Investment Decision-making Using Optional Models [1 ed.] 9781119687511, 1119687519

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Investment Decision-making Using Optional Models

Modern Finance, Management Innovation and Economic Growth Set coordinated by Faten Ben Bouheni

Volume 2

Investment Decision-making Using Optional Models

David Heller

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2019 The rights of David Heller to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019948429 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-522-0

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter 1. Risk and Flexibility Integration in Valuation . . . . . . . . .

1

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . 1.2. The scope of real options . . . . . . . . . . . . . 1.2.1. The concept of real options . . . . . . . . . 1.2.2. Empirical use of real options . . . . . . . . 1.2.3. Paradigms in options . . . . . . . . . . . . . 1.3. Valuation of investments by real options . . . 1.3.1. Optional valuation of investments in a discrete-time approach. . . . . . . . . . . . . . . . 1.3.2. Optional valuation of investments in a continuous-time approach . . . . . . . . . . . . . . 1.4. Option model extensions by incorporating new parameters (Levyne and Sahut 2008) . . . . . 1.4.1. Stochastic volatility. . . . . . . . . . . . . . 1.4.2. Transaction costs and models with jumps 1.4.3. Option pricing . . . . . . . . . . . . . . . . . 1.5. Conclusion . . . . . . . . . . . . . . . . . . . . .

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35 36 39 41 44

Chapter 2. Optional Modeling of Investment Choices and Surplus Value Linked to the Option to Invest . . . . . . . . . . . . . . . .

47

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2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Framework of optional interactions and option to develop an investment project . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Real investment opportunity . . . . . . . . . . . . . . . . . . . . . . .

47 48 50

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Investment Decision-making Using Optional Models

2.2.2. Opportunity to postpone decision-making to infinity . 2.2.3. Development cycle and taking into account new information within dependent projects and focusing on research and development . . . . . . . . . . . . . 2.3. Option to exchange and abandon an investment project . 2.3.1. Real options within the replacement cycle and disinvestment alternatives . . . . . . . . . . . . . . 2.3.2. The value of an investment project in the natural resources sector . . . . . . . . . . . . . . . . . . 2.3.3. Valuation of the abandonment option by investors . . 2.4. Growth option resulting from investment decisions and acquisition strategies . . . . . . . . . . . . . . . . 2.4.1. Company profiles justifying growth option value . . . 2.4.2. Growth option value related to interactions between financing and investment decisions. . . . . . . . . . 2.4.3. Acquisition strategies by the real options approach . . 2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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52

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62 65

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88 89

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90 98 106

Chapter 3. Data Generation Applied to Strategic and Operational Option Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Determining the right time to invest . . . . . . . . . . . 3.2.1. Application to the carry option . . . . . . . . . . . . 3.2.2. Application of the Dixit and Pindyck model . . . . 3.3. Flexibility of asset exchange, abandonment and temporary shutdown of projects . . . . . . . . . . . . . . 3.3.1. Application to the exchange option . . . . . . . . . 3.3.2. Application to the abandonment option . . . . . . . 3.3.3. Application to the temporary shutdown option . . 3.4. Incorporation of development phases . . . . . . . . . . 3.4.1. Implementation of a two-stage investment project 3.4.2. Valuation of a sequential project . . . . . . . . . . .

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107 107 108 110

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113 113 115 116 121 121 122

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139

Appendix 1. Demonstration of the CRR Formula . . . . . . . . . . . . .

141

Appendix 2. Stochastic Differential Calculus . . . . . . . . . . . . . . . .

147

Contents

vii

Appendix 3. Test of the Black and Scholes Formula and Return on the Log–Normal Distribution . . . . . . . . . . . . . . . . . . . .

155

Appendix 4. Demonstration of the Black and Scholes Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173

Introduction

In order to create value, companies must allocate their resources effectively and evaluate investment alternatives. By focusing a study on quantitative techniques such as net present value (NPV), capital recovery time or potential market evolution scenarios, analysts fail to consider intangible investment-related factors such as future growth opportunities, management flexibility or strategic value. In fact, the decision to invest or not based on the NPV criterion, or more generally on the DCF (Discounted Cash Flows), assumes that the projected cash flows are deterministic. While these traditional methods incorporate systematic risk using the discount rate, they do not consider the total risk of changes in cash flows as measured by their volatility. In practice, random factors lead the company to generate cash flows other than those originally expected at the time of the investment decision. Increased competition may lead, for example, to a downward revision of the price policy in a spirit of competitiveness. In addition, the increase in costs of raw materials in the industrial process can force the company to produce only if market conditions are synonymous with profitability. The possibility of splitting an investment into several sequences is also conceivable. By relying on this last criterion, the company certainly relinquishes economies of scale, but has the advantage of interrupting certain investment phases when market conditions prove to be less sustainable than those expected. Finally, a business plan considering positive cash flows can, in fact, depart considerably from the reality, leading the company to abandon an unprofitable activity. Consequently, the operational flexibility required for a company in an uncertain context must be associated with any decisionmaking process.

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Investment Decision-making Using Optional Models

This notion of risk is essential in the valuation method by the real options. The term “real” distinguishes this type of options from financial options of which the underlying is a financial asset. Financial options are specific contracts traded on organized or over-the-counter markets, while real options are found in risky, more difficult to identify and more specific projects with an operational or strategic flexibility. Real options are of particular interest when alternative investment management choices are available to executives or when competitive advantages allow them to wait before investing. Just like a financial option, a real option is asymmetrical since it grants the holder the right and not the obligation to exercise it, that is to say, the right to undertake an investment or abandon it, the opportunity to take advantage of sustainable changes and to leave behind unfavorable situations. Thus, an investment opportunity is similar to a call option to the extent that the company has the right to purchase the equivalent of the underlying – namely, the assets required for the operation of the project – at a future date or before a future date (respectively, the equivalent of a European and American option) – at a certain price. According to Leshchiy (2015), real options are the junction between finance and business strategies to the extent that corporate finance is focused on the valuation of risky assets in order to maximize enterprise value, while the business strategy focuses on studying market potential and finding favorable competitive sources. By highlighting the value of flexibility in investment decisions, both in terms of its operational and financial aspects, the real options approach models the creation and maximization of value. Armstrong (2015) argues that when a capacity constraint exists within an asset acquisition project, the NPV method may be insufficient in terms of the quality of the valuation and therefore the budgeting, as opposed to real options which allow us to consider the values of underlying variables, such as volatility, which impacts the value of the project. According to Levyne and Sahut (2008), “investment projects are for the most part real options pools that are rooted in the following three elements”: – the irreversibility of an investment means that the investment generates a non-recoverable cost (sunk cost) and, once started, it is difficult to stop

Introduction

xi

it without losing part of the expenses incurred. Its origin may be related to the specificity of the investment1, the obsolescence of equipment2 and the legislative or institutional constraints3; – the risk associated with real options may be endogenous or exogenous if, for example, there is a fluctuation in demand, a change in interest or exchange rates, the arrival of a new competitor, a regulatory change, etc. The company can influence the risk by carrying the investment project or undertaking an initial investment phase to gather information for the continuation of the project. In the event that the project generates cash flows that can be anticipated, the real options approach is of no use; – flexibility is defined as the opportunity to benefit from favorable circumstances and to prevent adverse circumstances. This perfectly reflects the situation of the holder of an option at maturity. Flexibility can be strategic4 or operation-related5. Therefore, for a given level of investment irreversibility, the real options value associated with a project will depend on risk and managerial flexibility. The more the company executives can favorably change the course of the investment project and/or the higher the uncertainty, the more the associated options will be expensive and vice versa. If a project is inflexible and uncertainty is low, considering a real options approach is unlikely to be useful. Low

High Risk

Managerial flexibility

High

Average value of real options

High value of real options

Low

Negligible value of real options

Low value of real options

Table I.1. Real options value

1 Specific to the company or industry to which it belongs. 2 The liquidation value of used goods is generally lower than its purchase value. 3 For example, capital restrictions may prohibit foreigners from liquidating their assets in a market. 4 This will include the carry option and the growth option. 5 For example, it will be a question of choosing between different raw materials according to their price.

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Investment Decision-making Using Optional Models

In this context, to what extent can the notions of risk and managerial flexibility be integrated into the valuation of an investment project? How does the real options approach make it possible to effectively value an investment project of a company? Chapter 1 will be devoted to the development of a theoretical framework to present the concept of real options. Indeed, it will be interesting to model this approach by integrating the fundamental parameters, as well as extensions in order to tend towards a fair value. Chapter 2 will address the study of financial literature by focusing, on the one hand, on the interactions of the different categories of options present within the same investment project and, on the other hand, by developing the conceptual aspects and academic extensions of the carrying, exchange, abandonment and growth options and the resulting acquisition strategies. Chapter 3 will be dedicated to the practical applications as for the different models exposed in the first two parts.

1 Risk and Flexibility Integration in Valuation

1.1. Introduction The investment must be identified as an entry fee that provides access to future opportunities. Thus, the value of a project is not limited to the present value of anticipated cash flows, but must capture all the growth opportunities that will arise in the future. For this, real options offer a long-term vision. They have the advantage of incorporating future upside and downside cash flow opportunities through volatility representing the risk and, consequently, make it possible to incorporate the notion of flexibility into project management. In fact, depending on the cash flows, the project can, among other things, be carried, abandoned, strengthened or developed in sequence. Volatility is the key parameter of options, whether financial or real. Its usefulness lies in the fact that the value of derivative financial products or investment projects depends on the possibility of benefiting from favorable conditions or, otherwise, reducing losses. In practice, real options remain less used than the NPV criterion for determining the value of a project. However, Graham and Harvey’s (2001) and Hartmann and Hassan’s (2006) studies indicate that about a quarter of Chief Financial Officers (CFOs) surveyed use the real options approach to help them make investment decisions. Black and Scholes (1973), on the one hand, and Cox, Ross and Rubinstein (1979) models, on the other hand, form a basis for the valuation of investment projects by real options. Initially intended to enhance the value of financial options, these models are particularly relevant to evaluate a project taking into account a range of opportunities characterizing it.

Investment Decision-making Using Optional Models, First Edition. David Heller. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

2

Investment Decision-making Using Optional Models

After the realization of a pragmatic analogy as to the different parameters constituting these paradigmatic models, it is possible to apply the valuation of projects assimilated to real options. Thus, projects with growth options, abandonment options, combined options, sequential development options, or options for expanding or reducing the activity can be the subject of a dynamic and at least complementary analysis to that of the NPV criterion. New parameters have been incorporated into the initial models in order to improve them by making them more precise. Thus, the notion of constant volatility established by Black and Scholes is questioned by some researchers who prefer a stochastic volatility with the objective of anticipating future developments in the price of the underlying. In addition, transaction costs complement the models by promoting a better definition of hedging strategies. Models with jumps are expected to consider important macroeconomic events. Finally, taking into account the payment of a discrete-time dividend within a continuous-time model would make it possible to refine the value of the project even better. 1.2. The scope of real options Unlike financial options, real options focus on valuing “real” assets, i.e. investment projects. In this context, an analogy between financial and real options can be seen. In fact, an investment opportunity is similar to a call option because the company has the right, not the obligation, to invest in an asset, on a fixed date or during a given period, on a price known in advance. Thus, the five fundamental parameters that make it possible to evaluate a financial option can be similar when it comes to valuing an investment project. Amran and Kulatilaka (1999), who also believe that real options are an extension of the theory of financial options applied to real assets (nonfinancial), manage to distinguish seven categories of options. As a result, real options offer some flexibility in the management of an investment project, unlike traditional methods, as decisions can be made throughout project implementation. In practice, Graham and Harvey (2001) find that, while the most commonly-used method of valuing an investment project is based on the NPV criterion, real options are used by almost a quarter of the CFOs surveyed. In other words, this method is more in demand than profitability index or value-at-risk. By refining the conclusions of these two researchers’

Risk and Flexibility Integration in Valuation

3

study, we see that real options particularly convince large industrial companies, whose debt and growth are moderate and whose management is the responsibility of shareholders who pay little or no dividend(s). By focusing on the use of real options in the pharmaceutical industry, Hartmann and Hassan (2006) reach the same types of conclusions: about 25% of the pharmaceutical groups surveyed say they use real options. The valuation models of real options rely on paradigms regarding the valuation of the option premium initially established for financial options. Black and Scholes (1973) set out to establish a continuous-time model, and Cox, Ross and Rubinstein (1979) consider a discrete-time model. A convergence between these two formulas allowed us to arrive at extensions such as taking into account the distribution of dividends. 1.2.1. The concept of real options Real options involve real activities unlike financial options that exclusively incorporate financial assets. Luehrman (1998) attempts to draw an analogy between these two categories of options by presenting an analytical framework aimed at reconciling options theory with valuations of investment projects. He compares the similarities between the DCF method and the options approach by stating that an investment opportunity is similar to a call option, in that the company can, without being forced to, buy the assets necessary for the operation of a new activity. He also demonstrates that the value of the option linked to an investment opportunity has the same characteristics as that present on the financial markets and that, consequently, it is possible to evaluate an option falling within the investment process of real assets. The nuance, however, lies in the fact that investment opportunities only come once. Synthetic options need to be established with the ability to replicate payments from investment projects. In this context, there is an analogy between the characteristics of an investment project and the five parameters used to calculate the price of an option. In fact, an investment opportunity is similar to a call option to the extent that the company has the right to purchase the equivalent of the underlying – namely, the assets required for the operation of the project – at a future date or before a future date (respectively, the equivalent of a European and American option) – at a certain price. The term “real” distinguishes this type of options from financial options of which the

4

Investment Decision-making Using Optional Models

underlying is a financial asset. Financial options are specific contracts traded on organized or over-the-counter markets, while real options are more difficult to identify and specify. Just like a financial option, a real option is asymmetrical since it gives the holder the right and not the obligation to exercise it, that is to say, the right to undertake an investment or abandon it, the opportunity to take advantage of favorable developments and to leave behind unfavorable situations. Thus, by discounting the value of the assets held, we obtain the value of the underlying. If the investment action corresponds to the exercise of the option, the amount of the investment will be the exercise price of the option. The maturity of the option is the period during which the investment decision can be carried out. The risk-free interest rate remunerates the time value of money. The volatility of the underlying asset is the risk attached to the value of future cash flows of the project. Thus, the analogy is summarized in Table 1.1. Investment considered in terms of real option

Financial purchase option

Variable

Current value of assets to be purchased for the operation of the project

Price of the underlying asset

S

Investment to make to realize the project

Exercise price of option

E

Period during which the decision can be carried

Time remaining until the maturity of the option

τ

Time value of money

Risk-free rate

r

Risk measurement of project assets

Volatility (standard deviation) of returns on financial assets

σ

Table 1.1. Analogy between the parameters of the real option valuing an investment project and the financial option

Amran and Kulatilaka (1999) distinguish seven categories of real options: – the growth option gives the company the opportunity to expand into new markets or new activities. In other words, it is the junction between a current project and future opportunities. It is often essential to make an initial investment that can lead to new future growth opportunities if the circumstances are favorable. These are typically investments in a research and development project. The growth option is therefore similar to a European call option, of which the underlying asset is the current value of

Risk and Flexibility Integration in Valuation

5

the cash flows generated by a future project and which depends on an initial investment. The exercise price corresponds to the additional investment to be undertaken to continue the project. Maturity is the time remaining until the project implementation. To proceed with the initial investment, the overall NPV of the project, i.e. the NPV of the original project, plus the growth option of the future project, must be positive. Then, at the maturity of the option, the additional project is realized, that is to say that the option is exercised, if the state of nature is favorable; in other words, if the current value of the cash flows is greater than the investment or if the value of the underlying is higher than the exercise price; – the carry option provides the flexibility to wait for the right moment to invest. This is certainly the most used option. In this case, the company can wait before exercising its call option linked to a land or a valuable resource. By remaining attentive to changes in the price of the output, the company will build or exploit a building or a field, respectively. To proceed with a traditional valuation, it is necessary to agree on an implicit assumption: either the investor can immediately undertake the project or definitively renounce it. In other words, he/she cannot wait. However, in this case, if the carrying of the investment decision deprives him/her of a portion of the profits, it may allow him/her to observe the evolution of the environment and choose the most favorable time to invest. The investor then adapts his/her investment choices according to this evolution, which allows him/her to limit the consequences of a possible irreversibility of the investment expenditure. This possibility of carrying provides additional room for maneuver with respect to the NPV. This additional value is calculated by a call option. If the investor exercises his/her right to build or exploit, he/she then appropriates the net present value of the investment project. If he/she decides to carry out his/her investment decision, he/she implicitly assumes that the lost profit due to waiting, having considered the possibility of obtaining new information, is greater than its cost; – the learning option makes it possible to defer a project or investment in the expectation of new information affecting the demand, the amount of the investment or the production costs (customer expectations, regulatory standards or technological developments); – the abandonment option finds its interest in the possibility of stopping the development of an unprofitable activity, which is equivalent to an insurance policy for the company. The latter may decide to interrupt at

6

Investment Decision-making Using Optional Models

any time a project because it is disappointing. The standard valuation methods, consisting of calculating the net present value of future flows at each period and stopping the project as soon as the NPV is negative, ignore, however, the asymmetry linked to the fact that by continuing the project today, we preserve the possibility of abandoning it later if the situation does not improve. The abandonment decision must therefore take into account this asymmetry present in the method by the real options. In fact, this is a put option that makes it possible to either receive the resale amount of the project assets, or cancel the costs related to its maintenance; – the sequential development option has the advantage of being able to divide a project into several stages, at the end of which the company may or may not have the option of continuing the development of the project if the phase that has just ended is considered a success. In fact, projects requiring significant investments are carried out at several time intervals depending on the performance achieved. By deciding to inject (or not) money at the end of each step, the company continues (or does not continue) the development of the project. Each step of the latter generates, consequently, an option that is exercised or not according to the results of the previous step. For example, before commercializing a new drug, companies in the pharmaceutical industry follow a process that begins with the research and development phases. Then, they perform clinical tests and approach; finally, the authorities to have their approval. In addition, by making sequential investments, mining and oil companies adopt this type of approach. After having obtained and analyzed the results of seismic studies and after having conducted the first drilling estimating the quantity of the resource to be extracted, they decide or not to continue their exploitation and production; – the exchange option of inputs or outputs confers the right to have a choice between different factors of production and finished products, by definition substitutable between them, forming part of an industrial process. A company having gas or coal-fired power plants has exchange options to the extent that either consumable is exploited depending on its production costs. In addition, farmers have exchange options on their finished products, because they will choose to grow one commodity over another depending on the season and also the price trends. Finally, car manufacturers who produce different models of vehicles from the same assembly line will adjust the production volume of each of them according to the evolution of demand and the margins obtained;

Risk and Flexibility Integration in Valuation

7

– the option of expanding or reducing production favors the variability of production throughput depending on market conditions that can lead to the temporary cessation of production after having begun by reducing activity. Thus, companies operating oil wells save part of their investment by stopping production, if the gross selling price of this fossil energy is below its extraction cost. This classification is taken up by Smit and Trigeorgis (2004), who stress the importance of the optional approach in choosing to carry out an investment project because of their contribution to managerial flexibility. In fact, the options take into account that decision-making can occur throughout the life of the project to maximize the expected returns while minimizing the likely losses. If, theoretically, the innovative concept of real options for estimating the value of a project proves relevant due to risk and flexibility integration, the question is whether practitioners concretely use this method, which may seem difficult to establish. 1.2.2. Empirical use of real options Graham and Harvey (2001) conducted a survey of 392 CFOs regarding, among other things, the techniques they use to evaluate an acquisition or a project to be carried out. The sample of companies having responded to their questionnaire is heterogeneous. In fact, 26% of them have a turnover of less than $100 million and 42% have a turnover of more than $1 billion. 44% of the companies in the sample have a production activity, 15% are financial companies, 13% are specialized in transport and energy, 11% have a retail activity, 9% belong to the high-tech sector and 8% are geared towards other activities. The median PER of the sample is 15. As a result, the authors consider that companies that meet or exceed this median PER (60% of the sample) are growing. In addition, the indebtedness of the sample is fairly uniform insofar as one-third of companies have a debt to assets ratio below 20%, one-third have the same ratio of between 20% and 40% and the last third have a debt to assets ratio of more than 40%.

8

Investment Decision-making Using Optional Models

Graham and Hervey estimate that companies with a ratio above 30% are in debt. Thus, the tables presented below identify the responses of the CFOs of the companies in the sample to the question: “How often does your company use the following techniques when deciding whether to pursue a project or an acquisition?” Even though the survey reveals that the most used methods in these contexts are, by far, the internal rate of return and the NPV, with nearly 75% of CFOs using them always or almost always, the real options approach is more solicited than the profitability of assets, value-at-risk or even the profitability index. In fact, more than a quarter of the CFOs surveyed say that they always or almost always rely on this method in the investment decision support process, at a frequency equivalent to that of the discounted payback period. According to this study, the frequency of its application is greater in large companies with industrial activity, with moderate growth and debt, managed by executive shareholders and with a low or no dividend distribution rate. Size

% always and almost always

Average

Internal rate of return

75.61

NPV

PER

Small

Large

Growth

No growth

3.09

2.87

3.41

3.36

3.36

74.93

3.08

2.83

3.42

3.30

3.27

Payback period

56.74

2.53

2.72

2.25

2.55

2.41

Barrier rate

56.94

2.48

2.13

2.95

2.78

2.87

Sensitivity analysis

51.54

2.31

2.13

2.56

2.35

2.41

Multiples

38.92

1.89

1.79

2.01

1.97

2.11

Discounted payback period

29.45

1.56

1.58

1.55

1.52

1.67

Real options

26.59

1.47

1.40

1.57

1.31

1.55

Risk and Flexibility Integration in Valuation

Profitability of assets

20.29

1.34

1.41

1.25

1.43

1.19

Value-at-risk

13.66

0.95

0.76

1.22

0.84

0.86

Adjusted NPV

10.78

0.85

0.93

0.72

0.97

0.69

Profitability index

11.87

0.83

0.88

0.75

0.73

0.81

9

Table 1.2. Inventory of responses from the CFOs in the Graham and Harvey’s (2001) sample to the question: How often does your company use the following techniques 1 when deciding whether to pursue a project or an acquisition ?

Dividend payment

Debt

Sectors

Executive shareholders

Low

High

Yes

No

Prod.

Other

Low

High

Internal rate of return

2.85

3.36

3.43

2.68

3.19

2.94

3.34

2.85

NPV

2.84

3.39

3.35

2.76

3.23

2.82

3.35

2.77

Payback period

2.58

2.46

2.46

2.63

2.68

2.33

2.39

2.70

Barrier rate

2.27

2.63

2.84

2.06

2.60

2.29

2.70

2.12

Sensitivity analysis

2.10

2.56

2.42

2.17

2.35

2.24

2.37

2.18

Multiples

1.67

2.12

1.88

1.88

1.85

2.00

1.85

2.04

Discounted payback period

1.49

1.64

1.54

1.62

1.61

1.50

1.49

1.76

Real options

1.50

1.41

1.37

1.52

1.49

1.45

1.40

1.52

Profitability of assets

1.34

1.32

1.40

1.27

1.36

1.34

1.30

1.44

Value-at-risk

0.78

1.10

1.04

0.82

0.95

0.92

0.95

0.86

1 The CFOs answered the question with a score on a scale of 0 (never) to 4 (always). Score 3 represents “almost always”.

10

Investment Decision-making Using Optional Models

Adjusted NPV

0.87

0.80

0.80

0.91

0.78

0.92

0.79

0.99

Profitability index

0.74

0.96

0.81

0.83

0.90

0.76

0.81

0.98

Table 1.3. Inventory of responses from the CFOs in the Graham and Harvey’s (2001) sample to the question: How often does your company use the following techniques 2 when deciding whether to pursue a project or an acquisition ?

The attractive results of this study carried out in the 1990s are not unlike the passion of some researchers of the same decade such as Coy (1999) who anticipate a “revolution of real options”. In fact, given their recent existence – Myers (1977) invented the term “real options” four years after the Black and Scholes (1973) article – it seems, according to Graham and Harvey’s (2001) research results, that the new real options approach has some success. However, not all empirical research goes in the same direction. In 2000, Bain & Company conducted a study of a sample of 451 companies in which only 9% of them said they used real options analysis when evaluating a project. Similarly, 11.4% of the 205 companies in Ryan and Ryan’s (2002) study sample report that they always, often or sometimes, refer to the real options technique. In order to recognize their potential utility, Hartmann and Hassan (2006) attempted to test the application of real options in a given sector – that of the pharmaceutical industry – potentially subject to this type of analysis by the amounts committed to research and development. In fact, the two researchers insist on the notion of risk and flexibility involved in these expenses in the field of health. It is a sort of productivity crisis in this sector contributing to higher R&D costs. Firstly, neurodegenerative diseases and some cancers remain without effective treatment. Then, many traditional pharmaceutical groups fail to integrate newly acquired knowledge – such as genome-related information – into their R&D process. In addition, competition has increased due to patent expiries of leading drugs and the emergence of profitable generics. Finally, the authorities are more cautious in extending their safety requirements taking into account the risks observed with regard to certain drugs already on the market. Their study focuses on each step of the R&D process. 2 The CFOs answered the question with a score on a scale of 0 (never) to 4 (always). Score 3 represents “almost always”.

Risk and Flexibility Integration in Valuation

Research Biological validation

Preliminary development

Chemical optimization

Preclinical phase

11

Later development

Clinical phases 1–3

Recording

Table 1.4. The pharmaceutical process of R&D according to Hartmann and Hassan (2006)

R&D steps

NPV/DCF ROE

Internal Real rate of Scoring options return

Value of net assets

Multiples Other

Research

59%

6%

18%

47%

0%

6%

6%

6%

Preclinical

76%

12%

24%

24%

12%

4%

0%

4%

Clinical phase 1

85%

15%

27%

19%

23%

4%

0%

4%

Clinical phase 2

100%

19%

22%

11%

26%

7%

0%

7%

Clinical phase 3

100%

22%

30%

11%

26%

7%

4%

11%

Recording

96%

21%

29%

8%

21%

8%

4%

13%

Table 1.5. The inventory of the responses of pharmaceutical companies to the evaluation methods used in a project according to Hartmann and Hassan’s study (2006)

Based on a questionnaire submitted in 2004, Hartmann and Hassan analyze, among other things, the evaluation methods used by about 20 pharmaceutical companies. According to Table 1.5, 23–26% of pharmaceutical groups use options when evaluating clinical trials. The fact remains that the most common method remains that of DCF (up to 100% of the sample uses this method in the clinical phase). After addressing the practical use of real options, the question is how valuations are driven and by what types of models. In fact, risk and flexibility integration leads us to consider fundamental parameters that it is essential to relate to each other, as well as the space-time in which they fit.

12

Investment Decision-making Using Optional Models

1.2.3. Paradigms in options The financial option is a derivative product used to build speculative strategies, arbitrage and risk hedges in the event of unfavorable developments – upwards or downwards – in the price of an asset. It is a right to buy (a call) or sell (a put) a financial asset, which is called an underlying, on a given date (European option) or for a given period (American option) at a price known in advance, which is called the exercise price. The price of the option, which is called the premium, is based on the exercise price E, the price of which the underlying is the subject S, the risk-free rate3 r, the maturity or the remaining term to maturity of the option τ and the volatility of the asset4 σ. Black and Scholes (1973) and Cox, Ross, and Rubinstein (1979) have established evaluation methods of the option premium. The former considered it in a continuous-time approach, while the latter demonstrated a discrete-time approach. The existence of a convergence between these two formulas has resulted in extensions such as, among others, taking into account the distribution of dividends5. The intrinsic value (VI) and the time value (VT) make up the price of the underlying: – the intrinsic value represents the value of the contract at the moment “t”. VI of a call = max (S - E, 0) and VI of a put = max (E - S, 0); – the time value corresponds to the surplus of the VI in view of the time remaining to maturity. As long as a probability of exercising the option at maturity exists, investors are ready to pay a surplus to hold the option. This surplus reaches its maximum value when the option is at the money (S = E)6. At maturity, the time value is zero, which implies that the option premium is equal to the intrinsic value.

3 Since the purchase of a call requires a lower investment than the purchase of the underlying, during the contract, the option holder may invest the remaining capital at the risk-free interest rate (r). 4 The volatility of the underlying (σ) is characterized by the annualized standard deviation of the price returns of the underlying asset. 5 The distribution of dividends at a discrete or continuous rate reduces the value of the option. Equivalent to a cash outflow, it encourages the exercise of the option as soon as possible. 6 According to the log-normality of the price of the underlying, the probability that S decreases or increases with respect to E is the same.

Risk and Flexibility Integration in Valuation

13

An option is called in the money when its exercise price is lower than the price of the underlying (for a call) or higher than the price of the underlying (for a put); out the money in the opposite case and at the money if both prices are equal. The value of an option is thus an increasing (call) or decreasing (put) function of the price of the underlying. The probability of a call being exercised increases as E is low. Its value is a decreasing function of E. The probability of exercising a put is all the more important as E is high. Consequently, the put value is an increasing function of E. Cox, Ross and Rubinstein (1979) valuation model is based on the assumption that the value of the underlying asset (shares) follows a discrete-time multiplicative binomial law7. The stock price may, in each period, either increase and go to uS at the end of the first period, or decrease and go to dS8 with the respective probabilities q and 1-q. In other words, the rate of return of the share in each period is either u-1 with the probability q, or d-1 with the probability 1-q. If the value of the asset increases (or decreases), the call premium also increases (or decreases) with the probability q (or with the probability 1-q). t=0

t=1

t=0

uS S

t=1 Cu = max (0, uS-E)

C dS

Cd = max (0, dS-E)

Table 1.6. Representative tree of the Cox, Ross and Rubinstein model (for t = 0 and for t = 1)

Based on the composition of a hedging portfolio P consisting of the purchase of a call of which the premium is C and the sale of H shares, we have: C = . p. C + (1 − p)C , where p = q =

7 Appendix 1. Demonstration of the CRR formula. 8 “θ” for upward and “d” for downward.

and r = (1 + r)

[1.1]

14

Investment Decision-making Using Optional Models

In this case, using a discount rate in continuous time: . E max(S − E), 0

C=e

[1.2]

where St is the value of the underlying asset at maturity. Suppose that a represents the minimum number of upward movements that the share will undergo during the next n periods so that the call is in the money and F(a, n, p) the corresponding binomial distribution function: C = SF(a, n, p ) − Er

F(a, n, p)

[1.3]

where F(a, n, p) is the probability that the share will follow at least a upward movements so that the call is in the money at the maturity date. If the call is in the money, it will be exercised. In other words, F(a, n, p) is the probability that the call is exercised at maturity. If n – namely, the number of periods (or subintervals) between the valuation date and the expiry date – is very high, the stock price multiplicative binomial law follows a log–normal distribution and the Cox, Ross and Rubinstein formula (CRR) converges to that of Black and Scholes: C = SΦ(d ) − Ee

Φ(d )

[1.4]

with: d +

[1.5]

√

d = d − σ√τ Φ(x) =

√

[1.6] e

dt

[1.7]

F(a,n,p) is replaced by Φ(d2), and Φ(d2) is the probability that the call is exercised at maturity. Black and Scholes (1972, 1973) presented a model for evaluating European options. It is established in continuous time according to

Risk and Flexibility Integration in Valuation

15

the principle of log–normality of the price of the underlying S9,10. As part of the valuation of options, S defines a geometric Brownian motion: dS = μSdt + ρSdz

[1.8]

where: – μ designates the expected return of the share (obtained from CAPM); – σ its volatility; – dz = ε dt with ε following a standard normal distribution. Ito’s lemma makes it possible to obtain an expression of a two-variable function: – x which defines an Ito process: dx = a(x, t)dt + b(x, t)dz

[1.9]

– t which represents the time divisible into infinitely many periods dt: dF(x; t) =

+

a(x; t) + b (x; t) +

dt + b(x; t)

dz [1.10]

Noting C(S, t) the premium of the call, we replace in the formula of Ito’s lemma a(x, t) by μS and b(x, t) by σS. So: dC(S; t) =

+

μS + S . σ

dt + σS

dz

[1.11]

The Black and Scholes formula is derived from the constitution of an arbitrage portfolio (risk-free). However, in [1.11], only the term σS dz has a random component. Consequently, assuming that the achievements of dz are identical for the option and for the underlying shares, the portfolio will be risk-free if the terms in dz offset each other, namely: – σS



from the formula of the variation of the option premium;

– σS.dz from the formula of the variation of the price of the underlying shares. 9 Appendix 2. Stochastic differential calculus. 10 Appendix 3. Test of the Black and Scholes formula and return on the log-normal distribution.

16

Investment Decision-making Using Optional Models

It is then necessary to compose the portfolio of a sold option and purchased shares. Let P be the value of the created portfolio: P = −C +

.S

[1.12]

. dS

[1.13]

Since: dP = −dC + +

= −

μS + S . σ

Simplifying by µS. dP = −



− S .σ

dt − σS and µS



dz +

μS. dt + σS. dz

[1.14]

, we obtain:

dt

[1.15]

This risk-free portfolio returns r, homogeneous with dt. As a result: − S .σ

−

dt = −C +

. S r. dt

[1.16]

Simplifying by dt: − S .σ

−

= −C +

.S r

[1.17]

Finally: + r. S

+ S .σ

= r. C

[1.18]

The partial differential equation, the resolution of which results in the risk-neutral universe of Black and Scholes formula11, excludes μ, which corresponds to investors’ risk aversion. Cox and Rubinstein (1985) show that when the number of periods between the valuation date and

11 Appendix 4. Demonstration of the Black and Scholes formula.

Risk and Flexibility Integration in Valuation

17

the maturity date of the option tends to infinity, their formula converges to that of Black and Scholes. The premium C of the call gives: C = SΦ(d ) − Ee ℎ d =

( √

Φ(d ) )

[1.19]

and d = d − σ√τ

[1.20]

where: – S: spot price of the underlying share; – E: exercise price of option; – τ: term to maturity (in years); – r: risk-free rate; – r’: continuous risk-free rate, with: r’ = ln(1 + r); – σ: volatility of the underlying; – Φ(.) : distribution function of the standard normal distribution. C = SΦ(d ) − Ee

Φ(d ) = e

Se

Φ(d ) − E. Φ(d )

[1.21]

Since Φ(d2) is the probability that the call is exercised at maturity, EΦ(d2) is the expected cash outflow at maturity. Since Φ(d1) is also a probability and since S er’τ is the future value of S at the maturity date, the amount corresponding to S er’τ Φ(d1) is the expected value of the cash receipt at maturity, assuming that the call was exercised and the underlying asset (the shares) is immediately sold on the market. Finally, the premium of a call is the current value of the net cash flow expected at maturity. Black (1975) presents an option valuation method by considering the fall of the price of the underlying shares at the time of the payment of the dividend. The holder of a call does not benefit from a right to dividends as long as the option is not exercised. If the underlying distributes a dividend before the maturity date of the option, it is necessary to substitute S by S’ = S - De-r’t’, in the Black and Scholes formula, with r’ = ln(1 + r). Merton (1973) assumes a dividend payment in continuous time at annualized rate q (if a is the discrete rate of return observed, then

18

Investment Decision-making Using Optional Models

q = ln(1 + a)). Considering a distribution of dividends between the date t = 0 and t = T, the share price changes from S to S T. eqt. Correlatively, in the absence of distribution, the price ST is obtained in t = T if the share price in t = 0 is equal to S. e-qt. This implies integrating the effect of distribution of dividends into the original Black and Scholes formula and replacing S by S. e-qt: C = S. e

. Φ(d ) − Ee

. Φ(d )

[1.22]

This result of C is obtained after solving a partial differential equation integrating the continuous payment of dividends at the rate q. The assumption of the constitution of an arbitrage portfolio in the absence of the distribution of dividends previously retained resulted in: −

− S σ

dt = −C +

. S r. dt

[1.23]

If a dividend is paid at the continuous rate q, the holder of the arbitrage portfolio acquires, in addition to the equation on the left of [1.23], the sum qS.dt per share and per time unit dt, i.e. qS.dt for shares. The risk-free portfolio reports the risk-free rate r. So: −

− S σ

dt +

q. S. dt = −C +

. S r. dt

[1.24]

Simplifying by dt, we have: −

− S σ

By grouping

+

qS and

+ (r − q)

q. S = −C +

.S r

[1.25]

. S , and then simplifying, we find:

S+ S σ

= Cr

[1.26]

In this context, as soon as the real options type is identified, its value is calculated using a continuous-time valuation model, such as that of Black and Scholes, or in a discrete-time model, such as that of Cox, Ross and Rubinstein. However, Levyne and Sahut (2008) insist on caution in adopting them as “some projects include several options that may be exclusive”.

Risk and Flexibility Integration in Valuation

19

The authors take as an example the exercise of an abandonment option resulting in the nullity of the value of this temporary stop option. They conclude by stating that the value of the options of a project does not equate to the sum of the options evaluated separately. As for the decision rules, we will retain the adjusted net present value (ANPV) criterion, i.e. the increased net present value (of the value of the real options of the project). Thus, the project is deemed profitable if the ANPV is positive12. In the case of a carry option, the methodology differs to the extent that the value of the real options has to be compared to the NPV. If this is greater than the value of the real options, the project can be completed immediately. Suppose a company plans to invest in the installation of offices in order to develop a new market by offering new services. The proposed project would take place in two stages. The first phase would require an investment of €1,176,000 specifically for the construction and layout of the offices. The NPV resulting from this “first project” is negative, with a value of - €398,000. If we did not consider the second phase of the project, it goes without saying that the latter should be abandoned. Thus, the second step is for the company to hold the possibility of proceeding to a new investment of €2,600,000, in two years, to offer new services. The growth option in question can therefore be exercised if this additional investment is made in two years. Inputs

Correspondence with a financial option to purchase

Value

S or Current value of cash flows

Price of the underlying asset

€2,580,000

E or Additional investment

Exercise price of option

€2,600,000

τ or Time in days

Time remaining until the maturity of the option

730

r or Time value of money

Risk-free interest rate

3%

σ or Standard deviation of cash flow returns*

Volatility (standard deviation) of returns on financial assets

35%

*Obtained from similar investment returns in the same sector.

Table 1.7. Valuation of the growth option

12 Even though the NPV is negative.

20

Investment Decision-making Using Optional Models

Applying Black and Scholes (1973) formulas [1.19] and [1.20] for the call price of a share, the value of this growth option is: €560,000. Consequently, the ANPV of the project is: - 398 + 560 = €162,000. Consideration of the growth option leads to the acceptance of the project. As a risk management tool, investment projects can be valued through options, assuming that managers are able to react to the risks of their environment. Consequently, financial and real options may be subject to an analogy by the method used: an investment project can be considered as a sequence of decisions over time and the risks are taken into account in the valuations. However, even though real options are prized by large companies subject to high volatility in their inputs and/or outputs, the fact remains that their scope is complicated to implement by the restrictive assumptions of which they are the object, which can distance the values obtained from reality. It is finally important that the project asset is correlated with its underlying. After explaining the two approaches for valuing an investment project by real options, I found it interesting to put these models into practice by applying them to the different categories of options that were listed. 1.3. Valuation of investments by real options The valuation of an investment project by real options can be performed either in a discrete-time or continuous-time approach. It is a question here of succinctly putting into practice these two types of methods in the form of examples of growth options, abandonment options, linked options, sequential development options and options for expanding or reducing the activity. 1.3.1. Optional valuation of investments in a discrete-time approach Valuations of investment projects using the discrete-time real options method are based on the model developed by Cox, Ross and Rubinstein (1979).

Risk and Flexibility Integration in Valuation

21

1.3.1.1. Valuation of a growth option Considering an investment project of which the current cash flow value is 2,200 with an investment of 2,500, the NPV is negative and therefore the project should be abandoned. Nevertheless, it presents an opportunity for growth to the extent that, in three years, if the state of the market allows it, it could be extended. If necessary, an increase in cash flows of 50% is expected with a cost of 400. Using the Cox, Ross and Rubinstein binomial model, we design the cash flow tree over a chosen number of periods, regardless of the number of years of life of the option. This allows us to determine the quality of the estimate. Here, we will take six periods13. The parameters of the binomial model are those presented in Table 1.8. S

2,200

T

3

σ

35%

r

3%

No. of periods

6

t (no. of years/no. of periods)

0.5

Adjusted risk-free rate

1.0149

up

1.2808

down

0.7808

Pu

0.4682

Pd

0.5318

Table 1.8. Parameters of the CRR binomial model to value a growth option

13 In order for the binomial model to provide an unbiased estimate, a number of periods close to 100 should be chosen. The values of this model converge to those of the Black and Scholes model when the number of periods tends to infinity. A trick is to perform the calculation of the option for two numbers of periods (e.g. 6 and 7) and to average it to get a price close to that obtained by Black and Scholes.

22

Investment Decision-making Using Optional Models

In the binomial tree that follows, at each period, the value of cash flows increases by 1.280 in the favorable case and decreases by 0.780 in the unfavorable case.

0

0

1

2

3

4

5

6

2,200

2,818

3,609

4,622

5,920

7,583

9,712

1,718

2,200

2,818

3,609

4,622

5,920

1,341

1,718

2,200

2,818

3,609

1,047

1,341

1,718

2,200

818

1,047

1,341

638

818

1 2 3 4 5 6

498 Table 1.9. Binomial tree to value a growth option

For period 1, line 0: 2,200 x 1.280 = 2,818. For period 1, line 1: 2,200 x 0.780 = 1,718. To obtain the call value, it is necessary to start with the terminal value at period 6. The final payment is the maximum between 0 and the revenue related to the expansion of the project. For period 6, line 0: Max (0; 9,712 x 50% - 400). This is to multiply the cash flows from period 6 to the growth rate and subtract the additional cost to obtain the revenue related to the expansion for line 6. Then, we go up the tree in the opposite direction towards period 0. In a neutral-risk case, the value at each node is calculated as the discounted expectation of the two possible values of the option. Assuming the indexes i and j, respectively, for columns and rows, we have: C, = where

,

,

is the adjusted risk-free rate.

[1.27]

Risk and Flexibility Integration in Valuation

23

Thus, for C5.0 (period 5, line 0), we have: C

,

0

=

0.4682 × 9,712 + 0.5317 × 5,920 1.0148 0

1

2

3

4

5

6

737

1,037

1,427

1,929

2,572

3,397

4,456

493

723

1,026

1,416

1,917

2,560

305

476

712

1,015

1,405

163

282

465

700

62

129

271

4

9

1 2 3 4 5 6

0 Table 1.10. Value of the growth option obtained by the discrete-time CRR binomial model

By adding the value of the call option to the NPV, the ANPV obtained is positive. Considering this possibility of expansion, the initial decision not to invest can be modified. Project NPV

−300

Call value

737.06

ANPV

437.06

Table 1.11. ANPV of the project taking into account the value of the discrete-time growth option

1.3.1.2. Valuation of an abandonment option Considering the same investment project as that developed above, but assuming that it can be resold at any time for a value of 1,800, the project is offered flexibility comparable to an American put option. It is then easier to value cash flows with the option, by obtaining an increased current value (ICV), than to value the option alone. To proceed with this valuation, two steps are necessary. Firstly, we must value the option as if it were European

24

Investment Decision-making Using Optional Models

considering that it is exercisable only at maturity. Then, it is necessary to introduce the possibility of reselling the project at any time to give the put option its “American” character. Assuming that the put option is European, the project cash flows and resale value are compared only at the maturity date (period 6). If this value does not exceed 1,800, the company sells its project and cash 1,800. For period 6, line 0: Max (1,800; 9,712) = 9,712. For period 6, line 6: Max (1,800; 498) = 1,800. Then, by moving the tree in the opposite direction towards period 0, in a neutral-risk framework, the value of each node is defined as the discounted expectation of the two possible values of cash flows (CF). The value of CF5.0 (period 5, line 0) is: CF

,

=

0.4682 × 9,712 + 0.5317 × 5,920 = 7,583 1.014 0

ICV 0 1 2 3 4 5

2,445

1

2

3

2,936

3,644

4,622

2,080

2,396 1,861

4

5

6

5,920

7,583

9,712

2,884

3,609

4,622

5,920

2,033

2,326

2,818

3,609

1,761

1,833

1,958

2,200

1,748

1,774

1,800

1,774

1,800

6

1,800 Table 1.12. Increased value of the cash flows of the discrete-time abandonment option (European)

The value of the project can be less than 1,800 at intermediate dates, while the project can be resold at any time. By browsing the decision tree, going this time from period 0 to period 6, the project is sold each time its value is less than 1,800.

Risk and Flexibility Integration in Valuation

25

We notice that this starts at period 3, line 3. However, it goes without saying that once the project is sold, it no longer generates cash flows. Its value becomes zero. 0

ICV 0

2,453

1

1

2

3

2,938

3,644

4,622

5,920

7,583

9,712

2,095

2,400

2,884

3,609

4,622

5,920

1,885

2,041

2,326

2,818

3,609

1,800

1,847

1,958

2,200

0

1,800

1,800

0

0

2 3 4 5

4

5

6

6

0 Table 1.13. Increased value of the cash flows of the discrete-time abandonment option (American)

The increased value of cash flows with the put option (ICV) is 2,453. It is higher than that calculated by considering a European put14. The project ANPV (American put) is shown in Table 1.14. Project NPV ICV

−300 2,453.14

ANPV

−46.86

Value of the American put

253.14

Table 1.14. Value of the project with the discrete-time abandonment option (American put)

The ANPV15 is negative even considering the put16 value. The project cannot be undertaken on the basis of this criterion. Moreover, considering the project by integrating a European put does not change the situation. The 14 This result is in line with the theoretical principle that the price of an American option is always greater than or equal to the price of a European option because of the possibility of exercising it at any moment. 15 ANPV = ICV – Investment. 16 Put value: ANPV – NPV.

26

Investment Decision-making Using Optional Models

difference between the two types of options is small (3.4%). To illustrate this, Table 1.15 shows the ANPV of the project with a European put. Project NPV

−300

ICV

2,445

ANPV Value of the European put

−55 244.80

Table 1.15. Value of the project with the discrete-time abandonment option (European put)

1.3.1.3. Valuation of a linked option It is possible to consider that a company combines different options within the same project. After being identified, the options of the project in question must be valued together knowing that certain opportunities are exclusive to each other. Suppose the company resells its investment project to the extent that profitability is too low. As a result, the company cannot subsequently benefit from growth opportunities. Using the assumptions of the two examples above (those of the growth option and those of the discrete-time abandonment option), the project has a negative NPV of −300 and confers the possibility for its decision-makers to exercise a growth option in 3 years (consisting of a 50% cash flow increase for a cost of 400) and a resale option for 1,800 at any time. The project must be valued in two stages integrating both options. Firstly, the assumption is made that the put option is European. At the final date chosen (period 6), we choose the situation that generates the largest cash flows depending on the situation where the project is kept as is, where it is extended or where it is resold. For period 6, line 0: Max (9,712; 9,712 × 1.5 – 400; 1,800) = 14,168. For period 6, line 6: Max (498; 498 × 1.5 – 400; 1,800) = 1,800. Then, by moving the tree in the opposite direction towards period 0, in a neutral-risk framework, the value at each node is defined as the discounted expectation of the two possible values of the cash flows (CF): CF

.

=

0.4682 × 14,168 + 0.5317 × 8,481 = 10,980 1.014

Risk and Flexibility Integration in Valuation

ICV 0

0

1

2

3

4

5

6

3,116

3,926

5,051

6,551

8,492

10,980

14,168

2,489

3,046

3,871

5,025

6,540

8,481

2,069

2,406

2,963

3,833

5,014

1,830

1,982

2,281

2,900

1,748

1,774

1,800

1,774

1,800

1 2 3 4 5 6

27

1,800 Table 1.16. Increased value of the cash flows of the discrete-time linked option (European)

Since the value of the project can be less than 1,800 at intermediate dates, we go up the tree from period 0 to period 6, and the project is dropped each time its value is less than 1,800. We start at column 4, line 4, and post-sale cash flows are eliminated since they are not collected. 0

ICV 0 1 2 3 4 5

3,123

1

2

3

4

5

6

3,928

5,051

6,551

8,492

10,980

14,168

2,502

3,050

3,871

5,025

6,540

8,481

2,090

2,413

2,963

3,833

5,014

1,864

1,996

2,281

2,900

1,800

1,800

1,800

0

0

6

0 Table 1.17. Increased value of the cash flows of the discrete-time linked option (American)

The ANPV of the project is positive thanks to this linked option. The impact of the growth opportunity on the total value of the project

28

Investment Decision-making Using Optional Models

is greater than the profit obtained since the possibility of reselling the project. Project NPV ICV

−300 3,123.43

ANPV

623.43

Value of both options

923.43

Table 1.18. Value of the project with the discrete-time linked option

The value of the two linked options (923.43) is less than the sum of the call option corresponding to a growth and sale option corresponding to an American type abandonment option (737.06 + 253.14 = 990.2), because the abandonment of the project at intermediate dates limits the possibilities of benefiting from a trend reversal, that is to say, growth opportunities. 1.3.2. Optional valuation of investments in a continuous-time approach Valuations of investment projects using the discrete-time real options method are based on the model developed by Black and Scholes (1973). 1.3.2.1. Valuation of a growth option and an abandonment option and empirical equivalence between the binomial model and that in continuous time Using the same assumptions as the examples presented to value a discrete-time growth option and an abandonment option, we propose to find an equivalent call value in the first scenario and put value in the second scenario17 by applying the Black and Scholes model (in continuous time).

17 And beyond the ANPV.

Risk and Flexibility Integration in Valuation

29

For the growth option, the parameters and the value of the continuous-time call option are those presented in Table 1.19. Current value of project cash flows

2,200

Current value of cash flows related to expansion S

1,100

Cost of investment E

400

Life of the call (year)

3

Risk-free rate

3%

Volatility (standard deviation of S)

35%

CF increase

50%

d1

2.1203

d2

1.5141

F(d1)

0.9830

F(D2)

0.9350

Call value

739.50

Table 1.19. Value of the growth option obtained by the continuous-time B&S model

As cash flow growth in the third year is 50%, the underlying S is worth 50% of the initial value of the cash flows of the project (50% × 2,200 = 1,100). Moreover, volatility can be determined from the historical variance of similar projects of the company in question, from the estimation of the distribution of the NPV carried out either since several simulations of the NPV (by changing the parameters) or from distributions of model parameters or from the variance of returns of companies having developed the same type of project. It is important to emphasize that the call value is similar to that obtained with the CRR binomial model (in discrete time). In fact, with seven iterations, a value of 737.06 was obtained. The ANPV becomes positive thanks to the call value which compensates for the negative NPV of the project. Based on this criterion, the company can undertake the project.

30

Investment Decision-making Using Optional Models

Current value of project cash flows

2,200

Investment

2,500

NPV (1)

−300

Call value (2)

739.50

ANPV (1 + 2)

439.50

Table 1.20. ANPV of the project taking into account the value of the continuous-time growth option

For the abandonment option, the parameters and the value of the continuous-time put option are those presented in Table 1.21. Current value of the project S

2,200

Receipts in case of abandonment E

1,800

Duration to abandon (year)

3

Risk-free rate

3%

Volatility (standard deviation)

35%

d1

0.7826

d2

0.1764

F(d1)

0.7831

F(d2)

0.5700

F(-d1)

0.2169

F(-d2)

0.4300

Put value

230.13

Table 1.21. Value of the abandonment option obtained by the continuous-time B&S model

Assuming that the project can be sold for 1,800 at maturity, it implies a valuation of a European-style put by the Black and Scholes model: P = Ee

Φ(−d2) − S. Φ(−d1)

The difference in value of the put obtained by the binomial model and the Black and Scholes model varies by approximately 6%. To reduce this gap, the number of periods in the CRR discrete-time approach should be increased.

Risk and Flexibility Integration in Valuation

Current value of project cash flows

2,200

Investment

2,500

NPV (1)

−300

Put value (2)

230.13

ANPV (1+2)

−69.87

31

Table 1.22. ANPV of the project taking into account the value of the continuous-time abandonment option

As with the binomial approach, the project should not be undertaken on the basis of this criterion. 1.3.2.2. Valuation of a sequential development option The purchase of a patent is an investment in a project, even though it is not immediately viable. In fact, the patent gives the right to the company to develop a product and market it. But the company will not exploit it if the current value of the cash flow expected from the sale of the product does not exceed the cost of development. And if this never happens, the company will bury the patent and no additional cost will be incurred. Note E = the product development cost and S = the current value of the expected cash flows. The patent gain is max (0, S − E), and the patent can be considered as a call on future incoming cash flows. δ is the dividend yield or the annual cost of carrying the project, since each year of delay results in a year of less value creation of cash flows. The Merton formula, including the dividend yield, is to replace S by S.e-δτ: .

C = S. e

. Φ(d ) − Ee

Φ(d ) and d =

√

[1.28]

Suppose a pharmaceutical company wants to buy a patent to manufacture and sell a new drug for 10 years. If the drug is produced today, the estimate of the current value of the incoming cash flow is 800, while the development cost of the drug for its commercialization is 1,000. The risk-free rate is 2% and the expected cash flow volatility, based on industry benchmarks, is 40%. The expected cost of delay is 1/10 = 10%. Based on the NPV traditional approach, the project is not worthy of interest since the NPV is - 200. However, due to the volatility of industry earnings

32

Investment Decision-making Using Optional Models

assumed at 40%, incoming cash flows are likely to increase in the future based on current estimates. Based on the option pricing model, its value is 65. S

800

q

10%

S.exp(-qt) E discrete r continuous r Volatility

294 1,000 2% 1.98% 40%

Valuation date

01/04/2015

Maturity date

01/04/2025

Duration (year)

10.01

d1

−0.178

d2

−1.443

F(d1)

0.429

F(d2)

0.074

Call value

65.19

Reimbursement

0

Table 1.23. Value of the pharmaceutical patent after the investment project

1.3.2.3. Valuation of an option of expanding or reducing production It is possible to consider that the valuation of an oil well as part of its concession cannot be based on a DCF approach given the high uncertainty on future cash flows resulting from the high volatility of the oil barrel. Concession can be valued as a portfolio of options for opening/closing the well according to its profitability prospects over given periods. For a 10-year concession, it may be considered to compare at the beginning of each year the selling price of the barrel at its unit cost. Thus, the concessionaire has 10 options for putting into operation. In the following example, we also assume the price per barrel at €104, the production costs at €80, the volatility at 80%, the risk-free rate at 3% and the installed capacity of 2 million barrels per year. Table 1.24 details the calculation of the call options available to the company in order to value the concession of the oil well:

80%

80%

3%

3%

σ

discrete r

continuous r

3%

3%

80%

80

104

5

3%

3%

80%

80

104

6

3%

3%

80%

80

104

7

3%

3%

80%

80

104

8

3%

3%

80%

80

104

9

2,000,000 barrels/year

20,000,000

13,026

Installed capacity

Production over the entire period (10 years)

Value of the concession (€M)

54

0.3888

0.8024

−0.28

0.85

2.01

62

0.3299

0.8281

−0.44

0.95

3.01

68

0.2868

0.8504

−0.56

1.04

4.01

73

0.2527

0.8696

−0.67

1.12

5.01

77

0.2248

0.8859

−0.76

1.20

6.01

Table 1.24. Example of the valuation of an oil well concession

651

43

0.4852

0.7779

−0.04

0.77

1.01

Sum of option premiums

24

1.0000

C

1.0000

6.25

d2

F(d2)

6.29

d1

F(d1)

0.00

τ

80

0.2012

0.8998

−0.84

1.28

7.01

83

0.1810

0.9119

−0.91

1.35

8.01

86

0.1634

0.9223

−0.98

1.42

9.01

13/03/2015 13/03/2016 13/03/2017 13/03/2018 13/03/2019 13/03/2020 13/03/2021 13/03/2022 13/03/2023 13/03/2024

3%

3%

80%

80

104

4

maturity date

3%

3%

80%

80

104

3

12/03/2015 12/03/2015 12/03/2015 12/03/2015 12/03/2015 12/03/2015 12/03/2015 12/03/2015 12/03/2015 12/03/2015

3%

3%

80%

80

104

2

valuation date

3%

3%

80

80

104

104

1

E

0

S0

Rank of the option

Risk and Flexibility Integration in Valuation 33

34

Investment Decision-making Using Optional Models

– The first option may be exercised immediately if the spot price S of the barrel is higher than the unit cost. This first option has no time value, and its intrinsic value is 104 − 80 = $24. – The second option may be exercised in one year if the price per barrel exceeds $80. – ... – The tenth option may be exercised in nine years if the price per barrel is $80. The value of the concession will be the sum of the value of these options multiplied by the total installed capacity (20 million barrels). The simulation presented above can be refined by linking spot prices to futures prices. Hedging transactions or investment strategies may be carried out by adjustments resulting from stockpiling or the arbitrage relationships between the physical market and the futures market. The analysis of these relationships makes it possible to understand that the level of the carrying18, on a futures market of commodities, is limited to the storage cost of the merchandise between the current date and the expiry of the contract. The level of deferral19 is determined by the price that buyers are ready to agree on to obtain the merchandise. In the presence of surplus stocks, prices cannot be offset. If this is the case, it would become profitable and risk-free to sell stocks in the spot markets and, simultaneously, buy them back in the futures market20. The multiplication of these arbitrage transactions would lead to a fall in the spot price, as a result of the massive sales of physical stocks, and to a rise in the futures price following the purchase of contracts. These transactions would only cease when the futures price would be higher than the spot price, by an amount representing the storage cost. The existence of surplus stocks thus leads to a carrying situation. This logic is correlatively found in the presence of carrying situations. Reverse cash and carry transactions are all the more unlikely as the shortage is recognized: operators have actually no interest in disposing their stocks as long as they anticipate a further increase in the spot price.

18 Positive difference between the future price and the spot price. 19 Negative difference between the future price and the spot price. 20 Reverse cash and carry transaction.

Risk and Flexibility Integration in Valuation

35

According to Keynes (1930), the stocks at the origin of the carrying situations in the futures markets are surplus stocks (redundant stocks) constituted following an error of appreciation. As soon as they appear, mechanisms for their elimination are established: the price of merchandise decreases until the increase in consumption or the decrease in production is sufficient to absorb them. Storage costs (costs of carrying) differ according to the commodity considered. They are based on various variables, such as deterioration and obsolescence costs, storage costs and insurance premiums, the risk of the monetary value of products, financial expenses and so on. As long as the storage capacities are not saturated, the costs are stable. Consequently, so is the level of carrying21. The storage costs theory, originally set in the context of surplus stocks, does not, a priori, explain the relationship between the futures price and the spot price when stocks become scarce. In fact, if the futures price is equal to the spot price plus positive storage costs, how can it become less than the spot price? In addition, if the futures price does include the storage cost of the merchandise, an operator, holding stocks and selling in the futures market his/her merchandise in a carrying situation, does not have to bear the storage cost. How then explain that in an offset situation, stocks are held without being hedged by a forward sale, even though this transaction is expensive? The concept of convenience yield, introduced by Kaldor (1939), provides an answer to these different questions. It is developed in the second part of the literature review. Black and Scholes (1973) and Cox, Ross and Rubinstein (1979) models are fundamental and considered as paradigms. However, the fact remains that the assumptions that constitute them are restrictive and perfectible. 1.4. Option model extensions by incorporating new parameters (Levyne and Sahut 2008) Authors have proposed incorporating new parameters into existing models to provide greater precision and practicality. Thus, while Black and Scholes retain constant volatility, other authors determine stochastic 21 In the oil market, when onshore storage capacities have reached saturation, oil is stored at sea. This storage method is much more expensive than the previous one; the level of carrying can have two levels: the first one is determined by the onshore storage costs, and the second one by the storage costs at sea.

36

Investment Decision-making Using Optional Models

volatility aimed at anticipating the evolution of future volatilities. The effect of volatility smile described by Aboura (2006) focuses on changes in implied volatility based on the position of the exercise price relative to the price of the underlying. Hull and White’s (1987) discrete-time model predicts the future volatilities that the underlyings will take. In addition, transaction costs or, more generally, exogenous phenomena or the payment of dividends can interact on the value of the project. In fact, Leland (1985) includes the effect of transaction costs in the volatility of the underlying. Boyle and Vorst (1992) define a hedging strategy compatible with this type of costs, and Lapied and Kast (1995) construct a model integrating the possibility of separating from its position at intermediate dates. Merton (1976) is a model with jumps in which discontinuities in price trajectories are integrated. Finally, Roll et al. (1981) incorporate a known discrete dividend into a continuous-time model. 1.4.1. Stochastic volatility Black and Scholes assume, in their model, constant volatility. However, the study conducted by Aboura (2006) leads us to consider that there is an effect of “volatility smile”. The latter results from the observation of changes in implied volatility with the exercise price of the options. In fact, in the case where the option is at the money, the implied volatility is the lowest; in cases where the option is in the money or out the money, the implied volatility is the highest. This is presented in the form of a smile. In addition, the smile is more pronounced in the case of puts, because stakeholders are more sensitive than for downside risk calls, requiring a higher premium. The behavior of arbitragers who focus on volatility challenges the reliability of the Black and Scholes model. They have at their disposal three types of volatility (including mixed volatility22). Historical volatility is defined as the variability of past returns on a financial asset23. In other words, this is the standard deviation of the returns of the underlying, calculated over a period of time expressed as an annualized percentage.

22 Mixed volatility results from a combination of historical and implied volatilities developed later and has no real theoretical base. 23 Since historical volatility is calculated on the basis of fluctuations, it has no forecast content, unlike implied volatility.

Risk and Flexibility Integration in Valuation

37

In practice, the number of observation days retained is between 45 and 6024. It is interesting to note that intraday volatility is disregarded, since the analyst generally relies on closing prices. Moreover, historical volatility proves intrinsically unstable to the extent that the probability distribution associated with the price movements of the underlying asset is non-stationary25. Implied volatility is defined as the determination of the standard deviation of the returns of the underlying asset in the resolution of the valuation model formula knowing the price of the option. In theory, historical and implied volatilities must be identical, but reality through the law of supply and demand varies the prices of the premiums of the options. Implied volatility is therefore calculated from option prices. Its value results from the equality obtained between the price of the effective option on the market and the value calculated using an evaluation model such as that of Black and Scholes. In this case, we rely on the Newton–Raphson algorithm (Gibson 1993), initially giving a value to the volatility and then seeking by iterations a refined result minimizing the difference between the observed price of the option and its price derived from the model in question. The resulting implied volatility is expected to express the future volatility of the underlying over the life of the option26. In absolute terms, it conveys uncertainty about the future foreseen by market operators. High implied volatility can be interpreted as the result of a price instability that leads investors to pay more for an option. Conversely, options will be cheaper if the emerging trend seems distinct. In other words, implied volatility reveals consensus or divergence on future prices between market participants. However, De la Bruslerie’s work (1988) goes against this reasoning. According to the researcher, implied volatility is not a sound estimator of future price volatility. Consequently, discrete-time stochastic volatility models focus on predicting future volatilities of the underlyings. Continuous-time volatility

24 There is a method problem in determining the number of days to consider: the larger the number of days, the greater the influence of the past. Conversely, if the number of days retained is small, the sample is not representative. 25 For example, historical volatility obtained from a calm period is not useful for estimating future volatility within a crisis period. 26 This assumes that market participants determine the price of the option using the same model.

38

Investment Decision-making Using Optional Models

models are used to evaluate options27. Hull and White (1987) are part of this dynamic by introducing two independent Wiener (Wy and Wv) processes, and a volatility following a Cox, Ingersoll and Ross-like dynamic (1978) oscillating approximately 0: = μdt + σdW

[1.29]

dσ = φσ dt + δσdW

[1.30]

where: – µ: expected return of the security; – σ: volatility of the security; – φ: expectation of the variance of the instantaneous returns of the security; – δ: volatility of the variance of the instantaneous returns of the security. Hestion (1993) exposes the process of volatility as a square-root-in-shock process correlated with the price of the underlying28: = μdt + V dW dV = κ(θ − V )dt + σ cov dW , dW

= ρdt

[1.31] V dW

[1.32] [1.33]

27 Several generations of models have succeeded each other: Geske’s (1979) first-generation deterministic models and Cox and Ross’s (1976) constant elasticity of variance (CEV) models emphasize volatility as a function of the price of the option; the second-generation or stochastic pure volatility models such as that of Hull and White (1987), Stein and Stein (1991) or Heston (1993) focus on volatility following a fully fledged exogenous process; finally, third-generation mixed models such as that of Bates (1996), Bakshi et al. (1997) and Duffie et al. (2000) incorporate jumps and stochastic volatility. The latter have a real theoretical interest. However, they do not propose an analytical solution, and their complexity limits the empirical tests. 28 In addition, it proves that the correlation between volatility and the support price is essential to obtain asymmetry in the distribution of returns.

Risk and Flexibility Integration in Valuation

39

where: – µ: expectation of the rate of return of the security; – κ: speed of return to the average level; – θ: average long-term level of variance; – σv: coefficient of variation of the instantaneous variance; – ρ: correlation coefficient between price and volatility of the security. 1.4.2. Transaction costs and models with jumps Arbitrage theory and the assumption that financial markets are perfect govern the valuation of derivative assets using conventional methods. Frictions such as transaction costs that annihilate or delay perpetual portfolio adjustments – which are necessary to take into account in hedging risks – are not part of the assumptions used. In fact, the valuation model of a call with the Black and Scholes method requires the creation of a risk-free portfolio consisting of shares and an option on the same shares sold. It is necessary to continuously readjust the number of shares held in order to manage as best as possible a neutral delta position, that is to say, to obtain a balance from the random variations of the option price. The incorporation of transaction costs makes the continuous portfolio adjustment very expensive. Calculated on the basis of the spread between the sell and the ask price at which the shares are traded, they are larger when the hedging errors are small and the periods are short. Hedging strategies are thus more difficult to conduct, and the arbitration supposed to be conducted by the agents is even more topical. In order to resolve this contradiction, Leland (1985) designed an optimal duplication model of an option, incorporating the effect of transaction costs into price volatility. After being calculated, this volatility is incorporated into the Black and Scholes model to determine the value of the option. However, Leland model remains debatable, in that replication portfolio adjustments are not endogenous. Even though the readjustments of the portfolio are continuous, the costs of recomposition are infinite, and therefore the call value should tend towards that of the underlying. Bensaid et al. (1992) proved that a strategy of super-replication of an option can be more efficient than a strict strategy of replication, insofar as there is a compromise between the precision of a hedging and its cost. The longer the hedging strategy in terms of time, the higher the transaction costs. This strategy must be stopped from the moment when the gain in the accuracy of the hedging is neutralized

40

Investment Decision-making Using Optional Models

by the transaction costs incurred. It is then possible to define a domain within which it is not advantageous to recompose its portfolio. In this context, Dumas and Luciano (1988) focused on considering that an option can only be evaluated within an optimal portfolio adjusted in continuous time by the stochastic optimal control technique. Merton (1989), however, questioned the conclusions of the two researchers because of the complex and unworkable nature of the proposed solutions. By defining a hedging compatible with proportional transaction costs, Boyle and Vorst (1992) determined a range of bid-ask prices within which an option can be traded without an arbitrage opportunity. Although they partly solved the problem described above, there is still a more general difficulty. In fact, Lapied and Kast (1995) demonstrated a paradox concerning the price rule. To answer this, they formulated a model that includes the possibility of selling the position at intermediate dates. Discrete-time models make it possible to choose the number of recompositions. The question then is which optimal number to choose in the sense that the bigger it is, the more the relative ranges are explosive. To achieve this, it seems possible to empirically choose a number of periods such that the relative ranges obtained converge to those observed on the market. Moreover, Merton (1976) integrates discontinuities in price trajectories by proposing a modeling by a diffusion with jumps using a geometric Brownian motion and a Poisson process. In fact, the Gaussian model makes it possible to obtain formulas to evaluate certain derivative products, but without incorporating the non-stationarity of the variance of the price returns nor the sudden variations or peaks in these prices. Empirically, there are jumps in market dynamics. Jorion (1988), in particular, tried to prove it on the exchange rate indexes. Levy’s processes establish models summarizing the presence of jumps, leptokurtic distribution and market incompleteness. They also include crisis phenomena, stock market crashes, risks of default or bankruptcy. Jump spreads can also be used to apply models with numerical resolution algorithms using Monte Carlo simulations to calibrate, for example, volatility surfaces. Cont and Tankov (2004) managed to statistically and analytically exploit this type of model.

Risk and Flexibility Integration in Valuation

41

1.4.3. Option pricing If there are two main types of models to price an option – continuoustime on the one hand, from the Black and Scholes model, and discrete-time on the other hand, based on Cox, Ross and Rubinstein’s work – it is necessary to consider two major obstacles: to retain the most relevant model and to choose good inputs to introduce into the model in question. The choices mainly result from the type of option considered – whether it is an American or a European option – of the nature of the underlying and its characteristics. Options on equity index futures are widespread during portfolio allocation and more commonly as a hedging instrument. At the exercise date, the call buyer (respectively, the put seller) is in a long position (respectively, in a short position) of futures contracts. These do not lead to the delivery of futures, but to a settlement in cash. The amount cashed or disbursed is the difference between the value of the spot index and the value of the future on the index. This type of options for index futures generally corresponds to options on an asset generating a continuous dividend rate (without considering whether the underlying pays a continuous or discrete dividend). In this context, the Black model applied to European options with its extensions is the most judicious to use to value this type of options. Share option participants have the choice between American-style short-term options and European-style long-term options. In addition, a dividend issue affects the choice of a model to consider. In the case of European options, the adjusted Black and Scholes model (taking into account a continuous dividend rate) is the one applied by practitioners. In the case of American options, professionals focus on two models: a continuous-time model derived from Roll, Geske and Whaley’s work (1981), incorporating a known discrete dividend, and the discrete-time CRR model, which takes into account discrete dividends (the iterative nature of the calculation is slow, but allows us to check for each intermediate date between the valuation date and the maturity of the option a possible early exercise of the option). The value of the underlying S is then: S−D×e where D is the amount of the dividend paid on the date tD.

[1.34]

42

Investment Decision-making Using Optional Models

Equity index options act like stock options. The index point is assigned a certain value in national currency. It is assumed that the index follows a general geometric Brownian motion, while each share composing the index follows a specific Brownian motion. To value the option, the index is equivalent to a share S distributing a continuous dividend throughout its life. The distribution rate q results from the average dividend yield for payments that occur during the term of the option. For European-style options, their price is calculated using the Black and Scholes model and taking into account a continuous dividend. The value of the underlying S is then: S ×e

[1.35]

For American options, the model chosen will be more than that of CRR extended to the consideration of discrete dividends distributed on each share making up the index during the life of the option. Regarding the inputs, it is necessary to take into account the value of the underlying, as well as the elements likely to modify its value between the initial date and the date of maturity of the option. Although initially suitable in some models, dividend assumptions should be considered pending the confirmation of the exact amount by the companies or tax credits to which the holding of securities gives entitlement. In addition, implied volatility reflects financial market expectations by taking into account the magnitude of future changes in the underlying. The Black and Scholes model assumes constant volatility while, in reality, the risk to be considered is different, depending on how the underlying moves. The volatility smile that this implies can be reproduced by models like GARCH, but it is more calculated by the investors according to the anticipated risk on the underlying. It is therefore essential to associate the level of implicit volatility from which these prices are established if we want to compare the prices present on a market by market-makers. Finally, the Black and Scholes model assumes that the interest rate is constant, regardless of the maturity of the option. However, in practice, the chosen rate is selected on the yield curve depending on the maturity of the option. To estimate volatility for real options, there are three methods: – the cash flow method is based on their variability. It is similar to that of historical volatility by substituting the prices observed on the market by the estimated cash flows (CF). The calculations are as follows:

Risk and Flexibility Integration in Valuation

- the returns:

=

43

;

- the logarithm of the returns: Ln Rt; - the yield spread relative to the average (ERM); - the ERM squared: deviation²; - the sum of the ERM squared: the sum of the values of “Deviation² between the initial date and the maturity date”; - the number of observations; - the unbiased average of squared deviations (ME); the year

- the estimated volatility: ME x (Number of periods)Period/Number of periods in .

The simplicity of this method has two flaws, namely that negative cash flows should not be inserted because of the logarithm and a limited number of values are required: – the proxy approach is an indirect method to find the “true value” of volatility. The proxy may concern a project similar to the level of performance and risk profile already achieved. It is also possible to rely on changes in the returns of companies that are in the same industry as the project to be valued through the calculation of their historical volatility. However, it can be considered that a company is not limited to a project that listed companies have a certain leverage effect because of their debt, which amplifies their risk and impacts their stock prices, and that the chosen calculation period often leads to bias. For example, during a stock market crisis, the historical volatility of companies (financial risk) is not necessarily correlated with the volatility of their project; – the Monte Carlo method is a technique identical to that of the NPV. The objective differs, however, insofar as it is a question of apprehending the most probable value for volatility. The method is based on simulations of the project parameters and then the average and the standard deviation of volatility, in order to define possible profiles for cash flows. It is difficult to proceed to interpretations, because only the average value of volatility is retained. However, the standard deviation can then be used to analyze the sensitivity of the value of the real option to fluctuations in volatility.

44

Investment Decision-making Using Optional Models

1.5. Conclusion The valuation of projects by real options is based on the same models as for financial options. After the parameters analogous to the Cox, Ross and Rubinstein discrete-time approaches (1979) and/or the Black and Scholes continuous-time approaches (1973) have been defined, it is therefore possible to value the seven categories of real options determined by Amran and Kulatilaka (1999) related to investment projects. However, the assumptions that constitute these models are perfectible. In fact, the consideration of stochastic volatility instead of constant volatility, the integration of transaction costs and the payment of dividends are supposed to improve the value obtained from the option premium, namely the value of the project. In the case of financial options, the underlying asset and the option are generally quoted, while in the case of real options they are not. The underlying asset of a financial option is a financial asset, such as shares in companies, bonds or the foreign exchange value of a currency, while it is a real asset for real options. They make it possible to highlight the value that is integrated in the traditional calculations of DCF. They make it possible to increase the NPV thanks to the time premium recognized by the optional valuation models. In other words, their advocates believe that a premium must be paid on DCF estimates. Real options are therefore useful for: – budgeting for an investment: - when the investor has a deferred option; - when the investment provides growth potential; - when the members of a joint venture have an option to abandon the project; – valuation: - of a patent; - a concession of oil well, forest, gold mine. Four years after the article by Black and Scholes (1973), Myers (1977) argued that growth opportunities can be considered as real options, the value of which depends on future investment. This suggests that the value of a company can be broken down by the value of assets in place and the value of

Risk and Flexibility Integration in Valuation

45

growth options. Copeland and Antikarov (2001) insist that a real option is a right and not the obligation to make a decision at a known price, equivalent to the exercise price, for a specified period of time that corresponds to the life of the option. This decision may concern, for example, the delay of an investment, the expansion, the reduction or the abandonment of an activity. Finally, Luenberger (1998) stated that any process of taking control of an activity can be analyzed as a series of real options. Trigeorgis (1993a) stated that real options involve discretionary decisions or rights to acquire or exchange an asset at a predetermined price. In this context, the financial literature proposes to combine different types of real options related to investment projects. In addition, it defines the parametric specificities of the different categories of real options by offering models for each of them and inspired by the paradigms of the financial options.

2 Optional Modeling of Investment Choices and Surplus Value Linked to the Option to Invest

2.1. Introduction In most cases, a valuation by the real options allows us to include the value of flexibility. From a practical point of view, the option valuation model is generally used by applying the Black and Scholes formula (1973), which corresponds to the solution of a stochastic differential equation. However, if the company has an option to defer its investment to infinity, the = 0 is not time premium does not decrease gradually. In other words, consistent with the Black and Scholes model. In this case, the Dixit and Pindyck formula (1994) must be considered. In addition, the notion of flexibility attached to an investment project may justify the combination of several option categories. Trigeorgis (1993b) shows that the interactions between different options lead to the conclusion that the added value of an additional option to other real options, exercisable within an investment project, is generally less than its isolated value. Under these conditions, the optional valuation of projects that include significant research and development expenses leads to the consideration of specificities. Moreover, the valuation of the optimal moment to exercise a real option is the real challenge relative to any investment decision. Thus, Mauer and Ott (1995), who are interested in the abandonment option, just like

Investment Decision-making Using Optional Models, First Edition. David Heller. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Investment Decision-making Using Optional Models

Damaraju et al. (2015) who focus on the analysis of investment and disinvestment decisions within joint ventures and alliances, stress the reluctance of companies to exercise their options as the information level is not sufficient. Within the natural resources sector, the investment decision largely depends on the uncertainties of the production policy. The highly volatile underlyings, the opening and closing costs of non-negligible concessions and the presence of convenience yield are all parameters that Brennan and Schwartz (1985) then Siegel et al. (1985) take into account in their respective models. They consist of defining an optimal policy regarding the decision to invest, which can also be influenced by changes in strategic orientation such as the debt policy and possible diversification of activities. In this context, the financial literature reveals that the more companies diversify, the smaller the growth options. In fact, the agency problems and the prospects for growth in the long term, less important in diversified companies, alter the value of options linked to future opportunities. For this reason, Long et al. (2004) explain that companies with growth options have incentives to delay their decision to invest. In addition, growth options may arise from interactions between investment and financing decisions. In fact, according to Childs et al. (2005), depending on the structure of the debt of the company, agency conflicts would encourage shareholders to adopt different action plans. To be in line with their optional exercise strategy, the holders of the capital of the company would have an interest in overinvesting to transfer creditors’ wealth or in underinvesting to avoid the development of wealth for the benefit of creditors. Finally, growth options may justify takeover transactions. In fact, beyond the considerations of power and executive compensation1, acquisition strategies would allow the buyer to benefit from growth options of the target who would be unable to find ways to implement them. Thus, Smith and Triantis (1995) evoke the common development biases of which two companies can benefit thanks to a double contribution: that of the resources of the initiator of the offer, on the one hand, and that of the real options of the target, on the other hand. 2.2. Framework of optional interactions and option to develop an investment project Trigeorgis (1993) examines the nature of the interactions between different types of real options – the carry option, the abandonment option, 1 Reasons mentioned in Chapter 1.

Surplus Value Linked to the Option to Invest

49

the contract option, the exchange option and the expansion option – by considering the notion of flexibility relative to the valuation of an investment project. He identifies situations in which interactions between options may be weak or high, negative or positive. These interactions depend in particular on the nature of the options involved in the study and the exercise price. He demonstrates that the added value of an additional option (within a project that already has certain options) is generally lower than its value taken in isolation. In addition, this phenomenon is accentuated as different types of options are considered. He concludes that financial analysts’ valuation errors due to the exclusion of optional features in investment projects may be weak. Dixit and Pindyck (1994) propose an optional valuation model that can both determine the right time to invest and help support decision-making in itself. The maturity date of such a hold option is designed to be deferred indefinitely. Thus, to invest in a project, the waiting time, which is continuous, must lead to choosing the optimal moment. The latter is the date on which the sum of the expected and determinable future discounted cash flows reaches a critical value. Dixit and Pindyck consider, therefore, that these cash flows define a geometric Brownian motion. Then, instead of considering the cash flow of the investment project as a variable, Dixit and Pindyck (1994) replace it with the price of the manufactured product. In addition, they assume the existence of production costs and try to obtain the critical value of a project for which the company can afford to wait or invest immediately. Childs et al. (1998) study the interactions between several investment projects and their values as part of a real option valuation. The choice of the development policy, which can be seen as a starting phase, depends on the values of the real options resulting from the chosen strategy. The Grenadier and Weiss model (1997) focuses on determining the right time to migrate to another technology. New information on a research and development project is incorporated into the Lint and Pennings (1998) model with jumps. In this context, other researchers are developing models specific to industries where research is paramount.

50

Investment Decision-making Using Optional Models

2.2.1. Real investment opportunity After considering an investment project requiring a series of expenditures at specific periods, Trigeorgis (1993) designs the following opportunities that can be converted into real options: – postpone project implementation (US call option); – abandon the project without recovering the expenditures made (US call option); – reduce project activity by reducing planned investment expenditure (European put option); – expand the scope of the project by incurring an additional investment expenditure (European call option); – change project with the possibility to recover a specified value (US put option). The presence of options exercisable at a later date relative to other options increases the value of the underlying asset, as these future options increase the value of the options that precede them. In other words, the value of the options prior to the others is a sum of the gross value of the project to which must be added the value of the future options. However, exercising one option prior to another may change the value of the project and, consequently, the value of the options that succeed it. Trigeorgis deduces that here lies a second-order interaction. In fact, if the first option is a put, the value of the project will be lower since there is a negative interaction, and if it is a call, the value of the underlying will be higher given a positive interaction. The greater the joint probability of exercising two options2, the greater the degree of change in the value of the option prior to the others and, beyond that, its degree of interaction. The probability of exercising a future option is impacted to a greater or lesser degree by the probability of exercising an option that precedes it. As a result, real options interact to different degrees because of the probability of their joint exercise.

2 This depends on the similarity of the linked options.

Surplus Value Linked to the Option to Invest

51

It is in these terms that the value of an option in the presence of other options may differ from its value taken in isolation. If the two options are of the opposite type, such as a put and a call, so that they are perfectly exercised in opposite circumstances (negative correlation), the probability of exercising the second option knowing that the exercise of the first has taken place is very small and, in any case, lower than the marginal probability of exercising the option if it had been on its own. The degree of interaction would therefore also be small as well as the additive value of these two options. Conversely, if both options are of the same type, such as a pair of put or call, the probability of exercising them would be higher, as well as the extent of their interaction. In this context, it would be interesting to study the possibility of postponing the investment decision, that is to say, to be able to determine the right moment to launch a project. Assumed data: V = 100; r = 5%; Var = 25%; T = 15 years; POSTPONEMENT = 2 years NPV of the project (discounted investment expenditure3 of 114.7): - 14.7 Value of a real option POSTPONE (PS)4 26.35; 416

Abandon (A) 22.1; 36.8

Reduce (RD) -7.8; 6.9

Expand (EX) 20.3; 35

Exchange (EC) 24.6; 39.3

Value of two real options PS & A 36.4; 51.1

PS & RD 27.7; 42.4

PS & EX 54.7; 69.4

PS & EC 38.2; 52.9

A & RD 22.6; 37.3

A & EX 50.6; 65.3

A & EC 24.6; 39.3

RD & EX 27.1; 41.8

RD & EC 25.5; 40.2

EX & EC 54.7; 69.4

Value of three real options PS & A & RD 36.8; 51.5

PS & A & EX 68.2; 82.9

PS & A & EC 38.2; 52.9

PS & RD & EX 57.1; 71.8

PS & RD & EC 38.7; 53.4

PS & EX & EC 71; 85.7

A & RD & EX 51.9; 66.6

A & RD & EC 25.5; 40.2

A & EX & EC 54.7; 69.4

RD & EX & EC 55.9; 70.6

3 Trigeorgis discounts at the rate of 5% (r) of the assumed investment expenditure: in year 1, an initial investment expenditure of 10; in year 3, an investment expenditure for works of 90; and in year 5, an investment expenditure for the construction of new infrastructures of 35. 4 The carry option with a value of 26.3 increases the value of the project by 41 (compared to the value of the project calculated by the NPV method). 5 Value of the project, including the value of the option or options. 6 Value of one or more options.

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Investment Decision-making Using Optional Models

Value of four real options PS & A & RD & EX 69.3; 84

PS & A & RD & EC 38.7; 53.4

PS & A & EX & EC 71; 85.7

PS & RD & EX & EC 71.9; 86.6

A & RD & EX & EC 55.9; 70.6

Value of five real options PS & A & RD & EX & EC 71.9; 86.6

No interaction between five types of real options

Table 2.1. Interactions between different types of real options

2.2.2. Opportunity to postpone decision-making to infinity The Dixit and Pindyck model (1994) allows for the postponement of investment decision-making taking into account future cash flows, the selling price of the product and production costs. 2.2.2.1. Sensitivity of the value of a hold option to the value of future cash flows Dixit and Pindyck assume the value V (positive) of discounted future cash flows: dV = α. V. dt + σ. V. dz

[2.1]

where: – α: FCF expected growth rate; – σ: FCF volatility. In addition, the amount I of the investment is fixed, regardless of the date on which it is completed. Then: – V*: critical value of future cash flows; – μ: expected rate of return of the share (excluding dividends), the price of which is correlated with V. In this context, the return of the project is the same as that of the share; – δ: future cash flow rate that would be generated by the project.

Surplus Value Linked to the Option to Invest

53

The return of the share (μ) can be considered as the sum of the future cash flow rate (δ) and the rate of appreciation (α): μ = α + δ or δ = μ − α

[2.2]

Suppose F the option premium to invest, no matter when. This US (call) option on the share price V has an exercise price corresponding to I. If the call is exercised, the company that holds it pays I and receives the underlying asset at price market V. In addition, the call is worth, in this case, V – I or the intrinsic value equal to the NPV. If the call is not exercised, the premium F – which includes the time premium – is greater than V – I. Then: F ≥ V – I, where F + I ≥ V

[2.3]

Consequently, the call must be retained as long as the full cost of the project (i.e. the investment and the carry option premium) is higher than the expected current value of the cash flows. The call can be exercised when F = V – I. Let V* be the abscissa of this point, which corresponds to the critical value of future cash flows: F(V*) = V* – I (necessary condition no. 1: corresponding value)

Figure 2.1. The hold option premium based the current value of future cash flows

As F ≥ V – I, the curve and the line F = V – I are tangent at their point of contact and the slope is equal to 1. An angle of 45° is formed with the abscissa axis. As a result: F’(V*) = 1 (necessary condition no. 2: smooth pasting)

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Investment Decision-making Using Optional Models

In addition, if V = 0, the current value of the cash flows cannot increase. Then, the project must be abandoned and the option premium is null: F(0) = 0 (necessary condition no. 3) The partial differential equation for a call C, the underlying asset of which is S, gives: + r. S.

+ σ S .

= rC

[2.4]

To apply this formula to the carry option of a project, it is necessary to replace: C with F, S with V and r with r – δ to include the project cash flow rate that corresponds to the dividends paid and to find that C does not have a time derivative, since the maturity date can be deferred to infinity. Then: (r − δ). V. σ V .

+ σ V .

= rF

(V) + (r − δ). V.

[2.5]

(V) − r. F(V) = 0

[2.6]

The second-order differential equation has the following conditions: F(V*) = V* – I

[2.7]

F’(V*) = 1

[2.8]

F(0) = 0

[2.9]

We thus recognize the form: F (x) + aF (x) + bF(x) = 0

[2.10]

By replacing F(x) with eλx: λ .e

+ α. λ. e

+ be

=0

[2.11]

And, by dividing by eλx: λ . +α. λ + b = 0

[2.12]

Surplus Value Linked to the Option to Invest

55

It is thus a second-degree trinomial, the roots of which will be denoted as and . It is then possible to determine A and B such that: F(x) = Ae

+ Be

[2.13]

F is a linear combination of two particular solutions of the differential equation. If the partial differential equation has a solution for a type of , the characteristic equation is: σ V . ( − 1).

+ (r − δ). V. β. V

By dividing all the terms by

− r. V = 0

:

. σ β. (β − 1) + (r − δ). β − r = 0 + Then: ∆ =

−

− . −

[2.14]

.

−

=0

+ 2 .

−

[2.15] [2.16] = 0.

[2.17]

Let β1 and β2 be the solutions of the characteristic equation: √

β =

[2.18]

is assumed to be the largest value. √

β =

[2.19]

is assumed to be the smallest value. Recall: ax + bx + c = 0 est égal à

[2.20]

Then: β β =

< 0,



é

β > 0 β < 0

[2.21]

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Investment Decision-making Using Optional Models

It is possible to prove that β1 > 1. In fact, the function: f(B) = σ β + r − δ − σ

.β − r

[2.22]

is represented graphically by a parabola, knowing that: – lim f(β) = + ∞ when β tends to +∞; – f(0) = –r and f(1) = -δ < 0, δ being a positive number. Thus, we find Figure 2.2.

Figure 2.2. Curve of f

The graph shows that f(1) is negative if β1 > 1. The necessary conditions make it possible to solve the general form of the partial differential equation: F = A. V

+ B. V

As: F(0) = 0, β > 0 Then: F = A. V .

[2.23] β < 0, it is necessary that: B = 0.

[2.24] [2.25]

Moreover, F(V*) = V* – I (corresponding value condition) and F’(V*) = 1 (smooth pasting condition). Therefore: A. V ∗

= V∗ − I

[2.26]

Surplus Value Linked to the Option to Invest

A. β V ∗ ∗

=1

= V ∗ − I ou V ∗ V∗ = A=

[2.27] − 1 = −I ou ∶ V ∗

.I ∗ ∗

57

=

=I

[2.28] [2.29] [2.30]

The Dixit and Pindyck assumptions to implement their model are as follows: – r = 4%; – δ = 4%; – σ = 20%; – I = 1. Thus, β1 = 2, V* = 2I and A = 0.25. Knowing that the decision to invest is indefinitely deferred and that as V* = 2I, the investment should be made when the sum of the future cash flows is at least twice the amount of the initial investment7. Subsequently, Dixit and Pindyck extend their model by including the price of the manufactured product. 2.2.2.2. Extension of the Dixit and Pindyck model: consideration of the price of the manufactured product In the absence of taxation and assuming the cost of production as zero, Dixit and Pindyck (1994) consider the profit P exactly corresponding to the sale of the product. Thus: dP = α. P. dt + σ. P. dz

[2.31]

Note: – α: profit growth rate; 7 Under the NPV criterion, it would be sufficient for the sum of the future cash flows to be at least equal to one time the amount of the initial investment.

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Investment Decision-making Using Optional Models

– µ: profit discount rate (calculated from the Medaf); – V: value of the project. Assuming that µ is greater than α: V=

[2.32]

Let δ = µ - α. In other words, δ corresponds to the convenience yield of the project. This is the benefit of holding the product. Thus: V=

[2.33]

The partial differential equation is as follows: ( ) + (r − δ). P. F − rF(P) = 0

σ P .

[2.34]

With the following conditions: F(P*) = P*- I

[2.35]

F’(P*) = V’(P*)

[2.36]

F(0) = 0

[2.37]

Then, Dixit and Pindyck establish: P∗ =

. δI

[2.38]

P* is the product price for which the company is indifferent to investing or waiting. Thus: V∗ =

∗

=

.I

Then, Dixit and Pindyck include production costs.

[2.39]

Surplus Value Linked to the Option to Invest

59

2.2.2.3. Extension of the Dixit and Pindyck model: the critical value of the investment project Assuming that production costs exist, Dixit and Pindyck (1994) seek to find the critical value of the project for which the company is indifferent to investing immediately or waiting. Recall: – P: price of the manufactured product; – V: value of the project. Note: – π: instant profit generated by the investment project; – C: production cost. Knowing that the company starts production if P is at C: π = max (0, P − C). [2.40] After creating an arbitrage portfolio, the differential equation is as follows: σ P .

+ (r − δ). P. V − rV + π = 0

[2.41]

In this context, two situations arise: – if P < C, π = 0 and: σ P . V = E .P

+ (r − δ). P. V − rV = 0 + E .P

[2.42] [2.43]

V is the value of the investment carry option, i.e. the value of an option to use production capacity when the market will allow the company to make a profit, knowing that β1 is positive and β2 is negative. Consequently, the project holds value even though the company is unable to generate profit instantly. In addition, the probability that the investment becomes profitable is also decreases as P tends to 0. In this case, V also tends to 0 and so E . equal to 0; knowing that β2 is negative, it is necessary that E2 is zero. Thus:

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Investment Decision-making Using Optional Models

V = E .P

[2.44]

– if P > C, π = P – C and: σ P .

+ (r − δ). P. V − rV + P − C = 0

[2.45]

Knowing that − is a particular solution of the above equation: V = B .P

+ B .P

+ −

[2.46]

As a result, in the absence of any possibility of postponement, the value of the project is equal to: V= −

[2.47]

In other words, it is a solution similar to that of the NPV. It results from the difference between the sum of future revenues discounted at cost of equity and a perpetual annuity of discounted operating expenses at the risk-free rate. Since P tends to infinity, the hold option would no longer exist. Thus, its value would tend to 0. In other words: V = B .P with

+ B .P

=0

[2.48]

tending to 0; β1 to be equal to 0. Thus: V = B .P

+ −

[2.49]

If P is equal to C, the company would be indifferent to waiting or investing. In that case: V = E .C

= B .C

+ −

[2.50]

Since F (V) has an identical slope, it does not matter that P is less than or greater than C (at the point P = C, we then see the same derivative): β .E .C

= β .B .C

+ −

[2.51]

Surplus Value Linked to the Option to Invest

61

Solving the system of equations [2.50] and [2.51], we have: E =

−

[2.52]

B =

−

[2.53]

The option to invest F is as follows: F(P) = A . P

+ A .P

[2.54]

Since F(0) = 0 and A2 = 0: F(P) = A . P

[2.55]

Considering the investment I to be made according to the contingencies, the selling price P* reflecting indifference as to whether to invest or to wait verifies: A . P∗

= B . P∗

+

∗

− −I

[2.56]

By deriving F at the critical point P*, we have: β . A P∗

= β B . P∗

+

[2.57]

After having multiplied equation [2.56] by β1 and equation [2.57] by P*, we have the system: β . A P∗

= β . B P∗

β . A P∗

+

∗

= β B . P∗

− +

− Iβ ∗

4

[2.58]

By subtracting the second equation from the first, a single equation is obtained with only an unknown P*. This can be found using a numerical calculation: β (β − β )P ∗

+ (β − 1)

∗

−β

−I =0

[2.59]

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Investment Decision-making Using Optional Models

Targiel (2015) insists on the importance of considering the real options approach in sustainable development projects where the question of temporality with regard to investment decision-making is paramount. In fact, by taking an overview of the options that can be considered as the carry option, the abandonment option or the compound options, Targiel asserts that the multi-criteria valuation makes it possible to decide when to start the project as such or approach the next phase. In addition, Franklin (2015) applies the Dixit and Pindyck model (1994) to investment decisions in the mobile telecommunications network sector. The investment project is defined as the purchase, installation and launching of a network where each element provides a hypothetical network service. These services are combined by a number of elements and are subject to different demand and uncertainty, so that an optional value is calculated for each element of the main network. This range of real options is calculated to favor investment decision-making. The impact of the hold option disappears as end-point investments are completed. As a result, the model would enable mobile operators to make better investment decisions, considering not only its cost but also the value of the integrated real option. In addition, the model is intended to help regulators to better assess the costs of mobile services. In fact, these costs often determine tariff rules. It is understandable then that real options can easily be a relevant tool for the valuation of specific projects focused on research and development. 2.2.3. Development cycle and taking into account new information within dependent projects and focusing on research and development According to Childs et al. (1998), project development can occur in parallel or in sequence. In other words, during the development phase, the company learns information about the state of the market, as well as about the interaction that can exist between several projects. Beyond the choice of the development of a single project, the investment decision can be motivated by comparing two possible solutions or a sequential development. Real options are combined to the extent that it is based on the success of a first investment that the company will choose to make a second investment. Childs, Ott and Triantis show, therefore, that when projects are highly correlated, are feasible over a short period of time, require significant fundraising and have low volatility, sequential development turns out to be a better choice than parallel project development. Conversely, parallel project

Surplus Value Linked to the Option to Invest

63

development is of greater interest when development costs are low because of high uncertainty, because it generates large cash flows and because it requires long periods of putting them in practice. Childs et al. (1998) rely on two examples from the aeronautics sector to justify their remarks. Based on McDonnell Douglas’ company project development, the authors envision investing in a cargo aircraft designed from a passenger aircraft model. Unlike airplanes carrying passengers, aircraft carrying goods must not have windows. A first possibility would be to modify the body of the device. The second possibility would be to replace the windows of the original aircraft. The investment expenditure of the first case is higher than the second because it is necessary to carry out a study phase for the design. McDonnell Douglas chose to develop both projects in parallel because the correlation between them is close to zero. The second application case deals with the manufacture of a new aircraft model. It is then recommended to proceed with a sequential development of several prototypes because the information revealed by a prototype can be used for the design of another one. Grenadier and Weiss (1997) develop a model to determine the right time to migrate to a new technology. It is, in fact, an option to invest for a company according to several possibilities: – either the company invests in the current technology at an exercise price Ce from the beginning of the project (i.e. t = 0)) and, in t = T, it has the possibility to keep its technology or to invest in a new technology by spending the amount Cu; – either the company does not invest in t = 0, but in t = T, it has the choice to opt for the current technology at the price of Cd or for the new technology at the price of C1. In this case, the authors assume that investment in current technology is less expensive in t = T than in t = 0. In addition, the expense C1 realized in t = T is less than the sum of the amounts invested by the company adopting the current technology project in t = 0 and the new technology in t = T. After an empirical study on business behavior, the two researchers observe that, as uncertainties about technological change are significant, companies are moving towards direct investment in new technologies or prefer to wait for the arrival of an innovation to invest in the previous one.

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Investment Decision-making Using Optional Models

Lint and Pennings (1998) integrate the knowledge of new information from a research and development project into a discrete Poisson process, leading to no longer considering the value of the option in continuous time, but according to a model with jumps. Thus, decision-making can be delayed because the information that reaches the company includes the appearance, within the sector, of new technologies that can lead to a reduction of production costs and the arrival of new competitors. In this context, the variance of the underlying project is as follows: σ = λ. γ

[2.60]

with: – λ: number of jumps planned per year; – γ: anticipated variation of the value of the project at each jump. We then have the value of the following investment option c(t): c(t) = S(t)N d + σ λ. γ. τ − I. e

N(d)

[2.61]

with: ( )

d=



.

. .

[2.62]

where: – S(t): value of the project; – I: amount of the investment; – r: risk-free rate; – τ: remaining maturity of the project. In addition, Wesseh and Lin (2015) evaluate the viability of wind energy projects in China through the real options approach. The latter allows them to take into account the flexibility offered by the deployment of this technology in response to a stochastic cost of non-renewable energies. According to the authors, the difference between this cost and research and development expenses for this type of more ecological project is to take

Surplus Value Linked to the Option to Invest

65

advantage of the wind. On the other hand, the Chinese government subsidizes this industry to encourage its development. The work of the two authors therefore shows that wind energy is an economically attractive activity, because there is a substantial optional value in the Chinese program in full development – a value that is all the greater if the cost of carbon dioxide emission is internalized. Based on the real options approach, Lund and Jensen (2016) question the profitability of an activity related to the development and commercialization of an animal vaccine. They find that the project value of this type of investment, which requires the integration of several phases, is more important when real options are integrated. In other words, when the various steps required to commercialize a vaccine (i.e. research, design, testing, approval, and launch) are designed as abandonment options using a binomial approach because of their flexibility – the completion of a step depends on the success of the one that precedes it – the real options framework can improve the financial analysis of investment decisions. In particular, according to the authors, the real options approach is more relevant for assessing vaccine development costs. In addition, they consider that the use of real options offers a more nuanced assessment in taking into account alternative policies by encouraging the abandonment of projects with poor prospects or by giving the possibility of reselling projects at intermediate stages. Finally, based on the Dixit and Pindyck model (1994), Baldenius et al. (2016) develop a dynamic congruence model in which information arrives over time, allowing the manager to anticipate, relative to the shareholder, investment and operating decisions. The authors then show that if demand follows a stochastic process, the expected optimized profits slow down. This decline does not occur in the event that the company is unable to adjust the use of its capacity. This is justified by the fact that the value of the carry option decreases if the company is affected by a series of adverse demand shocks impacting the profit level. Thus, valuation models of investment projects in the presence of abandonment or exchange options have been developed. 2.3. Option to exchange and abandon an investment project Magrabe (1978) designs an asset exchange option valuation model. The value of the project then corresponds to the value of a European call option. By looking at the optimal time to renew an asset, Mauer and Ott (1995) find

66

Investment Decision-making Using Optional Models

that, on the one hand, the volatility of operating costs as well as the purchase price of the replacement asset and the corporate tax rate increase and, on the other hand, companies are waiting to gather as much information as possible before making all types of decisions, which lengthens the renewal cycle. The uncertainty of decisions to invest in joint ventures or alliances encourages Damaraju et al. (2015) to integrate the presence of options in the choices of corporate governance modes. For example, divestment is similar to exercising the sale of a put option. Here again, companies are encouraged to wait until they have enough information before deciding on a possible abandonment. In the natural resources sector, production is subject to uncertainty. The integration of real-option-based models, such as that of Brennan and Schwartz (1985), makes it possible to define policies that optimize the decision to invest by retaining the effects of time, costs, and convenience yield, i.e. the profit a producer derives from his/her stocks. Siegel et al. (1985) integrate volatility in their valuation model of an operation of an oil well concession. In order to explain the parameters influencing the optimality of the investment decision, they make an analogy with stock options. The evolution of the strategic orientation, considered as a policy of debt reduction or refocusing activity, can also justify the exercise of abandonment options. Clark et al. (2010) even inquire as to whether investors value abandonment options when they separate from their securities, especially in a context of non-efficient markets (if not, as information is public, valuation cannot be biased). They observe that abandonment options are undervalued by about 1% due to early exercise. 2.3.1. Real options within disinvestment alternatives

the

replacement

cycle

and

Drawing on the Black and Scholes model, Magrabe (1978) suggests an exchange option valuation model in which a company can replace an asset with X2 as a random price with another asset with X1 as a random price. In this context, the value of the investment project w(X1, X2, t), which corresponds to the value of a European call option exercisable at the date t*, is as follows:

Surplus Value Linked to the Option to Invest

w(X , X , t) = Max(0, X − X )

67

[2.63]

The exercise price is thus equated with the value of the asset X2 in order to acquire the asset X1. The value of X1 expressible in unit of asset 2 corresponds to the ratio of X1 on X2 knowing that asset 2 is equal to 1 unit (ratio of X1 on X2). The volatility of the underlying w is as follows: σ

=

σ

+σ

− 2cov σ

σ

=

σ

+σ

− 2ρ. σ

;σ .σ

[2.64] [2.65]

Assuming that the linear correlation coefficient ρ is equal to: ρ=

(

,

)

[2.66]

.

so: cov(X , X ) = ρσX . σX .

[2.67]

By reasoning from a market equilibrium situation, Magrabe assumes a zero risk-free rate. In this context: w(X , X , t) = X ϕ(d ) − X ϕ(d )

[2.68]

where: .

d =

. √

d = d − σ √τ

[2.69] [2.70]

with: – τ: remaining maturity of the option; – Φ: distribution function of the standard normal distribution. On the other hand, if the assets X1 and X2 are dividend-yielding shares, at the respective rates q1 and q2, the value of the exchange option C must be

68

Investment Decision-making Using Optional Models

taken into account in the Black and Scholes formula the distribution rate and the compensation differential of the shares concerned. We then have: C=X

Φ(d ) − e

e

Φ(d )

[2.71]

where the risk-free rate r is assumed to be zero and: .

.

d =

√

d = d − σ √τ

[2.72] [2.73]

The Mauer and Ott (1995) asset renewal cycle concludes that the optimal time to replace it increases the volatility of operating costs, the purchase price of new assets and the corporate tax rate. However, they note that this right time is consistent with a reduction in the systematic risk of costs, the salvage value of the asset and the tax credit related to the investment. The optimal cycle can also evolve based on the depreciation rate. Finally, the appearance and adoption of a new technology reduces operating costs. However, in response to the work of Childs et al. (1998) and Grenadier and Weiss (1997), Mauer and Ott observe that companies choose to wait before investing in new technologies to give themselves time to gather additional information to limit uncertainties. The renewal cycle is thus extended. Damaraju et al. (2015) examine the decision to divest from companies in joint ventures or alliances through the real options approach as these shares are sensitive to uncertainty. As a result, choices in governance modes must incorporate the presence of optional values. In fact, the ownership of a business unit has a conceptual analogy with the holding of a put option. Thus, disinvestment corresponds to the exercise of the sale of this put option. Moreover, in an uncertain environment, the real value of a business unit cannot be precisely appreciated. In other words, its current value cannot be a good indicator of its future value. Under such conditions, by divesting, i.e. by exercising its put options, the company could incur significant losses that prevent it from taking advantage of new opportunities.

Surplus Value Linked to the Option to Invest

69

It is in these terms that, in accordance with Dixit and Pindyck’s findings (1995), it may be more appropriate, in the face of great uncertainty about the market, to wait rather than deciding too quickly to divest. Based, in particular, on a sample of 230 divestitures and 153 splits between 1980 and 2003, Damaraju, Barney and Makhija empirically conclude on the accuracy of this prediction: when the uncertainty in the environment of company business units is high, the latter are reluctant to sell their shares, preferring to preserve their options for disinvestment. Given the significant uncertainty that dominates the natural resources sector, researchers have focused on the value of investment projects relating to commodities. 2.3.2. The value of an investment project in the natural resources sector Price variability does not affect the valuation of certain projects in which the underlying is relatively predictable, but it is of extreme importance in the natural resources sector where securities vary between 25 and 40% per year8. The Brennan and Schwartz model (1985) focuses on defining an arbitrage portfolio, the cash flows of which replicate those to be valued (Harrison and Kreps 1979). The construction of such a portfolio assumes that the convenience yield of the return of the commodities is expressed as a function of the spot9 of the latter and that the interest rate is non-stochastic. These assumptions lead to a relationship between the spot price and the future price of the resource. Thus, project cash flows are adjusted by an arbitrage portfolio consisting of risk-free assets and future contracts. The first step in analyzing an investment project is to determine the current value of its future cash flows. It will then be necessary to compare it with that of the required investment in order to know whether to invest immediately or wait. 2.3.2.1. Farm valuation and optimal production policy Convenience yield, according to Kaldor (1939), is defined as the profit that a producer withdraws from his/her stocks, without bearing the cost associated with frequent orders or waiting for deliveries. This comfort, linked to the availability of stocks, justifies that the spot price may become higher than the futures price. When the market is in short supply, 8 Bodie and Rosansky (1980) estimated, over the period 1950–1976, the standard deviation of annual price variations of about 25% for silver, 47% for copper and 25% for platinum. 9 Spot means “price of the underlying” or “spot price”.

70

Investment Decision-making Using Optional Models

convenience yield can become higher than the storage cost, and an offset situation can be established. Brennan (1958) points out that the uncertainty of future demand is an incentive to hold stocks in a deportation situation. The model studied retains the spot price as sole explanatory variable of the futures price considering a constant convenience yield. A project using a single natural resource is assumed. The spot price, S, of the latter follows a geometric Brownian motion: = μ dt + σdz

[2.74]

with – dz: increment of the Brownian motion associated with S = ε dt and ε → N(0, 1); – σ: volatility of S; – µ: anticipated instantaneous return of S. Before developing the model, it is necessary to study the relation between the spot, its future evolution and the convenience yield of the commodity. Convenience yield is “the flow of services that increases the value of the stock of the physical resource holder” (Working 1948; Telser 1958). It is held by those for whom the net marginal rate is high. The latter evolves in a proportionally inverse way to the amount of the stored resource and the spot. Thus, the net marginal rate of convenience yield is a function of the price of the underlying at time t. It is noted C (S, t). However, convenience yield as such is proportional to the spot of the resource. Suppose a constant interest rate ρ relating the spot to its future prices. F (S, τ) represents the future price, at time t, of the delivery of a unit of the resource at time T (τ = T – t). By Ito’s lemma10, the instantaneous evolution of future prices is: dF = −F + F σ S

dt + F dS

[2.75]

Consider the instantaneous return of a portfolio consisting of the purchase of a resource unit and the sale of (Fs)-1 in the futures market. Using [2.75],

10 The concept of Ito’s lemma is explained in Appendix 2.

Surplus Value Linked to the Option to Invest

71

the instantaneous return that includes a net marginal rate of convenience yield is: +

( )

− (SF ) dF = (SF )

F C(S) − F σ S + F dt [2.76]

Let C (S) be the convenience yield (net) of a marginal unit of stock. If the instantaneous return is non-stochastic, then it must be equal to the risk-free rate ρ dt. We obtain the following differential equation: F σ S + F (ρS − C) − F = 0

[2.77]

Future prices represent the solution of [2.77]: F(S, 0) = S

[2.78]

Future prices are based on spot and time to maturity. The parameters of the convenience yield function can be estimated directly by the relationship between the spot and the future prices. If the convenience yield is proportional to the spot: C(S, t) = cS

[2.79]

Using equations [2.75], [2.77] and [2.78], the instantaneous variation of the future price of the resource is expressed using convenience yield: dF = F [S(μ − ρ) + C]dt + F Sσ dz

[2.80]

The operation value depends on the production rate q, the spot S, the stocks of the concession Q, the time t and the operating policy φ. Mathematically: H ≡ H(S, Q, t ; j, Φ)

[2.81]

The variable j is equal to 1 if the operation is open and 0 if it is closed. In the first case, the operating policy is expressed by the production rate function q(S, Q, t) and: – S1(Q, t): price of the underlying at the moment the operation is stopped or abandoned;

72

Investment Decision-making Using Optional Models

– S2(Q, t): price of the underlying at the time of the resumption of operations while it was stopped or abandoned; – S0(Q, t): price of the underlying at the time of the abandonment of the operation while it was stopped. Applying Ito’s lemma to [2.80], the variation in the value of the operation is: dH = H dS + H dQ + H dt + H (dS)²

[2.82]

where: dQ = −q dt

[2.83]

Cash flows (of the operation) after tax are: q(S − A) − M(1 − j) − λ H − T

[2.84]

where: – A(q, Q, t): production flow cost rate; – M(t): after-tax fixed maintenance cost rate of the operation at time t while it is closed; – λj(j = 0, 1): proportional rate of taxes on the value of the operation, whether open or closed; – T(q, Q, S, t): corporate tax when the farm is operated. In order to derive [2.81], let us consider the return of a portfolio consisting of the purchase of a concession and the sale of future contracts. The return of the operation is [2.82]–[2.83] and the future price variation is [2.80]. Combining with [2.74], the return of the portfolio: σ S H − qH + H + q(S − A) − M(1 − j) − T − λ H + (ρS − C)H [2.85] If the return is non-stochastic, then it must be equal to the risk-free rate, ρH, of the value of the investment. Using expression [2.85], the value of the operation must satisfy the following differential equation:

Surplus Value Linked to the Option to Invest

73

σ S H + (ρS − C)H − qH + H + q(S − A) − M(1 − j) − T − ρ + λ H = 0

[2.86]

The value of the property satisfies [2.86] for each operating policy ϕ = {q, S0, S1, S2}. When the latter is maximized ϕ* = {q*, S0*, S1*, S2*}, the values of the operation when it is open, V(S, Q, t), and when it is closed, W(S, Q, t), are: V(S, Q, t) ≡ max H(S, Q, t ; 1, Φ)

[2.87]

W(S, Q, t) ≡ max H(S, Q, t ; 0, Φ)

[2.88]

In addition, the value maximizing production and the value of the operation satisfy two equations (Merton 1971; Fleming and Rishel 1975; Cox et al. 1978): max

∈

,

[ σ S V + (ρS − C)V − qV + V + q(S − A) − T −

( ρ + λ )V] = 0 σ S W + (ρS − C)W + W − M − ( ρ + λ )W = 0

[2.89] [2.90]

If the opening, closing and abandonment policies are known to investors, we have: W(S ∗ , Q, t) = 0

[2.91]

V(S ∗ , Q, t) = max [W(S ∗ , Q, t) − K (Q, t), 0]

[2.92]

W(S ∗ , Q, t) = V(S ∗ , Q, t) − K (Q, t)

[2.93]

where K1(.) and K2(.) are, respectively, the costs of closing and opening the operation. Finally, according to Merton (1973) and Samuelson (1965): W (S ∗ , Q, t) = 0 V (S ∗ , Q, t) =

W (S ∗ , Q, t)si W(S ∗ , Q, t) − K (Q, t) ≥ 0 0 si W(S ∗ , Q, t) − K (Q, t) < 0

[2.94] [2.95]

74

Investment Decision-making Using Optional Models

W (S ∗ , Q, t) = V (S ∗ , Q, t)

[2.96]

The value of the operation depends only on time because the costs A, M, K1 and K2 and convenience yield C depend on it themselves. If the inflation rate π is constant for each of these parameters and if C (S, t) can be written κS, then [2.89]–[2.96] can be simplified. Let us define: a(q, Q) = A(q, Q, t)e

[2.97]

f = M(t)e

[2.98]

k (Q) = K (Q, t)e

, k (Q) = K (Q, t)e

[2.99]

s = Se

[2.100]

v(s, Q) = V(S, Q, t)e

[2.101]

w(s, Q) = W(S, Q, t)e

[2.102]

Then, the deflated value of the concession satisfies: max

∈

,

[ σ s v + (r − ĸ)sv − qv + q(s − a) − τ −

( r + λ )v] = 0 σ S w + (r − ĸ)sw − f − ( r + λ )w = 0

[2.103] [2.104]

where: r = ρ − π is the real interest rate

[2.105]

τ = t qs + max t q[s(1 − t ) − a], 0

[2.106]

w(s ∗ , Q) = 0

[2.107]

v(s ∗ , Q) = max[w(s ∗ , Q) − k (Q), 0]

[2.108]

w(s ∗ , Q) = v(s ∗ , Q) − k (Q)

[2.109]

w(s, 0) = v(s, 0) = 0

[2.110]

w (s ∗ , Q) = 0

[2.111]

Surplus Value Linked to the Option to Invest

w (s ∗ , Q) si w(s ∗ , Q) − k (Q, t) ≥ 0 0 si w(s ∗ , Q) − k (Q, t) < 0

v (s ∗ , Q) =

w (s ∗ , Q) = v (s ∗ , Q)

75

[2.112] [2.113]

Equations [2.103]–[2.113] constitute the general model of the value of an operation, but there is no analytical solution, although it is conceivable to solve it numerically. Brennan and Schwartz (1985) then present a simplified model. To do this, the stocks of the resource, Q, are assumed to be infinite, the purpose being to simplify the equations [2.103] and [2.113] of the value of the operation. It is assumed that there are only two exploitation rates: q* when the operation is open, and 0 when it is closed. Costs are incurred to move from one to the other. Thus, the deflated value of the concession when it is opened and operated at the rate q* satisfies the differential equation: σ s v + (r − ĸ)sv + ms − n − ( r + λ)v = 0

[2.114]

where: m = q*(1 – t1)(1 – t2) and n = q*a(1 – t2). Assuming that f (maintenance cost for a closed operation) is zero, the deflated value of the closed concession satisfies the differential equation: σ s w + (r − ĸ)sw − (r + λ)w = 0

[2.115]

The solutions of equations [2.114] and [2.115] are: w(s) = β s

+β s

v(s) = β s

+β s

[2.116] +

ĸ

−



[2.117]

where the β are constants and: γ = α + α , γ = α − α α = −

ĸ ²

, α = [α +

[2.118] (

)

]

[2.119]

76

Investment Decision-making Using Optional Models

If (r + λ) > 011, then β2 = 0 since γ2 is negative and w(s) is finite (s tends to zero). And, if γ1 > 1, then β3 = 0. The value of the closed concession is w(s) = β1Sγ1 and the open one is: v(s) = β s

+

ĸ

−

[2.120]



The remaining constants β1 and β4, as well as the optimal closing and opening policy represented by ∗ and ∗ are: ∗ (

β =

∗

∗

[2.121]

∗

[2.122]

)

(

)

)

(

s∗ = γ ∗

)

∗ (

β =

and

)

(

) (

(

[2.123]

)

=x

[2.124]

where e = k1 – , b = - k2 – , d = and x, the ratio of the price of the raw material for which the operation is closed and open, is the solution of the following equation: )(

( (

) )

=

)(

( (

) )

[2.125]

When γ < 0 and the spot is higher, the value of the closing option tends to zero. When the price is low and no policy has been used to counteract it, the losses recorded on the open operation are more profitable than the disbursements related to its closure. However, when prices are high and under the same condition, it is the opposite that is observed and, during s*2, it is the worst time to open the operation since one would spend money k2, which represents the farthest distance between the curves W (value of the closed concession) and V (value of the open concession). When operation opening and closing costs tend to zero, s*1 and s*2 approach the same value. When the cost of closing the operation becomes high, the closing option is futile and, ultimately, the goal is to ensure that the opening value of the operation approaches the current value of cash flows. Variations in closing 11 This is necessary for the current value of future costs to be finite.

Surplus Value Linked to the Option to Invest

77

costs, due, for example, to a government policy, alter the optimal policy of closure of the operation, s*1. However, this also affects the investment decision because of the change in the current value of future cash flows. Such effects, or those induced by changes in a tax system, must be immediately analyzed. 2.3.2.2. The investment decision of a project in the natural resources sector The investment decision12 requires comparing the current value of the cash flows of the project to the initial investment. V(S, Q*, t) is the value of the operation at time t of the property, Q* its stocks and S the current spot. V(.) is the current value of the cash flows assuming an optimal operating policy. Similarly, I(S, Q*, t) is the investment required to build a farm with a stock Q*. The NPV of the construction of the property is: NPV(S, Q∗ , t) = V(S, Q∗ , t) − I(S, Q∗ , t)

[2.126]

The investment decision can be postponed. Thus, it is not necessarily optimal to build the future farm only because the NPV is positive. In anticipation of its increase, a timing option boosts the investment decision. X(S, Q*, t) is the value of the property right to develop the operation. The stochastic process X (.) is obtained by Ito’s lemma, using [2.86]. X (.) must satisfy the following differential equation: σ S X

+ (ρS − C)X + X − ( ρ + λ)X = 0

[2.127]

with X(0, Q∗ , t) = 0

[2.128]

X(S, Q∗ , T) = 0

[2.129]

and

The optimal investment strategy can be assessed as a function depending on the returns of the spot SI(t): X S , Q∗ , t = V S , Q∗ , t − I(S , Q∗ , t) 12 Problem initially raised by Tourinho (1979).

[2.130]

78

Investment Decision-making Using Optional Models

X S , Q∗ , t = V S , Q∗ , t − I (S , Q∗ , t)

[2.131]

According to equation [2.130], the value of the holding right is the NPV at the time of the investment. Equation [2.131] (the Merton–Samuelson formula) maximizes SI. If the stock Q * depends on the initial investment, the value maximizing the size of the initial stock Q* is: V S , Q∗ , t = I (S , Q∗ , t)

[2.132]

The optimal investment strategy is obtained by solving equation [2.127] subject to conditions [2.128]–[2.132]. The optimal time to invest corresponds to the series of critical spots S1(t) described in [2.130] and [2.131]. The optimal amount for investing is [2.137]. The valuation of cash flows is a prerequisite for analyzing the investment decision. Consequently, to determine the right time to invest, Brennan and Schwartz have developed a valuation method for a natural resource operation, the cash flows of which depend on the price of a volatile underlying. An optimal management model through an arbitrage portfolio analyzes the investment decision in highly variable situations in which the distribution of future cash flows is not given exogenously but can be determined by managerial future decisions. However, because of the amplitude of its variations, the price parameter of the underlying implies certain instability as to the value of the concession. By focusing on defining the value of an operation in order to establish an appropriate moment to invest in its development, it is possible to agree on a valuation model focusing on the volatility of the underlying. 2.3.2.3. Valuation based on the volatility of the underlying in oil concessions: offsets and timing of development The profitability of operating an oil well faces uncertainty and the continuous study of new information. The options theory approach allows the notion of volatility to be incorporated (Mason and Merton 1985). Siegel et al. (1985) draw an analogy between oil well reserves and share calls to portray the parameters influencing the optimality of the investment decision13. Just as the call option gives the holder the right to buy a share by paying the exercise price, the oil well reserve gives its owner the right to 13 The authors take as an empirical example the values applied to the United States in the Gulf of Mexico in the late 1980s.

Surplus Value Linked to the Option to Invest

79

acquire and operate a concession by paying the cost of development. Thus, each parameter of the financial option corresponds to a variable characterizing a petroleum reserve. Stock call options

Reserve in development

Current share price

Current reserve value

Variance in the rate of return of the share Variance in the rate of change of the value Exercise price

Development cost

Duration before maturity

Conditions of abandonment

Risk-free rate

Risk-free rate

Dividend

Net profit (deficits)

Table 2.2. Comparison of valuation model variables for stock options and developing oil reserves

Reserves of natural resources are relatively liquid assets. Transaction prices can be assimilated to the market value of the reserve. Even though the latter is not listed on the stock market, estimates are provided by discounting the expected profits, deducting taxes and endowments. Over the 1981–1982 period, Gruy et al. (1982) estimated the average value of oil wells at $12 per barrel. As with equities, the standard deviation of reserve market prices is estimated based on past data. The standard deviation of the rate of change in crude oil prices is a reference for calculating the rate of change in concession values. This implies that the ratio of the value of the crude oil price to that of developed reserves is relatively constant14. Over the 1974–1980 period, the annual standard deviation (σ) of the cost of refining crude oil was 14.2%15. In order to predict statistically the evolution of the crude oil price from 1981, Table 2.3 assumes two reference volatility rates: the one actually observed over the 1974–1980 period and an upward-thought rate assumption.

14 A study by Gruy shows that the value of developed reserves tends to be equal to one-third of the price of a barrel of crude oil. This relationship has been true for a number of years. 15 Calculation based on monthly data; the confidence interval adopted is 95%.

80

Investment Decision-making Using Optional Models

σ = 14.2%

σ = 25%

Year

Low

High

Low

High

1

27.37

48.30

21.56

58.62

2

24.57

54.86

17.31

71.21

3

22.67

60.63

14.58

82.44

5

20.04

71.36

11.05

103.45

10

16.18

97.51

6.54

154.42

Table 2.3. Forecast of the crude oil price per barrel calculated according to standard deviation

Year 0 is 1980; the price per barrel of the year was $36. Note: Future prices are estimated from a 95% confidence interval assuming the log-normality of the barrel price: ln

~

−

,

√

[2.133]

where: – N(.,.): normal distribution; – x: expected rate of increase in the price of crude oil; – St: price of the crude oil at time t. The development cost is calculated by applying a low discount rate16,17 after tax and then deducting depreciation expenses from tangible assets. The risk-free rate is the after-tax return on OAT securities. Skelton (1983) shows that, in the case of long-term state bonds, the marginal tax rate for the 1974–1980 period is 25%. The marginal tax rate on short-term bonds is 50%. Taking an average rate of 37.5%, the risk-free rate is 1.25% (Trigeorgis 1986).

16 Developing a reserve of natural resources takes time, hence the need to update expenditures. 17 Development costs are relatively low risk, but the risk-free rate is not appropriate, as there is a risk associated with the cost of labor and materials.

Surplus Value Linked to the Option to Invest

81

Just like a holder of a dividend-paying share, the owner of an oil well concession receives income. Taking the dividend rate into account reduces the value of a share call. Thus, oil reserve investors must estimate the rate of profit (p) as a percentage of the market value of the reserve. The estimate of this rate requires the following assumptions: – the value per barrel of a reserve (V) is equal to one-third of the crude oil price (S), i.e. V/S = 0.33; – operating costs per barrel (OC) are on average 30% of the crude oil price (S), i.e. OC/S = x = 0.3 (assumed constant); – endowments (D) represent 20% of the crude oil price (S), i.e. D/S = y = 0.2 (assumed constant); – the production rate (g), i.e. the percentage of resources extracted each year from the total remaining in the reserve, is 10%, i.e. g = 0.1; – P = net profit per barrel; – t = corporate tax rate = 0.46. Knowing: Profit = [(production rate) × (profit per barrel)] – [(production rate) × (value of the reserve)]

[2.134]

p=

[2.135]

and





We have P = S – OC – (S – OC – D)t

[2.136]

P = S(1 – x) – 0.46S(1 – x – y)

[2.137]

By replacing S with 3V: P = 3V(1 – x) – (0.46.3V(1 – x – y)

[2.138]

P = g[0.62 – 1.62x + 1.377y]

[2.139]

82

Investment Decision-making Using Optional Models

P = 0.1[0.62 – 1.62 × 0.3 + 1.377 × 0.2] P = 4.1%

[2.140]

Thus, holding a concession means holding a share with a dividend entitlement of 4.1%. The holder of stock options receives the share(s) immediately after paying the exercise price. The owner of concessions must wait for the completion of his/her development before enjoying his/her investment. Thus, the exercise price is modeled as a payment equal to the current value of development expenses. In fact, the time lag leads one to think that one receives the current value of the developed concession after having paid the current value of the development expenses. They are calculated by discounting the value of the reserve developed by the rate of profit. If the rate of return of a concession is r*, the expected return for the investor is r* – p. Suppose the development lag at t’ years. The current value of the reserve V’ in t’ years is: V = V × e(

∗

)

 e

∗

= V  e



[2.141]

Thus, if the development lag is about 1 year as in the Gulf of Mexico, the current value of the reserve developed when the barrel is worth $12 is $11.52: V’ = e–0.041 × 1.0 × 12.2 = 11.52

[2.142]

The longer the development lag, the lower the value of the concession. On the other hand, the owner of a reserve in development decides whether to exercise the option or not, i.e. to operate the concession or not. If the ratio18 C = is above the volatility, it must be exercised. Thus, the authors provide an optimal timing rule similar to that which determines the exercise of a similar call option entitling to a continuous-time dividend at a given rate.

18 V’ is the current value of the developed reserve after the time lag, and D is the development cost.

Surplus Value Linked to the Option to Invest

83

The different parameters studied make it possible to refer to an example of oil well concession valuation based on a stochastic differential equation. The characteristics of the reserve are: – the expected output is 100 million barrels of oil; – the current value of the development cost is $11.79 per barrel; – the development lag is 3 years; – the abandonment of the reserve is possible after 10 years; – the expected standard deviation of the value of the reserve is 14.2%; – the profit ratio is 4.1%; – the value of the concession today is $12 per barrel. ,

The current value of the concession V’ is V = 10.61 $per barrel. The ratio of development costs is

=

=

. .

×

× 12 =

= 0.90.

Value of concession in development = Value of the option for $1 of development cost × Total development cost = 0.05245 × 11,179,000 = 61,840,000. σ = 14.2% V*/D

T=5

T = 10

σ = 25 % T = 15

T=5

T = 10

Out the money 0.7

0.00655

0.01322

0.01704

0.04481

0.07079

0.75

0.01125

0.01966

0.2410

0.05831

0.08650

0.80

0.01810

0.02812

0.3309

0.07394

0.10392

0.85

0.02761

0.03894

0.04430

0.09174

0.12305

0.90

0.04024

0.05245

0.05803

0.11169

0.14390

0.95

0.05643

0.06899

0.07458

0.13380

0.16646

In the money 1.00

0.07661

0.08890

0.09431

0.15804

0.19071

1.05

0.10116

0.11253

0.11754

0.18438

0.21664

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Investment Decision-making Using Optional Models

1.10

0.13042

0.14025

0.14464

0.21278

0.24424

1.15

0.16472

0.17242

0.17599

0.24321

0.27349

Table 2.4. Value of the option for $1 of development cost

Thus, even though a concession may not be profitable today, the right to develop it in the future has a positive value of approximately $62 million. Using the same data as in the previous example and applying variations in standard deviation, maturity periods and discounted values, the value of the concession in development shown in Table 2.5 is observed. σ = 14.2%

σ = 25%

V*/D

T=5

T = 10

T = 15

T=5

T = 10

0.7

7.72

15.59

20.09

52.83

83.46

0.80

21.34

33.15

39.01

87.18

122.52

0.90

47.44

61.84

68.42

131.68

169.66

1.00

90.32

104.81

111.19

186.33

224.85

1.10

153.77

165.35

170.53

250.87

287.96

Table 2.5. Value of the concession in development (in millions of $)

The longer the maturity period and the greater the standard deviation of the underlying, the higher the value of the concession. On the other hand, the higher the out the money values, the higher the change in the value of the concession between two maturity periods, especially if the volatility also increases. For example, with a C of 0.7 for a T of 5, the shift of σ from 14.2 to 25% increases the value by almost 600%. Consequently, the more the price of the barrel is variable, the more variable the value of the concession. As a result, the authors focused on adapting the theory of financial options to operating options in the oil industry. By excluding the forecasts of barrel prices that are too uncertain and discounted (except for the calculation of development expenses), Siegel, Smith and Paddock provided a timing response to the investment problem in oil well concessions based on price

Surplus Value Linked to the Option to Invest

85

volatility. According to Kaplan and Weisbach (1992), the reasons for deciding to abandon an investment project may come from a shift in strategic direction. 2.3.3. Valuation of the abandonment option by investors Lang et al. (1995) justify the exercise of the abandonment option to the extent that the company enters a debt reduction logic. Moreover, as suggested by John and Ofek (1995) as well as Comment and Jarrel (1995), the abandonment can result from a desire to refocus the activity on the part of the company. In order to estimate the exit value of a company, Berger et al. (1996) explain that US investors incorporate the abandonment option price into their equity valuation. However, if the approach is relevant, it turns out that, in a pragmatic way, the question of the accuracy of the pricing of this option remains unresolved. In fact, the exit value is usually negotiated confidentially between the various stakeholders involved in the transaction; the market knows precisely the price once the transaction is concluded. Clark et al. (2010) study 144 divestitures of British companies listed in the LSE19 between 1985 and 1991. The objective is to find out whether UK investors are valuing options for abandoning the assets of their company (similar to US put options) in exchange for an exit value. If necessary, the authors wish to take an interest in the methodology of their pricing. Finally, Clark, Gadad and Rousseau wonder if a possible bad pricing is the consequence of a lack of information related to the fact that the negotiations concerning the takeover are private and/or if other reasons can justify this phenomenon. The three researchers consider that the valuation of this type of abandonment options results from a process of spreading the value of an asset that can be sold. This value Vt, which is the current value of future cash flows, follows the following geometric Brownian motion: dV(t) = α V(t)dt + σ V(t)dz(t) where: – α: growth rate of the value of the asset;

19 London Stock Exchange.

[2.143]

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Investment Decision-making Using Optional Models

– dz(t): the Wiener process having a zero average and variance equal to dt; – σ: standard deviation of the asset. Also, the exit price S follows a geometric Brownian motion: dS = π S(t)dt + ω S(t)dw(t)

[2.144]

where: – π: growth rate of the value of the asset; – dw(t): the Wiener process having a zero average and variance equal to dt; – ω: standard deviation of S. Since both Wiener processes are correlated, it is possible to write: dz(t). dw(t) = ρ. dt

[2.145]

Consequently, the value of the abandonment option denoted F is a function of V and S. After taking into account the change of variable = and using Ito’s lemma, F is the result of the following second-order differential equation: F=

.

+

+ SK g

[2.146]

with –

=

–

=

− , where Rg is the required rate of return for g; −

−

+

; (

–

=

–

=

–

=−

–

∗

=

)

; −2 (

+ )

∗

(

)

(

)

; ;

, which corresponds to the optimal exit value;

Surplus Value Linked to the Option to Invest

87

– γ: probability of occurrence of an event causing the exercise of the abandonment option. This event is random and rare. It takes place, in principle, before the report V over S reaches, for the first time, the level of g*. It may be a crisis of liquidity, reorganization, hostile takeover bid or a change in regulation. According to Clark, Gadad and Rousseau, γ is proportional to the variance of the return of the company to the extent that the large variations in return are the result of situations such as those that characterize the existence of this variable. The three researchers find that an increase of α, r and/or ρ increases the value of g* and decreases that of F. On the other hand, an increase of π and/or κ decreases the value of g* while increasing F. In addition, at the time of the announcement of the abandonment of the asset, F is equal to S*. In response to their question about a possible error regarding the pricing of the abandonment option, Clark, Gadad and Rousseau distinguish two scenarios: – in a situation of market efficiency, the asset selling price is public information. As a result, investors are able to make a fair valuation; – otherwise, the information about the exit value remains confidential. Thus, the market is likely to make valuation errors. The three researchers’ empirical study focuses first on calculating the abnormal returns of their sample. They define them as the difference between a return observed at a specific moment corresponding to an event date and the return obtained from the formula of the market line: the coefficients resulting from a linear regression over the observation period. They retain an observation period of 118 days in addition to a 61-day event period. The latter begins 30 days before the announcement date of a disinvestment and ends 30 days after it has taken place. Then, the authors add the abnormal returns of the event period to obtain cumulative abnormal returns. In addition, they calculate abnormal capital gains by applying the accumulated abnormal returns to the market capitalizations recovered on the eve of the respective exit announcements. Thus, Clark, Gadad and Rousseau observe that abandonment options are undervalued by about 1% due to early exercise and that the 22% rate of abnormal capital gains is justified by this undervaluation. On the other hand,

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Investment Decision-making Using Optional Models

the options approach could make it possible to value investment projects that could generate growth opportunities. 2.4. Growth option resulting from investment decisions and acquisition strategies The financial literature reveals that the diversification of activity, resulting from investment decisions and strategic transactions in the control market, generates a diversification discount for the company at the initiative of the offer. The reasons are mainly related to agency problems and lower future growth prospects than those of companies operating in a single business segment. As a result, Long et al. (2004) explain that companies with growth options prefer to delay decision-making. Interactions between financing and investment decisions favor the existence of growth options. In fact, Childs et al. (2005) argue that financing choices regarding debt decisions and possibilities to increase debt or possibilities to reduce it give shareholders the opportunity to overinvest or underinvest according to the application of an exercise strategy. Under these conditions, the shareholders wish either to recover on their behalf the wealth of the creditors (over-investment) or to avoid that their wealth is developed for the benefit of the creditors (under-investment). Resulting agency conflicts reduce enterprise value, as well as optimal leverage effect while increasing credit risk. The growth options are found, if any, within the acquisition strategies to the extent that the potential buyer would express the wish to buy those that are missing but that the target has. Smith and Triantis (1995) develop this idea by conceiving that two companies are able to benefit from a common development resulting from a simultaneous sharing. In fact, the potential buyer can provide the missing resources to the target company that offers its opportunities. Value creation occurs when synergies related to these interactions are found. Thus, research carried out after the application of this theory empirically proves that this type of growth potential is related to acquisition strategies.

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89

2.4.1. Company profiles justifying growth option value From their various studies, Montgomery (1994), Scharfstein and Stein (2000), as well as Rajan et al. (1998) believe that companies with diverse activities encounter problems with agency theories. These managerial difficulties then explain why, in this type of company, a diversification discount is noted. However, Chevalier (1999) seeks to prove that diversification of activity is not synonymous with value destruction. In fact, by examining the behavior of companies that have merged, it shows that it does not evolve before and after the transaction. The announcement of a change aimed to diversify the activity even results in positive returns. Campa and Kedia (1999) argue that it is not diversification that is destructive of value. According to them, it is the specific characteristics of the companies that justify their decision to diversify and, beyond that, the application of a haircut. In this context, they show that once a company is controlled, the diversification discount becomes so low that it disappears in some cases. Bernardo et al. (2000) focus on proving that real options can explain all or part of the existence of the diversification discount. Through their empirical study, it appears that the market value of companies operating in more than one business segment is less than the sum of the market values of companies operating only in one of the corresponding industries. In fact, the market value of companies in an industry includes the value of diversification options while companies with several businesses have exhausted their growth options. Three results support this conclusion: – they find a positive correlation between the value of the growth options of a company and the number of new industries in which the company could invest; – they observe that companies in more than one business segment have fewer growth options than companies with only one business in the same segments. They find a justification in the fact that the former have, compared to the latter, lower research and development and tangible fixed asset expenditures, as well as higher cash flows and structural size; – with reference to the parameters of real options, they find that the diversification discount increases, as research and development expenditures and market volatility increase and decrease with the age of the company. Long et al. (2004) explain that the current value of growth options is even greater as the company invests its expenses in research and development,

90

Investment Decision-making Using Optional Models

when it is subject to high growth rates and past volatility levels and when it is listed on NASDAQ. To justify this, they find a negative correlation between the level of investment and the current value of real options. This implies that companies with more growth options further delay their decision to undertake investment projects. Moreover, they observe a low level of the value of the growth option when the company has diversified activities. In this context, they find that a company operating in a weakly competitive environment is more willing to postpone the decision-making time to invest, certainly because there is little risk that competitors will question the value of the real option. According to the researchers, the value of a company is equal to the valuation of its assets, to which we must add the value of future growth prospects related to the launch of new products or to the improvement of market conditions. Companies with large PERs are considered to have some growth potential. Thus, their market capitalization incorporates the value of this type of real option, i.e. the possibility of investing in value-creating investors in the future. They carry out a study based on financial data of 619 companies for the year 1992 and 871 companies for the year 1997. It consists of determining the implicit value of the growth options held by the listed companies in their sample. To do this, they subtract the market capitalization value from the current value of future operating cash flows (after deducting the amount of the debt). As a result, the interactions between financing and investment decisions can also explain the presence of growth options. 2.4.2. Growth option value related to interactions between financing and investment decisions Childs et al. (2005) examine the interactions between financing and investment decisions by integrating the concept of growth options. In fact, they consider that a company holding a portfolio of assets (the value is noted A) has the possibility to exercise or not a growth option allowing it to invest in a new portfolio of assets (the value is noted G). The underlying therefore corresponds to the value of the assets already held, the exercise price to the amount of the debt and its maturity date. Shareholders are then encouraged to invest more or less in the business, depending on whether the option increases or decreases. This is similar to the possibility of postponing the

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91

investment. Investment distortions reduce the value of companies as well as their optimal leverage and increase credit spreads. The authors find that the reduction in debt maturity mitigates these investment distortions. In this context, we have () () () ()

= (μ − δ )dt + σ dZ (t)

[2.147]

= (μ − δ )dt + σ dZ (t)

[2.148]

and dZ dZ = ρdt

[2.149]

where: – µA and µG: respective expected constant rates of return for asset portfolios A and G; – δA and δG: respective constant dividend rates for asset portfolios A and G; – σA and σG: respective volatilities of asset portfolios A and G; – R: correlation coefficient between asset portfolios A and G. The value of the asset portfolio in place (A) represents the current value of the cash flows generated by these assets after deduction of taxes and without taking into account the cash outflows related to the repayment of the debt. Moreover, the authors assume that δA is not affected by the level of indebtedness. Thus, if the total amount payable δA.A is less than the after-tax debt service amount. The deficit is financed by a new contribution from the shareholders. The previous interpretation is the same for asset portfolio G, except that shareholders would forego the additional cash flows of new investments if the growth option relating to asset portfolio G is not exercised. Childs, Mauer and Ott further explain that the optimal amount of debt and its maturity are determined by a compromise that takes into account the tax benefits related to interest, the various costs related to this means of financing, and the relative costs of agency conflicts between shareholders and bondholders. They specify that the shareholders have the possibility of abandoning the company to its creditors, which also supposes taking into

92

Investment Decision-making Using Optional Models

account costs of bankruptcy. In fact, in the event of default, equity is assumed to be zero and creditors take control of the structure. In the event of exercise of the growth option, the model suggests replacing new assets with a fraction γ of the assets held at the base by the company. In this context, if γ = 0, the exercise of the real option leads to replacing “old” assets with new ones. The exercise policy of the growth option maximizing equity value consists of a sequence of decisions made over time on the A, G, F and m values. Assuming this option is not yet exercised; it is defined by a set of exercise policies such as: Φ = ϕ(G, A, F, m, t) ∈ O, I ; G ≥ 0, A ≥ 0, F ∈ L, m ∈ M, t ∈ 0, T [2.150] where: – Φ: value of the growth option; – F: amount of the debt; – m: maturity of the debt; – t: time, i.e. the maturity of the option. Thus: ϕ(G, A, F, m, t) = O if the option is not exercised

[2.151]

ϕ(G, A, F, m, t) = I if the option is exercised

[2.152]

The choice of financing policies according to the level of indebtedness (F ∈ L) and its maturity (m ∈ M) depend on the following equations: L = F(G, A, m, t) ≥ 0; G ≥ 0, A ≥ 0, m ∈ M, t = 0, m, 2m, … , T − m [2.153] and M = m(G , A ,

, 0) ∈

,

,…,

; G ≥ 0, A ≥ 0,

≥ 0, = 0 [2.154]

Note that E ( , , , , ) is the market value of equity if the growth option is not yet exercised, and E ( , , , , ) is the market value of

Surplus Value Linked to the Option to Invest

93

equity if the option is exercised. By applying the partial differential methodology, we then have

Max ϕ ∈Φ

1 σ A E 2

+ 2AGσ σ ρ E

+σ G E =0

+(r − δ )A E

+(r − δ )G E

+ E

− rE + δ A − cF(1 − τ) [2.155]

Max ϕ ∈Φ

1 2

+2

+

+( − +( −

)

)

=0

+ +

+ (1 − )

− −

(1 − ) [2.156]

where: – the S index: exercise policy of the growth option; – τ: corporate tax rate; – if the option is not exercised, flows attributable to the shareholders;

A − F(1 − ) represents the cash

– if the option is exercised, (1 − ) A + the cash flows attributable to the shareholders;

G − F(1 − ) represents

Solving the above two equations, we have E G A, F , m , t , A, F , m , t = E G A, F , m , t , A, F , m , t − K , A ≥ 0, t ∈ [0, T]

[2.157]

E G E G

[2.158]

A, F , m , t , A, F , m , t = A, F , m , t , A, F , m , t − K , A ≥ 0, t ∈ [0, T]

where: – K represents the exercise price of the option;

94

Investment Decision-making Using Optional Models

–G ( , , , ) and G ( , , , ) represent the critical values of the underlying assets of the growth option for which shareholders exercise optimally their right; – G and G are not fixed exercise points, but exercise limits that depend on A, FS, mS and t; – FS and mS indicate that the policy of exercising shareholder growth options has an influence on the choice of financial policy. The level of debt and the associated maturity are the solutions of the following maximizations: Max E G ,A ,F ,m ,0 + F ≥ 0, m ∈ M (1 − κ)D G , A , F , m , 0 Max E G, A, F , m , t + (1 − κ)D G, A, F , m , t − F F ≥0

[2.159] [2.160]

where: – κ: share of debt issuance costs in the market value of the new debt issued; – t = m, 2m,…, T – m; – j = O, I. In equation [2.159], the company chooses the initial value of the debt F as well as its maturity to maximize, at time t = 0, the sum of the market value of equity E and the market value of net debt issuance of issuance costs D . Assuming that the company chooses mS < T and depending on the status of the exercise policy (O or I) at maturity dates of the debt, in equation [2.160], the company chooses a sequence of face values that maximizes the sum of the equity value and the market value of net debt issuance of issuance costs minus the repayment of the nominal amount of the maturing loan.

Surplus Value Linked to the Option to Invest

95

The values of equity and debt must also meet the boundary conditions in bankruptcy. In this case, the shareholders must choose the right time to declare the company in default and transfer ownership to the creditors. Thus: E G, A

G, F , m , t , F , m , t = 0, G ≥ 0, t ∈ [0, T], j = O, I

D G, A G, F , m , t , F , m , t = (1 − b)V G, A 0, t ∈ [0, T], j = O, I

[2.161]

G, F , m , t , t , G ≥ [2.162]

where: – the B index: bankruptcy situation; – b: share of the costs of bankruptcy in the enterprise value; – for a certain level of G, A ( , , , ) ≥ 0: the optimal value of the assets in place for the shareholders to abandon the company to the creditors; – ( , ( , bankruptcy.

,

, ), ): market value of the indebted company in

Finally, the values E rs and E following conditions: E

G, A, F

, m , T = Max A − F

must satisfy on the horizon T the + Max G − γA − K , 0 , 0 [2.163]

, m , T = Max (1 − γ)A + G − F E G, A, F G − K ,0 ,0

+ Max γA − [2.164]

If the growth option is not exercised at time T, condition [2.163] indicates that the shareholders will hold the maximum of the net value of the assets in , to which we must add the value of the payment of the place, A − F exercise of the growth option. If the company is bankrupt, the equity value is zero. The interpretation is identical with respect to condition [2.164], except that there is a change in the value of the exercise price of the shareholders if the option is exercised. Childs, Mauer and Ott then examine the costs associated with maximizing the equity value. To do this, they design a model in which an idealized situation is taken into account. In fact, they assume that the signing of new investment contracts involves no cost, making it easier for the

96

Investment Decision-making Using Optional Models

company to make decisions. The latter can then consider its investment policy in terms of growth options and maximize its enterprise value. Considering the index f as the best choice of investment policy, let us note ( , , , , ) is the market value of the company that the function (equity and debt) if the growth option is not exercised and the function ( , , , , ) is the market value of the company if the growth option is exercised. By applying the partial differential methodology, we then have

Max ∈ L, ϕ ∈ Φ,

+2

1 2 ∈ M

+ + (r − δ )G V

+(r − δ )A +

=0

− rV + δ A + τcF [2.165]

Max ϕ ∈ Φ, ∈ L,

+2

1 2 ∈ M

+

+(r − δ )A V + V

+ (r − δ )G V

=0

− rV + (1 − )δ A +

G + τcF [2.166]

where ϕ refers to all of the exercise policies that maximize enterprise value and F and are the corresponding financial policy choices. Enterprise values V and V meet the following conditions: V G A, F , m , t , A, F , m , t = V G A, F , m , t , A, F , m , t − K , A ≥ 0, t ∈ [0, T] [2.167] V G A, F , m , t , A, F , m , t = V G K , A ≥ 0, t ∈ [0, T]

A, F , m , t , A, F , m , t − [2.168]

The boundary conditions of the initial debt issuance, as well as its maturity dates are

Surplus Value Linked to the Option to Invest

V G ,A ,F ,m ,0 = E G , A , F , m , 0 − (1 − κ)D

G ,A ,F ,m ,0

97

[2.169]

V G, A, F , m , t = E G, A, F , m , t + (1 − κ)D G, A, F , m , t − , t = m, 2m, … , T − m, j = O, I [2.170] F Finally, the enterprise value satisfies the following conditions: V G, A, F

, m , T = 1 − bI

Max[A, (1 − γ)A + G − K ]

V G, A, F

, m , T = (1 −

)Max[A, (1 − )A + G −

[2.171] ]

[2.172]

The authors then perform numerical simulations and find that shareholders would benefit from overinvesting in the growth option in the case where the investments that can be conducted (underlying assets) are more risky than the assets in possession of the company and even more so if they are given the opportunity to substitute a portion of existing assets with new assets. The value of equity then increases through a transfer of wealth between creditors and shareholders. The exercise of this option at a sub-optimal level therefore reduces the enterprise value and increases the credit spread. As a result, the authors identify an agency conflict between different stakeholders of the company. The agency cost of this overinvestment negatively affects the optimal level of indebtedness. In addition, shareholders would tend to underinvest in the growth option if the decision is motivated by an expansion of the asset portfolio. They will prefer not to exercise the option when the company holds risky debt for not having to share wealth generation with creditors. As a result, the enterprise value decreases, the debt margin increases and the optimal debt ratio decreases. In other words, the financing decisions of the company include the choice of the initial level of debt and its maturity, as well as the option of increasing or reducing the level of indebtedness in the future. Depending on the characteristics of the growth option, shareholders have the opportunity to overinvest or underinvest according to an exercise strategy aimed to maximize enterprise value. These investment distortions are motivated by the shareholders who wish to either transfer, in their favor, the wealth of the creditors (overinvestment) or avoid the development of wealth for the benefit of the creditors (underinvestment). These agency conflicts significantly reduce the value of the company, as well as the optimal

98

Investment Decision-making Using Optional Models

leverage effect and increase the credit risk. Interactions and the birth of growth options can also occur in acquisition strategies. In fact, a target company may hold growth options that are lacking to the potential buyer who wishes to recover them. 2.4.3. Acquisition strategies by the real options approach According to Smith and Triantis (1995), two companies may share common development bias within a given sector. In fact, acquisitions are often synonymous with growth options through the interactions between the buyer and the target. In other words, the target may have growth options that the buyer lacks, while the latter may have growth options that he/she cannot exploit without taking control of another company. It is thus possible to buy a company present in a particular business segment, but it does not have the means to maintain its level of growth. The potential buyer – with the ability to provide the target company with the missing resources – can then invest less than the estimated value of the company once bought. Thus, synergies or new acquired distribution chains can lead to value creation through new investment opportunities. Kester (1984) provides the first set of empirical estimates of the value of growth options. It measures the growth value of a company as the difference between its market value and the current value of its cash flows. The growth value expressed as a percentage of the market value can, according to him, reach the threshold of 90%. McDonald and Siegel (1986) believe that the carry option of an investment also relates to corporate acquisition strategies. The model of choice they build should allow the company to define the ideal time frame for investing in a project of a given size. By performing simulations, they analyze the value lost during the waiting period compared to the value gained during the same period. They conclude that temporal considerations are quantitatively important for a multitude of parameters. In other words, a too hasty time frame can lead to a loss of value of the order of 10 to 20%. Ideally, it is optimal to wait for the moment when the gains are twice the investment costs knowing that the greater the uncertainty related to the income of the investment project, the more the company will require a high current return of the project to invest.

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99

Folta and Miller (2002a) examine the factors that influence the decision to acquire shares from a sample of 285 companies specializing in the biotechnology industry where research is dominant. The decision is to choose between flexibility and commitment. The options theory they propose to consider incorporates the effects of uncertainty, the assessment of developing technologies and the threat of intense competition. The resolution of uncertainty about the decision to invest in high added-value technologies is all the more motivated if the underlying growth option is subject to competition. They conclude that although it seems appropriate to delay investment in uncertain conditions (share purchases are more easily operable when uncertainties are low), there may be opportunity costs to waiting for exercise real options: Companies can forego cash flows or learning opportunities in addition to being outpaced by competitors. The opportunity costs of delaying the investment correspond to the dividends not received during the term of the option. Thus, in the absence of a dividend, Foltat and Miller (2002b) observe that the best strategy is always to wait until the maturity of the option. In other words, dividends or opportunity costs are the main reason for early exercise. The learning process is also considered by the works of Dapena and Fidalgo (2003). The latter study the acquisitions with an approach consistent with that of real options, that is to say, by considering the value of the control premium sequentially and not uniquely in time. The authors consider a waiting option in which the learning process lies and a growth option because the acquisition offers growth opportunities. Thus, a significant minority takeover gives access to information and offers the possibility of being present in the management bodies while a majority shareholding confers control rights. Inspired in particular by the work of Kester (1984), Reuer and Tong (2007) empirically study the values of the growth options of 293 companies divided into 19 industries between 1989 and 2000. On average, the real options represent 43% of the market value. However, both researchers find a significant heterogeneity in their results. For example, they report an average of 54% in the electrical equipment sector and an average of 22% in the textile sector. The major conclusions of their analysis concern the nature of investments that may contain more value related to growth options. In fact, Reuer and Tong distinguish that the investments of companies in research and development on the one hand, by forming joint ventures on the other hand, most integrate the potential of values of growth options. These

100

Investment Decision-making Using Optional Models

observations stem from the fact that research and development projects operate in stages and are therefore likely to generate value based on future business investment opportunities. With regard to joint ventures, the authors note that, for the entire population studied, only those that are minority in the capital of the investee company can capture the most value related to growth options. They explain this phenomenon by starting by recalling that, according to the work of Reuer and Leiblein (2000), joint ventures do not reduce the risk of companies because of the organizational complexity they generate. Then, Reuer and Tong suggest that small joint ventures can be a particularly rewarding means, heralding future commitments. This interpretation is similar to the work of Hurry et al. (1992). The latter noted that Japanese investors are following an optional strategy with small investments to capture a wider range of future growth opportunities. Collan and Kinnunen (2009) illustrate the strategic interest of real options in the acquisition process by relying on the acquisition in 2002 of Partek Inc. by Kone Inc. The company initiating the offer was able to create value by anticipating the possibility of splitting the assets and activities of the target while considering sequential abandonment options. In addition, the providers of securities, namely the initial shareholders of Partek (including the Finnish government), neglected in their assessment the flexibility of this type of development without paying attention to the value of strategic options. This case echoes the article by Alvarez and Stenbacka (2006). In fact, they expose different types of real options present in mergers and acquisitions, including the sale of units of the acquired company. The resulting divestiture option is considered sequential. The synergies resulting from the reorganizations must be a factor to be taken into account in the definition of the optimal time frame and acquisition prices. Before the Kone and Partek transaction takes place, Collan and Kinnunen (2009) note that the company initiating the offer has historically grown through external growth within a relatively mature industrial sector (elevators and escalators). In this context, the group is inclined to look for targets to restructure its business. Partek is a conglomerate of engineering companies whose stock price is, at this time, less than the substantial value of the company. In 2002, Kone launched a hostile takeover bid for all of Partek’s activities (including those that did not correspond to its core business) and, under the terms of the negotiations, the price paid was same as €1.45 billion. The workforce of the target was approximately 12,450 and the turnover was about €2.74 billion, which is about the size of Kone. The

Surplus Value Linked to the Option to Invest

101

activities that Kone considers compatible with its core business – namely elevators and escalators with their handling activities – have been valued between €960 and €1.04 billion and have integrated a separate division of the group, renamed in 2004 Kone Cargotec. This concerns Hiab companies (valued between €440 and €490 million) and Kalmar Industries (valued between €520 and €550 million). Then, between 2003 and 2004, Kone “exercised abandonment options” by selling, for a total amount of €1.15 billion, various units of Partek considered non-strategic. At the end of 2004, Kone purchased MacGregor Inc. to recover its marine handling and maintenance services. In June 2005, Kone divided the group into two entities: – Kone Corporation, which grouped together the group’s flagship activities (elevators, escalators and automated doors); – Cargotec Corporation, which encompassed the activities of newly acquired companies, namely Hiab, Kalmar and McGregor. The stated objective of this split was to form two more efficient synergistic companies with greater growth potential that enhance the shareholder value appreciation of the group. In 2007, the total workforce of the Kone Group was approximately 32,500 and the turnover was €4.08 billion. Thus, unlike Partek, Kone’s acquisition experience has allowed it to make rapid structural and strategic changes. Consequently, as mentioned by Boer (2002), the value of a company does not only include the cash flows generated by its assets, namely the economic value, but also the strategic value including the human and intellectual capital that allows us, thanks to know-how, to turn plans into economically viable operations. Agliardi et al. (2016) develop a model for analyzing changes in capital structure after a merger–acquisition transaction that incorporates the effects of synergies and operational growth options. Then, they test their model on a large panel of 1,121 acquisitions in the United States between 1980 and 2010. Their work shows that there are implications and adjustments between financial leverage, growth opportunities and merger gains within the merger–acquisition process. Their model results in new predictions for acquisitions that have been made in industries other than those of the acquiring company by taking into account changes in levels of indebtedness and merger gains related to the effects of diversification and growth options for the buyer and the target. The value of the latter depends on the differences in volatility between companies and the inclusion of the costs of

102

Investment Decision-making Using Optional Models

bankruptcy. In particular, the model predicts that, after the transaction, companies that merge and have a low correlation between their cash flows will see more merger gains while seeing an increase in their leverage. In addition, the authors note that because of their merger, companies that experience lower volatility and lower costs of bankruptcy have higher merger gains and levels of indebtedness. Moreover, companies with less correlated activities, significant growth option values, volatilities and low costs of bankruptcy have greater leverage, which increases after the transaction. This type of company also benefits from higher stock market returns around merger listings. Finally, the three researchers distinguish the existence of a “U”-shaped relationship between changes in levels of indebtedness (between before and after the transaction) and the value of growth options. In this context, they assume that the turnover levels for a potential buyer A and a potential target T follow the following correlated geometric Brownian motion: A, T = i et

= a dt + σ dz

[2.173]

where: – ai and σi > 0 are constant parameters; – dz is the Wiener process for the company i. The revenue processes of both companies are correlated with: dZ dZ = ρdt et − 1
1, where eM: relative share of synergies linked to the growth of the turnover of the entity M. If eM = 1, then there is no synergistic surplus linked to the taking of market shares. In that case: C = (1 − s)(C + C ) and P = e (P + P )

[2.175]

The authors also assume that merged companies may hold growth options that have a common maturity T1. The growth factor related to growth options is: e , , i = A, T, M

[2.176]

The exercise price of the growth option is as follows: I , , i = A, T, M knowing that

[2.177] >

,

for the growth option to be exercised.

Thus, in the event of exercise of the option, the company M receives, for a cost equal to IM,G: P

,

=e

,

P

,

+P

,

[2.178]

In the case where the companies operate one without the other, after the exercise of their growth option, they receive (individually), for a cost equal to Ii,G,i = A, T: P , = e , P , avec e , > 1

[2.179]

Considering the foregoing, Agliardi, Amel-Zadeh and Koussis specify that it is possible to study cases where the growth options exist before the merger and cases in which they are created at the end of the merger. Moreover, the authors assume that there are no merger costs and incorporate the cost of bankruptcy equal to bi, i = A, T, M. Thus, to model the process of income trends described in equation [2.173], the authors use a two-dimensional binomial model. Decisions are assumed to be made at each step Δt. The parameters of bullish (u) trends, bearish (d) trends and joint probabilities for the variations of turnover of the company initiating the offer and the target are the following:

104

Investment Decision-making Using Optional Models

u =e

√∆

et d = e

√∆

[2.180]

P

= 0,25 1 + ρ + √∆t

+

[2.181]

P

= 0,25 1 − ρ + √∆t

−

[2.182]

P

= 0,25 1 − ρ + √∆t

+

[2.183]

P

= 0,25 1 + ρ + √∆t

−

[2.184]

c and δ is the opportunity cost parameter, such as where = ( − ) − waiting according to the Dixit and Pindyck model (1994). The correlation between the turnover levels of the two companies is integrated in the probabilities. At each stage of the two-dimensional tree, the incomes of the two companies can move jointly upwards (with the probability Puu) or jointly downwards (with the probability Pdd) or upwards for the buyer and downwards for the target (with the probability Pud) or downwards for the buyer and upwards for the target (with the probability Pdu). A stronger correlation between the turnover of the two companies increases the probabilities of jointly upward and downward movement, respectively, Puu and Pdd, thereby reducing the probabilities of opposing movements, respectively, Pud and Pdu. The current leverage of the company i is defined by the level of the coupon Ri, i = A, T, M. It is assumed that companies choose optimal coupon levels at t = 0, thus maximizing the value of the indebted company. As a result, they no longer modify their financial structure. The optimal solution is the one that maximizes the value of the indebted firm = A, T, M à = 0. Thus, by proceeding inductively from the end horizon, the nodes of the tree are calculated, namely the values of the company initiating the offer, the target company and the company resulting from the merge using the following expressions: E , = max P , − C − R (1 − τ)∆t + E , , 0

[2.185]

If E , > 0: V , = P , − C (1 − τ)∆t + V ,

[2.186]

Surplus Value Linked to the Option to Invest

105

BC , = 0 + BC

[2.187]

TB , = τR ∆t + TB

[2.188]

D , = R ∆t + D

[2.189]

V, = E , + D ,

[2.190]

If E , = 0: V , = P , − C (1 − τ)∆t + V ,

[2.191]

BC , = b V ,

[2.192]

TB , = 0

[2.193]

D , = (1 − b )V ,

[2.194]

V, = E , + D ,

[2.195]

where: – VU: non-indebted company; – VL: indebted company; – TB: tax benefits related to indebtedness; – BC: cost of bankruptcy; – E: equity value; – D: value of the debt; – ( ): current expected value of the variable x, namely: x (t) = P X

,

+P X

,

+P X

,

+P X

,

e [2.196]

In the presence of growth options, the logic is the same, except that the value of equity includes the value of a compound option E with a maturity of T1:

106

Investment Decision-making Using Optional Models

E,

= max E ,

P

− I , ,E , P

[2.197]

2.5. Conclusion Dixit and Pindyck (2001) consider that the investment opportunity that a company seizes is akin to a financial call option. It embodies the right and not the obligation to acquire an asset corresponding to the right of access to the profit streams generated by a project on a date chosen in the future. If the investment is made, the exercise of the call option is considered to be undertaken. The managerial flexibility that real options offer is a major asset in helping decision-making. Beyond designing models specific to each category of options, the financial literature, which has assimilated all the complexity that governs investment projects, also seeks to associate them. The combination of options (and not considering them in isolation) gives less value to the project, but gives greater flexibility to the company’s reaction possibilities. Taking into account specific projects, based on development phases requiring significant research and development expenditure or on particularly volatile sectors, leads to inventing models that optimize the right time to invest. Empirically, companies tend to wait and see future prospects only when they find that their level of information is more accurate. In addition, companies can use the optional approach when it comes to external growth and financing policy. In fact, acquisition strategies can be founded when the buyer considers the potential of the target’s growth options. However, the diversification of activities reduces the value of these options. Thus, companies are still encouraged to delay their decision-making. The financial structure and agency conflicts between the various stakeholders encourage shareholders to prefer strategies equivalent to option exercises. Depending on the debt situation of the company, shareholders may wish to overinvest to recover value or, on the contrary, under-invest in order not to enrich the creditors. After focusing on a literature review of the optional modeling of investment choices and of the surplus value associated with the option to invest, I considered it useful to put into practice various optional models, such as those of Dixit and Pindyck and Magrabe, by establishing different simulations of investment projects.

3 Data Generation Applied to Strategic and Operational Option Models

3.1. Introduction The different applications presented in this third chapter are intended to verify the practicality of the methods developed in the literature review. Since the companies maintain the confidentiality of their investment projects internally and, as a result, these elements are not available to the general public, in order to carry out my calculations, I had to make prior assumptions regarding amounts, duration and rate. They justify the interest and usefulness of applying the real options approach as a tool to assist investment decision-making.

3.2. Determining the right time to invest The carry option allows the company to postpone the decision date of a planned investment (before its maturity) provided that the elements are favorable.

Investment Decision-making Using Optional Models, First Edition. David Heller. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Investment Decision-making Using Optional Models

3.2.1. Application to the carry option Assuming that a company sees the possibility of acquiring a portfolio of assets, such as machines worth €3,600 million, the investment should generate each year from the following year: – either with a probability of 50%, annual cash flows of €390 million; – or with a probability of 50%, annual cash flows of €210 million. This difference in flow is explained by the fact that the company is waiting for the results of the call for bids in which it has just participated. In addition, to the extent that the investment is launched at the beginning of the year (i.e. before the response to the call for bids and related flows in 1 year), cash flows would be €300 million. Figure 3.1 summarizes the projected cash flows. 300

390 where 210

390 where 210

390 where … 210

t=0

t= 1

t= 2

t=3

…

Figure 3.1. Evolution of cash flows of an investment project

By postponing its 1-year investment project, the company will know for certain the cash flows generated by the project. With a discount rate of 6%, if the call for bids is unsuccessful for the company, its cash flows will be 210 million per year. The project will then have a value of €3,500 million1. In this case, the value of the project will be lower than the cost price (€3,600 million) and the investment will have to be abandoned. Considering a strategic flexibility related to the fact that the company can postpone its investment in 1 year, the binomial approach makes it possible to obtain another value. The project will be worth:

1 That is, 210/6%.

Data Generation

109

In t = 0: 390 × 50 % + 210 × 50% = €5,000,000 6% In t = 1: – either: %

= €6,500,000 with a probability of 50% 

– or: 210 = €3,500,000 with a probability of 50% 6% In this context, the project holds: – a capital gain of 6,500–5,000 = €1,500 million, or 30% of the value of the project; – a capital loss of 5,000–3,500 = €1,500 million, or −30% of the value of the project. Considering the cash flow rate of 6%2, the return of the project is as follows: – either 30% + 6% = 36%; – or −30% + 6% = −24%. By retaining a risk-free rate of 2%, the probability of risk-neutral p can be calculated: 36%. p − 24%. 1 − p = 2% and ≈ 43.3% Consequently, the value of the option is as follows: 43.3% × 6,500 − 3,600 + 56.7% × 3,500 − 3,600 1 + 6%

2 That is, 300/5,000.

110

Investment Decision-making Using Optional Models

= €1,132.08 million If the company invests immediately, it waives this optional value which represents an opportunity cost of €1,132.08 million. The overall cost price is therefore 3,600 + 1,132.08 = €4,732.08 million, and the adjusted net present value (ANPV) is 5,000 − 4,732.08 = €267.92 million. The project can be realized. 3.2.2. Application of the Dixit and Pindyck model It is assumed that an investor wishes to subscribe to a call option of a share of a company with infinite maturity at an exercise price equal to the current stock price, i.e. €100. On the day of the valuation, we admit that the price is liquid enough to constantly reflect the analysts’ consensus. The valuations they realize are supposed to come from the DCF approach. To carry out a valuation using the options method, the following assumptions are used: – the 10-year Treasury bill rate is 2%; – the market risk premium is 4%; – the beta of the company is 1.2; – the last dividend paid is €5; – the volatility of the annualized share is 25%. It is then possible to determine the expected return of the share μ from the Capital Asset Allocation Model (CAPM) based on the Sharpe model (1964): μ = 2% + 1.2 × 4% = 6.8% The dividend rate δ is: δ=

5 = 5% 100

Data Generation

111

The rate of appreciation α, which corresponds to the difference between the rate of return and the dividend rate, is: α = 6.8% − 5% = 1.8% In order for the option to be exercised, the stock price must reflect a new value, obtained by the DCF method, which is called V*. According to the Dixit and Pindyck model, we have: V∗ =

β .I β −1

where: 1 − r−δ− σ 2 β = σ

+ √Δ

with: 1 Δ= r−δ− σ 2 Δ = 2% − 5 −

1 × 25% 2

+ 2. r. σ

+ 2 × 2% × 25% = 0.006

and: β =

− 2% − 5% −

1 × 25% 2 25%

+ √0.006

= 2.25

Thus: V∗ =

2.25 . I = 1.803 2. I 2.25 − 1

The exercise price of the option is represented by I. As a result, the call option with infinite maturity will have to be exercised when the value of the underlying derived from the DCF method (reflecting the stock price) will be 1.8 times higher than the current price. Here, I = €100, which means that the

112

Investment Decision-making Using Optional Models

call option will be exercised when the stock price reaches the value of €180.32. The premium F of the option is then: F = A. V and A =

V∗ − I V∗

Then: A=

180.32 − 100 = 0.001. 180.32 .

and: F = 0.001 × 100

.

= 21.38

Since the option is on the valuation day at the money3, the intrinsic value is 0. Thus, the option premium exclusively consists of the time value. A sensitivity analysis on the value of the option can then be constructed. The latter is based on the value of the underlying and the volatility. Volatility

Price of the underlying (in €) 100

110

120

130

140

25%

21.38

25.44

29.76

34.32

39.10

26%

22.49

26.71

31.19

35.90

40.83

27%

23.59

27.96

32.59

37.46

28%

24.69

29.21

33.98

38.99

29%

25.78

30.44

35.36

30%

26.85

31.66

36.72

V-I

0

10

20

150

170

180

190

200

44.09 49.27

54.62

60.14

65.82

71.64

45.97 51.29

56.79

62.46

68.28

74.23

42.54

47.82 53.29

58.93

64.73

70.69

76.78

44.22

49.64 55.25

61.03

66.97

73.05

79.27

40.51

45.87

51.44 57.18

63.09

69.15

75.36

81.71

42.00

47.50

53.20 59.07

65.11

71.30

77.63

84.10

30

40

70

80

90

100

50

160

60

Table 3.1. Sensitivity of the premium of an option to the value of the underlying and the volatility

3 The exercise price I is equal to the value V obtained using the DCF.

Data Generation

113

3.3. Flexibility of asset exchange, abandonment and temporary shutdown of projects Margrabe’s (1978) model of valuing a project with an exchange option is modeled after Black and Scholes by replacing the price of the underlying with the price relationship of two interchangeable assets. In addition, the abandonment option is similar to a put option to the extent that the company plans to make a risky investment by anticipating the eventual resale of the project if an adverse external event, such as a less advantageous regulation, occurs. Finally, when calculating the value of an investment project, the temporary shutdown option makes it possible to take into account any interruptions in the operation of a machine or a concession during a given period. This implies knowing the volatility of the market price of the marketed product. Depending on the state of the market, the company can choose, over a period of time, to continue or stop the operation. The option value resulting from this operational flexibility finds its essence in the fact that the operation is launched or interrupted, on the date of decision, provided that the selling price of the product is, respectively, higher or lower than its cost price. 3.3.1. Application to the exchange option Assuming that a company is considering a project to operate a production system consisting of turning an asset 2 into an asset 1: – asset 2 has a value of €200 and asset 1 has a value of €400; – the volatility of asset 2 is 25% and that of asset 1 is 35%; – the correlation coefficient is 45%; – the production system must operate for 10 years and be implemented annually if the operation is profitable by observing the beginning exercise price, i.e. if the price of asset 1 is higher than that of asset 2. Under such conditions, the company has an option to exchange asset 2 for asset 1. This option must, at the beginning of these considerations, be exercised. The intrinsic value is then 200. The value of the option can then

114

Investment Decision-making Using Optional Models

be calculated using Margrabe’s formula (1978). The latter replaces, in the Black and Scholes formula, the price of the underlying (here asset 2) by the ratio between the price of asset 1 and that of asset 2 and the volatility of the underlying by the variable relating assets 1 and 2. According to Margrabe, the risk-free rate is zero, because the company is supposed to intervene on a market at equilibrium. The volatility of the combined variable of the two assets is as follows: σ σ

=

=

σ

+σ

− 2ρ. σ

.σ

25%² + 35%² − 2 × 40% × 25% × 35% = 33%

The value of the project is then equal to the sum of the 10 call options with maturities ranging from 0 to 9 years. Table 3.2 summarizes the value of these options by showing, in addition to the total values obtained from Margrabe’s formula (1978), the values of d1, d2, F(d1) and F(d2), where F represents the distribution function of the standard normal distribution. Maturity

d1

d2

F(d1)

F(d2)

0

Value of the option 200

1

2.29 1.96

0.99

0.98

200.55

2

1.73 1.27

0.96

0.90

203.72

3

1.51 0.95

0.93

0.83

208.24

4

1.39 0.74

0.92

0.77

213.14

5

1.32 0.59

0.91

0.72

218.08

6

1.27 0.47

0.90

0.68

222.91

7

1.23 0.37

0.89

0.65

227.58

8

1.21 0.29

0.89

0.61

232.08

9

1.20 0.22

0.88

0.59

236.39 _________ 2,162.68

Table 3.2. Value of an investment project with an exchange option according to Margrabe’s formula (1978)

Data Generation

115

3.3.2. Application to the abandonment option Assuming that a company wishes to make an investment worth €3,600 million, the investment should generate each year: – either with a probability of 50%, annual cash flows of €390 million; – or with a probability of 50%, annual cash flows of €120 million. This difference in flow comes from the fact that the company is waiting for a call for bids in which it has just participated. In the case of resale of the initial investment (if the company does not win the call for bids), it would recover €3,300 million. In addition, a discount rate of 6% and a risk-free rate of 2% are used. Then, the value of the project in t = 1 will be: – either: 390 = €6,500,000 with a probability of 50% 6% – or: 120 = €2,000,000 with a probability of 50% 6 % The expected value is therefore: 50% × 6,500 + 50% × 2,000 = €4,250,000 In this context, the project holds: – a capital gain of: 6,500–3,600 = €2,900 million or 81% of the value of the project; – a capital loss of: 3,600–2,000 = €1,500 million or −44% of the value of the project. Considering the cash flow rate of 6%, the return of the project is as follows: – either 81% + 6% = 87%;

116

Investment Decision-making Using Optional Models

– or −44% + 6% = −38%. By retaining a risk-free rate of 2%, the probability of risk-neutral p can be calculated: 87%. p − 38%. 1 − p = 2%

≈ 32%

Consequently, the value of the option is as follows: 32

%× ,

,

%× , %

,

= − €136 million

3.3.3. Application to the temporary shutdown option The investment project is similar to a portfolio of n calls (where n represents the number of dates during which the company decides whether or not to continue operating). The options then have the following characteristics: – value of the underlying asset: cash sales price of the manufactured product; – exercise price: production cost; – maturity date: date of decision to pursue or shut down the operation. Assuming that a company is thinking of responding to a call for bids on the operation of a raw material concession for a period of 10 years, the market price of the resource is currently €30 per unit extracted which can be resold. Its cost price is €22, and the volatility of its market price is estimated at 35%. The annual production capacity is 500,000 units. The continuous risk-free rate is 2%, and it is assumed that start-up or shutdown costs are negligible. First application: the company can make the decision to stop the operation once a year. Thus, the project is similar to a portfolio of 10 options with an exercise price of €22 and a price of the underlying asset of €30. In this context, the first option must be exercised immediately. As it has no time value, its premium is equal to its intrinsic value, i.e. €8 per unit extracted, which is €4 million for the entire production of the year. The other nine options are valued by the Black and Scholes (1973) formula.

Data Generation

117

The value of the project is the sum of the 10 options. The details of the calculation are given in Table 3.3. Rank of the option S0 E σ Continuous r Valuation date Maturity date τ d1 d2 F(d1) F(d2) Value of the call option per unit produced Sum of call option values Installed capacity Production over the entire period (10 years) Value of the concession (in million €)

0

1

2

3

4

5

6

7

8

9

30 22 35%

30 22 35%

30 22 35%

30 22 35%

30 22 35%

30 22 35%

30 22 35%

30 22 35%

30 22 35%

30 22 35%

2%

2%

2%

2%

2%

2%

2%

2%

2%

2%

1/1/ 2016 1/1/ 2016 0.00

1/1/ 2016

1/1/ 2016

1/1/ 2016

1/1/ 2016

1/1/ 2016

1/1/ 2016

1/1/ 2016

1/1/ 2016

1/1/ 2016

1/1/ 2017 1.00 1.12 0.77 0.8683 0.7788

1/1/ 2018 2.00 0.95 0.46 0.8302 0.6772

1/1/ 2019 3.00 0.91 0.31 0.8196 0.6208

1/1/ 2020 4.00 0.91 0.21 0.8179 0.5821

1/1/ 2021 5.00 0.92 0.13 0.8200 0.5528

1/1/ 2022 6.00 0.93 0.07 0.8239 0.5291

1/1/ 2023 7.00 0.95 0.02 0.8287 0.5092

1/1/ 2024 8.00 0.97

1/1/ 2025 9.00 0.99

8.00

9.25

−0.02 −0.06 0.8340 0.8394 0.4920 0.4768

10.59 11.73 12.71 13.60 14.39 15.12 15.79 16.42

127.61 500,000 /year 5,000, 000

638.045

Table 3.3. Valuation of a temporary shutdown option (possibility of exercise once a year for 10 years)

118

Investment Decision-making Using Optional Models

Rank of the option

0

1

2

3

4

S0

30

30

30

30

30

E

22

22

22

22

22

σ

35%

35%

35%

35%

35%

Continuous r

2.0%

2.0%

2.0%

2.0%

2.0%

Valuation date

01/01/2016

01/01/2016

01/01/2016

01/01/2016

01/01/2016

Maturity date

01/01/2016

01/01/2018

01/01/2020

01/01/2021

01/01/2023

0.00

2.00

4.00

6.00

8.00

d1

0.95

0.91

0.93

0.97

d2

τ

0.46

0.21

0.07

−0.02

F(d1)

0.8302

0.8179

0.8239

0.8340

F(d2)

0.6772

0.5821

0.5291

0.4920

10.59

12.71

14.39

15.79

Value of the call option per unit produced

8.00

Sum of call option values

61.49

Installed capacity

500,000/year

Production over the entire period (10 years)

5,000,000

Value of the concession (in million €)

307.467

Table 3.4. Valuation of a temporary shutdown option (possibility of exercise once every 2 years for 10 years)

Second application: the option is exercisable every 2 years (5 times in total during 10 years).

Data Generation

119

Third application: the option can be exercised every 5 years, i.e. twice in total for a period of 10 years. σ

35%

35%

continuous r

2.0%

2.0%

valuation date

01/01/2016

01/01/2016

maturity date

01/01/2016

01/01/2018

0.00

2.00

τ d1

0.95

d2

0.46

F(d1)

0.8302

F(d2)

0.6772

Value of the call option per unit produced

8.00

Sum of call option values

61.49

Installed capacity Production over the entire period (10 years) Value of the concession (in million €)

10.59

500,000/year 5,000,000 307.467

€30,746,710.53 Table 3.5. Valuation of a temporary shutdown option (possibility of exercise once every 5 years for 10 years)

Consequently, the lower the flexibility, the lower the value of the option. In absolute terms, if the option is exercised only once every 10 years, the value of the project corresponds to its intrinsic value, i.e. €8 per unit. In fact, in this case, it is exercised immediately and has no time value. Thus, a concession whose production capacity is 500,000 units per year, or 5 million units produced over 10 years, will have a value of €40 million.

Investment Decision-making Using Optional Models

Value of the investment project (in million €)

120

700 600 500 400 300 200 100 0 1

2

5

10

Number of exercise dates of call opons Figure 3.2. Sensitivity of the value of an investment project composed of call options with the number of call exercise dates

Value of the project (in million €)

In addition, the value of the project is higher as the price volatility of the underlying is also higher. This is justified by the fact that the vega of the calls is positive. 1,000 800 Value of the project with 2duexercise dates2 Valeur projet avec

600

dates d'exercice Value of the project Valeur projet avec with 5duexercise dates5 dates d'exercice Value of the project Valeur du projet avec 10 withd'exercice 10 exercise dates dates

400 200 0 1% 15% 25% 35% 45% 55% 65% 75% 85% 95%

Volality Figure 3.3. Sensitivity of the value of the project to the volatility of the underlying

Data Generation

121

3.4. Incorporation of development phases An option to carry out the second phase of an investment project is equivalent for a company to commit funds of a relatively small amount initially. This allows it to test its market in order to know if it is appropriate to make a second, more significant, investment. The latter, intervening at the end of the first period, may depend on factors such as deregulation or a reduction in operating costs. In addition, the uncertainty relating to each investment project leads us to consider the notion of risk during their analysis, especially if they are of a sequenced nature, i.e. each phase depends on the success or failure of the preceding one. In the event of failure, the capital invested until then is definitively lost, but the decision to stop the project makes it possible to avoid depreciating it further. In this context, the consideration of market uncertainty, i.e. market acceptance of the proposed product or service, depends on the ability of the latter to meet consumer expectations. 3.4.1. Implementation of a two-stage investment project Assuming that a company plans a first investment project such as the acquisition of an asset, i.e. a machine, it also considers the possibility of investing in a second project, provided that at the end of the economic and fiscal life of this machine, the market is deregulated. The characteristics of the two projects are presented in Table 3.6. The NPV of the first project is: − 3,600 + 390.

1 − 1 + 6% 6%

= − €1,682,000

In addition, it turns out that as the cash flows generated by the second investment project (if realized) are €12,000 million at the date of completion of this second phase; the current value of the underlying at the time of the valuation, namely, today, is as follows: 12,000 1 + 6%

= €8,459.5 million

122

Investment Decision-making Using Optional Models

First investment phase Investment

3,600

Annual cash flow

390

Maturity of the project

6 years

Retained discount rate

6%

Second investment phase Investment

20,000

Discounted future cash flows

12,000

Investment implementation horizon

6 years

Volatility of the shares of the business sector

25%

Risk-free rate

2%

Table 3.6. Characteristics of two related investment projects (amounts in million €)

In this context, the option can be valued according to the Black and Scholes formula: 25% 8,459.5 + 2% + 20,000 2 d = 25 % × √6

.6

= −0.90

d = −0.9 − 25% × √6 = −1.51 Therefore: C = 8,459.5 × Φ −0.90 − 20,000 × e

%×

× Φ −1.51

= 400 million € By adding the NPV of project 1 and the value of the option of project 2, we note that the project must be abandoned: − 1,682 + 400 = − 1,282 million € 3.4.2. Valuation of a sequential project The probability of success or failure of a project can be defined in a risk-neutral framework, as thought by the Black and Scholes model (1973)

Data Generation

123

or the Cox, Ross and Rubinstein model (1979). On the other hand, uncertainty may be closely related to the actual events, that is to say, to the possible successes or failures relative to the realization of a real innovation, the regulation of which will authorize the commercialization of the product or the service. This is for managers to determine a more or less optimistic subjective probability about the chances of success or failure of the corresponding event. The determination of this probability is based on the experience and knowledge of the executives. Assuming that Sysrev’s business consists of designing flight reservation management systems for airlines (flight reservation, addition of services, change of flights and cancellation of reservation), a project to market their new system interests three companies. The investment includes a server, reservation systems, the cost of the installation and the operating license. Sysrev begins by analyzing the needs of its potential customers, and then the company defines the price structure of its systems. In fact, the invoices will be established based on a simple matrix, according to a fixed price for each system type. After carrying out a market study, Sysrev prepares its initial prices and provides the traffic structure of the type of actions performed on each of the reservations (simple reservation, addition of services, change of flight, cancellation of reservation). The company also expects to lower prices by 10% per year starting in the second year to encourage demand and to eliminate the entry of competition into this market. Type Service 1: standard reservation Service 2: adding of service Service 3: change Service 4: cancellation

Rate

Weighting

0.5

45%

1

25%

1.5

20%

2

10%

Table 3.7. Rates according to possible actions in €

Year

1

2

3

4

5

Traffic

5

7

14

19

21

Table 3.8. Traffic forecast (in millions of possible actions)

124

Investment Decision-making Using Optional Models

The investment expenditure is as follows: – Server: €12,000,000. – Reservation system, or RS (€5,000,000 each): - a first system is proposed by Sysrev the first year (year 0): it is a reservation system for the travel agents of the airlines; - an additional reservation system is proposed in the second year (year 1): it is a B2C e-commerce solution, namely, a reservation system available directly on the company’s website; - an additional reservation system in the third year (year 2): this is a system for offering rental cars and hotels. – The cost of installing, programming and configuring the server: €1,000,000. – The cost of the license for a period of 5 years (€1,500,000). Investments are amortized on a straight-line basis over the remaining life of the project, as there is no second-hand market for this type of service. The residual value is thus zero. Commercial and general operating costs break down as follows: – maintenance costs: €7,000; – personnel expenses: €4,000 per month; – billing fees: 0.3% of gross income; – overhead costs: €3,000,000 plus 2% of gross income; – commissions on sales: 8% of gross income. In addition, the following information is available: – the average tax rate on profits: 40%; – the risk-free rate: 2%; – the cost of the pre-tax debt: 4%; – the cost of equity: 15%; – the financing of the company: 35% per debt.

Data Generation

125

To begin, it is necessary to calculate an average price per possible action, i.e. according to the services offered. Prices and consumption patterns are assumed to be stable over the 5 years. Type

Weighting

Price/minute

Adjusted price/action

Service 1

45%

0.5

0.225

Service 2

25%

1

0.25

Service 3

20%

1.5

0.3

Service 4

10%

2

0.2

Average price/action

–

–

0.975

Table 3.9. Calculation of price/possible action

The turnover is obtained by multiplying the traffic (volume in millions of actions) by the average price per action. The latter is supposed to fall by 10% per year from the second year. Year Traffic (in millions of actions) Price Turnover (in million €)

1

2

3

4

5

5

7

14

19

21

0.975

0.88

0.79

0.71

0.64

4.875

6.14

11.06

13.50

13.43

Table 3.10. Turnover (in millions) of the Sysrev project

14 12 10 8 6 4

20 15 10 5

2

0

0 1

2

Year Traffic (in millions of acons)

3

4

5

Turnover (in million €)

Figure 3.4. Evolution of traffic and turnover

126

Investment Decision-making Using Optional Models

The “S” shape of the traffic curve is specific to the externalities of the service sector. According to Goolsbee and Klenow (2006), at the launch of an Internet project of this type, the users gradually increase until reaching a point of inflection synonymous with greater growth. After having reached its peak, the rate of use then slows down or even decreases when the market is saturated, because of the adoption of the innovation. As a result, in year 5, the continuous decline in prices (10% per year) to counter competition and encourage new users to enter the market is such that the turnover declines despite the positive (lower) trend of the traffic. We then guess the difficulty of evaluating this type of risky investment project, with network effects, with conventional methods such as the NPV, insofar as the forecasts of the users and, beyond, the sales prove to be a delicate exercise. In fact, the success of the project depends on achieving a critical mass of users that must generate not only positive cash flows but also sufficient profitability. Investment expenditure is made from the beginning (year 0) and for the first 3 years. 0 12,000 5,000 1,500 1,000 19,500

Year Server RS License Installation Total investments

1 0 5,000 0 0 5,000

2 0 5,000 0 0 5,000

Table 3.11. Investment expenditure in K €

Depreciation is linear and carried out over the remaining life of the project, assuming that as this market is very specific, it is impossible to sell the investments over time. Year I1

1

2

3

4

5

3,900

3,900

3,900

3,900

3,900

I2

0

1,250

1,250

1,250

1,250

I3

0

0

1,667

1,667

1,667

Total depreciation

3,900

5,150

6,817

6,817

6,817

Table 3.12. Depreciation in K €

Data Generation

127

Table 3.13 summarizes the forecasts of operational costs. Year

1

2

3

4

5

Number of RS

1

2

3

3

3

Maintenance cost

7

14

21

21

21

Personnel expenses

48

48

48

48

48

Total OPEX

55

62

69

69

69

Table 3.13. Operational costs in K €

In this context, it is possible to create a projected income statement. The latter shows that the operating result is negative the first 2 years, when the company reaches a critical threshold of users. Between the third and fourth years, REX increases significantly before falling back to year 5, due to lower prices. Year

1

2

3

4

5

Turnover

4.875

6.143

11.057

13.505

13.434

OPEX

55

62

69

69

69

Depreciation

3.900

5.150

6.817

6.817

6.817

Operating income

920

931

4.171

6.619

6.548

Operating margin

18.9%

15.1%

37.7%

49.0%

48.7%

Commercial costs

390

491

885

1,080

1,075

Billing costs

15

18

33

41

40

General and administrative costs

3.098

3.123

3.221

3.270

3.269

Subtotal

3.502

3.633

4.139

4.391

4.384

REX

−2.582

−2.702

32

2.228

2.164

Operating margin

−53.0%

−44.0%

0.3%

16.5%

16.1%

Table 3.14. Income statement in K €

In the first 4 years, the company would not pay a corporate tax, because of the carry forward of the previous deficits accumulated during the first years of the project. The postponement also improves the profitability of the

128

Investment Decision-making Using Optional Models

company, because it pays a very low tax in year 5, for a much higher operating result. Year

1

2

3

4

5

Tax base

−2.582

−5.284

−5.252

−3.024

−860

Corporate tax

0

0

0

0

−344

EBIAT

−2.582

−2.702

32

2.228

2.508

EBIAT margin

−53%

−44%

0%

16%

19%

Table 3.15. Taxation in K €4

In order to carry out the business plan and calculate the NPV of the project, it is necessary to calculate, beforehand, according to the information available, the weighted average cost of the capital of the company. Cost of debt (before tax)

5.0%

Cost of debt (after tax)

3.0%

Cost of CP

15.0%

% of debt

35.0%

% of CP

65.0%

WACC

10.8%

Table 3.16. WACC calculation

where WACC: K = kCP

+ kD

, with:

– kD: the rate of return required by the creditors after tax; – kCP: the rate of return required by the shareholders (previously, Ri); – VD: the market value of net indebtedness; – VCP: the market value of equity. In order for the above calculation to be relevant, that is to say, so that the assumed risk is the result of weighting the debt burden and the equity taking 4 EBIAT: Earning Before Interest After Taxes, or the result before taking into account the interest, but after taking into account the tax.

Data Generation

129

into account their cost, it is necessary to specify that the project must be of such a nature to carry the same economic risk as that of the company as a whole. In concrete terms, this means that the investment project is closely linked to the major business sector of the company, that the GOS is more or less constant and that the project is financed in the same proportions as the overall level of indebtedness of the company. Year

0

1

2

3

4

5

EBIAT

0

−2.582

−2.702

32

2.228

2.508

Depreciation

0

3.900

5.150

6.817

6.817

6.817

Cash flows

0

1.318

2.448

6.849

9.045

9.325

Discounted cash flows

0

1.189

1.994

5.035

6.001

5.584

Investment

−19.500

−5.000

−5.000

0

0

0

Discounted investment

−19.500

−4.902

−4.806

0

0

0

Discounted free cash flows

−19.500

−3.713

−2.812

5.035

6.001

5.584

Discounted cumulative FCF

−19.500

−23.213

−26.025

−20.990

−14.988

−9.404

Discounted cash flows

19.803

Discounted investment

−29.208

NPV

−9.404

Table 3.17. Business plan and calculation of the NPV of the project in K €

The discounted free cash flow at the WACC is the result of EBIAT reprocessing, which has been increased by amortization charges (non-disbursable expenses), reduced investments (expenses not taken into account in the income statement) and discounted at the risk-free rate. In fact, according to Luehrman’s work (1998), it is assumed that the company, which defers certain investment expenditures, places its funds at the risk-free interest rate. We note that the NPV is negative. Consequently, according to this valuation criterion, the project must be abandoned.

130

Investment Decision-making Using Optional Models

However, Sysrev wants to incorporate flexibility by incorporating possible additional development phases. In fact, in 3 years, the company would have, by its positioning on the market, exploitable information. In other words, the company plans to make an additional investment to set up passenger data services (frequent flyer management) to offer personalized solutions to travelers. Moreover, in 5 years, Sysrev sees the possibility of developing solution services (management of taxis, hotels, etc.) in connection with impromptu events such as natural disasters (volcano eruptions, etc.). The second project would require an additional investment expenditure of €90 million and would generate €80 million in cash. Similarly, the third project would require an additional investment expenditure of €150 million and would generate €125 million in cash. Data in million €

Inv.

Dis. inv. in 0

Dis. CF in 0

NPV

First phase

−29.21

−29.21

19.803

−9.404

Second phase in 3 years

−90

−84.81

80

−4.809

Third phase in 5 years

−150

−135.86

125

−10.860

Total

−269.21

−249.88

224.80

−25.07

Table 3.18. Incorporation of two other development phases in the investment project in K €

The term “Dis. inv.” means that, as before, the investment is discounted at the risk-free rate. The two additional projects envisaged have a negative NPV, leading to an amplification of the idea of abandoning the project. However, as it stands, the valuation of these projects does not take into account the potential for flexibility. In fact, if the state of the market is unfavorable for the company in 3 years, then in 5 years, it is reasonable to think that it will not undertake to make such investments. It is then a question of considering two composite growth options insofar as the third project will be realized only if the second one is a success. The maturities are, respectively, 3 and 5 years. We thus find the idea of the ANPV, which consists of adding to the NPV of the first project the value of the call composed of the second and third projects, according to the Cox, Ross and Rubinstein binomial model (1979).

Data Generation

131

Variable

Description

Second phase

Third phase

S

Current value of cash flows

80

125

T

Life of the option (in years)

3

5

s

Standard deviation of cash flow (volatility)

45%

45%

r

Risk-free rate

2%

2%

Nb per

Number of periods

3

5

t

Nb of years/Nb of periods

1

1

r'

Adjusted risk-free rate (1+r)^t

1.02

1.02

up

Exp(s × t^0.5)

1.5683

1.5683

down

1/up

0.6376

0.6376

Pu

(r'-down)/(up-down)

0.4109

0.4109

Pd

1-Pu

0.5891

0.5891

Table 3.19. Characteristics of the investment project according to the Cox, Ross and Rubinstein binomial model

0

0

1

2

3

80

125.46

196.77

308.59

51.01

80

125.46

32.53

51.01

1 2 3

20.74

Table 3.20. Evolution of the cash flows of the second phase of the project in K €

0 1 2 3 4 5

0

1

2

3

4

5

125

196.04

307.45

482.18

756.21

1,185.97

79.70

125

196.04

307.45

482.18

50.82

79.70

125.00

196.04

32.41

50.82

79.70

20.66

32.41 13.17

Table 3.21. Evolution of the cash flows of the third phase of the project in K €

132

Investment Decision-making Using Optional Models

In order to obtain the value of the option of the third phase, for example, it is necessary to start by calculating the terminal value in year 5, namely, the final payment, which corresponds to the maximum between 0 and the cash flows of the fifth year reduced by the exercise price. Thus, for year 5, line 0: Max (1,185.97-150; 0) = 1,035.97. Then, the tree must be reassembled in reverse order towards year 0. By reasoning in a risk-neutral context, it is then necessary to calculate the value at each node as the discounted expectation of the two possible values of the option. Thus, for year 4, line 0, the value of the option is: C

.

=

609.15 × 0.4109 + 160.39 × 0.5891 = 338.00 1.02

Call

0

1

2

3

4

5

0

46.27

92.38

179.65

338.00

609.15

1,035.97

15.70

34.65

75.32

160.39

332.18

3.01

7.47

18.54

46.04

0.00

0.00

0.00

0.00

0.00

1 2 3 4 5

0.00 Table 3.22. Value of the option of the third phase of the project in K €

Call 0 1 2 3

0

1

2

3

24.26

51.97

108.53

218.59

5.75

14.29

35.46

0.00

0.00 0.00

Table 3.23. Value of the option of the second phase of the project in K €

In order to valuate the composite option, it is necessary to rely on the binomial tree of the growth option of the third phase. Years 4 and 5 are reported without modification. In addition, for year 3, the payment of the option includes the second phase of the project and the discounting of cash flows corresponding to the third phase, considering that the second phase

Data Generation

133

was undertaken. Indeed, if the project is stopped at the end of the second phase, it no longer generates any cash flow thereafter (years 4 and 5). In this context, for example, for year 3, line 0, the value of the composite option is: C

.

308.59 − 90 + 609.15 × 0.4109 + 160.39 × 0.5891 ;0 1.02 = 556.60

= MAX

Call 0 1

0

1

2

3

4

5

67.52

141.85

288.18

556.60

609.15

1,035.97

17.97

44.62

110.78

160.39

332.18

0.00

0.00

18.54

46.04

0.00

0.00

0.00

0.00

0.00

2 3 4 5

0.00 Table 3.24. Value of the composite option of the second and third phases of the project in K €

The composite option thus has a value of €67.52 million and makes it possible to consider that the project must be undertaken, insofar as, by integrating managerial flexibility, the ANPV of the investment is positive up to €58 million. The additional phases in 3 and 5 years will then be conducted taking into account the state of the market. NPV 1

−9.404

Composite option

67.52

ANPV

58.114

Table 3.25. Overall value of the project in K €

Conclusion

The traditional calculation of the net present value (NPV) makes it possible to know if, considering the characteristics of a project, the company must invest immediately or never. However, when it has exclusive rights to a project for a given period, it may decide to delay it and realize it at a later date at its convenience. Thus, the question of investing is not simple. Beyond the fact of abandoning the investment, it is a question of knowing if it is necessary to provide for it immediately or to reject it. As a financial decision support tool for investment in low-visibility contexts, the real option applies to an underlying non-financial asset taking into account all the possibilities of the project. The process consists of discerning, step by step, market growth opportunities while limiting downside risks. The reasoning differs from traditional analyses insofar as active management is advocated on all continuous future flows. Managerial flexibility favors changes or optimization of operations according to newly available information. The tool provides a broader picture of the project than a traditional financial valuation. Finally, there is no discount rate assumption and the risk by taking into account volatility is modeled, since, in case of a wrong trajectory, the project can be abandoned. Two approaches to estimating risk can be used. The first, based on a market vision, is based on the volatility of the share of the company holding the project, on the implied volatility of the option on the security or on stochastic volatility. The second, focused on the project itself, estimates its own volatility using a simulation of the distribution of values by the Monte Carlo method. The standard deviation obtained is used as an estimator of volatility.

Investment Decision-making Using Optional Models, First Edition. David Heller. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

136

Investment Decision-making Using Optional Models

Different types of real options exist: those that value investment projects (growth option, abandonment option, carry option), those that value the production process (option of temporary cessation of production, choice of inputs, outputs), those that value the flexibility of fixed capital, etc. These may be call options (strategic investment, merger, acquisition, etc.) or put options (transmission, split, etc.). The valuation parameters are similar to those of the financial options modeled in continuous time by Black and Scholes (1973) and in discrete time by Cox et al. (1979). Volatility represents the level of project uncertainty measured by the dispersion of the project value. The latter is the current value of the probable cash flows. The exercise price is equal to the amount to be invested. The maturity of the option is the life of the project, and the risk-free rate depends on the level of solvency of the issuer. Models can also include project costs (such as transaction or production costs), dividend payments, or the revenue generated by the project. Thus, the real option appreciates as the price of the underlying is large, volatile and not subject to dividends. A low exercise price, high interest rates and a long expiration period also contribute to its valuation. The flexibility of adaptation increases the chances of success and limits the risk of loss. Absent from traditional valuation techniques and yet valuable, time gives decision-makers the opportunity to make new decisions or postpone them; flexibility makes it possible to respond or react to new information by changing the organization of a project. In addition, traditional methods, based on discounting, penalize risky projects, since the value decreases with risk (the discount rate includes a risk premium). The DCF method may lead to the rejection of a project with a high investment, low cash flows, but possibly future growth opportunities. On the other hand, the value of a project estimated by the real options is all the greater as the uncertainty and the time remaining before the end of the opportunity are considerable. Thus, the real options approach makes it possible to value a company’s investment project. If companies tend to wait before investing to gather more information about the market in general, they have the opportunity to combine different categories of options to better understand their investment project by giving them more flexibility and by framing even more precisely the notion of risk. In addition, the real options go beyond the scope of internal growth projects. In fact, they find their place on the control market insofar as the acquisition strategies can integrate growth options, beyond the considerations of

Conclusion

137

influence and remuneration of the executives developed in Chapter 1. In fact, synergies can take place in the implementation of growth options. The offerer who owns the resources may, after having redeemed its target holding growth options impossible to implement due to lack of sufficient means, make them available to its target. In this context, it is conceivable to consider that the real options approach can directly evaluate the liability structure – and no longer only the value of an asset – and thus be complementary to traditional valuation methods, such as that of DCFs or multiple stocks. In fact, the intrinsic value of equity is usually the result of valuation via the DCF approach. In addition, the net debt that forms part of the cost of capital calculation (which serves as a discount rate) is the amount present in the company’s reference documents. But the net debt, which is subsequently deducted from the enterprise value obtained in order to obtain equity, should also be based on an economic value.

Appendices

Investment Decision-making Using Optional Models, First Edition. David Heller. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

Appendix 1 Demonstration of the CRR Formula

The Cox, Ross and Rubinstein (1979) option valuation model is based on the assumption that the value of the underlying asset follows a discrete-time multiplicative binomial law. The stock price may, in each period, either increase and go to uS at the end of the first period, or decrease and go to dS with the respective probabilities q and 1−q. Thus, the rate of return of the share in each period is either u−1 with the probability q, or d−1 with the probability 1−q. If the value of the asset increases (or decreases), the call premium also increases (or decreases) with the probability q (or with the probability 1−q). Maturity dates correspond to periods. At t = 1, the value of the call corresponds to its payment. We then have: Cu = max(0,uS - E) and Cd = max(0,dS - E)

[A1.1]

Based on the constitution of a hedging portfolio P consisting of the purchase of a call of which the premium is C and the sale of H shares: P = HS – C

[A1.2]

Since P is a hedging portfolio, it is risk-free. Thus, its return corresponds to the risk-free rate r and its value, at each period (i.e. when t = 1), is unique: HS − C =

=

[A1.3]

142

Investment Decision-making Using Optional Models

Then: HuS − C = HdS − C and HS =

[A1.4]

Note: ̂ = 1 + . Thus: HS − C =

HuS − C 1+r

and: C = HS − =

HuS − C 1 = rHS − uHS + C r ̂

1 C −C r r u−d

=

−u

C +

C −C u−d C

+

u−d C u−d [A1.5]

Suppose: p= C=

; then 1 − p = 1 −

=

̂

pC + 1 − p C

=

[A1.6] [A1.7]

The probability q does not appear in the formula. This means that even though different investors have different subjective probabilities about an upward or downward movement of the share, they might agree on the relationship of C with S, u, d and r. C does not depend on the attitude of investors according to their respective risk. The formula would be the same, regardless of the investors’ risk aversion. So, if investors are in a risk-neutral environment, the required rate of return of the share is the risk-free rate, so: quS + 1 − q dS = rS and q =

=p

[A1.8]

Appendix 1

143

If the expiry date corresponds to two periods:

S

u2S

uS

udS

dS t=0

d2S t=2

t=1

Cuu

Cu

Cuu

C Cd t=0

Cdd t=2

t=1

At t = 1, the value of P is: HuS − C = Hu S − C

=

[A1.9]

= HudS − C

[A1.10]

HuS =

[A1.11]

HuS − C =

[A1.12]

Hu S − C C = HuS − r 1 = rHuS − Hu S + C r 1 C −C = r u−d r u−r 1 r−d C + C = u−d r u−d = pC + 1 − p C

−u

C

−C u−d

+

u−d C u−d [A1.13]

In the same way: Hds − C = Huds − C Hds =

=

[A1.14]

= Hd S − C

[A1.15] [A1.16]

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Investment Decision-making Using Optional Models

C = HdS − 1 r 1 = r = =

HudS − C r

=

1 rHdS − HudS + C r

C −C −C −u u−d u−d u−r r−d C + C u−d u−d pC + 1 − p C r

C

+

u−d C u−d [A1.17]

If Cu and Cd are, respectively, replaced by [A1.14] and [A1.18] in [A1.7]: 1 1 1 pC + 1 − p C p pC + 1 − p C 1−p r r r 1 = p C +p 1−p C +p 1−p C +p 1−p C r + 1−p C p C + 2p 1 − p C + 1 − p C [A1.18] = C=

If the maturity date is t = 2, the premium at t = 2 corresponds to the payment of the call: C=

C=

1 p . max 0, u S − E + 2p 1 − p . max 0, udS − E r + 1 − p . max 0, d S − E

∑

Cn p 1 − p

. max O, u d

S−E

[A1.19]

If the maturity date is n periods. Suppose X is the random variable corresponding to the number of upward movements of the share after n periods. X is the binomial variable with n and p parameters: X  B(n, p) P[X = k] = Cnk pk(1−p)n−k. Suppose that a represents the minimum number of upward movements that the share must perform after n periods so that the call ends in the money. Then: PX≥a = +⋯+ P X = n

C p 1−p

=P X=a +P X=a+1 [A1.20]

Appendix 1

145

Moreover, the value of C given in [A1.20] can be split into two terms: C=

1 r

C p 1−p

. max 0, u d

S−E [A1.21]

+

1 r

C p 1−p

. max 0, u d

S−E

If 0 < k < a, the call is out of the money and its payment is zero: max(0,ukd n−k S – E) = 0. In addition: max(0, ukd n−k S – E) = ukd n−k S – E. Et:

C=

C=

∑

∑

C p 1−p

C = S ∑

C p 1−p u d

∑

p=1− =

C p 1−p

C p 1−p

[A1.23] [A1.24]

1−p

.

r u−d −u r−d r u−d =

= C p

∑

[A1.22]

.

Thus: 1 − p = 1 −

C = S∑

S−E

S − Er − Er

C

Suppose: p =

u d

− Er

1−p ∑

C p 1−p

[A1.25] [A1.26]

Let F(a, n, p) be the complementary binomial distribution function: C = SF a, n, p − Er

F a, n, p

[A1.27]

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Investment Decision-making Using Optional Models

If n – namely, the number of periods (or subintervals) between the valuation date and the expiry date – is very high, the stock price multiplicative binomial law follows a log–normal distribution and the CRR formula converges to that of Black and Scholes: C = SΦ d

− Ee

Φ d

[A1.28]

with: d =

√

, d = d − σ√τ and Φ x =

√

e dt.

[A1.29]

Appendix 2 Stochastic Differential Calculus

A2.1. Introduction to the diffusion process The modeling of the share price used, in particular, in the context of the determination of option valuation formulas in continuous time is based on the use of stochastic differential calculus. EXAMPLE.– Let xt be the price of a share on date t and x0 its price today. It is assumed that the price increases by €3 per unit of time and that x0 = €1. Thus: xt = xt-1 + 3, i.e. xt – xt-1 = 3 that we can still note: Δx = 3.Δt, where Δt is 1 unit of time at the end of which the price has risen by €3. EXAMPLE.– Let xt be the price of a share on date t and x0 its price today. It is now assumed that the price increases by €18 per unit of time equal to 1 year and that x0 = €1,000. Thus: xt = xt-1 + 18, i.e. xt – xt-1 = 18 that we can still note: Δx = 18. Δt, where Δt is 1 unit of time at the end of which the price has risen by €18. If the unit of time is shortened and equal to 1 month, the change in the price of this new reference period is equal to 12: 18/12 = €1.5. Thus, we note: dx = 1.5.dt. A2.2. Simple Brownian motion or Wiener process Let Δx be the variation of the share price over a small time interval, noted Δt.

Investment Decision-making Using Optional Models, First Edition. David Heller. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

148

Investment Decision-making Using Optional Models

It is assumed that Δx = Δz with Δz = ε√ and ε follows a standard normal distribution which is classically noted: ε → N(0, 1). The symmetry of the curve of the standard normal distribution (representative of the density of probability of ε) shows that the area under the curve to the left of the ordinate axis is equal to the area under the curve to the right of the ordinate axis. Thus: P[ε > 0] = P[ε < 0]. In other words, at the end of each small time interval dt, ε is as likely to take a positive value as to take a negative value. PROPERTY.– It is then possible to characterize Δz using its expectation and its standard deviation: . 0 = 0

E(Δz) = E( √

)= √

V(Δz) = V( √

) = Δt .V(ε) = Δt .1 = Δt therefore Δz σ = √

E(ε) = √

[A2.1] [A2.2]

Finally: Δx = Δz → N(0, √

)

[A2.3]

Validity of properties for a large time interval: the property that has just been established remains valid for a large time interval denoted T, corresponding to n small intervals Δt. In other words: T = n.Δt

[A2.4]

Time interval = T Δt

x0

Δt

x1

Δt

Δt

x2

xT-1

xT

Figure A2.1. Breakdown of time intervals

In this context, Δx should be replaced by x(T) – x(0). However: x(T) – x(0) = ∑

Δ xi = ∑

Δzi= ∑

√Δt

[A2.5]

Appendix 2

√

x(T) – x(0) = x(1) – x(0) = x(T)-x(T-1) = √

+x(2) – x(1) =

√

149

+…+ [A2.6]

Thus: x(T) – x(0) = 1 √Δt + 2 √Δt +… + n √Δt = ∑

i √Δt

[A2.7]

As in the case of a price evolution over a small time interval, it is possible to characterize x (T) – x(0) using its expectation and its standard deviation: E[x(T) - x(0)] = E∑ V[x(T) = Δ .∑

 √Δ = √Δ ∑

x(0)] . 1 = n. Δ = T

=

V∑

(i) = √Δ . 0 = 0  √Δ =

√Δ ∑

[A2.8] (i) [A2.9]

We then find for a large time interval T: x(T) – x(0) → N(0, T)

[A2.10]

It is also possible to write that: x(T) → N[x(0), T]

[A2.11]

If Δt tends to 0, which amounts to considering a subdivision of time T into extremely small intervals, the share price undergoes on the period T an infinitely large number of variations. In other words, the process of evolution of the share price is continuous, which leads to replacing Δt by dt, Δx by dx and Δz by dz. In this case, dx → N(0, √ ) which defines a Wiener process (or Brownian motion with tendency). A2.3. Brownian motion with tendency DEFINITION.– In this case, the evolution of the price depends not only on a random process but also on a central tendency parameter, or drift denoted below by a.

150

Investment Decision-making Using Optional Models

In other words: dx = a.dt + b.dz with dz = ε dt and ε → N(0, 1)

[A2.12]

PROPERTY.– On a small time interval Δt, the process, in discrete time, is written: Δx = a. Δt + b. Δz

[A2.13]

In that case: E (Δx) = E (a. Δt + b. Δz) = a. Δt + b.E(Δz)

[A2.14]

since only Δz has a random component. Thus: E (Δx) = a. Δt + b.E(ε√Δdt ) = a. Δt + b. √Δt E(ε) = a. Δt + 0 = aΔt [A2.15] V (Δx) = V (a. Δt + b. Δz) = 0 + b2 .V (Δz) = b2 .V(ε.√Δt ) = b2 Δt V(ε) = b2 Δt

[A2.16]

Finally: Δx →N (a Δt, b √Δt )

[A2.17]

By subdividing a period T into n time intervals Δt (i.e. T = n. Δt the variation of the price becomes over this period T: x(T) - x (0) = ∑

Δxi

[A2.18]

Since: E[(x(T)- x(0)]=E(∑

Δxi) = ∑

E(Δxi) = n.aΔt = a.T

[A2.19]

and: V[(x(T)- x(0)] =V(∑ n.b²Δt = b²T Finally:

Δxi) = ∑

V(Δxi) = ∑

b √Δt =b².Δt∑ 1 = [A2.20]

Appendix 2

151

x(T) – x(0) → N (aT, b√T ) or else: x(T) → N[x(0) + aT, b√T ] [A2.21] A2.4. Ito process and the special case of geometric Brownian motion This process corresponds to a variation of x in continuous time defined by: dx = a(x, t).dt + b(x, t).dz

[A2.22]

a and b being then functions of the two variables x and t. It is possible to calculate the expectation and the variance of dx: E(dx) = a(x, t).dt because E(dz) = E(ε√Δt ) = √Δt E(ε) = 0

[A2.23]

V(dx) = b² (x, t).dt.V(ε) = b² (x, t).dt.1= b2 (x, t).dt

[A2.24]

Therefore: dx → N[a(x, t).dt, b(x, t) √dt ]

[A2.25]

with: – a(x,t) = instantaneous tendency; – b(x,t) = instantaneous variance. The geometric Brownian motion that defines the evolution of the return of a share is a special case of Ito process assuming that: a(x, t) = μ.x

[A2.26]

and b(x, t) = σ.x

[A2.27]

Since: dx = μ.x.dt + σ.x.dz

[A2.28]

Thanks to Ito’s lemma, it is possible to establish that such a process defines a log–normal law.

152

Investment Decision-making Using Optional Models

A.2.5. Ito’s lemma If a variable x follows an Ito process [dx = a(x, t).dt + b(x, t).dz], then a function of this variable and of time [F(x, t)] also follows an Ito process. Ito’s lemma is established from the Taylor formula with two variables x and t: ΔF =

Δt +

Δx +

² ²

Δt² +

² ²

Δx² +

²

Δt. Δx +…

[A2.29]

I.e.: Δx = a(x, t). Δt + b(x, t). Δz = a(x, t). Δt + b(x, t). ε. √t with ε → N(0, 1) [A2.30] To reduce the notations, it is noted, later: Δx = a.Δt + bε √t

[A2.31]

Ito’s lemma leads us to consider only the terms in Δx and Δt of degree equal to 1, which leads naturally to eliminate (by truncation) all the terms of the development of ΔF from the fourth one. On the other hand, the third term must be preserved. In fact: Δx2 = (a.Δt + bε √t )² = a² Δt² + b²ε² Δt+2ab Δt 3/2= b²ε² Δt

[A2.32]

by truncation. However: E(b2ε2.Δt) = b2.Δt .E(ε2) and E(ε2) = V(ε) +[E(ε)]2 = 1 + 0 = 1

[A2.33]

so: E(b2ε2.Δt) = b2.Δt

[A2.34]

In addition: V(b2ε2.Δt) = b4 .Δt2.V (ε2) which tends to 0 when Δt tends to 0.

[A2.35]

Appendix 2

153

Therefore: lim b2ε2.Δt = b2.Δt when Δt tends to 0

[A2.36]

Considering a subdivision of time into incredibly small intervals dt, and therefore in continuous time, the application of the Taylor formula becomes: dx +

²

b²dt

dF(x, t) =

dt +

dF(x, t) =

dt + (a. dt + b. dz) +

²

[A2.37] ² ²

b²dt

[A2.38]

Consequently, returning to the original notations, namely: – a = a(x, t) ; – b = b(x, t). It follows that: dF(x, t) = [

+

a(x, t) + b² (x, t) +

² ²

]dt + b (x, t)

dz

[A2.39]

A2.6. Log–normal distribution of the stock price If dx =μ.x.dt +σ.x.dz, then Ito’s lemma can be used to find the following process: F(x, t) = ln(x) = F(x)

[A2.40]

Thus: ( ,)

=

( )

( ,)

= and

=

[A2.41] ( )

=−

[A2.42]

As: a(x, t) = μ x. dt and b(x, t) = σx. dt

[A2.43]

dF = 0 +

[A2.44]

−

. dt + σx dz

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Investment Decision-making Using Optional Models

dF = μ −

dt + σ. dz

[A2.45]

and: dlnx = lnx − lnx = μ − lnx = lnx + μ −

dt + σ. dz

[A2.46]

dt + σ. dz

[A2.47]

.

x = x .e

[A2.48]

As dF defines a geometric Brownian motion with the derivative: dF → N

μ−

. dt, σ√dt or dlnx → N

μ−

. dt, σ√dt

[A2.49]

then dx is a random variable with the parameters of a normal distribution that are: μ−

. dt and σ√dt

[A2.50]

By taking the rate of return μ equal to the interest rate r, the risk-neutral parameters of the stock price are implicitly described. Then: S = S .e

.

+ σ. ε. √dt

[A2.51]

Appendix 3 Test of the Black and Scholes Formula and Return on the Log–Normal Distribution

A3.1. Monte Carlo simulations and test of the Black and Scholes formula The premium of a call is the current value of the scheduled repayments at maturity: C=e

. E max 0; S − E

[A3.1]

If the spot of the underlying asset defines a geometric Brownian motion: S = S .e

.

.

[A3.2]

where: dz = ε√τ

[A3.3]

from where: S = S .e

.

. √

Investment Decision-making Using Optional Models, First Edition. David Heller. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

[A3.4]

156

Investment Decision-making Using Optional Models

then: C=e

.

. E max 0; S . e

. √

−E

[A3.5]

The value of C can be approximated thanks to the simulations obtained by the Monte Carlo method: – simulations of many values of ε. Since ε is a random variable that is normally distributed with a mean of 0 and a variance t, its values can be obtained on Excel thanks to the following function: = RAND() – calculation of the corresponding values of ST for each value of ε: .

S = S .e

. √

in a risk-neutral environment; – discounted S =e

repayment

. max 0; S . e

calculation .

. √

for

each

value

of

−E ;

– calculation of the average of the different discounted refunds which depends on the premium of the call: C=e

. E max 0; S . e

.

. √

−E .

A3.2. Return on the normal–log distribution Assuming Y = lnX → N m, σ , it is possible to determine the density and the mean of X. Since Y = ln X, X = eY. Let us set the distribution function F and its density f: FX(x) = P[X < x] = P[eY < x] = P[Y < lnx] = FY(lnx)

[A3.6]

Appendix 3

157

Then: fX(x) = F’X(x) = F’Y(lnx) = f lnx

[A3.7]

Since lnx is defined for x > 0: – if x < 0, fX(x) = 0; – if x > 0, fX(x) =

[A3.8] .e

. .√

xf x dx =

E X =

1

=0+ E X =

.

.√

.

x. σ. √2π

e

[A3.9] xf x dx +

xf x dx

. xe

dx

dx

[A3.10]

Note: u=

[A3.11]

then: =ϕ u

x=e

[A3.12]

from where: ϕ u = σe

[A3.13]

ϕ

[A3.14]

and: u =

Therefore: E X =

1 σ. √2π

.

e

. σe

. du =

1 √2π

.

e

. du

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Investment Decision-making Using Optional Models

E X =

E X =

√

.

e

e √2π

.

e

. du

. du

[A3.15]

Let: v=u−σ

[A3.16]

u = v + σ = ϕ u

[A3.17]

then:

from where: ϕ u =1

[A3.18]

ϕ v =v−σ

[A3.19]

and:

Finally: E X =

√

.

e

. dv and E X = e

[A3.20]

Appendix 4 Demonstration of the Black and Scholes Formula

C=e

. E max(0; S − E)

[A4.1]

It is assumed that lnS follows a normal distribution of parameters m and s where: m = ln S + μ −

. τ and S = σ√τ

[A4.2]

In this case, S follows a log–normal law. The density of S is: f(x) =

. √

.e

si x > 0

(x) = 0 if x ≤ 0

[A4.3]

The mean of S is: E(S) = e

[A4.4]

Moreover, for a continuous variable X: E(X) =

xf(x). dx

[A4.5]

so: C=e

.

max(0; S − E) . f(S). dS

Investment Decision-making Using Optional Models, First Edition. David Heller. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

[A4.6]

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Investment Decision-making Using Optional Models

where f is the density of the log–normal law. It is possible to break down this integral into two terms: max(0; S − E) . f(S). dS + e C=e . E) . f(S). dS

.

max(0; S − [A4.7]

Now, if x is less than E, then max (0, S – E) = 0, so the first integral is zero. If x is greater than E, then max (0, S – E) = S – E. Thus: C=e

.

(S − E) . f(S). dS

C=e

.

S. f(S). dS − E. e

[A4.8] .

f(S). dS

[A4.9]

Now S, as an integration variable, is a dummy variable. In other words, it will not appear in the final result. We can then replace S, for example, by x. Thus: .

C=e

x. f(x). dx − E. e

.

f(x). dx

[A4.10]

Replacing f with the expression of density: C=e E. e

.

. √

.

x.

.

√

x. S√2π

.e

C=e −E. e

1

. dx −

. dx .

.e

.e

1 S√2π

[A4.11] e

. dx

. dx

[A4.12]

Appendix 4

161

Either the change of variable: u=

[A4.13]

Since: x=e

= ϕ(u)

[A4.14]

ϕ (u) = S. e

[A4.15]

and: ϕ (u) =

[A4.16]

Thus: C=e

E. e

.

.

1

.e

√

S. e

e

S√2π . S. e

du −

du

[A4.17]

By simplifying the first integral by S and the second by S and esu+m: C = e

−E. e

.

.e

√

.

1

du

e

√2π . du

[A4.18]

However: 1

e

√2π

−

1 √2π

du =

1 √2π

.e

.e

. du

. du

162

Investment Decision-making Using Optional Models

=1−ϕ

=ϕ

[A4.19]

Thus: C = e

.e

By taking C = e

.e

(

e

√

)

du − E. e

.ϕ

[A4.20]

.ϕ

[A4.21]

out of the first integral: (

e

√

)

du − E. e

Let a new change of variable for the first integral: v = u − S then: u = v + S = ϕ(v)

[A4.22]

Thus: ϕ (v) = 1 and − ϕ (v) = v − S

[A4.23]

Since: C = e

.e

C = e

e

√

.e

.ϕ S −

dv − E. e − E. e

.ϕ

[A4.24]

.ϕ

[A4.25]

Replacing S and m with the values recalled in equations [A4.2]: .e

C = e E. e C=e

.

.

.

. ϕ σ√τ −

√

.ϕ .S e .ϕ

− [A4.26]

√

− E. e

.ϕ

In a risk-neutral environment, we can consider that μ= r.

[A4.27]

Appendix 4

163

Thus: C=e

.S e .ϕ

C = S .ϕ

− E. e

√

− E. e

√

.ϕ

.ϕ

[A4.28]

√

[A4.29]

√

I.e.: d =

and d =

√

d − σ√τ =

√

− σ√τ =

[A4.30]

√

√

=

√

[A4.31]

Conclusion: C = S ϕ(d ) − E. e

ϕ(d )

[A4.32]

with: d =

√

[A4.33]

and: d = d − σ√τ

[A4.34]

Bibliography

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Investment Decision-making Using Optional Models, First Edition. David Heller. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Index

A, B, C acquisition strategies, 48, 88, 98, 106, 136 arbitrage portfolio, 15, 18, 59, 69, 78 binomial tree, 22, 132 Black and Scholes, 1–3, 10, 12, 14–18, 20, 21, 28, 30, 35–37, 39, 41, 42, 44, 47, 66, 68, 113, 114, 117, 122, 136, 146, 155, 159 Brownian motion geometric, 15, 40, 42, 49, 70, 85, 86, 102, 151, 154, 155 specific, 42 call, see also put, 12–15, 17, 20, 22, 28, 29, 31, 39, 41, 50, 51, 53, 54, 81, 130, 141, 144, 145, 155, 156 convenience yield, 35, 48, 58, 66, 69–71, 74 costs(s) of bankruptcy, 103, 105 opportunity, 104, 110 production, 5, 49, 52, 58, 59, 64 storage, 35 transaction, 2, 36, 39, 44, 136

Cox, Ross and Rubinstein (CRR), 1, 3, 12–14, 18, 20, 21, 23, 29, 30, 35, 41, 42, 44, 123, 130, 131, 136, 141, 146 D, E, F differential equation, 47, 54, 55, 59, 71, 72, 75, 77, 83, 86 Discounted Cash Flows (DCF), 3, 11, 32, 44, 110–112, 136, 137 diversification discount, 88, 89 dividend, 2, 3, 17, 18, 36, 41, 42, 81, 82, 99, 110, 111 exercise price, 4, 5, 12, 13, 17, 36, 45, 49, 53, 63, 67, 78, 82, 90, 93, 95, 103, 110–112, 116, 132, 136 flexibility, 1, 2, 5, 7, 10, 11, 23, 47, 49, 64, 99, 100, 106, 108, 113, 119, 130, 133, 135, 136 futures contracts, 41 I, J, M in the money, see also out the money, 13, 14, 36, 144 indebtedness, 7, 91, 92, 97, 129 Ito’s lemma, 15, 70, 72, 77, 86, 151–153

Investment Decision-making Using Optional Models, First Edition. David Heller. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

174

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joint ventures, 48, 66, 68, 99 Model binomial, 21, 23, 28–30, 103, 130, 131 with jumps, 49, 64 Monte-Carlo, 40, 43, 135 O, P, R option premium, 3, 12, 15, 44, 53, 54, 112 optional interactions, 48 options(s) abandonment, 5, 19, 23–26, 28, 30, 31, 44, 47, 85–87, 113, 115 American, 25, 41 carry, 5, 19, 48, 51, 54, 59, 62, 65, 98, 107, 108 closing, 76 combined, 2 development, 6, 31 divestiture, 100 European, 3, 25 exchange, 6, 65–67, 113, 114 expanding, 7, 32 financial, 1–3, 20, 44, 84, 136 growth, 4, 19–23, 26, 28–30, 90–97, 99, 103, 132, 136 of learning, 5 real, 1–4, 6–8, 10, 11, 19, 20, 28, 42, 44, 45, 47–52, 62, 64–66, 68, 89, 90, 98–100, 106, 107, 136, 137 resale, 26 out the money, see also in the money, 13, 36, 84 partial differential, 16, 18, 54, 55, 58, 93, 96 pricing, 41, 85, 87 process of spreading, 85, 147 Poisson, 40, 64

put, see also call, 12, 13, 25, 26, 28, 30, 41, 50, 51 rate distribution, 8, 42, 68 growth, 22, 52, 57, 85, 86, 90 of return, 13, 17, 31, 39, 42, 52, 79, 82, 86, 111, 141, 142, 154 risk-free, 17, 18, 22, 31, 32, 60, 64, 67, 68, 71, 72, 80, 109, 114–116, 124, 129, 130, 136, 141, 142 return, 15, 31, 38, 52, 53, 69–72, 80, 82, 87, 98, 109, 110, 115, 141, 151 right moment, 5, 49, 63, 68, 78, 95, 106, 107 S, T, U, V smile, 36, 42 spot, see also underlying, 41, 69–71, 76, 77, 155 spread, 91, 97 standard deviation, 4, 12, 19, 36, 37, 43, 69, 79, 80, 83, 84, 86, 135, 148, 149 time agency theories, 89 continuous, 2, 3, 12, 14, 17, 18, 20, 28–31, 36, 37, 40, 41, 44, 64, 82, 136, 147, 151, 153 discrete, 2, 3, 12, 13, 18, 20, 23–30, 36, 37, 40, 41, 44, 136, 141, 150 Timing, 77, 78, 82, 84 underlying, see also spot, 2–4, 12–15, 17, 19, 20, 29, 36, 37, 39, 41, 42, 44, 48, 50, 53, 54, 64, 67, 69–72, 78, 84, 90, 94, 97, 99, 111–114, 116, 120, 121, 135, 136, 141, 155

Index

value adjusted (ANPV), 19, 20, 23, 25, 27–31, 110, 130, 133 intrinsic, 12, 34, 53, 112, 113, 116, 119, 137 net present (NPV), 1, 2, 5, 6, 8–11, 19, 21, 23, 25, 26, 29, 31, 43, 44, 51, 53, 57, 60, 77, 78, 110, 121, 122, 126, 128–130, 133, 135

175

time, 4, 12, 34, 112, 116, 119 volatility, 1, 2, 4, 12, 15, 17, 29, 31, 32, 35–40, 42–44, 52, 62, 66–68, 70, 78, 79, 82, 84, 85, 89, 90, 101, 110, 112–114, 116, 120, 131, 135, 136 historic, 36, 37, 42, 43 implicit, 36, 37, 42, 135 stochastic, 2, 36–38, 44, 135

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