Real Options and Private Equity Valuation

Transcription

1 Real Options and Private Equity Valuation Master thesis M.Sc. in Finance & Strategic Management Author Mikael Sahlin Olesen Advisor Thomas Einfeldt Institution Copenhagen Business School Department of Finance Solbjerg Plads 3, A5.09 DK-2000 Frederiksberg Date September 30, 2009

2 Executive Summary The special construction of Private Equity (PE) investments with high leverage and rapidly changing capital structure has meant that a new valuation model was developed The LBO model in order to take this onto account. The LBO model is constructed to follow each step of a Private Equity investment by separating that valuation of debt and equity. It is, however, still a static valuation method which does not take managerial flexibility into account. As practitioner and theorists of real option theory states, this results in unrealistic values. The purpose of this thesis it therefore to examine the applicability on real option theory to PE valuation in order to establish weather PE firms are overlooking potential value by not focussing on real options in relation to their exit strategy. Because the main value creation in a PE investment comes from the sale of the portfolio companies, the focus is on the flexibility related to the PE firms exit strategy (PE Exit Option). Real option theory is the most recognised method for valuing flexibility. It uses the theory from financial options to value real assets. By linking the variables from financial options to characteristics of real assets it can be established that the valuation models for financial models can be used for real options. For the purpose of valuing PE Exit option the binomial model is the most appropriate because it can easily be adjusted for the specific characteristic of the PE Exit Option. The PE Exit option is divided into two separate options. One for the period where the target is owned by the PE firm (Holding period) PE Holding option, and another for the period after the expected exit of the target (Post Exit Period) PE Post Exit option. The most noticeable specifications when applying real option theory to the Exit options is: (1) it is the targets market value which is used as the underlying asset (2) that it is the required return calculated as a enterprise value that is the exercise price, which entails a variable exercise price (3) that the Exit option does not have the choice of not exiting at expiration of the Post Exit option. Using the theory from real options does add value to the PE firm by incorporating the flexibility of exiting at different times, but this value can only be realised if a potential buyer values the flexibility the same. The value is only as high as a buyer will pay. 2

3 Table of contends 1 Introduction Problem Statement Delimitations Methodology Structure Private Equity Business Model Structure of PE firm Strategy of PE firms Valuation of PE targets The starting point Discounted Cash Flow model (DCF) Adjusted Present Value (APV) Multiples for Valuation LBO valuation method Inadequacies of LBO model Real Option Theory Classic Real Options Valuing Real options Option Pricing Models Binomial Model Risk-neutral Probabilities Setting up the binomial lattices The Black & Scholes model Model choice PE Exit options and the general framework PE firms exit option (PE option) Expanded LBO valuation

4 7.1 Expanded LBO Static LBO valuation Debt Capacity Expected future cash flow for debt repayment Expected exit value Required rate of return Enterprise Value (at time zero) Volatility estimation Historic data vs. Forecasts Simulating Volatility PE Option valuation (Binomial model) The Variables Setting up the lattices Total value TDC A/S valuation using expanded LBO Case Company TDC A/S Pro Forma Income Statement Static LBO valuation Volatility estimation PE Option valuation (Binomial model) Total value Criticism of Real options Financial Issues Corporate Governance Issues Discussion The PE Exit options Choice at Exit

5 10.3 Interpreting the value Conclusion Further research References Appendices

6 1 Introduction They have been called everything from Barbarians at the gate to financial pyramid games, and have been thoroughly discussed in the media. But the Private Equity market has been a booming industry for several decades, and up until the credit crunch, the trend suggested an increasing influence in the merger and acquisition market. In the years leading up to the credit crunch, the financial climate was characterised by low interest rates, global economic boom with little risk of bankruptcy, high liquidity, and increasing competitiveness between banks, resulting in a very favourable debt market. These factors created a very suitable environment for Private Equity firms (PE firms) and their highly levered investments. All this has now dramatically changed, and the PE firms are now challenged by a very difficult debt market and low liquidity, and some experts that many PE firms will default in the coming years (Børsen 2009a). The credit crunch has also meant the there is a higher uncertainty about the future, and valuation techniques which incorporate this uncertainty is maybe now more than ever a area where potential value can be created. Every day, companies value projects using a technique that implicitly assumes the world stands still. Markets never change. Consumer demand never rises or falls. New technologies rarely emerge. And lessons are hardly ever learned. Why? Because one of the best ways to value projects real options analysis is flying somewhere under the corporate radar. AT Kearney (2009) PE firms valuation procedure is no different from the companies referred to here. It is tailored completely to the steps in a PE acquisition (Baldwin, 2001A, Baldwin, 2001B and Bonnerup et al 2007). It is, however, based on traditional valuation theory and the discounted cash flow model (DCF model), which in the past decade has been increasingly criticised in modern corporate finance theory for being static and not applicable in real world situations where change and uncertainty is inevitable. The uncertainty can also be beneficial to PE firms. Practitioners of Real Option Theory hypothesize that using real option theory as a complementary valuation tool results in a more realistic valuation which exploit the opportunities of uncertainty and might create a competitive advantage (Copeland and Antikarov, 2001). 6

7 There is little literature available on the subject of applying real option theory to the valuation of PE targets, and examinations of the connection between the basics of value creation in PE firms and real options as well as practical implications are virtually nonexistent. This thesis will therefore examine the prospect of applying real options to the valuation of PE targets from a theoretical as well as a practical point of view, in order to determine weather PE firms are overlooking potential value in their exiting valuation prices. 1.1 Problem Statement The main return from PE investments comes from the selling (exiting) of portfolio companies (targets). The flexibility of exiting when conditions are most favourable could therefore have great undiscovered value to PE firms. The focus of this thesis is thus to examine if and how Real Option Theory can be used as a tool for valuing PE targets by valuing this flexibility. Based on that, the main hypothesis is: Are PE firms overlooking potential (real) value by not focusing on real options in relation to their exit option? The overall problem statement will be answered through the following research questions, which provides theoretical as well as practical assessment of real options in PE valuation: - How is a Private Equity investments valued? - What is Real Options and how are real options valued? - What options are available to PE firms? - How can real option theory be applied to the valuation of PE targets? - How does the general criticism of Real options apply to PE Exit option? 7

8 1.2 Delimitations The scale and scope of the thesis naturally limits some aspects of the analysis, and I have prioritised the areas which I find most relevant for the analysis. I have chosen to highlight these specific limitations in the sections where they are relevant, as I feel this gives a better overview and a more fluent presentation. There are some general limitations, however, which will be presented next. The handling of input data has a great influence on the result of the valuation, as the real option analysis is (as any other model) subject to the concept of garbage in garbage out, meaning that the result is not better than its inputs. Therefore, the main focus of the thesis is a theoretical and technical background combined with a practical analysis and less emphasis on the strategic consideration regarding real option. This should not be interpreted as the strategic considerations are not important, but merely as necessary for the extent of the thesis. The receivers of the thesis are practicians and theorists of financial theory which can be assumed to have a basic knowledge of theory and method in finance. It is therefore assumed that a presentation of general valuation methods (DCF, multiples and APV) is not necessary, as their application and underlying assumptions a known by the reader, and only a brief description of the main characteristics is presented. The conclusions in the thesis will be based on theoretical review and a single case study. A large empirical study would of cause give a better foundation for drawing general conclusions. But because of the need for a thorough analysis of the theoretical applicability, the scope of the thesis does not allow a large empirical study. Even though many real options would be available to PE firms in relation to the target companies, the focus of this thesis will be on the options associated with exiting the target. This is partially because of the scope of the paper and partially because it is a general option which can be applied to most PE targets, while other option will be more specific, such as launching projects or firm specific investments. 8

9 1.3 Methodology The analysis of applying option theory to the valuation of PE targets is first of all a theoretical analysis. That is, since there are no theory that describes this particular combination of real option and private equity theory, it first needs to be established that application is theoretical feasible and can be done in coherence with existing valuation and real option theory. This implies using the most relevant theory from acknowledged text books and articles that can assist in making a modified practice that is consistent with general theory. After it is established that it is theoretical feasible the next step is to examine the practical feasibility. This is done by developing a practice based on real option and private equity valuation theory, but modified to account for practical implications. Subsequently this practise is challenged by a single case study of a PE target valuation. The aim of the case study is to satisfy the three tenets of the qualitative method: describing, understanding, and explaining (Yin 1994), and I find it reasonable to assume that a single case study is sufficient to fulfil this aim if the three tenets are fulfilled. Yin (1989) supports this by pointing out that generalization of results, from either single or multiple designs, is made to theory and not to populations. Based on that, I find that the similarity of the characteristics of PE acquisitions, the thorough theoretical analysis combined with one qualitative case study is sufficient to make adequate general conclusions. The case study will be based on NTC s acquisition of TDC A/S in 2006, and will be build on both qualitative and quantitative data. The qualitative data will primarily consists of articles and market reports of TDC from before the takeover, while the quantitative data is primarily TDC annual reports from 2006 to 2008 and market reports of the telecommunications and internet industry, including projections of future developments in the industries. Finally, because there is no prior research on this specific use of real options in PE valuation, I find it important to discuss the key findings and implications in order to give a more nuanced view of the applicability. This will be an ongoing process throughout the thesis, meaning that a large part of the thesis is to discuss the different issues in order to determine the best solution. 9

10 1.4 Structure Figure 1 Structure Introduction Problem Statement Method & Limitations Theoretical review Private Equity Real Option Theory Applayng Theory PE Exti Options Holding Option Expanded LBO LBO Model Post Exit Option TDC A/S Case Study General Critique Considerations Discussion Conclusion 10

11 2 Private Equity Business Model Private equity is a broad term, which in general signifies the source of money. As the name suggest the money is private and cannot be reached from public markets, such as stock exchanges. Private equity investments are typically divided into three categories: Leveraged Buy-Out (LBO) Venture Capital Other special investments e.g. mezzanine capital Furthermore they are typically characterised by a group of large investors which, through for example a Private Equity firm (PE firm) or Venture Capital firm (VC firms), invest in different firms in order to get a higher return than they would get by investing in public traded stocks 1. While venture capital firms tend to invest in earlier stage growth companies, PE firms tend to focus on more mature businesses. One of the key differences in the two types of companies is their funding of acquisitions. Where VC firms transactions are primarily funded with equity, due to uncertain cash flows, the primary funds for PE firm are usually debt, with up to % debt (Vinten & Thomsen 2008a). For this reason the typical PE transaction is called a leveraged buyout (LBO). The focus of this thesis will be PE firms. 2.1 Structure of PE firm The concept of PE firms can be compared to that of investment- or mutual funds. Instead of each investor invests in separate companies, they invest in a PE firm, which then invest in the different companies (targets). The reason for this structure is that the investors can reap all of the benefits from the excessive expertise, which the PE firm has accumulated through all its transactions. The formal structure of PE firms can be illustrated as follows. 1 Weather PE firm actually perform better than the market is questionable, and will not be discussed in this thesis. But no studies have so far proven that PE firms outperform public traded companies, when looking at a risk-adjusted return (Vinten and Thomsen 2008a). 11

12 Figure 2 - Structure of Private Equity Company Pension funds, Insurence companies, Private inestors etc. General Partners Wages Investors Management Company Management fee Counselling Capital Marketing return Private Equity Company (PE firm) Carried Interest Equity Sales price/ target return Target companies (portfolio companies) Debt Costof debt Debt providers Source: Andersen and Frigast 2008 p. 12 and own construction A PE firm consists of one or more PE funds. Each investor invests in a fund, and not in the PE firm itself. Each fund is usually formed as a limited partnership, with the management company as the general partner (GP), and the investors as limited partners (LP) (Bennedsen et. al. 2008). The reason for using a limited partnership is that it is easy to make capital increases and decreases, which is what is done each time the investors (LP) deposits into the fund. Once investors have been found the PE firms get the investors to commit to the funds with a predetermined level of investment. It should be noted that the investors do not actually transfer the money to the fund until financing is needed. The capital can be called with short notice, usually within two weeks, when new investments are found. This makes the system very effective and flexible for the PE firm, and ensures instant access to investors capital (Bonnerup et al., 2008). Investors, on the other hand, receive the returns when the fund s portfolio companies (targets) are exited, and it is therefore a relative long term and illiquid investment for the investors (Andersen & Frigast 2008). The managing of each fund is done by a management company with extensive knowledge in Private Equity takeovers. The management company finds the potential targets, which should be within the characteristics, such as firm size or industry, agreed upon with investors when they committed to the fund, (Bonnerup et al., 2008). When investment in a portfolio company is completed, the management company is also involved in the overall management of the portfolio company. I order to align the interests of the management company and the investors, the partners of the management company is required to invest in the PE fund they are managing. 12

13 This is supplemented with a carried interest, which is a percentage of the return above a given hurdle rate. The hurdle rate is typically 8 % of the total invested capital, and the carried interest in typically around 20 % (Andersen & Frigast 2008). To keep it simple I will in the remainder of the thesis not distinguish between the Management Company and PE firm but see them as one entity. 2.2 Strategy of PE firms The cornerstone in the strategy of a PE firm is what Andersen and Frigast (2008) calls active ownership. This means that the PE firm does not only provide capital, but actively works together with the management of the acquired firms to increase its value over the holding period by making firm specific changes.. Another vital part of the active ownership strategy is optimization of the capital structure, which for practically all PE firms means a high use of leverage. For that reason a PE investment is specified as a leveraged buyout (LBO). This is primarily because of the possibility of a higher return on equity for investors, combined with the tax deductions in the acquired firm (Christensen and Christensen 2007). Furthermore, high leverage entails high debt obligations, which in turn lead to a need for high earnings to fulfil the debt obligations. Because of that, PE firms target mature companies with high cash flow. The amount of leverage a PE firm uses on its investments is normally determined by the target firm s ability to service the debt with its operational cash flow, asset liquidity, management s skills, etc. (Christensen & Christensen 2007). The ability to generate cash allows the investor, in our case the PE firm, to take on more debt. Because of the extensive use of leverage almost all the cash generated must be used to repay the debt obligations. Baldwin (2001) goes as far as calling the net cash flow in an LBO for cash flow available for debt repayment. This entails a rapidly changing capital structure which is one of the main characteristics of the LBO, and the main challenge when valuing a PE target (See chapter 3.4). For these reasons, PE firms are intensely focused on the cash flow of the business, and investors often do not receive a return of their investment until the exit year (when the target is exited), while there is little or even negative return in the early years. This makes an investment in a PE fund very illiquid and the return profile of PE funds will typically follow a j-curve (Andersen & Frigast 2008). 13

14 Figure 3 - Value creation over time in PE Fund (J-curve) Value Time Source: Andersen and Frigast 2008 p. 13 and own construction Following Miller and Modigliani propositions, the high leverage is not in itself a direct value creating activity, as capital structure has no effect on the return on assets. The higher return is due to higher financial risk. No studies have so far proven that PE firms outperform public traded companies, when looking at a risk-adjusted return (Vinten and Thomsen 2008a). When a PE firm invest in a company it is typically with the intent to improve the company over a 2-7 years period (holding period) depending on the target (Andersen and Frigast 2008), after which the value of the company will have increased (if successful). In order to realize the gain, PE firms choose one of three possible exit strategies: An outright sale to a strategic buyer A public offering (i.e. sell the company through an IPO) A recapitalization either by the existing PE firm or by a new one Often PE firms choose to sell to either a strategic buyer or a financial buyer (Andersen and Frigast). However when market conditions are in favour of a public offering the PE firm exits through an IPO. The downside of an IPO is the restrictions of the lock down period, which forces the seller to hold a fraction of the initial shares over a certain period of time. This forces the PE firm to exit over a longer period of time. In the remainder of the thesis I will not distinguish between the different exit strategies, but simply refer to it as the target is exited. Life of a PE fund While the PE firm is an ongoing concern, the PE funds are limited life entities, with a lifespan normally of around 10 years, but often with the possibility to extend it with three yeas extra 14

15 (Spliid, 2007, p. 34). In order to explain the actions of a PE fund, Cendrowski et. al. (2008) divides the life of a PE fund into four stages: Organising/Fund-raising (0-1.5 years), Investment (1-4 years), Management (2-7 years) and Harvest (4-10 years) Figure 4 - Stages in the 'life' of a PE Fund Organizing & Fund Raising Investment Management Harvest (Exit) Harvest 10 Years +3 years Source: Cendrowski et. al p. 11 and own construction In the organising/fund-raising stage, the focus of the fund s investments is determined and investors are recruited. After the investors are found, the management company begin to scout for potential targets, and when suited targets are found, invest in them. This leads to the management stage, where managing of the acquired firms is commencing, which, as mentioned earlier, is one of the key features of the PE business model. Because the returns on the investments are relatively high, the time value of money is crucial to the funds. The funds will therefore try to realize their investments as soon as feasibly possible, which are done in the Harvest stage. 15

16 3 Valuation of PE targets The starting point Before I analyse the applicability of real option to PE valuation, a starting point is needed in order to see where we are coming from. In this chapter I will go through traditional valuation models and relate them to valuing a LBO in order to determine why these are not suitable for valuing a PE target and consequently how they should be valued. This will be followed by an assessment of how this model incorporated the value of the uncertain future and thereby flexibility. 3.1 Discounted Cash Flow model (DCF) In this section I will examine the Discounted Cash Flow (DCF) model s applicability to valuing PE investments. The DCF valuation method is the most widely used valuation method in practice because its applicability is perhaps the most profound within the different valuation methods. The DCF model takes an enterprise approach, where it focuses on the operating cash flows of the company, with both debt and equity holders as residual claimants. One of the most critical parts in the DCF model is the discount factor. The Weighted Average Cost of Capital (WACC) formula used for discounting the free cash flow (FCF) in DCF valuation assumes the company has a target capital structure, meaning that it keeps debt levels or debt equity ratios constant. Small short-term fluctuations in this ratio are accepted, as long as the company relatively fast returns to the pre-determined level of debt and equity. In addition to this, the WACC formula has fixed the cost of debt and equity given the capital structure chosen by the company (Brealey, Myers & Allen, 2006, p ). However in relation to valuing a potential PE target some of the models assumptions are challenged. The capital structure is not fixed due to large debt repayments, and the changes cannot be considered as small fluctuation in spite of the company returning to a normal debt-to-equity ratio after the exit of the PE firm. Because of this the DCF will be very cumbersome to use, as many calculations are needed to adjust for the challenged assumptions. 3.2 Adjusted Present Value (APV) In this section I will examine the Adjusted Present Value model s (APV) applicability to valuing PE investments. The APV approach takes a different route to the enterprise value than the DCF model. The big difference between the DCF method and the APV method is their way of handling the effects from financial leverage (Koller et al. 2005). APV in contrast to the Enterprise DCF method does not try to account for the leverage by weighing it in the cost of 16

17 capital but tries to value it as a separate entity, by dividing the value of the company into an all equity financed part and the value of the tax shield by having debt. Adjusted Present Value = Enterprise Value as if company was all equity financed + Present Value of Tax Shields Source: Brealey, Myers & Allen, 2006, p. 521 The basic concept of dividing the value of a company into an all-equity financed part and a part, which captures the value of financial leverage, is, in principal, suitable for valuing a potential PE target. By doing so we get to see how much the leverage creates in value for the LBO. However the model does not actually come up with a direct answer for the market value of debt and equity, which makes it difficult to value the initial investment of the equity holders. Furthermore the model does not have an intuitive explanation for the development in equity value over the lifetime of the LBO, because the value of an all equity-financed company is an arbitrary size. The final problem, when valuing a LBO using APV is how it addresses the value of the tax shields. The tax shields are definitely not certain and must be weighted by the expected default probability. In relation to this discounting of tax shield should be done with a diminishing rate, which needs to be estimated each year, because required return by debt holders decreases as the firm gets more unlevered. 3.3 Multiples for Valuation An alternative to the DCF and APV model is using multiples for valuation (MV). This approach uses the intuitive easy-to-understand proposition that similar companies should have similar value. Weston et al., 2001 calls it a quick and dirty way to value companies, and because of that it is a widely used approach to valuation. The basic idea behind MV is to value a firm by comparing it with equivalent firms or transactions (Koller et al., 2005). A multiple is the ratio of a fundamental (enterprise value or equity value) to the value driver (sales or EBITDA etc.). The valuation is done by multiplying a value driver, such as sales, earnings or cash flow, by a multiple from a selected peer group. If the valuation of a company is related to a potential merger or acquisition (M&A), multiples based on similar transactions might be a better estimator of value than peer group multiples, because they incorporate market premiums and synergies. By averaging many similar transactions it is possible to give a rough estimate of the expected premium of the particular target. 17

18 The main drawback when using multiples as a valuation tool is the problem of finding suitable peers. This drawback is enhanced when applied to a PE target valuation. Because of the aggressive use of leverage and changing capital structure, it is even harder to find appropriate peers. One way to solve this is by using transaction multiples. But even though this may give a more precise valuation, it still does not eliminate the difficulty of finding appropriate transactions. Furthermore it is problematic to determine, which multiples captures value in the best possible manner. When this is mentioned MV is an easy to understand measure, which can give a good indication of the value of an LBO, and can be a good support to other valuation models. 3.4 LBO valuation method As described above, the three most common valuation methods all have different characteristics which are useful for different scenarios. The one thing they have in common though is that they all have shortcomings when applied to a PE target, mainly because of the LBO s changing capital structure. A different valuation method is therefore needed. In this section I will give an overview of the model used in practise when valuating PE targets The LBO valuation model (Bonnerup et al. 2007). The purpose of the LBO valuation model is to identify how much equity the PE firm can justify investing in the target company. It is different from valuation methods like the DCF method, which captures all of the enterprise value in one calculation. In the LBO model four steps is needed to calculate the value of a company (Bonnerup et al 2007). 18

19 Figure 5 - LBO Valuation method EV 0 Step 4 Equity is discounted back using Internal Rate of Return (IRR) to et total value at t=0 EV 5 EQT 0 CF 1 Step 1 Level of debt Debt 0 CF 2 Total Enterprice value CF 3 CF 4 CF 5 EQT 5 Step 3 Total value of the company is estimated using: - DCF - Multiples etc. Debt 5 t = 0 t = 5 Step 2 - CF lowers debt levels Source: Own construction The structure of the LBO valuation can be seen from figure 4. The valuation is divided in two parts. The period where the target is owned by the PE target (Holding Period), and the period after the expected exit where it is owned by the potential buyer/buyers (Post Exit Period). The reason for this division is the characteristics of target in the two periods. In the holding period the PE firm makes firm specific changes (active ownership) and has a dramatically changing capital structure. In the post Exit periods the is assumed to be in a more steady state, as it is the value to potential buyers which is interesting as the target is exited (For elaboration se chapter 7.2 and 7.3). The reasoning is that first the optimal level of debt is calculated, and the cash flow available for debt repayments is calculated for the holding period in order to get the debt level at exit. Next the total enterprise value at exit is calculated using traditional valuation methods. This is the value the company will to the owners after the PE firms exits (Post Exit Period). The equity is found as the residual of the remaining debt and the total value. Finally the equity is discounted back using an IRR to get the total value by adding the discounted equity value to the debt level estimated at the beginning. The main reason for this approach is the changing capital structure of the company, which make a full enterprise valuation in one step difficult, and furthermore by using this approach, the highest possible price that the PE firm can justify paying for the company is found (Baldwin 2001a). 19

20 The model is bound by some of the same assumptions as the DCF and Multiples valuation because the Enterprise Value in the exit year is determined using these models. However the assumptions are not challenged as much as they are in the holding period, because it is assumed that the company returns to a normal rather steady capital structure at exit and in the following post exit period. In order to determine the equity value in the entry year the LBO models uses an IRR as discount factor. This reflects the changing required return to equity holders during the years of ownership, as well as a liquidity premium and a carried interest for the PE firm. The LBO valuation model is thus designed to follow every process in an actual Buyout. It therefore has obvious advantages compared to the traditional valuation methods as it captures the risks and value creation of every process in a LBO. Each step of the valuation will be examined in detail in chapter Inadequacies of LBO model Even though the LBO model is the superior valuation tool for PE targets, it is still a static valuation tool which does not incorporate the value of flexibility. The main problem with the LBO model, as well as all other static valuation models, is that it assumes that the estimated scenario regarding future cash flows will not be revised at a later point in time. Hereby the model implies that a one-off decision is made at t=0 on the basis of the strategy and expected investment plan under the assumption that the decision makers do not have any possibility of future actions. Because the future is uncertain managers often react in accordance with how the future develops, this assumption is in contrast to reality. For PE firms this could be exiting earlier or later than planned. The flexibility associated with being able to change strategy in the future is not incorporated in the LBO model, and simple sensitivity analysis using different exit points is inadequate for valuating this flexibility. This means that the LBO value only partially captures the total value of the company, and that another valuation tool is needed to assist when valuing PE targets. The focus in the next chapters will be on valuing the flexibility available to PE firms when planning their exit. 20

21 4 Real Option Theory In the previous chapter I stated that the model used by many PE firms (LBO model) has shortcomings when assessing the flexibility of exiting when conditions are most favourable. It is widely recognized in financial literature that Real Option Theory can be useful for analyzing and valuing flexibility. In order to analyze Real Options Theory s applicability to PE firm s exit planning, this chapter will give a general description of real option theory and the underlying assumptions. An option is a concept that has its background in financial theories where it is defined as a right, but not an obligation, to take action at a given time or time period at a predetermined price or price basis. Financial options relate to marketable priced assets, such as stocks, bonds, currencies or other similar assets, with two basic options as the foundation of option theory, call options and put options. A Call Option is the right but not the obligation to buy the underlying asset, while the Put Option is the right but not the obligation to sell the underlying asset (Allan et al., 2006). Based on the mindset of financial options, the concept of real options was first introduced by Stewart C. Myers in 1977, where he applied it to capital budgeting and allocation of R&D resources. Noting that some investment opportunities give the right, but not the obligation, to use a specific operating action in the future (Leiblein, 2003). Since then real option theory has been thoroughly discussed in both financial and management literature, with numerous different definitions suggested. Andersen (2000) describes real options as: the company s opportunity to use tangible and intangible assets in completely new or alternative ways in the future without having the obligation to do so, and can be related to all resource-committing actions in an organization, and thereby influence the company s flexibility to react on new opportunities or threats. 4.1 Classic Real Options Options can be distinguished along different dimensions such as ownership, the source of value, the complexity, and the degree to which options is available. But the most common typology, and the one I will use here, refers to the type of managerial action available (Mun 2006). Copeland and Antikarov (2001), Trigeorgis (1996), and Mun (2006) have all divided these actions into explicitly defined real options. In the following I will describe the most common Real options: 21

22 Option to defer As the name suggests this option gives the option to defer an investment. Examples of this could be oil companies with license to drill for oil or recover other natural resources or development of real estate. The option is an American call option on the value of the project. Option to contract/expand This is an option to either decrease the investment without abandoning it completely or expand the current asset in place, if the asset develops either positive or negative. These are either put or call options on part of a project or investment. Option to abandon This is the option to stop a project or shut down a company. Typically this option is used by capital intensive companies, where an option to stop operations and realize some kind of salvage value could have a significant value. Switching option Switching option are the right to close down an open operation by paying a fixed shutdown cost and the right to open it again for another fixed cost. This gives a portfolio of puts and calls. An example could be a temporary shutdown of mines due to deceasing mineral prices. Compound options These types of options are compounded call options on parts of the combined project, sometimes with an abandoning option at the end. Examples could be R&D projects within medical industry, where product development typically is divided into stages, where each stage is dependent on the previous stages. This option is in some contexts also known as a growth option. 4.2 Valuing Real options Since real options originates from financial options, the most apparent choice for valuing Real Options must be from existing options pricing models. This chapter will establish that there is a coherence between financial option valuation models and real assets. Even though Real option theory is based on the concept of financial options, they distinguish themselves in different ways. First of all, financial options are contractually based on financial assets which are commonly traded and priced in markets. Real options are not contractual based, but simply provides the holder with the opportunity to take an action, such as sell an asset or invest in a project. Furthermore, real options refer to the use of real assets which are not always 22

23 priced in a market nor have alternate value. The values of such assets are therefore subjective and depended on the utility of its holder and the cash flow it generates to the holder. Consequently, the underlying variables that determine real option values are less clearly defined and have to be approximated using reasonable assumptions. Therefore, in order to justify using financial models for valuing real options, it is first necessary to demonstrate coherence between the value drivers of financial options and real options. By doing so the characteristics of the real option can be mapped onto the structure of a financial option, and thereby use the option pricing models. Table 6 link the six underlying variables of an option to characteristics of a real option and illustrate how an increase in the variables affects the value of the option. This is done using an option on a common stock as example. Riskiness Timeframe underlying opportunity Figure 6 Mapping Option Inputs to Real Asset Characteristics Call Option Variable Real option Influence Stock Price S PV underlying asset Exercise Price X Cost of exercising option Time To expiration t Timeframe of opportinity Risk free rate of return r f Time value of money Stock Volatility σ Riskiness of asset Dividends on stock b Dividends on asset Source: Luehrman (1998) and Copeland & Antikarov (2001) p. 6 Exercise Price (X): To exploit a business opportunity (exercise option) often costs money. The money spent on doing this can be compared to the price of the stock. This corresponds to the option s exercise price. Stock Price (S): The present value of the underlying real asset corresponds to the price of the stock in a traditional financial option on stocks, and is in real option denoted as the options Asset Price. 23

24 Time to Expiration (T): The lengths of time the company has before losing the opportunity correspond to the options time to expiration. Standard deviation (σ): The uncertainty about the future value of underlying asset s return i.e. the riskiness of the investment, is equivalent to the standard deviation of the options return. Risk-free rate of return (r f ): Finally, the time value of money for the underlying asset correspond to the risk-free rate of return. Besides these five general variables (Copeland and Antikarov, 2001) introduces a sixth variable, dividends (b) that may be paid out by the underlying asset or cash flow lost to competitors i.e. the cash inflows or outflows over the life of the option. The previous section has made it feasible that real options can be valued using existing option pricing models, which will be examined in the next chapter. I will go more into detail on each parameter when applying real option theory to PE firms exit possibilities in chapter 6. 24

25 5 Option Pricing Models Before I go into the specifics of the PE options, it is necessary to present the different option valuation methods and determine the most appropriate one for valuing a PE option. In this section I will therefore present the two most common option pricing models and compare them in terms of practical applicability to the valuation of a PE option, resulting in a choice of valuation method. For the purpose of valuing options the two most taught and common valuation methods are The Binomial Model and The Black & Scholes model (Allen et al., 2006). Both models are built on the same underlying principle of the market-replicating portfolio. The main assumptions behind the market-replicating portfolio are that there are no arbitrage opportunities and that there exist a number of traded assets in the market that can be obtained in order to replicate the existing asset s payout profile. The replicating portfolio must have the same price as the option due to the Law of One Price, which states that two assets having the same future payoffs must have the same price; otherwise, arbitrage opportunities would exist. The key assumption that differentiates the two methods is the price development of the underlying asset. Binomial model assumes a discrete development of price and can therefore be solved using relative simple algebra, while The Black & Scholes assumes a continuous development which calls for more technical stochastic calculus mathematics (Mun, 2002). 5.1 Binomial Model Cos, Ross and Rubinstein (1979) was the first to develop a model based on the discrete time approach, where the assumption is that price develops gradually and only at specific points in time. They use discrete mathematics to develop a binomial lattice approach to option pricing. Lattices are, broadly speaking, more versatile than stochastic calculus when pricing options, and can be used to solve almost all option problems 2 (Copeland and Antikarov, 2001). No matter what type of real option problem you are trying to solve using the binomial lattice approach, the solution can be found in one of two ways. The market-replicating portfolio, as mentioned above, or the use of risk-neutral probabilities. The results obtained will be identical for both methods. In this thesis I will use risk-neutral probabilities, as the underlying mathematics are easier to apply. Furthermore, some financial perfectionists will argue that because a market-replicating 2 The reason for this is that the present value of real assets follow a geometric Brownian process as modeled by binomial lattices. 25

26 portfolio is based on highly liquid assets, real assets and firm specific projects do not fulfil the assumptions behind the market-replicating portfolio Risk-neutral Probabilities The reasoning behind the risk-neutral probabilities approach is basically simple. Instead of discounting risky cash flows with a risk adjusted discount rate similar to the DCF models, one can easily risk-adjust the probabilities of specific cash flows occurring at specific times. Consequently, the cash flows are transformed into a certainty equivalent which can be discounted at the risk free rate (Mun, 2002, p ). The calculation of the risk-free probability is done using: Time to Expiration (T), Standard deviation (σ), Risk-free rate of return (r f ), and Dividends (b). Using the variables the up and down factors are calculated using: = = = Source: Mun, 2002, p. 144 Where (δt) is the time-steps =. Time-steps are the number of periods that the options time to expiration (T) is divided into. The risk-neutral probability (p) is then calculated as: Source: Mun, 2002, p. 144 = Note that these equations assume a continuous development of prices and dividends (b) are therefore also continuous dividend in percentage (Mun, 2002, p 144) Setting up the binomial lattices Binomial lattices are used under the assumption that the time intervals are of equal length and that the price in following period can only have two outcomes. The process can be either multiplicative or additive. The main difference of the two is that when the number of periods becomes sufficiently large the multiplicative method tends to follow a log-normal distribution and the additive tends to follow a normal distribution (Copeland and Antikarov, 2001). 26

27 In any binomial lattice a minimum of two lattices are needed. One with the price development of the underlying asset (event lattice) and one with the development of the option value (option lattice). A third lattice is often made in order to establish what actions is preferred in each point in time i.e. exercise the option or not. This has no effect on calculating the value of the option, but is used as information on if and when the option should be exercised. Step 1: Evolution of the underlying asset The u and d factors described above are used to make the event lattice. The process is multiplicative which entails that the value of the underlying asset at each interval (t n ) is multiplied with the u and d factors, starting with S 0 at t 0 and continuing to t 1, t 2 etc. (from left to right in the lattice) (Copeland and Antikarov, 2001). Figure 7 shows example of underlying asset lattice using the multiplicative process. Figure 7 Underlying asset lattice u 3 S 0 S 0 u S 0 d S 0 u 2 S 0 ud S 0 d 2 S 0 u 2 d S 0 d 2 u S 0 d 3 S 0 t 0 t 1 t 2 t 3 Source: Own construction The example in figure 7 is limited to three periods, but the lattice can be extended to include as many periods as required which likewise increases the accuracy of the outcome. Notice that the lattice is recombining meaning that the branches come back to the same point. This simplifies then calculation greatly and non recombining lattices can easily become very chaotic. Step 2: Option valuation lattice The second step is to calculate the option value lattice using the evolution of the underlying asset. The procedure for valuing an option using binomial lattice is called backward induction (Mun 2002). This implies that the value of holding the option is derived backward in the lattice (from right to left), starting with the terminal nodes (t n ) and then the remaining nodes (t n-1 to t 0 ). Figure 8 shows an example of an option value lattice. 27

28 Figure 8 Option value lattice C o = max(p C u +(1-p) C d ; S 0 -X) e -rf δt C u = max(p C uu +(1-p) C ud ; u S 0 -X) e -rf δt C d = max(p C ud +(1-p) C dd ; d S 0 -X) e -rf δt C uu = max(u 2 S 0 -X ; 0) C ud = max(ud S 0 -X ; 0) C dd = max(d 2 S 0 -X ; 0) t 0 t 1 t 2 The terminal nodes are calculated through the maximization between exercising the option and letting the option expire, based on the number from the corresponding values in the event lattice. That is, if the cost of exercising the option exceeds the benefits it is better to let the option expire. The opposite goes for a put option, as this is the right to sell the underlying asset. For a call option the general equation in the terminal node is (Mun, 2002): Source: Mun, 2002, p. 166 =,0 The value at the intermediate nodes is calculated using the risk-neutral probabilities and discounting them with the risk-free interest rate. The general equation for an American option when assuming continuous time, is then the maximum of the two previous values discounted back one period and the difference between asset value and exercise price. =max + 1,S X Source: Mun, 2002, p. 166 Using backward induction all the way back to the starting period t 0 gives the value of the real option for the firm today. Step 3: Action lattice As mentioned above a third lattice is sometimes made in order to determine if and when a option should be exercised. At the final node the decision is only based upon the value of the underlying asset compared to the exercise price. If the value of the underlying asset is higher than the exercise price, the option should be exercised (if call option). In the other nodes the action does not just depend on weather the value of the underlying asset is higher than the exercise price. If the present value of the option value in the next period is higher than the difference between the 28

29 value of the underlying asset and the exercise price, the option should not be exercised as it has greater value to keep the option for another period. Notice that the higher the number of time-steps in the lattices, the more accurate the result becomes. At the most extreme, when the number of time-steps approaches infinity, that is the time between steps approaches zero, the discrete binomial lattice approaches that of a continuous model (Closed for solution), such as the Black & Scholes model (Min, 2002, p. 158). In the next section I will take a closer look at the Black & Scholes model. 5.2 The Black & Scholes model With their article from 1973, Fisher Black, Myron Scholes, and Robert Merton were the first to give a closed form solution for the equilibrium price for a European call option, the Black & Scholes Model (B&S model). It has since been the basis for numerous studies and papers about the prizing of options and empirical testing hereof. In essence, the model is a special case of the binomial model where the underlying asset is assumed to follow a continuous stochastic process instead of a discrete. Otherwise, it is based on the same underlying assumptions of no arbitrage and market replicating portfolio and that the movement of the underlying asset follows a lognormal distribution (multiplicative binomial model) (Copeland & Antikarov, 2001). The Black & Scholes model is a so called closed form solution, meaning that a value can be found with an equation using a set of inputs. The inputs in the B&S model are the same as the binomial model, with dividends as the one exception. The value of a call option (C) is calculated as: Source: Copeland & Antikarov, 2001, p. 106 = where N(d 1 ) and N(d 2 ) is the cumulative normal probability of unit normal variable d 1 and d 2 respectively. d 1 and d 2 are calculated as: = ; = Source: Copeland & Antikarov, 2001, p

30 Other than the assumptions also applying to the binomial model mentioned above, the B&S model has several other restrictive assumptions embedded (Copeland & Antikarov, 2001 and Mun, 2002): The option can only be exercised at maturity it is a European option There is only one source of uncertainty It can only be used on a single underlying risky asset; ruling out compound options No dividends on the underlying asset The current market price and stochastic process of the underlying asset is known (observable) The variance of the underlying asset is constant over time The exercise price is known and constant over time No transaction costs It has to be noted that variants of the B&S model have been made, which relaxes some of these assumptions. Examples are a B&S model for American options and options with dividends (Trigeorgis, 1996). But B&S models are based on calculus of stochastic differential equation, also called Itô calculus, which is highly complex. So unless one can find a modified B&S model that fits one own specific situation, the process of deriving a B&S model that does is very cumbersome and complex. 5.3 Model choice As stated in the previous sections the Binomial model is more versatile than the B&S model and is therefore more applicable in practice. As the B&S model is a closed form solution and was developed for valuing financial option, many of the underlying assumptions is bound to be violated when dealing with real options. Real options are more specific than financial options and need individual specifications. This is one of the binomial models most distinguished advantages and is therefore easier done using the binomial model and. And as I will highlight later, the options available to PE firms are very specific with distinctive modifications to a general put or call option, and correcting the Black & Scholes model to them would require a very complicated use of algebra, if it is even possible. Moreover, the binomial model can be a close approximation of the Black & Scholes model and the advantages of using continuous time are therefore not decisive. The Binomial model is more intuitive and easier to explain to for example investors, decision makers and potential buyers than a closed form B&S model. So all in all, the binomial model is most appropriate to apply in practice and the model of choice in the further analysis. 30

31 6 PE Exit options and the general framework Now that the appropriate framework has been established, the options available to PE firms in relation to their exit strategy can be determined 6.1 PE firms exit option (PE option) Basically the option available to PE firms when planning their exit strategy is simple. Exit now or later, meaning that the PE firm can choose to keep the portfolio company until conditions for exit are more favourable. Similar to financial options, real options can either be exercised at a specific point in time (European option) or merely at any point before expiration (American option). In general there are no restrictions on when a PE firm can exit their investments as they can exit as soon as conditions are favourable of an exit. The PE option can therefore be seen as an American option. Even though the option to defer exit is the option to divest/sell and not invest, which is normally the characteristics of a put option, the option can still be seen as a call option. The reason for that is that an increase in the value of the underlying asset (Target Company) increases the value of the option. This is the characteristic of a call option, while a put option benefits from a fall in value of the underlying asset. Hence it is not the action (buy vs. sale) that determines the definition but the value creation in relation to the underlying asset. The PE option can therefore, as a starting point, be seen as an American call option with the PE target as the underlying asset. As described in chapter 3.4, the valuation of a PE target is divided in two parts, the holding period and post exit period. In the holding period the characteristics of the PE target change dramatically because of the PE firm s active ownership strategy and the rapidly changing capital structure, hence the LBO valuation method. After exit the target is assumed to take on a more steady state with more traditional characteristics (Chapter 3.4). Because of these two relatively different stages of a PE targets life it is also appropriate to divide the real option available to PE firms into two different options, as the options will also have different characteristics, resulting in, among other things but most important, different volatilities (For elaboration see chap. 7.4). Calculating the value of an option with changing volatility entails changing u and d factors, which, in turn, results in the binomial lattice, no longer beingrecombining. This complicates the valuation a lot, and even a small number of periods results in very cumbersome calculations and chaotic lattices. For example a binomial option valuation using 20 periods would have 21 different possible values at expiration using constant volatility. The same option, but with two volatilities changing in period 10 would end up having 111 different possible values 31

32 at expiration instead of 21. This of cause only gets more extreme as the number of periods increases, and it is, in my opinion, not a feasible method to value one option with different volatilities when using binomial lattices as valuation method. The two options are illustrated in figure 9. The first option is the option to exit before the estimated exit i.e. during the holding period (PE holding option) with maturity at the expected exit. In this option it is the flexibility of being able to exit before the expected exit which is valued. The second is postponing exit to some point after the expected exit (PE post exit option), and is hence the value of the flexibility to postpone exit. In the following I will name these options after their position in the LBO valuation model; the PE Holding option (Holding option) and the PE Post Exit option (Post Exit option) respectively (see figure 8 for illustration). It has to be noted that now in the PE Post Exit option refers to the point in time where exit is expected to be and will have to be discounted back to present (see figure 9). This division into two options also entail that the Post Exit option is only available if the Holding option is not executed. Otherwise it is assumed to be zero. 32

33 Figure 9 PE options PE Post Exit Option C 0 C δt C 2δt C δt C 2δt EQT 0 C 2δt CF EQT T CF Debt 0 CF CF Debt T t = 0 C 0 C δt C 2δt C δt C 2δt C 2δt Expected Exit C T C 3δt C T C 3δt C T C 3δt C T C 3δt C T PE Holding Option Source: Own construction C T C T C T C T 33

34 7 Expanded LBO valuation This chapter will examine the applicability of the real option theory to the PE firms Exit options, starting with putting the valuation into a general framework followed an examination of the static LBO valuation. Finally the customization of real option theory to the two PE Exit options is examined. 7.1 Expanded LBO Before the real option theory is applied to the PE firms Exit option, a general framework for the total value is needed. When using option theory to value a PE target it is important to notice that the option only values the flexibility available as a result of the uncertainty associated with the future. It therefore needs to be combined with a base-case static enterprise value given by for example the LBO- or DCF model. The combination of static- and flexible value is known as the expanded NPV (enpv) and it accounts for both the base case analysis and the added value of flexibility (Mun, 2002, p ). Source: Mun, 2002, p. 168 enpv = NPV + option Value In the case of a PE target, the static NPV is calculated using the LBO model (chapter 3) and the option value is calculated by two options; the PE Holding option and the PE Post Exit option (chapter 6.1). I will call this modification the expanded LBO (elbo). elbo = LBO value + PE Holding Option + PE Post Exit Option Source: Own construction Notice that the Post Exit option value is calculated at the planned exit, while the LBO and Holding option value is calculated at t=0. The Post Exit value must therefore be discounted to t=0 to be able to compare and add the three values. The discount factor should be the IRR as this represents the expected return and will therefore show the value the option has to the stakeholder. In the next section I will analyse how to use the expanded LBO on the valuation on PE targets including the applicability of real options on a PE target. 34

35 Before any valuation can be done a pro forma income statement is needed, as this is the basis for every valuation no matter what method is used. The income statement must include the years from the expected purchase of the target until at least the expiration of the Post Exit option. 7.2 Static LBO valuation Before the PE firms values the target s flexible value using Real Option Theory, the value of the target without flexibility needs to be estimated, as the valuations are closely related. As established in chapter 3 the most appropriate method to value a PE target is the LBO model. The model is based on the four steps illustrated in figure 4. This chapter will examine in detail how each step is best estimated as this is the foundation for both the static and flexible part of the Expanded LBO Debt Capacity First the maximum level of debt needs to be determined. In practice this is done using different estimates of the target s financial situation, such as credit history, expected free cash flow, current debt situation and this to negotiate with the debt providers (Bonnerup et al 2007). In practise a multiple on the company s estimated EBITDA (Earnings before interest and taxes) is sometimes used as a quick and dirty way to get a good indicator of the cash flow left for debt services (Christensen and Christensen (2007) Expected future cash flow for debt repayment Next the expected future cash flow is estimated in the holding period. Because of the aggressive use of leverage, the cash flow generated is almost entirely consumed by the debt service. The cash flow in the LBO model can therefore be seen as the cash flow available for debt repayment. Consequently the cash flow in the LBO model is calculated as the free cash flow in the DCF model, but corrected for interest payments. This means, that instead of starting out with EBIT or NOPLAT as in the free cash flow model, the LBO model starts with net income. Other than that it is calculated in the same way as the free cash flow: non cash depreciation and amortisation expenses are added to net income, capital expenditures and increases in Net Working Capital (NWC) is subtracted (Baldwin, 2001a). 35

36 7.2.3 Expected exit value In the exit year a complete enterprise valuation is conducted in order to establish the expected exit value. The Exit value should represent the value the company has to a potential buyer. Hence, it should be the valuation method that gives the best indication of the PE target s market value which should be used. Using the LBO model to calculate the exit value indicates that the buyer has the same conditions as the PE firm; same leverage and strategic possibilities etc. It is very unlikely that a potential buyer will have the same conditions as the PE firm, even if it is another PE firm. The use of a LBO valuation is therefore not recomended, and because the characteristics of the potential new owners are unknown a valuation based on traditional valuation models such as DCF enterprise or Multiples would be preferred (For further elaboration see chapter 7.3). The most common way for private equity firms to value their target at exit is by using multiples. This is done because of ease of implementation as well as adaptability to different settings and scenarios. Despite the widely use of multiples in the real world, I would briefly advocate for the use of a DCF valuation in a theoretical context. The DCF model is not dependent on finding comparable companies with identical characteristics on business areas, capital structure, and risk. Furthermore I believe that the DCF model can be applied to the exit year valuation because the capital structure must be assumed to have reached a normal level, which can be maintained by the new owners, and in many cases maybe even preferred. The equity is the found as the residual of the sales price and the debt level at exit. As described in chapter 1.3 the DCF and Multiple methods will not be described in detail in this thesis Required rate of return When determining the required rate of return (IRR) of the LBO the PE firm has to take both the investors and their own expectations into consideration (Baldwin 2001b): 1. First the expected return on the investment in the capital market, given the same risk, is estimated (CAPM IRR) to give the investors the market return. 2. Because a PE investment is an illiquid investment (chapter 2), investors must be compensated for the illiquidity of the investment, as this is not compensated for in the market capitalization rate (Marketing IRR) 3. Finally, managers of the PE firms expect a premium for their involvement, which also need to be incorporated in the IRR (Target IRR) 36

37 Figure 10 Construction of Target IRR Rates of return Target EQT IRR Marketing IRR Debt CAPM IRR EQT Debt EQT Debt Carried interest EQT Debt EQT illiquidity Premium Debt D/E Ratio Source: Own construction The CAPM IRR has to satisfy the investors opportunity cost of capital, which is attainable in the capital markets for an equally risky investment. (Baldwin 2001b) One way to estimate this required return is to use the Capital Asset Pricing Model. It is important to notice that in the LBO valuation significant change in capital structure has to be account for. The reduction in debt every year affects the risks associated with the debt and equity in the company. Baldwin (2001b) uses the relationship between the risk on the assets as well as on the liabilities, and the leverage of the company to determine the risk of equity in the capital market: Source: Baldwin (2001b) = + + In practice, however, it is often assumed that the risk of debt is insignificant (i.e. almost equal to zero) and hence the second term of the nominator is eliminated (Baldwin 2001b). Nevertheless one might argue, in light of the past years turmoil in the credit market, that debt is indeed risky and has a risk level significant enough to alter our risk on equity. Using the exit valuation of the target and the remaining debt at exit, the debt and equity ratios can be calculated and backwards induction is used to estimate each year s equity beta. The 37

38 equity beta is then used to calculate each year s associated required return to equity holders. In order to translate this into an IRR over the ownership period of the target, the geometric average of yearly returns to investors is found. This is equivalent to the CAPM IRR, which is what the investors would expect to for an equally risky investment in the capital market. Because investing in a PE fund is a very illiquid investment, investors need to be compensated for this illiquidity in the form of a higher return. Arzac (2008) describes this compensation as a PE discount which is the percentage discount there is on the value of private equity compared to value of publicly-traded (i.e. liquid) equity. This discount is added to the CAPM IRR to get the Marketing IRR. The average discount for PE firm is 20 % (Arzac 2008). Finally the Marketing IRR needs to be adjusted for the carried interest to the PE firm. As mentioned in chapter 2.1 the carried interest is normally 20 % of the return exceeding the hurdle rate of 8%. The LBO model uses an IRR as discount factor based on all stakeholders required returns to value the initial equity investment. In real life the IRR is very easy to communicate, but from a theoretical context the discount factor IRR is not a desirable measure because it does not capture the underlying volatility of the decreasing required return to equity holders over the ownership period Enterprise Value (at time zero) Finally the company value can be found by using the four steps presented in figure 4. The remaining debt is found by subtracting each year s cash flow and the equity is found as the residual of the enterprise value at exit and the remaining debt. The equity is then discounted back to t=0 using the Target IRR to get the enterprise value by adding the equity to the initial debt. 7.3 Volatility estimation Because volatility is the most important variable in a Real Option valuation I have chosen to give this a chapter of its own before I examine the valuation of the flexible part of the Expanded LBO. The first part will be a discussion of the data used to estimate the volatility of the PE option. This will be followed by an assessment of the theory for estimating volatility and how it is applied to a PE option Historic data vs. Forecasts The volatility for financial assets can be estimated using (1) historical market prices of the underlying asset (2) Simulation using management s prediction of future values. 38

39 When using historical market prices, an historical series of asset market prices is used to compute a series of one-period returns, from which the standard deviation is calculated and stated on an annual basis as the volatility parameter. Estimating the volatility for a real options model is more challenging because there are typically no historical returns or current market prices for the underlying asset. Furthermore, there are typically no historical returns for assets that are perfectly correlated with the project cash flows, which hinder the use of volatilities calculated on other assets. The underlying asset in PE firm s exit options is the target company. The obvious choice for calculating volatility of the underlying asset is to use the historical volatility of the value of the company, because it will include all relevant factors for valuing an option based on the value of the underlying asset. The amount of historical data from target companies differs a lot from company to company, but because PE firms acquire mature firms, some historical data is bound to be available. But unless the company was publicly traded before the acquisition, historical data of company value can be hard to find and at best very arbitrary. Even if the company was publicly traded before the takeover, the PE firm acquire the company with the intention to change and improve it, and a considerably change in key figures are therefore expected. It is therefore often even more questionable to use historical data when dealing with PE firms portfolio companies, but this again is very case specific. Another point is that even though there are some historical data, many observations are needed in order for the statistical models to be valid. Another approach is therefore needed. Copeland & Antikarov (2003) recommend using what they call Market Asset Disclaimer assumption. It is assumed that the present value of the underlying asset is the best measure of the market value when no market exists: What is better correlated with the project than the project itself? We are willing to make the assumption that the present value of the cash flows of the project without flexibility (i.e., the traditional NPV) is the best unbiased estimate of the market value of the project were it a traded asset. We call this assumption the Market Asset Disclaimer (MAD) Source: Copeland & Antikarov, 2001, p

40 This approach entails using management s estimates of the future, which may be more suitable, as their predictions are a more accurate measure of future development than historical data are when dealing with a PE target. Because an estimation of the future will have a very small amount of observations, a simulation using the estimates as variables will therefore be an effective tool for estimating volatility when historical data is inadequate or found too arbitrary to use. It require managers to provide individual cash flow assumptions and their correlations, rather than merely providing a single estimate of project risk combined into one volatility parameter. A third possibility is to calculate an implied volatility from the Black-Scholes or binomial model and the current market price of the option. As these marked prices almost never exists when dealing with Real Options the method is not relevant for PE options. Based on the above, I recommend using the MAD assumption and managements estimates, as I believe this gives the most realistic results Simulating Volatility Even though simulation requires more work than merely using historical data, I find that because of the high importance of the volatility parameter in real option valuation and the fact that historical data in many cases are inadequate, it is here that the PE firms should invest some time and effort to get as accurate an estimation as possible. Copeland and Antikarov (2001) present a method for estimating volatility in real option analysis using Monte Carlo simulation. Their simulation is based on the assumption of Market Asset Disclaimer (chapter 7.3 above) and on the management s predictions/expectations of the future, in the form of a static valuation method such as the DCF analysis. The method is consolidated which means the output is a single estimate of volatility, built up from the different variables (e.g. price, quantity, costs etc.) that contribute to it. The idea behind the method is illustrated in figure 11 40

41 Figure 11 Monte Carlo simulation of real option volatility Inputs Monte Carlo Simulation Output Year 1, Year 2,... Year T Uncertainty 1 Probability of PV Uncertainty 2 Uncertainty 3 Present Value Model Uncertainty N 0 PV Source: Copeland % Antikarov (2001) p.245 Their method is based on Monte Carlo simulations using Monte Carlo simulation programs such as Crystal Ball and At Risk. Doing a manual Monte Carlo simulation with multiple, and sometimes correlated, variables is a very cumbersome and extremely difficult mathematical process. The purpose of the thesis is not to present how Monte Carlo simulation works, but simply to use it as a tool for applying Real Options to PE firms. I therefore find it appropriate to use a specialized program for simulating volatility when historical data are insufficient, and this will save time and space for the analysis of Real Options and the result will be no different from a manual simulation. I therefore compromise by using Crystal Ball for the Monte Carlo simulation. The basic idea behind the method is to simulate the evolution of a few key variables which have a great influence on value of the company and thereby retrieving at large number of different company values using NPV valuation. The volatility based on these values is then used as an estimate of the volatility of the underlying asset. The method includes: Making a NPV estimation of the company value Determine the variables witch influence the company value For each variable determine: Mean, Standard deviation, Distribution, and auto-correlation as well as cross-sectional correlation Use Monte Carlo simulation to generate volatility based on the NPV and variables. 41

42 NPV estimation of firm value The first step is to set up a NPV valuation of the underlying asset using the appropriate valuation method. The Holding option and the Post Exit option have different characteristics and therefore different volatility, which is the main reason for separating the option value into two different options (chapter 6.1). When calculating the value of the underlying asset for the PE Exit options, the PE firm is interested in the value it can get from exiting i.e. what a potential buyer can pay. Hence, it should be the valuation method that gives the best indication of the PE target s market value which is used to value the underlying asset in the PE Exit options. But what model does that? The LBO model gives the maximum value that a buyer can pay, given the same conditions for leverage, active ownership etc. Using the LBO model is therefore likely to give a gap between the value that the target has for the PE firm, and could thereby give an unrealistic high value of the option. The DCF model, on the other hand, is maybe the most used and recognised valuation method. It gives a more general picture of the market value of the company by using the assumption from Capital Asset Pricing Model (CAPM). This entails using the WACC instead of a individual IRR as discount rate, and is therefore more consistent with capturing the true enterprise value (market value) of the company. Furthermore, contrary to using multiples, the DCF is not dependent on finding comparable companies with identical characteristics on business areas, capital structure, and risk. Based on the above, I find using a traditional DCF to estimate the value of the underlying asset to be the most appropriate. Furthermore capital structure must be assumed to have reached a normal level, which can be maintained by the new owners making a DCF valuation possible. It is also the model recommended by Copeland & Antikarov (2002) when simulating volatility this way, and is the method I recommend using. A DCF model is therefore the best way to calculate the value of the underlying asset in both PE options, and to distinguish them from each other I will call them DCF Holding and DCF Exit for the PE holding- and Post Exit options respectively. Basically it is a traditional DCF that is needed for both options. But depending on the variables that are simulated, the DCF must be build to include these variables, which is not necessarily the variables included in the traditional DCF. 42

43 Using the DCF as valuation tool raises the question of the capital structure, as this is assumed to be constant in a DCF (chapter 3.2) But since the underlying asset in the Holding Option is not considered a LBO but simply a normal market valuation, this is not an implicating factor here. The only question is what capital structure should be used, as this will affect the WACC and thereby the value. Since it is hard to precisely estimate the optimal capital structure of a company, I recommend using the capital structure reached at exit in the LBO valuation as the capital structure can be assumed to have reached a normal level by then, which can be maintained by the new owners. Hence, it would be the best estimate of a general capital for the company. Determining relevant variables Determining the relevant variables for the valuation is the critical point in this method. The PE firm will need to come up with variables which have a significant influence on the value of the portfolio company for both options. This of cause is again very case specific. Copeland & Antikarov 2001 mentions sales price, sales amount, and variable costs as the most common variables. Because the PE firm will make firm specific changes in the holding period, and the target is assumed to go into a steady stage after the planned exit, it influences which variables are most relevant in the PE Holding option compared to post exit period. Consequently the two options will have different volatilities, hence the division into two options. In order to determine which variables have the greatest influence on firm value a sensitive analysis is needed. One way to do this is to construct a Tornado diagram. Tornado diagram As mentioned above a tornado diagram help identify which variables in the DCF model has the greatest influence of firm value. A tornado diagram can be made in different ways, but the main purpose is to analyse the sensitivity of the variables in the DCF model one at a time. One way of doing this is to increase and decrease each variable with +/- 25 percent respectively holding the other variables constant, and thereby resulting in a diagram over the absolute change in firm value from a +/- 25 % change in the variables. A graphic illustration will show which variables the DCF is most sensitive to. Often it will be clear that the vast amount of sensitivity come from 2-3 variables (Copeland & Antikarov 2003, p 236). Most attention should of cause be on the variables with the greatest influence on firm value, as they contribute the most to volatility of firm value, ignoring any offsetting correlations. It is recommended though, that the analysis is not over complicated by using too many variables. 43

44 Characteristics of variables After the variables are determined, their probability distribution, mean and standard deviation must be determined in order for the simulation tool to simulate their values. This again raises the question of using historical data or not. This will be very dependent on each variable and the situation. Is it for example a variable which the PE firm can affect, it is best to use the management s subjective estimates. On the other hand if it is variables which are out of the control of the PE firm, and where they have little knowledge of the future development, estimation using historical data is preferred. Volatility Copeland & Antikarov s (2001) method for estimating the volatility of each parameter using managers predictions assumes that the variable follows a geometric Brownian motion. The procedure is to find the expected values of the variable at different points in time (V t ) and calculate the continuous growth factor (r i ) for each period. This is usually done when calculating the DCF model. In some cases, the development of the variable is determined using the growth factor r i which will be constant. The next step is to determine a confidence band where the value is certain to be within with 95% confidence, meaning the highest (upper V T ) and lowest (lower V T ) value the variable is expected to have. The volatility can then be estimated using: = Source: Copeland and Antikarov, 2002, p. 262, = It can be discussed whether these qualified guesses from management are to arbitrary to give a correct estimation of the variables volatility. But it has to be kept in mind that this method should only be used when it is assessed that the management has better knowledge of the future development of the variable than historical data can give. Furthermore it is a fast and simple way for the PE firm to get an estimate of the variables characteristics. Distribution The distribution of the variables can also be estimated, but I believe that the effort it takes to do this is not worth it. A simple assessment of the comparison of each variable to the general 44

45 distributions which I based on a standard deviation and mean value would be adequate. This is supported by Copeland and Antikarov (2001) which gives the example prices. Prices can be assumed to follow a log-normal distribution, as it is unlikely that prices will ever be negative. Correlation and Autocorrelation After each variable is determined and their general characteristics are estimated, the correlation between variables and their autocorrelation with themselves need to be determined. These correlations are also entered into Crystal Ball, and will increase accuracy of the estimated volatility. The correlation in Crystal Ball is measured using r-squared values, where a positive r- squared means that there is a positive correlation and a negative r-squared value means that there is a negative correlation 3. The correlation and autocorrelation can again either be estimated using historical data, or be set by management using their assumption of the correlation. For example, it is reasonable to assume that an increase in prices is followed by another increase or vice versa a decrease is followed by a decrease, which entails a positive autocorrelation. Management can then set the autocorrelation percentage after their assumption of the degree of correlation. Simulating the volatility After all the variables as set up, the volatility can be simulated using Crystal Ball. Notice that the volatility is estimated on the return by using the change from one year to the next, by holding the first year constant. Source: Copeland and Antikarov, 2001, p. 246 = Simulation using Crystal Ball will automatically generate some very useful data in the form of both statistical data and graphic illustrations. Example hereof can been seen in figure 12 below. 3 R-squares values lie between -1 and 1 (-100% and 100%) 45

46 Figure 12 - Examples of output from Crystal Ball Source: Own construction in Crystal Ball 7.4 PE Option valuation (Binomial model) The next step is to value the flexibility. I have chosen to use the binomial model, as this is more intuitive and flexible than the closed form solutions, such as the Black & Scholes formula. As the theory behind the binomial model was presented in chapter 5.1 it will only be explained briefly again here in order to give a better overview of the process The Variables In the pervious chapter the volatility was estimated using simulation. Before the binomial lattice can be set up, the other parameters need to be estimated. Underlying asset The underlying asset in the PE options is the PE target, but as mentioned above it is the underlying asset s market value which is relevant not the value it has to the PE firm. Nevertheless, the underlying asset is assumed to have different characteristics in the holding period and post exit period. The different characteristics are caused by active ownership of the PE firm. It could be argued that these changes made by the PE firm does not influence the value to the potential buyer, as they are specific actions made to increase the value for the PE firm, and should therefore not be included in valuation of the option. But since the vast majority of the return of an PE investment comes from the exit of the portfolio company, I believe, it is a fair assumption that the changes made by the PE firm in the holding period are focused on increasing the market value of the target, as this would increase the return. Therefore, the assumption that the underlying asset has different characteristics in the holding period and post exit period is valid. 46

47 In the binomial model, as I have chosen to use it, the changes made by the PE firm are indirectly implemented via the volatility (Standard deviation), and hence the different volatilities. Note the development of the underlying asset is estimated using the up and down factors, which in turn are determinate by the volatility. Because of that, and because the volatility has great influence on the value of the option, the estimation of the volatility is essential to the value of the underlying asset and thereby the option and potential improvements should be implemented in the volatility (chapter 7.3). In the Post exit period the company is assumed to enter a more steady state with no activities from the PE firm (Chapter7.2). Since the evolution of the underlying asset is determined by the u and d factors the value at of S 0 is essential for the value of the option. In chapter 7.3 I established that the DCF model was the most appropriate to use as valuation tool for the underlying asset. Therefore, it is also the DCF value which should be used as S 0 and not the value from the LBO model. Exercise Price The exercise price is the costs that the firm has when exercising their option. In many real options this is an actual cost associated with for example investing in new business, product or other investments needed in order to get a return from exercising the option. The PE firm s options do not have any significant direct cost of exercising the option (exiting the target). Of cause there are some transaction costs, but they are not large enough to influence the value of the option, and are therefore assumed to be zero. This does not mean that the exercise price is zero on the option. The cost for PE firms when exiting is the required return from all stakeholders in the investment. As described in chapter the IRR contains all stakeholders required return, and when the equity is discounted using the IRR we get the equity value needed to give the required return. Therefore, in the Holding option, the PE firm should only exercise the option, if the value of the underlying asset is higher than the firm value at the given point in time that gives the stakeholders their required return (required return value). Or said in another way, by exiting before maturity (expected exit) the PE firm gives up its right to sell at maturity which the cost of exercising the option i.e. exercise price. Hence the exercise price is calculated as the equity extrapolated with the IRR plus the remaining debt cf. the LBO model. 47

48 = + = 1+ + Source: Own construction 4 The same general principle goes for the Post Exit option. The return should at least be the IRR before exiting. The difference here is that the characteristic of the company has changed and the capital structure can be assumed to be constant (chapter 7.2). The debt obligations which consumed most of the cash flow in the holding period would now be reduced considerably, and the assumption that all cash flow goes to debt repayment can now be relaxed, hence, the company has cash flow available to pay out to investors. The PE firm is keeping the target longer than expected and unless any excess cash flow can be invested in positive NPV projects, they should be paid out to investors. Because the stakeholders are now receiving dividends they are realising some of their expected return (IRR), which in turn means that the required return value of the company is correspondingly lower. Thus, the equity is extrapolated with the IRR and each period s dividends are subtracted and the periods debt level is added to get the required return value in each period for the Post Exit option. = + = 1+ + Source: Own construction The dividends can be calculated directly from the financial statement using the expected dividend policy, while the debt is calculated using the assumption of the development of the debt. What the dividends policy and debt assumptions are will obviously vary from target to target. I believe though, that there are one scenario which can be assumed to be applicable in most cases: pay all net earnings in dividends and a constant debt-equity ratio. The dividend payments is due to the investors wanting some return realised, and the debt because, as the argumentation for using DCF as valuation tool, it is a reasonable assumption when the debt future debt structure is unknown. Hence, it is only the equity that changes according to the expected return and the total value is found using the debt-equity ratio and the formula can be written as: 4 Notice that n refers to the sub period in the binomial lattice (δt) and t is the relevant year in the total time to expiration (T). 48

49 = + = 1+ Source: Own construction Alternatively the debt level could be assumed to be constant, as a constant debt-equity ratio assumes that the debt raises and falls proportional with equity, causing a rapidly increasing/decreasing exercise price. The above of cause gives a changing exercise price in both options, which needs to be incorporate in the Option value lattice. Time to Expiration In many real options it is hard to determine the exact time to maturity. For example, it can be hard to determine for how long a business opportunity will be there, which depends on whether the option is proprietary or shared. When the option is shared, the value of the option is not necessary increasing with time to expiration, as competitors can also exercise the option. The PE firm s Exit options on the other hand are completely proprietary as the PE firm are the only one who can exercise the option and sell the portfolio company, which as a starting point entails an increase in the value when time to expiration increases. The Holding option follows the estimated holding period and the time to expiration is then simply the same as the estimated holding period. It is more complicated when dealing with the PE Post Exit option. In theory, there is no upper bound on the time horizon for the PE Post Exit option, as the PE firm in principle can keep the firm as long as it wants. This is no problem in the short run, but unless the PE firm neglect the fact that they are an on-going concern that needs investors for future investments, they need to keep their obligations to investors. One of those obligations is the time horizon of their investments. As mentioned earlier, because of the illiquidity of investments in PE firms, investors are highly dependent on the PE firm s ability to sell its portfolio companies and time value of money is essential. Therefore, because investors are crucial for the existence of PE firm, they need to show both current but also future investors that they fulfil their obligations. Hence, I find it reasonable to assume that the time horizon of the Post Exit option is determined by each fund s agreed lifespan, meaning that the length of the option on each portfolio company depends on when it was acquired in the life of the fund. (See 49

50 figure 4). As stated earlier, the lifespan of PE funds is 10 years with the possibility of extending it with up to a further three years to a total of 13 years. So a firm acquired in the third year with exit planned six years later, will have the option to postpone exit for four years. This is of cause very case dependant, and in some cases it could maybe be possible to extend it further into the future. But in the further analysis I will assume that life plus three years is the time horizon of the option. Another consequence of on going concern arises when the Post Exit option is at expiration. Normally when using real options, the holder can chose not to exercise the option at expiration, in this case exiting the portfolio company. But based on the same arguments as above, this is not necessarily the case for PE firms, as they again will not fulfil their obligations, and it is therefore questionable weather the fact that the PE firm can actually choose not to exit at expiration in realistic. A reasonably assumption here is that if the option goes all the way to maturity the PE firm will have to exit. This means that the equation for the value at the terminal node is no longer a max formula but changes to simply 5 : Source: Own construction = The reasoning behind this is that the PE firm has to incorporate the fact that because the choice to exit is not valid, the risk associated with the option is higher, and the option could have a negative value at expiration which should be discounted back like any positive value. Weather this results in a negative total value depends on the characteristics of the option. Of cause the two assumptions above can be relaxed in some situations, but as assumptions for a general case I find them reasonable and realistic. Risk-free rate of return Because of the use of risk-neutral probabilities, the time value of money for the underlying asset is the risk-free rate of return. The job here is to find an acceptable measure of what the risk free rate of return is. The closest to a risk free interest rate is a government bond or bond guaranteed by a government, and because the interest rate is used over the entire life of the option, a bond with the same maturity as the option is the most accurate to use. 5 Note that the equation has also been corrected to include the changing Exercise Price. 50

51 Dividends As mentioned in section 2.2, most PE firms do not pay dividends from their portfolio companies in the holding period, and it is therefore reasonable to assume that the PE Holding option has no dividends. As mentioned above, it is likely that dividends are paid in the Post Exit Option. Besides using the dividends to determine the exercise price, they should also be incorporated in the valuation of the option through the risk-neutral probabilities. In order to do that the dividends must be given as a continuous percentage (chapter 5.1) Setting up the lattices After the variables are estimated the binomial lattices can be set up. Since the theory of the binomial model was presented in chapter 5.1, the main focus here will be to relate it to a PE option and the general theory and formulas will only be presented again if it benefits the general overview. Step 1 Company value lattice The first step is to set up the lattice with the evolution of the underlying asset. The underlying asset in both PE options is the portfolio company/target. The evolution of the value of the target is determined using the u and d factors, which are calculated using the volatility and δt in the formulas presented in chapter = = = Source: Chapter 5.1 After the u and d factors are calculated the process is simple. The target values at t=0 (S 0 ) is multiplied with the u ad d factors to get and which in turn are also multiplied with the u ad d factors to get the S 2 values and so on (see figure 7 for graphic illustration). In the PE Holding option S 0 is the value from the DCF Holding is used as this is the market value of the company and the value they can expect to exit to (see chapter 7.2). Because the Post Exit Option is placed at the expected exit (in the static LBO) the value from DCF Exit is used for S 0 for the PE Post Exit option. Notice if it is another value which is used 51

52 Step 2 Option Value lattice The next step is to calculate the value of the PE option using the option value lattice. In order to make the option value lattice the risk neutral probabilities needs to be calculated using u and d factors, the risk free rate, δt and dividends if any. Source: Chapter = The method for setting up the option value lattice is called backward induction, meaning that you start by calculating the value at expiration and then using the risk neutral probabilities to discount the value back in time (See figure 7 for illustration). For both options the final node is calculated as a max formula of asset value minus the exercise price and zero, but because the exercise price is not constant, as in the general equation from chapter 5.1.1, a small modification of the formula is needed. Source: Chapter own construction,0 In terms of the PE options this means that if the value of the target (asset value) is higher than the required return value (exercise price) the target should be sold (option exercised) and the option value is the difference between the target value and the required return value. If, on the other hand, the value of the target is lower than the required return value the target should not be sold (option not exercised) and the option value is zero. The remaining nodes are then calculated using backward induction. Because the PE firm has the possibility to sell the target firm at every point in time, the option is American, and the calculation is again a maximization of the value of exiting and not exiting. =max + 1,S X Source: Chapter own construction The value of exiting is, again, the difference between the target s value and the required return value, and the value of not exiting is the value of the next period s up and down option values 52

53 multiplied with the risk neutral probabilities and discounted with risk free rate. A small modification of the variable exercise price is also needed here compared to the general equation from chapter 5.1. Step 3 Action lattice The third step is to make the action lattice. This has no effect on the value of the option, but gives an overview of the optimal actions in each period for the PE firm. The option should not be exercised unless the difference between the company value and the exercise price is higher than the present value of next period s option values, and in the final node the option should be exercised if the company value is higher than the exercise price. 7.5 Total value Finally the static LBO value and the flexible option values from the binomial valuations are combined to get the total value of the PE target. The LBO value and the PE Holding option value are both calculated at t=0, while the PE Post Exit option value is calculated at the time of the planned exit. The Post Exit option value therefore needs to be discounted back to t=0, in order to sum the three values. Target Value = LBO value + PE Holding Option + PE Post Exit Option Source: Chapter 7.1 The argument for discounting the equity at exit with the IRR was that it incorporates all stakeholders required return, and thereby getting the maximum value the PE firm can pay for the target. Therefore in order to compare and sum the different values the PE Post Exit value must also be discounted back from exit to t=0 using the IRR, and thereby getting the value it represents for all stakeholders. 53

54 8 TDC A/S valuation using expanded LBO In order to give the reader a better understanding of the process and to highlight possible pitfalls of applying real option to the valuation of PE targets, this chapter will go through a specific example of how a Real Option can be used in relation to PE firms and their option to delay exit of their investments. First I will value the case company using the LBO valuation method described in chapter 7.2, after which I will value the Real Option and combine the two to give a full value of the company. As case company I have chosen TDC A/S (former Tele Danmark A/S). The reasons for choosing TDC is that even though it was bought by PE firms, it is still listed on the OMX stock exchange, as the PE firms where unsuccessful in buying the whole company and thereby denoting it from the OMX. This means that the company is till obligated by the rules of a publicly traded company, which among other things include making much more information public. The information on TDC is therefore much more comprehensive than traditional PE firm s acquisitions. Furthermore, even though TDC has several different products, their main income comes from three markets (fixed telephone, mobile telephone and internet distributor) where a few, easily accessible data series gives a very good indication of the future development. The use of these information series will be described in detail later. 8.1 Case Company TDC A/S At the time of the takeover it was, with a value of 95 billion dkr., the largest PE takeover in Europe. The takeover was highly debated in the media due to some large investor s fight (with ATP as the main contender) to keep their shares and stop the takeover. The buyer was NTC, a consortium consisting of Permira, Blackstone, APAX, Providence and KKR. On November 30 th 2005 NTC made a public offer with deadline at January 12 th. NTC needed 90 % of the shares in order to denote from the OMX exchange, but only 88.2 % of the current stockholder accepted the offer. TDC are now in the unusual situation of being (primary) owned by a PE firm with all it includes but at the same time still publicly traded with all that includes in regards to information available etc (Spliid 2007, p 314ff). NTC financed the TDC takeover using 83 % debt resulting in TDC s equity to fall from 51 % before the takeover to 15 percent after, and 70 percent of EBIT are now used for interest payments. Out of the 95 bill raised, NTC s equity payments, Senior- and Bridge loans accounted for a total of 80.1 bill. The final 14.9 bill is divided between 0.8 bill from employees exercising their 54

55 employee options and because TDC at the time of the takeover was highly liquid 14.1 bill came from TDC s own cash reserve. The setup after the takeover is illustrated in figure 13. Figure 13 TDC takeover Temporary financing Final financing Consortium 100% NTC Holding Plc. 100% 15.2 bill bridge loan 15.2 bill bonds ATP 5.5% NTC Plc. 88.2% Others 6.3% TDC A/S 48.5 bill bank loans Source: Spliid (2007) p. 337 The senior debt is to be paid back over the holding period, while the Bridge loans are temporary loans which will be paid back as soon as the final bond structure is set. Because TDC was actually bought by a NTC in 2006 I will assume that it is now 2005 and value TDC on the basis of 2005 figures, and for simplicity assume takeover on january 1 st I do this, because the TDC acquisition is in its final years, and basing a valuation of a PE target on figures already influenced by a the active ownership initiatives as well as changed capital structure would mean I had to make many assumption my self and the calculation could become very arbitrary. By assuming it is 2005 I can use the actual financial figures of the acquisition and the following years after the acquisition, and thereby include some of the actual changes made by the PE firm in my analysis as well as using actual realised number as my assumptions. In my opinion this gives a more correct analysis, as it is beyond the scope of this thesis to make the necessary extensive market analysis needed to make a completely realistic valuation. The main sources for my analysis are therefore TDC s financial statements, public market analyses and research reports of TDC. I will elaborate on the sources as I go through the valuation. 55

56 8.2 Pro Forma Income Statement In reality a pro forma income statement is based on market analysis as well as internal firm knowledge. This is outside the scope of this thesis, and is furthermore not essential for the analysis of applying real options to the valuation of PE targets. But in order to make the income statement as realistic as possible I have made a few simple and realistic assumptions. One of my main sources for many assumptions is a LBO valuation made by Carnegie in November 2005 (Carnegie 2005), and I will refer to this report whenever I use their assumptions. The pro forma income statement is calculated using the yearly average changes (returns) from the actual figures, with 2006 as base case 6. By doing this, I can simulate on the yearly return instead of simulating each year s figure separately. This simplifies the simulation while it over a four year period (holding period) has little effect on the final valuations and the volatility estimate. Furthermore I assume that the development in the first three years of the holding period ( ) is the norm for the whole holding period, meaning that I use the same return for 2009 as well. For the post exit period, where there are no actual figures I have used market and industry reports, again with the estimation of a constant yearly return, as the estimator of future development. For example, the development of the fixed telephony sales beyond 2009 is calculated using the average yearly return on projected fixed telephone calls (in minutes) for the Nordic region and Switzerland, which are TDC main markets. By doing this I of cause assume that fixed telephone calls (in minutes) is a good estimator of TDC sales and that their market share is unchanged. Furthermore, I have assumed that the active ownership done by the PE firms have resulted in the yearly change in operational costs of -8%. Reduction of costs is an obvious choice for a firm specific change, and is likewise a good example of how the value of the active ownership can also be transferred to the potential new owners. This also entails that when the holding period ends, the company is assumed a more steady state and the reduction of costs falls to 1 %. I find these assumptions, reasonable, and fully adequate for the purpose of analysing the use of real options in PE valuation (For a full overview of yearly returns as well as the underlying data refers to appendix 3). 6 I have simplified the Pro forma Income statement, compared to TDC actual income statement and small insignificant items are disregarded, hence, the summarizations (EBITDA, EBIT and Net income) will differ slightly from the actual numbers. 56

57 . The pro forma income statement goes from beginning of 2006, when the acquisition of TDS was made, until the end of 2013, with 2006 to end 2009 being the holding period 7. In the years from I have used actual figures from TDC s annual reports, to the degree of which it is possible. The full pro forma income statement is presented in appendix 1, and from the income statement it can be seen that the income is rising from 2.48 bill to 7.62 bill in the holding period after which it stabilises with only a small increase for the remaining years. This is mainly due to a decrease in operational costs of 8% per in the holding period. 8.3 Static LBO valuation In this section I will calculate the LBO value of the TDC takeover (The full LBO valuation including assumptions is presented in appendix 2). Because TDC was bought by a consortium of PE firms, it is difficult to establish how long their expected holding period is. I have therefore used Carnegie s (2005) estimate of a holding period of 4 years, with exit at January 1 st Hence, a holding period of four years from 2006 to end This is supported by Børsen (2009c) which states that TDC could be nearing a sale. Obtainable leverage As mentioned in chapter 7.2 the obtainable leverage available to the PE firm is normally calculated using different financial predictions and by negotiating with the debt providers. This is naturally not possible here, but instead I have simply used the actual debt level of dkr bill which give a multiple of 7.1 times EBITDA in Expected future cash flow for debt repayment The cash flow for debt repayment is calculated as a traditional free cash flow, but because all cash flow is assumed to go to debt repayment net income is used instead of NOPLAT of EBIT. 7 The actual takeover was Feb. 2006, to keep things simple I assume the takeover was Jan. 1 st

58 Table 1 TDC Cash Flow to debt repayment Pro Forma Cash Flow Net income Depreciation & Amortisation Capex Change in Net Working Capital Cash flow to debt repayment Source: Pro forma Income Statement (appendix 2) As table 1 show, the cash flow available to debt repayment is increasing during the holding period. This is again mainly because of the decease in operational coats. Expected exit Enterprise value The expected exit value can be estimated using different valuation methods. I have chosen to use a multiple valuation, as this is the method mostly used in practice. I use an EV/EBITDA multiple of 5.0x (Carnegie, 2005), and, with an estimated EBITDA of bill in 2009, the exit value comes to bill. Required rate of return The required rate of return (Target IRR) is calculated using the method presented in chapter 7.2. First the CAPM IRR is calculated using the formula for the equity beta and backward induction to get each year s required return to equity holders and the geometric average these is the Target IRR (Table 2 shows the calculation). Table 2 TDC IRR calculation Target IRR Total Debt (DKKm) Equity Value (DKKm) Total Enterprise Value (DKKm) Debt ratio 81% 78% 74% 69% 64% Equity ratio 19% 22% 26% 31% 36% Equity beta 3,07 2,61 2,23 1,90 1,62 Cost of Equity 18,04% 16,15% 14,51% 13,10% CAPM IRR 15,44% Marketing IRR 18,52% Target IRR 20,63% Source: Own construction (appendix 2) 58

59 Notice that the equity and enterprise values in table 2 is just used for calculating the CAPM IRR, and are not the actual values of the company, as they are not adjusted for illiquidity and management s carried interest in the discount factor. The CAPM IRR is therefore adjusted for Illiquidity premium (20 %) and carried interest (20 % of the return over 8%) to get the Target IRR of 20.63% (For the complete calculation and assumptions see appendix 2). Company Value Finally the company value is found by using the steps from chapter 3.4 which is presented in table 3. Table 3 TDC LBO valuation LBO Valuation Initial Debt (Step 1) Debt repayments (Step 2) Debt at Exit Enterprice Value (Step 3) Equity at exit Initial Equity (Step 4) Firm Value Source: Own construction (appendix 2) The LBO valuation shows that in order to give all stakeholders their required rate of return, they will have to invest bill in equity, which combined with the initial debt of 78.9 bill gives a company value billion. This is therefore the maximum the PE firm can pay for the company, and a lower price will result in a higher return. 8.4 Volatility estimation As determined in chapter 6.1 the PE firm s Exit option is divided into two options with different volatilities, the PE Holding option and the PE Post Exit option, and hence two estimations/ simulations of the volatility is needed. NPV Estimation of Firm Value The first step is to set up the NPV estimation of the company value (underlying asset in Option). As established in chapter 7.3 the most appropriate method for estimating the value of the underlying asset in a PE option is a DCF model. The value of the company in the holding period (holding value) is bill (See appendix 4) and for the post exit period (post exit value) it is bill (See appendix 5). 59

60 Determining relevant variables One way to determine which financial figures has the highest influence on company value, and hence the most suitable variables to include in the simulation, is to set up tornado diagrams (Chapter 7.3). Figure 14 shows the tornado diagrams for TDC. Figure 14 TDC Tornado Diagrams with +/- 25% change (Holding Option) Variables' influence on company value Sales segments' influence on total sales Sales Operational Costs Depresiation & Amortisation Fixed Telephony Mobile Internet Other Source: Own Construction From the financial statement I have chosen sales, operational costs, and depreciation and amortisation as variables in the tornado diagram. As figure 14 shows (left figure) operational costs have the largest influence on the company value in the holding period and sales the second largest, while depreciation and amortisation have little influence. The two main variables for the simulation are therefore sales and operational costs. But TDC s sales are divided into four main segments; fixed telephony, mobile telephony, internet and other. These different segments have different developments and characteristics. I have therefore made another tornado diagram to determine which sales segments have the most influence on company. Not surprisingly it is mobile, and internet which has the greatest influence, as they account for the greatest part of total sales. Furthermore they are both increasing during the holding period, while fixed telephony is decreasing and other sales are constant. It could be argued that because of this, the selection of variables could easily be done without making a tornado diagram. But had the development been different with for example fixed telephony getting a larger part of total sales and mobile sales decreasing over time, it would have been much harder to determine their influence without making a comparison such as the tornado diagram. Because mobile telephony is the most influential sales segment and that the mobile industry is assumed to be the most volatile and competitive, I have chosen to divide mobile into quantity (represented by number of mobile customers) and price (represented by sales per customer) to give a more accurate estimate of the evolution of mobile sales. 60

61 Figure 15 TDC Tornado Diagrams with +/- 25% change (Post Exit Option) Variables' influence on company value Sales segments' influence on company value Operational Costs Sales Depresiation & Amortisation Mobile Internet Fixed Telephony Other Source: Own Construction The picture is the same for the Post Exit option (Figure 15), and the variables used in the simulation of both options are therefore: fixed telephony, internet sales, number of mobile customers, sales per mobile customer, and operational costs. Characteristics of Variables After the variables have been determined their characteristics must be estimated: mean, standard deviation and correlation. The mean is simply the yearly return used in the calculation of the pro forma income statement, whereas the standard deviation (volatility) is calculated using the volatility formula from chapter 7.3. =,,, = 0.20 Source: Chapter 7.3 and own construction (Appendix 4) The equation above is a calculation of the volatility of the fixed telephony sales for the holding period. The thing to notice here is that, because the volatility is calculated on the constant yearly return, the sum of returns is simply the return times T. Furthermore the lower level of the 95% confidence interval is assumed to be 4,500. I have based the lower values on how volatile I believe the variable is. For example, I believe the mobile industry to be the most volatile and hence the mobile customer and sales per mobile customer has a relatively smaller lower value and thereby higher volatility. Because I do not have the same insight in these figures as the management of TDC would have, the assessment of the lower values is quite arbitrary. I have chosen to disregards any correlations, as it here also will be difficult for me to give precise estimates. This would of cause be easier for the management of TDS, as they have much more 61

62 insight in the company and its variables. An overview of all variables and their characteristics can be seen in table 4. Table 4 Input Variables: Volatilities calculations Volatility Holding option Post Exit Option V0 r Lower Vol. (%) Vol. (actual) V0 r Lower Vol. (%) Vol. (actual) Fixed Telephony ,06% ,92% -1,21% ,00% ,54% -0,88% Internet ,32% ,14% 1,55% ,77% ,34% 0,68% Operational Costs (27.302) -8,02% (15.000) 10,34% -0,83% (21.288) -1,00% (10.000) 21,03% -0,19% Mobile Telephony: Mobile customers ,35% ,01% 2,98% ,39% ,62% 0,68% Sales pr. customer 3,00-5,18% 1 27,23% -1,41% 2,55-0,77% 1 26,34% -0,20% Source: Own construction (appendix 4 + 5) Simulating Volatility Now the data from table 4 can be typed in to Crystal Ball, wherethat it is the volatility of the return between t=0 and time 1 that is needed using the formula from chapter 7.3. Figure 16 Volatility simulation (Holding Option) Source: Own construction in Crystal Ball Figure 17 - Volatility simulation (Post Exit Option) Source: Own construction in Crystal Ball 62

63 From figures 16 and 17 it can be seen that after simulations the Holding option has a larger volatility, 13.68%, compared to the Post Exit option s 5.62 %. 8.5 PE Option valuation (Binomial model) The underlying asset The underlying asset for both options is TDC and the development of the value in the holding period and the post exit period starts with the DCF values of bill for the Holding option and the multiple value of bill for the Post Exit option. The reason for using the multiple value, from the LBO valuation as the starting value for the Post Exit option (option is at the money), is that it is the value that gives stakeholders their required return (IRR) and a higher value than that in 2009 would entail exiting the target and hence no Post Exit option. Had the DFC value been used in the LBO valuation, it should have been the DCF value which was used. Therefore, in order to keep consistence between the two periods it is the multiple value that is used as staring value, and the DCF valuation in the Post Exit option is merely used for estimating the volatility. Dividends In accordance with the general assumptions of the LBO model I will assume no dividends in the holding period, and because investors are not getting their return realised until the target is sold due to the high leverage in the holding period, I will assume that all net earnings are paid to dividends in the post exit period (see calculation of exercise price for elaboration). Because dividends are paid in the Post Exit option, the continuous dividend ratio needs to be estimated. The actual dividends are close being equal to the IRR value each period. I will therefore assume that dividend ration is equal to the IRR value as a percentage of total value resulting in a continuous dividend ratio of 6.81%. Time to expiration The Holding option s time to expiration is the holding period from 2006 until end 2009, i.e. a time to expiration of four years. Normally it is reasonable to assume that the Post Exit option s duration is as long as the remaining years of the fund plus three years (chapter 2.2). As mentioned above, TDC is a special PE case with many PE firms involved and information about the duration of the fund/funds by which TDC is acquired is not available. I have therefore assumed that the fund closes at the planned exit and hence that the time to expiration is the three extra three years the fund can be prolonged (2010 to end 2012). For both options I have chosen 63

64 to divide each year into four periods resulting in 16 periods in the Holding option lattices and 12 periods in the Post Exit option lattices and a δt of The exercise price The exercise price is calculated using the results from the LBO valuation, as this is the values which gives stakeholders their required returns and therefore the values on which the PE firm should decide on exiting or not. For the Holding option the exercise price in the first period is calculated as the equity in 2005 of bill extrapolated one period with the Target IRR of 20.64% plus the remaining debt, which is the debt of bill from 2005 subtracted with 2006 s cash flow to debt repayment of 3.08 bill equally divided into the four periods 8 (chapter 7.4 and appendix 6), which in the first period of 2006 gives an exercise price of bill. 0,25 = =78.13 bill Source: Own construction (appendix 6) The remaining periods exercise prizes is calculated in the same way using the previous periods values of equity and debt. The Post Exit option s exercise price is calculated using the same general mindset, but because of dividends are now paid and, to keep things consistent with DCF, the debt-equity ratio is assumed to be constant. The equity minus the dividends is thus determining the total value by multiplying with the debt-equity ratio from The first period s exercise price is therefore calculated as the equity of bill from 2009 extrapolated one period with the Target IRR of 20.64% and subtracted the period s dividend of 1.93 bill 9. By using the debt-equity ratio from 2009 of 1.79 it give an exercise price in the first period of 2010 of 90.1 bill. 0,25 = =90.1 bill Source: Own construction (appendix 7) Because TDC s expected net earnings are higher than the IRR value of the equity, the exercise price of the Post exit option is decreasing over the lifespan of the option. For a complete overview of the exercise prices of the two options see appendix or figure 18 and 20 below. 8 I have used the assumption that debt repayments are divided equally over each year. 9 Using the assumption that each year s dividend is paid out equally over each year. 64

65 Risk-free rate of return I have assumed the same risk-free rate of return of 5% on both options (Carnegie, 2005). Setting up the lattices Next the lattices can be set up using the input variables from above to calculate the u and d factors and the risk neutral probability, starting with the holding option. =.. =1.070 =. =0.935 =..... =0,596 Source: Chapter 7.4 and appendix 18 Figure 18 shows the evolution of TDC s value under the assumptions and variables determined above. Figure 18 PE Holding option: Underlying lattice Calculations Periods in grid 16 S Lenght of periods, dt 0,25 Volatility 13,44% rf 6,00% u 1,070 4 T 4 years d 0,935 R 1,053 IRR 20,63% Equity qu 0,596 Debt qd 0,404 Input Exercise Price Equity Debt X TDC value S Source: Own construction (appendix 6) 65

66 Using the underlying asset lattice the Holding option value can be calculated using the risk neutral probabilities and backward induction, see figure Figure 19 PE Holding option: Option Value lattice option Value C Source: Own construction (appendix 6) The option value lattice shows that the Holding option, at ultimo 2005, has a value of bill, meaning that the value of having the possibility to exit before the planned exit is worth 25 bill for the PE firm. Furthermore, from the action lattice in appendix 6 it can be seen that under the current assumptions and estimations the option should not be exercised (TDC exited) until the option matures. The same is done for the Post Exit option to get the value of postponing exit after the planned exit. =.. =1.027 =. =0.974 =.,.... =0,455 Source: Chapter 7.4 and appendix 7 10 Note the in horizontal movement in the lattice represents an upward movement. This is done in order to make calculations in excel easier. 66

67 Figure 20 Post Exit option: Option value Calculations Input Periods in grid 12 S Lenght of periods, dt 0,25 Volatility 5,32% rf 6,00% u 1,027 4 T 3 d 0,974 R 0,998 IRR 20,63% Equity qu 0,455 Debt qd 0,545 Debt/Equity 1, Dividends 0, Exercise Price Equity Debt X TDC value S option Value C (72) (1.847) (3.578) (1.555) 525 (5.265) (3.291) (6.909) Based on the underlying asset lattice the option value lattice in figure 20 shows a Post Exit option value of bill, meaning that the flexibility of postponing exit has a value of around 16 bill (ultimo 2009) to the PE firm at the planned exit. Notice that the exercise price decreases over time due to the fact that dividends are higher than IRR value, and the equity needed to give stakeholder their expected return is thus decreasing. Furthermore, this leads to that there are few states at expiration where the option has negative value. Hence, the assumption of having to exit at maturity has very little effect on the value of the option. Finally, like with the Holding option, the action lattice in appendix 7 shows that the Post Exit option should not be exercised until the option matures. 8.6 Total value Finally the total value of TDC can be calculated using the elbo formula from chapter 7.5, but before the values can be added they need to be comparable. The LBO value and Holding option 67

68 value are both calculated as ultimo 2005 values while the Post Exit value is ultimo 2009 value. Therefore the Post Exit value of bill must be discounted back to 2005 using the IRR in order to get the combined value of TDC. Source: Own construction (chapter 7.1) TDC elbo = = The LBO value combined with the two option values give a total value of TDC of bill compared to the bill without the options. The flexibility of either exiting before or postponing to after the planned exit therefore has a value of 30 bill, and increase of around 30%. The size of the values themselves are not the most interesting in the analysis, as they are based on the specific case of TDC and they can be very different for other companies. What is interesting, though, is by using real options in the valuation process, PE firms are able to value the flexibility of variable exit, which the static LBO model is unable to. But as the above indicates there are many factors, influences which if changed would, alter the results, and further research is needed in order to make definitive conclusions. I do, however, believe my findings can make the preliminary conclusion that using real options does create value to the PE firm. 68

69 9 Criticism of Real options In order to give a complete assessment of the real option theory s applicability to PE valuation I feel it is important to look at general critique of real optionas and examine how it applied to PE valuation. The literature about the challenges and limitations in connection with real options can generally be divided into two main groups. One group deals with problems of using methods for valuation of real options, which was originally developed for valuing financial options. This category includes for example the challenges regarding the lack of market data on the real assets. The second group deals with corporate governance perspectives, including for example, principal- agent problem of whether management is capable of/willing to take advantage of available real options. 9.1 Financial Issues Financial models are typical characterised by the output being dependant on the inputs as well as the assumptions about the models and the inputs. This is no different for real option valuation models, which are also subject to en number of assumptions. Some of these assumptions are bound with critical issues. The most important will be presented next. Assumption about distributions Real option models do not estimate future values of uncertain variables, but rather assumes that they follow established processes. Numerical methods are then used to estimate these processes. In my analysis I chose to use the binomial model, which assumes that the underlying asset follow a multiplicative binomial process. But what are the consequences if the value of the underlying asset (target) does not follow such a process. It would be relevant to examine weather such uncertainties have significant effect on the value of the option (Lander & Pinches, 1998, p 546). There is no sufficient research available in this area in relation to PE acquisitions, but in relation to a oil project Smith & McCardle (1997) show that the value of the project changes significantly depending on if the valuation followed a Geometric Brownian motion or a mean reverting process. It can therefore be stated that the choice of distribution has an effect on the option value, but until further research can clarify the subject more, it is something that should be kept in mind when using real options in PE valuation. Assumption about variables No matter what distribution a factor is assumed to follow, there will also be issues regarding the input variables in the model i.e. interest, volatility etc. The general assumption is that the variables are constant during the life of the option (Lander & Pinches, 1998, p. 548), which was 69

70 also the assumption in this thesis. This was also the main reason for dividing the exit option into two options, Holding and Post Exit option (chapter 6.1). McDonald & Siegel (1986) address the issue of constant variables by concluding that the option value would be lower had the volatility and dividend ration been modelled as a function of price instead of being constant, which would give a better reflection of reality. The estimation of certain variables can also be problematic especially when there is no existing market for the underlying asset, with the volatility generally being the most problematic. I have compensated for this by simulating the volatility using Copeland & Antikarov s (2003) Market Asset Disclaimer assumption (chapter 7.3). This however leads to the same issues, as the variables are bound with some of the same assumptions as the variables in the option valuation. Therefore, it is necessary to acknowledge that the valuation of real options becomes arbitrary because of the missing markets. 9.2 Corporate Governance Issues In the classic corporate finance literature the value of a real option is naturally allocated to the owners of the company. This is however one of the main critiques of modern literature which questions weather (1) the value of the real option can be allocated to other stakeholders than the owners and (2) the managers ability to handle real options in such a way that they maximize owners (shareholders) value (Phillip, 2005). To answer this we have to look at behavioural incentives for other stakeholders than the managers as well as the principal-agent relationship between the owner (principal) and the managers (agent) Allocation of value In many traditional real options, there can be many different stakeholders who have the right to the option. In R&D projects it is often key employees who have necessary knowledge to exercise a real option. The incentives of these employees must be assessed in order to determine weather they can exercise the option themselves and thereby retrieving the value of the option themselves, or they have the incentives to keep the option within the company. When it comes to PE firms and their exit options, it is generally only the managers (Management Company) who have the rights to exercise the option, as no other stakeholder can make the decision to exit a target. Since the underlying asset has clearly defined ownership rights it is not possible for the managers to keep the option for themselves, as it is with for example more intangible real options where knowledge is a key value driver. 70

71 Owners versus managers Now that it is established that the value of the option can only be allocated to the company i.e. the shareholders, the second questions is thus weather the managers have the incentives to use the real options in the way that maximized the return on equity. Many factors can prevent managers from making the rational financial choice. Psychological factors such as human strive for personal benefit like for example empire building, personal reputation or professional recognition can affect the manager s decisions, and projects which should have been stopped can continue unjustified due to the manager s personal interests (Mun, 2006, p. 135). In PE firms the management company is often part of the daily management of the target, and could therefore affected by the personal factors mentioned above, and hence make irrational financial decisions in regards to the exit strategy. Therefore the manager s incentives should be aligned with the owner s interests. Phillip (2005) argues that financial incentive programmes are an effective way of aligning managers and owners. PE firms do that by including the management company as investors in the target and thereby aligning their incentives with the other investors (owners) and by paying the management company according to the return using the carried interest (chapter 2.1) In conclusion, the question of allocation of the real option value is therefore not applicable on PE firm s exit options, as the ownership rights of the option is well defined, and the principal-agent problematic is minimized by having management as investors. The corporate governance issues regarding real options do therefore not present ground for rejecting the use of real options in PE valuation. Overall, there are many areas that can be criticized within real options, especially regarding the assumptions of real option models. It is important to note that this criticism not only exists, but it is also essential in relation to continue to develop solutions to the problems. How much uncertainty these problems cause is not easy to assess. It may simply be noted that there are areas that form the basis for further development, but no immediate criticisms which makes the use of real options for PE valuation infeasible. 71

72 10 Discussion Real Option Theory is widely recognised in corporate finance theory as being the best way to incorporate flexibility and uncertainty in the valuation process. Therefore, it is also the obvious choice when analysing weather PE firms are overlooking potential value by not including uncertainty in their valuation. In chapter 8 I argued that the general criticism of real options do not give reason for dismissing the use of real option as a tool for valuing PE targets. There are, however, more specific issues in regards to the applicability which also needs to be discussed in order to further justify or reject) the use of real options on PE targets. There are especially three areas which I feel needs further discussion: The division into two options, the question of the degree of flexibility at expiration in the Post Exit option, and the interpretation of the option values The PE Exit options The division into two options entails some implications. First of all there is the discussion of weather a division is necessary or a one option solution would be just as applicable. The question here is weather the active ownership actions by the PE firm affects the market value of the target, resulting in different volatilities. If the assumption is (as in this thesis) that it does, it will entail the division into two option, as valuing one option with changing volatility using the binomial model is not recommended. This, however, also raises the question of the connection between the two options. The assumption in the thesis is that the two options are completely independent of each other. In practise this is not correct. The Post Exit option should be dependant on the evolution in the Holding period as it is the same underlying asset. This would entail that the value of the underlying asset in the Post Exit option should start from where the Holding option ends. But where that is, is not possible to predict and a compensation that the Post Exit option starts at the expected value (IRR value) is in my opinion the best solution. If on the other hand it is assumed that active ownership has little or no effect on the market value, a one option solution with constant volatility would be adequate. It therefore boils down to deciding between accepting the fact that the two periods are different and hence accepting the problems of the two options not being connected. Or neglecting the fact that the periods are different and hence valuing using the same volatility. Non of the two are theoretically correct, but I believe that because the sole purpose of a PE firm is to increase the market value in order to get a return, the actions they make must be influencing the market value and neglecting that would be more problematic than the issue of non connecting options. 72

73 10.2 Choice at Exit Another assumption, which challenges the basic understanding of the real options, is the issue of weather the PE firm has the choice to exit or not at expiration in the Post Exit option. The assumption is that they do not, as PE funds have finite live spans. This assumption is problematic as it undermines the fundamental idea behind real options, that the holder has flexibility to make his/hers decision depending the development of the underlying asset i.e. the right to but not the obligation. On the other hand, the PE firm has the flexibility up until the expiration, and the negative value of not having a choice at expiration is incorporated in the valuation by assuming that the option can undertake a negative value. In the case of TDC this has very little influence, as most states at expiration suggested an exit, and the few that did not only undertook small negative values. But this is not necessarily the same in other situation, and this is, in my opinion, a large problem to the applicability when applying real option theory to PE valuation Interpreting the value The TDC case confirmed that using real options in the valuation process does add value, but the question is. Is it real? AT Kearney (2009, title) states: Real Options = Real Value, which is based on the fact that real options merely is a NPV valuation which incorporate flexibility. The PE exit option valuation is based on the same financial figures as the LBO valuation and the values are discounted using probabilities based on the volatility calculated on the basis of the management s estimates, the value of the option is as real as the value from the LBO valuation, but to whom? The Exit option values are the values of the flexibility that the PE firm has in relation to their exit strategy. It is the value of the possibility that the market value increases more than the required return. But unless there is a buyer which is willing to pay for this possibility as well, the value can not be realised. On that basis it can be concluded that the value is not real. On the other hand, because the value of the underlying asset is the market value, the same possibility should be there for the potential new owners. Or said in another way, the underlying asset lattice is the same for the PE firm as well as a potential buyer. The question is, if they rate the possibility the same i.e. are the option value lattice the same. This of cause depends on the remaining variables in the valuation, and they are probably different from the PE firm. Especially the exercise price would be different, as it is based on the PE firms IRR. Consequently, the option value for the PE firm is not what value the flexibility has to them but what value is has to a potential buyer, and 73

74 unless the two valuations can be aligned the value is not real, and should not be used when valuing a potential target. Alternatively the option values could be used as a tool for comparing potential investments as a higher option value would entail higher probability of increased value in the future, hence investments with relative higher option value should be ranked higher than investments with lower option values. 74

75 11 Conclusion The main hypothesis of the thesis was to examine weather PE firms are overlooking potential value by not focussing on real option in their valuation process. I order to answer that a series of research question was set up in order to thoroughly examine the main hypothesis. How is a Private Equity investments valued? A PE investment is characterised by being highly levered, hence the name levered buyout, and are therefore using virtually the entire cash flow on debt service, resulting in a significant change in the capital structure during the holding period. This characteristic is also the main reason why the traditional Discounted Cash Flow and multiple models are inadequate for the purpose of valuing PE investments. The Adjusted present value is suited for valuing with changing capital structure, but its failure to give explicit market values of debt and equity makes it difficult to estimate the initial equity investment. The LBO valuation model has clear advantages over the rest of the models, mainly because of the way the valuation process is constructed. The model follows every step of a natural process in a LBO and values these separately. First optimal level of debt is determined, and the cash flow available for debt repayments is calculated for the holding period. The total enterprise value at exit is calculated using traditional valuation methods. The equity at exit is found as the residual of the remaining debt and the total value. Finally the equity is discounted back using an IRR to get the initial equity investment, and the total enterprise value at t=0 is the equity and the initial debt level. By construction this gives the maximum value the PE firm can pay for the target in order to give investors their required return. Even though the LBO model is the superior valuation tool for PE targets, it is still a static valuation tool which does not incorporate the value of flexibility, and another valuation method is needed to incorporate the value of flexibility. What is Real Options and how are real options valued It is widely recognized that Real Option Theory is useful for analyzing and valuing flexibility. It uses the mindset of financial options of having the right but not the obligation to some action in the future. Instead of financial traded assets it is based on real assets such as projects, possibilities or as in the case of a PE firm a company. It was shown, by linking the valuation models inputs (Exercise Price, Stock Price, time to expiration, volatility, Risk-free rate of return and dividends) to characteristics of real assets, that the valuation models normally used for financial options can be used for real options. The choice of valuation model was between black 75

76 & Scholes model and the binomial model. The binomial model was chosen because of its easier adaption to individual characteristics of different real options, which proved to be essential when dealing with the PE Exit options. Furthermore it is intuitively easier to understand than the closed form solution of the Black & Scholes model. The binomial model values option under the assumption that the value of the underlying asset will either go up or down in the next period, which is modelled using binomial lattices. The up and down factors are calculated using the volatility, and the option value is found by discounting the difference between asset value and the exercise price at expiration using the risk-neutral probability and the risk free rate. What options are available to PE firms? In relation to the exit strategy the PE firm have two options, the PE Holding option and the PE Post Exit option. The holding option is the option to exit in the holding period before the expected exit. The Post Exit option is the option to postpone exit after the expected exit, given that the holding option is not exercised. The reason for dividing the Exit option into two separate options is the significant different characteristics of the target in the holding and post exit period respectively, which results in different volatilities. This division presents some issues for discussion. The optimal solution would be to only use one option stretching over both periods, but valuing a option with different volatilities is not desirable. Using the same volatility will violate the idea behind the LBO valuation, so the optimal solution is to it divide into two options. How can real option theory be applied to the valuation of PE targets? In order to value a PE investment using the Exit options available the Expanded LBO framework was presented, which based on the mindset of the Expanded NPV, divides the valuation of PE targets into two parts. One part values the static part of the target value by using the LBO valuation model, while the second part values the flexible part using real option theory. It was found that the special characteristics of the LBO lead to specialized use of real option theory. First, the value of the underlying assts (target) is the market value of the asset, as it the value to potential new owners which creates value to the PE firm. This entails that the underlying volatility should also be estimated on the market value. This is done using Monte Carlo simulation on the management s estimates under the Market Asset Disclaimer assumption. The cost of exercising an option is the exercise price. The cost of exercising the Exit options is thus the required return value at the point in time, as the option should not be exercised unless the value is higher than expected value. This leads to a changing exercise price which is incorporated in the binomial valuation. If the target is assumed to pay dividends, these should be 76

77 subtracted from the exercise price, as they are realised return and the asset value needed to give the stakeholders their required return is therefore lower. The time to expiration was determined by the expected holding period for the Holding option and the lifespan of the fund for the Post Exit option. How does the general criticism of Real options apply to PE Exit option? The two main areas of criticism of real options are financial and corporate governance. Within corporate governance it is especially the principal agent problem between the management and the shareholder and the question regarding ownership tights which are criticised. But because the ownership rights are well defined and the management is included as investors, the critique is not justified when applied to PE Exit options. The financial critique regarding the distribution of the underlying asset and the assumption regarding the variables are one the other more substantiated. But as there are no definite conclusion on weather these issues preclude the use of real options, the use of real option on PE Exit options is neither made infeasible. Are PE firms overlooking potential (real) value by not focusing on real options in relation to their exit option? From The TDC case it was found that using the framework presented in the thesis does have a value to the PE firm. The possibility of Exiting when conditions are right is valuable to the PE firm. This is, however, a specific value to the PE firm, which has no value to a potential buyer. But any potential buyer is bound to have a similar option as the value of the underlying asset is based on the market value. The question is then if the option has the same value to the buyer as it has to the PE firm, because unless the buyer value the option the same (or higher), the value can not be realised, and hence is not real. So answer to the hypothesis is that there are value associated by using real option in the valuation of PE firms, but the value is only real (can be realised) if a potential buyer values his equivalent option the same. Otherwise the option value is more suitable as a strategic tool for comparing different investment opportunities. 77

78 12 Further research After analysing real option theory s applicability to the valuation of PE investments, I find that there are several areas where further research could be very beneficial. The fact that the thesis was a mainly theoretical analysis with a single case study raises the need for a broader empirical study to confirm the findings from the thesis. Besides this more general study, I find that there are especially two specific areas where further research is needed. The first is a sensitivity analysis of key variables in the valuation and the second is about the options value the PE firm versus the value to potential new owners. Sensitivity Analysis of option value There are many inputs and assumptions (factors) which needs to be estimated and established, all more or less influential on the value of the option. But how influential each individual factor is on the total value can be difficult to determine without a sensitivity analysis. By making a sensitivity analysis the PE firm can, among other things, establish where the PE firm should use its resources to get as correct measures as possible because of a high influence and where simple qualified guesses and assumptions are sufficient as the influence is small in the big picture. Besides the general factors directly influencing the option value, I find that extra research is needed on the variables used for volatility simulation/estimation. How does it influence the volatility of adding or removing extra variables? How influential is the characteristics (distribution, volatility and correlation) of each variable on the volatility? This is both questions which would be very valuable to answer, as the volatility is the most influential factor of the option value. In the thesis I have used variables in the volatility estimation which are specific to the PE firm. In these financial difficult times there could be many indicators in the general economy which could have a large influence on company value, and research on how the general economy can be included in the volatility estimation would add to the realism of the volatility. Division into two options The second place where I believe more research is needed is with regard to the combining the options in the two periods. I believe that there should be some connection between the two periods, and because I do not rate the method of valuing one option with changing volatility as a valid option (without a special option valuation programme), research about how the two periods can be connected would be beneficial to the valuation process. 78

79 13 References Andersen, Ole Steen and Frigast, Christian Aktivt ejerskab og åbenhed i kapitalfonde, Danish Venture Capital and Private Equity Assosiation, 2008 Andersen, T. J. (2000). Real Options Analysis in Strategic Decision Making: an applied approach in a dual options framework. Journal of Applied Management Studies, Vol. 9, No. 2 Allen, Franklin, Brealey, Richard A., and Myers, Stewart C. (2006): Corporate Finance, McGraw-Hill & Irwin, 8 th edition Arzac, Enrique R. (2008): Valuation for Mergers, Buyouts, and Restructuring John Wiley & Sons Inc. AT Kearney (2009): Real Option = Real Value AT Kearney online publications. Baldwin, Carliss (2001 A): Technical note on LBO Valuation (A) LBO Structure and the Target IRR Method of Valuation, Harvard Business School (Publishing), July (06), pp Baldwin, Carliss (2001 B): Technical note on LBO Valuation (B) The Equity Cash Flow Method of Valuation Using CAPM, Harvard Business School (Publishing), July (06), pp. 1 6 Bennedsen, Morten, Thomsen, Steen, Nielsen, Søren Bo, Bundgaard, Jakob, Nielsen, Kasper M., and Poulsen, Thomas (2008) Private Equity i Danmark, Center for Economic and Business Research, June 2008 Børsen (2009a): Finanskrisen truer flere kapitalfonde på livet, Børsen Business Net, february Børsen (2009b): Pengene fosser ud af kapitalfonde, Børsen Business Net, march Børsen (2009c): Kapitalfonde på vej ud af TDC, Børsen Business Net, August Bonnerup, Henrik, Morgan, Christian, and Petersen, Steffen Wulff (2007): Leveraged Buyout Modeller En gennemgang af kapitalfondenes transaktionsstrukturer og værdiansættelses-metoder, Børsens Ledelseshåndbøger, Køb og salg af virksomheder Chapter 4 (section 2) Carnegie (2005): TDC Private Equity takeover intact ; Thomson Financials, Cendrowski, Harry, Martin, Sames P., Petro, Louis W., Wadecki, Adam A. (2008): Private Equity: History, Govarnance and Operations, John Wiley & Sons, Inc,

80 Christensen, Troels and Christensen, Jørn (2007): Finansiering af kapitalfondes virksomhedskøb, Børsens Ledelseshåndbøger, Køb og salg af virksomheder Chapter 7 (section 2) First Year project Cox, J., Ross, S., and Rubinstein, M. (1979): Option Prizing: A Simplified Approach, Journal of Financial Economics. Copeland, Tom and Antikarov, Vladimir (2001): Real options: A Practitioner s Guide, Texere Publishing LCC, 2001 Hill, Andrew: Private equity-backed IPOs will need a new look, Financial Times, June Koller, Tim, Goedhart, Marc and Wessels, David (2005): Valuation Measuring and managing the value of companies, John Wiley & Sons, Inc. (In association with McKinsey & Co.), 4 th edition Lander, D.M. and Pinches, G.E., Challenges to the Practical Implementation of Modeling and Valuing Real Options. Quarterly Review of Economics & Finance, 38(4), pp Leiblein, Michael J. (2003), The Choice of Organizational Governance Form and Performance: Predictions from Transaction Cost, Resource-based, and Real Options Theories, Journal of Management (Vol. 29(6)), pp Luehrman, Timothy A. (1998): Investment Opportunities as Real options: Getting Started on the Numbers, Harvard Business Review, July-August McGrath, R.G., Ferrier, W.J., and Mendelow, A. (2004). Real Options as Engines of Choise and Heterogeneity. Academy of Management Review, 29, Mun, Johnathan (2006) Real Option Analysis second edition, John Wiley & Sons, Inc. Second Edition. Philippe, H., Corporate Governance: A New Limit to Real Options Valuation? Journal of Management & Governance, 9(2), pp Spliid, Robert (2007): Kapitalfonde: Rå pengemagt eller aktivt ejerskab, Børsens Forlag, Trigeorgis, Lenos (1996), Real Options: Managerial flexibility in strategy and resource allocation, The MIT Press,

81 Vinten, Frederik, and Thomsen, Steen (2008 a): A Review of Private Equity, Danish Corporate Governance in Practise Working Paper no. 1, 2008 Yin, Robert. (1989). Case study research: Design and methods Beverly Hills, CA: Sage Publishing. Yin, Robert. (1994). Case study research: Design and methods (2nd ed.), Beverly Hills, CA: Sage Publishing. Internet & Databases 81

82 14 Appendices Appendix 1 TDC A/S Pro Forma Income Statement TDC Pro Forma Income Statement E 2010E 2011E 2012E 2013E Sales , , , , , , , , ,32 Denmark Fixed Telephony , , , , , , , , ,09 Mobile Telephony , , , , , , , , ,14 Internet 7.757, , , , , , , , ,09 Other 8.193, , , , , , , , ,00 - Operational Costs (29.581,32) (27.307,00) (25.207,54) (23.269,49) (21.480,45) (21.287,99) (21.097,26) (20.908,23) (20.720,90) EBITDA , , , , , , , , ,42 - Depreciation & Amortisation (6.491,00) (6.227,00) (5.819,00) (5.558,60) (5.549,66) (5.561,11) (5.572,63) (5.584,22) EBIT 6.142, , , , , , , ,19 - Net Interests (2.697,00) (3.396,00) (2.412,00) (2.500,00) (2.500,00) (2.500,00) (2.500,00) (2.500,00) Pre-tax Profit 3.445, , , , , , , ,19 - Taxes (964,85) (1.342,71) (1.969,64) (2.541,31) (2.575,70) (2.640,97) (2.705,91) (2.770,55) Net income 2.481, , , , , , , ,64 Data E 2010E 2011E 2012E 2013E Mobile customers 3.953, Sales pr. customer 3,35 3,00 2,85 2,71 2,57 2,55 2,53 2,51 2,49 Source: TDC Annual rapport (2006, 2007, 2008) and appendix 3 82

83 Appendix 2 TDC LBO Valuation Pro Forma Cash Flow Net income Depreciation & Amortisation Capex Change in Net Working Capital Cash flow to debt repayment Target IRR Total Debt (DKKm) Equity Value (DKKm) Total Enterprise Value (DKKm) Debt ratio 81% 78% 74% 69% 64% Equity ratio 19% 22% 26% 31% 36% Equity beta 3,07 2,61 2,23 1,90 1,62 Cost of Equity 18,04% 16,15% 14,51% 13,10% CAPM IRR 15,44% Marketing IRR 18,52% Target IRR 20,63% LBO Valuation Initial Debt (Step 1) Debt repayments (Step 2) Debt at Exit Enterprice Value (Step 3) Equity at exit Initial Equity (Step 4) Firm Value Assumptions Net Working Capital (% of sales) 16% 16% 16% 16% Net Working Capital CAPEX (% of sales) 15% 15% 15% 15% Depreciation (% of sales) -14% -14% -14% -14% Assumptions New Net Debt Risk free rate 5% Net Debt/EBITDA (04e) 6,2x Market risk premium 5% EBITDA Outstanding shares (m) 195,4 New Debt Price 3. nov Beta debt 0,00 Enterprice Value at Exit Asset beta 0,58 EV/EBITDA (09e) 5,0x Net Debt/EBITDA (04e) 6,2x EBITDA (09e) EV/EBITDA (09e) 5,0x Exit Enterprise Value Growht Factor 0,00% Return Equity 13,10% Return Debt 5% Working Capital (% of sales) 16% CAPEX (% of sales) 15% Depreciation (% of sales) -14% 83

84 Appendix 3 Data Yearly change Yearly change Internet og Netværk ,32% Fastnettelefoni ,06% Operational Costs (27.301) (23.255) 8,02% Mobile customers ,35% Sales per customer 3,00 2,87 2,71 5,18% Source: TDC Annual rapport (2006, 2007, 2008) Fixed telephone calls - mn minutes Yearly change National fixed telephone calls Denmark , , , , , , , , ,9 Finland , , , , , , , , ,3 Norway , , , , , , , , ,5 Sweden , , , , , , , , ,7 Switzerland , , , , , , , , ,7 International outgoing fixed telephone calls Denmark 707,0 740,0 795,4 641,0 663,3 637,2 595,9 577,4 558,9 Finland 467,9 540,8 469,4 231,0 215,0 187,0 155,3 126,4 99,2 Norway 558,7 626,3 573,6 557,5 516,1 562,2 621,8 632,3 642,8 Sweden 1.086, , , , , , , , ,3 Switzerland 2.624, , , , , , , , ,0 Total fixed telephone calls ,00% Sources: Euromonitor International from International Telecommunications Union/national statistics 2009 Euromonitor International 84

85 Internet subscribers - ' Yearly change Denmark 2.114, , , , , ,6 Finland 1.922, , , , , ,1 Norway 1.712, , , , , ,9 Sweden 4.262, , , , , ,8 Switzerland 2.849, , , , , ,2 Total Internet Subscribers , , , , , ,6 0, Sources: Euromonitor International from trade sources/national statistics 2009 Euromonitor International Mobile telephone calls per subscriber minutes Denmark 694,8 684,3 674,3 666,5 660,9 657,0 654,5 652,9 652,2 652,0 652,3 652,9 653,8 Finland 1.253, , , , , , , , , , , , ,9 Norway 991,7 977,7 962,9 945,9 936,2 923,6 916,7 906,4 901,5 892,5 888,7 880,8 877,5 Sweden 688,0 673,5 663,1 655,9 650,9 647,7 645,7 644,7 644,4 644,5 645,0 645,7 646,6 Switzerland 638,5 609,8 587,5 570,9 558,7 549,7 543,3 538,5 535,1 532,8 531,2 530,2 529,6 Total 4.266, , , , , , , , , , , , ,4-0, Sources: Mobile telephone calls per mobile telephone subscriber: Euromonitor International from national statistics 2009 Euromonitor International Yearly change Mobile telephone subscriptions - ' Denmark 6.550, , , , , , , , , , , , ,1 Finland 6.830, , , , , , , , , , , , ,0 Norway 5.353, , , , , , , , , , , , ,0 Sweden , , , , , , , , , , , , ,9 Switzerland 8.780, , , , , , , , , , , , ,9 Sources: , , , , , , , , , , , , ,9 0, Mobile telephone subscriptions: International Telecommunications Union/World Bank/Trade Sources/Euromonitor International 2009 Euromonitor International 85

86 Appendix 4 Volatility Simulation (Holding Option) Pro Forma Income Statement E 2010E Variable Sales Fixed Telephony ,06% Mobile Telephony Internet ,32% Other Operational Costs (29.581) (27.302) (25.197) (23.256) (21.463) (21.271) -8,02% EBITDA Depreciation & Amortisation (6.491) (6.227) (5.819) (5.559) (5.550) EBIT Net Interests (2.697) (3.396) (2.412) (2.500) (2.500) Pre-tax Profit Taxes (966) (1.345) (1.973) (2.546) (2.580) Net income Data Variable Mobile customers 3.953, ,35% Sales pr. customer 3,16 3,00 2,85 2,71 2,57 2,55-5,18% DCF Valuation (Holding Period) EBIT Depreciation & Amortisation Capex Net Working Capital Taxes Free Cash Flow Contunuation value PV PV (2006) Return 10,73% Simulated variables Output Volatility estimation V0 r Lower Vol. (%) Vol. (actual) Fixed Telephony ,06% ,92% -1,21% Internet ,32% ,14% 1,55% Operational Costs (27.302) -8,02% (15.000) 10,34% -0,83% Mobile Telephony: Mobile customers ,35% ,01% 2,98% Sales pr. customer 3,00-5,18% 1 27,23% -1,41% Assumptions Growht Factor 0,00% Return Equity 14,10% Return Debt 6% WACC 8,90% 86

87 Appendix 5 - Volatility Simulation (Post Exit Option) Pro Forma Income Statement 2009E 2010E 2011E 2012E 2013E Variable Sales Fixed Telephony ,00% Mobile Telephony Internet ,77% Other Operational Costs (21.480) (21.288) (21.097) (20.908) (20.721) -1,00% EBITDA Depreciation & Amortisation (5.559) (5.550) (5.561) (5.573) (5.584) EBIT Net Interests (2.500) (2.500) (2.500) (2.500) (2.500) Pre-tax Profit Taxes (2.541) (2.576) (2.641) (2.706) (2.771) Net income Data 2009E 2010E 2011E 2012E 2013E Variable Mobile customers 5.096, ,39% Sales pr. customer 2,57 2,55 2,53 2,51 2,49-0,77% DCF Valuation (Holding Period) 2009E 2010E 2011E 2012E 2013E EBIT Depreciation & Amortisation Capex Net Working Capital Taxes Free Cash Flow Contunuation value PV PV (2010) Return 10,55% Simulated variables Output Volatility estimation V0 r Lower Vol. (%) Vol. (actual) Fixed Telephony ,00% ,54% -0,88% Internet ,77% ,34% 0,68% Operational Costs (21.288) -1,00% (10.000) 21,03% -0,19% Mobile Telephony: Mobile customers ,39% ,62% 0,68% Sales pr. customer 2,55-0,77% 1 26,34% -0,20% Assumptions Growht Factor 0,00% Return Equity 14,10% Return Debt 6% WACC 8,90% 87

88 Appendix 6 Holding option value Calculations Periods in grid 16 S Lenght of periods, dt 0,25 Volatility 13,44% rf 6,00% u 1,070 4 T 4 years d 0,935 R 1,053 IRR 20,63% Equity qu 0,596 Debt qd 0,404 Input Exercise Price Equity Debt X TDC value S option Value C Action lattice Action Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep 88

89 Appendix 7 Post Exit option valuation Calculations Input Periods in grid 12 S Lenght of periods, dt 0,25 Volatility 5,32% rf 6,00% u 1,027 4 T 3 d 0,974 R 0,998 IRR 20,63% Equity qu 0,455 Debt qd 0,545 Debt/Equity 1, Dividends 0, Exercise Price Equity Debt X TDC value S option Value C (72) (1.847) (3.578) (1.555) 525 (5.265) (3.291) (6.909) Action lattice Action Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Keep Sell Keep Keep Keep Keep Sell Keep Keep Keep Sell Keep Keep Sell Keep Keep Keep 89