 2. Multi-period MAPS models.
Preference functional model Mean-risk model
Objective function
( ), ,
maxU Tx z CSCS
, ,max ( )
T
p CSωω
ω∈Ω
⋅∑x z CS
Budget constraints
00 0( , , ) ( )CF CS bωω ω− = −x z
( ) ( ) 0( , , ) (1 ) ( ) \ B BCF r CS CS bω ω ω ωω ω ω ω→+ + ⋅ − = − ∀ ∈Ωx z
Decision consistency constraints
0
1 1,...,dk
aa A
z k m∈
= =∑
0( ) \ 1,...,
d
a ap d k ka A
z z d D d k m∈
= ∀ ∈ =∑
Risk constraints
( ), Rρ − +∆ ∆ ≤
( ) 0 TCSω ω ωτ ω+ −− − ∆ + ∆ = ∀ ∈ΩTCS
Variables
0,1 1,...,a d kz a A d D k m∈ ∀ ∈ ∀ ∈ =
, free 1,...,ix i nω ω∀ ∈Ω =
freeCSω ω∀ ∈Ω
0,1 1,...,a d kz a A d D k m∈ ∀ ∈ ∀ ∈ =
, free 1,...,ix i nω ω∀ ∈Ω =
freeCSω ω∀ ∈Ω
0 Tω ω−∆ ≥ ∀ ∈Ω
0 Tω ω+∆ ≥ ∀ ∈Ω
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10
Three types of constraints are imposed on the model: (i) budget constraints, (ii) decision
consistency constraints, and (iii) risk constraints, which apply to risk-constrained models only.
The multi-period MAPS model under both a preference functional and a mean-risk model is
given in Table 2. The constraints are explained in more detail in the following sections.
3.3.1 Budget Constraints
Budget constraints ensure that there is a nonnegative amount of cash in each state. They can be implemented using continuous cash surplus variables CSω , which measure the amount of cash in
state ω . These variables lead to the budget constraints
00 0( , , ) ( )CF CS bωω ω− = −x z
( ) ( ) 0( , , ) (1 ) ( ) \ B BCF r CS CS bω ω ω ωω ω ω ω→+ + ⋅ − = − ∀ ∈Ωx z
Note that if CSω is negative, the investor borrows money at the risk-free interest rate to cover a
funding shortage. Thus, CSω can also be regarded as a trading variable for the risk-free asset.
3.3.2 Decision Consistency Constraints
Decision consistency constraints implement the projects’ decision trees. They require that (i) at
each decision point reached, only one action is selected, and that (ii) at each decision point that is
not reached, no action is taken. Decision consistency constraints can be written as
0
1 1,...,dk
aa A
z k m∈
= =∑
0( ) \ 1,...,
d
a ap d k ka A
z z d D d k m∈
= ∀ ∈ =∑ ,
where the first constraint ensures that one action is selected in the first decision point, and the
second implements the above requirements for other decision points.
3.3.3 Risk Constraints
A risk-constrained model includes one or more risk constraints. We focus on the single constraint case. When ρ denotes the risk measure and R the risk tolerance, a risk constraint can be expressed
as ( ) Rρ ≤TCS .
In addition to variance (V), several other risk measures have been proposed in the literature on
portfolio selection. These include semivariance (Markowitz 1959), absolute deviation (Konno
and Yamazaki 1991), lower semi-absolute deviation (Ogryczak and Ruszczynski 1999), and their
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fixed target value counterparts (Fishburn 1977). Semivariance (SV), absolute deviation (AD) and
lower semi-absolute deviation (LSAD) are defined as
SV: ( )2 ( )X
X X Xx dF xµ
σ µ−∞
= −∫ ,
AD: ( )X X Xx dF xδ µ∞
−∞
= −∫ , and
LSAD: ( ) ( ) ( )X X
X X X X Xx dF x x dF xµ µ
δ µ µ−∞ −∞
= − = −∫ ∫ ,
where Xµ is the mean of random variable X and FX is the cumulative density function of X. The
fixed target value statistics are obtained by replacing Xµ by some constant target value τ. All of
these measures can be formulated in a MAPS program through deviation constraints. In general,
deviation constraints are expressed as ( ) 0 TCSω ω ωτ ω+ −− − ∆ + ∆ = ∀ ∈ΩTCS ,
where ( )τ TCS is a function defining the target value from which the deviations are calculated,
and ω+∆ and ω
−∆ are nonnegative deviation variables which measure how much the cash surplus
in state Tω∈Ω differs from the target value. For example, when the target value is the mean of
the terminal wealth level, the deviation constraints are written as
( ) 0 ,T
TCS p CSω ω ω ωω
ω ω+ −′
′∈Ω
′− − ∆ + ∆ = ∀ ∈Ω∑
Using these deviation variables, some common dispersion statistics can be written as follows: AD: ( ) ( )
T
p ω ωω
ω − +
∈Ω
⋅ ∆ + ∆∑ .
LSAD: ( )T
p ωω
ω −
∈Ω
⋅∆∑ .
V: 2( ) ( )
T
p ω ωω
ω − +
∈Ω
⋅ ∆ + ∆∑
SV: 2( ) ( )T
p ωω
ω −
∈Ω
⋅ ∆∑ .
The respective fixed-target value statistics can be obtained with the deviation constraints 0 TCSω ω ωτ ω+ −− − ∆ + ∆ = ∀ ∈Ω ,
where τ is the fixed target level. EDR, for example, can then be obtained from the sum
( )T
p ωω
ω −
∈Ω
⋅∆∑ .
It is also possible to set a limit for the (critical) probability of getting an outcome below some
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target level. Mathematically, the critical probability is defined as ( ) ( )crit
X XP F Xτ τ= < ,
where FX denotes the cumulative distribution function of random variable X and τ is the desired
target level. This risk measure can be implemented by using the following linear constraints:
TCS Mω ωτ ξ ω− ≤ ∀ ∈Ω , and
0,1 Tωξ ω∈ ∀ ∈Ω ,
where τ is the target value from which critical probability is calculated and M is some very large
number. Critical probability can then be constrained from above using the constraint
T
p Rω ωω
ξ∈Ω
≤∑ .
3.3.4 Other Constraints
Other types of constraints can also be included in the model, such as non-negativity restraints that
prevent short selling, upper bounds for the number of shares bought, and credit limit constraints
(Markowitz 1987). In the ensuing sections, we assume, for the sake of simplicity, that no such
additional constraints have been imposed.
4 Project Valuation
4.1 Breakeven Buying and Selling Prices Because we consider projects as non-tradable investment opportunities, there is no market price
that can be used to value the project. In such a situation, it is reasonable to define the value of the
project as the amount of money at present, time 0, that is equally desirable to the project. In a
portfolio context, this can be interpreted so that the investor is indifferent between the following
two portfolios: (A1) a portfolio with the project and (B1) a portfolio without the project and cash
equal to the value of the project. However, we may alternatively define the value of a project as
the indifference between the following two portfolios: (A2) a portfolio without the project and
(B2) a portfolio with the project and a debt equal to the value of the project. The project values
obtained in these two ways will not, in general, be the same. Analogously to Luenberger (1998),
Smith and Nau (1995), and Raiffa (1968), we call the first value the “breakeven selling price”
(BSP), as the portfolio comparison can be understood as a selling process, and the second type of
value the “breakeven buying price” (BBP).
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A crucial element in BSP and BBP is the definition of equal desirability of two different
portfolios. In preference functional models, the investor is, by definition, indifferent between two
portfolios whenever their utility scores are equal. In a mean-risk setting, an investor is indifferent
between two portfolios if the means and risks of the two portfolios are identical. More generally,
when the risks are considered as constraints, the investor is indifferent between two portfolios if
their expectations are equal and they both satisfy the risk constraints. Hence, we obtain the pairs of optimization problems shown in Table 3. Here, the variable *az is the action variable
associated with not starting project j; thus, * 0az = indicates that investor invests in the project,
whereas * 1az = means that the project is not started. In the case that the project starting decision
is a binary “go / no go” decision, we can alternatively impose the restrictions on the variable
indicating project starting instead. Finding a BSP and BBP is an inverse optimization problem: one has to find the values for the parameters s
jv and bjv so that the optimal value of the second
optimization problem matches the optimal value of the first problem.
4.2 Inverse Optimization Procedure Inverse optimization problems have recently attracted interest among researchers, such as Ahuja
and Orlin (2001). In an inverse optimization problem, the challenge is to find the values for a set
of parameters, typically a subset of all model parameters, that yields the desired optimal solution.
Inverse optimization problems can broadly be classified into two groups: (i) finding an optimal
value for the objective function, and (ii) finding a solution vector. Ahuja and Orlin (2001) discuss
problems of the second kind, whereas the problem of finding a BSP or BBP falls within the first
class.
In principle, the task of finding a BSP is equivalent to finding a root to the function ( ) ( )s s s
j s j sf v W v W− += − , where sW + is the optimal value of the portfolio optimization problem in
the status quo and ( )ss jW v− is the corresponding optimal value in the second setting as the
function of parameter sjv . Similarly, the BBP can be obtained by finding the root to the function
( ) ( )b b bj b b jf v W W v− += − . Note that these functions are increasing with respect to their
parameters. To solve such root-finding problems, we can use any of the usual root-finding
algorithms (see e.g. Belegundu and Chandrupatla 1999) such as the bisection method, the secant
method, and the false position method, which have the advantage that they do not require the
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knowledge of the functions’ derivatives, which are not typically known. If the first derivatives are
known, or when approximated numerically, we can use the Newton-Raphson method to obtain
quicker convergence.
4.3 General Analytical Properties
4.3.1 Sequential Consistency
Breakeven selling and buying prices are not, in general, equal to each other. While this
discrepancy is accepted as a general property of risk preferences in expected utility theory (Raiffa
1968), it may also seem to contradict the rationality of these valuation concepts. It can be argued
that if the investor were willing to sell a project at a lower price than at which he/she would be
prepared to buy it, the investor would create an arbitrage opportunity and lose an infinite amount
of money when another investor repeatedly bought the project at its selling price and sold it back
at the buying price. In a reverse situation where the investor’s selling price for a project is greater
than the respective buying price, the investor would be irrational in the sense that he/she would
not take advantage of an arbitrage opportunity – if such an opportunity existed – where it would
be possible to repeatedly buy the project at the investor’s buying price and sell it at a slightly
higher price which is below the investor’s breakeven selling price.
However, these arguments neglect the fact that the breakeven prices are affected by the budget
and that therefore these prices may change after obtaining the project’s selling price and after
paying its buying price. Indeed, it can be shown that in a sequential setting where the investor
first sells the project, adding the selling price to the budget, and then buys the project back, the
investor’s selling price and the respective (sequential) buying price are always equal to each
other. This observation is formalized as the following proposition and it holds for any preference
model accommodated by the model in Section 3. The proof is given in the Appendix.
PROPOSITION 1. A project’s breakeven selling (buying) price and its sequential breakeven
buying (selling) price are equal to each other.
4.3.2 Consistency with Contingent Claims Analysis
Option pricing analysis, or contingent claims analysis (CCA; Luenberger 1998, Brealey and
Myers 2000, Hull 1999), can be applied to value projects whenever the cash flows of a project
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15
can be replicated using financial instruments. According to CCA, the value of project j is given
by the market price of the replicating portfolio (or trading strategy) less the investment cost of the
project: 0 0
0
nCCAj j i i
iv C S x∗
=
= − +∑ ,
where ix∗ is the amount of security i in the replicating portfolio and 0jC is the investment cost of
the project in at time 0.
It is straightforward to show that, when CCA is applicable, i.e. if there exists a replicating
portfolio, then the breakeven buying and selling prices are equal to each other and yield the same result as CCA (Smith and Nau 1995). For example, when CCA
jv is positive, we know that any
investor will invest in the project, since it is possible to make money for sure by investing in the
project and shorting the replicating portfolio. Furthermore, any investor will start the project even when he/she is forced to pay a sum b
jv less than CCAjv to gain a license to invest in the project,
because it is now possible to gain CCA bj jv v− for sure. On the other hand, if b
jv is greater than CCAjv , the investor will be better off by investing the replicating portfolio instead and hence
he/she will not start the project. A similar reasoning applies to breakeven selling prices. These
observations are formalized in Proposition 2. The proof is obvious and hence omitted. Due to the
consistency with CCA, the breakeven prices can be regarded as a generalization of CCA to
incomplete markets.
PROPOSITION 2. If there is a replicating portfolio for a project, the breakeven selling price and
breakeven buying price are equal to each other and yield the same result as CCA.
4.3.3 Sequential Additivity
The BBP and BSP for a project depend on what other assets are in the portfolio. The value
obtained from breakeven prices is, in general, an added value, which is determined relative to the
situation without the project. When there are no other projects in the portfolio, or when we
remove them from the model before determining the value of the project, we speak of the
project’s isolated value. We define the respective values for a set of projects as the joint added
value and joint value. Figure 2 illustrates the relationship between these concepts.
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16
Isolated Value
Joint Value Joint Added Value
Added Value
No other projects Additional projects
Portf
olio
of
proj
ects
Sing
le
proj
ect
Num
ber
of p
roje
cts b
eing
val
ued
Number of other projects in the portfolio Figure 2. Different types of valuations for projects.
Isolated project values are, in general, non-additive; they do not sum up to the value of the project
portfolio composed of the same projects. However, in a sequential setting where the investor buys
the projects one after the other using the prevailing buying price at each time, the obtained project
values do add up to the joint value of the project portfolio. These prices are the projects’ added
values in a sequential buying process, where the budget is reduced by the buying price after each
step. We refer to these values as sequential added values. This sequential additivity property
holds regardless of the order in which the projects are bought. Individual projects can, however,
acquire different added values depending on the sequence in which they are bought. These
observations are proven by the following proposition. The proof is in the Appendix.
PROPOSITION 3. The breakeven buying (selling) prices of sequentially bought (sold) projects add
up to the breakeven buying (selling) price of the portfolio of the projects regardless of the order
in which the projects are bought (sold).
4.4 Analytical Properties of Mean-Risk Preference Models
4.4.1 Equality of Prices and Solution through a Pair of Optimization Problems
When the investor is a mean-risk optimizer and the risk measure is independent of an addition of
a constant to the portfolio, like variance, breakeven selling and buying prices are identical,
provided that unlimited borrowing and lending are allowed. In contrast, they are typically
different under expected utility theory and under most non-expected utility models. Also, for
mean-risk optimizers the breakeven prices can be computed directly by solving the expectations
of terminal wealth levels when the investor invests and does not invest in the project and
discounting their difference back to its present value at the risk-free interest rate. Therefore, with
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17
mean-risk models, there is no need to resort to possibly laborious inverse optimization. We
formalize these claims in Proposition 4. The proof is in the Appendix.
PROPOSITION 4. Let the investor be a mean-risk optimizer with risk measure ρ that satisfies [ ] [ ]X b Xρ ρ+ = for all random variables X and constants b. When limitless borrowing and
lending are allowed, the breakeven selling price and the breakeven buying price of any given
project are identical. Moreover, the prices are equal to
( )
1
0
1 ( )T
ft
W Wvr t
+ −
−
=
−=
+∏,
where W + is the expectation of the terminal wealth level when the investor invests in the project,
W − is the expectation of the terminal wealth level when the investor does not invest in the
project, and rf,(t) is the risk-free interest rate from time t to t+1.
Proposition 4 is striking both in its generality and simplicity, since neither the given valuation
formula, nor the equality of breakeven prices, generally holds under expected utility theory. The
results are also intuitively appealing: the investor places only a single price on a given project,
and this price is related to the investor’s terminal wealth levels with and without the project. This
indicates that mean-risk models exhibit a very reasonable type of pricing behavior.
4.4.2 Relationship to the Capital Asset Pricing Model
According to the CAPM, the market price of any asset is given by the certainty equivalent
formula (see, e.g., Luenberger 1998):
[ ][ ]1 1
0cov ,
1 var 1j j M M fCAPM
j jf M f
E C C r E r rv C
r r r
⎡ ⎤ ⎡ ⎤ −⎣ ⎦ ⎣ ⎦= − + − ⋅+ +
,
where 0jC is the investment cost of the asset, 1
jC is the random value of the asset at time 1, and
Mr is the random return of the market portfolio. For non-market-traded assets like projects, the
outcome of the formula can be interpreted as the price that the markets would give to the asset if
it were traded.
In general, breakeven selling and breakeven buying prices are inconsistent with CAPM
valuations, because some of the CAPM assumptions do not hold in a MAPS setting. For example,
private projects are not included in the derivation of the CAPM market equilibrium, yet they may
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18
be strongly correlated with the securities. Therefore, results that hold for the CAPM do not
necessarily hold here. In particular, the optimal financial portfolio for the investor is not
necessarily a combination of the risk-free asset and the market fund consisting of all securities in
proportions according to their market capitalization.
However, there is a special case when the investor’s optimal financial portfolio is what the
CAPM predicts, namely when the project portfolio is uncorrelated with the market securities and
the investor is a mean-variance optimizer. This is proven in Proposition 5. Nevertheless, even
though the optimal financial portfolio falls on the capital market line with and without the project
being valued, the CAPM valuation is incorrect even in this case, as an extra risk premium term
appears in the breakeven selling and buying prices, as shown in Propositions 6 and 7. Proofs for
the propositions can be found in the Appendix. Note that this is a particularly interesting special
case, because project outcomes do not usually depend on the fluctuations of financial markets.
Proposition 5 is to be understood as a mixed asset portfolio extension of the usual Separation
Theorem (Tobin 1958), applicable when projects are uncorrelated with securities.
PROPOSITION 5. If the investor is a mean-variance optimizer and projects are uncorrelated with
securities, the optimal financial portfolio is a combination of the market fund and the risk-free
asset.
PROPOSITION 6. If the conditions of Proposition 5 hold and the composition of the optimal
project portfolio is the same with and without project j, the breakeven selling price of project j is
[ ]1 1
1 1 10
1
1
var var
1 11 1var
m m
k k k kk k
j k j M fsj j Mn
f fi i
i
C z C zE C E r r
v C br rS x
= =≠
=
⎛ ⎞⎡ ⎤⎜ ⎟⎡ ⎤ ⎢ ⎥−⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦⎡ ⎤ ⎜ ⎟ −⎢ ⎥⎣ ⎦ ⎣ ⎦= − + − + −⎜ ⎟
+ +⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟
⎝ ⎠
∑ ∑
∑,
where Mr is the random rate of return of the market portfolio and bM is the amount of money
spent on securities in the status quo. That is, 00
1
m
M k kk
b b x C z=
= − −∑ .
PROPOSITION 7. If the conditions of Proposition 5 hold and the composition of the optimal
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19
project portfolio is the same with and without project j, the breakeven buying price of project j is
[ ]1 1
1 1 10
1
1
var var
1 11 1var
m m
k k k kk k
j k j M fbj j Mn
f fi i
i
C z C zE C E r r
v C br rS x
= =≠
=
⎛ ⎞⎡ ⎤⎜ ⎟⎡ ⎤ ⎢ ⎥−⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦⎡ ⎤ ⎜ ⎟ −⎢ ⎥⎣ ⎦ ⎣ ⎦= − + − − −⎜ ⎟
+ +⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟
⎝ ⎠
∑ ∑
∑,
where 00
1
m
M k kkk j
b b x C z=≠
= − −∑ .
Note that bM is computed here slightly differently than in Proposition 6 due to a different situation
in the status quo. Despite the differences in the formulas in Propositions 6 and 7, we know from
Proposition 4 that the formulas will give the same result. The following proposition generalizes
Propositions 6 and 7 to a case where the optimal project portfolio changes with and without the
examined project.
PROPOSITION 8. If the conditions of Proposition 5 hold, the difference in budget required to
make two portfolios with different projects (the decision variables of the first are denoted by , 1,...,kz k m= , and those of the second by , 1,...,kz k m′ = ) equally desirable is
( )( ) [ ]
1 11
1 10 1
1 1
1
var var1 1
1 1var
m mm
k k k kk k kmM fk kk
k k k Mnk f f
i ii
C z C zE C z z E r rC z z b
r rS x
= ==
=
=
⎛ ⎞⎡ ⎤ ⎡ ⎤′⎡ ⎤ −⎜ ⎟′− ⎢ ⎥ ⎢ ⎥⎣ ⎦ −⎣ ⎦ ⎣ ⎦⎜ ⎟′∆ = − − + − + −⎜ ⎟+ +⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠
∑ ∑∑∑
∑
,
where 00
1
m
M k kk
b b x C z=
= − −∑ .
As the CAPM predicts that the value of an uncorrelated project is equal to its NPV discounted at
the risk-free interest rate, this suggests that breakeven buying and selling prices are, in general,
different from CAPM values, even when the optimal financial portfolio falls on the capital market
line. However, the prices approach CAPM values when the amount of securities in the portfolio
increases (i.e. when the investor’s risk tolerance increases). This can be verified by multiplying
bM inside the parentheses. The last term in Proposition 8 now becomes
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20
[ ]1 1
1 12
var var
1var
m m
k k k kM fk k
M MfM
C z C z E r rb b
rr= =
⎛ ⎞⎡ ⎤ ⎡ ⎤′−⎜ ⎟⎢ ⎥ ⎢ ⎥ −⎣ ⎦ ⎣ ⎦⎜ ⎟+ −⎜ ⎟ +⎡ ⎤⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠
∑ ∑,
which goes to 0 as bM goes to infinity.
Convergence of the project values to CAPM prices as the investor’s risk tolerance goes to infinity
applies in general to all projects, regardless of their correlation with each other or with the
market. As long as the optimal project portfolio is the same with and without the project being
valued, every project’s BSP and BBP will converge towards the CAPM price of the project, as
shown in Proposition 9; otherwise, they will converge towards the CAPM value of a difference
portfolio ′z - z specified in Proposition 10. The proofs of the propositions are in the Appendix.
Note that these limit results are valid regardless of whether the markets actually abide by the
CAPM. If there is a discrepancy between securities’ expected returns and the capitalization
weights that a mean-variance investor would find appropriate, or vice versa, the market portfolio
in Propositions 5–10 is to be understood as the portfolio that would be the market portfolio if all
investors in the market were mean-variance optimizers.
PROPOSITION 9. If the investor is a mean-variance optimizer and the composition of the optimal
project portfolio is the same with and without project j, the project’s breakeven buying and
selling prices converge towards the project’s CAPM price
[ ]
[ ]1 10
cov ,
1 var 1j j M M fb s CAPM
j j j jf M f
E C C r E r rv v v C
r r r
⎡ ⎤ ⎡ ⎤ −⎣ ⎦ ⎣ ⎦= = = − + − ⋅+ +
as the investor’s risk tolerance goes to infinity.
PROPOSITION 10. If the investor is a mean-variance optimizer, a project’s breakeven buying and
selling prices converge towards
( )( ) ( )
[ ][ ]
11
10 1
1
cov ,
1 var 1
mm
k k k Mk k kmM fkb s CAPM k
k k kk f M f
C z z rE C z z E r rv v v C z z
r r r==
=
⎡ ⎤′⎡ ⎤ −′− ⎢ ⎥⎣ ⎦ −⎣ ⎦′= = = − − + − ⋅+ +
∑∑∑
as the investor’s risk tolerance goes to infinity. Here, , 1,...,kz k m= , denote the decision
variables for the optimal project portfolio with the project and , 1,...,kz k m′ = those without the
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project.
Apart from this limit behavior, another case where breakeven buying and selling prices coincide
with CAPM recommendations is when a replicating portfolio exists for the project. This is a
direct result of the fact that each of the three valuation methods is consistent with CCA.
4.5 Valuation of Opportunities and Real Options When valuing a project, we can either value an already started project or an opportunity to start a
project. The difference is that, although the value of a started project can be negative, that of an
opportunity to start a project is always non-negative, because a rational investor does not start a
project with a negative value. While BSP and BBP are appropriate for valuing started projects,
new valuation concepts are needed for valuing opportunities.
Since an opportunity entails the right but not the obligation to take an action, we need selling and
buying prices that rely on comparing the situations where the investor can and cannot invest in
the project, instead of does and does not. The lowest price at which the investor would be willing
to sell an opportunity to start a project can be obtained from the definition of the breakeven
selling price by removing the requirement to start the project in the status quo, i.e. by removing the constraint * 0az = in the top-left quadrant of Table 3. We define this price as the opportunity
selling price (OSP) of the project. Likewise, the opportunity buying price (OBP) of a project can be obtained be removing the starting requirement in the second setting, i.e. the equation * 0az =
in the bottom-right quadrant of Table 3. It is the highest price that the investor is willing to pay
for a license to start the project. Opportunity selling and buying prices have a lower bound of
zero; it is also straightforward to show that the opportunity prices can be computed by taking a
maximum of 0 and the respective breakeven price. Table 4 gives a summary of breakeven and
opportunity selling and buying prices.
Opportunity buying and selling prices can also be used to value real options (Trigeorgis 1996)
that may be contained within the project portfolio. These options result from management’s
flexibility to adapt later decisions to unexpected future developments. Typical examples include
possibilities to expand production when markets are up, to abandon a project under bad market
conditions, and to switch operations to alternative production facilities. Real options can be
valued much in the same way as opportunities to start projects. However, instead of comparing
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portfolio selection problems with and without a possibility to start a project, we will compare
portfolio selection problems with and without the real option. This can typically be implemented
by disallowing the investor from taking a particular action (e.g. expanding production) in the
setting where the real option is not present. Since breakeven prices are consistent with CCA, also
opportunity prices have this property, and hence they can be regarded as a generalization of the
standard CCA real option valuation procedure to incomplete markets.
Table 4. A summary of an investor’s buying and selling prices.
Valuation concept Idea Difference from breakeven prices
Properties
Breakeven buying and selling prices
Comparison of settings where the investor does and does not invest in the project.
- Breakeven buying price and selling price are in general different from each other.
Opportunity buying and selling prices
Comparison of settings where the investor can and cannot invest in the project.
The investor is not obliged to invest in the project.
Lower bound of 0. Equal to maximum of 0 and the respective breakeven price.
5 Numerical Experiments In this section, we demonstrate the use of a MAPS model in project valuation through a series of
numerical experiments. We employ the multi-period MAPS model but construct only a single-
period model, because this allows creating several insights, is easier to follow and replicate, and
because we can contrast the results with the CAPM, also a single-period model. We have also
conducted further experiments featuring real options and a multi-period MAPS model in two
working papers (De Reyck et al. 2004, and Gustafsson and Salo 2004). The present experiments
confirm some of the theoretical results obtained in the previous sections and also cast light on the
following issues:
1 How are project values affected by the presence of other projects in the portfolio?
2 How are project values affected by the opportunity to invest in securities?
3 How are project values affected by the investor’s risk tolerance (maximum level for
risk)?
4 How are project values related to the CAPM?
5 How does the presence of twin securities affect the value of a project?
5.1 Experimental Set-up The experimental set-up includes 8 equally likely states of nature, four projects, A, B, C, and D
(Table 5), and two securities, 1 and 2, which together constitute the market portfolio (Table 6).
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23
The setting can be extended to include more securities, but for the sake of simplicity we limit our
market portfolio to two assets only. This does not influence the generality of our results. Note that
projects C and D and security 2 are the same that Smith and Nau (1995) use in their examples.
Table 7 shows the correlation between the assets’ cash flows. Numbers in italic in Tables 5 and 6
are computed values. The risk-free interest rate is 8%.
Table 5. Projects.
Project A B C D Investment cost $80 $100 $104 $0.00 State 1 $150 $140 $180 $67.68 State 2 $150 $140 $180 $67.68 State 3 $150 $150 $60 $0.00 State 4 $150 $110 $60 $0.00 State 5 $50 $170 $180 $67.68 State 6 $50 $100 $180 $67.68 State 7 $50 $90 $60 $0.00 State 8 $50 $90 $60 $0.00 Expected outcome $100 $123.75 $120 $33.84 St. dev. of outcome $50 $28.26 $60 $33.84 Beta 0.000 0.431 1.637 3.683 Market price (NPV) $12.59 $11.33 -$4.00 $25.07
The market prices in Tables 5 and 6 are obtained by using the CAPM, where the expected rate of
return of the market portfolio has been chosen so that the price of security 2 is $20, the price used
by Smith and Nau (1995). Technically, this can be accomplished by including all of the assets
into the market portfolio, with projects having zero issued shares, and by finding the market
prices, excluding projects’ investment costs, that minimize the sum of squared errors (SSE)
between the rate of return given by the CAPM formula and the real expected rate of return, as
computed from the market price. The prices converge, resulting in SSE equal to zero. The desired
expected rate of return is 15.33%. The standard deviation of the market portfolio is then 35.32%.
In Table 6, the security capitalization weights represent the ratio between the market
capitalization of the security and that of the entire market. These weights are of interest, because
the CAPM predicts that an MV investor will always invest in a combination of the risk-free asset
and the market fund, where securities are present according to their capitalization weights.
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Table 6. Securities.
Security 1 2 Market price $39.56 $20.00 Shares issued 15,000,000 10,000,000 Capitalization weight 74.79% 25.21% State 1 $60 $36 State 2 $50 $36 State 3 $40 $12 State 4 $30 $12 State 5 $60 $36 State 6 $50 $36 State 7 $40 $12 State 8 $30 $12 Beta 0.79 1.64 Expected return 13.76% 20.00% St. dev. of return 28.26% 60.00%
Table 7. Correlation matrix.
A B C D 1 2 A 1 0.398 0 0 0 0 B 0.398 1 0.487 0.487 0.653 0.487 C 0 0.487 1 1 0.894 1 D 0 0.487 1 1 0.894 1 1 0 0.653 0.894 0.894 1 0.894 2 0 0.487 1 1 0.894 1
The experiment comprises several steps. We start with a setting where the investor can invest
only in the project being valued, which is then extended to cover a portfolio of all projects. We
then add securities 1 and 2. Apart from the very first case, we assume that limitless borrowing at
the risk-free rate and shorting of securities are allowed. As predicted by Proposition 4, the
breakeven selling and buying prices are identical in this case and hence we display both of the
prices in one entry. Unless otherwise noted, in each of the steps we use a budget of $500. The
maximum risk level is defined in terms of the rate of return of the portfolio. For example, a 50%
risk level implies that the maximum standard deviation for the terminal wealth level is $250.
5.2 Project Values without Securities In general, the value of a project depends on what would happen to the invested funds if the
project were not started. In the absence of other projects and securities, three cases described in
Table 8 are possible. These cases both provide a benchmark for project valuations obtained later,
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and show that even apparently small differences in available investment opportunities can make a
significant difference in the values of projects. In the first case, where unused funds are lost, the
net present value of the project is undefined, and hence we give the future value (FV) of the
project instead.
Table 8. Projects values when only the project being valued is available.
Project What happens to unused funds: A B C D Funds are lost (FV) $100.00 $123.75 $120.00 $33.84 No interest gained $20.00 $23.75 $16.00 $33.84 Deposited at risk-free interest rate $12.59 $14.58 $7.11 $31.33
Let us next assume that the investor is able to invest in all of the projects and in the risk-free
asset. The values of the projects are described in Table 9 as a function of the investor’s risk
tolerance. In the optimal policy, the investor starts the projects with positive value.
Table 9. Projects values when all of the projects are available.
Mean-Variance Risk level A B C D 15% -$1.99 $1.99 -$38.81 $18.74 20% $12.59 $14.58 -$20.06 $24.22 25% $5.48 $7.47 -$5.48 $24.22 30% and up $12.59 $14.58 $7.11 $31.33
-$60.00
-$40.00
-$20.00
$0.00
$20.00
$40.00
12% 15% 18% 21% 24% 27% 30%
Risk level
Proj
ect v
alue A
B
C
D
Figure 2. Project values for MV investor.
Table 9 and Figure 2 show that the project values behave rather erratically when the risk tolerance
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is varied. For example, at a 20% risk level, the MV values for projects A and B are identical to
the values they have in isolation ($12.59 and $14.58), but then they drop to $5.48 and $7.47 at
25% and rise back to the values in isolation for higher risk tolerance levels. Intuitively, one might
expect project values to rise as the risk constraint is relaxed, because more projects can be
included into the portfolio so that fewer projects impose an opportunity cost. At the limit, this is
correct: when the risk constraint is relaxed enough to allow the inclusion of all profitable projects,
the project values coincide with the prices on the last row of Table 8. For intermediate risk
values, however, another effect confounds this result: the value of a project depends on the
projects that fit into the portfolio when the project is started and when it is not. For example, the
decrease of the price of projects A and B when the risk constraint is increased from 20% to 25%
is due to the fact that, when the risk constraint is 25%, project C can be started if either of the
projects is not included into the portfolio, but this is not the case at 20%. So at a 25% risk
constraint, project C imposes an opportunity cost on these projects, but not at 20%, since the
project does not fit into the portfolio regardless the decision on projects A and B. This
counterintuive result is typical for project values in a MAPS setting.
5.3 Project Values with Securities We will now investigate what happens to the project values when securities are also available. In
this case, the risk constraint will always be binding as long as the rate of return of the optimal
security portfolio for the investor is higher than the risk-free interest rate: expectation of the
whole portfolio will be maximized by purchasing as much of the optimal security portfolio as
possible and by borrowing the necessary funds at the risk-free interest rate. When the investor
does not invest in the projects, i.e. in a pure CAPM setting, the optimal security portfolio for a
MV investor is to buy 75% of security 1 and 25% of security 2 at all risk levels. Note that these
weights are the capitalization weights of the securities in Table 6.
Columns 2–5 of Table 10 describe the project values assuming that the investor can only invest in
the project being valued and in securities. The presence of securities changes the project values
for two reasons. On the one hand, securities impose an additional opportunity cost, which lowers
project values. On the other hand, it is possible to hedge against project risks by buying
negatively correlated or shorting positively correlated securities, which increases the project
values.
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Table 10. Project values when securities 1 and 2 are available.
Single project available All projects available Risk level A B C D A B C D 15% $8.92 $10.78 -$4.00 $25.07 $6.36 $8.22 -$4.00 $25.07 20% $10.02 $10.92 -$4.00 $25.07 $8.61 $9.51 -$4.00 $25.07 30% $10.94 $11.06 -$4.00 $25.07 $10.14 $10.26 -$4.00 $25.07 40% $11.37 $11.13 -$4.00 $25.07 $10.80 $10.55 -$4.00 $25.07 50% $11.62 $11.17 -$4.00 $25.07 $11.17 $10.72 -$4.00 $25.07 60% $11.79 $11.19 -$4.00 $25.07 $11.42 $10.82 -$4.00 $25.07 70% $11.90 $11.21 -$4.00 $25.07 $11.59 $10.90 -$4.00 $25.07 80% $11.99 $11.23 -$4.00 $25.07 $11.72 $10.95 -$4.00 $25.07 90% $12.06 $11.24 -$4.00 $25.07 $11.81 $11.00 -$4.00 $25.07 100% $12.11 $11.25 -$4.00 $25.07 $11.89 $11.03 -$4.00 $25.07 200% $12.35 $11.29 -$4.00 $25.07 $12.24 $11.18 -$4.00 $25.07 500% $12.50 $11.31 -$4.00 $25.07 $12.45 $11.27 -$4.00 $25.07 2000% $12.57 $11.32 -$4.00 $25.07 $12.56 $11.31 -$4.00 $25.07 10000% $12.59 $11.33 -$4.00 $25.07 $12.59 $11.33 -$4.00 $25.07
Since project A is uncorrelated with the market portfolio, we can use its values to verify
Propositions 6, 7, and 8. For example, let us calculate the breakeven buying price for project A at
50% risk level. The variance of the project portfolio with the project (including only project A) is
502 = 2500 and 0 without the project. The variance of the optimal security portfolio in the status
quo (no projects) is 2502 = 62500 and the amount of money spent on securities is $708.3. As the
market rate of return is 15.33%, we get from Proposition 7 that
100 2500 0.073380 1 1 708.3 12.59 0.0202 48.07 11.621.08 62500 1.08
bAv
⎛ ⎞= − + − − − = − ⋅ =⎜ ⎟⎜ ⎟
⎝ ⎠,
which coincides with the value in Table 10. Also, as predicted by Proposition 9, when the risk
tolerance approaches infinity, the project values converge to their CAPM prices, as shown in the
row “Market price” in Table 6. Intuitively, one may expect that the values would approach risk-
neutral values as the risk-constraint is being relaxed more and more, but this is not the case.
Notice that security 2 is a twin security for projects C and D. This explains why these projects are
priced constantly to their CCA value across all risk levels. To replicate the cash flows of project
C, one needs 5 shares of security 2, costing $100. Since the investment cost of project C is $104,
we obtain the value of –$4 for the project, regardless of the investor’s risk tolerance. The cash
flows of project D can be replicated by constructing a portfolio of 2.82 shares of security 2 and
borrowing of $31.33 at the risk-free interest rate. Therefore, the value of this project is
$0 2.82 $20 31.33 $1 $25.07− + ⋅ − ⋅ = .
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28
Let us next investigate how the presence of other projects, as well as that of securities, affects
project values. Columns 6–9 of Table 10 give the results. First, we observe that the values of
projects A and B still increase monotonically with increasing risk tolerance without any erratic
behavior. We see also that the values still converge to CAPM prices. Additionally, the values of
projects A and B decrease in comparison with the single-project case, because the optimal
security portfolio with these projects is less profitable than what it was with a single project only.
5.4 Summary of Experimental Results Table 11 summarizes our findings with respect to the research questions posed at the beginning of
this section. Further, we have results on the valuation of real options and multi-period projects in
two working papers, De Reyck et al. (2004) and Gustafsson and Salo (2004). The results indicate
that (i) the present approach can readily be extended to the valuation of real options in incomplete
markets, and that (ii) it will also be practically feasible in a multi-period setting. In addition, when
valuing a single project, the present approach will yield the same results as Smith and Nau’s
(1995) method, provided that the assumptions of Smith and Nau’s method are satisfied and the
investor’s preference models in both approaches are identical.
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29
Table 11. Summary of research questions.
Research question Answer Influence of other projects on project values Alternative projects can lower project values by imposing an opportunity cost.
Influence of securities on project values
Securities can both lower and raise project values. On the one hand, they impose an opportunity cost. On the other hand, they enable better diversification of risk.
Influence of risk tolerance on project values
When securities are not available, project values may rise non-monotonically as the risk tolerance rises. When securities are available, project values for MV investor may rise (or decrease, see De Reyck et al. 2004) monotonically by risk tolerance.
Relationship to CAPM prices For an MV investor, project values converge towards CAPM prices as the risk tolerance goes to infinity.
Influence of twin securities on project values
A twin security makes the MAPS project value consistent with the project’s CCA value at all risk levels, provided that limitless borrowing is allowed. As ordinary securities for other projects, twin securities also influence values of other projects.
6 Summary and Conclusions In this paper, we analyzed the valuation of private projects in a mixed asset portfolio selection
(MAPS) setting, where an investor can invest in a portfolio of projects as well as securities traded
in financial markets, but where the replication of project cash flows with financial securities is not
necessarily possible. We developed a valuation procedure based on the concepts of breakeven
selling and buying prices. This inverse optimization procedure requires the solution of portfolio
selection problems with and without the project being valued and finding a lump sum that makes
the investor indifferent between the two situations. To make the solution of these portfolio
selection problems possible in a multi-period setting, we developed a multi-period MAPS model
using the Contingent Portfolio Programming framework (Gustafsson and Salo 2005). Finally, we
produced several theoretical results relating to the analytical properties of breakeven prices, and
studied the pricing behavior of mean-variance (MV) investors through a set of numerical
experiments.
Our theoretical results indicate that the breakeven prices are, in general, consistent valuation
measures, exhibiting sequential consistency, consistency with contingent claims analysis, and
sequential additivity. The results also show that MV investors have several notable pricing
properties. First, when limitless borrowing and lending are allowed, breakeven buying and selling
prices are identical, which is not the case, in general, under expected utility theory (Smith and
Nau 1995, Raiffa 1968). Also, the prices can be computed by solving the investor’s terminal
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30
wealth level when he/she invests and he/she does not invest in the project and by discounting the
difference back to its present value at the risk-free interest rate. In addition, we derived analytical
formulas to calculate the breakeven prices for an MV investor when the optimal portfolio at
present is known and projects are uncorrelated with securities. Finally, we showed that the project
values given by an MV investor converge towards the projects’ CAPM prices as the investor’s
risk tolerance goes to infinity.
Overall, our results suggest that alternative investment opportunities have a significant impact on
the value of a project. Therefore, valuation of projects in isolation may potentially lead to biased
estimates of the values of projects. This emphasizes the argument, which appears for example in
the real options literature, that it is crucial to also consider financial instruments in project
valuation. However, it is also important to recognize other projects in the portfolio.
Managers can draw several conclusions from our analysis. First, it is of central importance to
clearly recognize the real investment alternatives to the projects. The presence of the possibility
to borrow or short, or in general to invest in financial markets, may significantly influence the
value of a project. Second, project values are non-additive. To obtain the value of a portfolio of
two or more projects, it is necessary to calculate the terminal wealth levels when the investor
starts and does not start the projects. This can be accomplished by solving the appropriate
portfolio selection problems. Third, projects influence, in general, the optimal financial portfolio
for the investor, so that when the projects are started, the optimal financial portfolio does not
typically fall on the capital market line. Therefore, the allocation of funds to projects and
securities separately may potentially lead to suboptimally diversified portfolios.
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––––––. 2005. Contingent Portfolio Programming for the Management of Risky Projects.
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Appendix PROOF OF PROPOSITION 1: Let the breakeven selling price for the project be s
jv . The time-0
budget used to determine the sequential buying price is then sjb v+ in the status quo and
s sbj jb v v+ − in the second setting, where sb
jv is the sequential buying price. Let s sW W− += be the
optimal value for the optimization problems with the breakeven selling price and sb sbW W− += be
the optimal values for the optimization problems with the sequential buying price. By definition,
sbW − = sW − and therefore also sbW + = sW + . Thus, it follows that s sbj jb v v b+ − = , or s sb
j jv v= . A
similar reasoning shows the equality between the breakeven buying price and the respective
sequential selling price. Q.E.D.
PROOF OF PROPOSITION 3: Let us begin with the breakeven buying price and a setting where the
portfolio does not include projects and the budget is b. Suppose that there are n projects that the
investor buys sequentially. Let us denote the optimal value for this MAPS problem by W0.
Suppose then that the investor buys a project, indexed by 1, at his or her breakeven buying price,
1bv . Let us denote the resulting optimal value for the MAPS problem by W1. By definition of the
breakeven buying price, W0 = W1. Suppose then that the investor buys another project, indexed by 2, at his or her breakeven buying price, 2
bv . The initial budget is now 1bb v− , and after the second
project is bought, it is 1 2b bb v v− − . By definition, the optimal value for the resulting MAPS
problem W12 is equal to W1. Add then the rest of the projects in the same manner. The resulting budget in the last optimization problem, which includes all the projects, is 1 2 ...b b b
nb v v v− − − − .
We also know that W12…n = W12…n–1 = … = W0, and therefore, by definition, the breakeven buying price of the portfolio including all the projects is 12... 1 2 ...b b b b
n nv v v v= + + + . By re-indexing the
projects and using the above procedure, we can change the order in which the projects are added
to the portfolio. In doing so, the projects can obtain different values, but they still sum up to the
same joint value of the portfolio. Similar logic proves the proposition for breakeven selling
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33
prices. Q.E.D.
PROOF OF PROPOSITION 4: Let us first examine the breakeven selling price. Let us denote the optimal expectation for the given risk constraint in the status quo with s
SQµ and the one in the
second setting when 0sv = with s s sSS SQµ µ= + ∆ . The expectations can be made to match by
increasing the time-0 budget by sδ , which reduces the amount of money borrowed by the same
amount, leaving the effective budget unchanged. This increases the expectation of the portfolio
by ( )1
0
1 ( )T
sf
t
r tδ−
=
+∏ , where rf(t) is the risk-free interest rate from time t to t+1. This chain of
logic is valid on the condition that increasing the budget does not influence the composition of the
optimal risky portfolio, which is ensured when the risk measure is independent of an addition of a constant, i.e. it satisfies [ ] [ ]X b Xρ ρ+ = for all random variables X and constants b. Since we
desire to make the expectations in the status quo and in the second setting under a modified budget (denoted by s
SSµ ∗ ) equal, we have the equation
( )1
0
1 ( )T
s s s s s sSQ SS SQ SQ f
t
r tµ µ µ µ δ−
∗
=
= ⇔ = + ∆ + +∏ , from which it follows that
( )1
0
/ 1 ( )T
s sf
t
r tδ−
=
= −∆ +∏ , which is also the breakeven selling price of the project.
Let us then examine the breakeven buying price and denote the optimal expectation in the status quo by b
SQµ and the one in the second setting when 0bv = by b b bSS SQµ µ= + ∆ . As before, the
expectations can be matched by lowering the budget by bδ and increasing the amount of money
borrowed by the same amount. The expectation in the second setting is now
( )1
0
1 ( )T
b b b bSS SQ f
t
r tµ µ δ−
∗
=
= + ∆ − +∏ . By requiring that
(1 )b b b b b bSQ SS SQ SQ frµ µ µ µ δ∗= ⇔ = + ∆ − + , we get ( )
1
0
/ 1 ( )T
b bf
t
r tδ−
=
= ∆ +∏ , which is also the
breakeven buying price of the project.
Finally, we observe that s b
SS SQµ µ= and b sSS SQµ µ= , from which it follows that
s b s s b b s bSS SQ SQ SSµ µ µ µ= ⇔ +∆ = −∆ ⇔ ∆ = −∆ . Hence, .s bδ δ= To obtain the valuation
formula, let us denote bSS Wµ += and b
SQ Wµ −= , and we get the formula directly from the
equation for the breakeven buying price. Q.E.D.
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34
PROOF OF PROPOSITION 5: Since projects are uncorrelated with securities, we know that
1 1 1 1
0 1 0 1var var var
n m n m
i i k k i i k ki k i k
S x C z S x C z= = = =
⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ = +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦∑ ∑ ∑ ∑ and hence the risk constraint can be
rewritten as 1 1
0 1var var
n m
i i k ki k
S x R C z= =
⎡ ⎤ ⎡ ⎤≤ −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑ ∑ . Similarly, the objective can rewritten as
1 1 1 1
0 1 0 1max max
n m n m
i i k k i i k ki k i k
E S x C z E S x E C z= = = =
⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ ⇔ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦∑ ∑ ∑ ∑ . For any project portfolio with
fixed , 1,...,kz k m∗ = , the optimal financial portfolio will be acquired by solving an optimization
problem 1
0max
n
i ii
E S x a=
⎡ ⎤ +⎢ ⎥⎣ ⎦∑ subject to 0 0
0 1
n m
i i k ki k
S x C z b∗
= =
+ =∑ ∑ and 1
0var
n
i ii
S x R∗
=
⎡ ⎤ ≤⎢ ⎥⎣ ⎦∑ ,
where 1
1
m
k kk
a E C z∗=
⎡ ⎤= ⎢ ⎥⎣ ⎦∑ is a constant and 1
1var
m
k kk
R R C z∗ ∗
=
⎡ ⎤= − ⎢ ⎥⎣ ⎦∑ is a constant. The parameter
b does not influence the xi’s that yield the maximum to the problem, and R* is just an adjusted
maximum risk level for the investor. Thus, for any given project portfolio, the optimal financial
portfolio is obtained by the solution of a usual mean-variance problem. We know from the
Separation Theorem (Tobin 1958) that a mean-variance investor will invest in the combination of
the market fund and the risk-free asset. Q.E.D.
PROOF OF PROPOSITIONS 6, 7, AND 8: Let us prove first Proposition 8. Let the project portfolio
in the status quo be 1
1
m
k kk
P C z=
= ∑ and the security portfolio 1
1
n
i ii
M S x=
= ∑ . The respective
portfolios in the second setting are 1
1
m
k kk
P C z=
′ ′= ∑ and 1
1
n
i ii
M S x=
′ ′= ∑ . The amount of money
lent in the status quo is denoted by 0x and the respective amount in the second setting by 0x′ . By
Proposition 4 we know that the portfolio weights in the security portfolios M and M’ are identical, i.e., that / , 1,...,i ix x a i n′ = = , where a is the positive ratio of funds invested in risky securities in
the second setting and in the status quo. That is, 0
01
00
1
m
k kkm
k kk
b x C za
b x C z
=
=
′ ′ ′− −=
− −
∑
∑, (A1)
where b′ is the budget in the second setting. This implies that M aM′ = . Let us require next
that
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35
0 0
0 0
(1 ) (1 )
var (1 ) var (1 )
f f
f f
E r x M P E r x M P
r x M P r x M P
′ ′ ′⎡ ⎤ ⎡ ⎤+ + + = + + +⎣ ⎦ ⎣ ⎦′ ′ ′⎡ ⎤ ⎡ ⎤+ + + = + + +⎣ ⎦ ⎣ ⎦
By using the relationship M aM′ = , the equivalence of expectations yields us [ ] [ ] [ ] [ ]0 0(1 ) (1 )f fr x E M E P r x aE M E P′ ′+ + + = + + +
[ ] [ ] [ ]0 0
(1 )1 f
a E M E P E Px x
r′− + −
′⇒ = ++
(A2)
Similarly, from the equivalence of variances we obtain [ ] [ ] [ ] [ ]2var var var varM P a M P′+ = +
[ ] [ ][ ]
var var1
varP P
aM
′−⇒ = ± + (A3)
The minus sign alternative can be dropped, because we know that a has to be positive. Let us
define b b′∆ = − . By the definition of a in (A1), we get
0 00 0
1 1
m m
k k k kk k
b b a b x C z x C z b= =
⎛ ⎞′ ′ ′∆ = − = − − + + −⎜ ⎟⎝ ⎠
∑ ∑ .
By substituting (A2) into this, we obtain [ ] [ ] [ ]0 0
0 01 1
(1 )1
m m
k k k kk kf
a E M E P E Pb b a b x C z x C z b
r= =
′− + −⎛ ⎞′ ′∆ = − = − − + + + −⎜ ⎟ +⎝ ⎠∑ ∑ .
By denoting 00
1
m
M k kk
b b x C z=
= − −∑ and (1 )M MM r b= + , we get
[ ]( ) [ ] [ ] 00
1
(1 ) 11
mM M
M k kkf
a E r B E P E Pab x C z b
r =
′− + + −′∆ = + + + −
+ ∑ .
Let us then add and subtract 0
1
m
k kk
C z=∑ from the right-hand side of the equation to obtain
00
1
m
M k kk
b b x C z=
= − −∑ on the right-hand side. Thus we get
[ ]( ) [ ] [ ] 0 0
1 1
(1 ) 1( 1)
1
m mM M
M k k k kk kf
a E r b E P E Pa b C z C z
r = =
′− + + −′∆ = − + + −
+ ∑ ∑ .
By adding and subtracting rf inside [ ]( )1 ME r+ we are able to eliminate the term (a – 1)bM from
the beginning and obtain [ ]( ) [ ] [ ] 0 0
1 1
(1 )1
m mM f M
k k k kk kf
a E r r b E P E PC z C z
r = =
′− − + −′∆ = + −
+ ∑ ∑ .
By denoting [ ] 1
1
m
k kk
E P E C z=
⎡ ⎤= ⋅⎣ ⎦∑ and [ ] 1
1
m
k kk
E P E C z=
⎡ ⎤′ ′= ⋅⎣ ⎦∑ and by rearranging the terms,
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36
we get
[ ]
1
0 1
1
( )( ) ( 1)
1 1
m
k k kmM fk
k k k Mk f f
E C z z E r rC z z a b
r r=
=
⎡ ⎤ ′⋅ −⎣ ⎦ −′∆ = − − + − −
+ +
∑∑ . (A4)
Finally, by substituting (A3) into (A4), we obtain the desired formula
[ ]1 11
1 10 1
1 1
1
var var( )( ) 1 1
1 1var
m mm
k k k kk k kmM fk kk
k k k Mnk f f
i ii
C z C zE C z z E r rC z z b
r rS x
= ==
=
=
⎛ ⎞⎡ ⎤ ⎡ ⎤′⎡ ⎤ −⎜ ⎟′⋅ − ⎢ ⎥ ⎢ ⎥⎣ ⎦ −⎣ ⎦ ⎣ ⎦⎜ ⎟′∆ = − − + − + −⎜ ⎟+ +⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠
∑ ∑∑∑
∑.
Proposition 6 is obtained by assuming that k kz z k j′= ∀ ≠ and 1jz = and 0jz′ = . We get now
[ ]1 1
1 1 10
1
1
var var
1 11 1var
m m
k k k kk k
j k j M fjs j Mn
f fi i
i
C z C zE C E r r
v C br rS x
= =≠
=
⎛ ⎞⎡ ⎤⎜ ⎟⎡ ⎤ ⎢ ⎥−⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦⎡ ⎤ ⎜ ⎟ −⎢ ⎥⎣ ⎦ ⎣ ⎦= ∆ = − + − + −⎜ ⎟
+ +⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟
⎝ ⎠
∑ ∑
∑.
Similarly, Proposition 7 is obtained by assuming that k kz z k j′= ∀ ≠ and 0jz = and 1jz′ = . Then,
[ ]1 1
1 1 10
1
1
var var
1 11 1var
m m
k k k kk k
j k j M fjb j Mn
f fi i
i
C z C zE C E r r
v C br rS x
= =≠
=
⎛ ⎞⎡ ⎤⎜ ⎟⎡ ⎤ ⎢ ⎥−⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦⎡ ⎤ ⎜ ⎟ −⎢ ⎥⎣ ⎦ ⎣ ⎦= −∆ = − + − − −⎜ ⎟
+ +⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟
⎝ ⎠
∑ ∑
∑.
Q.E.D.
PROOF OF PROPOSITIONS 9 AND 10: Let us prove first Proposition 10. The notation is as in the
proof of Propositions 6–8. Observe first that when the investor’s risk tolerance increases, she
invests more and more in securities and the effect of projects on the total mixed asset portfolio
diminishes. At the limit, the investor invests in an infinitely large security portfolio whose
composition matches that of the market portfolio regardless of what projects there are in the
portfolio. Therefore, the portfolio weights in the security portfolios M and M’ are identical, so
that M aM′ = , where a is some constant. Now, we have by definition
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37
0 0var (1 ) var (1 )f fr x M P r x M P′ ′ ′⎡ ⎤ ⎡ ⎤+ + + = + + +⎣ ⎦ ⎣ ⎦ ,
from which we obtain [ ] [ ] [ ] [ ] [ ] [ ]2var var 2cov , var var 2 cov ,M P M P a M P a M P′ ′+ + = + +
[ ] [ ] [ ] [ ] [ ] [ ]2 var 2 cov , var var var 2cov , 0a M a M P M P P M P′ ′⇒ + − + − − = .
Solving this for a gives
[ ] [ ] [ ] [ ] [ ] [ ] [ ]( )[ ]
[ ][ ]
[ ][ ]
[ ] [ ] [ ] [ ]( ) [ ] [ ][ ]
2
22
2 2
2cov , 4cov , 4 var var var var 2cov ,
2 var
var var var var 2 var cov ,cov , cov ,var var var
M P M P M M P P M Pa
M
M M P P M M PM P M PM M M
′ ′ ′− ± + − + +⇒ =
′+ − +′ ′−= ± +
[ ][ ]
[ ][ ]
[ ] [ ][ ]
[ ][ ]
[ ][ ]
[ ][ ]
[ ][ ]
[ ][ ]
[ ][ ]
[ ] [ ][ ]
[ ][ ]
[ ][ ]
2, ,
2 2 2, , , ,
2
,
varcov , var var1 2
var var var
var var varcov , var var2 1
var var var var var
(cov ,1
var
M P M P
M P M P M P M P
M P
P stdev PM P P PM M M stdev M
P stdev P P PM P P PM M stdev M M M M
stdev PM PM stdev M
ρ ρ
ρ ρ ρ ρ
ρ ρ
′
′
′′ ′− −= ± + + +
′′ ′− −= ± + + − + +
⎛ ⎞′−= ± + +⎜ ⎟⎜ ⎟
⎝ ⎠
[ ][ ]
[ ][ ]
[ ]
[ ][ ]
[ ][ ]
[ ][ ]
2 2, ,
var
1) var ( 1) varvar var
cov , cov , cov ,1 1
var var var
M P M P
M
P PM M
M P M P M P PM M M
ρ′
→∞
′− −−
′ ′− −→ + + = +
The minus sign alternative can be dropped, because we know that a has to be positive. Now let us
substitute this into Equation (A4) in the proof of Propositions 6–8 to obtain
[ ]
[ ]11
10 1
1
cov , ( )( )( )
1 var 1
mm
k k kk k kmM fkk
k k k Mk f f
M C z zE C z z E r rC z z b
r M r==
=
⎡ ⎤′⎡ ⎤ ⋅ −′⋅ − ⎢ ⎥⎣ ⎦ −⎣ ⎦′∆ = − − + −+ +
∑∑∑ . (A5)
For the BSP of project j, we have 1jz = and 0jz′ = . Since ∆ is the budget increment required to
make two portfolios equally desirable, ∆ is, by definition, the BSP of the project. Formula for the
BBP, which is a budget reduction, is obtained by multiplying both sides of Equation (A5) by –1.
Finally, since (1 )M MM r b= + , we obtain the CAPM formula
[ ][ ]
11
10 1
1
cov , ( )( )( )
1 var 1
mm
M k k kk k kmM fkCAPM k
k k kk f M f
r C z zE C z z E r rv C z z
r r r==
=
⎡ ⎤′⎡ ⎤ ⋅ −′⋅ − ⎢ ⎥⎣ ⎦ −⎣ ⎦′= ∆ = − − + −+ +
∑∑∑ . (A6)
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38
Note that the definition of iz ’s and iz′ ’s in Proposition 10 is different from the definition in
Proposition 8. Here, Proposition 10 for the BSP case can be obtained directly using Equation
(A6). The BBP case is obtained by changing the notation so that iz′ indicates the status quo (no
project) and iz indicates the second setting (project present). Proposition 9 is obtained for the
BSP case from (A6) by assuming that k kz z k j′= ∀ ≠ and 1jz = and 0jz′ = . Proposition 9
for the BBP case is again obtained by changing the meaning of iz′ (to no project) and iz (to
project present) and assuming that k kz z k j′= ∀ ≠ and 1jz = and 0jz′ = . Q.E.D.
Acknowledgements We would like to thank an anonymous referee and an anonymous associate editor for the
comments on an earlier version of this paper. Grants from the Finnish Cultural Foundation, the
Foundation of Jenny and Antti Wihuri, and the Foundation of Emil Aaltonen are gratefully
acknowledged.
92
Helsinki University of Technology Systems Analysis Laboratory Research Reports E17, June 2005
PROJECT VALUATION UNDER AMBIGUITY
Janne Gustafsson Ahti Salo
AB TEKNILLINEN KORKEAKOULUTEKNISKA HÖGSKOLANHELSINKI UNIVERSITY OF TECHNOLOGYTECHNISCHE UNIVERSITÄT HELSINKIUNIVERSITE DE TECHNOLOGIE D’HELSINKI
93
Title: Project Valuation under Ambiguity Authors: Janne Gustafsson Cheyne Capital Management Limited Stornoway House, 13 Cleveland Row, London SW1A 1DH, U.K. [email protected] Ahti Salo Systems Analysis Laboratory Helsinki University of Technology P.O. Box 1100, 02015 HUT, FINLAND [email protected] www.sal.hut.fi/Personnel/Homepages/AhtiS.html Date: June, 2005 Status: Systems Analysis Laboratory Research Reports E17 June 2005 Abstract: This paper examines the valuation of risky projects in a setting where the
investor’s probability estimates for future states of nature are ambiguous and where he or she can invest both in a portfolio of projects and securities in financial markets. This setting is relevant to investors who allocate resources to industrial research and development projects or make venture capital investments whose success probabilities may be hard to estimate. Here, we employ the Choquet-Expected Utility (CEU) model to capture the ambiguity in probability estimates and the investor’s attitude towards ambiguity. Projects are valued using breakeven selling and buying prices, which are obtained by solving several mixed asset portfolio selection (MAPS) models. Specifically, we formulate the MAPS model for CEU investors, and show that a project’s breakeven prices for (i) investors exhibiting constant absolute risk aversion and for (ii) investors using Wald’s maximin criterion can be obtained by solving two MAPS problems. We also show that breakeven prices are consistent with options pricing analysis when the investor is a non-expected utility maximizer. The valuation procedure is demonstrated through numerical experiments.
Keywords: project valuation, capital budgeting, portfolio selection, ambiguity,
mathematical programming, decision analysis
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1
1 Introduction Recently, Gustafsson et al. (2004) proposed a framework for valuing risky projects in a setting
where the investor can invest in a portfolio of private projects and market-traded securities.
This framework assumes that the investor knows – by subjective estimation, quantitative
analysis, or otherwise – the probability distribution of returns for each asset. Yet, in practice, it
can be difficult to obtain well-founded probability estimates for real-world events, such as for
the success of a research and development (R&D) project or the increase in the stock price of a
newly listed company.
Several approaches for dealing with incomplete information about probabilities (or, ambiguity)
have been developed over the past few decades. Prominent approaches include (i) Wald’s
(1950) maximin analysis, which can be applied when probabilities are unknown, (ii) Dempster-
Shafer theory of evidence (Dempster 1967, Shafer 1976), where beliefs can be assigned to sets of
states, (iii) second-order probabilities (Marschak 1975, Howard 1992), where a (second-order)
probability distribution is associated with each probability estimate, and (iv) Choquet-Expected
Utility (CEU) theory, where the expectation is taken with respect to a nonadditive probability
measure (Schmeidler 1982, 1989; Gilboa 1987). Among these, the CEU approach has attracted
substantial attention among researchers and found several applications in economic analysis,
not least due to its robust theoretic foundation (see, e.g., Camerer and Weber 1992, Sarin and
Wakker 1992, Wakker and Tversky 1993, Salo and Weber 1995).
In this paper, we examine the valuation of risky projects when the investor’s probability
estimates for future states of nature are ambiguous. This setting is relevant to investors who
allocate resources to industrial R&D projects or make venture capital investments in new
technologies. We use the CEU approach to capture the ambiguity of probability estimates as
well as the investor’s attitude towards ambiguity. We also assume that the investor is
consistent with first-degree stochastic dominance, which is widely accepted as a plausible
requirement of rational decision making. In this case, the general CEU model reduces to a more
amenable form, which is effectively identical to the Rank Dependent Expected Utility (RDEU)
model (Quiggin 1982, 1993; see Wakker 1990).
The CEU/RDEU model, hereafter referred to as the CEU model for short, has several
intuitively appealing implications. For each portfolio, an ambiguity averse investor adjusts the
initial probability estimates, if such are given, so that the probabilities of the states in which
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2
the portfolio performs poorly are adjusted upwards, while the probabilities of the states with
high outcomes are adjusted downwards. Thus, each portfolio is evaluated by using a dedicated
set of state probabilities, obtained by adjusting each state’s initial probability estimate
according to the investment portfolio’s performance in that state.
The main contributions of this paper are (i) the explicit formulation of a mixed asset portfolio
selection (MAPS) model for CEU investors and (ii) the development of a computational
framework for applying this model in project valuation. We consider three special cases of the
general CEU model: (i) the quadratic ambiguity aversion model, which leads to linear
probability distortion, (ii) the exponential ambiguity aversion model, and (iii) Wald’s (1950)
maximin model, which can be obtained as a limit case of the exponential ambiguity aversion
model. We demonstrate these models through numerical experiments and contrast the results to
expected utility models with exponential utility functions.
For a practitioner, this paper offers (a) a model for selecting an investment portfolio that is
robust with respect to probability estimation errors and (b) a framework for that uses this
model in project valuation. The framework can be particularly beneficial in the valuation of
high technology investments in large corporations, because in these firms resources are often
diversified across financial instruments and several technology projects, whereby the related
probabilities entail ambiguities.
The remainder of this paper is structured as follows. Section 2 reviews CEU theory and its link
to RDEU theory. Section 3 presents the MAPS models based on CEU theory. Section 4
describes how projects can be valued in a MAPS setting when the investor is a CEU maximizer.
Section 5 uses the framework in a series of numerical experiments. Section 6 summarizes our
findings and discusses their managerial implications.
2 Choquet Expected Utility Since the publication of the Ellsberg paradox (Ellsberg 1961), which questioned the empirical
validity of Savage’s (1954) subjective expected utility (SEU) theory, several ways of dealing
with incomplete information about probabilities have been developed. The CEU approach is
based on the hypothesis that the investor’s response to ambiguity can be captured through
nonadditive probabilities. An analogous idea of probability distortion was utilized already in
prospect theory, where Kahneman and Tversky (1979) introduced a probability distortion
function (called the weighting function) expressing the investor’s chance attitude. While
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3
prospect theory was motivated by empirical evidence about human behavior under risk, several
axiomatic models featuring distorted probabilities, or decision weights, were developed later on.
These approaches include weighted utility theory by Chew and MacCrimmon (1979), rank
dependent utility theory by Quiggin (1982, 1993), subjectively weighted linear utility of Hazen
(1987), dual theory of Yaari (1987), and cumulative prospect theory axiomatized by Wakker
and Tversky (1993), among others.
The key difference between the early decision weight models and the CEU theory is that most
of the earlier models deal with decision making under risk (see Starmer 2000), where
probabilities are either subjectively or objectively known. In contrast, the CEU approach is
concerned with decision making under uncertainty, where probabilities are not precisely known
(see Camerer and Weber 1992). The approach is largely based on the works of Schmeidler
(1982, 1989) and Gilboa (1987) who show that a nonadditive capacity measure (“nonadditive
probability measure”; Choquet 1955) represents the investor’s preferences when certain axioms
hold. Further axiomatizations of nonadditive probabilities can be found in Wakker (1989) and
Sarin and Wakker (1992).
A capacity measure is a function [ ]Ω →: 2 0,1c which satisfies ∅ =( ) 0c , Ω =( ) 1c , and
⊂ ⇒ ≤( ) ( )A B c A c B , where Ω is the set of states of nature and Ω2 is the power set of Ω . If
c is also additive, i.e. ∪ = +( ) ( ) ( )c A B c A c B for all disjoint A and B, then it is also a
probability measure. For an act Ω → R:X , the Choquet-integral with respect to c, called the
Choquet-expectation of X, is
[ ] ( )( ) ( )∞
−∞
= > − + >∫ ∫0
0
1cE X c X x dx c X x dx . (1)
Although a capacity measure is not, in general, linked to a probability measure, it can be
shown (Wakker 1990) that if the investor is consistent with first-degree stochastic dominance
with respect to probability measure P, capacity is of the form
( ) ( )( ) ( )ϕ ϕ> = > = ( )Xc X x P X x G x , where = −( ) 1 ( )X XG x F x is the decumulative
distribution function of X and ϕ is nondecreasing with ϕ =(0) 0 and ϕ =(1) 1 . For example,
such a probability measure may reflect the investor’s best but ambiguous probability estimate
or, in the case of total uncertainty, the maximum entropy (uniform) distribution. In the
following, we assume that such a probability measure exists, and that ϕ is differentiable.
Using the transformation function and integrating by parts, the Choquet-expection in (1) can
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4
be expressed as the Lebesgue-Stieltjes integral
[ ] ( )( ) ( )
( )( ) ( )
( ) ( )
( )
0
00
0
00
( ) 1 ( )
( ) 1 ( ) ( )
( ) ( ) ( )
( ) ( ) (1 ( )) ( )
c X X
X X X
X X X
X X X X
E X G x dx G x dx
x G x x G x dG x
x G x x G x dG x
x G x dG x x F x dF x
ϕ ϕ
ϕ ϕ
ϕ ϕ
ϕ ϕ
∞
−∞
−∞−∞
∞∞
∞ ∞
−∞ −∞
= − +
′= − − +
′−
′ ′= − = −
∫ ∫
∫
∫
∫ ∫
This shows that the Choquet-expectation with respect to c is effectively the same as the
expectation with regard to P where probabilities are distorted by the factor ϕ′ −(1 ( ))XF x .
Note that when ϕ( )x is increasing and convex, ϕ′ −(1 )x is nonnegative and decreasing,
expressing ambiguity aversion. Thus, the Choquet-expected utility can be written as
[ ] [ ] ϕ∞
−∞
′= = −∫ ( ) ( )( ) (1 ( )) ( )c u X u XCEU X E u X s F s dF s .
With a strictly increasing u, we obtain ( ) ( )1( )( ) ( ) ( )u XF s P u X s P X u s−= ≤ = ≤
1( ( ))XF u s−= . A change of variables → ( )s u x gives
[ ] [ ]( ) ( ) (1 ( )) ( )c X XCEU X E u X u x F x dF xϕ∞
−∞
′= = −∫ . (2)
This is essentially the same preference model as in the RDEU theory (Quiggin 1982, 1993; see
especially Wakker 1990). Note that since RDEU applies to decision making under risk, the
transformation function ϕ can alternatively be interpreted to be an additional component of
the investor’s risk preferences, rather than to describe aversion to ambiguous probabilities.
However, regardless of the interpretation of ϕ , a convex ϕ always influences the investor’s
behavior by increasing the probabilities of states with low utility outcomes and decreasing those
with high utility outcomes.
3 Mixed Asset Portfolio Models Following Gustafsson et al. (2004), we consider a two-period setting with discrete states. Let
there be n securities, m projects, and l states of nature at time 1. The price of security
= 0,...,i n at time 0 is denoted by 0iS and the respective random price at time 1 by 1
iS . The
outcome of 1iS in state ω = 1,...,l is ω1( )iS . The 0-th asset is the risk-free asset with =0
0 1S
and 10( ) 1 fS rω ≡ + , where rf is the risk-free interest rate. The amounts of securities in the
investor’s portfolio are indicated by continuous decision variables =, 0,...,ix i n . The
investment cost of project = 1,...,k m at time 0 is 0kC , and the respective random cash flow at
time 1 is 1kC . The outcome of 1
kC in state ω = 1,...,l is 1( )kC ω . Binary variable zk indicates
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5
whether project k is started or not.
An investor’s preferences over risky mixed asset portfolios can be captured with a preference
functional U, which gives the utility of each risky portfolio. For example, under expected utility
theory, the investor’s preference functional is [ ] [ ]= ( )U X E u X , where u is the investor’s von
Neumann-Morgenstern (1947) utility function. A MAPS model using a preference functional U
can thus be formulated as follows:
= =
⎡ ⎤+⎢ ⎥⎣ ⎦∑ ∑1 1
,0 1
maxn m
i i k ki k
U S x C zx z
(3)
subject to
= =
+ =∑ ∑0 0
0 1
n m
i i k ki k
S x C z B (4)
∈ =0,1 1,...,kz k m (5)
= free 0,...,ix i n , (6)
where B is the budget. The expression inside U in (3) is the investor’s terminal wealth level.
The budget constraint (4) is written as equality, because we know that in the presence of a
risk-free asset a rational investor always spends her entire budget in the optimum.
3.1 Binary Variable Model
In general, the MAPS model for the maximization of CEU can be expressed by adapting (2) as
the preference functional in (3). Let Fx,z be the cumulative distribution function for the
terminal wealth level of mixed asset portfolio ( , )x z , i.e., for the random variable
= =
= +∑ ∑1 1
0 1
n m
i i k ki k
W S x C zx,z . Letting 1 1
0 1
( ) ( ) ( )n m
i i k ki k
W S x C zω ω ω= =
= +∑ ∑x,z denote the investor’s
terminal wealth level in state ω , the objective function maximizing CEU can be written as
1 1 1 1,,
1 0 1 0 1
max 1 ( ) ( ) ( ) ( )l n m n m
i i k k i i k ki k i k
p F S x C z u S x C zωω
ϕ ω ω ω ω= = = = =
⎛ ⎞⎛ ⎞ ⎛ ⎞′ − + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
∑ ∑ ∑ ∑ ∑x zx z. (7)
A challenge in this formulation is to calculate the values for the cumulative distribution
function, because the function Fx,z depends on the portfolio ( , )x z . For the given target level t,
we can calculate 1
( )l
F t pω ωω
ξ=
= ∑x,z for each portfolio ( , )x z using the following linear
constraints:
1 1
0 1
( ) ( ) 1,...,n m
i i k ki k
t S x C z M lωε ω ω ξ ω= =
+ − − ≤ =∑ ∑ , (8)
ωω ω ε ξ ω= =
+ − − ≤ − =∑ ∑1 1
0 1
( ) ( ) (1 ) 1,...,n m
i i k ki k
S x C z t M l , and (9)
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6
ωξ ω∈ =0,1 1,...,l , (10)
where M is some large number and ε some small positive number. The variable ωξ indicates
whether the value of the portfolio is less than or equal to t in state ω ; (8) ensures that ωξ is
equal to 1 when the terminal wealth level is less than t ε+ ; (9) ascertains that ωξ equals 0
when the terminal wealth level is greater than t ε+ . Note that the use of a non-zero ε is
necessary, because (8) and (9) do not define the value of ωξ at t ε+ .
In the CEU model, we need to calculate the values of Fx,y for each state, using the state-
specific portfolio outcome 1 1
0 1
( ) ( ) ( )n m
i i k ki k
W S x C zω ω ω= =
= +∑ ∑x,z as the target value. Extending
our earlier notation, let ωωξ ′ , ω′ = 1,...,l , be a binary variable indicating whether the portfolio
outcome in state ω′ is less than or equal to the target level ( )W ωx,z . The associated
constraints can now be expressed with (8)–(10) by replacing t with the appropriate target
value.
With the help of binary variables ωωξ ′ , the CEU model can be written as the following mixed
integer non-linear programming (MINLP) model:
ωω ω ω
ω ω
ϕ ξ ω ω′ ′′= = = =
⎛ ⎞ ⎛ ⎞′ − +⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
∑ ∑ ∑ ∑ξ
1 1
, ,1 1 0 1
max 1 ( ) ( )l l n m
i i k ki k
p p u S x C zx z
(11)
subject to
= =
+ =∑ ∑0 0
0 1
n m
i i k ki k
S x C z B (12)
ωωω ω ω ω ε ξ
ω ω
′= = = =
′ ′+ − − + ≤
′= =
∑ ∑ ∑ ∑1 1 1 1
0 1 0 1
( ) ( ) ( ) ( )
1,..., , 1,...,
n m n m
i i k k i i k ki k i k
S x C z S x C z M
l l (13)
ωωω ω ω ω ε ξ
ω ω
′= = = =
′ ′+ − − − ≤ −
′= =
∑ ∑ ∑ ∑1 1 1 1
0 1 0 1
( ) ( ) ( ) ( ) (1 )
1,..., , 1,...,
n m n m
i i k k i i k ki k i k
S x C z S x C z M
l l (14)
∈ =0,1 1,...,kz k m (15)
ωωξ ω ω′ ′∈ = =0,1 1,..., , 1,..., ,l l (16)
= free 0,...,ix i n (17)
Equivalently, the objective function can be expressed in a form similar to the certainty
equivalent formula in expected utility theory. This form has some useful properties, which are
discussed later in this paper. This “certainty equivalent” form is given by
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7
ωω ω ω
ω ω
ϕ ξ ω ω−′ ′
′= = = =
⎛ ⎞⎛ ⎞ ⎛ ⎞′ − +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠
∑ ∑ ∑ ∑ξ
1 1 1
, ,1 1 0 1
max 1 ( ) ( )l l n m
i i k ki k
u p p u S x C zx z
.
The model (11)–(17) has n continuous decision variables, + 2m l binary variables, and + 21 2l
linear constraints. Such a model can usually be solved in a reasonable time for models with less
than 30 states.
3.2 Rank-Constrained Model
Even when the MINLP model (11)–(17) has a reasonable number of binary variables, it can be
relatively hard to solve numerically due to the complex relationships defining the variables ωωξ ′
and the normalization problems implied by the large values with M. Our experiments with the
GAMS software package (see www.gams.com) indicate that standard MINLP algorithms do not
necessarily find the optimal solution for models of this type. Therefore, we propose an
alternative method for solving the CEU MAPS model. The method relies on solving a separate
MINLP model for each possible rank order of states (i.e., ranking is based on the outcome of
the portfolio in each state). In each model, portfolios must satisfy the given rank-order.
Consequently, distorted probabilities (decision weights) in each model are constant and can be
used as parameters in the optimization problem. The decision weight of state ω is
( )( )1 ( )p p F Wω ωϕ ω∗ ′= − x,z x,z , where ( )( )F W ωx,z x,z is the sum of the probabilities of the states
where the portfolio outcome is less than or equal to the portfolio outcome in state ω ,
1 1
0 1
( ) ( ) ( )n m
i i k ki k
W S x C zω ω ω= =
= +∑ ∑x,z .
Assuming that hω denotes the state with h:th rank, the rank-constrained model can be
formulated as
ωω
ω ω∗
= = =
⎛ ⎞+⎜ ⎟⎝ ⎠
∑ ∑ ∑1 1
,1 0 1
max ( ) ( )l n m
i i k ki k
p u S x C zx z
(18)
subject to
= =
+ =∑ ∑0 0
0 1
n m
i i k ki k
S x C z B (19)
1 1 1 11 1
0 1 0 1
( ) ( ) ( ) ( ) 0 1,..., 1n m n m
i h i k h k i h i k h ki k i k
S x C z S x C z h lω ω ω ω+ += = = =
+ − − ≤ = −∑ ∑ ∑ ∑ (20)
∈ =0,1 1,...,kz k m (21)
= free 0,...,ix i n , (22)
where ( )( )1 ( )p p F Wω ωϕ ω∗ ′= − x,z x,z . This model is solved for each possible rank-order of states,
which leads to a total of !l models. The rank-order with the highest optimum gives the optimal
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8
solution to the problem (11)–(17).
An advantage of this approach is that its complexity does not depend on the transformation
function ϕ . However, because the calculation of the optimal solution requires the solution of !l
rank-constrained models, this approach is computationally attractive only for problems with
some 10 states or less.
3.3 Specific CEU Models
In this section, we consider three interesting special cases of the general CEU model, (i) the
quadratic ambiguity aversion model, (ii) the exponential ambiguity aversion model, and (iii)
Wald’s (1950) maximin model. The pricing behavior of these models is studied in Sections 4
and 5.
3.3.1 Quadratic Ambiguity Aversion
An investor with quadratic ambiguity aversion is characterized by a convex quadratic
transformation function ϕ = + +2( )x ax bx c , a > 0. The normalization ϕ =(0) 0 , ϕ =(1) 1
imply that c = 0 and b = 1 – a, and hence ϕ = + −2( ) (1 )x ax a x and ϕ′ = + −( ) 2 1x ax a . To
ensure that ϕ is nondecreasing over the interval [0,1], the constant a must be less than or
equal to 1. This also ensures that distorted probabilities (decision weights) will be nonnegative.
The appeal of the quadratic transformation function comes from the linearity of the probability
distortion, which potentially facilitates the solution of the resulting models. Figure 1 illustrates
the decision weights obtained by applying a quadratic transformation function in a setting with
six equally likely states of nature. For the sake of comparison, the decision weights are
normalized so that they to sum up to 1.
Together with a linear utility function =( )u x x , a quadratic transformation function leads to a
mixed integer quadratic programming model, where the objective function is
1 1 1 1
,1 0 1 0 1
max 1 1 2 ( ) ( ) ( ) ( )l n m n m
i i k k i i k ki k i k
p a F S x C z S x C zωω
ω ω ω ω= = = = =
⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞+ − ⋅ + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎣ ⎦
∑ ∑ ∑ ∑ ∑x,yx z.
This model is interesting, because (i) it can be computationally appealing and (ii) it falls within
the scope of Yaari’s (1987) dual theory, expressing the properties observed under this theory,
such as linearity in the payments.
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9
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Rank 6 Rank 5 Rank 4 Rank 3 Rank 2 Rank 1
State
Nor
mal
ized
Dec
ision
Wei
ght 0
0.2
0.4
0.6
0.8
1
Figure 1. Quadratic ambiguity aversion: Normalized decision weights for different levels of a
when there are 6 equally likely states. Rank i indicates the i:th best state.
3.3.2 Exponential Ambiguity Aversion
When the investor’s transformation function is ( ) xx a beγϕ = + , 0γ > , the investor is said to
express exponential ambiguity aversion. The condition ϕ =(0) 0 , ϕ =(1) 1 imply that
1/(1 )a b eγ= − = − so that ( ) ( )( ) 1 / 1xx e eγ γϕ = − − and ( )( ) / 1xx e eγ γϕ β′ = − . Figure 2
illustrates the decision weights implied by applying exponential ambiguity aversion in a setting
with six equally likely states of nature. Again, the decision weights are normalized to sum up to
one.
In numerical experiments (Section 5), we also consider investors who exhibit both constant
absolute risk aversion and exponential ambiguity aversion. In this case, the investor’s
preferences are captured by a concave exponential utility function ( ) xu x e α−= − , α > 0 , and a
convex exponential transformation function. This implies a CEU objective function
1 1 1 1
0 1 0 1
( ) ( ) ( ) ( )
,1
max1
n m n m
i i k k i i k ki k i k
F S x C z S x C zl
p e ee
γ ω ω α ω ω
ω γω
γ= = = =
⎛ ⎞ ⎛ ⎞− + − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠−
=
⎛ ⎞ ⎛ ⎞∑ ∑ ∑ ∑⎜ ⎟ ⎜ ⎟−⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠
∑x,y
x z. (23)
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10
0
0.1
0.2
0.3
0.40.5
0.6
0.7
0.8
0.9
1
Rank 6 Rank 5 Rank 4 Rank 3 Rank 2 Rank 1
State
Nor
mal
ized
Dec
isio
n W
eigh
t
0.001
0.01
0.1
0.5
1
2
3
5
10
Figure 2. Exponential ambiguity aversion: Normalized decision weights at different γ -levels
when there are 6 equally likely states. Rank i indicates the i:th best state.
3.3.3 Wald’s Maximin Criterion
With Wald’s (1950) maximin criterion, the investor maximizes the utility of the least preferred
state. Wald’s (1950) model is a special case of the general CEU model, as it is obtained with a
capacity measure that gives 0 to all proper subsets of Ω ; it is also an extreme case of the
exponential CEU model when γ goes to infinity.
A MAPS model using the maximin criterion can be formulated as the following mixed integer
linear programming (MILP) problem:
ππ
, ,maxx z
(24)
subject to
= =
+ =∑ ∑0 0
0 1
n m
i i k ki k
S x C z B (25)
π ω ω ω= =
− − ≤ =∑ ∑1 1
0 1
( ) ( ) 0 1,...,n m
i i k ki k
S x C z l (26)
∈ =0,1 1,...,kz k m (27)
= free 0,...,ix i n (28)
π free (29)
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11
In contrast to nonlinear CEU models, this model does not require additional binary variables or
the solution of several MINLP models. Hence, maximin models can be solved relatively rapidly
even with a large number of states.
4 Valuing Projects: CEU Investors In a MAPS setting, an individual project can be valued by comparing the values of the optimal
portfolios when the investor does and does not invest in the project. Formally, the comparison
is carried out according to the concepts of breakeven selling and buying prices (De Reyck,
Degraeve and Gustafsson 2003, Luenberger 1998, Smith and Nau 1995). Specifically, breakeven
selling price is the lowest price at which a rational investor would be willing to sell the project
if he or she had it, while the breakeven buying price is the highest price at which the investor
would buy the project if he or she did not have it. Hence, when determining the breakeven
selling price, we assume that the investor invests in the project at the beginning (referred to as
the status quo) and compare the result to the situation where he or she does not have the
project (referred to as the second setting). The breakeven selling price is the budget increment
which makes the portfolio in the second setting equally preferred to the portfolio in the status
quo. Similarly, when calculating the breakeven buying price, we assume that the investor does
not have the project in the status quo and that he or she has it in the second setting. The
breakeven buying price is the budget reduction that makes the portfolio with the project
equally desirable to the portfolio without the project.
Optimization problems for calculating breakeven selling and buying price of project j are given
in Table 1. For CEU maximizers, we can use the explicit portfolio optimization model (11)–(17)
or (18)–(22), adding the relevant constraint on project j to the model in each setting (i.e.,
= 1jz or = 0jz ). The breakeven prices are optimized iteratively by changing the parameter sjv or b
jv , as appropriate, until the optimal values of the portfolio selection problems in the
status quo and in the second setting are identical. In general, the breakeven selling price and
the breakeven buying price are not equal.
When the investor can borrow funds beyond any limit and no risk constraint is imposed on the
portfolio, the optimization problems in Table 1 may yield unbounded solutions; if so, the
breakeven prices are undefined. For example, this may happen when the utility function of a
CEU maximizer is linear. For example, the investors abiding by Yaari’s (1987) dual theory are
CEU maximizers with a linear utility function. There are also other classes of CEU models
which may lead to unbounded solutions, such as models using logarithmic utility functions.
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12
Indeed, if limitless borrowing is possible, it may be necessary to use a risk or borrowing
constraint with several types of preference models in order to avoid unbounded solutions.
Table 1. Definitions of the value of project j. For CEU investors U is as in Section 3.
Breakeven selling price Breakeven buying price
Definition sjv such that − +=W W b
jv such that − +=W W
Status quo +
= =
⎡ ⎤= +⎢ ⎥⎣ ⎦∑ ∑1 1
,0 1
maxn m
i i k ki k
W U S x C zx z
subject to
= =
+ =∑ ∑0 0
0 1
n m
i i k ki k
S x C z B
= 1jz
∈ =0,1 1,...,kz k m
= free 0,...,ix i n
−
= =
⎡ ⎤= +⎢ ⎥⎣ ⎦∑ ∑1 1
,0 1
maxn m
i i k ki k
W U S x C zx z
subject to
= =
+ =∑ ∑0 0
0 1
n m
i i k ki k
S x C z B
= 0jz
∈ =0,1 1,...,kz k m
= free 0,...,ix i n
Second
setting
Project sold
−
= =
⎡ ⎤= +⎢ ⎥⎣ ⎦∑ ∑1 1
,0 1
maxn m
i i k ki k
W U S x C zx z
subject to
= =
+ = +∑ ∑0 0
0 1
n ms
i i k k ji k
S x C z B v
= 0jz
∈ =0,1 1,...,kz k m
= free 0,...,ix i n
Project bought
+
= =
⎡ ⎤= +⎢ ⎥⎣ ⎦∑ ∑1 1
,0 1
maxn m
i i k ki k
W U S x C zx z
subject to
= =
+ = −∑ ∑0 0
0 1
n mb
i i k k ji k
S x C z B v
= 1jz
∈ =0,1 1,...,kz k m
= free 0,...,ix i n
On the other hand, portfolio optimization problems typically remain bounded when a CEU
investor has a concave exponential utility function. Moreover, for such an investor, there exists
a pricing formula analogous to the one introduced in Gustafsson et al. (2004) for mean-risk
investors, provided that the preference functional is expressed in expected utility theory’s
“certainty equivalent” form. The proof of the following proposition is in Appendix.
PROPOSITION 1. Let the investor be a CEU maximizer with α−= −( ) xu x e , α > 0 , and the
preference functional be [ ] ( )[ ]( )−= 1cU X u E u X . Then, breakeven selling and buying prices
are identical and can be computed using the formula + −−
=+1 f
V Vvr
, where V + and V − are the
optimal preference functional values with and without the project, respectively.
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13
It can also be shown that, in arbitrage-free markets, an investor who applies Wald’s (1950)
maximin criterion never invests in securities, if he or she does not invest in projects
(Proposition 2). However, if he or she does invest in projects, it may be optimal to invest in
securities as well. The breakeven selling and buying prices for a maximin investor can be
computed by calculating the value of the portfolio when the investor does and does not invest
in the project and by discounting the difference to its present value at the risk-free interest rate
(Proposition 3). The difference from analogous formulas in Gustafsson et al. (2004) is that the
value of the portfolio is here defined as the value obtained in the worst possible state, as given
by decision variable π in (24). The proofs of the propositions are given in Appendix.
PROPOSITION 2. In arbitrage-free markets, if a maximin investor does not invest in projects, she
does not invest in securities, unless the resulting portfolio replicates the risk-free asset. If a
maximin investor invests in projects, she may also invest in a risky security portfolio.
PROPOSITION 3. If the investor abides by Wald’s maximin criterion, breakeven selling and
buying prices for any project are identical and can be computed by the formula + −−
=+1 f
V Vvr
,
where V + and V − are the values of the portfolio in the worst state (given by decision variable
π ) with and without the project, respectively.
Breakeven selling and buying prices for a CEU investor (including a maximin investor) exhibit
an important consistency property. Namely, a project’s breakeven selling and buying price are
consistent with options pricing analysis, or contingent claims analysis (CCA), which is used in
the theory of finance (see, e.g., Brealey and Myers 2000, Hull 1999, Luenberger 1998). An
analogous property was proven by Smith and Nau (1995) for expected utility maximizers; the
following proposition generalizes their result to non-expected utility maximizers. The
consistency property holds in the sense that CCA and the breakeven prices yield the same
result whenever CCA is applicable, i.e. when there is a replicating portfolio for the project. The
CCA price of project j is given by
∗
=
= − + ∑0 0
0
nCCAj j i i
i
v C x S ,
where ix ∗ is the amount of security i in the replicating portfolio. Due to this property,
breakeven buying and selling prices can be regarded as generalizations of CCA for incomplete
markets. The proof of the proposition is in Appendix.
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14
PROPOSITION 4. If there is a replicating portfolio for a project, breakeven selling and buying
prices for the project are identical and equal to the value given by options pricing analysis.
5 Numerical Experiments In this section, we examine CEU-based project valuation through numerical experiments. We
first offer comparative results with expected utility and expected value maximizers and then
analyze investors with varying degrees of ambiguity aversion. In particular, we consider the
following research questions:
Q1. When an expected utility maximizer becomes less risk averse, do the project values
converge to the values given by an expected value maximizer?
Q2. To which values do project values converge with increasing risk aversion?
Q3. When a CEU maximizer becomes less ambiguity averse, do the project values
converge to the values given by the respective expected utility maximizer?
Q4. To which values do project values converge with increasing ambiguity aversion?
5.1 Experimental Setup
The experimental setup consists of 6 states of nature, 4 projects (Table 3), and 2 securities
(Table 2) which constitute the market. Initial state probabilities are generated by assuming
maximum entropy, implying a probability of 1/6 for each state. The security prices are
obtained from the Capital Asset Pricing Model (CAPM; Sharpe 1964, Lintner 1965; see also
Luenberger 1998), where the expected rate of return of the market portfolio has been set to
13.62%, so that the price of security 2 is $20. This price is used by Smith and Nau (1995),
Trigeorgis (1996), and Gustafsson et al. (2004) for this security. The standard deviation of the
market portfolio is then 19.68%. Betas are calculated with respect to the resulting market
portfolio. The market prices in Table 3 are the prices that the CAPM would give to the
projects if they were traded in financial markets, assuming infinite divisibility and a negligible
market capitalization. Italicized text indicates a computed value. The risk-free interest rate is
8%.
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15
Table 2. Securities.
Security
1 2
Shares issued 15,000,000 10,000,000
State 1 $60 $36
State 2 $50 $36
State 3 $40 $36
State 4 $60 $12
State 5 $50 $12
State 6 $40 $12
Beta 0.66 2.13
Market price $44.75 $20.00
Capitalization weight 77.05% 22.95%
Expected rate of return 11.72% 20.00%
St. dev. of rate of return 18.24% 60.00%
Table 3. Projects.
Project
A B C D
Investment cost $80 $100 $104 $40
State 1 $150 $140 $180 $20
State 2 $50 $140 $180 $40
State 3 $150 $150 $180 $40
State 4 $150 $110 $60 $20
State 5 $50 $170 $60 $60
State 6 $150 $100 $60 $60
Beta 0.000 0.240 2.13 -1.679
Market price $28.02 $23.46 -$4.00 $0.58
Expected outcome $116.67 $135 $120 $40
St. dev. of outcome $47.14 $23.63 $60 $16.33
We consider investors who exhibit (i) constant absolute risk aversion and (ii) exponential,
quadratic, or maximin ambiguity aversion. By Propositions 2 and 4, breakeven selling and
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16
buying prices are identical, and hence both prices are displayed in a single entry. Unless
otherwise indicated, we assume that the investor’s budget is $500. Optimizations were carried
out with the GAMS software package and the rank-constrained formulation in (18)–(22), except
for the maximin model, for which the model (24)–(29) was employed.
5.2 Expected Utility, Expected Value, and Maximin Investors
Before considering the impacts of ambiguity aversion, we first examine three cases without
ambiguity aversion, namely, (i) expected utility (EU) maximizers, (ii) expected value (EV)
maximizers (risk-neutral investors), and (iii) maximin investors. Table 4 gives the project
values for an expected utility maximizer as a function of the risk-aversion coefficient α , as well
as the optimal mixed asset portfolios at each level of risk aversion. At each level, the investor
starts projects A, B, and D. Weights of securities 1 and 2 in the security portfolio are denoted
by 1w and 2w , respectively. The amount of funds invested in the risk-free asset is given in the
column “ 0x ”.
Table 4. Breakeven prices and optimal portfolios for an EU maximizer.
A B C D Expectation St.dev.
0.000001 $28.67 $23.06 -$4.00 $0.64 $83,218.7 $289,177.2 76.99% 23.01% -$1,468,382.6
0.00001 $28.67 $23.07 -$4.00 $0.64 $8,858.8 $28,919.6 77.01% 22.99% -$146,659.7
0.0001 $28.68 $23.10 -$4.00 $0.65 $1,422.8 $2,894.1 77.14% 22.86% -$14,487.5
0.001 $28.72 $23.40 -$4.00 $0.79 $679.3 $293.5 78.38% 21.62% -$1,271.5
0.005 $28.65 $24.78 -$4.00 $1.44 $613.7 $69.8 82.08% 17.92% -$102.1
0.010 $27.77 $26.54 -$4.00 $2.28 $606.1 $47.7 83.89% 16.11% $37.7
0.015 $26.77 $28.26 -$4.00 $3.09 $603.9 $42.5 83.96% 16.04% $80.0
0.020 $25.74 $29.85 -$4.00 $3.81 $603.1 $40.8 83.18% 16.82% $98.4
0.030 $23.90 $32.42 -$4.00 $4.96 $602.6 $40.1 80.79% 19.21% $114.0
0.040 $22.50 $34.02 -$4.00 $5.77 $602.6 $40.4 78.43% 21.57% $121.1
Project value Optimal mixed asset portfolio
α 1w 2w 0x
In Table 4, we observe that when α approaches zero, the values of projects A–D converge to
$28.67, $23.06, -$4.00, and $0.64, which are close to the projects’ CAPM market prices (Table
3). This is surprising, because one would expect that the project values would converge to the
values given by an EV maximizer, i.e., to $17.22, $12.50, –$4.00, and –$6.67. These values are
obtained by discounting the project cash flows at a 20% discount rate, the expected rate of
return of the security with the highest expected return (security 2). Also, the financial portfolio
converges close (but not exactly) to the CAPM market portfolio, containing 76.99% of security
1 and 23.01% of security 2 when α −= 610 . A risk-neutral portfolio would contain 100% of
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security 2. Even though the amount of funds borrowed and the monetary worth of the mixed
asset portfolio approach infinity with decreasing risk aversion, project values and the weights of
the investor’s financial portfolio converge to specific values. Our experiments also show that,
when the investor cannot invest in projects, he or she invests in a security portfolio with
weights (76.99%, 23.01%) at all levels of risk aversion. Taken together, these observations
indicate that the pricing behavior of EU maximizers with constant absolute risk aversion is
similar to mean-variance investors whose project values converge to CAPM market prices with
increasing risk tolerance, and who always invest in the CAPM market portfolio when there are
no projects (De Reyck, Degraeve and Gustafsson 2003).
Because the optimal security portfolio for the EU investor is near to the CAPM market
portfolio, the expected rate of return of the mixed asset portfolio converges to a value that is
close to the excess rate of return of the CAPM market portfolio. Likewise, the volatility of the
mixed asset portfolio approaches a value near to the volatility of the market portfolio. Notice
that volatility does not decrease monotonically with increasing risk aversion. This is because
standard deviation does not exactly measure the risk perceived by an EU maximizer. Hence, it
may happen that, means being equal, a portfolio with more standard deviation is preferred to
the portfolio with less standard deviation.
On the other hand, it can be shown that, when the risk aversion parameter α goes to infinity,
project values converge towards the values for a maximin investor, who prices the projects at
$17.69, $25.37, -$4.00, and $8.15. In the optimum, a maximin investor invests in a security
portfolio with weights (76.32%, 23.68%) and lends $104.3. Table 4 gives project values and
mixed asset portfolio statistics up to the α -level of 0.04. At higher α -levels optimization
problems become computationally unstable due to the very large negative values in the
exponent. Notice that the expected convergence behavior is not clear from Table 4. This
suggests that some solutions at α -levels below 0.04 may be distorted by computational issues
or convergence to a local optimum, for example.
Between the two extremes, the value of project A decreases with increasing risk aversion. In
contrast, the values of projects B and D grow when risk aversion is increased. The behavior
with project D can be explained by the negative correlation of project D with the rest of the
portfolio; the negative correlation is partly highlighted by the project’s negative beta (Table 3).
As indicated by the results from Gustafsson et al. (2004), the value of a nearly-zero-correlation
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project may decrease by growing risk aversion, explaining the convergence behavior of project
A. However, the value of project B increases with α , even though the beta of the project is
positive. This may be explained by the project’s negative correlation with project A (–0.599).
Note that the value must start to decrease at some point, because the maximin value of project
B is $25.37, which is less than the price of project B when 0.04α = , $34.02. Hence, the
convergence to maximin values need not be monotonic. Project C is priced at a constant –$4
level, because there exists a replicating portfolio: it is possible to replicate the cash flows of
project C by buying 5 shares of security 2. Since the replicating portfolio costs $100, it implies a
CCA price of –$4 for project C.
In summary, when the risk aversion parameter α approaches zero, the project values given by
an EU maximizer with constant absolute risk aversion converge to values near to CAPM
market prices, rather than to the values for an EV maximizer [Q1]. Also the optimal financial
portfolio converges close to the CAPM market portfolio. With increasing risk aversion project
values approach those of a maximin investor [Q2]. The results also indicate that project values
may either decrease or increase with growing risk aversion, and that the convergence behavior
is not necessarily monotonic. Finally, a replicating portfolio implies a constant pricing behavior.
5.3 Choquet-Expected Utility Maximizers
In this section, we examine investors with exponential and quadratic ambiguity aversion. The
risk aversion parameter α is 0.005 in each experiment. Table 5 describes project values and
mixed asset portfolio statistics for exponentially ambiguity averse investors as a function of
ambiguity aversion parameter γ . At each value of γ , the investor invests in projects A, B, and
D. When γ approaches zero, the values converge, as expected, towards project values given by
an expected utility maximizer. The prices for such an investor are given on the row where
0γ = . At the other extreme, when γ approaches infinity, project values converge towards the
values given by a maximin investor. These values are described on the row with γ = ∞ .
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Table 5. Breakeven prices and optimal portfolios for a CEU investor with exponential ambiguity aversion.
A B C D Expectation St.dev.
0 $28.65 $24.78 -$4.00 $1.44 $613.7 $69.8 82.08% 17.92% -$102.1
0.1 $28.51 $24.96 -$4.00 $1.53 $612.4 $65.9 82.75% 17.25% -$81.1
0.5 $27.98 $25.89 -$4.00 $1.84 $607.7 $52.1 85.12% 14.88% $0.0
1 $27.35 $27.41 -$4.00 $2.14 $604.2 $43.4 81.08% 18.92% $84.0
2 $25.25 $31.05 -$4.00 $3.67 $603.8 $43.5 76.32% 23.68% $104.1
3 $22.86 $34.03 -$4.00 $4.96 $603.8 $43.5 76.32% 23.68% $104.1
5 $19.78 $29.90 -$4.00 $8.05 $603.8 $43.5 76.32% 23.68% $104.1
$17.69 $25.37 -$4.00 $8.15 $603.8 $43.5 76.32% 23.68% $104.1
Project value Optimal mixed asset portfolio
γ
∞
2w1w 0x
Similarly to increasing risk aversion, the value of project A decreases and the value of project D
increases with increasing ambiguity aversion. This is because project A has a near-zero
correlation with the rest of the portfolio, whereas project D has a negative correlation, which
reduces the aggregate risk of the portfolio. Also, because it is possible to construct a replicating
portfolio, project C has a constant price which is equal to its CCA value. Note also that that
the value of project B increases at low ambiguity aversion levels and decreases at higher levels.
This suggests that a similar behavior is likely to happen with growing risk aversion, as
conjectured in the previous section.
For values of γ higher than and equal to 2, the optimal portfolio is almost identical with a
maximin investor’s portfolio, even though project values at these γ -levels are different. The
reason for this is that, even though the optimal portfolios without restrictions on what projects
can be started are the same at each level, optimal portfolios when the investor must or must
not start a project (the settings in Table 1) are different, which implies different project values.
Table 6 presents project values and optimal mixed asset portfolios under quadratic ambiguity
aversion. As before, the investor invests in projects A, B, and D at all levels of ambiguity
aversion considered. Project values converge towards those for an expected utility maximizer
when a goes to 0. At the other extreme, however, project values do not anymore reach maximin
values, because a is bounded from above by 1. When a = 1, the level of ambiguity aversion
roughly corresponds to exponential ambiguity aversion when γ is in the range [ ]1.5,4 . Overall,
quadratic ambiguity aversion gives lower values to projects A and B and a higher value to
project D than exponential ambiguity aversion.
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Table 6. Breakeven prices and optimal portfolios for a CEU investor with quadratic ambiguity aversion.
A B C D Expectation St.dev.
0 $28.65 $24.78 -$4.00 $1.44 $613.7 $69.8 82.08% 17.92% -$102.1
0.2 $27.95 $25.64 -$4.00 $1.79 $608.8 $54.4 85.15% 14.85% -$16.7
0.4 $26.97 $26.70 -$4.00 $1.99 $605.6 $45.9 83.45% 16.54% $53.6
0.6 $25.84 $28.11 -$4.00 $2.57 $604.4 $43.3 80.14% 19.86% $88.4
0.8 $24.19 $29.05 -$4.00 $3.31 $604.1 $43.5 76.32% 23.68% $104.1
1 $22.03 $29.33 -$4.00 $4.10 $603.8 $43.5 76.32% 23.68% $104.1
Project value Optimal mixed asset portfolio
a 1w 2w 0x
In summary, when ambiguity aversion approaches zero, the project values and the optimal
portfolios are approach those of the expected utility maximizer [Q3]. When exponential
ambiguity aversion goes to infinity, the project values approach the values given by a maximin
investor [Q4]. Under quadratic ambiguity aversion, however, this occurs only to a certain
extent, because the ambiguity aversion parameter a is limited by 1 from above. Also, the
convergence to the extreme values is not necessarily monotonic.
6 Summary and Conclusions In this paper, we have examined project valuation in a setting where the investor’s probability
estimates are ambiguous and where she can construct an investment portfolio that includes
both projects and securities. We have used the CEU theory to capture the investor’s
preferences for portfolios characterized by ambiguity. In this framework, projects are valued
using the concepts of breakeven selling and buying prices, which require the solution of MAPS
models with and without the analyzed project.
We formulated two alternative MINLP models for solving MAPS problems for CEU investors,
(i) the binary variable model, where the values of the portfolio’s cumulative distribution
function are captured by dedicated binary variables, and (ii) the rank-constrained model, which
is solved separately for each possible rank-order of states. Although the former model is suitable
for a larger number of states than the latter, our experience with standard MINLP algorithms
suggests that models of the former type may be difficult to solve. The latter model is
computationally more tractable in this respect.
We showed that a project’s breakeven selling and buying prices for (i) CEU maximizers who
exhibit constant absolute risk aversion and for (ii) maximin investors can be computed by
solving MAPS models with and without the project and by discounting the difference of the
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obtained portfolio values back to its present value at the risk-free interest rate. This makes it
possible to calculate breakeven selling and buying prices from two optimization problems only.
We also showed that breakeven selling and buying prices give consistent results with CCA for
non-expected utility maximizers whenever the method is applicable, i.e., when a replicating
portfolio exists for the project. Therefore, these pricing concepts can be regarded as
generalizations of CCA. In our numerical experiments, we examined investors who exhibit (i) constant absolute risk
aversion and (ii) either quadratic, exponential, or maximin ambiguity aversion. The
experiments show that when an expected utility maximizer becomes less risk averse, project
values approach values that are close to the projects’ CAPM prices, rather than the values
given by a risk-neutral investor. Hence, it appears that expected utility maximizers with
constant absolute risk aversion behave similarly to risk-constrained mean-risk investors. When
the investor becomes increasingly averse to risk or ambiguity, project values approach the
values given by a maximin investor. Under quadratic ambiguity aversion, project values do not
reach maximin values, because the ambiguity aversion parameter is bounded from above. Our analysis has several managerial implications. The present framework makes it possible to
(i) select an investment portfolio that is relatively insensitive to changes in probability
estimates and to (ii) calculate defensible values for risky projects whose success probabilities are
ambiguous. Our results also indicate that when ambiguity is taken into account, assets that
perform well when the rest of the portfolio performs poorly often gain higher values than when
ambiguity is not accounted for. Therefore, hedging through relevant derivative securities is
particularly useful under ambiguity, because many hedging instruments are costless zero-value
contracts that make the portfolio outcomes more uniformly distributed and thereby lead to
higher values. Also, diversification possibilities can be expanded by using – in addition to usual
market-traded securities – over-the-counter (OTC) instruments, such as commodity and credit
derivatives, that many investment banks provide for their corporate clients. This work suggests several avenues for further research. In particular, efficient MINLP
algorithms for solving the binary variable model (11)–(17) are called for. Computational issues
could also be relaxed if the CEU model were changed to the weighted utility model, which may
be computationally more amenable than the CEU model. Also, since many real-world project
valuation settings involve multiple time periods, the present model should be extended to a
multi-period setting, for example, in the spirit of Gustafsson and Salo (2005).
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Appendix PROOF OF PROPOSITION 1: We know that an expected utility maximizer with an exponential
utility function satisfies the delta property, i.e. ( )[ ]( ) ( )[ ]( )− −+ = +1 1u E u X b u E u X b . Since
Choquet-expectation satisfies [ ] [ ]=c cE aX aE X , we also have [ ] ( )[ ]( )1cU X b u E u X b−+ = + =
( )[ ]( ) [ ]1cu E u X b U X b− + = + . Let us next consider the breakeven buying price. Let µb
SQ be the
optimal value of the objective function in the status quo, and µ µ= + ∆b b bSS SQ be the optimal value
in the second setting, when the budget is B. To obtain an optimal value in the second setting
which is equal to µbSQ , we can lower the budget by δ b , whereby the amount borrowed effectively
increases by the same amount, lowering the optimal value by δ +(1 )bfr . When
[ ] ( )( )∗ − ∗= ⎡ ⎤⎣ ⎦1
cU X u E u X is the optimum to the portfolio selection problem with the budget B,
( )( )δ δ∗ − ∗⎡ ⎤− + = − +⎡ ⎤⎣ ⎦⎣ ⎦1(1 ) (1 )b b
f c fU X r u E u X r is the optimum to the portfolio selection
problem with budget δ− bB . By requiring that δ + = ∆(1 )b br we obtain δ = ∆ +/(1 )b bfr . By
denoting that µ −=bSQ V and µ +=b
SS V , we immediately obtain the desired formula. With
breakeven selling price, we have the optimum value µsSQ in the status quo and µ µ= − ∆s s s
SS SQ in
the second setting when the budget is B. Again, we can increase the budget by δ s to obtain an
optimal solution ( )( )δ δ∗ − ∗⎡ ⎤+ + = + +⎡ ⎤⎣ ⎦⎣ ⎦1(1 ) (1 )s s
f c fU X r u E u X r . By requiring that
δ + = ∆(1 )s sfr we obtain δ = ∆ +/(1 )s s
fr . By observing that µ +=sSQ V and µ −=s
SS V , we get
again the desired formula. Q.E.D.
PROOF OF PROPOSITION 2: In arbitrage-free markets, each risky portfolio composed of securities
must have a state in which it yields less than the risk-free interest rate, as otherwise it would be
possible, by buying the portfolio and borrowing at the risk-free interest rate, to create an arbitrage
opportunity, i.e. a portfolio with a chance of yielding a positive amount of money without an initial
capital outlay. Since a maximin investor values each portfolio by its worst state, such an investor
prefers the risk-free asset to all risky security portfolios. However, in a combined project-security
portfolio it may occur that the rate of return of the combined portfolio exceeds the risk-free
interest rate in all states, because projects are not bound by no-arbitrage conditions. For example,
consider a setting with two equally likely states, a project yielding $100 in state 1 and $0
otherwise, and a security yielding $10 in state 2 and $0 otherwise. The cost of the project is $45
and the security is priced at $4.5. The risk-free interest rate is 8% and the budget is $90. The
optimum is to start the project and buy 10 shares of the security. Q.E.D.
PROOF OF PROPOSITION 3: Let us first consider the breakeven buying price. Let π bSQ be the
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25
optimal value in the status quo and π π= + ∆b b bSS SQ the optimal value in the second setting when
the budget is B. Because the investor invests in the optimal mixed asset portfolio regardless of the
budget (as long as it yields more than the risk-free interest rate in the worst state), lessening the
budget by δ b will reduce 0x by the same amount, and consequently decrease the optimal value
(the value in the worst state) by δ +(1 )bfr . By requiring that δ + = ∆(1 )b b
fr we obtain,
δ = ∆ +/(1 )b bfr . By denoting that π −=b
SQ V and π +=bSS V , we obtain the desired formula. Let
us then consider the breakeven selling price. Here, we have the optimal value π bSQ in the status quo
and π π= − ∆s s sSS SQ in the second setting. As above, we can increase the optimal value in the
second setting by δ +(1 )sfr by increasing the budget by δ s . By setting δ + = ∆(1 )s s
fr and
denoting π +=sSQ V and π −=s
SS V we again obtain the desired formula. Q.E.D.
PROOF OF PROPOSITION 4: Let us first consider the breakeven buying price and let j indicate the
project being valued. Let ( , )SQ SQx z be the optimal portfolio in the status quo (with 0jz = ).
Notice that when there is a replicating portfolio for project j and the investor invests in the project,
it is possible to construct a shorted replicating portfolio, which nullifies the cash flows of the
project at time 1, and which jointly with project j yields money equal to ∗
=
− + ∑0 0
0
n
j i ii
C x S at time 0
(x* is the amount of security i in the replicating portfolio). Therefore, in the second setting, a
portfolio ( , ) ( , )∗= − +SS SS SQ SQjx z x x z e , where je is an m-dimensional vector with a zero value for
each element except for the j:th element which has the value of 1, will yield the same optimal value
with budget 0 0
0
( )n
j i ii
B C x S∗=
− − + ∑ as the portfolio ( , )SQ SQx z yields with budget B in the status
quo. The portfolio ( , )SS SSx z is the optimum to the problem in the second setting, because
otherwise ( , )SQ SQx z would not be the solution to the problem in the status quo. Therefore, the
breakeven buying price for project j is, by definition, equal to ∗
=
− + ∑0 0
0
n
j i ii
C x S , which is also the
CCA price of the project. A similar logic proves the theorem for breakeven selling prices. Q.E.D.
119
120
Helsinki University of Technology Systems Analysis Laboratory Research Reports E18, June 2005
VALUING RISKY PROJECTS WITH CONTINGENT PORTFOLIO
PROGRAMMING
Janne Gustafsson Ahti Salo
AB TEKNILLINEN KORKEAKOULUTEKNISKA HÖGSKOLANHELSINKI UNIVERSITY OF TECHNOLOGYTECHNISCHE UNIVERSITÄT HELSINKIUNIVERSITE DE TECHNOLOGIE D’HELSINKI
121
Title: Valuing Risky Projects with Contingent Portfolio Programming Authors: Janne Gustafsson Cheyne Capital Management Limited Stornoway House, 13 Cleveland Row, London SW1A 1DH, U.K. [email protected] Ahti Salo Systems Analysis Laboratory Helsinki University of Technology P.O. Box 1100, 02015 HUT, FINLAND [email protected] www.sal.hut.fi/Personnel/Homepages/AhtiS.html Date: June, 2005 Status: Systems Analysis Laboratory Research Reports E18 June 2005 Abstract: This paper examines the valuation of multi-period projects in a setting
where (i) the investor maximizes her terminal wealth level, (ii) she can invest in securities and private investment opportunities, and (iii) markets are incomplete, i.e. the cash flows of private investments cannot necessarily be replicated using financial securities. Based on Gustafsson and Salo’s (2005) Contingent Portfolio Programming, we develop a multi-period mixed asset portfolio selection model, where project management decisions are captured through project-specific decision trees. This model properly captures the opportunity costs imposed by alternative investment opportunities and determines the appropriate risk-adjustment to the projects based on their effect on the investor’s aggregate portfolio risk. The project valuation procedure is based on the concepts of breakeven selling and buying prices, which require the solution of mixed asset portfolio selection models with and without the project being valued. The valuation procedure is demonstrated through numerical experiments.
Keywords: project valuation, mixed asset portfolio selection, contingent portfolio
programming
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1 Introduction The valuation of risky multi-period projects is one of the most fundamental topics in corporate
finance (Brealey and Myers 2000, Luenberger 1998). Among the numerous methods developed
for this purpose, the risk-adjusted net present value (NPV) method (Brealey and Myers 2000,
Chapter 2) and the decision tree technique (Raiffa 1968; Brealey and Myers 2000, Chapter 10)
are among the most popular ones. These two methods are complementary in the sense that a
decision tree can be used to structure the management decisions and uncertainties related to a
project, while the risk-adjusted NPV method gives the present value of each risky cash flow
stream that can be acquired from the decision tree, provided that the appropriate risk-adjusted
discount rate is known.
For projects of market-traded companies, the appropriate discount rate can, in principle, be
derived using the Capital Asset Pricing Model (CAPM; Sharpe 1964, Lintner 1965; see
Rubinstein 1973). However, there are no direct guidelines for determining the discount rate
when the investor is not a public company, e.g., when the investor is an individual, a
governmental agency, or a non-listed firm, which do not have a share price to maximize. In a
series of recent papers, Gustafsson et al. (2004), De Reyck et al. (2004) and Gustafsson and
Salo (2004) have studied the valuation of risky projects in this setting under the assumptions
that (i) the investor maximizes her terminal wealth level, (ii) she can invest in securities in
financial markets as well as in a portfolio of private investment opportunities, and (iii) markets
are incomplete, i.e. the cash flows of private investments cannot necessarily be replicated using
financial securities. However, due to the complexity of modeling a portfolio of multi-period
projects, the developed models have been limited to two time periods only and hence cannot be
applied to the valuation of a multi-period project whose management strategy can be altered
over the course of the project’s life cycle.
In this paper, we extend the valuation framework presented in Gustafsson et al. (2004) and
Gustafsson and Salo (2004) to multiple time periods and intermediate project management
decisions. Both the original framework and its multi-period extension are based on a mixed
asset portfolio selection (MAPS) model – a portfolio model including both projects and
securities – and the concepts of breakeven selling and buying prices, which compare the values
of optimal portfolios including and not including the analyzed project. We employ Gustafsson
and Salo’s (2005) Contingent Portfolio Programming (CPP) to develop a multi-period MAPS
model that captures the management decisions and uncertainties related to risky multi-period
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projects through project-specific decision trees. We extend CPP to include securities and
generalize its objective function to several types of preference models, ranging from the
expected utility model to various non-expected utility models.
The main contribution of this paper is the development of a generic framework for the
valuation of risky multi-period projects in a setting where the investor maximizes her terminal
wealth level and where the possible investment assets include both projects and securities. Also,
we discuss how the results of the earlier MAPS-based project valuation papers extend to the
multi-period setting. In particular, we show that breakeven prices remain consistent with
contingent claims analysis when the investor is a non-expected utility maximizer, and that the
resulting prices are, under certain conditions, the same as in Hillier’s (1963) method. The use of
the resulting framework is demonstrated through numerical experiments.
The remainder of this paper is structured as follows. Section 2 reviews earlier approaches to the
valuation of multi-period projects and discusses their assumptions and shortcomings. Section 3
gives an overview of the model developed in this paper. Section 4 discusses the modeling of
states, securities, and projects in the present approach, and Section 5 gives an explicit
formulation of the portfolio selection model. Section 6 describes how the developed portfolio
model can be used in project valuation, and Section 7 illustrates the approach with numerical
experiments. Finally, Section 8 draws conclusions.
2 Earlier Approaches The literature on corporate finance contains a large number of apparently rivaling methods for
the valuation of risky multi-period projects. In what follows, we briefly review some of the most
widely used approaches, namely, (i) decision trees (Hespos and Strassman 1965, Raiffa 1968),
(ii) expected utility theory (von Neumann and Morgenstern 1947, Raiffa 1968), (iii) the risk-
adjusted NPV method (Brealey and Myers 2000), (iv) real options (Dixit and Pindyck 1994,
Trigeorgis 1996), (v) Robichek and Myers’ (1966) certainty equivalent method (see also Brealey
and Myers 2000, Chapter 9), (vi) Hillier’s (1963) method, and (vii) Smith and Nau’s (1995)
method. These methods are summarized in Table 1.
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Table 1. Methods for the valuation of risky multi-period investments.
Use
Method CE PV ST Formula / explanation
Risk-adjusted NPV X [ ]
1 (1 )
Tt
tt adj
E cNPV I
r=
= − ++∑
Decision tree X A flow chart with decision and chance nodes
Expected utility theory X [ ] [ ]( )1 ( )CE X u E u X−=
Contingent claims analysis X 0
n
i ii
NPV I S x ∗
=
= − + ∑
Robichek and Myers (1966) X [ ]
1 (1 )
Tt
tt f
CE cNPV I
r=
= − ++∑
Hillier (1963) X 1 (1 )
Tt
tt f
cNPV I CEr=
⎡ ⎤= − + ⎢ ⎥
+⎢ ⎥⎣ ⎦∑
Smith and Nau (1995) X
NPV = breakeven selling or buying price
Preference model for cash flow streams:
( ) ( )1 2 1 2[ , ,..., ] , ,...,T TU c c c E u c c c∗= ⎡ ⎤⎣ ⎦
Gustafsson et al. (2004)
Gustafsson and Salo (2004) X
NPV = breakeven selling or buying price
Preference model for terminal wealth levels
Key: CE = Certainty equivalent for a risky alternative, PV = Present value of a risky cash flow stream,
ST = Structuring of decision opportunities and uncertainties, I = investment cost, ct = risky cash flow at
time t, radj = risk-adjusted discount rate, u = utility function, Si = price of security i, ix ∗ = amount of
security i in the replicating portfolio, rf = risk-free interest rate, u* = intertemporal (multi-attribute)
utility function.
2.1 Decision Trees and Related Approaches
In general terms, a decision tree describes the points at which decisions can be made and the
way in which these points are related to unfolding uncertainties. Conventionally, decision trees
have been utilized with expected utility theory (von Neumann and Morgenstern 1947) so that
each end node of the decision tree is associated with the utility implied by the earlier actions
and the uncertainties that have resolved earlier. This decision tree formulation does not
explicitly include the time axis or provide guidelines for accounting for the time value of money.
In corporate finance, decision trees are used to describe how project management decisions
influence the cash flows of the project (see, e.g., Brealey and Myers 2000, Chapter 10). Here,
decision trees are typically applied together with the risk-adjusted NPV method, whereby an
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explicitly defined time axis is also constructed. However, the selection of an appropriate
discount rate for the NPV method is often problematic, mainly because the rate is influenced
by three confounding factors, (i) the risk of the project, which depends on the project’s
correlation with other investments, (ii) the opportunity costs imposed by alternative investment
opportunities, and (iii) the investor’s risk preferences, which affect the two other factors by
determining (a) the degree of risk-adjustment for the project and (b) the optimal alternative
investment portfolio when the investor does not invest in the project.
Several methods for selecting the discount rate have been proposed in the literature. However,
most of them have problematic limitations. For example, the weighted average cost of capital
(WACC) is appropriate only for average-risk investments in a firm, whereas discount rates
based on expected utility theory do not account for the opportunity costs imposed by securities
in financial markets. The real options literature suggests the use of contingent claims analysis
(CCA) to derive the appropriate discount rate by constructing replicating portfolios using
market-traded securities. Still, it may be difficult to construct replicating portfolios for private
projects in practice. Last, the use of a CAPM discount rate is appropriate only for market-
traded companies.
2.2 Robichek and Myers’ and Hillier’s Methods
Robichek and Myer’s (1966) and Hillier’s (1963) methods are two alternative ways of
determining a risk-adjustment to a discount rate in a multi-period setting. These methods have
been widely discussed in the literature on corporate finance (see, e.g., Keeley and Westerfield
1972, Bar-Yosef and Mesznik 1977, Beedles 1978, Fuller and Kim 1980, Chen and Moore 1982,
Gallagher and Zumwalt 1991, Ariel 1998, and Brealey and Myers 2000). They both employ
expected utility theory or a similar preference model to derive a certainty equivalent (CE) for a
risky prospect.
In Robichek and Myers’ method, the investor first determines a CE for the cash flow of each
period, and then discounts it back to its present value at the risk-free interest rate. Yet,
because CEs are taken separately for each cash flow, the method does not account for the effect
of cash flows’ temporal correlation on the cash flow stream’s aggregate risk; hence, it may lead
to an unnecessarily large risk-adjustment. For example, consider a cash flow stream that yields
a random cash flow X at time 1 and (1 )fX r a− + + at time 2 (a is a constant and rf is the
risk-free interest rate). Assuming that the investor invests the funds obtained at time 1 in the
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risk-free asset, the investor will acquire cash equal to a for sure at time 2. To avoid arbitrage
opportunities, this cash flow stream has to be discounted at the risk-free interest rate. Still,
since cash flows at times 1 and 2 are separately risky, Robichek and Myers’ method leads to a
discount rate that is higher than the risk-free interest rate, which is incorrect. The importance
of recognizing intertemporal correlation of cash flows is further discussed in Fuller and Kim
(1980).
On the other hand, in Hillier’s (1963) method, which is in the context of certainty equivalents
also referred to as the single certainty equivalent (SCE) method (Keeley and Westerfield 1972),
we first determine the cash flow streams that can be acquired with the project in different
scenarios and then calculate the NPVs of these streams using the risk-free interest rate. The
result is a probability distribution for risk-free-discounted NPV, for which a CE is then
determined. However, the use of the risk-free interest rate essentially means that any money
received before the end of the planning horizon is invested in the risk-free asset. Yet, it might
be more advantageous to invest the funds in risky securities instead. Therefore, Hillier’s method
is, strictly speaking, applicable only in settings, where the investor cannot invest in risky
securities.
Perhaps the most restrictive assumption used in Hillier’s method as well as in Robichek and
Myers’ method is that the risk-adjustment to the discount rate is determined only using the
investor’s risk preferences without considering alternative investment opportunities. Yet, in
general, the appropriate discount rate depends also on the investment portfolio in which the
investor would invest if she did not invest in the project. For example, consider a risk-neutral
investor who can invest in one project and in risky securities in financial markets. We know
that, in financial markets, such an investor would invest all her funds in the security with the
highest expected return. Therefore, the appropriate discount rate for the NPV of the investor’s
project is equal to the expected rate of return of this security. Yet, both of the methods
discussed in this section would give a discount rate equal to the risk-free interest rate for the
project. The fact that Robichek and Myers’ and Hillier’s methods do not adequately consider
alternative investment opportunities limits their applicability to a setting where the investor
can invest only in the analyzed project and in the risk-free asset.
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2.3 Smith and Nau’s Method
The idea behind Smith and Nau’s (1995) method, which Smith and Nau call “full decision tree
analysis,” is to explicitly account for security trading in each decision node of a decision tree.
The main advantage of the approach is that it appropriately accounts for the effect that the
possibility to invest in securities has on the discount rate of a risky project. However, the
method does not consider alternative projects, which impose an opportunity cost on the project
being valued. Also, the method relies on the assumption that the investor maximizes the
(intertemporal) utility of the project’s cash flows rather than the utility of the investor’s
terminal wealth level. Therefore, it does not necessarily lead to an investment portfolio with
maximal NPV, when NPV is defined as the present equivalent of the investment’s future value
(Luenberger 1998).
Also, practically appealing forms of the method rely on several restrictive assumptions: in order
to develop a useful rollback procedure, Smith and Nau (1995) assume (i) additive independence
(Keeney and Raiffa 1976), (ii) constant absolute risk aversion (CARA), and (iii) partial
completeness of markets. Yet, as pointed out by Keeney and Raiffa (1976), additive
independence entails possibly unrealistic preferential restrictions. The CARA assumption seems
also questionable, because it leads to an exponential utility function with utility bounded from
above. This is known to result in an unrealistic degree of risk aversion at high levels of
outcomes (see, e.g., Rabin 2000). In practice, it may also be difficult to create a replicating
portfolio for market-related cash flows of a project, as assumed in partially complete markets.
In view of the limitations of earlier approaches, we develop a valuation method for risky multi-
period projects where project management decisions are structured as project-specific decision
trees. The method relies on fewer assumptions about the investor’s preference structure than
Smith and Nau’s integrated procedure, allowing the use of a wide range of preference models.
We also employ the objective of maximization of the investor’s terminal wealth level, which
ensures consistency with NPV maximization.
3 Model Overview In a MAPS model, there are two kinds of assets, projects and securities. Projects produce cash
flows according to the chosen project management strategy; cash flows from securities are
realized through trading decisions. Similarly to CPP, uncertainties are modeled using a state
tree, which depicts the structure of future states of nature (Figure 1). The state tree need not
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7
be binomial or symmetric; it may also take the form of a multinomial tree that has different
probability distributions in its branches. In each non-terminal state, securities can be bought
and sold in any, possibly fractional quantities. As usual in the financial literature, we assume
that there are no transaction costs or profits tax.
Using the CPP framework, projects are modeled using decision trees that span over the state
tree. Figure 2 describes how project decisions (the figure on the left), when combined with the
state tree in Figure 1, lead to project-specific decision trees (the figure on the right). The
specific feature of these decision trees is that the chance nodes – since they are generated using
the common state tree – are shared by all projects. Security trading is implemented through
state-specific trading variables, which are similar to the ones used in financial models of
stochastic programming (e.g., Mulvey et al. 2000) and in Smith and Nau’s (1995) method.
0 1 2
Stat
es
Time
ω0
50%
30%
70%
40%
60%
50%
15%
35%
20%
30%
ω1
ω2
ω12
ω21
ω22
ω11
Figure 1. A state tree.
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8
Start? Continue?Yes
No
Yes
No
0 1 2Time
0
Start?Yes
No
2
Continue?Yes
No
Yes
No
1Time
ω11
ω12ω11
ω12ω21
ω22ω21
ω22
ω1
ω2
Figure 2. A decision tree for a project.
The investor seeks to maximize the utility of her terminal wealth level. When the investor’s
preferences are captured by the preference functional U, this objective can be expressed as
max [ ]U X ,
where the random variable X represents the amount of cash at the end of the planning horizon.
When the investor is able to determine a certainty equivalent for any X, U can be expressed as
a strictly increasing transformation of the investor’s certainty equivalent operator CE. Hence,
the objective can be written as
max [ ]CE X ,
which is the general objective function used in CPP.
Alternatively, it is possible to use a risk-constrained mean-risk model (Gustafsson et al. 2004),
where the investor maximizes the mean of X and constrains the risk level for X. Let ρ be the
investor’s risk measure and R her risk tolerance, i.e. the maximum level for risk. Then, the
objective function is
max [ ]E X
and the risk constraint is
[ ]X Rρ ≤ .
A risk-constrained model is often relevant, because without a risk constraint several preference
models may yield unbounded solutions when there is a possibility to purchase securities and
borrow without limit (see Gustafsson and Salo 2004).
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4 Formulation of States, Assets, and Cash Flows
4.1 States Let the planning horizon be 0,...,T . The set of states in period t is denoted by tΩ , and the
set of all states is 0
T
tt=
Ω = Ω∪ . The state tree starts with base state 0ω in period 0. Each non-
terminal state is followed by at least one state. This relationship is modeled by the function
:B Ω → Ω which returns the immediate predecessor of each state, except for the base state,
for which the function gives 0 0( )B ω ω= .
The probability of state ω , when ( )B ω has occurred, is given by ( )( )Bp ω ω . Unconditional
probabilities for each state, except for the base state, can be computed recursively from the
equation ( )( ) ( ) ( ( ))Bp p p Bωω ω ω= ⋅ . The probability of the base state is 0( ) 1p ω = .
4.2 Assets
Let there be n securities available in financial markets. The amount of security i bought in
state ω is indicated by trading variable ,ix ω , 1,...,i n= , ω ∈Ω , and the price of security i in
state ω is denoted by ( )iS ω . Under the assumption that all securities are sold back in the next
period (they can immediately be re-bought), the cash flows implied by security i in state
0ω ω≠ is ( ), ( ) ,( )i i B iS x xω ωω ⋅ − . In base state 0ω , the cash flow is 00 ,( )i iS x ωω− ⋅ .
The investor can invest privately in m projects. The decision opportunities for each project are
structured as a decision tree which is formed of decision points. For each project 1,...,k m= ,
the decision tree is implemented using a set of decision points kD and the function ( )ap d that
gives the action leading to decision point 0\k kd D d∈ , where 0kd is the first decision point of
project k. Let dA be the set of actions that can be taken in decision point kd D∈ . For each
action a in dA , a binary action variable ,k az indicates whether the action is selected or not.
These variables are bound by the restriction that, at each decision point, only one of them can
be equal to one at a time. The action in decision point d is chosen in state ( )dω .
For a project k, the vector of all action variables ,k az relating to the project, denoted by kz , is
called the management strategy of project k. The vector of all action variables of all projects,
denoted by z, is termed the project portfolio management strategy. The pair ( , )x z , composed
of all trading and action variables, is the (mixed asset) portfolio management strategy.
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4.3 Cash Flows and Cash Surpluses
Let ( , )pkCF ωkz be the cash flow of project k in state ω when the project management strategy
is kz . When , ( )k aC ω is the cash flow in state ω implied by action a , this cash flow is given by
, ,:
( ) ( )
( , ) ( )k dB
pk k a k a
d D a Ad
CF C z
ω ω
ω ω∈ ∈∈Ω
= ⋅∑ ∑kz ,
where the restriction in the summation of decision points guarantees that actions yield cash
flows only in the prevailing state and in the future states that can be reached from the
prevailing state. The set ( )B ωΩ is defined as
( ) | 0 such that ( )B kk Bω ω ω ω′ ′Ω = ∈Ω ∃ ≥ = , where 1( ) ( ( ))n nB B Bω ω−= is the n:th
predecessor of ω ( 0( )B ω ω= ).
The cash flows from security i in state ω ∈Ω are given by
( ),
, ( ) ,
( )( , )
( )
i is
ii i B i
S xCF
S x x
ω
ω ω
ωω
ω
− ⋅⎧⎪= ⎨⋅ −⎪⎩
ix 0
0
if
if
ω ω
ω ω
=
≠
Thus, the aggregate cash flow ( , , )CF ωx z in state ω ∈Ω , obtained by summing up the cash
flows for all projects and securities, is
( )
1 1
, , , 01 :
( ) ( )
, ( ) , , , 01 :
( ) ( )
( , , ) ( , ) ( , )
( ) ( ) , if
( ) ( ) , if
k dB
k dB
n ms p
i ki k
n
i i z a k ai d D a A
d
n
i i B i z a k ai d D a A
d
CF CF CF
S x C z
S x x C z
ω
ω ω
ω ω
ω ω
ω ω ω
ω ω ω ω
ω ω ω ω
= =
= ∈ ∈∈Ω
= ∈ ∈∈Ω
= +
⎧ − ⋅ + ⋅ =⎪⎪⎪= ⎨⎪ ⋅ − + ⋅ ≠⎪⎪⎩
∑ ∑
∑ ∑ ∑
∑ ∑ ∑
i kx z x z
Together with the initial budget of each state, cash flows define cash surpluses that would
remain in state ω ∈Ω if the investor chose portfolio management strategy ( , )x z . Assuming
that excess cash is invested in the risk-free asset, the cash surplus in state ω ∈Ω is given by
( ) ( )
( ) ( , , )
( ) ( , , ) (1 )B B
b CFCS
b CF r CS
x z
x zωω ω ω
ω ω
ω ω →
+⎧⎪= ⎨ + + + ⋅⎪⎩
0
0
if
if
ω ω
ω ω
=
≠, (1)
where ( )b ω is the initial budget in state ω ∈Ω and ( )Br ω ω→ is the short rate at which cash
accrues interest from state ( )B ω to ω . Cash surplus in a terminal state is the investor’s
terminal wealth level in that state.
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5 Optimization Model When using the preference functional U, the objective function for the MAPS model is written
as a function of cash surplus variables of the last time period, i.e.
( ), ,
maxU Tx z CSCS
where CST denotes the vector of cash surplus variables related to period T. Under the risk-
constrained mean-risk model, the objective is to maximize the expectation of the investor’s
terminal wealth level, viz.
, ,max ( )
T
p CSx z CS ω
ω
ω∈Ω
⋅∑ .
Three types of constraints are imposed on the model: (i) budget constraints, (ii) decision
consistency constraints, and (iii) risk constraints, which apply to risk-constrained models only.
Different versions of the MAPS model are summarized in Table 2.
5.1 Budget Constraints
Budget constraints ensure that there is a nonnegative amount of cash in each state. They can
be implemented using continuous cash surplus variables CSω , which measure the amount of
cash in state ω . Using (1), these variables lead to the following budget constraints:
00 0( , , ) ( )CF CS bωω ω− = −x z
( ) ( ) 0( , , ) (1 ) ( ) \ B BCF r CS CS bω ω ω ωω ω ω ω→+ + ⋅ − = − ∀ ∈Ωx z .
Note that if CSω is negative, the investor borrows money at the risk-free interest rate to cover
a funding shortage. Thus, CSω can also be regarded as a trading variable for the risk-free asset.
5.2 Decision Consistency Constraints
Decision consistency constraints implement the projects’ decision trees. They require that (i) at
each decision point at which the investor arrives, only one action is selected, and that (ii) at
each decision point at which the investor does not arrive, no action is taken, implying that the
point does not incur any cash flows. Decision consistency constraints can be written as
0
, 1 1,...,dk
k aa A
z k m∈
= =∑ (2)
0, , ( ) \ 1,...,d
k a k ap d k ka A
z z d D d k m∈
= ∀ ∈ =∑ , (3)
where (2) ensures that one action is selected in the first decision point, and (3) implements the
above requirements for other decision points.
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12
Table 2. Multi-period MAPS models.
Preference
functional model
General mean-risk
model
Deviation-based mean-risk model
Objective
function ( )
, ,maxU Tx z CS
CS , ,
max ( )T
p CSωω
ω∈Ω
⋅∑x y CS
Budget
constraints 00 0( , , ) ( )CF CS bωω ω− = −x y
( ) ( ) 0( , , ) (1 ) ( ) \ B BCF r CS CS bω ω ω ωω ω ω ω→+ + ⋅ − = − ∀ ∈Ωx y
Decision
consistency
constraints
0
, 1 1,...,dk
k aa A
z k m∈
= =∑
0, , ( ) \ 1,...,d
k a k ap d k ka A
z z d D d k m∈
= ∀ ∈ =∑
Risk
constraints - ( ) Rρ ≤TCS
( ), Rρ − +∆ ∆ ≤( ) 0 TCSω ω ωτ ω+ −− − ∆ + ∆ = ∀ ∈ΩTCS
Variables
, 0,1 1,...,k a d kz a A d D k m∈ ∀ ∈ ∀ ∈ =
, free 1,...,ix i nω ω∀ ∈Ω =
freeCSω ω∀ ∈Ω
, 0,1 1,...,k a d kz a A d D k m∈ ∀ ∈ ∀ ∈ =
, free 1,...,ix i nω ω∀ ∈Ω =
freeCSω ω∀ ∈Ω
0 Tω ω−∆ ≥ ∀ ∈Ω
0 Tω ω+∆ ≥ ∀ ∈Ω
5.3 Risk Constraints
A risk-constrained model includes one or more risk constraints. For the sake of analogy with
Gustafsson et al. (2004), we focus here on the single constraint case. When ρ denotes the risk
constraint and R the risk tolerance, the risk constraint can be expressed as
( ) Rρ ≤TCS .
Many common dispersion statistics such as variance (V; Markowitz 1952, 1987), semivariance
(SV; Markowitz 1959), absolute deviation (AD; Konno and Yamazaki 1991), lower semi-
absolute deviation (LSAD; Gotoh and Konno 2000), and expected downside risk (EDR; Eppen
et al. 1989) can be formulated through deviation constraints introduced in Gustafsson and Salo
(2005) (see also Gustafsson et al. 2004). In general, deviation constraints are expressed as
( ) 0 TCSω ω ωτ ω+ −− − ∆ + ∆ = ∀ ∈ΩTCS ,
where ( )τ TCS is a function defining the target value from which the deviations are calculated,
and ω+∆ and ω
−∆ are nonnegative deviation variables which measure how much the cash surplus
in state Tω ∈Ω differs from the target value. For example, when the target value is the mean
of the terminal wealth level, the deviation constraints are written as
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13
( ) 0 ,T
TCS p CSω ω ω ωω
ω ω+ −′
′∈Ω
′− − ∆ + ∆ = ∀ ∈Ω∑
Using these deviation variables, some common dispersion statistics can be written as
AD: ( ) ( )T
p ω ωω
ω − +
∈Ω
⋅ ∆ + ∆∑ .
LSAD: ( )T
p ωω
ω −
∈Ω
⋅ ∆∑ .
V: 2( ) ( )
T
p ω ωω
ω − +
∈Ω
⋅ ∆ + ∆∑
SV: 2( ) ( )T
p ωω
ω −
∈Ω
⋅ ∆∑ .
The respective fixed-target value statistics can be obtained with the deviation constraints
0 TCSω ω ωτ ω+ −− − ∆ + ∆ = ∀ ∈Ω ,
where τ is the fixed target level. EDR, for example, can then be obtained from the sum
( )T
p ωω
ω −
∈Ω
⋅ ∆∑ .
6 Project Valuation The value of a project can be defined as the lowest price at which a rational investor would be
willing to sell a project, if she had the project, and as the highest price at which a rational
investor would be willing to buy the project, if she did not have it. These prices are referred to
as the breakeven selling price and breakeven buying price of the project, respectively
(Gustafsson et al. 2004, Luenberger 1998, Smith and Nau 1995). In this section, we extend the
definitions of these prices, as presented in Gustafsson et al. (2004), to a multi-period MAPS
setting.
The breakeven prices for a project can be determined from optimization problems where the
investor invests and does not invest in the project (see Table 3). Let the project being valued
be indexed with j. The action associated with not starting the project is denoted by a*. When
calculating the breakeven selling price, we first determine the optimal investment portfolio with
the project. Then, we iteratively solve another optimization problem where the investor does
not invest in the project but where an amount sjv is added to the budget at time 0 instead. The
breakeven selling price is the amount sjv that makes the portfolios with and without the project
equally preferable. In a preference functional model, this means that the utilities of the two
portfolios are identical. In a mean-risk model, two portfolios are equally preferred if they are
equal in terms of the expected terminal wealth level and they both satisfy the risk constraint.
The breakeven buying price for a project is determined similarly as its breakeven selling price.
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14
The difference is that, in the first setting, the investor does not invest in the project, and in the
second setting, she invests in the project and deducts an amount bjv from the budget at time 0.
The breakeven selling price is the amount that makes the investor indifferent between the
portfolios with and without the project. In general, the breakeven buying price and the
breakeven selling price for a project are not equal to each other.
Table 3 describes how breakeven selling and buying prices can be determined based on the
optimization problems in Table 2. In each setting, an additional constraint is added to the
optimization model. Also, the budget at time 0 is modified in the second setting. The objective
function values of the resulting models at optimum are denoted by W + and W − depending on
whether the investor invests in the project or not.
Table 3. Definitions of the value of project j. Each setting is based on the model in Table 2.
Breakeven selling price Breakeven buying price
Definition sjv such that s sW W+ −= b
jv such that b bW W+ −=
Status quo Additional constraint:
, * 0j az = , i.e. invest in project
Budget at time 0:
0( )b ω
Optimal objective function value:
sW +
Additional constraint:
, * 1j az = , i.e. do not invest in project
Budget at time 0:
0( )b ω
Optimal objective function value:
bW −
Second
setting
Additional constraint:
, * 1j az = , i.e. do not invest in project
Budget at time 0:
0( ) sjb vω +
Optimal objective function value:
sW −
Additional constraint:
, * 0j az = , i.e. invest in project
Budget at time 0:
0( ) bjb vω −
Optimal objective function value:
bW +
Breakeven selling and buying prices exhibit a notable property: they yield the same result as
CCA whenever the method is applicable, i.e. whenever the cash flows of the project can be
replicated by trading financial securities. In general, the CCA value of the project is defined as
the value of the portfolio that is required to create a trading strategy replicating the future cash
flows of the project minus the project’s investment cost at time 0 (see Table 1). Smith and Nau
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(1995) show that this property holds for investors with an intertemporal utility function over
cash flow streams, whereas Gustafsson and Salo (2004) prove the same for non-expected utility
investors who maximize their terminal wealth level in a two-period setting. The following
proposition generalizes the latter result to a multi-period setting.
PROPOSITION 1. If there is a replicating trading strategy for a project, the breakeven buying
price and breakeven selling price for the project are equal and yield the same value as CCA.
PROOF: See Appendix.
Gustafsson and Salo (2004) and Gustafsson et al. (2004) also discuss valuation formulas that
facilitate the computation of project values for (i) certain types of mean-risk investors, (ii)
Choquet-expected utility maximizers exhibiting constant absolute risk aversion, and (iii)
investors using Wald’s (1950) maximin criterion. These formulas are applicable in the two-
period case and are of the form
1 f
V Vvr
+ −−=
+,
where V + and V − are the optimal objective function values with and without the project,
respectively, and rf is the risk-free interest rate. Since the propositions rely on the examination
of the last time period with no consideration for the number of intermediate periods, these
formulas can also be used in a multi-period setting with the modification that the objective
function values are discounted using the time-T spot-rate, sT, viz.
(1 )TT
V Vvs
+ −−=
+.
The formal proof of the formulas in a multi-period setting is almost identical to their proof in
the two-period setting and is hence omitted.
As shown by Proposition 2, there exists also an analogous formula for the breakeven selling
price when the investor can invest only in a single project and the risk-free asset. This
proposition holds for any rational preference model accommodated by the portfolio models in
Table 2. If we further assume that the investor exhibits constant absolute risk aversion, i.e. the
investor’s certainty equivalent operator satisfies [ ] [ ]CE X b CE X b+ = + for all random
variables X and constants b, breakeven selling and buying prices are equal to each other; they
are also independent of the budget, and they can be computed from the certainty equivalent of
the project’s future value (Proposition 3). The proofs of these two propositions are in Appendix.
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16
PROPOSITION 2. When the investor can invest in a single project and in the risk-free asset, the
breakeven selling price for the project can be computed using the formula (1 )TT
V Vvs
+ −−=
+, where
V + and V − are the investor’s certainty equivalents for her terminal wealth level with and
without the project, respectively, and sT is the time-T spot rate.
PROPOSITION 3. When the conditions of Proposition 2 hold and the investor’s certainty
equivalent operator satisfies [ ] [ ]CE X b CE X b+ = + for all random variables X and constants
b, breakeven buying and selling prices are equal and can be computed using the formula
(1 ) (1 )T
T TT T
V V CEvs s
+ −−= =
+ +, where CET is the certainty equivalent for the project’s future value
and sT is the time-T spot rate.
If all short rates are equal to rf, the formula in Proposition 3 can be expressed as
1
(1 )
(1 )
TT t
t ft
Tf
CE c rv I
r
−
=
⎡ ⎤+⎢ ⎥⎣ ⎦= − ++
∑,
where, following the notation in Table 1, I is the investment cost at time 0 and ct is the random
time-t cash flow implied by the project under the optimal project management strategy. Notice
that if the investor’s certainty equivalent operator further satisfies [ ] [ ]CE aX aCE X= for all
random variables X and constants a (that is, the investor exhibits also constant relative risk
aversion), the formula reduces to Hillier’s method (Table 1). This observation is formalized
with the following proposition. The proof is obvious from the above and is hence omitted.
PROPOSITION 4. When the conditions of Proposition 2 hold and the investor’s certainty
equivalent operator satisfies [ ] [ ]CE aX b aCE X b+ = + for all random variables X and
constants a and b, the breakeven buying price and breakeven selling price are equal and yield the
same value as Hillier’s method, given by the formula 1 (1 )
Tt
tt t
cNPV I CEs=
⎡ ⎤= − + ⎢ ⎥+⎣ ⎦
∑ , where I is
the investment cost of the project, ct is the cash flow at time t, and st is the time-t spot rate.
For example, Proposition 4 can be applied with risk-neutral investors, some mean-risk models
(see, e.g., Gustafsson and Salo 2005), and preference models based on Yaari’s (1987) dual
theory. However, no risk-averse expected utility maximizer exhibits both constant absolute and
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17
constant relative risk aversion, so that no risk-averse expected utility maximizer is, in general,
consistent with Hillier’s method.
While Hillier’s method can be obtained as a special case of breakeven selling and buying prices,
Robichek and Myers’ (1966) method does not comply with these prices, unless restrictive
assumptions about the analyzed cash flow streams are made. For example, when a project’s
cash flows are perfectly correlated with each other and the conditions of Proposition 4 hold,
Robichek and Myer’s method reduces to Hillier’s method, and hence gives consistent results
with the breakeven prices. On the other hand, Smith and Nau’s (1995) method coincides with
the present approach when the chosen intertemporal utility function rewards for the
maximization of the expected utility of the terminal wealth level, i.e., when the utility functions
for periods other than the last are identically zero. Indeed, maximization of intertemporal
utility and that of the utility of terminal wealth level are not equivalent objectives: in effect, it
can be shown that an intertemporal utility function can lead to an investment portfolio, which
is stochastically dominated by another investment portfolio in terms of terminal wealth levels
(Proposition 5). Since the terminal wealth level is widely regarded as the future value
equivalent of NPV, care should be taken when applying Smith and Nau’s method in a setting
where the investor aims to maximize the NPV of the investment portfolio. Table 4 summarizes
the conditions under which other project valuation methods give the same results as the present
approach.
PROPOSITION 5. Maximization of intertemporal utility may lead to a portfolio that is
stochastically dominated by another portfolio in terms of the terminal wealth level.
PROOF: See Appendix.
Table 4. The conditions when the present approach coincides with other project valuation methods.
Method Preferences Cash flows Available investments
Hillier (1963) [ ] [ ]CE aX b aCE X b+ = + – One project, risk-free asset
Robichek and Myers (1966) [ ] [ ]CE aX b aCE X b+ = + Perfectly correlated One project, risk-free asset
Smith and Nau (1995) Utility function for each period
except the last is identically 0
– One project, risk-free asset,
securities
Finally, it is worth noting that the results based on the CAPM in Gustafsson and Salo (2004)
and Gustafsson et al. (2004), such as the convergence of project values to the CAPM market
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18
prices, may not hold in a multi-period setting. The reason for this is that the mean-variance
model, on which the CAPM is based, is a dynamically inconsistent preference model (Machina
1989). Therefore, the optimal portfolio for a mean-variance investor maximizing her terminal
wealth level is, in general, different from the portfolio obtained using the two-period CAPM
consecutively in a multi-period setting.
7 Numerical Experiments In the previous sections, we developed a framework for valuing risky multi-period projects in
incomplete markets, and showed that the framework gives consistent results with CCA and,
under certain conditions, with Hillier’s (1963) method. In this section, we demonstrate the
approach through a set of numerical experiments and show that most of the phenomena
observed in a two-period MAPS setting also occur in the multi-period setting. In particular, we
cast light on the following issues:
Q1. How are the values of multi-period projects influenced by the possibility to invest
in risky securities?
Q2. How are the values of multi-period projects influenced by the possibility to invest
in other projects?
Q3. How do the values of multi-period projects depend on the investor’s risk attitude?
7.1 Experimental Setup
The experimental setup involves 3 time periods, 4 projects, and 2 securities that together
constitute the market. Uncertainties related to projects and securities are captured through a
state tree which consists of 8 equally likely time-1 states and 64 equally likely time-2 states
(Figure 3). Each state is formed of two underlying states, (i) the market state (denoted by M),
which determines the prevailing security prices, and (ii) the private state (denoted by P) that
influences project outcomes. Security prices at time 2 are given in Figure 3. Security prices at
time 1 are computed from these prices using the CAPM (Table 5), and the prices at time 0 are
obtained similarly from the time-1 security prices (Table 6).
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Security 1 Security 2 Project A Project B Project C Project D
State 0 State M1/P1 State M11/P11 $60 $36 $150 $150 $100 $180State M12/P11 $50 $12 $150 $100 $120 $60State M13/P11 $40 $36 $150 $50 $90 $180State M14/P11 $30 $12 $150 $0 $80 $60State M11/P12 $60 $36 $50 $250 $170 $180State M12/P12 $50 $12 $50 $200 $200 $60State M13/P12 $40 $36 $50 $150 $40 $180State M14/P12 $30 $12 $50 $100 $50 $60
State M2/P1 State M21/P11 $70 $36 $150 $200 $130 $180State M22/P11 $60 $12 $150 $150 $70 $60State M23/P11 $50 $36 $150 $100 $130 $180State M24/P11 $40 $12 $150 $50 $110 $60State M21/P12 $70 $36 $50 $300 $100 $180State M22/P12 $60 $12 $50 $250 $180 $60State M23/P12 $50 $36 $50 $200 $40 $180State M24/P12 $40 $12 $50 $150 $50 $60
State M3/P1 State M31/P11 $80 $36 $150 $250 $190 $180State M32/P11 $70 $12 $150 $200 $45 $60State M33/P11 $60 $36 $150 $150 $60 $180State M34/P11 $50 $12 $150 $100 $100 $60State M31/P12 $80 $36 $50 $350 $110 $180State M32/P12 $70 $12 $50 $300 $50 $60State M33/P12 $60 $36 $50 $250 $140 $180State M34/P12 $50 $12 $50 $200 $90 $60
State M4/P1 State M41/P11 $90 $36 $150 $300 $65 $180State M42/P11 $80 $12 $150 $250 $105 $60State M43/P11 $70 $36 $150 $200 $100 $180State M44/P11 $60 $12 $150 $150 $115 $60State M41/P12 $90 $36 $50 $400 $175 $180State M42/P12 $80 $12 $50 $350 $90 $60State M43/P12 $70 $36 $50 $300 $210 $180State M44/P12 $60 $12 $50 $250 $180 $60
State M1/P2 State M11/P21 $60 $36 $80 $220 $50 $180State M12/P21 $50 $12 $80 $170 $40 $60State M13/P21 $40 $36 $80 $120 $110 $180State M14/P21 $30 $12 $80 $70 $105 $60State M11/P22 $60 $36 $40 $260 $125 $180State M12/P22 $50 $12 $40 $210 $115 $60State M13/P22 $40 $36 $40 $160 $235 $180State M14/P22 $30 $12 $40 $110 $170 $60
State M2/P2 State M21/P21 $70 $36 $80 $270 $250 $180State M22/P21 $60 $12 $80 $220 $215 $60State M23/P21 $50 $36 $80 $170 $180 $180State M24/P21 $40 $12 $80 $120 $190 $60State M21/P22 $70 $36 $40 $310 $130 $180State M22/P22 $60 $12 $40 $260 $100 $60State M23/P22 $50 $36 $40 $210 $70 $180State M24/P22 $40 $12 $40 $160 $90 $60
State M3/P2 State M31/P21 $80 $36 $80 $320 $240 $180State M32/P21 $70 $12 $80 $270 $150 $60State M33/P21 $60 $36 $80 $220 $120 $180State M34/P21 $50 $12 $80 $170 $100 $60State M31/P22 $80 $36 $40 $360 $230 $180State M32/P22 $70 $12 $40 $310 $100 $60State M33/P22 $60 $36 $40 $260 $120 $180State M34/P22 $50 $12 $40 $210 $240 $60
State M4/P2 State M41/P21 $90 $36 $80 $370 $100 $180State M42/P21 $80 $12 $80 $320 $160 $60State M43/P21 $70 $36 $80 $270 $250 $180State M44/P21 $60 $12 $80 $220 $290 $60State M41/P22 $90 $36 $40 $410 $200 $180State M42/P22 $80 $12 $40 $360 $100 $60State M43/P22 $70 $36 $40 $310 $50 $180State M44/P22 $60 $12 $40 $260 $45 $60
Figure 3. State tree with security prices and project cash flows at time 2.
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20
Table 5. Securities in states M1–M4.
M1 M2 M3 M4
1 2 1 2 1 2 1 2
Beta 0.821 1.527 0.782 1.795 0.755 2.064 0.735 2.332
Market price $39.32 $20.00 $48.58 $20.00 $57.83 $20.00 $67.09 $20.00
Capitalization weight 74.68% 25.32% 78.46% 21.54% 81.26% 18.74% 83.42% 16.58%
Expected rate of return 14.46% 20.00% 13.23% 20.00% 12.39% 20.00% 11.78% 20.00%
St. dev. of rate of return 28.44% 60.00% 23.02% 60.00% 19.33% 60.00% 16.66% 60.00%
Market expected return 15.86% 14.68% 13.82% 13.15%
Market st. dev. 31.15% 26.49% 23.05% 20.39%
Table 6. Securities in state 0.
Security
1 2
Market price in M1 $39.32 $20.00
Market price in M2 $48.58 $20.00
Market price in M3 $57.83 $20.00
Market price in M4 $67.09 $20.00
Beta 1.264 0.000
Market price in 0 46.85 18.52
Capitalization weight 79.14% 20.86%
Expected rate of return 13.58% 8.00%
St. dev. of rate of return 22.10% 0.00%
Market expected return 12.41%
Market standard deviation 17.49%
The CAPM requires us to specify the amounts of outstanding shares and the expected rate of
return of the market portfolio in each state. We have assumed here that there are a total of
15,000,000 shares of security 1 and 10,000,000 shares of security 2. In each state, the expected
rate of return of the market portfolio is determined so that the market price of risk – the excess
rate of return of the market portfolio divided by the standard deviation of the market –
remains constant at 0.252. This is a reasonable way of selecting the expected rate of return of
the market, because hereby it is, in each state, proportional to the standard deviation of the
market (see Tables 5 and 6). Also, this implies that security 2 is priced at $20 in each time-1
market state, which is desirable, because the time-2 prices for security 2 are independent of the
time-1 market state (Figure 3). The risk-free interest rate is 8%.
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21
Table 7 presents the investments required to start projects A, B, C and D. The investments are
made in two rounds; the first financing round takes place at time 0 and the second at time 1. If
the investment is made in both rounds, the project produces a cash flow at time 3. Figure 3
gives this cash flow as a function of the time-2 state. The specific characteristics of these cash
flows are summarized in Table 7. The cash flows of project A depend only on the private states,
implying that the project is uncorrelated with the market. On the other hand, the cash flows of
project B can be separated into private and market related components; the private component
depends on the private state and the market component on the market state. Furthermore, the
markets are partially complete (see Smith and Nau 1995) with respect to project B in the sense
that the market component of project B can be replicated by buying 5 shares of security 1; the
private component is obtained by taking the negative of project A. Similarly, the outcome of
project C depends on both private and market states but it cannot be separated into market
and private components. In contrast, the cash flows of project D are entirely market-dependent
and they can be replicated by purchasing 5 shares of security 2.
Table 7. Projects.
Project
A B C D
Private state dependence Yes Yes Yes No
Market state dependence No Yes Yes Yes
Market component replication No Yes No Yes
Investment cost at time 0 $10 $20 $30 $40
Investment cost at time 1 $60 $150 $70 $40
Expected outcome $80.00 $220.00 $127.42 $120.00
St. dev. of outcome $43.01 $90.00 $62.66 $60.00
We examine two preference models, (i) the risk-constrained mean-variance (MV) model, with
which we use the deviation-based mean-risk model in Table 2, and (ii) the expected utility
(EU) model with constant absolute risk aversion (CARA). As the base case, the latter model
employs the preference functional [ ] [ ]( ) XU X E u X E e α−= = − ⎡ ⎤⎣ ⎦ with 0.003α = unless
otherwise noted, and the former the risk tolerance of $250 for the standard deviation of the
terminal wealth level. For both preference models, breakeven selling and buying prices are equal
(see Gustafsson et al. 2004, Gustafsson and Salo 2004) and therefore they are displayed in a
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22
single entry. We begin our experiments by analyzing project values when securities are not
available and then continue to the setting with securities. The optimizations were carried out
using the GAMS software package with SBB and CONOPT algorithms (www.gams.com).
7.2 Single Project without Securities
We first solve project values in a setting where the investor can invest only in a single project
and the risk-free asset. Under both preference models, each of the projects is started at time 0.
The optimal management strategies for the projects at time 1 are described in the first eight
data rows of Table 8. The mark “X” indicates that a further investment is made in the project,
while the mark “–“ means that the project is terminated. The first mark is the optimal strategy
for the MV model; the second one indicates the optimal action for the EU investor. In the MV
model, the risk constraint is not binding, wherefore it now corresponds to expected value
maximization.
Table 8. Optimal project management strategies and project values.
Project
Time-1 State A B C D
M1/P1 X/X –/– X/X X/X
M2/P1 X/X X/X X/X X/X
M3/P1 X/X X/X X/X X/X
M4/P1 X/X X/X X/X X/X
M1/P2 –/– X/– X/X X/X
M2/P2 –/– X/X X/X X/X
M3/P2 –/– X/X X/X X/X
M4/P2 –/– X/X X/X X/X
Project value, MV/EV model $5.09 $33.69 $14.43 $25.84
Project value, EU model $3.17 $25.59 $9.58 $21.24
In Table 8, observe that a further investment is made in project A only if private state P1
obtains. This is because the outcomes of project A in the states ensuing P1 are substantially
higher than in the states following P2 (Figure 3). Project B is terminated in state M1/P1 under
both preference models. In contrast, only the MV investor chooses to continue the project in
state M1/P2. The termination decisions are explained by the low outcome with the market
component of project B (i.e., security 1); furthermore, project B is more profitable in private
state P2 than in P1, explaining the continuation decision with the (risk-neutral) MV investor.
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23
Both MV and EU investors choose to continue projects C and D in all states. With project D
this is because the realized time-1 state does not convey any information about the future cash
flows of project D (Figure 3); with project C the future outcome distribution in each time-1
state is different but in each case it warrants a continuation decision.
Since we analyze here only a single project with the risk-free asset, we can use the project
values in Table 8 to numerically verify Proposition 3. Let us take project A, for example. The
certainty equivalent for an EU investor’s terminal wealth level with project A is $586.90, while
it is $583.20 without the project. Discounting the difference of these values back to its present
value yields 2($586.90 $583.20)/1.08 $3.17Av = − = , which coincides with the project’s
breakeven selling and buying prices given in Table 8.
7.3 Single Project with Securities
We next consider a setting where the investor can invest in a single project, in the risk-free
asset, and in securities 1 and 2. We first examine the investor’s trading strategy in a case where
the investor cannot invest in projects, because we can consequently examine how the starting of
a project influences the optimal trading strategy. Table 9 gives the optimal trading strategy for
the MV investor. Note that purchase of security 2 at time 0 is equivalent to investment in the
risk-free asset; for the sake of clarity, we have assumed here that the investor invests in the
risk-free asset and not in security 2. The column CS gives the amount of money lent in each
state.
Table 9. Optimal trading strategy for the MV investor without projects.
Portfolio weights Securities bought
State 1 2 1 2 CS
0 100% 0% 16.557 0 -$275.602
M1/P1 74.7% 25.3% 14.203 9.468 -$394.475
M2/P1 78.5% 21.5% 11.809 7.872 -$224.445
M3/P1 81.3% 18.7% 9.414 6.275 -$10.034
M4/P1 83.4% 16.6% 7.019 4.679 $248.724
M1/P2 74.7% 25.3% 14.203 9.468 -$394.475
M2/P2 78.5% 21.5% 11.809 7.872 -$224.445
M3/P2 81.3% 18.7% 9.414 6.275 -$10.034
M4/P2 83.4% 16.6% 7.019 4.679 $248.724
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24
In comparison, an EU investor would buy 7.634 shares of security 1 at time 0, and 5.169 shares
of security 1 and 3.478 shares of security 2 in each time-1 state. Since the price of security 1
changes between the time-1 states, the portfolio weights for an EU investor do not remain
constant across time-1 states. An EU investor lends $142.032 in state 0, implying that the EU
model with 0.003α = is a more risk averse preference model than the mean-standard
deviation model with R = $250.
When an MV investor invests in project A, the portfolio weights of the optimal security
portfolio remain the same as in Table 9, but absolute amounts of securities in the portfolio
change. The constant portfolio weights are explained by the fact that project A is uncorrelated
with the market; this was proven in a two-period setting by Gustafsson et al. (2004). For the
same reason, an EU investor preserves her investment behavior: even when she invests in
project A, the optimal trading strategy is, as before, to buy 5.169 shares of security 1 and 3.478
shares of security 2 in each time-1 state.
Recall that the cash flows of project B can be formed by buying 5 shares of security 1 and
deducting the cash flows of project A from the resulting portfolio. The weights of the optimal
security portfolio for an MV investor now change, because project B counts for 5 shares of
security 1; if these 5 shares were added to the security portfolio, the portfolio weights would
match with the weights in Table 9. With the EU investor, we clearly observe this behavior: she
invests in 0.169 shares of security 1 in each time-1 state where the project is continued (see
Table 10) and 5.169 shares otherwise.
In contrast, the cash flows of project C cannot be separated into private and market
components, wherefore the weights of the security portfolio change without a clear pattern.
Project D can be replicated using 5 shares of security 2. Therefore, in the optimum, an EU
investor shorts 5 3.478 1.522− = shares of security 2 and buys 5.169 shares of security 1.
When these 5 shares are added to the security portfolio of the MV investor, the weights of the
portfolio correspond to the ones in Table 9.
Table 10 presents the optimal project management strategy and the values of the projects for
MV and EU investors. The optimal strategies for project A, C and D are the same as in Table
8, whereas project B is now terminated in states M2/P1 and M1/P2 under both preference
models due to opportunity costs imposed by the securities, i.e. it is more profitable to invest in
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25
the securities in states M2/P1 and M1/P2 than in project B. In particular, this implies that the
opportunity costs imposed by securities can have a major effect on projects’ optimal
management strategies.
Table 10. Optimal project management strategies and project values.
Project
A B C D
M1/P1 X/X –/– X/X X/X
M2/P1 X/X –/– X/X X/X
M3/P1 X/X X/X X/X X/X
M4/P1 X/X X/X X/X X/X
M1/P2 –/– –/– X/X X/X
M2/P2 –/– X/X X/X X/X
M3/P2 –/– X/X X/X X/X
M4/P2 –/– X/X X/X X/X
Project value, MV model $4.13 $14.53 $8.54 $15.56
Project value, EU model $3.17 $14.15 $7.07 $15.56
In each case, except for project A under the EU model, the opportunity costs lower the
resulting project values. Notice also that project D is priced at $15.56 due to the existence of a
replicating portfolio: the price of 5 shares of security 2 minus the risk-free discounted NPV of
the costs is 5 $20/1.08 $40 $40/1.08 $15.56⋅ − − = .
In summary, securities typically lead to lower project values by imposing additional
opportunity costs. However, as noted by Gustafsson et al. (2004), they may also increase
project values by providing more efficient diversification possibilities. Furthermore, if a
replicating portfolio exists for a project, the project will be priced at its CCA value. Also, the
introduction of securities may, at times, change the optimal management strategies for projects.
[Q1]
7.4 Project Portfolio with Securities
In this section, we consider the case where the investor can simultaneously invest in projects A–
D as well as in the risk-free asset and in the two securities. Here, the optimal project
management strategies are the same as in Table 10, except that both MV and EU investors
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26
make here a further investment in project B in states M2/P1; this is largely due to the fact
that projects A and B are negatively correlated with each other. Note also that, apart from
project D, the values of the projects increase, because with more investment assets available,
the risks of each project can be diversified more efficiently than previously.
Table 11 presents the optimal security portfolio of the MV investor when the projects are
managed using the optimal project management strategy. This portfolio combines the effects
that were described in the previous section: in states where project B is continued (in all states
except M1/P1 and M1/P2), the investor buys five shares of security 1 less than she would
otherwise; on the other hand, the starting of project D implies that the investor buys five
shares of security 2 less in each state. With these modifications, the optimal trading strategy is
very close to the optimal trading strategy when only project C was started. The small
differences are due to correlations between projects.
Table 11. Optimal trading strategy for the MV investor with projects A–D.
Portfolio weights Securities bought
State 1 2 1 2 CS
0 100.0% 0.0% 11.136 0.000 -$121.69
M1/P1 74.7% 25.3% 9.495 6.319 -$363.27
M2/P1 76.2% 23.8% 5.167 3.912 -$239.70
M3/P1 105.3% -5.3% 4.731 -0.685 -$67.29
M4/P1 116.5% -16.6% 4.019 -1.915 $64.41
M1/P2 95.1% 4.9% 17.451 1.769 -$525.09
M2/P2 77.3% 22.7% 4.180 2.989 -$113.30
M3/P2 94.5% 5.5% 2.046 0.343 $127.43
M4/P2 114.8% -14.8% 2.488 -1.076 $210.33
Tables 12 and 13 present sensitivity analyses for project values with respect to the MV
investor’s risk tolerance R and the EU investor’s risk aversion parameter α , respectively. At
low risk tolerance levels (high degree of risk aversion), an MV investor gives high values for
projects A and B, whereas the values of the projects decrease as the investor becomes more risk
tolerant. In contrast, the price behavior of project C is opposite: project C obtains low values at
small risk tolerance levels and the values increase as the investor becomes more risk tolerant.
Project D is constantly priced at $15.56 due to the existence of a replicating portfolio. An EU
investor prices the projects similarly: values of projects A and B decrease by diminishing risk
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27
aversion, while the value of project C increases. Also in this case, the value of project D is
constant due to the existence of a replicating portfolio.
In both cases, project values converge to limit values as the investor becomes less risk averse.
However, these limit values are different, and in neither case they coincide with a risk neutral
investor’s project values ( $4.13, $9.35, $1.86, $12.83)− , obtained by using a short rate
13.58% in period 0 (the expected rate of return of security 1; Table 6) and 20.00% in period 1
(the expected rate of return of security 2; Table 5). Gustafsson et al. (2004) showed in their
numerical experiments that, in a two-period setting, the project values for an MV investor
converge towards the CAPM prices of the projects; the experiments in De Reyck et al. (2004)
showed a similar behavior for EU investors with CARA. The observed converge behavior for
both types of investors shows that a similar phenomenon can happen also in a multi-period
setting.
Table 12. Project values for an MV investor.
Risk level A B C D
75 $11.70 $16.87 $4.12 $15.56
100 $8.49 $16.21 $6.20 $15.56
125 $7.55 $15.96 $7.20 $15.56
150 $7.05 $15.82 $7.80 $15.56
175 $6.73 $15.73 $8.21 $15.56
200 $6.50 $15.67 $8.51 $15.56
225 $6.34 $15.62 $8.74 $15.56
250 $6.21 $15.58 $8.92 $15.56
300 $6.02 $15.53 $9.19 $15.56
350 $5.89 $15.49 $9.38 $15.56
400 $5.79 $15.46 $9.52 $15.56
450 $5.71 $15.44 $9.63 $15.56
500 $5.65 $15.42 $9.71 $15.56
1000 $5.38 $15.35 $10.10 $15.56
1500 $5.28 $15.32 $10.23 $15.56
2500 $5.21 $15.30 $10.33 $15.56
5000 $5.15 $15.28 $10.41 $15.56
50000 $5.10 $15.27 $10.48 $15.56
Project
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28
Table 13. Project values for an EU investor.
α A B C D
0.005 $8.03 $16.67 $5.97 $15.56
0.004 $7.43 $16.41 $6.87 $15.56
0.003 $6.83 $16.15 $7.79 $15.56
0.002 $6.24 $15.90 $8.74 $15.56
0.001 $5.67 $15.66 $9.71 $15.56
0.0001 $5.15 $15.46 $10.59 $15.56
0.00001 $5.10 $15.44 $10.68 $15.56
0.000001 $5.09 $15.44 $10.69 $15.56
Project
In summary, project values when the investor can invest simultaneously in projects A–D can be
different from the values obtained with single projects due to correlations between projects and
better diversification possibilities implied by a greater number of investment assets in the
portfolio. Also, the optimal management strategies for the projects may change. [Q2] Sensitivity
analysis reveals that the value of a project may either decrease or increase with increasing risk
aversion depending on its correlation with the rest of the investment portfolio. Furthermore, a
project with a replicating portfolio is constantly priced at its CCA value. Also, with decreasing
level of risk-aversion, project values converge towards prices that are different from the
projects’ risk-neutral values. These values are analogous to the CAPM prices and EU
convergence prices observed in a two-period setting. [Q3]
8 Summary and Conclusions In this paper, we have presented a model for selecting a multi-period mixed asset portfolio and
a framework that uses this model in the valuation of multi-period projects. In the MAPS model,
project management decisions are structured as project-specific decision trees. The project
valuation framework is based on the concepts of breakeven selling and buying prices
(Gustafsson et al. 2004, Luenberger 1998, Smith and Nau 1995); the breakeven selling price is
the lowest price at which a rational investor would sell the project if she had it, whereas the
breakeven buying price is the highest price at which the investor would buy the project if she
did not have it.
The present project valuation framework differs from Smith and Nau’s (1995) “full decision tree
analysis” in that (i) it maximizes the investor’s terminal wealth level rather than the
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29
intertemporal utility implied by the project and in that (ii) it takes into account the
opportunity costs imposed by alternative projects in the portfolio. The framework also (iii)
allows the use of a wide range of preference models, and it leads to an explicit optimization
problem, which can readily be solved using standard techniques of nonlinear programming. In
comparison with Smith and Nau’s (1995) integrated rollback procedure, which is also an
explicit project valuation method, the present framework relies on fewer assumptions about the
investor’s preference structure and the nature of project cash flows.
In addition to formulating the project valuation framework, we showed in this paper that, when
the investor uses either a preference functional or a mean-risk model, the framework gives
consistent values with CCA, wherefore it can be regarded as a generalization of CCA to
incomplete markets. In addition, we showed that the framework leads to Hillier’s (1963)
method when the investor can invest only in a single project and in the risk-free asset, and
when she exhibits both constant relative and constant absolute risk aversion.
In our numerical experiments, we examined the pricing behavior of mean-variance and expected
utility maximizers when markets were assumed to abide by the standard CAPM. We observed
that many of the phenomena occurring in a two-period setting also took place in the multi-
period setting. For example, we observed that, when compared to the situation without
securities, the presence of securities typically resulted in lower project values due to imposed
opportunity costs. Furthermore, replicating portfolios led to project values that were
independent of the investor’s risk attitude. We also observed that the value of a project, when
it did not have a replicating portfolio, could either increase or decrease with increasing risk
aversion, depending on its correlation with the rest of the investment portfolio. When the
investor became less risk averse, project values converged towards prices that were different
from the prices that a risk-neutral investor would give to the projects.
This work suggests several areas for further research. For example, it would be relevant to
examine the effect of alternative security pricing models, such as the multi-period models of
Merton (1973), Stapleton and Subrahmanyam (1978), and Levy and Samuelson (1992), on
project values and optimal trading strategies. Also, the convergence behavior of project values
demands more analysis: the results from Gustafsson et al. (2004) indicate that in a two-period
setting project values for a mean-variance investor converge towards the projects’ CAPM prices
as the investor becomes less risk averse; a similar behavior could potentially be proven in the
151
30
multi-period case.
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Appendix
Proofs of Propositions 1, 2, 3 and 5 Proof of Proposition 1: Let ω
*x denote the portfolio of securities in state ω that replicates the cash
flow pattern of the next period. Let ,0
( ) ( )n
j i ii
P S xωω ω ∗
=
= ∑ be the price of this replicating portfolio.
Let ω**x be the portfolio whose cash flow is equal to the value of the replicating portfolio in each
state of the next period. (For convenience, let us assume that ω =**x 0 if ω belongs to the last or
the second last period and that ω =*x 0 if ω belongs to the last period.) Then, * *ω ω ω= +** * *x x x is
the portfolio that replicates the cash flows of the next period and which can be used to construct
replicating portfolios for the period after the next period. Let us then define **ω**x so that it is
equal to *ω**x if ω belongs to the last or the second last period, and in other periods it is the
portfolio whose cash flow is equal to the value of the replicating portfolio **ω′**x in each state ω′ of
the next period. If there is a replicating trading strategy for the project, there exists portfolios **
ω**x for all states ω . Thus, the time-0 portfolio
0
**ω**x can be used to construct a replicating
strategy for the project. Its price is 00 ,
0
( )n
i ii
S xωω ∗∗∗∗
=∑ . Since a shorted replicating trading strategy,
which can be created using portfolio 0
**ω− **x , nullifies the cash flows of the project, the setting
where the investor invests in the project and shorts 0
**ω**x is identical with the situation where the
investor cannot invest in the project and receives money equal to 0
00 ,
0
( )n
j i ii
C S xωω ∗∗∗∗
=
− + ∑ , where 0jC
is the investment cost of the project. When the time-0 budget in the setting without the project is
( )0b ω , the respective budget is ( )0
00 0 ,
0
( )n
j i ii
b C S xωω ω ∗∗∗∗
=
− + ∑ with the project and portfolio 0
**ω− **x .
Hence, the breakeven buying price of the project is given by
0
00 0 , 0
0
( ) ( ) ( )n
bj i i j
i
b C S x v bωω ω ω∗∗∗∗
=
− + − =∑ , or 0
00 ,
0
( )n
bj j i i
i
v C S xωω ∗∗∗∗
=
= − + ∑ , which is also the CCA
price of the project. Conversely, in the setting with the project and budget ( )0b ω , we can nullify
the project cash flows by shorting 0
**ω**x and obtain an effective budget of
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33
( )0
00 0 ,
0
( )n
j i ii
b C S xωω ω ∗∗∗∗
=
− + ∑ . Hence, in the setting without the project, we will obtain an equally
desirable portfolio by increasing the budget by 0
00 ,
0
( )n
sj j i i
i
v C S xωω ∗∗∗∗
=
= − + ∑ , which is both the
breakeven selling price and the CCA price of the project. Q.E.D.
Proof of Proposition 2: Since the investor can invest only in a single project and in the risk-free
asset, in the setting without the project the investor invests her entire budget in the risk-free asset;
consequently, the portfolio is riskless and is its own certainty equivalent, i.e.
0(1 ) ( )TTV s b Oω− = + + , sT is the time-T spot rate and O is the future value of the budgets at
states other than 0ω . Also, changing the time-0 budget to 0( )b ω δ+ will give a portfolio value of
( )0(1 ) ( )TTV s b Oω δ∗ = + + + without the project. Setting V V+ ∗= we obtain
(1 )TTV V s δ+ −= + + , wherefrom it follows (1 )TT
V Vs
δ+ −−
=+
, which is also, by definition, the
breakeven selling price of the project. For a mean-risk investor, the proposition follows from the
observation that, when the risk-constraint is not binding, the preference model corresponds to a
preference functional model with CE[X] = E[X]. If the risk constraint prevents the starting of the
project, then 0(1 ) ( )TTV V s b Oω+ −= = + + , implying a zero value for the project. Q.E.D.
Proof of Proposition 3: When the investor can invest only in a single project and in the risk-free
asset, we have 0(1 ) ( )TTV s b Oω− = + + , where sT is the time-T spot rate and O is the future value
of the budgets at states other than 0ω . Also, we can separate the optimal terminal wealth level
with the project to random variable X, which equals to project’s future value (project’s cash flows
with the accrued interest), and to f that is the future value of the budgets. Since
[ ] [ ]CE X f CE X f+ = + , and denoting CE[X] by CET , we have TV CE f+ = + and V f− = .
Increasing the budget by δ both with and without the project will now increase the respective
certainty equivalent by (1 )TTs δ+ . Let V ∗ be the certainty equivalent of the optimal portfolio
when the investor invests in the project and the time-0 budget is 0( )b ω δ− . Then,
(1 )TTV V s δ∗ += − + . Setting V V− ∗= , we have (1 )TTV V s δ− += − + , and hence we have
(1 )TT
V Vs
δ+ −−
=+
, which is also the breakeven buying price of the project. Noticing that
TV V CE+ −− = , we immediately obtain the desired formula. Using further this observation and
Proposition 2, the proposition is also proved for the breakeven selling price. For a mean-risk
investor, the proposition follows from the observation that, when the risk-constraint is not binding,
the preference model corresponds to a preference functional model with CE[X] = E[X]. If the risk
constraint prevents the starting of the project, the project will have a zero value. Q.E.D.
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34
Proof of Proposition 5: The proposition is proven with a three-period counterexample. Consider an
investor with an intertemporal utility function 1 2 31 0.95 0.90 1 2( , , ) 5 c c cu c c c e e e∗ − − −= − − − , where we
have assumed CARA and additive independence for the sake of analogy with Smith and Nau
(1995). Note that the parameters of the utility function reflect the investor’s risk aversion and the
investor’s subjective perception about the relative importance of the cash flows received at different
periods, and hence they can be selected independently of the risk-free interest rate or available
securities. Consider then a project that yields a cash flow stream (–1, 1.1, 0) with 50% and a cash
flow stream ( 1,0,1.22)− with 50%. There is also a riskless zero-coupon bond yielding the cash flow
stream ( 1,0,1.21)− for sure. The risk-free interest rate is 10%, the time-0 budget is 1, and no other
securities are available. The terminal wealth level with the project is 1.21 with 50% and 1.22 with
50%. The bond yields a terminal wealth level of 1.21 for sure. Note that the terminal wealth level
implied by the project stochastically dominates the one implied by the bond. The expected
intertemporal utility for the project is 0.939 and for the bond 0.945. Thus, the bond is preferred to
the project even though the project’s terminal wealth level stochastically dominates that of the
bond. Q.E.D.
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MULTIPLE CRITERIA DECISION MAKING IN THE NEW MILLENNIUM, M. KÖKSALAN, S. ZIONTS (EDS.), LECTURE NOTES IN ECONOMICS AND MATHEMATICAL SYSTEMS 507, 2001, SPRINGER-VERLAG, BERLIN. REPRINTED WITH THE PERMISSION OF THE PUBLISHER.
PRIME Decisions: An Interactive Tool for Value Tree Analysis
Janne Gustafsson, Ahti Salo, and Tommi Gustafsson Helsinki University of Technology, Systems Analysis Laboratory P.O. Box 1100, FIN-02015 HUT, Finland Abstract. Several methods for the processing of incomplete preference information in additive preference models have been proposed. Due to the lack of adequate decision aiding tools, however, only a few case studies have been reported so far. In this paper, we present PRIME Decisions, a decision aiding tool which supports the analysis of incomplete preference information with the PRIME method. PRIME Decisions also offers novel features such as decision rules and guided elicitation tours. The application of PRIME Decisions is illustrated with a case study on the valuation of a high-tech company.
Keywords. Decision analysis, interval judgement, incomplete information, linear programming, decision aiding tools.
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Systems Analysis Laboratory Research Reports, Series A A91 Electricity Derivative Markets: Investment Valuation, Production May 2005 Planning and Hedging Erkka Näsäkkälä A90 Optimal pilot decisions and flight trajectories in air combat March 2005 Kai Virtanen A89 On foresight processes and performance of innovation networks November 2004 Jukka-Pekka Salmenkaita A88 Systems Intelligence - Discovering a hidden competence in human October 2004 action and organizational life Raimo P. Hämäläinen and Esa Saarinen, eds. A87 Modelling studies on soil-mediated response to acid deposition and October 2003 climate variability Maria Holmberg A86 Life cycle impact assessment based on decision analysis June 2003 Jyri Seppälä A85 Modelling and implementation issues in circuit and network planning May 2003 tools Jukka K. Nurminen A84 On investment, uncertainty, and strategic interaction with April 2003 applications in energy markets. Pauli Murto A83 Modeling of microscale variations in methane fluxes November 2002 Anu Kettunen A82 Modelling of water circulation and the dynamics of phytoplankton October 2001 for the analysis of eutrophication Arto Inkala Systems Analysis Laboratory Research Reports, Series B B25 Systeemiäly 2005 May 2005 Raimo P. Hämäläinen ja Esa Saarinen, toim. B24 Systeemiäly - Näkökulmia vuorovaikutukseen ja kokonaisuuksien June 2004 hallintaan Raimo P. Hämäläinen ja Esa Saarinen, toim. B23 Systeemiäly! April 2003 Tom Bäckström, Ville Brummer, Terhi Kling ja Paula Siitonen, toim. B22 Loruja - Esseitä uoman ulkopuolelta April 2002 Tommi Gustafsson, Timo Kiviniemi ja Riikka Mäki-Ontto, toim.
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Systems Analysis Laboratory Research Reports, Series E Electronic Reports: www.e-reports.sal.hut.fi E18 Valuing Risky Projects with Contingent Portfolio Programming June 2005 Janne Gustafsson and Ahti Salo E17 Project Valuation under Ambiguity June 2005 Janne Gustafsson and Ahti Salo E16 Project Valuation in Mixed Asset Portfolio Selection June 2005 Janne Gustafsson, Bert De Reyck, Zeger Degraeve and Ahti Salo E15 Hydropower production planning and hedging under inflow and May 2005 forward uncertainty Erkka Näsäkkälä and Jussi Keppo E14 Models and algorithms for network planning tools - practical May 2003 experiences Jukka K. Nurminen E13 Decision structuring dialogue April 2003 Sebastian Slotte and Raimo P. Hämäläinen E12 The threat of weighting biases in environmental decision analysis April 2003 Raimo P. Hämäläinen and Susanna Alaja E11 Timing of investment under technological and revenue related March 2003 uncertainties Pauli Murto E10 Connecting methane fluxes to vegetation cover and water table November 2002 fluctuations at microsite level: A modeling study Anu Kettunen E9 Comparison of Hydrodynamical Models of the Gulf of Finland in 1995 September 2001 - A Case Study Arto Inkala and Kai Myrberg E8 Rationales for Government Intervention in the Commercialization of September 2001 New Technologies Jukka-Pekka Salmenkaita and Ahti A. Salo E7 Effects of dispersal patterns on population dynamics and synchrony April 2000 Janica Ylikarjula, Susanna Alaja, Jouni Laakso and David Tesar Series A and E reports are downloadable at www.sal.hut.fi/Publications/ Orders for Helsinki University of Technology paper copies: Systems Analysis Laboratory P.O. Box 1100, FIN-02015 HUT, Finland [email protected]
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