The Pennsylvania State University

The Graduate School

Department of Industrial and Manufacturing Engineering

ESSAYS ON INVESTMENT DECISIONS UNDER UNCERTAINTY

A Dissertation in

Industrial Engineering

by

Yongma Moon

© 2010 Yongma Moon

Submitted in Partial Fullfillment of the Requirements

for the Degree of

Doctor of Philosophy

May 2010

The dissertation of Yongma Moon was reviewed and approved* by the following:

Tao Yao Assistant Professor of Industrial Engineering Dissertation Advisor Chair of Committee

Terry L. Friesz Harold and Inge Marcus Chaired Professor of Industrial Engineering

Jose A. Ventura Professor of Industrial Engineering

Robert D. Weaver Professor of Industrial Engineering Professor of Agricultural Economics

Charles (Quanwe) Cao Smeal Chair Professor of Finance

Paul Griffin Professor of Industrial Engineering Head of the Harold and Inge Marcus Department of Industrial and Manufacturing Engineering

*Signatures are on file in the Graduate School

iii

ABASTRACT

This dissertation aims at modelling uncertainty and deriving managerial implications in

different areas. Due to uncertainty and limited information, a firm may encounter difficulties in

decision making and sometimes make wrong decision. For this reason, I provide models to

incorporate uncertainty, investigate the impacts of the uncertainty, and provide more appropriate

decisions and new interpretations.

More specifically, this dissertation studies investment decision problems under uncertainty in

negotiation, outsourcing vs. joint venture contract, investment of dual channel and portfolio

optimization (Chapter 2, 3, 4, and 5, respectively). To model uncertainty in four areas, two

methodologies are considered in this dissertation. For the first three chapters, the real option

theory which became widely used in both academia and practice is applied to study uncertainty.

For the portfolio optimization, a robust optimization method is used.

iv

TABLE OF CONTENTS

LIST OF FIGURES .....................................................................................................vi LIST OF TABLES.......................................................................................................vii ACKNOWLEDGEMENTS.........................................................................................viii Chapter 1. Overview and Organization of a Dissertation ............................................1 Chapter 2. Price Negotiation Under Uncertainty.........................................................3

2.1. Introduction....................................................................................................3 2.2. Literature Review ..........................................................................................5 2.3. Backgrounds for a Negotiation Agreement ...................................................8

2.3.1. Relation between a Negotiation and Real Option ...............................8 2.3.2. Optimal Buying Timing as a Call........................................................10 2.3.3. Optimal Selling Timing as a Put .........................................................12

2.4. The Implicit Zone of Possible Agreement and Equilibrium Probability.......13 2.5. Negotiation Agreement Probability...............................................................19 2.6. Extension: Negotiation Power .......................................................................22 2.7. Conclusions....................................................................................................25 APPENDIX...........................................................................................................28

Chapter 3. Outsourcing versus Joint Venture from Vendor’s Perspective ..................38 3.1. Introduction....................................................................................................38 3.2. Literature Review ..........................................................................................40 3.3. Model Settings ...............................................................................................42

3.3.1. Model for an Outsourcing Contract.....................................................43 3.3.2. Model for a Joint Venture Contract.....................................................44

3.4. Contract Feasibility with Optimal Investment Thresholds............................45 3.5. Contract Feasibility and Selection .................................................................49 3.6. Conclusion and Further Studies.....................................................................52 APPENDIX...........................................................................................................54

Chapter 4. Investment Timing for a Dual Channel Supply Chain...............................60 4.1. Introduction....................................................................................................60 4.2. Literature Review ..........................................................................................62 4.3. Model.............................................................................................................65

4.3.1. Firm’s Price Decisions and Necessary Conditions for Dual Channels .................................................................................................69

4.3.2. Firm’s Investment and Timing Decision.............................................72 4.4. Analysis of Investment Timing and Customer Behavior ..............................76

4.4.1. Analysis and Numerical Examples......................................................76 4.4.2. Comparison with the Traditional Net Present Value Method .............78

4.5. Conclusions and Further Studies ...................................................................80 APPENDIX...........................................................................................................82

Chapter 5. A Simple Robust Mean Absolute Deviation Model for Portfolio Optimization .........................................................................................................91 5.1. Introduction and Literature Review...............................................................91

v

5.2. Mean Absolute Deviation Model...................................................................95 5.3. Robust Formulation of MAD Model .............................................................96 5.4. The Performance of a Robust Portfolio .........................................................102

5.4.1. RMAD Resutls with Market Conditions .............................................103 5.4.2. RMAD Results with Standard Deviations...........................................106 5.4.3. RMAD Results with Financial Elasticity ............................................108

5.5. Conclusion and Further Studies.....................................................................109 References....................................................................................................................111

vi

LIST OF FIGURES

Figure 2.1. Relationship between negotiation and real option.....................................9 Figure 2.2. Implicit Zone of Possible Agreement (IZOPA) ........................................15 Figure 2.3. The impact of options on the condition for agreement..............................17 Figure 2.4. Comparative studies on agreement threshold *

thλ with respect to volatility and mean-drift ...........................................................................19

Figure 2.5. Probability with and without option values...............................................21 Figure 2.6. (a) IZOPA with negotiation power and (b) the impact of options on the

condition for agreement............................................................................25 Figure 3.1. The contract feasibility by ROT and NPV ................................................47 Figure 3.2. Feasibility and utility for outsourcing vs. joint venture: Case (i)..............50 Figure 3.3. Feasibility and utility: Case (ii) .................................................................51 Figure 3.4. Feasibility and utility: Case (iii) ................................................................51 Figure 3.5. Feasibility and utility by NPV for outsourcing vs. joint venture ..............52 Figure 4.1. Demand from distribution of customer value...........................................66 Figure 4.2. The impact of (a) cost efficiency, (b) customer behavior, (c) volatility,

(d) volatility on optimal investment thresholds. ......................................77 Figure 4.3. Thresholds by real option (ROT) vs. NPV method...................................79 Figure 5.1. The historical data of Dow Jones and S&P index .....................................104 Figure 5.2. All stocks in growth market condition ......................................................105 Figure 5.3. All stocks in steady state market condition ...............................................105 Figure 5.4. All stocks in decline market condition ......................................................106 Figure 5.5. Portfolio of high (upper) and low (lower) standard deviation stocks........107 Figure 5.6. Portfolio of high (upper) and low (lower) beta stocks ..............................109

vii

LIST OF TABLES

Table 2.1 Estimated parameters...................................................................................21 Table 3.1 Parameters for numerical examples.............................................................46 Table 4.1. Customers acceptance index (θ ) for the internet channel..........................67 Table 4.2 Inequality conditions to derive demand functions......................................67 Table 4.3. Percentage of a firm’s value uncaptured by NPV.......................................79

viii

ACKNOWLEDGEMENTS

“I would like to give a sincere gratitude to every people who I have known. They are the people

who made me who I am now, and they are the best reason for me to be happy and smile”

I would like to express the deepest appreciation to my committee chair and advisor, Professor Tao

Yao. Without his guidance and persistent help, this dissertation would not have been possible and

I could not feel any interest in the research. Also, I would like to thank my committee members,

Professor Robert D. Weaver, Professor Terry L. Friesz, Professor Jose A. Ventura, and Professor

Charles Cao. Their comments always gave me a new way of thinking on my works.

In addition, I would like to thank my family for everything that they have done for me all the time.

Especially, my wife Yoojin Hong always has been with me and has believed in me through my

Ph.D. study. The belief was my happiness. Also, the happiest thing during my study was to have

my son, Max Terang Moon. He is my pride and joy. Furthermore, I am deeply indebted to my

parents, parents-in-laws, brothers (Yongwoong Moon, Yonghan Moon) and sisters-in-law

(Myungene Hong, Kyungene Hong) who have supported my family and encouraged me to

achieve higher goals throughout my work. I would like to share this achievement with them.

Thank you, my friends: Lab-mates, PSUIEKSA, Kyunghee, Shingu, Chungdam, SNU, and

Joonseock Lee.

1

Chapter 1. Overview and Organization of a Dissertation

Recently, the study about uncertainty has been highlighted in economics, business and

engineering area. Many researchers recognized that a limited knowledge or information of future

outcome or even the exact current state can bring biased results or lead us to wrong decisions.

Thus, huge efforts to incorporate uncertainty in existing models and to derive more realistic

results have been made. Besides, as many studies have done, the consideration of uncertainty can

bring new interpretations regarding to the uncertainty and provide better performances by

reducing an estimation bias. This dissertation studies investment decision problems under

uncertainty in negotiation, outsourcing vs. joint venture contract, investment of dual channel and

portfolio optimization (Chapter 2, 3, 4, and 5, respectively). To model uncertainty in four areas,

two methodologies are considered in this dissertation. For the first three chapters, the real option

theory which became widely used in both academia and practice is applied to study uncertainty.

For the portfolio optimization, a robust optimization method is used.

Investment policy has been central to economic theory or corporate finance for a long time.

Alfred Marshall first formulated investment decision criterion as a “net present value” rule, where

investment cost is compared to the present value of certain cash flows that result from investment.

More recently, the Real Option Theory has extended to encompass uncertainty with respect to

future investment productivity. Dixit and Pindyck (1994) discuss that investment opportunities

can be interpreted as American call options with irreversibility of sunk cost and managerial

flexibility of investment timing. Besides, many literatures showed that the real option theory is an

appropriate tool to analyze investment problem. For these reason, investment problems in the

2

negotiation (Chapter 2), different contracts (Chapter 3), and dual channel strategy (Chapter 4) are

analyzed by a real option theory.

Besides, quantitative techniques to manage a financial portfolio, portfolio optimizations have

become common and grown in the investment industry. As the use of modeling techniques has

become widespread among portfolio managers, however, the issue of how much confidence

practitioners can have in theoretical models and data has grown in importance because of

uncertainty. Consequently, there is an increased level of interest in the subject of robust

estimation in modern portfolio management. Recently, robust optimization methodologies have

been developed actively in the optimization fields. Besides, a few studies show good performance

of a robust optimization methodology. For these reasons, the simple robust portfolio optimization

model is proposed in the Chapter 4.

Therefore, in this dissertation, the Chapter 2, 3, 4 and 5 discuss negotiation, contracts,

dual channel and portfolio optimization.

3

Chapter 2. Price Negotiation Under Uncertainty

This chapter examines supply contract negotiation when buyer’s revenue and seller’s cost are uncertain. In these circumstances, both the seller and the buyer have an option to determine when to sell and buy, which may influence negotiation outcomes. Thus, we developed a bilateral negotiation model to derive the optimal selling (buying) rule considering the option. Our results show that the options of waiting to sell and to buy 1) narrow the traditional zone of possible agreement and 2) lower the probability of negotiation agreement. It is also shown that impasses can occur due to uncertainty, even when a purchase price is lower than the buyer’s future revenue and higher than the seller’s future cost.

2.1. Introduction

Negotiation is a normal approach to determine a purchase price in a supply chain contract. In a

bilateral negotiation process, the contract (purchase) price is negotiated between a buyer (retailer)

and a seller (supplier). This paper examines the negotiation contract when buyer’s revenue and

seller’s cost are uncertain and describes the impact of the uncertainties on negotiation outcomes.

As time elapses, the buyer's revenue and the seller's cost for a procured product evolve

dynamically and continuously. Under these circumstances, both parties determine whether or not

to negotiate a supply contract. However, since the decision to buy or sell the product is

irreversible and the revenue and the cost are uncertain, each party also has to decide when to buy

or sell. In the case of a buyer, the optimal purchasing time will be determined by comparing the

value of buying now to the value of purchasing in some future time. In other words, the buyer has

an option to wait. Herein, as discussed in Pindyck (1991), because buying products now gives up

the possibility of waiting for new information to arrive which might influence future revenue, the

opportunity cost of waiting should be included as a part of the investment costs. Also, we need to

4

note that the opportunity cost cannot be ignored and can be very large (Bernanke (1983,

McDonald and Siegel (1986, Trigeorgis (1993)).

Similarly, the seller can have an option to delay selling until a contract price compensates the

loss of the waiting option in addition to the cost and margin. Moreover, the values of options to

wait depend upon how uncertain the revenue and the cost are. For example, a buyer may want to

purchase later for a better payoff in more faster-growing and more volatile market conditions.

Even though the option value caused by uncertainty as well as irreversibility is an important

factor to be considered, few studies in the supply chain contract negotiation literature have

discussed such option. Motivated by the oversight, this research aims to fill this gap through a

real option approach which allows us to explain buyer’s and seller’s options.

In this paper, we explore the significance of the value of waiting to buy and sell, and

investigate the role of the option and uncertainty in the negotiation outcome. A proposed bilateral

negotiation model to consider the value of option to sell (buy) is derived by the optimal selling

(buying) rule under cost and revenue uncertainty. Based on the model, we introduce novel

concepts in supply contract negotiation, an implicit reservation price and an implicit zone of

possible agreement (IZOPA), and derive the corresponding negotiation agreement probability.

Then, by comparing the traditional concept of the reservation price and the zone of possible

agreement, the role of uncertainty and an option in a negotiation outcome will be discussed. This

paper contributes to the bilateral negotiation theory in supply chain by providing a real option

model to consider each party’s timing flexibility (option) and by investigating how the options

and uncertainties affect the negotiation outcome. This research will help both parties to appreciate

the highly intertwined relationship between participants’ options and negotiation agreement and

to make appropriate decisions accordingly.

The next section discusses literature related to supply chain, negotiation, and a real option and

then addresses the main findings. Section 2.3 explains the model set-up for both a buyer and a

5

seller and derives the associated stochastic dynamic programming formulation. In Section 2.4, we

will provide definitions of an implicit reservation price and IZOPA. In addition, the comparison

of the negotiation models with and without an option will be studied. Section 2.5 presents

analysis for negotiation agreement probability. Section 2.6 extends to a model incorporating each

party’s negotiation power. Section 2.7 concludes with brief discussions and limitations, as well as

directions for future research.

2.2. Literature Review

Negotiation in the supply chain arises between buyers (retailers) and sellers (suppliers). Even

though negotiation is ubiquitous in supply chain contracts, the detailed negotiation processes are

not generally considered in the supply chain literature (Cachon (2003)). Most supply chain

contract research has focused on how a certain type of contract coordinates players (supplier,

manufacturer, and/or retailer) in a supply chain (Bajari and Tadelis (2001, Giannoccaro and

Pontrandolfo (2004, Cachon and Lariviere (2005)). Only a few recent papers explored the

negotiation aspect based on the game theoretic approach. For example, Mieghem (1999)

discussed the role of bargaining power in the supply chain contract. Similarly, Bernstein and

Marx (2006) addressed the problem of supply chain performance relevant to negotiation such as

the effect of retailers’ bargaining power in the allocation of total supply chain profits. For the

negotiation process, Ertogral and Wu (2001) explicitly studied a bargaining process and an

outcome within a supply chain considering outside options and the breakdown probability.

However, these studies overlooked the timing issue, which is important and will be discussed

in this paper. When contract negotiation proceeds, determining when to sell or to buy can affect

supply chain members’ strategies. Several papers in the supply chain area considered this timing

issue and emphasized the importance of the timing. For example, Ferguson (2003) studied

6

quantity commitment timing in the environment of a stochastic demand. This research explained

environments under which supplier and manufacturer might prefer delayed or early commitment.

Taylor (2006) also discussed sale timing strategies, motivated by the fact that a retailer can sell

products at a higher price if market conditions are strong enough. In addition, the paper

investigated the impact of information asymmetry, retailer’s effort and different contract types on

the sale timing strategy, and explained conditions under which the manufacturer prefers to sell

either early or late. As shown in these studies, timing is crucial to decision in supply chain, and

thus we incorporate the timing flexibility into the price negotiation problem using a real option

theory in this paper.

The real option theory has been used to quantify the timing flexibility under uncertainty (see

Brennan and Schwartz (1985, McDonald and Siegel (1986, Dixit (1989, Schwartz and Zozaya-

Gorostiza (2003)). Pindyck (1991) showed that a real option approach is very appropriate to

model valuation of a project under uncertainty and irreversible investment. The real option

approach develops a quantitative framework to provide analytical solutions for option (flexibility)

values as well as optimal timing rules under uncertainty. Dixit and Pindyck 1994 point out the

parallels between the investment opportunities and perpetual American options and the existence

of opportunity costs which fundamentally influence decision-making behavior. In most cases,

negotiation time is not fixed. When uncertainty (revenue and cost) and irreversibility are

involved, both parties’ profits are hedged by their implicit option that generates managerial

flexibility: to exercise its opportunity immediately or to wait. Once each party contracts, however,

profits are no longer hedged, because the option to wait is gone. Thus, the real option theory is

applied to many applications in order to derive an optimal time to invest and this will be utilized

to our price negotiation problem.

General negotiation models have been studied from various points of view (see Harsanyi and

Selten (1972), Chatterjee and Samuelson (1983), Cramton (1984), Chatterjee and Samuelson

7

(1987), Cramton (1992)). Among those, from the fact that outcomes are constructed within a

certain range (or many, many possible equilibria, see Sebenius (1992)), negotiation analysts

defined the range, ‘Zone of Possible Agreement’ (ZOPA). This range is a fundamental concept in

negotiation research outlined by Walton and Mckersie (1965) and Raiffa (1982). Simply, this

traditional ZOPA is constituted by an overlap between the parties’ reservation prices.

However, the ZOPA still fails to incorporate a participant’s option; in other words, most

research has ignored the fact that a buyer and a seller can determine when to purchase and sell

respectively. In addition, the ZOPA might have limitations in quantifying and explaining the role

of uncertainty such as the amount of uncertainty and the rate of change of commodity value. For

these reasons, this paper addresses new concepts of a reservation price and the ZOPA, involving

participants’ options under uncertainty in the supply contract negotiation. Specifically, the new

concepts (implicit reservation price and implicit zone of possible agreement) will be defined by

deriving optimal timing rules for a buyer and a seller and will be used as a first step to interpret

the role of uncertainty in the negotiation.

Furthermore, unlike existing studies, this paper considers the fact that the revenue and the

cost of a product may change over time and each party thus has options to wait (timing

flexibility) which may play a significant role in the negotiation. We believe that the new model

and concepts that we will propose help us interpret how uncertainty and options affect a

negotiation outcome in a supply chain. In addition, our analytical model allows us to derive

implications by comparative statics.

Below we summarize the Main findings and implications from our proposed model:

i) Narrower ZOPA: The Zone of possible agreement considering an option (IZOPA) is

narrower than one ignoring the option. This shrinkage occurs because participants delay

strategically, in other words, because both will not buy or sell unless the opponent compensates

for the option value.

8

ii) Condition for Existence of IZOPA: We provide a necessary condition of negotiation

agreement that the IZOPA exists. Depending on environments, the negotiation can be agreeable

to each party. For example, intuitively, if revenue is larger than cost, then it seems that the

negotiation can be agreeable to both. However, unlike our intuition, the condition derived in this

paper implies that negotiation can be induced to breakdown even when a contract price is lower

than the buyer’s future revenue and higher than the seller’s future cost. Also, the analytical form

of the condition enables analyses of the impacts of uncertainty on the negotiation agreement

feasibility. For example, when revenue and cost become more uncertain, the changes of

environments make both parties more hesitant to negotiate.

iii) Overestimation of Agreement Probability: We study behaviors of possible negotiation

equilibrium with a probability function. Similar ideas using chance or probability of agreement

can be found in Raiffa (1982) and Zeng and Sycara (1998). We compare agreement probabilities

suggested by traditional approaches to ours. We show that the agreement probability without

option is always higher than one with the option and, hence overestimates the true probability of

negotiation agreement.

2.3. Backgrounds for a Negotiation Agreement

2.3.1. Relation between a Negotiation and Real Option

This section provides optimal timing strategies based on the real option theory for both buyer

and seller. In many real option studies (see, Pindyck (1991), Grenadier and Weiss (1997), Weeds

(2002), and Lambrecht (2004)), the researchers provided optimal thresholds either to invest or to

delay projects using perpetual American call options. Pindyck (1991) derived an optimal

investment timing using a real option approach when the irreversible and deferrable project has

an uncertainty. Similarly, an optimal selling timing problem can be formulated as a perpetual put

9

option, such as in Reuer and Tong (2005). However, for a negotiation procedure, we need to

consider both buyer and seller at the same time, because the negotiation can be successful only

upon the agreement of both parties. Therefore, as shown in Figure 2.1, we divided the negotiation

into two viewpoints: a buyer and a seller. For the buyer, the negotiation is the opportunity to

acquire a commodity and the duty to pay a contract price to a seller. However, obviously, the

buyer would like to defer the negotiation deal until the value of the commodity would be

estimated more than the contract price. In the same manner, a seller has an option to sell its own

commodity. Two parties trade at a contract price, which will play the role of connecting a buyer

and a seller when constructing a novel concept of ZOPA. This simple paradigm displays an

agreement model based on both an American call option and a put option in one framework.

Figure 2.1. Relationship between negotiation and real option

As aforementioned, the revenue and the cost are dynamically changing and the future values

are uncertain. Thus, valuations by two parties, the revenue for the buyer and the cost for the

seller, are assumed to follow the Geometric Brownian Motion (GBM). This is the standard setting

in a real option theory (Dixit (1989)) and also the good first approximation for uncertainties (Abel

and Eberly (1994), Dixit and Pindyck (1994), and Murto (2004)). This assumption has also been

used in the supply chain area (Kamrad and Lele (1998, Li and Kouvelis (1999, Caldentey and

Wein (2002, Marathe and Ryan (2005), Caldentey and Wein 2006). Empirical evidence

supporting the GBM assumptions for revenue have been found in literature (e.g., Marathe and

10

Ryan 2005). The GBM is stated, mathematically, for the revenue and the cost for a buyer and a

seller, 2S and 1S ,

iiiiii dztSdttStdS )()()( σμ += for 2,1=i

where iμ is the mean-drift of iS (rate of increasing or decreasing), and iσ is the volatility of

iS (amount of uncertainty). idz is the increment of a standard Wiener process where

),0(~ dtNdzi and dtdzdzE 1221 ][ ρ= as assumed in many researches (i.e. see McDonald and

Siegel (1986, Kamrad and Siddique (2004)). In addition, the index 1 and 2 represent the seller

and buyer respectively, while parameters ( iμ and iσ ) are assumed to be constant over time. In

the next subsection, we will derive an optimal buying and selling time using a real option theory.

2.3.2. Optimal Buying Timing as a Call

A buyer compares his or her revenue of a commodity with a contract price K (i.e.

transaction or wholesale price). Here, the contract price K is assumed to be exogenously given.

In Section 6, we extend to our consideration of a contract price as dependent negotiation power.

Naturally, the buyer wants to purchase when the revenue of a commodity at time t , )(2 tS , is

larger than the contract price K . However, the revenue of a commodity is uncertain. Also, the

value is usually discounted as time passes (so called, “time preference” in Rubinstein (1985) and

Cramton (1992)). The optimal investment rule for a buyer is determined by solving the following

stochastic optimal stopping problem, where the payoff functional, )( 22 SV , is given by

( ){ }KTSeESV rT

T−= − )(max)( 2222

2

2

,

where K= contract price, r= risk-free rate , and 2T =optimal buying time.

Based on Dixit and Pindyck (1994), the Bellman equation for this problem is

11

)]([)/1()( 2222 SdVEdtSrV = .

Using Ito’s lemma to manipulate )( 22 SdV ,

22,22,222

222,222 )()()2/1()()(

222dSSVdtSVSdtSVSdV

SSSt ++= σ .

By substitution of 222222 dzSdtSdS σμ += , we have

dtVdzSVSdtSVSdtSVSSdV tSSSS+++= 22222

22

2222222 )()()2/1()()(

2222σσμ .

Because 0)( 2 =dzE and the partial derivative with respect to time equals zero for the perpetual

American option, as in Merton (1973) and Dixit and Pindyck (1994), we have

)()2/1()()]([)/1( 2,222

222,22222 2

222SVSSVSSdVEdt

SSSσμ += .

This result and the Bellman equation lead us to have

0)()()()2/1( 222,2222,222

22 222

=−+ SrVSVSSVS SSS μσ , (2.1)

where 22,2 SSV =

2

2

2 dSdV

dSd

, 2,2 SV =

2

2

dSdV

and t

VVt ∂

∂= 2 .

Equation (2.1) is a differential equation that )( 22 SV must satisfy and )( 22 SV must satisfy the

following boundary, value-matching and smooth-pasting conditions (see Dixit and Pindyck

(1994)):

)0(2V =0,

)( *22 SV = KS −*

2 ,

)( *2,2 2

SV S =1,

where *2S is the optimal threshold to buy.

We assume 02 >− μr for the buyer’s value to be finite. Finally, after solving the above

differential equation, the value evaluated by a buyer is given by

12

⎪⎩

⎪⎨⎧

≥−<

=*222

*2222

22

2

)(SSifKSSSifSASV

β

(2.2)

where KS12

2*2 −=ββ , =2A ( )

2

1*2

2

β

β−S , and =2β

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⎟⎟⎠

⎞⎜⎜⎝

⎛−+− 2

2

2

22

222

2 8212121

σσμ

σμ r >1.

From these results, the advantageous investment for buying goods can be made when it is

exercised for *22 SS ≥ .

2.3.3. Optimal Selling Timing as a Put

Similar to the buying time, the seller’s decision-making problem can also be analyzed as an

optimal stopping problem. In this case, the valuation can be considered as a perpetual American

put option, as in Aase (2005). The seller’s payoff functional, )( 11 SV is:

( ){ })(max)( 11111

1

TSKeESV rT

T−= − ,

where 1T =optimal selling time

dztSdttStdS )()()( 11111 σμ += .

Using Ito’s Lemma to manipulate )( 11 SdV and the Bellman equation as we did in the case of a

buyer, we obtain the following differential equation:

0)()()()2/1( 111,1111,121

21 111

=−+ SrVSVSSVS SSS μσ , (2.3)

where 11,1 SSV =

1

1

1 dSdV

dSd

and 1,1 SV =

1

1

dSdV

Equation (2.3) is a differential equation that )( 11 SV must satisfy. Also, )( 11 SV must satisfy the

following boundary, value-matching and smooth-pasting conditions (see Dixit and Pindyck

(1994)):

13

0)(lim 111

=∞→

SVs

.

)( *11 SV = *

1SK − .

)( *1,1 2

SV S = 1− .

Solving this differential equation, we have:

⎩⎨⎧

≤−>

=*111

*1111

11

1

)(SSifSKSSifSASV

β

(2.4)

where KS11

1*1 −=ββ

, =1A( )

1

1*1

1

β

β−

−S

and =1β⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

21

2

21

121

1 221

21

σσμ

σμ r <0.

From these results, we know that a seller will exercise the put option strategically only

when *11 )( StS ≤ . Otherwise, the seller will delay selling a commodity until the inequality

condition is satisfied.

2.4. The Implicit Zone of Possible Agreement and Equilibrium Probability

In Section 2.3, we examined the optimal timing strategies for each party with the real option

approach. Based on these strategies, the conditions and probability for negotiation agreement will

be developed in this section. Also, related characteristics like comparative statics will be studied.

First, we need to recognize that negotiation agreement occurs only when both parties decide to

buy and sell a commodity. However, since valuations by two parties are uncertain, we develop

novel concepts of reservation price and zone of possible agreement under uncertainty in Theorem

1.

14

From the equation (2) and (4), we know that a seller will supply when

KStS1

)(1

1*11 −=≤ββ and a buyer will purchase when KStS

1)(

2

2*22 −=≥ββ . For given

)(1 tS and )(2 tS , we have

KtSK ≤−

≡ )(11

1

11 β

β for a seller,

KtSK ≥−

≡ )(12

2

22 β

β for a buyer.

Thus, we derive the following theorem.

Theorem 1. (Implicit reservation price and IZOPA) The optimal buying and selling strategies

are to sell when KK ≤1 and to buy when KK ≥2 , where 1K and 2K are defined as Implicit

Reservation Prices for a seller and a buyer in negotiation. Thus, we define the region

],[ 21 KK determined by implicit reservation prices as an Implicit Zone of Possible Agreement

(IZOPA).

Intuitively, when a buyer and a seller estimate the revenue and cost as 2S and 1S , a

traditional deterministic ZOPA (without an option to wait) implies that the agreement occurs

within a range of 1 2[ , ],S S i.e., 21 SKS ≤≤ . However, because two parties negotiate under

revenue and cost uncertainty, the possible agreement occurs in a narrower range. As shown in

Figure 2.2, two parties at time t would agree between two implicit reservation prices which are

derived by the thresholds of optimal selling and buying time. The implicit reservation prices by

real option models constitute the implicit zone of possible agreement (IZOPA).

15

Figure 2.2. Implicit Zone of Possible Agreement (IZOPA)

Moreover, the Figure 2.2 illustrates the relationship between ZOPA without an option and

IZOPA as in Theorem 2.

Theorem 2. (Width of IZOPA) The width of IZOPA is narrower than the width of ZOPA. (See the

proof in Appendix).

The option value affects reservation prices of both parties, and hence corresponding possible

agreement occurs within narrower ranges as shown in Figure 2.2. The top blue range stands for

the traditional ZOPA, while the bottom red one represents newly constructed IZOPA. The

difference results from the option caused by timing flexibility. For both parties to reach

agreement, each party has to compensate for the loss of option. As aforementioned, selling and

buying now implies that both parties should give up the possibility of waiting for new

information to arrive and the expectation for better future profits. For these reasons, unless each

party remunerates the other’s option value, the negotiation cannot be concluded successfully.

Thus, the range of agreement becomes narrower.

To see how the negotiation outcome changes when market conditions change, Proposition 1

provides comparative statics with respect to volatilities and mean-drifts.

16

Proposition 1. (Comparative statics for implicit reservation prices) For a given valuation at time

t, )(tSi for 2,1=i , the implicit reservation price of a seller (a buyer) increases (decreases)

monotonically as volatility of the cost (revenue increases), respectively. Also, the implicit

reservation prices are decreasing for increasing mean-drifts. (See the proof in Appendix.)

1

1

σ∂∂K 0> and

2

2

σ∂∂K 0< ,

01

1 <∂

∂

μK

and 02

2 <∂

∂

μK

.

When the revenue is faster-growing (higher 2μ ) and more volatile (higher 2σ ), a buyer asks a

lower contract price, because the buyer has to give up the opportunity of higher revenue in the

future by contracting now. Similarly, if a cost is more faster-decreasing (lower 1μ ) and more

volatile (higher 1σ ), a seller will not supply unless he or she is compensated with a higher price.

Next, we discuss the necessary condition so that both parties can agree. Because the existence

of IZOPA is not always guaranteed depending on valuations of two parties, Theorem 3 provides

the condition under which the agreement zone exists.

Theorem 3. (Condition for the existence of IZOPA) Two parties involved in negotiation can

reach an agreement with each other, when the following condition is satisfied

)()(

1

2

tStS≥ *

thλ , where *thλ =

( )( ) 1

11

21

21 >−

−ββ

ββ.

Note that 1β and 2β are obtained from Section 3. This result provides the negotiation

threshold *thλ which should be satisfied by the ratio of valuations of a buyer and seller at time t in

negotiation. When valuations at time t are observed or evaluated at )(2 tS and )(1 tS , the ratio of

17

these should be larger than or equal to a threshold *thλ . Otherwise, no agreement can be expected.

(See the proof and details in Appendix.)

Intuitively, a buyer’s estimated revenue should be larger than a seller’s estimated cost for

negotiation agreement to be feasible ( 12 SS ≥ ). However, interestingly, our result demonstrates

that this condition is not sufficient. Theorem 3 shows that buyer’s revenue should be larger than a

multiple of a seller’s cost and a certain threshold, )(2 tS ≥ )(1* tSthλ . Herein, the threshold is larger

than 1. This implies that revenue should be larger than a cost plus an option premium, because

both parties can strategically delay negotiation. This result is illustrated in Figure 2.3. A red

dotted line and a blue solid line represent thresholds for the agreement by a real option and a

traditional approach, respectively. Both parties consider negotiation on the left regions of each

line. As seen in the figure, the feasible (agreeable) region defined by considering options is

smaller than the region defined by ignoring options. The region between two lines represents the

amount of an option premium.

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Cost

Rev

enue

Comparison of Conditions for Agreement

Threshold with OptionThreshold without Option

Defined byReal Option

DefinedTraditionally

Figure 2.3. The impact of options on the condition for agreement

18

Since the threshold is affected by uncertainty factors, Proposition 2 studies the impact of

uncertainty, volatility ( 1σ and 2σ ) and rate of change of valuations ( 1μ and 2μ ). Note that the

traditional concept of ZOPA cannot explain these results.

Proposition 2. (Comparative statics for thresholds) The negotiation threshold *thλ is an

increasing function of the mean drift and volatility of the buyer’s revenue, and the volatility of the

seller’s cost, while a decreasing function of the mean drift of the seller’s cost:

1

21* ),(

σββλ

∂∂ th 0> and

*1 2

2

( , )thλ β βσ

∂∂

0> .

1

21* ),(

μββλ

∂∂ th 0< and

2

21* ),(

μββλ

∂∂ th 0> .

See the proof in Appendix.

This result shows that agreement is more difficult to reach when the market is more uncertain

or volatile, when revenue increases and when cost decreases. Proposition 2 is illustrated in Figure

2.4. These figures are obtained when 01.0=iμ for 2,1=i for the study of volatility effect

(right graph in Figure 2.4, 05.0=r ) and 1.0=iσ for 2,1=i for the study of mean-drift effect,

respectively.

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

10

50

100

150

200

Volatility of selling firm

Comparative Study on Volatility

Volatility of acquiring firm

Thr

esho

ld

−0.05

0

0.05

−0.05

0

0.050

20

40

60

80

100

120

Mean drift of selling firm

Comparative Study on Mean Drift

Mean drift of acquiring firm

Thr

esho

ld

19

Figure 2.4. Comparative studies on agreement threshold *thλ with respect to volatility and mean-

drift

2.5. Negotiation Agreement Probability

From this point, we study the probability of negotiation agreement similarly to Raiffa (1982)

and Zeng and Sycara (1998). The probability of agreement at a particular time t can be interpreted

as the probability that a final contract price is located between two implicit reservation prices,

which are andKKP ≤1( );2 tKK ≥ . However, the probability is difficult to calculate directly

because )(tSi in the implicit reservation prices, )(1 tSK ii

ii β

β −= , for 2,1=i is a random

variable varying from zero to infinity. To resolve this difficulty, the probability is calculated by a

transformation as stated in Theorem 4.

Theorem 4. (Probability of negotiation agreement) Probability of negotiation agreement can be

derived as an equation (2.5):

);( 21 tKKandKKP ≥≤ = ),;( *22

*11 tKSSandSSP ≥≤

∫∫∞

∞−−=

*2

*1

221 12

1X

X

NN ρσπσ

12

2

2

22

21

22112

1

112

))((2)1(2

1exp dXdXXXXX

N

N

NN

NN

N

N

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−

−σ

μσσ

μμρ

σμ

ρ (2.5)

where Niμ = tii ⎟⎠⎞

⎜⎝⎛ −

2

21σμ , tiNi σσ = , ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

)0()(ln

**

i

ii S

tSX , for 2,1=i

KS11

1*1 −=ββ , KS

12

2*2 −=ββ .

Here, the correlation

20

)1))(exp(2exp()1))(exp(2exp(

])2/12/1exp[(])exp[(),(

222

211

22

2121211221

21−−

−−+−++==

tttt

ttSScorr

σμσμ

σσμμσσρμμρ .

For detail and proof, see Appendix.

The probability with correlation can be obtained by numerical methods like the Gaussian

quadrature (Drezner (1978), ) or Monte Carlo simulation (Genz (1992)). However, at the outset of

negotiation or when correlation between revenue and cost is ignorable, we might expect low or

almost zero correlation. In that case, Theorem 4 can be induced into Proposition 3 as a special

case providing a closed form solution. The probability solution takes advantage of reduced

computational complexity because a special numerical method for double integral is not required.

Proposition 3. (No correlation) When the revenue and the cost are independent of each other, the

probability of negotiation agreement can have the following closed form solution:

),;( *22

*11 tKSSandSSP ≥≤

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −−

Φ−×⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −−

Φt

ttX

t

ttX

2

222

*2

1

211

*1 2

1)(12

1)(

σ

σμ

σ

σμ.

For detail and proof, see Appendix.

Based on Theorem 4 and Proposition 3, the probability can provide implications for

negotiation strategies. Before the discussion, more and simply, Theorem 5 explains why the

option values need to be considered.

Theorem 5. (Negotiation agreement probability) Negotiation agreement probability ignoring the

options to wait always overestimates the true probability (considering options):

);();( 2121 tKKandKKPtKSandKSP ≥≤>≥≤ , Kfor ∀ .

21

0 1 2 3 4 5 6

x 105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Contract Price

Pro

babi

lity

Uncertainty and Probability at Equilibrium

With OptionWithout Option

Figure 2.5. Probability with and without option values

Table 2.1 Estimated parameters

Parameter Value

1σ 0.127555

2σ 0.109667

1μ -0.02366

2μ 0.04082

R 0.05

T 60 )0(1S 62292 )0(2S 480318

For the detail and proof, see Appendix. A numerical example in Figure 2.5 under given

parameters in Table 2.1 shows that, like our intuition, the probability of agreement is higher in

the mid-range of contract prices. However, we can also see that the probability without

considering the option value is higher and is highly skewed to the right.

22

2.6. Extension: Negotiation Power

The model in the previous sections is developed in an environment of an exogenously given

contract price without considering a negotiation power. However, a buyer and a seller may

sometime have relative negotiation powers to negotiate the price. By extension, if a seller has

more negotiation power than a buyer, the contract will be negotiated at a higher price. To

accommodate this factor, this section develops a model incorporating individual party’s

negotiation power.

Let us assume that the seller’s negotiation power is ]1,0[∈γ , which implies that the buyer’s

negotiation power is )1( γ− . From the generalized Nash Bargaining game (Gurnani and Shi

(2006, Nagarajan and Sosic (2008)), a contract price is determined as follows (see Appendix for

the details and proof):

21)1( SSK γγ +−= (2.6)

For given negotiation powers for each party, the buyer’s value ( npV2 ) and the corresponding

optimal time to purchase can be derived as follows (See Appendix for the detailed proof): Let

12 / SS=λ ,

⎪⎩

⎪⎨⎧

≥−−−

<=

−

*212

*2

1122

212 )1()1(),(

22

λλγγλλββ

ifSSifSSA

SSVnpnpnp

np

npnp

np

A2

1*

2

)1()( 2

β

γλ β −=

−

, 12

2*2 −= np

np

ββλ ,

12

)(222

122

12112

22

12

2112

22

122112

22

122 >

+−−

+⎟⎟⎠

⎞⎜⎜⎝

⎛

+−−

−++−

−−=

σσρσσμ

σσρσσμμ

σσρσσμμβ rnp .

Also, the seller’s value ( npV1 ) and optimal time to sell ( *1λ ) are:

23

⎪⎩

⎪⎨⎧

≥−

<=

−

*112

*1

1122

211

22

),(λλγγλλββ

ifSSifSSA

SSVnpnpnp

np

npnp

np

A2

1*

1

2)(

β

γλ β−

= , * 21

2 1

np

np

βλβ

=−

.

Note that * *1 2λ λ= , so we define * * *2

1 22 1

np

np np

βλ λ λβ

≡ = =−

. From these results, we can see that

values for each party are different, but we have the same optimal threshold *2

*1 λλ = . This arises

from the fact that each party’s objective becomes aligned with the joint objective. Thus, both

parties reach agreement when *npλ λ≥ . Moreover, the threshold *

npλ is independent of the

negotiation power γ . From the above results, we can newly define an IZOPA and derive a

condition for existence of the IZOPA in the following theorem.

Theorem 6. (IZOPA with negotiation powers and condition for existence of IZOPA) The optimal

buying and selling strategies incorporating negotiation powers are to sell when KK np ≤1 and to

buy when KK np ≥2 , where K is defined in Eq. (2.6), npK1 and npK2 are given as follows:

21 1

2

(1 )1

npnp

npK Sβ γβ− −

=−

and 22 2

2

(1 )npnp

npK Sβ γβ− −

= .

Also, as in Theorem 2, we have a narrower agreement zone. Two parties incorporating

negotiation powers can make an agreement with each other, when the following condition is

satisfied:

22 1

2

/1

np

npS S ββ

≥−

.

See Appendix for proof.

Herein, when we do not consider the participants’ options, we know that the condition 12 SS ≥

should be satisfied so that a buyer and a seller can reach an agreement. However, our result

24

considering negotiation powers and options shows that the condition for agreement should be

necessarily stronger than the traditional condition because 2 2 2 1 1/ ( 1)np npS S Sβ β≥ − ≥ . Figure

6(a) illustrates that the IZOPA considering negotiation powers is narrower than the traditional

ZOPA similar to Figure 2.2. Figure 2.6(a) shows the change of the condition for negotiation

agreement when we consider participants’ options and negotiation powers. A red dotted line and

blue solid line represent thresholds for the agreement by a real option and a traditional approach,

respectively. Negotiation agreement is feasible when the revenue and the cost are on the left

regions of each line. As seen in the figure, the feasible region considering options is smaller than

the feasible region ignoring options. Thus, the feasible region defined by the real option model is

smaller than the traditional approach. Here, when we compare this IZOPA to the IZOPA without

the consideration of negotiation power in Theorem 1, consistent results unfortunately cannot be

found. Depending on parameters such as volatility and mean-drift, the one IZOPA can have the

other. However, main difference is that the IZOPA considering negotiation power is affected by

the opponent’s conditions, while the IZOPA without considering the power is not. However,

more important thing is that phenomena described in Section 4 still remain consistent when

individual’s negotiation powers are incorporated.

25

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Cost

Rev

enue

Comparison of Conditions for Agreement

Threshold with OptionThreshold without Option

Defined byReal Option

DefinedTraditionally

Figure 2.6. (a) IZOPA with negotiation power and (b) the impact of options on the condition for agreement

Moreover, from Theorem 6, we can derive the derivatives of the implicit reservation prices

with respect to negotiation powers as follows:

0/1 >γddK np , 0/2 >γddK np , 0)1(/1 <− γddK np , and 0)1(/2 <− γddK np .

Here, we know that γ and ( γ−1 ) stand for the negotiation power of a seller and a buyer

respectively. Thus, these derivatives imply that both parties implicitly raise reservation prices

when a seller’s negotiation power is lower than a seller’s.

2.7. Conclusions

In this paper, we have examined formal bilateral negotiation in a supply contract where the

buyer’s revenue and the seller’s cost are uncertain, and discussed the roles of the uncertainty in

negotiation outcomes. As aforementioned, both parties in negotiation have options to delay which

cannot be ignored and sometimes can be large, under the uncertainties and decision

irreversibility. Thus, in order to incorporate the option values in the supply contract negotiation,

we proposed novel concepts, implicit reservation prices and IZOPA, using a real option theory.

26

The new concepts enable us to analyze a negotiation outcome under uncertainty with the option

value. Our results help managers to appreciate the highly intertwined relation between

participants’ options and negotiation agreement and to make appropriate decisions accordingly.

First, we found that the IZOPA considering options is always narrower than the traditional

ZOPA ignoring options. If a buyer or a seller does not compensate for the opponent’s giving up

the opportunity to wait for a better future profit, agreement cannot be made. Therefore, the buyer

(the seller) asks lower (higher) contract price. Thus, the possible agreement zone becomes

narrower. Secondly, similar to the first result, the probability of agreement without considering

option value is always overestimated compared to the probability with an option value. Finally,

we found that, even though buyer’s revenue from the commodity is higher than the seller’s cost,

negotiation may not be agreed. The breakdown of negotiation has been explained in several ways,

such as the existence of third parties or outside options, as discussed in Muthoo (1995) and Li et

al. (2006). However, our result reveals that the breakdown can arise from each party’s option

values or timing flexibility. Furthermore, the proposed model provided quantitative analyses on

important parameters regarding to the uncertainty, such as volatility (σ ) and mean-drift ( μ ) and

the role of those in a negotiation outcome.

There are many areas where the concepts we suggested. For example, while this paper

focuses on a single commodity between two parties, we can apply the concepts in multi-

commodity, multi-buyers and multi-seller negotiation. Such extension will enable us to

investigate how uncertainty and option value affect the negotiation outcome under the existence

of the possible third party like Li, et al. (2006). Also, this paper assumed a fixed-price type of a

contract in supply chain. However, there are many different types of contract such as buy-back,

price-discount, quantity flexibility, and revenue sharing (see Pasternack (2008), Bernstein and

Federgruen (2005), Tsay et al. (1999) and Cachon and Lariviere (2005)). For these different

contracts, we can compare the role of uncertainty and option in negotiation. Also, decision

27

maker’s risk preference in negotiation is a critical component because timing is affected by the

preference. Consideration of risk preference can be another extension of this research (see

Henderson and Hobson (2002)).

28

APPENDIX

Appendix 2.A. Proof of Theorem 2

According to the IZOPA and the ZOPA by Theorem 1,

222

21

1

11 )(1)(1 KtSKtSK =

−≤≤

−=

ββ

ββ and )()( 21 tSKtS ≤≤ .

Since 01 <β and 12 >β , following conditions are derived as comparing both sides of conditions:

)()(11)(1

111

11

11 tStStSK >⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

−=

βββ ,

)()(11)(1

222

22

22 tStStSK <⎟

⎟⎠

⎞⎜⎜⎝

⎛−=

−=

βββ

. □

Appendix 2.B. Proof of Proposition 1

Implicit reservation prices for a buyer and a seller are )(11 11

1 tSK ⎟⎟⎠

⎞⎜⎜⎝

⎛−=β

and )(11 22

2 tSK ⎟⎟⎠

⎞⎜⎜⎝

⎛−=β

. Thus, comparative statics are developed from the first order conditions

of these with respect to 1σ and 2σ as well as 1μ and 2μ . From differential equation (3.1) and

(3.3), 01 <β and 12 >β should satisfy the following equation:

0)1(21 2 =−+− rμβββσ

Let =)( iQ β r−+− μβββσ )1(21 2 and take a derivative:

0=∂∂

+∂∂

∂∂

ii

i QQσσ

ββ

, for 2,1=i

29

where derivatives are evaluated at 1β and 2β .

Also, we know 0<∂∂βQ at 1β and 0>

∂∂βQ at 2β since Q is quadratic, 01 <β and 12 >β .

Moreover, 0)1( >−=∂∂

iiii

Q ββσσ

for 2,1=i . From these inequality conditions, we have

01

1 >∂∂σβ and 0

2

2 <∂∂σβ .

Thus, taking derivatives of the gaps with respect to 1σ and 2σ , the results are as follows:

011

1

12

11

1 >∂∂

=∂∂ SK

σβ

βσ and 01

2

222

22

2 <∂∂

=∂∂

σβ

βσSK .

Moreover, comparative studies with respect to 1μ and 2μ can be obtained similarly.

21

2

21

121

1

21

2

21

121

21

1

211

1

821821

2122

σσμσ

β

σσμσ

σμ

σμβ

rr+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−=∂∂ 0< ,

21

2

21

121

2

22

2

22

222

22

2

222

2

821821

2122

σσμσ

β

σσμσ

σμ

σμβ

rr+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−=∂∂ 0< .

Finally, we have

01

1

112

11

1 <∂∂

=∂∂

μβ

βμSK and 01

2

222

22

2 <∂∂

=∂∂

μβ

βμSK . □

Appendix 2.C. Proof of Theorem 3.

30

From Theorem 1, we got *11 )( StS ≤ and *

22 )( StS ≥ so that agreement occurs. And in Section 3,

the thresholds *1S and *

2S are derived. Therefore, we have

KStS1

)(1

1*11 −=≤ββ

KStS1

)(2

2*22 −=≥ββ

Changing these inequality conditions with respect to a contract price K leads to

KtS ≤− )(1

11

1

ββ

KtS ≥− )(1

22

2

ββ

These two inequality conditions give the following inequality conditions:

)(1)(12

2

21

1

1 tSKtSβ

ββ

β −≤≤

− .

Herein, we know that the condition, )(1)(12

2

21

1

1 tStSβ

ββ

β −≤

− , should be satisfied so that the

possible agreement zone exists. Since we know that 01 <β , 12 >β , and )(1 tS >0, the threshold

for the existence is:

*thλ = ( )

( )11

21

21

−−ββ

ββ ≤)()(

1

2

tStS .

Also, since we know 01 <β and 12 >β , we can derive 1)1(

1

1 >−

ββ and 1

)1( 1

2 >−β

β . Accordingly,

we have

)()(

1

2

tStS ( )

( ) 11

1 *

21

21 >=−

−≥ thλ

ββββ □

31

Appendix 2.D. Proof of Proposition 2.

From the Proof of Proposition 1, we have

01

1 >∂∂σβ and 0

2

2 <∂∂σβ .

Taking derivatives of function, ),( 21* ββλth = ( )

( )11

21

21

−−ββ

ββ , with respect to 1β and 2β , we have

=∂

∂

1

21* ),(

αββλth

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

− 1

12

12

2 11 σ

βββ

β 0> ,

=∂

∂

2

21* ),(

αββλth

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−−

−

2

22

21

1

)1(1)1(

σβ

βββ 0> .

Also, the proof of Proposition 1 leads to:

=∂

∂

1

21* ),(

μββλth

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

− 1

12

12

2 11 μ

βββ

β 0< ,

=∂

∂

2

21* ),(

μββλth ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−−

−

2

22

21

1

)1(11

μβ

βββ 0> . □

Appendix 2.E. Proof of Theorem 4.

Since iS for 2,1=i follows a lognormal distribution with a mean tii ⎟⎠⎞

⎜⎝⎛ − 2

21σμ , a standard

deviation tiσ , and dtdzdzE 1221 ][ ρ= , the probability density functions will be:

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−

−×

−=

2

2

22

21

22112

1

112

212

21

21

log))(log(log2log)1(2

1exp

12

1),(

σμ

σσμμ

ρσ

μρ

ρσπσ

NNNN

NN

xSSS

SSSSg

where Niμ = tii ⎟⎠⎞

⎜⎝⎛ −

2

21σμ , tiNi σσ = .

32

Herein, because we assumed that two correlated assets follow the Geometric Brownian

Motions with dtdzdzE 1221 ][ ρ= , as in many researches (i.e. see McDonald and Siegel (1986,

Kamrad and Siddique (2004)), we need to transform the correlation coefficient ( 12ρ ) to the one

of lognormal distribution ( ρ ). Therefore, we derive the correlation ρ in the following way.

By definition of correlation, we know

)()()()()(

)()(),cov(

),(21

2121

21

2121 SstdSstd

SESESSESstdSstd

SSSScorr

−===ρ .

When 1S and 2S follow the Geometric Brownian Motion, the 21 SS also follows the geometric

Brownian motion (See details in the chapter 3 in Dixit and Pindyck 1994). In short, the change in

the logarithm of 21 SS is normally distributed with mean t)5.05.0( 22

2121 σσμμ −−+ and

variance t)2( 2122

21 σρσσσ ++ .

[ ]tSS

ttSSSSE

)(exp)0()0(

])5.05.0()5.05.0exp[()0()0()(

21122121

2122

21

22

21212121

σσρμμ

σρσσσσσμμ

++=

+++−−+=

Besides, since both 1S and 2S follow the Geometric Brownian Motion, we know

)exp()0()( tSSE iii μ= for i=1,2

)1))(exp(2exp()0()( 2 −= ttSSstd iiii σμ for i=1,2.

By substitution, the correlation ρ is

)1()1()0()0(

])2/1exp[(])2/1exp[()0()0(])exp[()0()0(

)()()()()(

),(

222

211 22

21

222

2112121122121

21

212121

−−

−−−++=

−=

tttt eeeeSS

ttSStSS

SstdSstdSESESSE

SScorr

σμσμ

σμσμσσρμμ

Therefore, we can derive ρ from 12ρ as follows:

)1))(exp(2exp()1))(exp(2exp(

])2/12/1exp[(])exp[(222

211

22

2121211221

−−

−−+−++=

tttt

tt

σμσμ

σσμμσσρμμρ .

33

Also, we know that, because exponential function is an increasing function, ρ is increasing

in 12ρ .

From the above, the probability, ),;( *22

*11 tKSSandSSP ≥≤

∫∫∞

∞− −=

*2

*1

212

21 12

1S

NN

S

SSρσπσ

12

2

2

22

21

22112

1

11 log))(log(log2log)1(2

1exp dSdSSSSS

N

N

NN

NN

N

N

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−

−σ

μσσ

μμρσ

μρ

Changing variable ⎟⎟⎠

⎞⎜⎜⎝

⎛=

)0()(ln

i

ii S

tSX for 2,1=i , we have )()(

1 tdStS

dX ii

i = and

∫∫∞

∞−−=

*2

*2

221 12

1X

X

NN ρσπσ

12

2

2

22

21

22112

1

112

))((2)1(2

1exp dXdXXXXX

N

N

NN

NN

N

N

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−

−σ

μσσ

μμρσ

μρ

□

Appendix 2.F. Proof of Proposition 3.

Because we assumed independence between the revenue and the cost, the probability can be

simplified as follows:

),;( *22

*11 tKSSandSSP ≥≤ = ),;)(( *

11 tKStSP ≤ ),;)(( *22 tKStSP ≥

Similarly to the proof for Proposition 3, we can derive the probabilities for each case as follows:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −−

−=t

tS

tS

tSttSP

i

ii

i

iiii 2

22

2/12 221)

)0()(ln(

exp)()2(

1))((σ

σμ

σπ and

34

),;)(( *22 tKStSP ≥ = ∫

∞

)( 22*2

))((tS

dStSP = ∫∞

−−

KrdStSP

)(1

222

2

2))((

μββ

),;)(( *11 tKStSP ≤ = ∫ ∞−

)(

11

*1 ))((

tSdStSP =

( )∫

−−

∞−

KrdStSP1

1

1

111 ))((

μββ

Change variable ⎟⎟⎠

⎞⎜⎜⎝

⎛=

)0()(ln

i

ii S

tSX , )()(

1 tdStS

dX ii

i = , for 2,1=i

∫∞

)( 22*2

))((tX

dStSP = ∫∞

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −−

−)( 22

2

222

2

2

22/12

2*2 2

21)

)0()(ln(

exp)()2(

1tX

dSt

tS

tS

tSt σ

σμ

πσ

= ∫∞

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −−

−)( 22

2

2222

2/122

*2 2

21

exp)2(

1tX

dXt

tX

t σ

σμ

πσ

=⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −−

Φ−t

ttX

2

222

*2 2

1)(1

σ

σμ.

Similarly, we have

∫ ∞−

)(

11

*1 ))((

tXdStSP =

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −−

Φt

ttX

1

211

*1 2

1)(

σ

σμ.

Finally, we can have the probability ),;)()(( *22

*11 tKStSandStSP ≥≤ as follows:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −−

Φ−×⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −−

Φt

ttX

t

ttX

2

222

*2

1

211

*1 2

1)(12

1)(

σ

σμ

σ

σμ. □

35

Appendix 2.G. Proof of Theorem 5.

From Theorem 4, we know that probability considering option value

is );( 21 tKKandKKP ≥≤ , while probability without option value is obtained

by );( 21 tKSandKSP ≥≤ . For given K at time t , Theorem 4 gives us

⎟⎟⎠

⎞⎜⎜⎝

⎛≤

−−≤=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −≤≤

−=≥≤

)(1

,1

)(

)(1)(1);(

22

2

111

22

21

1

121

tSKKtSP

tSKtSPtKKandKKP

ββ

ββ

ββ

ββ

∫ ∫⎟⎟⎠

⎞⎜⎜⎝

⎛

−∞−

∞

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

=K

KdSdSSSg1

121

211

1

2

2),(β

β

ββ .

);( 21 tKSandKSP ≥≤ ∫ ∫∞−∞

=K

KdSdSSSg 1221 ),( .

Because KK <⎟⎟⎠

⎞⎜⎜⎝

⎛

−11

1

ββ

and KK <⎟⎟⎠

⎞⎜⎜⎝

⎛

−12

2

ββ

from Theorem 2, we have

∫ ∫⎟⎟⎠

⎞⎜⎜⎝

⎛

−∞−

∞

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

K

KdSdSSSg1

121

211

1

2

2),(β

β

ββ ∫ ∫∞−

∞<

K

KdSdSSSg 1221 ),(

Therefore, );( 21 tKKandKKP ≥≤ );( 21 tKSandKSP ≥≤< □

Appendix 2.H. Proof of an extension model in Section 6.

The Generalized Nash Bargaining game (GNB) provides a way to capture negotiation

powers. The generalized Nash solution ( ),;,;,( 2121 βαddxxB ) is obtained by solving the

following form (Nagarajan and Sosic (2008))

βαβα )()(maxarg),;,;,( 2211,

212121

dxdxddxxBxx

−−= .

where 1=+ βα .

36

The indices α and β describe the relative negotiation powers. Also, if Player 1 is more

powerful on the negotiation than Player 2, thenα is larger than β . When the negotiation is

agreed, each player will have utilities of 1x and 2x . Also, 1d and 2d are the disutilities when both

parties fail to reach an agreement. These disutilities are usually assumed to be zero, because both

have nothing as negotiation fails.

Applying the Generalized Nash Bargaining problem to our problem, we can determine the

contract price (K). When both parties agree the negotiation, each will make profits ( KS −2 ) and

( 1SK − ). As disagreement occurs, the negotiation means nothing to both parties, which implies

that disutilities are zero. Finally, we have the solution );,;( 12 γSSKB of our Generalized Nash

Bargaining game in the form

])()();,;([maxarg);,;( 11

21212γγγγ SKKSSSKGSSKB

K−−≡= − .

The first order condition of log );,;( 12 γSSKG leads us

[ ] 0)log()log()1());,;((log 1212 =−+−−= SKKSdKdSSKG

dKd γγγ ,

21)1( SSK γγ +−=

Based on the derived contract price K , we reformulate the equation (3.3) as follows:

( ){ } ( ){ })()(max)1()(max),( 1222122

2

2

2

tStSeEKtSeESSV rTtT

rTtT

np −−=−= −− γ .

Using Ito’s Lemma, we have

0

)2/1(2)2/1(

2,211,222

,221

21,21212,2

22

22

12

1111222

=−++

++

npnpS

npS

npSS

npSS

npSS

rVVSVS

VSVSSVS

μμ

σσρσσ

Corresponding value-matching and smooth-pasting conditions (see Dixit and Pindyck 1994) are:

0)0,0(2 =npV ,

37

),( *2

*12 SSV np = )1()1( *

1*2 γγ −−− SS .

)1()( *1,2 1

γ−−=SV npS

)1()( *2,2 2

γ−=SV npS

.

The homogeneity property of this problem allows us to reduce the above problem to an one-

dimensional problem by letting 12 / SS=λ and )(),( 1212 λvSSSV np = as follows:

0')(")2(2/1 1222

11222 =−−++− rvvv λμμλσσρσσ .

0)0( =v ,

)( *λv = )1()1(* γγλ −−− ,

)1()( *2

γλλ −=v .

Solving similarly to the derivations in Section 3, we have )(λv and then, by substitution ofλ , we

have ),( 212 SSV . Also, we can have ),( 211 SSV in the same way. □

Appendix 2.I. Proof of Theorem 5.

By substituting Eq. (3.6) into *1λ and *

2λ , we can derive the new implicit reservation prices,

npK1 and npK 2 . Since 11

)1(

2

2 ≥−−−

np

np

βγβ and 1)1(

2

2 ≤−−

np

np

βγβ , we know 11 SK np ≥ and 22 SK np ≤

implying a narrower zone. Also, for the necessary condition, since * 22 1

2

/ , 1

np

np npS S βλ λβ

= =−

,

then * npλ λ≥ leads to 22 1

2

/1

np

npS S ββ

≥−

.

□

38

Chapter 3. Outsourcing versus Joint Venture from Vendor’s Perspective

We consider a vendor’s contract decision facing price and cost uncertainty. Even though many studies discuss a contract from the client’s perspective, little attention has been paid to different contracts from vendor’s perspective. This paper develops real option models to investigate whether a vendor should sign an outsourcing contract from the client or establish a joint venture with the client. Our results show that, while the feasibility of an outsourcing contract can be increased by a higher contract price offered by a client, feasibility of a joint-venture depends on market conditions. We also find that there are loss-by-acceptance regions, in which either an outsourcing or a joint venture contract is currently feasible to start, but a vendor may sustain a loss by accepting such a contract.

3.1. Introduction

Strategic alliance among different firms has been highlighted in both literature and practice.

The reason is that cooperation can be another way to obtain many benefits, such as ease of market

access, cost reduction, operational efficiency, capacity pooling and access to external expertise or

technology advantages (Hamel et al. (1989, Li and Kouvelis (1999, Barthelemy (2001, Deloitte

(2005, Plambeck and Taylor (2005)). According to Hamel, et al. (1989), even competitive

collaboration between competitors can strengthen both firms against outsiders. For example, to

compete against Japanese companies Matsushita, Sanyo, and Sharp in 1980’s, GE contracted out

the production of microwave ovens to Samsung of South Korea because it could perform the

manufacturing operations at a lower cost. The outsourcing decision worked well and GE’s profit

margins significantly increased (Domberger (1998)). Also, in 2008, US steel, POSCO and SeAH

steel agreed to a joint venture with shares of 35%, 35% and 30% based upon their respective

ownership interests in proportion to invested capitals.

These collaborations were contracted in many different forms such as joint-venture,

outsourcing, product licensing and cooperative research. When a client suggests a joint venture

39

contract and its potential partner (vendor) agrees on it, these firms proportionally share some of

their resources, capabilities and profits. In an outsourcing contract, a client externalizes all

operational or technological activities to a vendor, who acquires the pre-contracted amount of

output or revenue by the client firm. However, even though a contract has two sides of

interaction, most research has mainly focused on the client’s side while overlooking the vendor’s

standpoint (Levina and Ross (2003) and Jiang et al. (2008)). The importance of considering

vendor’s side arises from the fact that it is connected to the feasibility of the contract agreement.

Unless the contract is expected to bring positive profits to the vendor, the contract cannot be

agreed upon by both parties. Nevertheless, little research examines a vendor’s strategic decision.

Moreover, evaluating a contract is challenging for a vendor, because a vendor’s costs are not

usually constant over the duration of a contract due to fluctuations in the exchange rate, labor

wage policy changes and hyperinflation conditions (Austin (2002), Li and Kouvelis (1999),

Chopra and Sodhi (2004)). Accordingly, the market price fluctuates substantially as market

conditions and sourcing costs change over time. Under such an uncertain environment, different

alliance contracts may result in different outputs to a vendor. This article develops valuation

models incorporating cost and price uncertainty and compares different contracts from the

vendor’s perspective.

As representative schemes of alliance contracts, a fixed-price outsourcing contract and a

joint-venture contract are compared in this research. In a fixed-price outsourcing contract, a client

externalizes its previous in-house activities to a vendor and acquires the relevant output or

revenue by paying a fixed price. However, for the joint-venture, a client is responsible for paying

the vendor a share of the cost as well as requiring a share of the revenue. More importantly from

the vendor’s perspective, in the case of an outsourcing contract, a certain amount of revenue is

always guaranteed from a client. However, after contracting a joint venture, a vendor must be

ready to suffer possible loss due to revenue and cost uncertainty.

40

For these reasons, we first develop a valuation model under uncertainties from a vendor’s

perspective and then compare the two contracts in regards to which contract becomes feasible and

how the vendor should make a decision. In Section 2, literature related to contracts and the real

option theory will be discussed. Section 3 provides model settings using real option theory.

Section 4 studies vendor’s value for each contract and discusses contract’s feasibility based on an

optimal contract threshold. Section 5 compares two contracts and explores the main difference in

vendor’s decision making between the traditional approach, Net Present Value (NPV) method,

and the real option theory (ROT) approach. Section 6 summarizes this research with managerial

implications and suggests further studies.

3.2. Literature Review

The majority of research on outsourcing is focused on client’s side, while the vendor’s aspect

is rarely examined. However, recent studies have discussed how organizational relationships and

different contractual structures affect outsourcing outcomes and emphasize the importance of

studies from vendor’s perspective (Kern et al. (2002), Levina and Ross (2003)). For example, the

case study by Kern, et al. (2002)) illustrated “the winner’s curse”: in order to win an outsourcing

contract bid from a client, a vendor will sometimes bid at a lower price, which may lead to a loss.

Jeffery and Leliveld (2004)) report that most vendors are using the standard net-present-value

(NPV) approach to evaluate outsourcing contracts. If the NPV is positive, the project is

worthwhile and should be pursued; if it is negative, the project should be turned down. For

example, Dayanand and Padman (2001)) use the standard NPV approach (i.e. NPV > 0) to study

the progress payments problem of outsourcing contracts. In recent years, many researchers have

criticized that NPV based decision making can miss the additional value of managerial flexibility,

which results from uncertainty. Firms may have the option to exercise the investment opportunity

41

or to hold it for some length of time (Brennan and Schwartz (1985), Dixit (1989), McGrath

(1997)). In other words, a vendor can ignore a client’s project now, leaving them free to invest in

other external or internal projects in the future or wait for another client. This freedom makes

investment timing an important factor in decision making. For this reason, Dixit and Pindyck

(1994)) apply the real options theory (ROT) to discuss optimal investment timing in the

framework of investment irreversibility and uncertainty, and also point out the parallels between

an investment opportunity and a “call option”. A call option gives an investor the right to acquire

an asset of uncertain future value. If conditions favorable to investing arise, the investor can

exercise the investment.

Beside whether to accept a client’s offer, a vendor has one more option in hand: if the offer is

accepted, when to exercise the contract (Jiang, et al. (2008)). Such an option is not obvious

because it is implicit. In fact, the timing of when to exercise a contract usually is out of the

vendor’s control. It is rare that a client would hold on to a contract opportunity to wait for the

vendor’s optimal time to exercise the agreement. If a vendor signs a contract, it is valid at that

given time. From the ROT perspective, however, the exercise at the given time enforces the

vendor to give up the option to wait. Because losing the option to wait could expose the investor

under a potential loss of money (Luehrman (1998), Zhu and Weyant (2003)), it is a commonly

agreed principle in financial economics that no investment should take place unless its net

benefits at least compensate for the loss of “value of waiting” (Huchzermeier and Loch (2001)).

In other words, if NPV is larger than zero, it is not the critical point of investment, but if NPV is

larger than the value of the lost option (Pindyck (1991), Trigeorgis (1993)). Since the vendor who

has to exercise a contract at the given time is no longer hedged by its option to wait, the vendor

must make up this lost option value.

Our research considers two different alliance contracts (an outsourcing and a joint

venture contract) with a vendor’s option to wait. Even though many studies have discussed

42

different contracts and compared the from the perspectives of incentive scheme, channel

coordination and risk hedging (Laffont and Tirole (1993, Cohen and Agrawal (1999, Cachon and

Lariviere (2005), Kouvelis et al. (2006)), the vendor’s options have rarely been discussed. Thus,

we develop real option models and investigate when the contract becomes feasible and how the

selection strategy should be established.

3.3. Model Settings

We consider a vendor who is facing two possible alliance contracts from a client: an

outsourcing contract and a joint venture contract. The vendor’s product cost is tC over the

contract duration D . The product is sold by the client to an exogenous market at a market

price tP . Here, tC and tP , evolve uncertainty over time as Geometric Brownian Motions (GBM),

which are the continuous-time formulation of the random walk. This is the standard setting in real

option theory and also a good approximation for uncertainties (Dixit (1989), Dixit and Pindyck

(1994), Abel and Eberly (1994),Alvarez and Stenbacka (2007)). Specifically,

CtCtCt dzCdtCdC σμ += ,

PtPtPt dzPdtPdP σμ += ,

where Cdz , Pdz = an increment of standard Wiener process for cost and price

Cμ , Pμ = the shift rate of expected future change for cost and price

Cσ , Pσ =the uncertainty rate of such a process for cost and price

with the correlation, vdtdzdzE PC =][ .

43

3.3.1. Model for an Outsourcing Contract

We represent the time when the vendor exercises the contract at time 0t . Suppose that under

the outsourcing contract, the client pays a fixed outsourcing contract price OP to the vendor over

the contract duration D , i.e., from the contract starting time 0t to ending time Dt +0 . Such an

outsourcing contract’s net present value (NPV) to the vendor is:

C

D

t

D

OTt

tO

Dt

tt

CeCePdteCPEμρρ

μρρρ

−−

−−

=−=Φ−−−

−−+

∫)(

)( 11])([0

0

00

where ρ is the discount rate. Assume Cμρ > , Pμρ > for convergence.

Real option theory enables us to consider that before undertaking a contract, a vendor has the

option to exercise or to wait. As a result, the real option value reflects more accurately the value

of an investment opportunity by considering a value of option to wait as well as a standard NPV

than the standard NPV does (Benaroch (2002)). The real option value at 0t , the time before the

vendor exercises the contract, can be described by the standard real options expression:

][max)( )(0

0

0

tTTtTO etF −−+

≥Φ= ρ ,

where )0,max(XX =+ , reflecting the essence of an option. By definition, there is no obligation

to exercise an option, the value of option to wait is always non-negative. For example, if the

standard NPV is negative at 0t , NPV method recommends to give up the investment while ROT

suggests that a firm needs to wait rather than to relinquish. The vendor hopes to maximize its real

option value at 0t by selecting the optimal investment time T in the future.

Solving this (See Appendix 3.A.), we have the vendor’s value function as follows:

⎪⎩

⎪⎨

⎧

>

≤−

−−

−

=

−−−

*1

*)(

0

1

0

00

0

,

,11

),(CCCA

CCe

Ce

PCPF

tt

tC

D

t

D

OtOO

O

C

β

μρρ

μρρ (3.1)

44

where DC

D

OO

OCe

ePC )(

1

1*

1

1

1 μρ

ρ μρρβ

β−−

−

−−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

−= ,

0221

21

2

2

221 <+⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=

CC

C

C

CO σ

ρσμ

σμβ ,

1

)()1(*

111

)( 1

OC

DC

Oe

CAβμρ

μρβ −

−

−=

−−− .

The threshold for an outsourcing contract implies that, if *0

CCt ≤ , the vendor can sign this

contract with a particular contract price OP . Otherwise, the vendor should not accept this contract

offer.

3.3.2. Model for a Joint Venture Contract

In a joint venture contract, a client and a vendor share the market price, tP , and the

operating cost tC . Then a vendor with a share α receives tPα and incurs cost tCα , for

10 <<α . (Lee (2004)). Thus, a profit function for a vendor during the contract will be:

∫+ −−−=Ψ

Dt

t

ttttt

dteCPE 0

0

0

0])([ )(ρα

=C

D

tP

D

t

CP eC

eP

μρα

μρα

μρμρ

−

−−

−

− −−−− )()( 1100

.

From definition of an option, the value function for a joint venture contract is:

][max),( )( 0

00

00

tTttT

ttJ eCPF −−+

≥Ψ= ρ .

We now omit the time index in expressions for convenience. Using homogeneity property of the

optimal decision with respect to a market price and a cost, we have a firm’s value function as

follows (See Appendix 3.B. for the proof):

45

⎪⎪⎩

⎪⎪⎨

⎧

≤

>−

−−

−

−=

−

−−−−

*12

*)()(

/,

/,11

),(22

J

JC

D

P

D

J

RCPCPA

RCPe

Ce

PCPF

JJ

CP

ββ

μρμρ

μρα

μρα

(3.2)

where ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−−

==−−

−−C

D

DP

J

JJ

C

P

e

eCPR

μρμρ

ββ

μρ

μρ

)(

)(2

2*** 1

1

)(1

/ ,

12

)(222

122

122

2

22222 >+−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛

+−−

−++−

−−=

PPCC

C

PPCC

RC

PPCC

PCJ vvv σσσσ

μρσσσσ

μμσσσσ

μμβ ,

)(

1)( )(

2

1*

2

2

P

D

J

JPJ eR

Aμρβ

α μρβ

−

−=

−−−

.

If the threshold for a joint venture, CPRJ /* < , then alliance will start to contract. In other words,

if a market condition is sufficiently good compared to a cost, the vendor can sign this contract

with a particular contract share,α . Otherwise, the vendor should not accept this contract offer.

3.4. Contract Feasibility with Optimal Investment Thresholds

A client is likely to think all contracts are feasible to start at a given time, when it makes a

contract offer. However, since different contracts have different levels of flexibility, contract

feasibility might vary according to a market condition and uncertainty (i.e., a market price and a

cost). For example, at a given time, t, a vendor can accept an outsourcing contract, but cannot

accept a joint venture contract. Therefore, we compare an outsourcing contract and a joint venture

contract from the contract feasibility perspective. The condition that a certain contract is feasible

can be verified by investigating whether or not a contract price for an outsourcing and a share

ratio for a joint venture reach the optimal investment thresholds. Furthermore, we compare the

46

contract feasibility with the NPV method versus the ROT method and show that the contract

feasibility can be reduced by a vendor’s embedded option to wait.

First, let us discuss the role of a vendor’s embedded option to contract feasibility for each

contract by comparing investment thresholds between a standard NPV evaluation method and our

ROT approach.

Theorem 1. Contract feasibility by NPV (and ROT). The contract feasible region by NPV, feaNPVC

(and feaNPVR ) is always larger than that by ROT, fea

ROTC (and feaROTR ). (See Appendix 3.C. for the

proof)

where }0{ *ROT

feaROT CCCC ≤≤= ∈ }0{ *

NPVfea

NPV CCCC ≤≤= and

}{ *ROT

feaROT RRRR ≥= ∈ }{ *

NPVfea

NPV RRRR ≥=

where ** CCROT = and **JROT RR = as defined in Eqn. (3.1) and (3.2).

In other words, an optimal threshold by NPV for an outsourcing and an optimal ratio

threshold for a joint venture are larger than one by ROT, respectively. These results reveal that

the standard NPV approach provides more favorable criteria to the vendor’s decision making than

that of ROT approach does. Under given parameters in Table 3.1, Figure 3.1 shows the

comparison of results by NPV and ROT. As shown in Figure 3.1 (a), a feasible region of an

outsourcing contract by ROT is a subset of a feasible region by NPV. Similarly, Figure 3.1 (b)

shows that a feasible region by ROT is more restricted to sign.

Table 3.1 Parameters for numerical examples

Parameter ValueCμ -0.04

Pμ 0.05 ρ 0.06 v 0.8

OP 10 D 3

47

Cσ 0.5

Pσ 0.4

0 50 100 150 2000

50

100

150

200

250

300

Cost

Ven

dor’s

Val

ue

Vendor’s Value with and without Option

ROT (with option)NPV (without option)

Feasible by NPV

Feasible by ROT

Figure 3.1. The contract feasibility by ROT and NPV

(a) Outsourcing contract (b) Joint venture contract

Theorem 1 explains a relation and a property between results by a ROT and NPV. From now

on, we study how the feasibility of each contract is affected by a client’s offer (Theorem 2) and

uncertainty factors (Proposition 1). When a client offers a contract price, P, for an outsourcing

contract and a share ratio,α , for a joint venture, the impacts of the offers to feasibility of a

contract are discussed in Theorem 2.

Theorem 2. (Vendor’s readiness) A higher contract price OP increases the feasibility of an

outsourcing contract, while a higher share ratioα does not improve the feasibility of a joint

venture contract.

Proof) Since D

CD

OO

O

Ce

ePC

)(1

1*

1

11 μρ

ρ μρρβ

β−−

−

−

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

−=

Feasible by ROT

Feasible by NPV

48

and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−−

=−−

−−C

D

DP

J

JJ

C

P

e

eR

μρμρ

ββ

μρ

μρ

)(

)(2

2* 1

1

)(1

is independent of a share ratio, α , we have

0*

>∂

∂

OPC

and 0*

=∂∂α

JR . Q.E.D.

For an outsourcing contract, when a vendor is not ready to undertake a contract at given time,

a client urges a vendor to start outsourcing by compensating for the hurriedness at a higher

contract price. In contrast, for a joint venture contract, even though a client offers higher share

ratio to a vendor, the vendor cannot hurry up to contract out, because the threshold is not

dependent of a share ratio. However, the profit from the joint venture contract is affected by the

share ratio as induced in an equation (3.2). In addition, the contract feasibility may be affected by

uncertain market conditions, such as degree of volatility of a cost and a price. Thus, Proposition 1

gives implications about the role of uncertainties to the contract feasibility.

Proposition 1. (Feasibility and Uncertainty) Thresholds for an outsourcing (a joint venture)

contract are decreasing (increasing) for increasing volatility. Simply, more uncertain market

conditions make a vendor more difficult to accept offers for both contracts. (See Appendix 3.D.)

0*

<∂

∂

C

Cσ

,

0*

>∂∂

P

JRσ

, 0*

>∂∂

C

JRσ

.

Proposition 1 states that uncertainties in the cost, C, and the market price, P , affect both

contracts. Theorem 1 and Proposition 1 conclude that the feasibility of an outsourcing contract is

affected by both of a client’s offer and market condition but a joint venture contract depends on

only market.

49

3.5. Contract Feasibility and Selection

A vendor needs to select a contract between an outsourcing and a joint venture, when it

received offers. For better profits, the vendor would compare profits from different contracts, i.e.,

comparing the functional ),( CPF OO with ),( CPFJ . Simply, a vendor will choose a joint venture

if ),( CPFJ > ),( CPF OO , while, otherwise, it selects outsourcing. Thus, we will exploit a selection

strategy based on the value function of each contract. Also, we will compare contract selection

strategies and show that the strategies might be different.

Theorem 3. There are loss-by-acceptance regions, in which a contract is currently feasible to

start an outsourcing or joint venture, but a vendor might have loss by accepting the feasible

contract.(See Appendix 3.E. for proof)

(i) If 12 1 OJ αββ −> , there exists a joint venture loss-by-acceptance region.

(ii) If 12 1 OJ αββ −< , there exists an outsourcing loss-by-acceptance region.

(iii) If 12 1 OJ αββ −= , there is no loss-by-acceptance region.

Theorem 3 explains that a vendor may have three different scenarios which are determined by the

relationship among 1Oβ , α and 2Jβ . For example of case (i), a vendor might have loss by

accepting a joint venture contract which is currently feasible to start. Therefore, depending on the

relationship, a vendor needs to establish a different strategy. Moreover, the result implies that

these different scenarios are determined by market conditions (i.e., volatilities and mean drifts of

price and cost) and a share ratio for a joint venture, but not by a fixed-price for an outsourcing

contract.

To illustrate Theorem 3 and to explain what the loss-by-acceptance regions implies to a

vendor, we provide numerical examples. In these numerical examples, we only change a share

ratio ( 95.0=α for case (i), 8013.0=α for case (ii)), but keep other parameters stable (see Table

50

3.1). We show the three cases in Theorem 3 by Figure 3.2, Figure 3.3 and Figure 3.4,

respectively. Figure 3.2 shows the optimal thresholds for each contract and a vendor’s value.

Dotted lines in the represent each contract threshold and a vendor would accept the client’s offer

on left of the dotted lines for each contract. In Figure 3.2 (b), a black region represents that an

outsourcing can return a higher value to a vendor, while a grey region shows that a joint venture

is higher. In this figure, we can find a particular strategic region, a joint venture loss-by-

acceptance region illustrated as a region inside red solid line. In this area, an outsourcing is more

beneficial to a vendor but only a joint venture is feasible or acceptable. Thus, if a vendor accepts

the joint venture contract because it is feasible, there exists a loss by not waiting another.

Therefore, a vendor should not accept a client’s offer if the client does not compensate for the

loss incurred as accepting a current feasible contract. Similarly, according to conditions given in

Theorem 3, a vendor has an outsourcing loss-by-acceptance region and a no loss-by-acceptance

region, which are shown in Figure 3.3 and Figure 3.4.

Figure 3.2. Feasibility and utility for outsourcing vs. joint venture: Case (i)

(a) Firm’s Values and Investment Timing (b) Feasibility and Utility in 2-D view

Loss-by-acceptance region

51

Figure 3.3. Feasibility and utility: Case (ii)

(a) Firm’s Values and Investment Timing (b) Feasibility and Utility in 2-D view

Figure 3.4. Feasibility and utility: Case (iii)

(a) Firm’s Values and Investment Timing (b) Feasibility and Utility in 2-D view

Without consideration of the option to wait, the NPV approach also can evaluate a contract in

such a way: exercising now or never. As aforementioned, the value of option to wait or the

managerial flexibility cannot be derived by the NPV approach. For example, the real options

method estimates an outsourcing contract’s value as 1

01

O

tCA β , while the NPV method’s value is

zero, when outsourcing does not reach the optimal threshold. The difference between the two

Loss-by-acceptance region

No Loss-by-acceptance region

52

evaluation methods is 1

01

O

tCA β , defined as the value of option. Thus, we need to examine whether

such an option value is significant to a vendor’s decision making. For this reason, we derive

feasible regions by NPV and compare the selection strategy as shown in Figure 3.5. Main

difference between NPV and ROT is that a strategy by NPV provides only one case (case (iii) in

Theorem 3) while ROT covers three different scenarios in Theorem 3. As shown in Figure 3.5,

NPV evaluates the same vendor’s value for unexercised contracts, which is zero (region shaded

by black dotted lines). This result shows that no consideration of a vendor’s embedded option

might be crucial and sometimes critical in strategic decision making.

Figure 3.5. Feasibility and utility by NPV for outsourcing vs. joint venture

(a) Firm’s Values and Investment Timing (b) Feasibility and Utility in 2-D view

3.6. Conclusion and Further Studies

This paper provides a model for evaluating a contract under price and cost uncertainty

fromvendor’s perspective. Even though many studies on different contracts are established, little

attention is paid to vendor’s perspective. A vendor can determine whether or not to accept a

client’s offer and implicitly when to accept the offer under uncertainty. Therefore, the option

No loss-by-acceptance region

53

value or managerial flexibility incurred by uncertainty is taken into account using real option

theory

First, we found that to the vendor, if a contract is evaluated by the NPV method, it will be

more feasible than the ROT method. Second, our results show that, while the feasibility of an

outsourcing contract can be increased by increasing a contract price offered by a client, feasibility

of a joint venture depends only on market conditions. The share ratio between both parties for a

joint venture contract affects only how much the contract makes profits. Finally, by comparing an

outsourcing and a joint venture contract, our real option framework provides two more strategic

scenarios, the outsourcing and the joint venture loss-by-acceptance regions, which cannot be

explained by the traditional NPV method. This implies that the NPV method fails to derive such

strategies and might lead to wrong contract selection.

This paper only focuses on price and cost uncertainty, but other factors that we have not

considered in this paper may influence the vendor’s decision. Simply, in the environment of a

global partnership, fluctuating exchange rates and the stability of the vendor’s country can be

other important factors. For example, Bergin et al. (2007)) reported that the outsourcing industries

in Mexico is roughly twice as volatile as the corresponding industries in U.S. These different

market conditions between a client and vendor should be considered. Moreover, as discussed in

Levina and Ross (2003)), most vendors focus on the reputation of clients. Vendors are concerned

about their long-term market position as well as their current contracting structure. In the long-

term, a contract problem can be viewed as a multi-stage problem, because a vendor can sign a

new contract with either a new client or a current client. At the next contract stage, a vendor’s

reputation might affect a client decision in a competitive market among multiple vendors.

Depending on the vendor’s reputation, a vendor can ask more favorable contract or less.

54

APPENDIX

Appendix 3.A. Derivations for outsourcing contract valuations

We represent the value of option to outsource as ),( CPF OO , simply denoting )(CF here.

Hence as shown in Dixit and Pindyck (1994) (Chapter 4), in the continuation region the Bellman

equation is

)()]([ dFEdtCF =ρ (A.1)

Using Ito’s Lemma, the Bellman equation becomes

0)()(')(21 ''22 =−+ CFCFCCFC C ρμσ (A.2)

In addition, )(CF should satisfy the following boundary conditions:

0)( =∞F (A.3)

C

D

t

D CeC

ePCF

μρρ

μρρ

−

−−

−=

−−− )(* 11)(

0

(A.4)

C

DCeCCF

μρ

μρ

−

−−=

∂

∂ −− )(* 1)( (A.5)

Condition (A.3) arises since the vendor cannot have any value when ∞=C . Conditions

(A.4) and (A.5) are smooth pasting and value matching conditions coming from optimality. The

general solution for equation (A.2) must take the form 21 21)( OO CACACF ββ += , where 21, AA

are constants to be determined, and 12)21(

21

22

222 >+−+−=CC

C

C

CO σ

ρσμ

σμβ ,

02)21(

21

22

221 <+−−−=CC

C

C

CO σ

ρσμ

σμ

β . To satisfy the condition (A.3), we must have 02 =A ,

so the general solution must have the form 11)( OCACF β= . The solution should satisfy the

conditions (A.4) and (A.5). Finally, we have:

55

⎪⎪⎩

⎪⎪⎨

⎧

>

≤−

−−

−

=

−−−

*1

*)(

0

1

0

00

0

,

,11

)(CCCA

CCe

Ce

PCF

tt

tC

D

t

D

Ot

O

C

β

μρρ

μρρ

where D

CD

OO

O

Ce

ePC

)(1

1*

1

1

1 μρ

ρ μρρβ

β−−

−

−

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

−= , 02

21

21

2

2

221 <+⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=

CC

C

C

CO σ

ρσμ

σμ

β ,

1

)()1(*

111

)( 1

OC

DC

Oe

CAβμρ

μρβ −

−

−=

−−− .

Appendix 3.B. Derivation of joint venture contract valuation

Let ),( CPFJ be the value of the option, simply denoting ),( CPF here. We find a differential

equation for it, as shown in Dixit and Pindyck (1994) (Chapter 6). We assume that both the risks

in a market price P and a cost C . Then, consider a portfolio consisting of one unit of the option,

m units short in the output, and n units short in capital.

Using Ito’s Lemma, we have

dCnFdPmFnCmPFd CP )()()( −+−=−−

dtCFPCvFPF CCCCPPCPPP )2(2/1 2222 σσσσ +++ .

Note that the dP and dC on the right-hand side are stochastic. However, we can choose

PFm = and CFn = to remove these terms and make the portfolio riskless. Then the holder of the

portfolio over the interval ),( dttt + will have a capital gain

dtCFPCvFPF CCCCPPCPPP )2(2/1 2222 σσσσ ++ ,

where PF

FPP ∂

∂=

2

, CF

PFPC ∂

∂∂∂

= and 2

2

CF

FCC ∂

∂= .

56

An investor must also make a payment, dtnCmP CP ))()(( μρμρ −+− , to hold the short position.

Equating the sum of these two components to the riskless return dtnCmPFr )( −− .

Collecting terms, we have

0)()()2(2/1 2222 =−+−++−+++ rFCFrPFrdtCFPCvFPF CCPPCCCCPPCPPP μρμρσσσσ.

),( CPF should satisfy the following boundary conditions over the region of the ),( CP :

P

D

P

D PP eC

ePCPF

μρα

μρα

μρμρ

−

−−

−

−=

−−−− )(

2

)( 1

)(

1),( ,

2

)(

)(

1),(

P

D

P

PeCPF

μρα

μρ

−

−=

−−

,

C

D

C

CeCPF

μρα

μρ

−

−−=

−− )(1),( .

Homogeneity property of CPR /≡ enables us to transform the above option value and

conditions:

)()/(),( RCfCPPfCPF == ,

where f is now the function to be determined.

Differentiation gives:

)('),( RfCPFP = , )(')(),( RRfRfCPFC −=

CRfCPFPP /)(''),( = , CRRfCPFPC /)(''),( −= , CRfRCPFCC /)(''),( 2= .

Substituting these differentiations to the original PDE and collecting terms, we have:

0)()()(')()()2(2/1 ''222 =−−+−−+++ RfRRfRfRv CPCCCPP μρμρμρσσσσ

And we have boundary conditions as follows:

57

C

D

P

D CP eeRRf

μρα

μρα

μρμρ

−

−−

−

−=

−−−− )(

2

)( 1

)(

1)(

2

)(

)(

1)('

P

DPeRf

μρα

μρ

−

−=

−−

, C

DCeRRfRf

μρα

μρ

−

−−=−

−− )(1)(')( .

Similar to one uncertainty case in Appendix A, we can find the solution.

Appendix 3.C. Proof of Theorem 1.

When a net present value method is applied ( 00≥Φ

t and 0

0≥Ψ

t), investment strategies for

an outsourcing and a joint venture will be:

DC

D

ONPVCe

ePC

)(*

1

1μρ

ρ μρρ −−

−

−

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −= and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−=

−−

−−C

D

DP

NPV

C

P

e

eR

μρμρ

μρ

μρ

)(

)(* 1

1

)( .

By comparing thresholds by a real option and NPV, we have

111

1*

*

<−

=O

O

NPV

ROT

CC

ββ and 11

2

2*

*

<−

=J

J

ROT

NPV

RR

ββ ,

because 01 <Oβ and 12 >Jβ . Q.E.D.

Appendix 3.D. Proof of Proposition 2

From differential equation (A.2), 01 <Oβ and 12 >Oβ should satisfy the following equation:

0)1(21 2 =−+− rμβββσ

Let =)( iQ β r−+− μβββσ )1(21 2

and take a derivative:

58

0=∂∂

+∂∂

∂∂

σσβ

βQQ i , for 2,1=i

where derivatives are evaluated at 1β and 2β .

And we know 0<∂∂βQ at 1β and 0>

∂∂βQ at 2β since Q is quadratic, 01 <β and 12 >β .

Moreover, 0)1( >−=∂∂

iiQ βσβσ

for 2,1=i . From these inequality conditions, we have

01 >∂∂σβ and 02 <

∂∂σβ .

Since 01 >∂∂σβ , then 0

*

<∂

∂

σ

C.

)()()1()2(2/1)( 22CJPCJJCCPPJJ vQ μρβμρμρββσσσσβ −−+−−+−++=

Similarly, since 0<∂∂

P

P

σβ , 0<

∂∂

C

P

σβ , we have 0

*

>∂∂

P

JRσ

, 0*

>∂∂

C

JRσ

. Q.E.D.

Appendix 3.E. Proof of Theorem 3.

Let us discuss case iii) first, and then verify the condition. The case iii) happens when

*C and *JR intersect at a value indifference curve. The value indifference curve holds following

condition:

01

)1()(

11

1

)(

111

)()(

)()()(

=−

−−−

−

−−

−=

−

−+

−

−−

−

−−

−

−−−−−

−−−−−−−

C

D

P

DD

O

C

D

P

D

C

DD

O

CP

CPC

eC

eP

eP

eC

eP

eC

eP

μρα

μρα

ρ

μρα

μρα

μρρμρμρρ

μρμρμρρ

Also, since D

CD

OO

O

Ce

ePC

)(1

1*

1

1

1 μρ

ρ μρρβ

β−−

−

−

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

−= and

59

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

−−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−−

==−−

−−−

−− DP

D

J

J

O

OO

C

D

DP

J

JJ

P

C

P e

ePe

eCPR

)(2

2

1

1)(

)(2

2***

1

)(1

11

1

1

)(1

/μρ

ρμρ

μρ

μρρβ

ββ

βμρ

μρββ ,

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

−−=

−−

−

DP

D

J

J

O

OO

Pe

ePP

)(2

2

1

1*

1

)(1

11 μρ

ρ μρρβ

βββ .

Substituting *C and *P into value indifference condition, we have

12 1 OJ αββ −= .

Also, case i) and case ii) can be determined depending on the intersection point and a value

indifference curve between two contracts. Q.E.D.

60

Chapter 4. Investment Timing for a Dual Channel Supply Chain

A crucial question in a supply chain is whether to construct an internet channel in addition to a traditional retail channel. Despite many studies on this issue, little attention has been paid to optimal investment timing for developing the internet channel. To address this gap in the literature, this paper proposes a model to help a firm determine the timing by considering a firm’s option value of waiting. Our model also incorporates customer’s preference between the retail and the internet channel to derive total profits of the vertically integrated channels. Based on the proposed model, we study the impacts of cost uncertainty, sunk cost, cost efficiency and customer’s behavior on an investment timing strategy. In addition, the comparison study of a strategy by our proposed model and that by the widely used net present value method implies that the traditional method underestimates an investment of a dual channel and leads to suboptimal decisions of a firm.

4.1. Introduction

The rapid development of the internet and e-commerce has led to a low-cost and ubiquitous

tool that can be used to facilitate business transactions. Many benefits of the internet channel,

such as cost efficiency, direct control of distribution, and closer contact with customers, have

attracted firms seeking to increase profits (Tsay and Agrawal (2004)). Many firms already use the

internet to sell products directly in addition to selling via a traditional retail channel, and still

other firms are seriously evaluating such investments. For example, Proctor & Gamble (P&G) is

considering constructing an online selling channel and testing (TheSeattleTimes (2010)). Also,

some wireless operator companies hesitated to sell online because they had recognized that even

leading US firms sell only 3 percent of its products online and are worried about channel conflicts

(Sarkar et al. (2006)). These firms face the need to design a dual channel strategy, managing both

the internet channel and a traditional retail store.

61

The decision to introduce an internet channel and operate a dual channel supply chain might

be inefficient for two reasons. First, since selling prices and corresponding costs on each channel

are different, total profits from integrated channels can be reduced. Second, there may also be

losses due to a channel conflict when the internet channel is developed (Ingene and Parry (1995),

Tsay and Agrawal (2004), Rangaswamy and Van Bruggen (2005), Mukhopadhyay et al. (2008)).

For example, in the late 1990s, Levi Strauss and Co. (Levi’s) decided to construct an internet

channel, but, in 1999 it had to cease internet selling through its website because the costs of the

internet channel were not affordable. Recently, the company reopened the internet channel. As

this example shows, evaluating a dual channel precisely and developing the internet channel at

the right time are very important to manager’s decisions.

When evaluating a dual channel investment, a firm needs to consider two main factors which

are usually overlooked in the literature. The first factor is that the investment is optional because

the firm is currently selling products through the retail channel. Because costs for each channel

are dynamically changing with time and sunk costs are incurred with the investment of

developing the internet channel (Borenstein and Saloner (2001)), the firm needs to determine

whether or not to develop the internet channel, and further when to develop (option to wait). The

case of Levi’s, which restarted the internet channel, emphasizes the importance of optimal

investment timing. Levi’s should have waited rather than invested in the late 90’s. The second

factor that a firm needs to consider is customer’s behavior under a dual channel. Depending on

the profits, a firm will determine a channel strategy. Especially for a dual channel strategy, we

need to notice that the customer’s behavior is an important factor to evaluate the profits.

Customers can purchase products from either a retail or an internet channel, depending on their

preference. From the firm’s perspective, total profits from a newly constructed dual channel can

be changed depending on the customer’s behavior and the different operating costs for each

channel due to channel conflict as discussed in Chiang et al. (2003) and Tsay and Agrawal

62

(2004). Also, different customer behavior will result in a different dual channel evaluation and

will change the firms’ investment timing strategies.

For these reasons, we have developed a dual channel valuation model using a real option

theory to include a firm’s option to wait under cost uncertainties and incorporating customer

behavior. In Section 2, literature related to dual channel and real option theory as an appropriate

way to value uncertain investment opportunities is discussed. Section 3 develops a real option

valuation model and derives an optimal threshold to launch the internet channel. Section 4

analyzes the impact of important factors on the investment and also explores the main difference

in a strategic decision using a traditional Net Present Value (NPV) method instead of a real option

theory (ROT). Section 5 describes extensions. Section 6 summarizes this research and suggests

further studies.

4.2. Literature Review

A significant body of literature discusses challenges of dual channel organizations (see Tsay

and Agrawal (2004) and Cattani et al. (2006) for surveys). Among those studies, a customer’s

preferences between a retail and an internet channel have been recently highlighted. Empirically,

Liang and Huang (1998) found that some customers prefer to buy some products like shoes and

toothpaste online. Furthermore, Chiang, et al. (2003) suggested a measure corresponding to a

customer preference between a retail and an internet channel. For six kinds of products, they

provided customers preference index based on experiments and Chiang, et al. (2003) studied the

effect of the customer behavior on a supply chain and provided strategic analysis under

competition. Cattani, et al. (2006) analyzed studies about the change of strategies according to an

introduction of a direct channel, especially focusing on a wholesale price. Similarly, Druehl and

Porteus (2006) considered a customer behavior on preference between the internet direct channel

63

and the traditional retail channel in a game theoretic setting. In addition to the above studies,

research about a customer behavior in E-business is reviewed in the handbook by Simchi-Levi et

al. (2004).

Despite the extensive body of literature considering retail and internet channels, most

literature overlooked the fact that costs for each channel are dynamically changing and uncertain

and that a firm has an option to wait until the internet channel addition becomes profitable. Like

the Levi’s case, the investment may not currently be optimal but may be profitable in the future.

Therefore, evaluation needs to include the firm’s option value. Despite these characteristics, most

firms have used the standard net-present-value (NPV) in order to evaluate dual channel additions

(Geyskens et al. (2002)). According to this approach, if the NPV is positive, the project is

worthwhile and should be pursued; if it is negative, the project should be turned down. Many

recent researchers have criticized that NPV decision-making can miss the additional value of

managerial flexibility, which results from uncertainty. Firms may have the option to exercise the

investment opportunity or to hold on to it for some length of time (Brennan and Schwartz (1985),

Dixit (1989), McGrath (1997)). Dixit and Pindyck (1994) apply the real options theory to discuss

optimal investment timing in the framework of investment irreversibility and uncertainty, and

also point out the parallels between an investment opportunity and an “American call option”. A

call option gives an investor the right to acquire an asset of uncertain future value. If conditions

favorable to investment arise, the investor can exercise the investment.

Many real option theoretical models have been developed in the R&D project area including

uncertainty (Dixit and Pindyck (1994), Grenadier and Weiss (1997), Schwartz and Zozaya-

Gorostiza (2003)). In supply chain literature, the application of a real option theory is also

growing. However, most literature applied a real option theory from the perspective of flexible

contracts or operational hedging strategies. For example, Li and Kouvelis (1999) study flexible

and risk sharing contracts under price uncertainty. They show how the flexible contracts can

64

reduce sourcing costs. Similarly, Van Mieghem (1999) values the option of subcontracting to

improve financial performance and system coordination. Kamrad and Siddique (2004) consider a

supplier’s reaction option in supply chain contracts. Nembhard et al. (2005) apply real options

theory to develop a supply chain model with exchange rate uncertainty by considering the time

lag of switching strategies. In addition to the aforementioned literature, one could refer to other

studies such as Kogut (1991), Alvarez and Stenbacka (2007), Van Mieghem (2007). However,

until now, little attention has been paid to supply chain integration, specifically the timing

decision to develop the internet channel.

In summary, our paper differs from the existing dual channel literature in that we consider an

investment timing issue, in other words, a firm’s option to wait which is caused by cost

uncertainty. This consideration helps a firm determine when to develop an internet channel.

Furthermore, we address the issue of customer choice behavior via customer acceptance index, as

studied in literature like Chiang, et al. (2003) and Bernstein et al. (2008). By using the real option

method, we derive the optimal timing threshold to develop the internet channel and investigate

the impact of uncertainty. Also, we examine the effect of customer’s behavior on the investment

policy. Our results show that when uncertainty is high, it is better to defer the launching of the

internet channel. Also, the results indicate that the higher sunk cost a firm has, the later the client

is recommended to introduce the internet channel. When the difference between a retail channel’s

and an internet channel’s operating cost increases, optimal investment thresholds will increase.

Also, a higher customer acceptance index decreases the thresholds. In addition, the comparison

study of strategies by our proposed model and by the NPV implies that the NPV underestimates

an investment of a dual channel and leads to suboptimal decisions of a firm.

65

4.3. Model

This section addresses demand functions incorporating customer behavior for a single

channel and dual channels. In the case of dual channels, we explain how the demand functions are

determined by a customer’s behavior between a retail channel and an internet channel. Based on

demand and cost functions, Section 3.1 derives the corresponding optimal price policies. Section

3.2 provides the model for an optimal timing decision, i.e., when the firm should develop the

internet channel. Throughout the paper, we follow basic notations in Chiang, et al. (2003). The

notations used in this paper are defined as follows:

v : Customer’s value (or wiliness to pay) for a product in a retail channel, which is assumed

uniformly distributed from 0 to 1.

)(tptr , )(tQtr : Price and demand of a retail channel when a firm operates a single traditional

retail channel.

( )rp t , ( )dp t : Price of a retail channel and an internet channel, when a firm operates dual

channels.

( )rQ t , ( )dQ t : Demand of a retail channel and an internet channel, when a firm operates dual

channels.

θ : Customer acceptance index.

( )rC t , ( )dC t : Cost by the manufacturer selling via a retail channel and an internet channel.

Demand for a Single Traditional Retail Channel

We first derive the demand function when there is a single traditional retail channel. As

shown Figure 4.1 (Chiang, et al. (2003)), customers with a value of v which is greater than retail

price ( ( )trp t ) will buy the product. Therefore, the demand can be represented as the shaded part

in Figure 4.1 and expressed as ( ) 1 ( )tr trQ t p t= − for 0 ( ) 1trp t≤ ≤ .

66

Figure 4.1. Demand from distribution of customer value (Chiang et al., 2003)

Demands for Dual Channels

Next, we provide demand functions where there are both retail channel and internet channel.

We introduce a customer acceptance index θ to reflect losses or benefits from the internet

channel purchase (Chiang, et al. (2003)). The customer index θ represents the relationship

between purchasing from the internet channel and retail channel. While a customer has a value v

for a product in a traditional retail store, he or she has a value of vθ for the same product bought

online. This customer index incorporates factors beyond price.

The customer may have a lower value for the same product purchasing online, that is,

10 <<θ . This happens since sensory evaluation such as taste, touch, and smell are limited and

immediate possession is not possible (Yan and Ghose (2009)). In the real world, when a customer

wants to purchase clothes from the Internet channel, there is the risk that the clothing might not

fit. When a customer wants to buy a TV, the video and sound qualities may be suspected. For

these reasons, even though the product from a retailer and an e-tailer is the same, the value of the

product from the latter may be lower to the customer. Therefore, it is necessary to introduce an

adjustment parameter in order to discount the customer’s value for the product from the internet

channel ( 10 <<θ ). For instance, if a book from a retail channel is worth 1 to a customer, then the

value of a book from an internet will be 0.904 on the average. This adjustment is reflected in the

customer acceptance index as shown in Table 4.1.

67

Table 4.1. Customers acceptance index (θ ) for the internet channel (Chiang, et al. (2003))

Category Book Toothpaste DVD player Flowers Food Shoes

Acceptance 0.904 0.886 0.787 0.792 0.784 0.769

Moreover, the value for the goods from an internet may be more beneficial than from the

retailer, therefore the customer’s value for the internet channel may be appreciated using the

higher customer acceptance index ( 1≥θ ). Purchasing from the internet becomes more intuitive

and convenient nowadays due to technological development and more tech-savvy consumers.

Many customers may want to download movie or music files instead of visiting a retail video or

music shop. Also, customers may prefer quoting insurance in the internet and purchasing, because

it is easier to compare to other companies’ quote and they do not need to spend time visiting local

retailers. Netflix, Amazon, Itunes, or GEICO can be good examples.

We now discuss how to derive the demand functions. When the retail price and the internet

price are ( )rp t and ( )dp t , customers will choose a traditional retail channel or an Internet

channel after comparing the utilities from each channel. In other words, if the customer buys the

product from the retailer, the utility would be ( ) ( )rv t p t− , while the utility for an internet

purchase would be ( ) ( )dv t p tθ − . Between these two options, the customer will choose one

channel with higher utility. Based on the comparison of utility functions as in Table 4.2, demand

functions can be derived as follows: (See Appendix 4.A for the detailed derivation).

Table 4.2 Inequality conditions to derive demand functions

Inequality conditions to derive demand Customer Behavior

(Utility from Retailer) ≥ (Utility from internet) and (Utility from Retailer)≥0 Buy from Retailer

(Utility from Retailer) ≤ (Utility from internet) and (Utility from internet)≥0 Buy from internet

(Utility from Retailer)≤0 and (Utility from internet)≤0 Will not buy

68

i) 10 << θ

( ) ( ) ( )1 ( )( ) 11 ( )

r d dr

r

r

p t p t p tif p tQ tp t otherwiseθ θ

⎧ −− ≤⎪= −⎨

⎪ −⎩

(4.1)

⎪⎩

⎪⎨⎧

≤−−

=otherwise

tptp

iftptp

tQ rddr

d

0

)()(

)1()()(

)( θθθθ

(4.2)

for 0 ( ) 1rp t≤ ≤ , 0 ( )dp t θ≤ ≤ .

ii) 1>θ ,

⎪⎪⎩

⎪⎪⎨

⎧

−−

≤=

otherwisetptp

tptpiftQ

dr

rd

r

θθ

θ

1)()(

)()(0)( (4.3)

( ) ( )1 ( )( )

( ) ( )11

d dr

dr d

p t p tif p tQ t

p t p t otherwise

θ θ

θ

⎧− ≤⎪⎪= ⎨

−⎪ −⎪ −⎩

(4.4)

for 0 ( ) 1rp t≤ ≤ , 0 ( )dp t θ≤ ≤ .

From Eqs. (4.1)-(4.4), we can notice that a firm may have different demand structures

depending on the value of θ and the relationship of ( )rp t and ( )dp t . For example, a firm can

have demand from both retail channel and internet channel when rd pp θ≤ and 10 <<θ , or

when 1≥θ and rd pp θ> . However, when rd pp θ> and 10 <<θ , or when rd pp θ≤ and 1≥θ ,

demand will be incurred only through either a single retail channel or a single internet channel.

Cost Uncertainty

If a firm makes a product, it will incur marginal costs ( )rC t and ( )dC t when it is sold through

a retail channel and an internet channel, respectively. By introducing a cost efficiency measure λ

69

between the two marginal costs, we have ( ) ( )d rC t C tλ= . Usually, the costs for the internet

channel are expected to be less than the costs for the retail channel, which implies 1<λ . A lower

value of λ implies that the internet channel is more efficient than a retail channel from the cost

perspective. To express the dynamically changing costs, rC evolves over time as Geometric

Brownian Motions (GBM). This is a standard setting in real option theory and also a good

approximation for uncertainties (Dixit (1989), Dixit and Pindyck (1994), Abel and Eberly

(1994)). Specifically,

( ) ( ) ( )r r rdC t C t dt C t dzμ σ= + ,

where dz = a standard Wiener process for a cost, μ = the mean drift for a cost, σ =the

uncertainty rate (volatility) of such a process for a cost. Here, the mean-drift and the volatility

imply how much a cost increases or decreases and how much a cost fluctuate, respectively.

Moreover, we suppose an upper reflecting barrier on the cost process at ( )rC t C= . This could

be a ceiling of cost imposed by the competitive market (e.g., see Dixit (1993)).

4.3.1. Firm’s Price Decisions and Necessary Conditions for Dual Channels

This section provides price policies for a single traditional retail channel ( rd pp θ> ,

10 <<θ ), for dual channels ( rd pp θ≤ , 10 <<θ or rd pp θ> , 1≥θ ), and for a single internet

channel ( rd pp θ≤ , 1≥θ ). First, for the single traditional retail channel, a firm wants to

determine a retail price )(tptr at each time t maximizing profits. Based on the demand function

described, we have the following profit-maximizing problem: for 0 ( ) 1trp t≤ ≤ ,

( ; ) max( ( ) ( )) ( ) max( ( ) ( ))(1 ( ))tr tr

tr tr tr r tr tr r trp pp t p t C t Q t p t C t p tΦ = − = − − .

70

In contrast, if a firm decides to launch the internet channel in addition to the existing retail

channel, the firm would make profits from both channels: the vertical integration of a dual

channel makes total profits );,(1 tpp drviΦ for 10 <<θ or );,(2 tpp drviΦ for 1≥θ in the form

of the sum of profits from a retail channel and the internet channel. From the demand functions in

Eqs. (4.1)-(4.4), we can derive a similar maximization problem as the traditional channel case.

Therefore, profits at time t are written as follows: for 0 ( ) 1,0 ( ) ,r dp t p t θ≤ ≤ ≤ ≤

i) 10 <<θ

1 ,

,

( , ; ) max( ( ) ( )) ( ) ( ( ) ( )) ( )

( ) ( ) ( ) ( )max ( ( ) ( )) 1 ( ( ) ( )) .1 (1 )

r d

r dd r

vi r d r r r d d dp p

r d r dr r d dp p

p p

p p t p t C t Q t p t C t Q t

p t p t p t p tp t C t p t C tθ

θθ θ θ

<

Φ = − + −

⎛ ⎞ ⎛ ⎞− −= − − + −⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

ii) 1θ >

2 ,

( ) ( ) ( ) ( )( , ; ) max ( ( ) ( )) ( ( ) ( )) 11 1r d

d r

r d r dvi r d r r d dp p

p p

p t p t p t p tp p t p t C t p t C tθ

θθ θ

>

⎛ ⎞ ⎛ ⎞− −Φ = − + − −⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

.

Moreover, the firm can decide to operate on a direct internet channel only. From the top of

Eqs (4.3) and (4.4), total profits for the internet channel );( tpdodoΦ can be derived, where dop is

the internet price in a single internet channel. The total profit can be written as follows: for

0 ( )dop t θ≤ ≤ ,

( ; ) max( ( ) ( ))(1 ( ))do

dodo do do d

p

pp t p t C t tθ

Φ = − − .

In Proposition 1, we check the optimality condition for each problem and then derive optimal

price decisions.

Proposition 1. The value functions for the single traditional retail channel, the vertically

integrated dual channel and the single direct internet channel, );( tptrtrΦ , );,(1 tpp drviΦ ,

71

);,(2 tpp drviΦ and );( tpdodoΦ , are concave functions with respect to trp , rp , dp and dop , and

the following optimal solutions are global optima. See Appendix 4.B for the detailed derivation.

⎩⎨⎧ ≤+

=otherwise

tCiftCtp rrtr 1

1)(2/))(1()(*

⎩⎨⎧ ≤+

=otherwise

tCiftCtp rrr 1

1)(2/))(1()(* , * ( ( )) / 2 ( ) /( ) r r

dC t if C t

p totherwise

θ λ θ λθ

⎧ + ≤= ⎨⎩

* ( ( )) / 2 ( ) /( ) r r

doC t if C t

p totherwise

θ λ θ λθ

⎧ + ≤= ⎨⎩

When a firm manages both selling channels, the firm will set a retail price and an internet

price to *rp and *

dp (We will omit time indices for simplicity). Also, for a single retail or a single

internet channel, the firm will determine a price to *trp or *

dop respectively. Here, we know that,

depending on profits, a firm will determine a channel strategy. Moreover, we need to notice that

the customer’s behavior is an important factor to evaluate the profits (Cattani, et al. (2006) and

Noble et al. (2005)). The reason is that, in a dual channel environment, customers can deviate

from one channel to the other channel and this incurs the competition between two channels.

Depending on how much customers prefer the internet channel, demands and profits from the

dual channel will be changed and the dual channel may fail to make more profits than a single

channel. Thus, we need to note that a channel strategy can be affected depending on some

exogenous factors such as the customer acceptance index level or the cost efficiency level. We

provide the results in Proposition 2.

Proposition 2. (See Appendix 4.C for proof)

Case I. If 0 1θ λ< < < , a firm should consider the traditional retail channel only.

Case II. If 0 1λ θ< ≤ < , a firm should consider dual channels.

Case III. If 0 1λ θ< ≤ < , a firm should consider the internet channel only.

72

Proposition 2 implies that when a cost efficiency level 1λ < , a firm would consider developing

the internet channel when λ is lower than customer acceptance indexθ .

4.3.2. Firm’s Investment and Timing Decision

In the previous section, we have derived price policies and discussed three different cases

depending on the exogenous factors. Now, we will discuss optimal investment timing policies.

First, we focus on case II in Proposition 2 for dual channels ( 0 1λ θ< ≤ < ) and then consider

case III for internet channel only ( 0 1λ θ< ≤ < ).

We develop a real option model to derive the optimal investment timing decisions. In

practice, a firm has an option to delay investment. In other words, the firm needs to determine

when it should develop the internet channel. The firm is currently selling products via a retail

channel and incurs cash flows )1)(( *,

*trrtr pCp −− τ at an instant time until an optimal time T to

develop an internet channel. After developing the internet channel at time T , the firm would

make profit );( TCrviπ by incurring an irreversible sunk cost K . Then, as in an American option,

a firm chooses the optimal investment time. Therefore, a value function for optimal developing

timing is derived as follows:

* *

0( ) max ( )(1 ) ( ; )

T Tr tr r tr vi rT

V C E e p C p d e C Tρτ ρτ π− −⎡ ⎤= − − +⎢ ⎥⎣ ⎦∫ ,

where

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−+⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−−= ∫∞ −− KdtppCpppCpeETC

Tdr

dddr

rrTt

rvi )1()(

11)();(

****

***)(

θθθ

θπ ρ ,

where ρ is a discount factor. We assume 22 σμρ +> for convergence.

73

Also, in Case III of Proposition 2, a firm might want to operate a single internet channel.

Similar to the derivation for a dual channel case, we can develop the value function to obtain

optimal investment time. We have the following optimization problem.

⎥⎦⎤

⎢⎣⎡ +−−= ∫ −−T

rdoT

trrtrTrdo TCedpCpeECV0

** );()1)((max)( πτ ρρτ ,

where *

( ) *( ; ) ( ) 1t T dodo r do rT

pC T E e p C dt Kρπ λθ

∞ − −⎡ ⎤⎛ ⎞= − − −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫

To simplify the analysis, we assume { }min 1, /C θ λ≤ 1. Then, the optimal prices are as follows:

* ( ) (1 ( )) / 2tr rp t C t= +

*( ) (1 ( )) / 2r rp t C t= + , * ( ) ( ( )) / 2d rp t C tθ λ= +

* ( ) ( ( )) / 2do rp t C tθ λ= + .

Therefore,

2 2 2( )

2 2 2

2 2

(1 ) ( )( ; )4 4 (1 )

( ) ( ) ( ) ( )1 1 24 2 (1 ) 2

t T r rvi r T

r r r

C CC T E e dt K

C T C T C T K

ρ λ θπθ θ

λ θρ ρ μ ρ μ σ θ θ ρ μ σ

∞ − −⎡ ⎤⎧ ⎫− −= + −⎨ ⎬⎢ ⎥−⎩ ⎭⎣ ⎦

⎛ ⎞−= − + + −⎜ ⎟− − − − − −⎝ ⎠

∫

2( )

2 2

2

( )( ; )4

( ) ( )1 24 ( 2 )

t T rdo r T

r r

CC T E e dt K

C T C T K

ρ θ λπθ

λ λθρ ρ μ θ ρ μ σ

∞ − −⎡ ⎤−= −⎢ ⎥

⎣ ⎦⎛ ⎞

= − + −⎜ ⎟− − −⎝ ⎠

∫

1 If 1)( >tCr ( λθ /)( >tCr ), from the proposition 1, the optimal price 1)()( ** == tptp rtr

( 1)()( ** == tptp dod ), a firm has zero demands and zero revenues from the retail channel (the internet channel). In this paper, we are interested in channel strategy when both channels are viable, therefore we focus on the case where }/,1min{ λθ≤C .

74

By solving these problems, we can see the optimal timing thresholds to develop the internet and

the firm’s values in Proposition 3.

Proposition 3. Suppose }/,1min{ λθ=C . The firm’s value for the dual channel with managerial

flexibility is as follows: (See the proof in the Appendix 4.D)

For Case II ( 10 <≤< θλ ),

⎪⎪

⎩

⎪⎪

⎨

⎧

≤≤−−−−

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+

−−+

<⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+

−−+

=

rrrrrr

r

rrrr

r

r

CCCifKCCCCa

CCifCCCa

CV*

2

22

2

2

2

*2

2

1

2)1(4)(

221

41)(

221

41)(

)(1

1

σμρθθθλ

σμρμρρ

σμρμρρ

β

β

(4.5)

where KCr 2

2

1

1*

)(

)2)(1(4

2 θλ

σμρθθ

ββ

−

−−−⎟⎟⎠

⎞⎜⎜⎝

⎛

−= , 22)

21(

21

22

221 >+−+−=σρ

σμ

σμβ

11

*2

22

21 2)1(4)( β

β

σμρθθθλ −

−

−−−−

−+= r

r KCC

aa ,

2

2

211

12 22

)1(4)(

2212)(

41

1

σμρθθθλ

σμρμρβ β

−−−

−−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−+

−−= −− rr

rCC

Ca .

For Case III ( θλ <≤< 10 ),

⎪⎪

⎩

⎪⎪

⎨

⎧

≤≤−⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+

−−+

<⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+

−−+

=

rrdorrr

rdo

dorrrr

r

rdo

CCCifKCCCa

CCifCCCaCV

*,2

22

,2

*,2

21

)2(2

41)(

221

41)(

)(1

1

σμρθλ

μρλ

ρθ

σμρμρρ

β

β

,

(4.6)

where

75

)2(

))(2(

)1(4)2(

))(2()1)(1()1)(1(

2

21

112

21

211

*,

σμρθ

λθβ

ρθβρβ

σμρθ

λθβμρ

λβμρ

λβ

−−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−−

−−−⎟

⎟⎠

⎞⎜⎜⎝

⎛

−−−

−−

−−

=

K

C dor ,

dodo

dddd

dodo CKCCCaa ,1

,1

)()2(

)1)2(()(

)1(214)( *

22

2*2**

,2,1β

β

σμρλθλ

μρλλ

ρθ −

−

−⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−−+

−−

−−

+= ,

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+

−−

=−−

2

211

,2 22

2

)( ,1

,1

σμρλ

μρ

β βdd

doCC

ado

do ,

22)21(

21

22

22,1 >+−+−=σρ

σμ

σμβ do

.

For Case II, the top of Eq. (4.5) can be interpreted as follows: when a firm’s current

operating cost ( rC ) is lower than a certain threshold ( *rC ), the firm will keep the single channel

by having the traditional net present value ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−+

−− 2

2

221

41

σμρμρρrr CC

and holding the

value of the option to add 1)(1β

rCa , which reflects the value of the firm’s managerial flexibility.

The bottom of Eq. (4.5) is just the typical NPV after the introduction of the internet channel. It

can be interpreted as: when the firm’s operating cost is high enough (higher than the threshold

*rC ), the firm’s option to develop the internet channel becomes exercisable, so the firm will

manage both the internet and retail channels by paying the sunk cost K . Finally, the firm can

develop the internet channel optimally at the following time,

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

−−−⎟⎟⎠

⎞⎜⎜⎝

⎛

−≥≥= KtCtT r 2

2

1

1

)(

)2)(1(4

2)(:0inf

θλ

σμρθθ

ββ

.

76

For Case III, the firm’s value function eq (4.6) can be interpreted in the same way. If

operating cost is higher than the threshold *,dorC , then the firm needs to develop internet channel

only at the following time

2 21 1 1 1 1

2

21

2

( 1)(1 ) ( 1)(1 ) ( 2)( ) 4 ( 1)( 2 )

inf 0 : ( )( 2)( )

( 2 )

r

K

T t C t

β λ β λ β θ λ ρβ β θρ μ ρ μ θ ρ μ σ ρ

β θ λθ ρ μ σ

⎡ ⎤⎛ ⎞ ⎛ ⎞− − − − − − − −⎢ ⎥+ +⎜ ⎟ ⎜ ⎟⎢ ⎥− − − −⎝ ⎠ ⎝ ⎠= ≥ ≥⎢ ⎥− −⎢ ⎥

⎢ ⎥− −⎢ ⎥⎣ ⎦

.

4.4. Analysis of Investment Timing and Customer Behavior

This section investigates the impacts of crucial factors on investment decisions and compares

the results of our real option approach with those by a traditional net present value method.

4.4.1. Analysis and Numerical Examples

We study the impacts of volatility of the operating cost, the investment sunk cost, the cost

efficiency level, and customer’s behavior (acceptance index). We summarize the analytical

solutions in the Proposition 4.

Proposition 4. If 0 1λ θ< ≤ < , a firm's optimal investment threshold *rC is an increasing

function of sunk cost K and cost efficiency level λ , and a decreasing function of the customer

acceptance index θ . If 0 1λ θ< ≤ < , a firm's optimal investment threshold *,r doC is an

increasing function of sunk cost K. (See proofs in the Appendix 4.E.)

When it is difficult to obtain analytical solutions, we use numerical methods for their

solutions. We have similar results for the impact of cost efficiency level λ and customer

behavior θ on the threshold of internet channel only case ( *,dorC ) . As shown in Figure 4.2(a) and

77

(b), a firm's optimal investment threshold *,r doC is an increasing function of cost efficiency level

λ and a decreasing function of the customer acceptance index θ . We have also studied the

impact of volatility on both thresholds. Figure 4.2 (c) and (d) show that the thresholds increase

when the volatility increases. This implies that a firm needs to introduce the internet channel at a

later time with higher operating cost when the cost is more volatile.

0.5 0.55 0.6 0.65 0.7 0.750.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

Cost Efficiency (λ)

Thr

esho

ld (

Cr,

do*

)

The Effect of Cost Efficiency

1.05 1.06 1.07 1.08 1.09 1.10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Customer Acceptance Index (θ)

Thr

esho

ld (

Cr,

do*

)

The Effect of Customer Behavior

0.05 0.1 0.15 0.20.55

0.6

0.65

0.7

0.75

0.8

Volatility (σ)

Thr

esho

ld (

Cr* )

The Effect of Volatility

0.05 0.1 0.15 0.20.2

0.22

0.24

0.26

0.28

0.3

Volatility (σ)

Thr

esho

ld (

Cr,

do*

)

The Effect of Volatility

Figure 4.2. The impact of (a) cost efficiency (Case III, upper left), (b) customer behavior (Case III, upper right), (c) volatility (Case II, lower left), (d) volatility (Case III, lower right) on optimal investment thresholds.

78

4.4.2. Comparison with the Traditional Net Present Value Method

The traditional NPV method assumes that the development of an internet channel is a one-

time decision, i.e., whether a firm should develop the internet channel now (by choosing a dual

channel strategy or an internet channel only strategy) or never (by choosing a single retail channel

strategy). However, the ROT model we present in this paper considers a more realistic situation

where the firm has managerial flexibility to wait and develop the internet channel later.

In this section, we examine what happens if we ignore managerial flexibility by comparing

the strategies by NPV and ROT, and also provide implications with reference to the case of

Levi’s.

Proposition 5. The feasible region to develop dual channels (the internet channel only) derived

by NPV, feaNPVC ( fea

doNPVC , ), includes the feasible region by the real option theory, feaROTC ( fea

doROTC , ).

**NPVROT CC > , }{ *

ROTfea

ROT CCCC ≥= ⊂ }{ *NPV

feaNPV CCCC ≥=

*,

*, doNPVdoROT CC > , }{ *

,, doROTfea

doROT CCCC ≥= ⊂ }{ *,, doNPV

feadoNPV CCCC ≥=

(See proofs in Appendix 4.F.)

As implied from Proposition 5, it may be possible that adding the internet channel is

appropriate from the NPV perspective but inappropriate from the ROT perspective. While the

NPV approach suggests dual channel investment, the ROT approach recommends a firm waits

and invests later. Specifically, this occurs between the NPV threshold and the ROT threshold as

shown in Figure 4.3. Interestingly, this result helps to explain the case of Levi’s which started the

internet channel, stopped soon after, and eventually restarted. When the company decided to

develop the internet channel, the NPV might have shown that the investment was profitable, but

the real option model might have suggested waiting.

79

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Operating Cost

Firm

’s V

alue

Real Option Vs. NPV

Real OptionNPV

Feasible by NPV

Feasible by ROT

Figure 4.3. Thresholds by real option (ROT) vs. NPV method

Now we discuss how much the firm’s value for dual channels changes if we do not consider

the firm’s option to wait. Table 4.3 shows the percentage of the option value that is not captured

by the NPV approach. Experiments are conducted for different uncertainty rates (σ ) as well as

for a customer acceptance index (θ ). The parameters for the volatility range from 0.01 to 0.1,

because, as shown in Loayza et al. (2007) and Jacks et al. (2009), commodity price volatility in

practice varies from 3.2% for industrialized economies to 12.2% for south Asia. Also, the values

for the customer acceptance index range from 0.6 to 0.95 based on information presented in Table

4.1. Since there are a limited number of empirical studies on a dual channel problem, we

conducted extensive experiments for these parameters. More precise experimental results might

be obtained by testing more empirical data in the future. Un-captured values that are lost by

adopting the NPV approach range from 6.32 to 148.02 percent, with more than half of them range

between 10 and 40 percent. This result implies that a firm might be able to underestimate a dual

channel investment and perhaps make suboptimal decisions.

Table 4.3. Percentage of a firm’s value uncaptured by NPV

θ σ 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

0.04 19.801 19.335 16.749 14.035 11.548 9.4337 7.7065 6.3218

80

0.06 32.324 31.518 27.302 22.878 18.824 15.377 12.562 10.305 0.08 55.035 53.706 46.435 38.912 32.016 26.154 21.366 17.527 0.10 91.374 89.238 77.05 64.565 53.124 43.397 35.452 29.082 0.12 148.06 144.6 124.81 104.58 86.052 70.296 57.426 47.108

4.5. Conclusions and Further Studies

Selling products via an Internet channel is increasingly prevalent and it has become an

important source of firm profits. Therefore, firms which do not have the Internet channel yet but

are interested in constructing it are seriously evaluating the investment. For these firms, when to

introduce an Internet channel in addition to a retail channel is very decisive, as shown in Levi’s

case. However, many studies have overlooked this issue. In this paper, we have developed and

demonstrated a model to help a firm to determine optimal development timing by considering a

firm’s option value of waiting. Based on the proposed model, we studied the impacts of important

factors on an investment timing strategy such as volatility, sunk cost, cost efficiency level and

customer acceptance index. When the volatility and the sunk cost are high, it is better to defer the

launching of the Internet channel. Also, when the difference between a retail channel’s and the

Internet channel’s operating cost decreases, an optimal timing threshold increases, which means

less cost saving from the internet channel delay launching. A higher customer acceptance index

decreases the thresholds. In addition, the comparison study of strategies by our proposed model

and by the NPV implies that the NPV underestimates an investment of a dual channel and leads

to suboptimal decisions of a firm.

This paper focuses on cost uncertainty, but other factors that we have not considered here in

this paper may influence the firm’s decision. In the environment of global transactions via the

internet channel, adding another internet channel in other countries or selling to other countries

81

can increase total demand. Thus, consideration of total demand uncertainty in the model would

lead to a more appropriate decision. Also, competition with other firms can be another extension

of this research. When many firms compete, the timing strategy of adding the internet channel

will be more important due to the effect of preemption. Even though the investment sunk cost for

the internet channel is somewhat higher, a firm might introduce earlier in order to preempt other

firms. Moreover, generalized demand and cost models incorporating dynamic customer

acceptance index and sunk costs will be interesting to study. However, it will be challenging to

obtain analytical solutions. Numerical methods such as Monte Carlo simulation (Longstaff and

Schwartz (2001)) can be developed to analyze the optimal pricing and timing strategies. While

this was beyond the scope of the present paper, it provides a fruitful avenue for future research to

explore. The authors have been working on research along some of these lines.

82

APPENDIX

APPENDIX 4.A. Derivation of Demand Functions in Dual Channel

The demand functions for a dual channel can be derived from the Hotelling model which

takes into account the customer preference and compares all utility functions. In our setting, a

customer may have two utilities, which are the utility by purchasing from a retail channel (v- rp )

and the utility from an internet channel ( dpv −θ ). By comparing these two utilities, the customer

might have these three possible decision cases as provided in Table 4.2.

For each case, we have:

Case 1: rpv − ≥ dpv −θ and 0≥− rpv

Case 2: rpv − ≤ dpv −θ and dpv −θ 0≥

Case 3: rpv − ≤ 0 and dpv −θ ≤ 0

However, we need to note that these inequality conditions changes depending on the value of θ .

Thus, when 10 <<θ , we obtain:

Case 1: θ−

−≥

1dr pp

v and rpv ≥ (A.1)

Case 2: θ−

−≤

1dr pp

v and θ

dpv ≥ (A.2)

Case 3: rpv ≤ and θ

dpv ≤ (A.3)

From (A.3), we further consider two possible cases of these inequalities:

(i) rd pp θ≥

83

Note that rdr p

pp≤

−−θ1

and we assume that customer’s valuation follows uniform

distribution [0, 1] (Chiang et al. 2003). From the condition (A.1), a demand for a retail channel

)( rQ will be rp−1 . Also, from (A.2) we can see that these two equalities contradict each other.

Therefore, a demand for the direct internet channel )( dQ will be zero.

(ii) rd pp θ≤

Likewise, because rdr p

pp≥

−−θ1

, we know that rQ is θ−

−−

11 dr pp

from (A.1). According to

(A.2), the demand for the internet )( dQ is θθ

ddr ppp−

−−

1. Thus, the demand function for the

internet )( dQ is )1( θθ

θ−− dr pp

.

According to (i) and (ii), we can have demand functions like (1) and (2) in Section 3.

When 1≥θ , we can derive demand functions in the same way.

Case 1: θ−

−<

1dr pp

v and rpv ≥ (A.4)

Case 2: θ−

−>

1dr pp

v and θ

dpv ≥ (A.5)

Case 3: rpv ≤ and θ

dpv ≤ (A.6)

(iii) rd pp θ≥

Since rdr p

pp≥

−−θ1

and θθ

ddr ppp≥

−−

1,

θθ

−−

=1

drr

ppQ and

θ−−

−=1

1 drd

ppQ .

(iv) rd pp θ≤

84

rQ is zero and θ

dd

pQ −=1 .

According to (iii) and (iv), we can have demand functions like (3) and (4) in Section 3. Q.E.D.

APPENDIX 4.B. Proof of Proposition 1

(1) First, we determine a retail and an internet price policy for a dual channel case.

By taking derivatives, we have

01

)1)(())()((21

1))()((

1)()(

1)()(

1)(

);,(1

=−

−+−+=

−−

+−−

−−−

−=∂

Φ∂

θλ

θθθ

tCtptp

tCtptCtptptptp

tpp

rrd

ddrrdr

r

drvi

2/))(1()(* tCtp rr += , 2/))(()(* tCtp rd λθ += .

01

)/1)((/)(2)(2

)1(/)(/)(

1)()(

)1(/)()(

)();,(1

=−

−−−=

−−

−−−

+−−

=∂

Φ∂

θθλθ

θθθ

θθθ

tCtptp

tCtptCtptptptp

tpp

rdr

ddrrdr

d

drvi

Similarly, we can derive the optimal solutions for 2 ( , ; )vi r dp p tΦ and the solutions for each

case are the same, which are:

2/))(1()(* tCtp rr += , 2/))(()(* tCtp rd λθ += .

Then, to check the concavity for optimality conditions, we construct the Hessian matrices

( ),(1 dr ppH and ),(2 dr ppH )of functions );,(1 tpp drviΦ and );,(2 tpp drviΦ with respect

to rp and dp as follows:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−

−−−

=

)1(2

)1(2

)1(2

)1(2

),(

θθθ

θθdr ppH

85

011)1(

2)()1(

2 22

221 <⎟

⎠⎞

⎜⎝⎛ −

−−−

−−= xxxHxxT

θθθ for Rxx ∈∀ 21 , .

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−

−−

−=

)1(2

)1(2

)1(2

)1(2

),(

θθ

θθθ

dr ppH

02)()1(

2 21

221 <−−

−= xxxHxxT

θ for Rxx ∈∀ 21 , .

Since the Hessian matrices are negative definite matrices, the functions );,(1 tpp drviΦ and

);,(2 tpp drviΦ are concave functions and the optimal solutions are globally optima.

Moreover, 0 ( ) 1rp t≤ ≤ , 0 ( )dp t θ≤ ≤ , then we have

⎩⎨⎧ ≤+

=otherwise

tCiftCtp rrr 1

1)(2/))(1()(* , * ( ( )) / 2 ( ) /( ) r r

dC t if C t

p totherwise

θ λ θ λθ

⎧ + ≤= ⎨⎩

(2) Next, the optimal retail price strategy for the single traditional retail channel and the single

direct internet channel at time t can be derived in a similar way to a dual channel analysis by the

first order condition, 2/))(1()(* tCtp rtr += and ))(1(2

)(* tCtp ddo +=θ . Also, since

02);(2

2

<−=Φ tpdpd

trtrtr

and 02);(2

2

<−=Φθ

tpdpd

dododo

, );( tptrtrΦ and );( tpdodoΦ are concave

with respect to )(tptr and )(tpdo . Finally, Together with 0 ( ) 1trp t≤ ≤ , 0 ( )dop t θ≤ ≤ ,

we have ⎩⎨⎧ ≤+

=otherwise

tCiftCtp rrtr 1

1)(2/))(1()(* ,

* ( ( )) / 2 ( ) /( ) r r

doC t if C t

p totherwise

θ λ θ λθ

⎧ + ≤= ⎨⎩

. Q.E.D.

86

APPENDIX 4.C. Proof of Proposition 2

The conditions for a dual channel are rd pp θ≤ and 10 <<θ from Eqs. (4.1) and (4.2), and

also rd pp θ> and 1≥θ from Eqs. (4.3) and (4.4). Based on the results in Proposition 1, we can

derive following conditions:

i) If 10 <<θ , the condition θλ ≤ is equivalent to **

21

222 rrr

d pCC

p θθλθ

=⎟⎟⎠

⎞⎜⎜⎝

⎛+≤+= . In other

words, a firm needs to construct a dual channel. When θλ > , a firm will prefer a single

traditional channel.

ii) If 1≥θ , the condition θλ > is equivalent to ⎟⎟⎠

⎞⎜⎜⎝

⎛+>+

21

222rr CC

θλθ suggesting a dual

channel. When θλ ≤ , a single internet channel is preferred.

However, the first case in ii) should be excluded, because the condition that θλ > and 1≥θ

violates the aforementioned assumption of 1<λ . Thus, if 1≥θ , then a single internet channel is

preferred when θλ ≤ .

Finally, by Case II and III, we can show that the condition to construct the internet channel is

θλ ≤ . Q.E.D.

APPENDIX 4.D. Proof of Proposition 3

The derivation of firm’s value is very similar to that of a financial American option as shown

in McDonald and Siegel (1986) and Pindyck (1991). The optimal investment rule for a firm is

determined by solving a stochastic optimal stopping problem. At time T, a firm will launch an

internet channel. Before introducing the internet channel, a single channel produces a net cash

flow dtpCp rtrr )1)(( *,

* −− . After developing the internet channel, the firm would makes profit

87

);( TCrviπ incurring an irreversible sunk cost K . We represent the firm’s value as )( rCV .

Finally, a value function for optimal developing timing is derived as follows:

* *

0( ) max ( )(1 ) ( ; )

T Tr tr r tr vi rT

V C E e p C p d e C Tρτ ρτ π− −⎡ ⎤= − − +⎢ ⎥⎣ ⎦∫ ,

Hence as shown in Pindyck (1991) and Dixit and Pindyck (1994) (Chapter 4), in the

continuation region the Bellman equation is

* *[ ( ) ( )( )] ( )r tr r trV C p C P p dt E dVρ − − − = (D.1)

Using Ito’s Lemma to manipulate dV (for the details, refer to Wilmott et al. (1995) and

Oksendal (2003)) we have

rrCrCCrrr dCCVdtCVCdtCVCdVrrr

)()()2/1()()( 22 ++= σ .

By substitution and manipulation (see Pindyck (1991)), the Bellman equation becomes

2 2 '' ' * *1 ( ) ( ) ( )( ) 02 r r r r tr r trC V C C V C V p C P pσ μ ρ+ − + − − = (D.2)

In addition, )( rCV should satisfy the following boundary conditions:

ρ41)0( =V (D.3)

KCCC

CV rrrr −

−−−

−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−+

−−= 2

2*2

2

2***

2)1(4)(

221

41)(

σμρθθθλ

σμρμρρ (D.4)

2

2

2

**

2)1(2)(

21

21)(

σμρθθθλ

σμρμρ −−−

−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−+

−−=

∂∂ rr

r

r CCCCV

(D.5)

022

)1(4)(

2212

41)(

)(2

2

21

121 =

−−−

−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−+

−−+=

∂∂ −

σμρθθθλ

σμρμρβ β rr

rr

r CCCa

CCV

(D.6)

Condition (D.3) implies profits that the a firm makes when 0=rC . Conditions (D.4) and

(D.5) are smooth pasting and value matching conditions coming from optimality. Condition (D.6)

88

arises from the ceiling barrier rC (See Dixit (1993)). The general solution for equation (D.2)

must take the form 2131)( ββ

rrr CaCaCV += , where 31 , aa are constants to be determined, and

,22)21(

21

22

221 >+−+−=σρ

σμ

σμβ (since 22 σμρ +> )

02)21(

21

22

222 <+−−−=σρ

σμ

σμ

β .

To satisfy the condition (D.3), we must have 03 =a , so the general solution must have the form

11

βCa . The solution should satisfy the conditions (D.4) and (D.5). Finally, we have Proposition

3. Q.E.D.

APPENDIX 4.E. Proof of Proposition 4

02

1)(

)2)(1(42 2

2

1

1*

>−

−−−⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

∂∂

KKCr

θλ

σμρθθ

ββ .

Moreover, since a necessary condition for a dual channel is that θλ ≤ ,

02

)2)(1(4)(

1

1

21

2

*

>⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−−−

−=

∂∂ KCr

βσμρθθβ

λθλ,

21

21

1

21

*

)()1(22

2)2(4

)()1(

2)2(4

λθθθλλθθ

β

σμρβλθθθ

θβ

σμρβθ −−

−+−⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−−=

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−−=

∂∂ K

ddKCr

0)()1(2)1(2)(

2)2(4

21

21 <

−−−+−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−−=

λθθθθλθλ

β

σμρβK

*, 1

2 21 1 1 1

2

2 0( 1)(1 ) ( 2)( ) 4 ( 1)

( 2 )

r doCK K

β

β λ β θ λ ρβ β θρ μ θ ρ μ σ ρ

∂= >

∂ ⎛ ⎞ ⎛ ⎞− − − − − −−⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠

. Q.E.D.

89

APPENDIX 4.F. Proof of Proposition 5

When the net present value method is applied, an investment strategy can be derived by

comparing );( TCrtrπ and );( TCrviπ . First, for the single channel, a firm makes profits over time

horizon ),[ ∞T at decision making time T . With optimal policies in Proposition 1, we have total

profits:

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+

−−=⎥⎦

⎤⎢⎣⎡ −=

⎥⎦⎤

⎢⎣⎡ −−=⎥⎦

⎤⎢⎣⎡ −=

∫

∫∫∞ −−

∞ −−∞ −−

2

2,

22)(

**)(*)(

2)()(

241)(

41

))(()();(

σμρμρρ

π

ρ

ρρ

TCTPCPdtCPeE

dtpPCpeEdtQCpeETC

rr

T rTt

T trrtrTt

T rrtrTt

rtr

Thus, from 02)1(4

)();();( 2

22

=−−−−

−=− KCTCTC r

rtrrvi σμρθθθλ

ππ , the threshold by the

NPV, *NPVC , becomes:

KCNPV 2

2*

)()2)(1(4

θλσμρθθ

−

−−−= .

And we know that the threshold by a real option theory is

CROT KC 2

2

1

1*

)()2)(1(4

2 θλσμρθθ

ββ

−

−−−⎟⎟⎠

⎞⎜⎜⎝

⎛

−= .

By comparing, we have 121

1*

*

>−

=ββ

NPV

ROT

CC

. Thus, the feasible region by NPV always includes

the feasible region by the real option theory.

Similarly, we have following

90

*,

2

2

2

22

2

2

221

211

2

1

1

1

1

2

21

112

21

2

11

*,

)2(

)(

)1(4)2(

)()1()1(

)2(

)(

)1(4)2()2(

))(2(

))(2()1)(1(

))(2()1)(1(

)2(

))(2(

)1(4)2(

))(2()1)(1()1)(1(

doNPV

doROT

C

K

K

K

C

=

−−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−−

−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

−−−

≥

−−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

−−−

−−−⎟

⎟⎠

⎞⎜⎜⎝

⎛

−−

−−−

−−

−−

=

−−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

−−

−−−⎟

⎟⎠

⎞⎜⎜⎝

⎛

−−−

−−

−−

=

σμρθ

λθ

ρθρ

σμρθ

λθ

μρλ

μρλ

σμρθ

λθ

ρθρ

σμρθβ

λθββ

μρβλβ

μρβλβ

σμρθ

λθβ

ρθβρβ

σμρθ

λθβ

μρλβ

μρλβ

because we know 01 >− λ , 02 >− λθ , ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

>⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−

−μρλ

μρλ

ββ 11

21

1

1 and

0)2(

1)1(1)2(

1)2()1()1(

)1(4)2(

)()1(

)1(4)2()2(

))(2(

))(2()1)(1(

21

2

1

1

2

1

12

2

22

221

211

2

1

1

<⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛

−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−

≤

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

−−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

−−−

−−−⎟

⎟⎠

⎞⎜⎜⎝

⎛

−−

−−

βμρλ

ββ

ββ

μρλ

ρθρ

σμρθ

λθ

μρλ

ρθρ

σμρθβ

λθββ

μρβλβ

K

K

Q.E.D.

91

Chapter 5. A Simple Robust Mean Absolute Deviation Model for Portfolio

Optimization

In this paper we develop a robust model for portfolio optimization. The purpose is to consider parameter uncertainty by controlling the impact of estimation errors on the portfolio strategy performance. We construct a simple Robust Mean Absolute Deviation (RMAD) model which leads to a linear program and reduces computational complexity of existing robust portfolio optimization methods. This paper tests the robust strategies on real market data from 1996 to 2003 and discusses performance of the robust optimization model empirically based on financial elasticity, standard deviation, and market condition such as growth, steady state, and decline in trend. Our study shows that the proposed robust optimization generally outperforms a nominal mean absolute deviation model. We also suggest precautions against use of robust optimization under certain circumstances.

5.1. Introduction and Literature Review

Portfolio optimization is the process of analyzing a portfolio and managing the assets within

it. It enables allocating and rebalancing portfolio assets based on an overall target return. The

portfolio optimization problem, as originally developed by Markowitz (1952), assumed that an

investor has two goals: expectation high return and minimum level of risk. The portfolio

optimization model by Markowitz is a quadratic programming problem and has difficulties: 1)

Quadratic programming problems are more difficult to solve (computational intensity) and to

apply than linear problems (applicability), and 2) for practical markets, the size of the covariance

matrix for solving portfolio selection model is very large and difficult to estimate. These kinds of

difficulties make Markowitz model less practical in the real market. To overcome these problems,

many trials have attempted to transform the problem into a linear programming problem. Konno

and Yamazaki (1991) suggested a Mean Absolute Deviation model (MAD) as a linear

programming model. The linear program of portfolio optimization is formulated by the

92

consideration of different risk measure, the so-called mean absolute deviation. The similar ideas

on the risk measure for a linear programming formulation extended the model such as Kellerer et

al. (2000), Mansini et al. (2003), Chiodi et al. (2003) and Papahristodoulou and Dotzauer (2004).

Moreover, the efforts to reduce computational intensity by developing heuristic algorithms have

attempt in such studies as Mansini and Speranza (1999), Kellerer, et al. (2000), Chang et al.

(2000), and Kim (2005).

However, most of the parameters defined in the earlier mentioned models are estimated or

approximated by using expected values. This means that parameters include estimation error or

uncertainty. In addition, importantly, solutions to optimization problems can show remarkable

sensitivity to uncertainty (Bertsimas et al. (2007)). In this paper we develop robust models for

portfolio optimization. The purpose is to consider parameter uncertainty by controlling the impact

of estimation errors on the portfolio strategy performance. We construct a simple Robust Mean

Absolute Deviation (RMAD) model which leads to a linear program and reduces computational

complexity of existing robust portfolio optimization methods.

To handle the parameter uncertainty problem, there is a recent research trend of development

of new robust optimization approaches. Traditional optimization methods require full knowledge

of parameters to allow transformation to a stochastic program. From full information,

assumptions of parameters following specific known distributions can sometimes be too strong

and criticized for their validity. Therefore, within a deterministic framework, Soyster (1973)

proposed initially the concept of a robust optimization considering simple perturbations on

parameters in a linear optimization system. The idea behind robust optimization is to consider the

worst case scenario without a specific distribution assumption. However, this approach

admittedly provides the highest protection or most conservatism. Ben-Tal and Nemirovski (2000)

addressed a model that can adjust the conservatism by defining an ellipsoidal uncertainty set.

Even though the proposed model is less conservative, the program still has nonlinear terms,

93

which can be problematic when extended to practical problems, such as extension to a discrete

optimization like a mixed integer program as mentioned in Bertsimas and Sim (2004). These

difficulties from nonlinear terms lead Bertsimas and Sim (2004) to propose a robust formulation

that is linear, applicable, deterministically-solvable, and extendable to discrete optimization

without the loss of feasibility of solution. Another advantage of their model is its ease of

controlling the level of conservatism. By adjusting this parameter, the solution set can contain

solutions of both a nominal deterministic model and the Soyster’s robust model. Thus, the model

proposed in this paper will be developed based on methodology by Bertsimas and Sim (2004).

Despite of intensive theoretical development, few papers discuss the detailed performance of

robust optimization, even in the financial area. El Ghaoui et al. (2003) proposed a robust portfolio

model under an uncertainty of covariance matrix which is developed by semidefinite

programming (SDP) and considers worst case Value-at-Risk (VaR). These researchers showed

that the robust solution significantly outperform nominal portfolios performed on real market data.

And Goldfarb and Iyengar (2003) performed detailed experiments on both simulated and real

market data and compared classical Markowitz portfolio model to their robust models using a

second order cone program (SOCP). With real market data, the robust portfolios did not always

outperform the classical portfolio approach. As another approach, Tutuncu and Koenig (2004)

developed a robust portfolio optimization problem formulated in a quadratic program (QP) and

based on efficient frontiers. For their model, using real-world market data, they found that the

robust portfolios provide a significant improvement in worst-case return versus nominal

portfolios at the expense of a much smaller cost in expected return. Also, Ben-Tal et al. (2006)

studied and compared various robust optimization problems according to the theory of convex

risk measure, which shows that different robustness settings can result in different performances.

However, these literatures comparing a robust method with a nominal one are based on only one-

94

time period analysis, in-sample analysis or simulation experiments which might mislead

comparison result of two methods.

In view of the cited weakness, the contributions of this paper arise from modeling and

empirical perspective. First, we develop a simple robust portfolio optimization model which can

consider parameter uncertainty. Moreover, in contrast to literatures, we consider a linear

formulation of robust portfolio optimization. The linear formulation takes advantage of less

computational effort and of more applicability to practice which could be suffered from by other

robust portfolio optimizations in the forms of a quadratic program, a second order cone program,

or a semidefinite program. Second, this research provides a more comprehensive empirical study

on an out-of-sample basis to compare a robust portfolio optimization with a classical method.

Moreover, this research considers three different time horizons with respect to market conditions

(growth, steady-state, decline), while most literatures studied over the only one time horizon.

Also, performances according to different characteristics of stocks (standard deviation and

financial elasticity) are investigated. The results show that the proposed robust approach

outperforms a classical method, generally, but cannot always guarantee superiority. Therefore, we

verify several circumstances requiring attention using a robust optimization such as declining

market condition and a portfolio of low standard deviation stocks.

The organization of this paper is as follows. The next section provides notations and explains

the mean absolute deviation (MAD) formulation. And then, the robust formulation of MAD

model to consider parameter uncertainty and the corresponding reformulation into a linear

program is derived. With the robust mean absolute deviation (RMAD) model, results of empirical

study based on historical data from January 1996 to September 2003 are discussed. The study

concludes with some discussion of extensions.

95

5.2. Mean Absolute Deviation Model

The notations for the mean absolute deviation model by Konno and Yamazaki (1991) are:

xj = units of asset j to be included in the portfolio

yt = deviation below the average rate of return at time period t

T = the length of the time horizon

t = each period over the time horizon, Tt ...,,2,1=

ρ = minimum rate of return required by an investor

Rj = a random variable representing the rate of return of asset j

rj = the expected return, rj = ][ jRE , of asset j

rjt = the observed return of asset j during the period t

uj = the maximum amount of asset j

C = the total portfolio expenditure

The Mean Absolute Deviation model is based on the Mean Variance model by Markowitz

(1952). Konno and Yamazaki (1991) redefined the risk of a portfolio to formulate portfolio

optimization in a linear system for the reasons explained in introduction. The objective function

deals with the deviation below average rate of return as a variance. Therefore, this model does not

require calculation of the covariance matrix among assets, which is usually considered drawbacks

of Markowitz model. And this model can work without putting assumptions on the distributions

of the uncertain parameters. The MAD model and robust optimization method have a connecting

thread since neither require assumptions. The MAD model is:

minimize ∑∑==

−n

jjjjt

T

txrr

T 11)(1

96

subject to ∑=

≥n

jjj Cxr

1ρ ,

∑=

=n

jj Cx

1,

jj ux ≤≤0 , Nj ,...,2,1= .

This original problem can be reformulated using auxiliary variable ty :

minimize ∑=

T

tty

T 1

1

subject to ∑=

≥−+n

jjjjtt xrry

10)( , Tt ,,1L= ,

∑=

≥−−n

jjjjtt xrry

10)( , Tt ,,1L= ,

∑=

≥n

jjj Cxr

1ρ ,

∑=

=n

jj Cx

1,

jj ux ≤≤0 , Nj ,...,2,1= .

5.3. Robust Formulation of MAD Model

While this section develops a robust reformulation of MAD model, an uncertainty set for the

robust optimization prior to the reformulation is necessary. Representative ways to define the

uncertainty categorize as three models developed by Soyster (1973), Ben-Tal and Nemirovski

(1998), and Bertsimas and Sim (2004). The first uncertainty set is too conservative and the

second creates a nonlinear problem, so this study follows the uncertainty set formation of the last

97

model (Bertsimas and Sim (2004)). In the MAD model, the expected return rj , of asset j is

approximated by E[Rj], which means that an actual return cannot be exactly obtained and has

uncertainty. Therefore, to express the uncertainty for an actual return vector r~ , we define as

follows: Each entry jr~ , j ∈ N stocks, takes values in ]ˆ,ˆ[ jjjj rrrr +− , where jr̂ represents

the deviation from the nominal estimated mean coefficient, jr .

Also, definition of the sets for the robust counterpart is based on the work by Bertsimas and

Sim (2004) as follows. Let 0J be the set of coefficients jr̂ , j ∈ 0J that are subject to parameter

uncertainty. As a way to control the level of robustness in constraints, we address a parameter

0Γ+ℜ∈ which value is in the interval ],0[ 0J . For the chosen 0Γ , consider a subset 0S satisfying

the conditions 00 JS ⊆ and ⎣ ⎦00 Γ=S .

From now on, we will explain derivation of a robust counterpart with the first constraint in

the formulation (2). For the given set 0S and a coefficient vr̂ where 00 \ JSv∉ , we would like to

allow a certain level of deviations in constraints, while guaranteeing the feasibility of solutions.

Here, we can define the deviations such that one variable can change up to ( ⎣ ⎦00 Γ−Γ ) vr̂ and

the ⎣ ⎦0Γ variables for 0S can change. Therefore, we can incorporate a certain level of deviations

for a given jr̂ , jx , vr̂ , vx and 0Γ where 0Sj∈ and 00 \ JSv∉ like

⎣ ⎦ vvSj

jj xrxr ˆ)(ˆ 00

0

Γ−Γ+∑∈

.

Also, since the feasibility of solutions should be guaranteed, the following condition should

hold:

⎣ ⎦ 0ˆ)(ˆ)()( 001 11 1 0

≤⎟⎟

⎠

⎞

⎜⎜

⎝

⎛Γ−Γ+++−−≤+−− ∑∑ ∑∑ ∑

∈= == =vv

Sjjj

n

j

n

jjjjjtt

n

j

n

jjjjjtt xrxrxrxryxrxry .

98

Furthermore, the solution sets should be robust against all scenarios of different 0S and vr̂

for a given 0Γ . Therefore, the robust counterpart is constructed in a way to maximally influence

which is:

{ } ⎣ ⎦⎣ ⎦ 0ˆ)(ˆmax)(

00000000

00}\,,{1 1

≤⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Γ−Γ+++−− ∑∑ ∑∈∈Γ=⊆∪= = Sj

jvjjSJvSJSvS

n

j

n

jjjjjtt xrxrxrxry

As in Ben-Tal and Nemirovski (2000) and Bertsimas and Sim (2004), we can transform the

above robust counterpart into a simplified form. More specifically for the transformation

technique, let jw be the upper bound for feasible jx which means jj wx ≤ (or equivalently

wx j ≤ and wx j ≤− ). We know that

{ } ⎣ ⎦⎣ ⎦

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Γ−Γ+∑∈∈Γ=⊆∪

00000000

ˆ)(ˆmax 00}\,,{ Sj

jvjjSJvSJSvS

xrxr

is equivalent to

{ } ⎣ ⎦⎣ ⎦

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Γ−Γ+∑∈∈Γ=⊆∪

00000000

ˆ)(ˆmax 00}\,,{ Sj

jvjjSJvSJSvS

wrwr

s.t. jj wx ≤ and jj wx ≤−

0≥jw .

Thus, we have the equivalent robust counterpart as follows:

{ } ⎣ ⎦⎣ ⎦ 0ˆ)(ˆmax)(

00000000

00}\,,{1 1

≤⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Γ−Γ+++−− ∑∑ ∑∈∈Γ=⊆∪= = Sj

vvjjSJvSJSvS

n

j

n

jjjjjtt wrwrxrxry .

with jjj wxw ≤≤− and 0≥jw .

Here, 0Γ = 0 enables us to completely ignore the influence of the approximated-return

deviations, while 0Γ = | 0J | realizes all possible conditions of the deviations, which is the most

99

conservative. In other words, a higher value of 0Γ increases the level of robustness. For this

reason, the parameter 0Γ is called a protection level (PL) in a robust optimization. Following

Bertsimas and Sim (2004) and manipulating the uncertainty set and MAD model, we propose the

robust optimization formulation of the MAD (RMAD) model below:

minimize ∑=

T

tty

T 1

1

subject to ∑ ∑= =

+−−n

j

n

jjjjjtt xrxry

1 1

)(

{ } ⎣ ⎦⎣ ⎦ 0ˆ)(ˆmax

00000000

00}\,,{

≤⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Γ−Γ++ ∑∈∈Γ=⊆∪ Sj

vvjjSJvSJSvS

wrwr , Tt ,...2,1= ,

∑ ∑= =

−+−n

j

n

jjjjjtt xrxry

1 1)( (5.2)

{ } ⎣ ⎦⎣ ⎦ 0ˆ)(ˆmax

00000000

00}\,,{

≤⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Γ−Γ++ ∑∈∈Γ=⊆∪ Sj

vvjjSJvSJSvS

wrwr , Tt ,...2,1= ,

{ } ⎣ ⎦⎣ ⎦∑ ∑

= ∈∈Γ=⊆∪−≤

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Γ−Γ++−n

j Sjvvjj

SJvSJSvSjj Cwrwrxr

100

}\,,{00000000

ˆ)(ˆmax ρ ,

∑=

=n

jj Cx

1,

jjj wxw ≤≤− , 0≥jw , jj ux ≤≤0 , Nj ,...,2,1= .

However, the above formulation to a robust optimization still has a nonlinear term. Thus, the

robust mean absolute deviation (RMAD) is reformulated into the linear programming problem

based on Bertsimas and Sim (2003) in the following way.

100

First, since each entry jr~ , j ∈ N takes values in ]ˆ,ˆ[ jjjj rrrr +− , we also know the

jjt rr ~− takes values in ]ˆ,ˆ[ jjjtjjjt rrrrrr +−−− , which means the uncertainty set can be

expressed in the same deviations as that of jr~ .

Let us consider a robust counterpart in the constraint, given an optimal solution vector

),,,( ***1

*Nj xxxX LL= :zzz

),( 0* ΓXβ =

{ } ⎣ ⎦⎣ ⎦

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Γ−Γ+∑∈∈Γ=⊆∪

00000000

*00

*

}\,,{ˆ)(ˆmax

Sjjvjj

SJvSJSvSxrxr

is equivalent to

maximize ∑∈ 0

*ˆJj

jjj zxr

subject to 00

Γ≤∑∈Jj

jz , (5.3)

,10 ≤≤ jz 0Jj∈∀ .

For the derivation of a dual problem, we change the problem (5.3) and define dual variables as

follows:

maximize ∑∈ 0

*ˆJj

jjj zxr

subject to 00

Γ≤∑∈Jj

jz , (dual variable: 0z )

,1≤jz 0Jj∈∀ . (dual variables: jp , 0Jj∈∀ )

0≥jz , 0Jj∈∀

Thus, the dual of the above problem is:

101

minimize ∑∈

Γ+0

00Jj

j zp

subject to *0 ˆ jjj xrpz ≥+ , 0Jj∈∀ , (5.4)

0≥jp , 0Jj∈∀ ,

0≥oz .

Since the primal problem (5.3) is a linear program and feasible, the strong duality theorem

guarantees the objective value of the dual problem (5.4) is equivalent to that of the problem (5.3)

which is ),( 0* ΓXβ . Therefore, manipulating this program (5.4) in the original problem (5.2),

the RMAD problem can be formulated to the linear program as (5.5).

minimize ∑=

T

tty

T 1

1

subject to 0)(0

001 1

≤Γ+++−− ∑∑ ∑∈= = Jj

j

n

j

n

jjjjjtt zpxrxry , Tt ,...2,1= ,

0)(0

001 1

≤Γ++−+− ∑∑ ∑∈= = Jj

j

n

j

n

jjjjjtt zpxrxry , Tt ,...2,1= , (5.5)

∑ ∑= ∈

≤+Γ++−n

j Jjjjj Czpxr

100 0

0

ρ ,

jjj wrpz ˆ0 ≥+ ,

∑=

=n

jj Cx

1,

0≥jp , 0Jj∈∀ ,

0≥oz ,

jjj wxw ≤≤− , 0≥jw , jj ux ≤≤0 , Nj ,...,2,1= .

102

Therefore, the RMAD model is solved within an LP framework which reduces computational

complexity, and at the same time considers parameter uncertainties.

5.4. The Performance of a Robust Portfolio

This section reports the results of empirical study for the robust portfolio, based on real

market data. The cumulative return results by RMAD under several circumstances are compared

with returns optimized by a nominal MAD model and are provided with attention to different

protection levels (PL).

Data Set Description

The 100 stocks for a portfolio investment are randomly selected from NYSE, NADAQ, and

AMEX. The data set are obtained from January 1996 to September 2003 in the

CRSP/COMPUSTAT merged database. This chosen time period includes recession data to allow

the data set to contain a variety of market conditions to clarify the role. Thus, the data of the

recession from March 2001 to November 2001, declared by the National Bureau of Economic

Research, are included. Also, unlike the literatures which usually compare return results over one

time horizon, this empirical study is conducted in the following three subsections as to three

important factors: market condition, standard deviation, and financial elasticity. Our belief that

market condition would affect the performance of RMAD very much leads to considering the

involvement of recession data. Moreover, because standard deviations of stocks are intimately

associated with a robust counterpart, the standard deviation is chosen as an analysis factor in this

paper. And we considered a financial elasticity which is regarded as a very important factor and

correlated with a stock market. For these all cases, we conduct out-of-sample analysis and set up

the time that the portfolio is constructed as 0=t . Moreover, the analyses are developed for two

103

parameter estimation periods, short term and long term. Also, in order to group stocks for analysis

of standard deviation and value of financial elasticity (β ), we collect the corresponding decile

data set from the CRSP/COMPUSTAT merged database. Alternatively, we might group stocks in

descending order from chosen 100 stocks. However, results by this rule can be quite relative

according to chosen stocks. For the reason, we adopt the deciles data of stocks as an absolute

index for a sorting criterion which are ranked from the whole stocks in the market.

5.4.1. RMAD Resutls with Market Conditions

This section considers the results of using a robust mean absolute deviation model with

respect to market condition. The market conditions in the data set include three categories: 1)

growth, 2), steady state and 3) decline in trend. These criteria were chosen based on the report of

Business Cycle Dating Committee (2003). For example, we define a steady state as the period

that several economic indices began to change after a market growth and a decline begins with a

trough defined by Business Cycle Dating Committee (2003). The relevant Dow Jones Index and

S&P 500 index for three periods appear in Figure 5.1. The three regions divided by dotted lines

represent each case. Therefore, the data for growth, steady state, and decline are from the time

before December 1999, from January 2000 to August 2001 and after September 2001. Three

constructed portfolios for each of the three time period allowed comparison of the performances

of the robust and nominal portfolio optimizations. Moreover, the cumulative actual returns based

on out-of-sample analysis were analyzed for two different parameter estimation periods: a short-

term period and a long-term period which is twice the length of the short term period. For

example, at each breakpoint time for a declining market, September 2001, 30 months of data is

used for vector jtr and nominal value, jr , while 60 months of data is employed for a long term

104

period. And the cumulative actual returns for the next 20 months were calculated from the

optimal portfolios by RMAD and MAD.

S&P500 and Dow Jones Index

0

2000

4000

6000

8000

10000

12000

14000

1/2/

1996

7/2/

1996

1/2/

1997

7/2/

1997

1/2/

1998

7/2/

1998

1/2/

1999

7/2/

1999

1/2/

2000

7/2/

2000

1/2/

2001

7/2/

2001

1/2/

2002

7/2/

2002

1/2/

2003

7/2/

2003

Date

Index

0

200

400

600

800

1000

1200

1400

1600

DOW SP500

Figure 5.1. The historical data of Dow Jones and S&P index

Furthermore, the role of the protection level (PL) which can adjust the level of robustness in a

robust portfolio optimization was considered in the empirical study. Higher protection levels

means that the solution set of a robust optimization accounts for more uncertainties on the return

vector, jr . The results with PL=0 represent a portfolio constructed by a nominal MAD model

with no uncertainty, while the portfolios with PL=5 is close to the solution by the most

conservative robust optimization method proposed by Soyster (1973). However, this study’s

results according to different market conditions, different protection levels, and different

estimation time horizons show interesting phenomena. Generally speaking, robust portfolios

show better performances than nominal portfolios for the most cases. However, the robust

portfolios constructed by RMAD for long-term in a declining market produced worse return than

the nominal portfolio, as shown in the right-hand graph of Figure 5.2. Compared to other

Steady-State Growth Decline

105

literatures, such as El Ghaoui, et al. (2003) and Tutuncu and Koenig (2004) which show that

robust models outperform the conventional model, the appearing results in Figure 5.2 to 4

contradict this in some areas. This implies that the parameter estimation period should be

carefully considered when applying the robust portfolio optimization method. An additional

discovery is that returns of different protection levels are not consistent or regular, unlike

simulation-based analysis results such as Bertsimas and Sim (2004). For example, the result in a

steady-state market in the short-term shows that a higher protection level sometimes provides

higher returns than a lower protection level, but sometimes had a lower return, even lower than a

nominal portfolio.

0 5 10 15 20−1

0

1

2

3

4

5

6x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Return with Different Protection Levels (Growth Market/Short)

PL=0PL=1PL=3PL=5

0 5 10 15 20−1

0

1

2

3

4

5

6x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Return with Different Protection Levels(Growth Market/Long)

PL=0PL=1PL=3PL=5

Figure 5.2. All stocks in growth market condition

0 5 10 15 20−1

0

1

2

3

4

5

6

7x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Return with Different Protection Levels (Steady Market/Short)

PL=0PL=1PL=3PL=5

0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

2.5

3x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Return with Different Protection Levels (Steady Market/Long)

PL=0PL=1PL=3PL=5

Figure 5.3. All stocks in steady state market condition

106

0 5 10 15 20−1

−0.5

0

0.5

1

1.5x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Return with Different Protection Levels (Decline Market/Short)

PL=0PL=1PL=3PL=5

0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

2.5x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Return with Different Protection Levels (Decline Market/Long)

PL=0PL=1PL=3PL=5

Figure 5.4. All stocks in decline market condition

5.4.2. RMAD Results with Standard Deviations

Robust portfolios sometimes provided performances contradicting previous literatures as

shown in the previous subsection and the standard deviation of stock has a close connection to the

robust counterpart. Therefore, this section considers standard deviations of stocks to see if the

robust optimization method performs better than the nominal method. For this analysis, stocks

were divided into two categories: 1) Low standard deviation (LSD) stocks and 2) High standard

deviation (HSD) stocks. Two divisions use the decile data of standard deviations from the

CRSP/Compustat merged database. Stocks positioned in 1 to 5 deciles of the whole market are

used to construct an LSD sorted portfolio, while stocks of 6 to 10 deciles become those in an

HSD portfolio. Accordingly, 39 LSD stocks and 61 HSD stocks constitute among the 100 stocks

in the two data sets. Based on these data sets, comparison of the cumulative actual returns for

these two different types of portfolios occurred for the time interval of the declining market since

a robust approach apparently works well for other market conditions. Again long and short term

periods were considered just like the previous subsection. As a result, the robust approach for

HSD sorted portfolios, especially for long term time horizon, appears to outperform nominal

portfolios as shown in the upper two graphs of Figure 5.5. For the LSD sorted portfolio, the

107

RMAD model works well for the short-term. This result is intuitive because the parameter

estimation of HSD stocks could be more biased and the biased parameter might produce a higher

probability of a misleading solution for a portfolio optimization. In other words, since RMAD

considers the uncertainty to cure this bias and the estimation errors can be reduced by using long

term data, a robust portfolio methodology can work better for an HSD sorted portfolio, especially

long-term parameter estimation. However, an LSD portfolio of long-term is dominated by a

nominal portfolio for the whole time horizon. This can similarly be explained by intuition that,

because LSD stocks of long-term do not have much uncertainty or the uncertainty can be treated

by using more data, a robust optimization considering the worst case might not work better.

0 5 10 15 20 25−8

−6

−4

−2

0

2

4

6x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Portfolio of High Standard Deviation Stocks (Short)

PL=0PL=1PL=3PL=5

0 5 10 15 20 25−1

0

1

2

3

4

5

6

7

8x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Portfolio of High Standard Deviation Stocks (Long)

PL=0PL=1PL=3PL=5

0 5 10 15 20 25−1

0

1

2

3

4

5

6x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Portfolio of Low Standard Deviation Stocks (Short)

PL=0PL=1PL=3PL=5

0 5 10 15 20 25−2

−1

0

1

2

3

4

5

6x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Portfolio of Low Standard Deviation Stocks (Long)

PL=0PL=1PL=3PL=5

Figure 5.5. Portfolio of high (upper) and low (lower) standard deviation stocks

108

5.4.3. RMAD Results with Financial Elasticity

Financial elasticity ( β ), as a key parameter in the capital asset pricing theory, implies a

relationship between the individual stock and the rest of market. Because a correlation with, or

dependence on, a systematic risk can work differently in portfolio optimization, the stocks in the

portfolios are next divided by the value of financial elasticity (β ) which is:

)(),(

m

mjj rVar

rrCov=β , for Nj ,...,2,1= .

where mr is a market return.

Similar to the case of standard deviation, the decile data of β is also obtained from

CRSP/Compustat merged database and stocks positioned in 1 to 5 deciles of the whole market are

constructed for a low beta (LB) sorted portfolio, while stocks of 6 to 10 deciles are done for high

beta (HB) sorted portfolio. For our data, 57 stocks are included in the LB portfolio, while 43 are

in the HB portfolio. Here this study conducts the out-of-sample analysis of data under the

declining market condition. Figure 5.6 shows that the robust method works well for all cases.

Interestingly, when comparing the gaps between RMAD and MAD results for HB and LB sorted

portfolio, the LB portfolio have relatively small differences. This implies that our RMAD model

is sensitive to beta as well. If the stocks are not correlated with a market much like low beta

stocks, it can be concluded that a significant performance is difficult to expect from RMAD.

109

0 5 10 15 20 25−3

−2

−1

0

1

2

3x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Portfolio of High Beta Stocks (Short)

PL=0PL=1PL=3PL=5

0 5 10 15 20 25−1

0

1

2

3

4

5x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Portfolio of High Beta Stocks (Long)

PL=0PL=1PL=3PL=5

0 5 10 15 20 25−2

−1

0

1

2

3

4

5

6x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Portfolio of Low Beta Stocks (Short)

PL=0PL=1PL=3PL=5

0 5 10 15 20 25−1

0

1

2

3

4

5

6

7

8

9x 10

6

Time Horizon (month)

Act

ual R

etur

n of

por

tfolio

s

Portfolio of Low Beta Stocks (Long)

PL=0PL=1PL=3PL=5

Figure 5.6. Portfolio of high (upper) and low (lower) beta stocks

5.5. Conclusion and Further Studies

This paper proposes a simple robust portfolio optimization model using mean absolute

deviation methods in a linear program framework and studied empirical results. This model takes

advantage of a reduction in a computational complexity and provides optimal solutions under

parameter uncertainty. Recently, some robust optimization methods developed to incorporate a

parameter uncertainty adopted modeling in forms of QP, SOCP, and SDP such as El Ghaoui, et al.

(2003), Ben-Tal, et al. (2006), Ben-Tal and Teboulle (2007), and Brown and Sim (2007).

However, these models encounter the problems that computational efforts to solve remain quite

110

challenging. Therefore, consideration and development of transformation to a linear programming

becomes the alternative.

Moreover, the empirical study of the proposed model shows that the RMAD generally

outperforms the nominal mean absolute deviation model. However, more importantly, the results

also show that sometimes the robust portfolio might not be successfully constructed, despite most

of literatures concluding that a robust portfolio optimization outperforms a nominal one. To

specify the conditions of the superiority of a robust optimization method, this study considers

several factors such as market condition, financial elasticity, standard deviation and parameter

estimation period. Specifically, the results show that a robust optimization approach could result

in poor performance under such a certain condition as parameters are estimated over long time

horizons in a declining market. And the RMAD significantly outperforms a nominal method in a

portfolio consisting of high standard deviation stocks or high beta stocks. Besides, for a low beta

sorted portfolio, the difference between a robust and a nominal approach is very small.

Involvement of the important factors as covered in this paper could improve the current

robust portfolio optimization models. For example, addressing new variables related to standard

deviations can amplify the advantage of the robust approach that functions well for a high

standard deviation portfolio. Also, an extension exists for practical application. For example,

investors are asked to purchase by the unit of a certain number of stocks (minimum lot). In

addition, transaction costs can be incorporated into the RMAD like Konno and Wijayanayake

(2001) and Lobo et al. (2007). However, by considering the minimum lot and transaction cost, the

LP Robust Mean Absolute Deviation model should be transformed to a MILP (mixed integer

linear programming) formulation of RMAD model. These ideas also can be extended with the

philosophy of a discrete robust optimization methodology.

111

References

Aase, K.K. 2005. The perpetual American Put Option for jump-diffusions with Applications. Working paper.

Abel, A.B., J.C. Eberly. 1994. A Unified Model of Investment Under Uncertainty. The American Economic Review. 84(5) 1369-1384.

Alvarez, L.H.R., R. Stenbacka. 2007. Partial outsourcing: A real options perspective. International Journal of Industrial Organization. 25(1) 91-102.

Austin, J.E. 2002. Managing in Developing Countries: Strategic Analysis and Operating Techniques. Free Press.

Bajari, P., S. Tadelis. 2001. Incentives versus Transaction Costs: A Theory of Procurement Contracts. The RAND Journal of Economics. 32(3) 387-407.

Barthelemy, J. 2001. The hidden costs of IT outsourcing. MIT Sloan Management Review. 42(3) 60-69.

Ben-Tal, A., D. Bertsimas, D.B. Brown. 2006. A flexible approach to robust optimization via convex risk measures. Working Paper.

Ben-Tal, A., A. Nemirovski. 1998. Robust Convex Optimization. Mathematics of Operations Research. 23(4) 769-805.

Ben-Tal, A., A. Nemirovski. 2000. Robust solutions of Linear Programming problems contaminated with uncertain data. Mathematical Programming. 88(3) 411-424.

Ben-Tal, A., M. Teboulle. 2007. An old-new concept of convex risk measures: The Optimized Certainty Equivalent. Mathematical Finance. 17(3) 449-476.

Benaroch, M. 2002. Managing Information Technology Investment Risk: A Real Options Perspective. Journal of Management Information Systems. 19(2) 43-84.

Bergin, P.R., R.C. Feenstra, G.H. Hanson. 2007. Outsourcing and Volatility. NBER Working Paper.

Bernanke, B. 1983. Irreversibility, uncertainty, and cyclical investment. The Quarterly Journal of Economics. 98(1) 85-106.

Bernstein, F., A. Federgruen. 2005. Decentralized Supply Chains with Competing Retailers Under Demand Uncertainty. Management Science. 51(1) 18.

Bernstein, F., L.M. Marx. 2006. Reservation Profit Levels and the Division of Supply Chain Profit. Working paper, Duke University, City.

Bernstein, F., J.-S. Song, X. Zheng. 2008. Bricks-and-Mortar vs. Clicks-and-Mortar: an Equilibrium Analysis. European Journal of Operational Research. 187(3) 671-690.

Bertsimas, D., D. Brown, C. Caramanis. 2007. Theory and applications of Robust Optimization Working Paper.

Bertsimas, D., M. Sim. 2003. Robust discrete optimization and network flows. Mathematical Programming. 98(1) 49-71.

Bertsimas, D., M. Sim. 2004. The Price of Robustness. Operations Research. 52(1) 35-53. Borenstein, S., G. Saloner. 2001. " Economics and Electronic Commerce". JOURNAL

OF ECONOMIC PERSPECTIVES. 15(1) 3-12.

112

Brennan, M.J., E.S. Schwartz. 1985. Evaluating Natural Resource Investments. The Journal of Business. 58(2) 135-157.

Brown, D.B., M. Sim. 2007. Satisficing measures for analysis of risky positions. Working Paper.

Business Cycle Dating Committee, N.B.E.R. 2003. The NBER's Business-Cycle Dating procedure. http://www.nber.org/cycles/july2003.html, City.

Cachon, G.P. 2003. Supply chain coordination with contracts. Handbooks in Operations Research and Management Science: Supply Chain Management.

Cachon, G.P., M.A. Lariviere. 2005. Supply Chain Coordination with Revenue-Sharing Contracts: Strengths and Limitations. Management Science. 51(1) 30-44.

Caldentey, R., L. Wein. 2002. Revenue management of a make-to-stock queue. Manufacturing & Service Operations Management. 4(1) 4-6.

Cattani, K., W. Gilland, H.S. Hees, J. Swaminathan. 2006. Boiling Frogs: Pricing Strategies for a Manufacturer Adding a Direct Channel that Competes with the Traditional Channel. Production and Operations Management. 15(1) 40-56.

Chang, T.J., N. Meade, J.E. Beasley, Y.M. Sharaiha. 2000. Heuristics for cardinality constrained portfolio optimisation. Computers and Operations Research. 27(13) 1271-1302.

Chatterjee, K., L. Samuelson. 1987. Bargaining with Two-sided Incomplete Information: An Infinite Horizon Model with Alternating Offers. The Review of Economic Studies. 54(2) 175-192.

Chatterjee, K., W. Samuelson. 1983. Bargaining under Incomplete Information. Operations Research. 31(5) 835-851.

Chiang, W.K., D. Chhajed, J.D. Hess. 2003. Direct Marketing, Indirect Profits: A Strategic Analysis of Dual-Channel Supply-Chain Design. Management Science. 49(1) 1-20.

Chiodi, L., R. Mansini, M.G. Speranza. 2003. Semi-Absolute Deviation Rule for Mutual Funds Portfolio Selection. Annals of Operations Research. 124(1) 245-265.

Chopra, S., M.S. Sodhi. 2004. Managing risk to avoid supply-chain breakdown. MIT Sloan Management Review. 46(1) 53-61.

Cohen, M.A., N. Agrawal. 1999. An analytical comparison of long and short term contracts. IIE Transactions. 31(8) 783-796.

Cramton, P.C. 1984. Bargaining with Incomplete Information: An Infinite-Horizon Model with Two-Sided Uncertainty. The Review of Economic Studies. 51(4) 579-593.

Cramton, P.C. 1992. Strategic Delay in Bargaining with Two-Sided Uncertainty. The Review of Economic Studies. 59(1) 205-225.

Dayanand, N., R. Padman. 2001. Project Contracts and Payment Schedules: The Client's Problem. Management Science. 47(12) 1654.

Deloitte. 2005. Calling a Change in the Outsourcing Market, available at www.deloitte.com/dtt/cda/doc/content/us_outsourcing_callingachange.pdf.

Dixit, A. 1989. Entry and Exit Decisions under Uncertainty. The Journal of Political Economy. 97(3) 620-638.

Dixit, A. 1993. The art of smooth pasting. Routledge.

113

Dixit, A.K., R.S. Pindyck. 1994. Investment under Uncertainty. Princeton University Press.

Domberger, S. 1998. The Contracting Organization: A Strategic Guide to Outsourcing. Oxford University Press.

Drezner, Z. 1978. Computation of the Bivariate Normal Integral. Mathematics of Computation. 32(141) 277-279.

Druehl, C.T., E.L. Porteus. 2006. Online versus Offline Price competition: Service differentiation and the Effect of Internet shopping Penetration. Working Paper.

El Ghaoui, L., M. Oks, F. Oustry. 2003. Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Operations Research. 51(4) 543-556.

Ertogral, K., S.D. Wu. 2001. A bargaining game for supply chain contracting. preprint. Ferguson, M.E. 2003. When to commit in a serial supply chain with forecast updating.

Naval Research Logistics. 50(8) 917-936. Genz, A. 1992. Numerical Computation of Multivariate Normal Probabilities. Journal of

Computational and Graphical Statistics. 1(2) 141-149. Geyskens, I., K. Gielens, M. Dekimpe. 2002. The Market Valuation of Internet Channel

Additions. Journal of Marketing. 66(2) 102-119. Giannoccaro, I., P. Pontrandolfo. 2004. Supply chain coordination by revenue sharing

contracts. International Journal of Production Economics. 89(2) 131-139. Goldfarb, D., G. Iyengar. 2003. Robust Portfolio Selection Problems. Mathematics of

Operations Research. 28(1) 1-38. Grenadier, S.R., A.M. Weiss. 1997. Investment in technological innovations: An option

pricing approach. Journal of Financial Economics. 44 397-416. Gurnani, H., M. Shi. 2006. A Bargaining Model for a First-Time Interaction Under

Asymmetric Beliefs of Supply Reliability. Management Science. 52(6) 865. Hamel, G., Y. Doz, C. Prahalad. 1989. Collaborate with your competitors and win.

Harvard Business Review. 67(1) 133-139. Harsanyi, J.C., R. Selten. 1972. A Generalized Nash Solution for Two-Person Bargaining

Games with Incomplete Information. Management Science. 18(5) 80-106. Henderson, V., D.G. Hobson. 2002. Real options with constant relative risk aversion.

Journal of Economic Dynamics and Control. 27(2) 329-355. Huchzermeier, A., C.H. Loch. 2001. Project Management Under Risk: Using the Real

Options Approach to Evaluate Flexibility in R&D. Management Science. 47(1) 85-101.

Ingene, C.A., M.E. Parry. 1995. Channel Coordination when retailers compete. Marketing Science. 14(4) 360-377.

Jeffery, M., I. Leliveld. 2004. Best Practices in IT Portfolio Management. MIT Sloan Management Review. 45(3) 41?49.

Jiang, B., T. Yao, B. Feng. 2008. Valuate Outsourcing Contracts from Vendors' Perspective: A Real Options Approach. Decision Sciences. 39(3) 383-405.

Kamrad, B., S. Lele. 1998. Production, operating risk and market uncertainty: a valuation perspective on controlled policies. IIE Transactions. 30(5) 455-468.

Kamrad, B., A. Siddique. 2004. Supply Contracts, Profit Sharing, Switching, and Reaction Options. Management Science. 50(1) 64-82.

114

Kellerer, H., R. Mansini, M.G. Speranza. 2000. Selecting Portfolios with Fixed Costs and Minimum Transaction Lots. Annals of Operations Research. 99(1) 287-304.

Kern, T., L.P. Willcocks, E. van Heck. 2002. The winner's curse in IT outsourcing: Strategies for avoiding relational trauma. California Management Review. 44(2) 47-69.

Kim, J.S. 2005. An Algorithm for Portfolio Optimization Problem. Informatica. 16(1) 93-106.

Kogut, B. 1991. Joint Ventures and the Option to Expand and Acquire. Management Science. 37(1) 19-33.

Konno, H., A. Wijayanayake. 2001. Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming. 89(2) 233-250.

Konno, H., H. Yamazaki. 1991. Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market. Management Science. 37(5) 519-531.

Kouvelis, P., C. Chambers, H. Wang. 2006. Supply Chain Management Research and Production and Operations Management: Review, Trends, and Opportunities. Production and Operations Management. 15(3) 449-469.

Laffont, J.J., J. Tirole. 1993. A Theory of Incentives in Procurement and Regulation. MIT Press.

Lambrecht, B.M. 2004. The timing and terms of mergers motivated by economies of scale. Journal of Financial Economics. 72(1) 41-62.

Lee, T. 2004. Determinants of the foreign equity share of international joint ventures. Journal of Economic Dynamics and Control. 28(11) 2261-2275.

Levina, N., J.W. Ross. 2003. From the Vendor's Perspective: Exploring the Value Proposition in IT Outsourcing. MIS Quarterly. 27(3) 331-364.

Li, C., J. Giampapa, K. Sycara. 2006. Bilateral negotiation decisions with uncertain dynamic outside options. Systems, Man and Cybernetics, Part C, IEEE Transactions on. 36(1) 31-44.

Li, C., P. Kouvelis. 1999. Flexible and Risk-Sharing Supply Contracts Under Price Uncertainty. Management Science. 45(10) 1378-1398.

Li, C.L., P. Kouvelis. 1999. Flexible and Risk-Sharing Supply Contracts under Price Uncertainty. Management Science. 45(10) 1378-1398.

Liang, T.-P., J.-S. Huang. 1998. An Empirical Study on Consumer Acceptance of Products in Electronic Markets: A Transaction Cost Model. Decision Support Systems. 24 29-43.

Lobo, M.S., M. Fazel, S. Boyd. 2007. Portfolio optimization with linear and fixed transaction costs. Annals of Operations Research. 152(1) 341-365.

Longstaff, F.A., E.S. Schwartz. 2001. Valuing American options by simulation: a simple least-squares approach. Review of Financial Studies. 14(1) 113-147.

Luehrman, T.A. 1998. Investment Opportunities as Real Options: Getting Started on the Numbers. Harvard Business Review. 76(4) 51-67.

Maglaras, C., J. Meissner. 2006. Dynamic pricing strategies for multi-product revenue management problems. Manufacturing and service operations management. 8(2) 136-148.

115

Mansini, R., W. Ogryczak, M. Grazia Speranza. 2003. LP solvable models for portfolio optimization: a classification and computational comparison. IMA Journal of Management Mathematics. 14(3) 187-220.

Mansini, R., M.G. Speranza. 1999. Heuristic algorithms for the portfolio selection problem with minimum transaction lots. European Journal of Operational Research. 114(2) 219-233.

Marathe, R., S. Ryan. 2005. On the validity of the geometric brownian motion assumption. The Engineering Economist. 50(2) 159-192.

Markowitz, H. 1952. Portfolio Selection. The Journal of Finance. 7(1) 77-91. McDonald, R., D. Siegel. 1986. The Value of Waiting to Invest. The Quarterly Journal of

Economics. 101(4) 707-728. McGrath, R.G. 1997. A Real Options Logic for Initiating Technology Positioning

Investments. The Academy of Management Review. 22(4) 974-996. Merton, R. 1973. Theory of rational option pricing. The Bell Journal of Economics and

Management Science 141-183. Mieghem, J.A.V. 1999. Coordinating Investment, Production, and Subcontracting.

Management Science. 45(7) 954-971. Mukhopadhyay, S.K., D.-Q. Yao, X. Yue. 2008. Information sharing of value-adding

retailer in a mixed channel hi-tech supply chain. Journal of Business Research. 61 950-958.

Murto, P. 2004. Exit in Duopoly under Uncertainty. The RAND Journal of Economics. 35(1) 111-127.

Muthoo, A. 1995. On the strategic role of outside options in bilateral bargaining. Operations research. 43(2) 292-297.

Nagarajan, M., G. Sosic. 2008. Game-theoretic analysis of cooperation among supply chain agents: Review and extensions. European Journal of Operational Research. 187(3) 719-745.

Nembhard, H.B., L. Shi, M. Aktan. 2005. A real-options-based analysis for supply chain decisions. IIE Transactions. 37(10) 945-956.

Noble, S., D. Griffith, M. Weinberger. 2005. Consumer derived utilitarian value and channel utilization in a multi-channel retail context. Journal of Business Research. 58(12) 1643-1651.

Oksendal, B. 2003. Stochastic Differential Equations: An Introduction with Applications. Springer.

Papahristodoulou, C., E. Dotzauer. 2004. Optimal portfolios using linear programming models. Journal of the Operational Research Society. 55(11) 1169-1177.

Pasternack, B.A. 2008. Optimal Pricing and Return Policies for Perishable Commodities. Marketing Science. 27(1) 133.

Perakis, G., A. Sood. 2006. Competitive Multi-period Pricing for Perishable Products: A Robust Optimization Approach. Mathematical Programming. 107(1) 295-335.

Pindyck, R.S. 1991. Irreversibility, Uncertainty, and Investment. Journal of Economic Literature. XXIX 1110-1148.

Plambeck, E.L., T.A. Taylor. 2005. Sell the Plant? The Impact of Contract Manufacturing on Innovation, Capacity, and Profitability. Management Science. 51(1) 133-150.

116

Raiffa, H. 1982. The art and science of negotiation. Belknap Press of Harvard University Press Cambridge, Mass.

Rangaswamy, A., G.H. Van Bruggen. 2005. Opportunities and challenges in multichannel marketing: An introduction to the special issue. Journal of Interactive Marketing. 19(2) 5-11.

Reuer, J.J., T.W. Tong. 2005. Real Options in International Joint Ventures. Journal of Management. 31(3) 403.

Rubinstein, A. 1985. A Bargaining Model with Incomplete Information About Time Preferences. Econometrica. 53(5) 1151-1172.

Sarkar, S., B. Klaseen, M. Fabel. 2006. Selling Wireless Services on the Internet: Turning Clickers into Buyers, A.T. Kearny, available at http://www.atkearney.com/index.php/Publications/selling-wireless-services-on-the-internet-turning-clickers-into-buyers.html.

Schwartz, E.S., C. Zozaya-Gorostiza. 2003. Investment Under Uncertainty in Information Technology: Acquisition andd Development Projects. Management Science. 49(1).

Sebenius, J.K. 1992. Negotiation Analysis: A Characterization and Review. Management Science. 38(1) 18-38.

Simchi-Levi, D., S.D. Wu, Z.M. Shen. 2004. Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era, . Kluwer Academic Publishers, Norwell.

Soyster, A.L. 1973. Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming. Operations Research. 21(5) 1154-1157.

Taylor, T.A. 2006. Sale Timing in a Supply Chain: When to Sell to the Retailer. Manufacturing & Service Operations Management. 8(1) 23.

TheSeattleTimes. 2010. Procter & Gamble jumping into retail online, testing new site, available online at http://seattletimes.nwsource.com/html/businesstechnology/2010798405_procteronline17.html?syndication=rss.

Trigeorgis, L. 1993. The Nature of Option Interactions and the Valuation of Investments with Multiple Real Options. The Journal of Financial and Quantitative Analysis. 28(1) 1-20.

Trigeorgis, L. 1993. Real Options and Interactions with Financial Flexibility. Financial Management. 22(3) 202-224.

Tsay, A.A., N. Agrawal. 2004. Modeling Conflict and Coordination in Multi-Channel Distribution System: A Review. Kluwer Academic Publishers.

Tsay, A.A., S. Nahmias, N. Agrawal. 1999. Modeling Supply Chain Contracts: A Review. Quantitative Models for Supply Chain Management 299-336.

Tutuncu, R., M. Koenig. 2004. Robust Asset Allocation. Annals of Operations Research. 132(1) 157?187.

Van Mieghem, J. 1999. Coordinating Investment, Production, and Subcontracting. Management Science. 45(7) 955.

117

Van Mieghem, J. 2007. Risk Mitigation in Newsvendor Networks: Resource Diversification, Flexibility, Sharing, and Hedging. Management Science. 53(8) 1269.

Walton, R.E., R.B. Mckersie. 1965. A Behavioral Theory of Labor Negotiations: An Analysis of a Social Interaction System. McGraw-Hill, New York.

Weeds, H. 2002. Straegic Delay in a Real Options Model of R&D Competition. Review of Economic Studies. 69 729-747.

Wilmott, P., S. Howison, J. Dewynne. 1995. The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press.

Yan, R., S. Ghose. 2009. Forecast information and traditional retailer performance in a dual-channel competitive market. Journal of Business Research. In Press, Corrected Proof.

Zeng, D., K. Sycara. 1998. Bayesian learning in negotiation. International Journal of Human-Computers Studies. 48(1) 125-141.

Zhu, K., J.P. Weyant. 2003. Strategic Decisions of New Technology Adoption under Asymmetric Information: A Game-Theoretic Model*. Decision Sciences. 34(4) 643-675.

VITA

Yongma Moon

DATE & PLACE OF BIRTH October 4, 1977 Seoul, Korea. RESEARCH INTERESTS • Investment Decisions under Uncertainty • Financial Engineering • Operations Management (Supply Chain, Contract) • Revenue Management and Dynamic Pricing • Stochastic Programming / Real Option / Robust Optimization / Game Theory EDUCATION 2005 – 2010 Pennsylvania State University, Industrial Engineering (Ph.D.) 2003 – 2005 Pennsylvania State University, Industrial Engineering (M.S.) Pennsylvania State University, Operation Research (M.S. Dual Degree) 1996 – 2000 Seoul National University, Industrial Engineering (B.S.)