Annals of the University of North Carolina Wilmington International Masters of Business Administration

Annals of the University of North Carolina Wilmington International Masters of Business Administration http://csb.uncw.edu/imba/

VALUATION OF CLOSELY-HELD COMPANIES CONSIDERING MANAGERIAL OPTIONS AND PROBABILITY DISTRIBUTIONS

Pavel Kochetkov

A Thesis Submitted to the University of North Carolina Wilmington in Partial Fulfillment of the Requirements for the Degree of Master of Business Administration

Cameron School of Business

University of North Carolina Wilmington

2012

Approved by

Advisory Committee

Cet Ciner Ravij Badarinathi

Joe Farinella Chair

Accepted by

______Dean, Graduate School

TABLE OF CONTENTS ABSTRACT ...... iii LIST OF TABLES ...... iv

LIST OF FIGURES ...... v

INTRODUCTION ...... 1v LITERATURE REVIEW ...... 9 PART 1. THEORY ...... 15 1.1. Probability distributions and why do we need to use them ...... 15 1.2. Discount Rates...... 18 1.3. Forecasts ...... 27 1.4. Classical and Extended Automata Theory ...... 30 1.5. Distribution of the sum ...... 32 PART 2. DATA ...... 34 2.1 Conceptual Analysis ...... 34 2.2. Available Options ...... 35 2.3. Variables...... 39 PART 3. ALGORITHM ...... 42 3.1 Creating an automaton ...... 42 3.2 Decision Rules and Constraints...... 53 3.3 Computations ...... 57 3.4 Stability Analysis ...... 59 3.5 Computation Time and Optimization ...... 61 CONCLUSION ...... 64 BIBLIOGRAPHY ...... 67

ABSTRACT

In this paper a new algorithm for valuing closely-held companies and investment projects is introduced. The algorithm considers not only the information on the current situation but also probability distributions of the future events and the managerial options which are already available or will appear in the future. The conceptual part of the algorithm is discussed along with some theoretical issues including dealing with probability distributions, classification of the real options, a section on the discount rates and several sections concerning various technical aspects of implementation of the algorithm to the real cases. Relevant examples are used to illustrate the model.

iii

LIST OF TABLES

Table Page

1. Real Options Classification Matrix ...... 36

LIST OF FIGURES

Figure Page

1. Forecasted Net Income on Various Companies ...... 16

2. Illustration of the Concept of Observer Horizon ...... 27

3. Difference Between Options ...... 39

4. The First Step in Automaton ...... 42

5. The Second Step in Automaton ...... 43

6. The Third Step in Automaton ...... 44

7. The Final Step in Automaton ...... 45

8. Trick Track – Automaton...... 48

9a. Stability Area ...... 60

9b A Stable Result...... 61

INTRODUCTION

The valuation of investment projects is one of the most widely discussed issues in the world of finance. Actually, we should not differentiate little projects inside a giant corporation or some tiny company as the whole because evaluation of their efficiency is very similar. Although the conditions are different, we still say that both a small company and a separate project have the value being equal to the present value of all the future cash flows.

The challenge with this definition is defining the discount rate and cash flows but still this is a common approach and it makes sense.

When we evaluate the investment project inside the big company we are supposed to present not the precise amount of money it will generate or cost but rather the overall conclusion – whether or not it should be performed. The final decision is based not only on the calculations but also the intuition and vision of the person or committee responsible for that type of investments, so even the projects with high NPV or IRR would sometimes be declined.

Valuing a closely-held company is a complex task which relies on estimating the appropriate discount rate and cash flows. Even the experts often disagree about exact value.

Dr. Shannon Pratt, a leading expert in the valuation of closely held companies, maintains that an unbiased experts‘ value of a closely held company might vary by 20%. This means that if the true value of a company is $1,000,000 the range would be from $800,000 to $1,200,000, and this makes huge difference. However, the value is considered to be a precise amount of money the business is worth of and is used in all kinds of estate taxes, equitable distribution, minority shareholder disputes, initial public offerings, and business sales. The question therefore is how to calculate this value with a maximum precision possible.

Comparing the two situations above we come to the conclusion that even though the process of evaluation is mostly the same for each of them the aim and the result of it differs a lot. However, in this paper we present an algorithm that considers the range of values and also the value of options available to investors.

Hundreds and thousands of articles, books and monographs are devoted to evaluation of the investment projects – both for particular industries and in general. A specialist without experience would be in a total loss when he‘d have to deal with the variety of approaches available nowadays; at the same time most of the managers are not acquainted with them well enough to see clearly the underlying assumptions - this might lead to misinterpretation of the calculations result and to wrong decisions concerning the effectiveness of the project.

Lots of approaches are currently being used for valuing private companies as well.

Each of them is operating with its own methodology and understanding of the business processes, so it is common for the results of these methods to vary greatly. After calculating all these figures the researcher must somehow determine the final result – either using the average (perhaps weighted) or just using the result of the most reliable method (in his opinion). Consequently what we see is that the value of the company depends on the personality, experience and the current mood of the researcher, which does not seem appropriate when the precise number is required.

Dr. Pratt (2008) indicates four basic and widely-used approaches for valuing the company: the asset approach, the capitalized excess earnings approach, the income approach and the market approach. Each of them has some advantages as well as certain flaws, but only a very few researchers and valuators use any techniques beyond these four ones.

From the conceptual point of view the easiest method is the income approach (or rather income approaches, as different quantitative techniques could be used). The idea is completely identical to the classic NPV-valuation of the investment projects – when the discount rate and the future cash flows are found, we should discount all of those to the present time and the result will be the value of the firm. However, the seeming simplicity of

2 this method is shattered when the real situation is considered: estimating both the future cash flows and the discount rate is a difficult and complex task, especially in case of unusual type of business or when the perspectives in general are unclear, like it is nowadays. In the main part of the thesis, we will look at the present value calculations closer from mathematical point of view and find out another problem with this method; here we should mention that the major advantages with this method are the strong theoretical fundament and the simplicity of the calculations comparing to the other methods. There is also one significant structural advantage – when using the income approach we are not relying on the estimates and guesses made by other experts – at least we understand clearly, why we use any quantity and therefore our model is self-consistent.

Closely related to the income approach but also having some important features of its own is the capitalized excess earnings approach. It was developed in 1920s by the US

Treasury specialists to value the intangible assets, and from the logical point of view the evaluator should add the value of the intangible assets to one of the tangible assets to find the total value of the company, but for different reasons this method not used very often. Perhaps, one of the reasons for it is that ―… This approach may be used in determining the fair market value of intangible assets only if there is no better basis available for making the determination‖ (direct quote from the Revenue Ruling 68-609, describing the capitalized excess earnings method). The main methodology is adding the market value of the tangible assets to the capitalized intangible assets, but lots of different problems arise when calculating both of these values, especially when the estimation of the value of intangible assets is performed. Another problem with this method is connected to the unclear terms in the regulation itself; altogether this leads to the fact that this method is used much less than other ones.

The third method is the asset approach (sometimes called the asset accumulation method which shows its core idea). The situation here is close to one we faced when discussing the income approach – there are several methods united under this label, but if all the income approaches differed from each other by the mathematical model the asset approaches differ by the assets included or excluded from the final calculation. The principle is as simple as possible – the total value of the company is the market value of all its assets together excluding all liabilities, but what is simple in theory is one of the hardest things to perform in practice. The first thing to be mentioned is ―all assets‖. Even for a very small business (the value is less than $1 million) all assets will contain cash, inventory, real estate

(if any), tangible personal property and intangible assets of different kinds, and for each of those the special evaluation is required, especially for the real estate and intangible assets. At the same time ―all liabilities‖ include long- and short-term debt along with contingent and special obligations, which also require some specific treatment. This method does not require any calculations except putting all the values together but the cost of it is that there might be a lot of problems in finding these values. This leads to the second problem – we need not the book but the market values of all the assets and liabilities, and in some cases it would be a difficult task to determine those, especially for some types of assets not actively traded; at the same time such intangible assets as ―good relationship with the supplier‖ cannot be measured at all and their value is determined using some specific procedures. Finally, there is a chance that the market over- or undervalues some asset and therefore its market price does not reflect its true value, so one should be careful when using this approach during the unstable economic period and all sorts of ―market panics‖. However, even though this method is complicated in the sense mentioned above, it is still a very useful one. Actually, we are not estimating anything in the future – we use the market value of the assets and liabilities today to find the equity today – the only source of mistakes is the possible inaccuracy in the

4 evaluation of the assets and they are much smaller than the errors of the prediction. The result of the evaluation is simple and clearly shows the value of each asset of the company, thus if the value seems unreasonable the source of the misevaluation can be quickly found. Finally, this method provides not only the value of the company itself but also shows what assets are the most effective and what are not used properly, which will help to improve the management. As the result, we can see that the asset approach is an acceptable method of the evaluation and therefore widely used for valuing the companies.

There is one more method which seems to have the leading position among all of the evaluation techniques. It is the market approach – the core idea of which is that we should look at the similar companies that have already been evaluated and make some adjustments to make an estimate for the company we are evaluating. Several problems arise from this statement. First of all, each company is unique and there is no explanation to what the word

―similar‖ means in the definition – the situation in which there would be at least three companies matching the requirements exactly is extremely unlikely and less than three companies are not considered as a representative sample, so we would one way or another have to make comparisons with somewhat similar, but not exactly corresponding companies.

But even if we were able to find three perfect copies of our target company there is still a problem – we are going to use the results of the evaluation performed for them, but can we really rely on those results? This is particularly important if we have three resembling companies and three completely different values. There is nothing a valuator could do because the valuation reports are mostly private and the information is not available, so this is an incorrigible bias of the market approach. These are the disadvantages of this method; still it is the most popular one, partly because it seems to be the easiest of all methods and partly because we do not need to make any forecasts, which is also a benefit.

Out of four methods we have observed there is one used very rarely because of its opacity (the capitalized excess earnings method), one which requires direct calculations based on the actual data and the forecasts (the income approach) and two based on the market values (the asset approach and the market approach). The latter share the same problem – when we use the market price of something, we implicitly assume it being correct, which is reasonable while the market is stable and predictable but dangerous during all sorts of crises.

If we make one more step and ask ourselves how does the market price any asset, we will either see that the price is determined via comparison with some other similar assets already been priced or the present value of the cash flows is taken, so actually all we have is the combination of the income approach and comparisons. The law of big numbers states that the higher the amount of different random factors influencing the result the less the probability of the high deviation from the mean is, but sometimes all these factors are unidirectional and in this case all the approaches based on comparison are doomed to be wrong, so only when using all the approaches together we can trace the mistakes and make the reasonable conclusion. At least, this is the modern convention of the valuation and it has not changed for quite a while.

Although modern methods used for evaluating the investment projects are sometimes beyond the imagination, not much has changed in the area of valuing the closely held companies. There are no obstacles to using the same, if not more complicated techniques in this case – after all, the giant corporation can afford a small loss due to misevaluation, but the owner of the small company having its earnings as his only source of income probably can‘t.

One can argue that the on average the estimated value should be equal to the true one but this is not a satisfactory explanation for any particular case. Another common opinion is that the algorithm of the evaluation should be simple enough for everyone to understand it, and this is definitely true, but when choosing from the simple but biased algorithm and complicated but

6 unbiased one the second seems to be preferable. Finally, there is a long history of using four approaches and most of the valuators have lots of experience in applying them to the actual problems, but still there is a lot of room for improvement in this area.

There are two important factors that most of the researchers and valuators ignore completely or assume some default values. The first one is the value of managerial decisions and another one is the stochastic nature of the future. When we imply that the company will maintain a steady 5% growth till perpetuity we, in fact, ignore both of these two factors, because the deviation of the income from its mean value could sometimes be not only very significant but also skewed, which is not shown by our analysis; at the same time our estimate is based on some virtual managing strategy which may not be close to any of the strategies that really exist. Thereby the final result of such calculation is both biased and not representing the real probabilities.

The real option method is widely used for evaluation of the investment projects, especially in the cases where the managerial opportunities are clearly seen and the final result can be calculated for each of the paths. As the real options are internal factors of the project, meaning that the decision to choose some path is made by the manager in charge of the project, this type of analysis is nothing more than just performing the calculations several times for each possible decision.

The analysis of the external factors is also widely recognized among the valuators, especially dealing with the projects in Research & Development. In most cases the probability of some turn of events is estimated, then the overall effect for this situation is calculated and at the very end the estimated effect of the project is the mathematical expectation of all the effects. Though simple it is a very useful procedure it helps to distinguish the projects in the case of uncertainty, however, it has its own flaws, the major one being using the point estimations of cash flows and events instead of continuous ones.

In some cases the joint analysis is performed, which is mostly called the real option analysis. The procedure consists of determining some external scenarios (in most cases three scenarios are taken – the best one, the worst one and the normal one) and then the outcomes are determined for each of them, the final result is the mathematical expectation of the outcome. This algorithm is discrete, but its accuracy is much higher than that of simple NPV- technique, the cost of it is the increased amount of calculations. Generally this type of valuation is used for the projects with large deviation of cash flows from the mean or if the project is extremely vulnerable to some stochastic factors and at the same time the management has some options concerning it.

The aim of this investigation is to introduce the methodology that could be used to value both the investment projects and the closely held companies; this algorithm will account for both internal factors (managerial decisions) and external ones (random events).

During the construction of this algorithm we will also discuss major differences between these types of evaluation and determine the parts of the evaluation process being heavily dependent on the personality of the evaluator or the research group. We will discuss several issues connected with our main study, including probability distributions and operations with them, handling the discount rate and different types of calculations using this quantity, extended automata theory, which will be the core of our investigations, various questions concerning the real options and managerial decisions, and, finally, some aspects of the algorithm itself and interpretation of its output. Several examples will be used to simplify the explanations and ensure clarity.

LITERATURE REVIEW

The approaches for evaluating the investment projects are widely known and recognized among both the theoreticians and practitioners. In their two-part work Remer and

Nieto (1995) discuss twenty five different classical approaches to evaluate the investment projects. They classify all the approaches into five groups: net present value methods, ratio methods, rate of return methods, payback methods and accounting methods. Comparing the methods using two simple examples they came to the following conclusions:

. the net present value group of methods provide the most correct and precise information though the flaw of all these methods is that the project requiring higher investment in most cases has higher NPV and therefore is preferable over the project with the low initial investment (if both of the projects have the NPV positive), so comparison is not always straightforward;

. the rate of return methods convert all the internal information given about the estimated cash flows into its ―efficiency‖ and the decision is made by comparing it with the required rate of return; the problem with the methods in this group is that in some cases the rates can be either not unique or not representing the actual situation, so additional analysis and deep understanding of the possible outcome of these methods is necessary;

. the ratio methods are either simple but ignore the time value of money or much more complicated than the previous groups; most of these methods are used to support the decision but never as the only criterion, also these methods do not provide the rule of accepting or declining some single project;

. the payback methods‘ main problem is that they do not show the real structure of the cash flows during the whole lifetime of the project, therefore these methods are misleading and should never be used solely; . the accounting methods require more calculations than the other methods and generally ignore the time value of money, the main problem with this group is that the numbers used are the book values, not the real cash flows, so the analysis requires deep knowledge of accounting to make a decision; these methods should be used to support the main ones.

The most venerable and disputable methods are the present value calculation and the internal rate of return calculation. Each of them is both praised and criticized by the theoreticians while the practitioners continue using them. The methods based on calculating the IRR provide the analysis of the internal structure of the project but at the same time apart from the mathematical problems of several IRR‘s and the opacity of the algorithm for the non-mathematicians there are also some hidden assumptions which in most cases do not hold.

For instance, Kierulff (2008) points out that the IRR in its initial form implicitly assumes that all the cash flows are reinvested at the same rate (equal to IRR), but if the internal rate of return is much higher than the investment interest rate there is no way to technically achieve such returns. Another trouble with IRR is it being a percentage, not a dollar amount, which in some cases complicates the choice of a project from some set of mutually exclusive ones; one more problem manifests when the projects with different life periods are compared. One of the solutions to the latter is offered by Arrow and Levhari (1969), the IRR being a percentage is no longer considered a problem but rather a benefit and to get rid of the abnormally high reinvestment rate the modified techniques (MIRR‘s) are suggested; however, none of those is flawless (see, for example, Bernhard, 1979). Therefore the IRR method is mostly used in its original version but the interpretation of its outcome is made very attentively.

On the other hand, the present value methods are not ideal as well. One of the obvious drawbacks is that if we rescale the project (multiply all the cash flows by the same amount) the final result will also be multiplied by this amount; this leads to the bigger projects being

10 always preferred to the smaller ones even if the yield on the former is actually lower, this problem is mostly solved by diving the NPV by the initial investment, but for the projects having negative cash flows in the periods other than the first one this might lead into a much more dangerous trap. Ross (1995) shows that the incorrect, mechanical use of NPV and different ―rules of thumb‖ (to accept all projects with the positive NPV and reject all the rest) often result in lamentable decisions.

Both IRR and NPV methods require calculating the future cash flows, which is a problem itself and sometimes can be the hardest part of the algorithm, but there is something else to be found to make the final decision. For the NPV-methods one needs the discount rate to find the present value, whereas in the IRR-methods the actual discount rate is used to find whether the project has higher yield than some alternative investment. The problem of using the appropriate discount rate is far from the solution because it is a purely mathematical term and in reality there is lots of confusion on which of the actual values should be taken; however, even when we consider purely risk-free projects there is no single discount rate.

Compton and Farinella (2012), utilizing actual court cases and the surveys conducted by

Brookshire, Luthy and Slesnick (2004), came to a conclusion that the valuators tend to use very different discount rates even in exactly same situations. Some of them prefer using the short-term Treasury rates while others use the rates for 30 years T-bonds; neither do they agree on whether the historical or current rates should be used. In the survey conducted in

1991 more than 57% of all the valuators dealing with the discounting used the historic averages for their calculations, whereas in 2003 only slightly less than 38% were acting the same way; the amount of valuators using the current rates grew from 25 to 47 percent during these twelve years. This example clearly shows that there is no agreement even on the risk- free rates, no need to talk about the risk adjustments, so the final result depends heavily on the personality of the researcher. As the result of such situation the judges, even though they

11 are not specialists in finance, have to make a decision and in some cases it does not have any sense at all except that there are no further debates – for example, in the state of Michigan the rate is set to 5% and it must be used in the cases connected with personal injury or wrongful death. In the closely-held companies and projects valuation the problem of the discount rate is much more complicated.

Partly to solve these problems and partly to incorporate some specific features or achieve more precise results lots of new methods appeared during the previous two decades.

Among those we should mention evaluation algorithms using the fuzzy logic principles (Irani et al., 2002), the stochastic scheduling methods (Yang et al, 1993; Etgar et al, 1995), genetic algorithms (Irani and Sharif, 1997) and different forms of continuous sensitivity analysis

(Bogronovo and Pecatti, 2003; Jovanovic, 1999). When looking closer at any specific industry we can find some algorithms showing good results in that particular area, whether it is agriculture (Wang and Tang, 2010), mining industry (Bhappu and Guzman, 1995), information technologies (Milis and Mercken, 2003) or any other field of activity. Looking at the investigations mentioned in this paragraph we can clearly see that the investment projects are now being evaluated much more meticulously than some couple of decades ago and with the growth of machinery calculation power available this trend will remain the same in future.

One wide set of the modern methods is based on the real options approach. Almost all of them are based on the present value calculation and their core idea is the fact that any possibility (option) the management has is equal to the non-negative value added to the present value of the non-flexible project (Yeo and Qiu, 2003). Most of the theoreticians try to apply some version of the Black-Scholes model to the managerial options in order to find their value (see, for example, Benaroch and Kauffman, 1999). However, the Black-Scholes model is not a simple mathematical instrument and if more than one option is considered at

12 the same time the complexity increases dramatically, so most of the researchers content themselves with three major alternatives: to abandon, to delay and to postpone (an example on how to calculate the value of these options in practice can be found in an article by

Kemna, 1993). In addition to the classical view lots of alternative ways of valuing options have been developed, as an example we can mention the evaluation using the fuzzy variables

(Wang and Hwang, 2007) and the decision lattices instead of classical decision trees (Smith,

2005). The implication of the real option analysis is undoubtedly extremely useful, especially when the project is highly dependent on unpredictable external factors, but in some cases the results of the calculations might be conflicting with the strategic decision making as shown by Bowman and Moskowitz (2001). Therefore one shouldn‘t blindly use the real option valuation without deep understanding of the nature of the options and the possible outcomes.

The real options are in fact deeply connected with the stochastic nature of the future.

When we say that an option has value we implicitly assume that it is the amount of money (or its equivalent) we are willing to pay for the possibility to react on the future. In their review,

Klinke and Renn (2002) use very vivid classification of risks, dividing them into six categories:

 The Sword of Damocles is the event which has very severe consequences but

low probability, most of the periodic natural hazards are in this category (this

risk is accepted on purpose);

 The Cyclops is the event which could, but was not paid attention to due to

various reasons, the example is any non-periodic natural hazard (this risk is

either invisible or ignored);

 The Pythia is the event with unpredictable consequences, it occurs mostly in

the technological sphere when no one knows what to expect from some action

(this risk is unpredictable);

 The Pandora‘s Box is the event with the consequences that are visible only

after some time will have passed, a good example of such type of risk would

be some new chemicals used in agriculture or in food industry – the

consequences can show up in several decades (this risk is unpredictable);

 The Cassandra is an event which is ignored because its consequences are

delayed, the global warming would have been an excellent example if

everyone were sure that it really does exist (this risk is ignored);

 The Medusa is something new which frightens people because they are not yet

acquainted to it, all items we have around us were once in this stage of the

evolution (this is purely psychological risk).

Based on this classification the authors present three algorithms to deal with these risks. The article does not consider the financial risks but later in this work we will see how these methods can be used for evaluation purposes.

The classic application of the probability analysis to the project evaluation can be found in the article by Hertz (1979). He points out that the person or people responsible for the decision of whether the project is acceptable or not should not make this decision based only on the mean values of the future cash flows or on the means and variances, but rather they should look at the plot of the probability distribution. This is especially important if the probability is skewed or the distribution is far from normal, which is, actually, not a very rare case. Of course, it is much more complicated process studying the graph than just comparing two numbers but the managerial work should not be easy. Managerial decisions should result in right decisions and examine all the information available.

PART 1. THEORY

1.1. Probability Distributions and Why We Need to Use Them

Instead of using the point estimates of the future cash flows or net incomes we should use the probability distributions. There are several reasons for this:

 The point estimates are in most cases biased relative to the actual expected value due

to the rounding errors and subjective factors whereas the distributions correctly

represent the actual information and the forecasts;

 The distributions represent all the information available, because the set of data is

converted into a function and therefore the amount of information does not decrease,

whereas the point estimate reduces the information from a set to a single point which

is highly undesirable;

 The point estimate does not represent the precision of the prediction. Some experts

having much more experience than the others can make more precise estimates and

those should be weighted higher than the others‘ ones (this is especially important if

several experts examined the situation and the final result is based upon all their

predictions and forecasts).

To make the reason for abandoning the point estimates clearer let us consider four companies and the probability distribution of their net income for one year. In the exhibit below we analyze four different probability distributions and show that the point estimates are frequently useless. From now on we consider only the probability distributions unless stated otherwise.

Figure 1. Forecasted Net Income for Various Types of Companies

Graph 0.1 Normal Distribution (wide) Graph 0.3 Right-Skewed Distribution

Graph 0.2 Normal Distribution (narrow) Graph 0.4 Peaked Distribution

The graphs above represent the forecast on the net income for various companies during the following year. All the industries and the business models are different; neither do they have the same size, but what is common is that all of them have the equal expected net income (denoted here as EΦ). If the point estimate is used only this value is considered and an investor has to suggest that all of them provide equal opportunities, which is apparently not true.

The first graph is a typical company, ignoring some minor issues we can assume the probability being distributed normally. We do not account for the seasonality or any other factors having periodic structure, so a small supermarket will be a good example.

The second graph is also typical, but the variation being much less means that the market the company is operating in is more stable. A company using some imported details to produce goods and selling it on a fixed price is a good example if it does not hedge its currency risks because the fluctuation in the currency market are usually quite low.

The third graph can represent, for instance, an internet start-up. These companies do not require a heavy initial investment, so the losses in this case are limited, but the chance of success is small (the graph is right-skewed). However, a small start-up has some chance of becoming a huge giant like Facebook, so the probability distribution has a fat tail on the right side.

The last graph is representing a company whose profit depends on the results of some research – pharmaceutics is fitting this template ideally. In case of some study showing successful result the company makes profit, in case of failure it has losses. For such situations the expected value is absolutely useless because it does not represent the situation at all – the probability of the net income being equal to its expected value or lying anywhere near it is zero.

Sometimes the standard deviation is used in addition to the expected value for completeness. It will definitely help to distinguish the two first companies, but for the rest it will be very troublesome. The problem with the third company is that such type of distributions can have no moments higher than the first one, so the mean is finite but the deviation is infinite; in this case there is nothing to be done except just looking at the graph.

For the company number four the deviation is close to half of the distance between the peaks, but if only the mean and the deviation are known there is no clue that the graph actually looks like this. Therefore the deviation does make some improvement only occasionally.

As the conclusion, we can say that neither the point estimate (supposing it is exactly equal to the expected value, which is the only reasonable suggestion but often is not in fact the case), nor the estimate together with the standard deviation can provide enough information to detect the real possible net incomes of the company. Only the full analysis, including the study of the probability distributions is acceptable if we want the result to be justified.

There are several ways to make the estimate of a probability distribution. If enough data is available the distribution histogram will be a good fundament for the probability distribution, but this is an exception rather than a normal case (rarely can a company be found which can provide weekly or even monthly data on their net income or cash flows; the yearly data is in most cases unacceptable because to make even a rough estimate at least 50 observations are required). If the data is unavailable or for some reasons cannot be used (for example the valuator might suspect that the sample is not random or the data was collected during the period which is going to end soon, so the future cash flows can be significantly higher or lower than the past ones), the main task for the valuator is to find out the relevant shape of the distribution and then determine the parameters of that function which, in his opinion, describe the future cash flows best of all.

1.2. Discount Rates

The problem of determining the correct discount rate remains unresolved no matter whether we are considering the private company, a project inside a corporation or the risk- free rate. However, apart from the problems of determining this quantity there are some methodological problems with its definition. The discount rate is tightly connected with calculations of the present value and the internal rate of return, but the essence of this value lies much deeper than most of the managers believe.

The discount rate is the change of the value of one dollar between different points in time. This definition itself does not afford an algorithm of calculating this value but it is very useful for theoretical purposes. In practice one of the best and most useful definitions of the discount rate is based on the cost of the alternatives. The discount rate for any project is the yield on the trailing alternative having the same risk, it states. For example, the pure risk free project which lasts ten years will have the discount rate equal to the yield on 10-year T-note

(we assume that the Treasury notes are risk-free). However, if some risk is associated with

18 the project the direct application of this rule is much more problematic: firstly no one knows how to measure the risk and therefore it is questionable what does ―having the same risk‖ mean; secondly, if the project has some risk associated with it there is some probability distribution of the final outcome and therefore it is unclear how the yield on this project should be measured. To overcome this difficulty the discount rate is commonly presented as the sum of the risk-free rate and the risk premium; in this case the risk-free rate is usually taken equal to the Treasury bond yield and the risk premium is calculated subjectively, so after all there is no clear and single way of dealing with the discount rate.

Apart from the financial problems lots of mathematical troubles are connected with the discount rate. The first and the main problem considering the discount rate is that it is often (in practice, always) considered discrete. All the cash flows are summed up to the end of each period and then the resulting sum is discounted. We will use the following example to show that this approach is not showing the appropriate results and should be replaced with something more effective.

Example:

Let us assume that we have the project which requires heavy investment at the very beginning ($1.6M) and at the end (utilizing costs, $300k). It lasts 15 years. Sales are highly dependent on the season and the cash flows are as follows: in spring (March, April, May) the cash flows are equal to $20k, in summer (June, July August) - $60k, in autumn (September,

October, November) - $20k and in winter (December, January, February) it is -$20k. The discount rate is 12% a year, the recalculation to other periods is made in a precise way, not by just dividing or multiplying (EAR). We will see how the classical NPV depends on various factors based on this example.

Let us first consider the dependence on the timeframe. Most of the managers will take annual data by default, which is simply the sum of the cash flows; one more reason for doing

19 this is that the discount rate is annualized. If we calculate the effect of the project (NPV), in will be equal to -$20k and the projects seems to destroy value, so it should be rejected without any hesitation. However, if we use the monthly data (which is available for us here) and calculate the effect based on monthly discounting the result will be equal to +$123k – more than enough to accept the project. The difference between these results has its roots in the methodology – when we calculate the result on annual basis it does not matter for us whether some particular cash flow happens in January or in December, which is clearly a fault of the method. One can argue that due to prediction errors we should rather use the conservative annual estimate than a more precise but also more dependent on the forecast precision monthly one; however, in this case the information given is in fact representing the presence of the periodical structure of the cash flows and we cannot simply ignore it even if the precision of the data is not quite acceptable. We indeed do neutralize some effect of the unsystematic errors when using the aggregate annual data, but the cost of it is a huge bias we introduce. To avoid this bias we can use the following scheme: instead of using end of period as a discrete point we can take middle of period; it will transform the common NPV formula

∑ ( )

into another one:

∑ ( ) ∑ ( ) ( )

Here the additional multiplier represents the half-year forward shift of all the points in time (the cash flows for the second year, for instance, will now be considered not at but at 5 and so on). As is positive (in most cases it actually is) this factor increases the effect of the project (for the regular projects, which have the positive cash flows, this holds true in all cases, but if some of the cash flows are negative the relationship can become the opposite). In our example the unbiased NPV is equal to $75k. This is the best estimate to be

20 done if we want to stick with the annual data, it is still lower than the NPV calculated on the monthly basis but now it is at least positive. We could do the same procedure with the monthly data and the result will therefore increase even more, but this correction will not be justified because we do not know the structure of the cash flows within each month and the conservative estimate is used to make sure that we are not exposed to any additional risk due to the calculations.

The result of our calculations when the monthly data is used is dependent on the current month. The one we have obtained earlier ($123k) refers to the situation when the first month is March. If it were September the result would be much different and equal to $20k.

This example shows that one should be very careful – if the calculations are performed on the monthly basis the starting date must be determined precisely, in case the precise date is unknown it would be reasonable to consider wider periods – for example, quarters instead of months. We should also notice that the unbiased effect of the project calculated on annual basis is close to the average of the two monthly effects we have (the average of $20k and

$123k is about $75k). This is one more reason to use the medium-year method instead of the end-year one.

Summarizing what we have said in this example we should point out that the period and method we use depend on the particular situation. The general rule should be as follows: the middle-year method is preferable to the end-year one and the period should represent the quality of data we have: if we are confident over the monthly data we should use it, if not we should check the quarterly data and so on, until we find some acceptable and reliable values.

This stage of analysis should be performed prior to the calculations but after building the full model including the automaton.

From this example we can see that the length of one period does affect the final result and the difference can be very significant. The general rule is simple – the smaller the period,

21 the less is the bias, so one should use the smallest of all possible periods for which he is still confident about the data. When the length of the periods tends to zero the discount rate becomes continuous. The following paragraph is the definition of the continuous discount rate.

Let us consider two moments of time close to each other and denote those as and

, where is a very small period of time comparing to the duration of the project (for example, it can be an hour or a day if the lifetime of the project is several years). The smaller is the gap between these time moments, the less events could happen that will affect our time value of money, and we can suppose that it remains constant during this interval. However, we do not assert that it remains the same from period to period; on the contrary we suppose that any variations in the discount rate occur only at the beginning or the end of each period.

Let us denote the discount rate for one period as ( ), this notation shows that the discount rate changes with time. If ( ) is the present value of one dollar at time , by definition

( ) ( ) ( ( ))

This equation can be transformed into the following:

( ) ( ) ( ) ( )

taking the limit we get

( ) ( ) ( )

This is the ordinary differential equation; its general solution is well-known:

( ) ( ) [ ∫ ( ) ]

The present value of one dollar today is exactly equal to one dollar, ( ) , so we have the following formula for determining the present value of one dollar at any future moment:

( ) [ ∫ ( ) ]

This formula reveals the meaning of the function ( ) being the accumulated loss of value from the starting point (considered as 0) till any point in future. From now on we will call it the discount index. The total decrease of the value between two points in time is calculated as:

[ ( ) ] ( ) ∫ [∫ ( ) ] ( ) [ ∫ ( ) ]

From these formulae we can directly obtain the present or future value of any set of cash flows:

̂ ̂ ∫ ( ) ( ) ̂ ( ) ( )

where ( ) is the cash flow function (continuous cash flows), ̂ ( ) is the value of the whole set of cash flows at time and ̂ ̂ ( ) is the present value. The first relation is the analog for the classical discrete sum of the discounted cash flows and the second one represents the main principle of time vale of money. The hat over the letter represents no adjustment for inflation in the calculations.

So far we did not actually use any specific information about the discount rate; neither had we made any assumptions on the structure of the object of evaluation. This formula is the most accurate of all we could possibly have; it considers each factor influencing the discount rate once it takes place. Let us now find out the properties of the continuous discount rate

( ) and discount index ( ).

1. ( ) is a stochastic function. This is the main reason why there are still discussions over the methodological issues – it is impossible to consider every single factor

23 influencing the discount rate and predict their behavior till the end of the project. It is still only an estimate, but a much better one than we usually use.

2. ( ) is an unobservable function. The situation with it is very similar to the one we have in quantum mechanics – some variables cannot be measured directly but only obliquely, by the effect caused on some measurable values. However, here we do not have anything of that sort – the only things that are real are our cash flows being spread in time and what is rest is our attempts to somehow collect all these cash flows into one quantity to simplify the decision making. In the mathematical sense the value being unobservable means that we cannot use statistics over the past data because we are not sure whether the data is relevant – an absolutely blank space with nothing to start with.

3. ( ) is not necessarily positive. There could be some moments when money tomorrow is much more valuable for us than today. We will consider only the regular projects and situations for which the discount rate is non-negative, but still we should remember that in some situations various irregularities can appear. The following example is showing one of such occasions.

Example.

Let us suppose that we have a loan with the floating rate. We will not consider all the conditions and peculiarities and assume that on January 1 each year the rate is recalculated.

All the information allows us to conclude that the rates will fall next year. The loan is repaid as follows: out of the company‘s operating income some fixed percentage (say, 40%) is used for this purpose at the end of each year. In this situation we will prefer receiving any amount of money in January, not in December, because we will cover larger amount of loan if the interest rate is lower and therefore each dollar has higher value in January than in December.

Even though this example is in some sense exotic, we can find such type of situations in the real world, in most cases when we are dealing with floating quantities and/or values

24 aggregated in time (one can clearly see that if taxes are paid on April 1st the company would rather receive any payments after this date, not before). When making a rough calculation most of these factors are excluded and the discount rate is normal, however, abnormal behavior sometimes appears is case of more thorough analysis of the same situations.

4. The absolute value ( ) is generally decreasing with time. To understand this concept one should ask himself: ―Does a month delay of some cash flow today have the same effect as the same delay in the same cash flow in, say, ten years?‖ The simple test to find out the trend of the discount rate is to ask someone to say how much money does he want to be paid instead of being paid $1000 today in a) 1 month, b) 10 years and c) 10 years and one month. If we denote the answers as and respectively, being equal to $1000 (the present value), then

( ) ( )

and we can directly see how the discount rate changes with time.

5. The more distant the cash flow is from the present day, the less we worry about it. In fact, it means that any payments done in infinite amount of time are equal to zero, so ( ) .

In this paper we will always use the inflation-adjusted discount rates. The reason for this choice is that in this case we do not need to adjust the cash flows to match the actual loss of money value; the prediction models will work with less bias and some trends will become more visible in this case. To make the inflation adjustment we split the discount rate into two parts:

( ) ( ) ( )

where ( ) is the continuous inflation and ( ) is the continuous interest rate.

Substituting this sum into the formulae above we obtain

( ) [ ∫ ( ) ]

[ ∫( ( ) ( )) ]

[ ∫ ( ) ] [ ∫ ( ) ] ( ) ( )

The functions ( ) and ( ) have the meaning similar to ( ) – they are showing the accumulated discount due to inflation ( ) and interest ( ). From now on we will not perform these calculations and always assume that we know the inflation-adjusted interest index ( ), which we will call simply the discount index when it cannot lead to confusion.

When we use theoretical methods and reasoning we can always use the continuous rates – they are both correct and simplify the calculations; however, the real data is discrete and we must take this into account. Let us assume, for example, that we have the monthly cash flows * + estimates for some project and the discount index ( ) calculated on the annual basis (which means that one year is taken as one unit of time for this quantity). To calculate the present value we cannot use the formulas above because our cash flows are discrete, so we need a discrete analog for those. By means of the generalized functions concept our integral transforms into

∑ ( ) ( ) ( )

here is the frequency factor being equal to the number of data points per basis of the discount rate (in this example we have monthly data and the annual basis, so is equal to amount of months in one year, which is twelve), is the inflation-adjusted present value and ( ) is the inflation-adjusted future value. These formulae form the basis of the computational algorithm.

1.3. Forecasts

Both the NPV and IRR calculations require the future cash flows as the income. This problem has the similar nature to determining the discount rate – the opacity of the future, but at the same time determining the cash flows is in some sense much simpler because there are no different definitions and measures of this quantity. The cash flows are both observable and in most cases much easier to predict, nevertheless forecasting requires some special knowledge and deep understanding of the processes in the industry. Let us have a look at these processes.

The first thing to do is to determine the observer horizon . The observer horizon is the duration of the period starting right now for which we can make any reasonable forecasts.

For example, the horizon of the commodity price forecasts can be several years, for the natural calamities – several centuries and for the stock markets it could be at most several days. The more complicated target object is the less its horizon will be; this value also depends on the researcher‘s confidence in his estimates. Depending on the way the forecast is made the horizon could be in some cases calculated.

Figure 2. Illustration of the Concept of Observer Horizon

Historic Area of Forecasts Unpredictable Future

Time

T = 0, T = H, Present Moment Observer Horizon

There are three general approaches to forecasting. Their essence can be reduced to the following:

 Extrapolation method assumes using the past data to make the forecasts;

 Subjective method transforms the beliefs and experience of the valuator into the forecast;

 Market method involves the use of the forecasts made by other experts.

The simplest and, perhaps, the most efficient method is the last one. Using the market method the researcher relies on the forecasts made by the specialists in the area – whether it is the oil market, the currency exchange market or any industry there are people and agencies having enough experience to make reasonable and justified forecasts. Of course there is no guarantee that they will not make a mistake in that particular forecast the researcher needs but if he has to make the forecast himself the result can be much worse. This method is omitted only in case the researcher suspects that the experts did not have all the necessary information to make the prediction or some new important information appeared since the time of the last forecast. If the market method is used the horizon should be set equal to the forecast period.

In case no forecast is available or none of the existing ones are reliable the researcher has to make the prediction himself. If the historic data is available (and it always is, except completely new ideas) the statistics can provide an estimate for the future data using the extrapolation methods and auto regression models. We should note, however, that even if the auto regression fits the curve with the close to 100% we should not be confident that the prediction will be correct. Without getting into mathematical details we note that the horizon of the extrapolation is finite even with a perfect auto regression because there is always a probability of some external event occurrence which will ruin this model. The researcher himself should choose the appropriate horizon relying on his knowledge and experience.

In some cases the auto regression provides unacceptable results and the only way left to make the forecast is using all the information available to make subjective estimate.

Generally this situation is not expected to happen when a private company is being valued because the historic data on the cash flows often provides enough information for the quite

28 specific predictions, but in project evaluating and the scenario analysis this is not a rare event.

There is no way to avoid this situation and the only solution is a thorough analysis of the data and similar historic events to make the reasonable forecast. In this situation the horizon is very small and the distribution usually has pretty large deviation.

In the algorithm developed in this paper we use the concept of the observer horizon as the upper bound for the calculations. However, we should also add the residual value to the final amount because it can be significant. The following sidebar illustrates this idea.

Sidebar: The residual value.

When we use the concept of the observer horizon we cut off the possible cash flows in the unpredictable part of the future. One of the ways of dealing with this is mandatory selling of all the assets for their market price at that time. We can use the following argument to justify this way of action. The observer horizon is the point in time beyond which no reasonable prediction can be made; this implicitly means that the cash flows beyond this point have infinite dispersion or it is so large that the NULL-hypothesis of the cash flows being equal to zero holds true in all cases. If this is true, the residual value being the sum of the discounted cash flows beyond the horizon is itself statistically indistinguishable from zero and therefore between two options – selling all the assets at some finite price and shutting the business down or continuing operations with the unknown result the first alternative should be used. This explanation does not imply, of course, that the business should actually be shut down when the horizon is achieved, this is only a mathematical operation used for making the computations finite. We can see from here that in fact we have the combination of asset and income approach – we use the income approach in the area of forecasts and the asset approach to estimate the residual value.

1.4. Classical and Extended Automata Theory

The analysis of the managerial options can be conducted by means of the automata theory. However, we will need to extend it to be able to incorporate some special types of options not used in the classic theory. In our definition the options an automaton is a mathematical object which is generally described by the following sets and values:

The States: a finite set of possible internal states of the automaton, denoted as .

The Transitions: a set (finite or countable) of the links between the states, denoted as

; the transition itself is characterized by the cost , length and factors * +.

The Initial State: one and only one of the states is initial, it is always denoted as .

The Final State: there is only one final state denoted as .

The Step: a non-negative quantity representing the length of one period (month, week, year, quarter or any other period of time), denoted as .

The Horizon: a non-negative quantity used to limit the algorithm (its meaning is described in the previous section), denoted as .

The Service Variables: these variables are used to distinguish different actual situations united under the same state, they can have different notations depending on their meaning and we will not pay much attention to these.

In this section we will describe the theoretical features of the automata we will use.

The next section will be devoted to the properties of these automata in general and their technical characteristics.

An automaton is an abstract machine which is capable of committing some transitions between the so-called states and as a side-effect with each of these transitions some additional actions are performed. This term appeared soon after Alan Turing created his theory of the artificial intelligence based upon the abstract machines called Turing machines after him. The

30 actual realization of an automaton is a computer program which performs all the actions required. The theory of the automata is extremely useful in programming; all new programming languages are written using these techniques.

The classical automaton works as follows: it reads a piece of external information and based on it performs some actions including possible transitions between the states; all these actions are included into the main element of the automaton – the transition function. The most important case of the automaton is the lexical analyzer within the compiler: it reads the line input symbol by symbol and depending on the current symbols determines what the meaning is. For example, the SAS program can contain some line like the following:

proc print data=newdata; run;

The automaton is initially in its zero-state, it starts reading the entering queue letter by letter. After letter ―p‖ it transits to the state usually called ―Letter‖, remains in it while the letters ―r‖, ―o‖ and ―c‖ are read but once the space sign is achieved the analysis itself begins.

When the automaton is in state ―Letter‖ it has a variable which adds the current symbol to its end, therefore when the analyzer is observing the space symbol this variable stores ―proc‖ in itself. This word is compared with the list of keywords stored inside the automaton and if it matches one of them this part of input is treated as the corresponding keyword, the analyzer removes the word ―proc‖ from the input and replaces it with the directions of handling it which are passed to the following parts of the compiler. The automaton itself then returns to the initial state. The next part of the input, ―print‖, is treated the same way. Then time comes to analyze ―data=newdata‖, which is more complicated. Here the text variable collects the letters until the equality sign is observed, then the automaton transits to a new state in which it is expecting the name of the variable only. After ―newdata‖ is read the analyzer checks for it being the variable name, if it is one this part of code it transformed into the machine language and the compilation continues, else (if, for example, the input were data=print or

31 data=4) the compilation is terminated and the error message appears. The semicolon as any punctuation mark is always the instruction to the compiler itself; in this case it means that the previous lexemes should be treated as one meta-lexeme. The keyword ―run‖ is another meta- lexeme. All the actions described above are inside the transition function. This is the brief description of the automaton and its implication.

For our purposes we need to change the classic definition significantly. First of all, there is no transition function – at least in its initial meaning. From each state there are transitions to other states, representing the options available for us, they are not mandatory and the only final criterion is maximization of some quantity. The input line of symbols also is no longer required, instead we use the time-model, which implies the discrete flow of time and the transitions happening at some points due to the decisions of the management and/or some other factors. In Chapter 2 we will describe in detail the process of creating an automaton, but for now we should point out that the classical automata theory is dealing only with the fixed objects; therefore to have an ability to deal with such flexible terms as possibility and probability we need to construct a complicated instrument which requires much more skill of the researcher to handle it.

1.5. Distribution of the sum

The problem we will be facing in our computations seems simple but in fact requires some non-trivial computation. Suppose we have the series of data (representing, in our case, the net income for some small periods of time in the future) - * + and the discount index for the corresponding period * +. The question is – if the data are random in nature and the probability distribution for each value is known, how to find the probability distribution for the present value ∑ ?

The most well-known result is the central limit theorem – if ,

and all the data points are independent we know that the distribution of converges into

32 the normal distribution. The central limit theorem is a very narrow result and we can expand it to cover much more various data – the price will be that the calculations will be no longer be able to be performed manually, but for our purposes it is suitable as other steps of the algorithm we present require computer calculations anyway.

To find the probability distribution for the sum of the series shown above we will use the characteristic functions. By definition,

( ) ∫ ( )

where ( ) is the probability distribution for the random variable , √ and

( ) is the characteristic function. It exists for any distribution possible and in fact is its inverse Fourier transform. The main reason for using it is the following equality:

( ) ( ) ( ) ( )

When extended by means of this equality to the whole series we obtain

( ) ( ∑ ) ∏ ( )

and the probability of the present value is the Fourier transformation of its characteristic function:

∫ ( )

We can write this algorithm as one formula:

∫ [∏ ∫ ( ) ] ( ∑ )

PART 2. DATA

2.1 Conceptual Analysis

We start the analysis with determining its aim and its outcome data. The algorithm we design in this paper can be used for the following three purposes:

 Evaluation of the closely-held companies (primary aim);

 Evaluation of the investment project (secondary aim);

 The analysis of the current situation (used by the managers to see the possible

outcomes of their decisions).

If our target is a private company we are expecting the outcome to be some quantity representing its value. In this case we must use some standardized methods which would be accepted by every court and all the participants of the deal. There are no adjustments for any reasons and the procedure itself is very formal.

If we are evaluating the investment project the outcome is either the Boolean

True/False value (in case we know all the available information this would be the advice for the final decision – whether or not the project should be accepted) or a probability of this project being successful (the criteria of ―success‖ are determined prior to the evaluation itself, not always do managers require only the NPV being positive). In case some additional information is required it can also be presented in the outcome but generally we will concentrate on the question whether the project should be undertaken or not.

For the purposes of helping the manager the final result should be presented in such a form that it clearly shows the possible outcomes of each strategy being tested. The outcome in this case is the probability distribution of the present value of the company or the department for each strategy, the final decision is then made comparing these outcomes. The evaluation algorithm is by no means a substitute for the skillful manager but it can help him with the calculations and visualize the outcomes of different decisions so that the process of choosing one of them and justification of this choice would be much easier.

In this paper we will mostly discuss the private company evaluation; the differences in the algorithm when it is applied for other purposes will be discussed within the main text unless they are changing some piece of the algorithm completely, in this case the separate subsections will be used.

2.2. Available Options

This part of the analysis should be conducted prior to defining the variables and collecting the data because it often specifies the type and amount of data required for the calculations. This process has several steps.

The first thing to do is to answer the question ―Do we need to incorporate the options to our analysis?‖ The answer is not always straightforward as it seems to be. For the purpose of valuing the company we use the standardized algorithm and always take the options into account, but it is a whole different story with the investment projects. In some cases the analyst is supposed to not only calculate the effect of the project in its current form but also has to show the outcome for several different scenarios. In this case the algorithm includes all these options, but if the project is immutable in nature or the manager does not have enough credentials to make any changes this stage is skipped. The example of such a situation is the government order – nothing can be changed in it and the decision is either to accept it as it is or decline it without any further discussion, in such situations the analysis continues from stage 3.

If the answer to the first question is positive the second step is putting down all the possibilities. To make this process easier and to structure the data we suggest using the following table:

Table 1. Real Options Classification Matrix

Grand Options: Major Options:

 Exercising this option changes  Exercising this option changes

completely the business or the significantly the business or the

project project

 Often such type of options are  Often such type of options are

connected with significant changes connected with changes in marketing

in marketing policy, structure of the policy, structure of the company or the

company or the type of goods or type of goods or services produced but

services produced they have much smaller magnitude

than the grand options

Examples: starting the production of new type of goods, expanding to new markets Examples: producing the new type of

(domestic or foreign), working on the goods under the existing brand (iPad 2, government order, Initial Public Offering Activia Light etc), changes in the price policy,

(for private companies). advertising campaign.

Note: there are usually very few grand Note: the major options are typical for the options for the investment projects whereas investment projects as well as the private most of the options for the small private companies. company fall into this category.

Table 1. Cont.

Minor Options: Blind Options:

 Exercising this option can change the  Exercising this option leads to

business or the project very slightly, unpredictable effects, opposite to the

usually no risk is connected with minor options the risk is very high

these options or the risk is  These options are very rare and occur

insignificant mostly in the areas which are

 These are mostly administrative developing quickly

changes and improvements to the

current project or business Examples: R&D projects, projects in IT,

pharmaceutics and telecommunications.

Examples: using new office technics, Note: These options almost never appear in changing the color of the production or the project evaluation but sometimes happen buying new furniture. in the company evaluation, they are placed in

Note: most of the options actually fall into a separate group because of the abnormal this group. level of risk.

After all the options are written down and classified each of the groups is treated separately. The minor options in most cases can be either ignored (if the expected value is close to zero) or the effect is added to the compound effect of the project or the value of the company (if the expected value is positive and the deviation is small). For example, the effect of using less paper for the office needs with new electronic software is predictable and can be just added to the effect of the project; at the same time changing the color of the product from red to blue has unpredictable consequences, but if the expected value of this action is zero then it should not be taken into account at all. The Law of Large Numbers states that the sum

37 of random values with the mean of zero and limited dispersion has the mean of zero and the smaller its dispersion is, more factors are in the model, so this way of treating the minor options is reasonable.

The grand options are the main element of the algorithm and are always considered along with the main project. In chapter 6 we will show the way of dealing with these options but for now we should mention that the more options have been detected, more time the calculations will take and more thorough the analysis will be.

The major options are treated depending on the amount of the grand options. The algorithm can become extremely slow if lots of options are considered in the same model so the grand options always have the priority and the major options take what is left. One of the ways to deal with too many major options is breaking them into several groups and conducting the analysis for each of these groups, we will return to this problem in chapter 2.5.

Finally, the blind options require some specific handling. In most of the cases these possibilities appear when both the valuator and the manager are not specialists in some area but it is clearly seen that there is an opportunity to step into this area. If it is possible the additional consultation from the expert in this area will transform the blind option into a grand, major or minor one; in case there are no specialists available or the cost of the consultation is unacceptable the blind options are in most cases ignored. In chapter 2.5 we will see an example of such situation and explain why it is the best solution in this situation.

Figure 3. Difference Between Options

The graph above roughly represents the difference between different types of options.

The horizontal axis is the estimated effect of an option and the vertical axis is the standard deviation (if it does exist and correctly represents the possibilities; if the situation with the option is similar to ones shown on pictures 0.3 and 0.4 in subsection 0.1 we consider this option to be located in the point (0 ; + ) on this graph. We see that minor options are not worth considering, major ones are more important, grand options are keys to the possible success and blind options are unreliable – their effect is unknown and therefore they can be called Black Swans (following the terminology introduced in the classical work by

N.N.Taleb).

2.3. Variables

To conduct the analysis we should first determine which data is required for it. Here we will not only describe the variables and their meanings but also discuss some differences between the applications of the algorithm for different targets.

The first variable is the discount rate. As we have previously seen the continuous discount rate should be used instead of the discrete analog in most cases; also we will always use the inflation-adjusted discount rate to exclude the inflation and represent only the pure interest. If the target is the investment project the company‘s WACC can be used as the discount rate (if there are no reasons to choose something else). If we are valuing the company determining the discount rate is not an easy process, lots of algorithms are developed to find the appropriate discount rate and they can provide very different results; in this case we should take the average of these and then conduct the sensitivity analysis on the discount rate to find out whether the result of the algorithm is reliable. This is also a good solution if the algorithm is used for the analysis of the possible consequences of some decision, however in this case the outcome will provide some additional information which will be beneficial for understanding the current situation. Anyway, to perform the calculations we need the inflation-adjusted continuous discount rate as the function of time:

( ).

The next question is what quantity is considered as the value generator – putting things in a simpler way, what quantity is the numerator for the NPV-calculation formula. Is it the EBITDA? The Net Income? The Cash Flows? The answer depends on the particular situation and the target of the investigation. When we consider the investment project the free cash flows (including the possible tax shelters) are used in most cases, but for the small companies the problem is much more complicated. We will not concentrate on the reasons for choosing one value or another, but one thing should definitely be mentioned: if for some reasons there are some doubts on what quantity to take it would be reasonable to make calculations for all of them and then compare the answers; in case of significant differences between the outcomes the task of the valuator is to pick one of them he thinks is the most reliable. Hereinafter we will call this value the net income and denote it as . The algorithm

40 itself will not change if instead of the net income the cash flows or any other measure is taken, but once it is chosen it should remain the same till the final result is obtained.

After determining the value generator we need to find out which external factors should be included in the calculation. Among these could be the following: the prices on any commodities and raw materials, the currency exchange rates, tax rates in different countries, the costs of different logistics operations and any other factors that are influencing the result of at least one of the grand options (or the initial state itself). To sum up, in each case we need the set of external data denoted as * +. This data can be of different kind and can be expressed as point values, sets or functions, but the criteria are that the result of our calculations does depend on it and the performance of the company itself does not affect these values (we require the absence of the backward linkage so that there is no information loop, in most cases this holds true and we will later see how to handle the cases where it does not).

PART 3. ALGORITHM

3.1 Creating an automaton

We have already discussed the extended automata theory, by means of which we are going to calculate the distribution of the company‘s or project‘s value. In this chapter we will describe the process of creating an automaton for some particular situation and then look at the example of its application.

Figure 4. The First Step in Automaton

Automaton: Step 0

횯 S

Step 0: Prior to constructing the automaton we put down all the detected managerial options in the order of decreasing of their influence on the project: first grand options, then major ones; if there are some blind options we put them in a separate group. Then we take a draft of the automaton, which always looks the same – see the picture on the right. Regardless of the options, any automaton has two states: the initial state, in which the company is at the moment of the evaluation, and the final state, which is a gravestone – the company does not exist anymore if it transits to this state.

Figure 5. The Second Step in Automaton

Automaton: Step 1 횯 S

Step 1: We look at the list of options and find out which of them are associated with the initial state . For example, the option of conducting an advertising campaign makes sense only if there is some possibility to meet the increased demand – if we have the opportunity to fulfill this requirement in the initial state (i.e. right now) then this option is associated with the initial state, else it is not. For all the options which are associated with the initial state we draw the arrows coming out from as shown on the picture to the right.

Sidebar: What is the essence of the options being associated with the states?

The association is the core feature of the extended automata theory because this is the point where the differences between the classical and extended theories appear. If we have the finite list of options in front of us we can clearly see which of them can be applied under some circumstances and which cannot. The classical automaton will terminate the process with the error if the input symbol or sequence is unacceptable in some state, but here the links represent the possibilities, not the obligations, and this is the main difference between the algorithms. We should notice also that the automaton we are going to create is used for context-dependent grammars – the effect of each input is indirectly influenced by all previous steps.

Figure 6. The Third Step in Automaton.

Automaton: Step 2

횯 S

Step 2: For each of the options represented as arrows on the graph we have to find the end state. The rule here is as follows: let us take some option and put a new state called – the auxiliary state, in which it ends. Then for this auxiliary state we find the associated options and compare them with the options associated with the initial state . If there is at least one difference between these sets the auxiliary state becomes a permanent one, we rename it to free the letter X and the arrow ends in it; if not, the arrow is a loop which begins and ends in the initial state. For each loop we create a new service variable (for the example shown on the picture we should create two variables and connected with the initial state

; generally the transition between the states does not require a new variable to be created but in some cases it should be done as well. These variables store the amount of loops and might affect the distribution of the cash flows.

Sidebar: Why is it done that way?

The process done in step 2 makes sense from the financial point of view and reveals the concept of a state. There are some options which do not affect other options – they could be exercised simultaneously or one after another but there is no qualitative relationship between them – only quantitative. The example is the advertising campaign and opening the new product line – these actions affect each other‘s effect but each of them could be

44 performed solely. These options are represented as loops on our graph – they do not change the state in the meaning that all the options that were available continue to be available and nothing that was not available became available due to this transition. However, there is another type of options which do affect the state. For example, if we shrink the production the advertising campaign is no more useful for us – even if it does increase the demand we cannot meet it, so the transition on the arrow leads us to a new state in which the option to conduct the advertising campaign no longer exists. This is shown by separate states on the graph.

Figure 7. The Final Step in Automaton.

Automaton: Step 3

횯 S

B A

Step 3: After this process is finished with state we take the first of the new states and repeat the actions described in steps 1 and 2. Note that if the auxiliary state has the same set of associated options as any of the currently existing states we do not need to make a new state, instead we use the already existing one as the end for this option. We continue this process until we find out that there are no more states that have not been considered yet, this means that we have finished the process and there are no other possibilities that could be derived from the options detected.

Sidebar: Notes.

We will put some notes on the algorithm here. If there are some options which have not been used at all till the very end it can mean two things: either this option is not relevant to the investigation object or on some stage this option has wrongly not been associated with some state. This mistake is corrected by analyzing the financial essence of the option and all the states. The options which are not incorporated into the algorithm are called dangling

(analogous to dangling pointers in the programming). The amount of states is always finite because the amount of sets constructed of the finite number of elements is finite and every two states have different set of associated options. The final state is the only one that has no associated options and represents selling all assets and quitting the business; see the sidebar to the previous section devoted to the residual value for dealing with this option.

Of course, there will almost never be an automaton like shown in the scheme with four states only. Depending on the particular object we can expect the automaton to have six to twelve states and twenty to thirty transitions which represent ten to twenty options (as we have seen, one option can generate several transitions, especially if these are loops; at the same time several options could sometimes be united under the same transition). If the object of an investigation is a small private company the amount of options is usually much less and therefore the automaton will be simpler and the computations will not require much time. On the other hand, for large companies and especially multinational corporations and highly- diversified businesses the projects often come along with an enormous amount of different options; in this case the researcher should at the very beginning range these possibilities on the scale of their importance and consider only the most crucial of them. We will return to this topic in section 2.6 when discussing the troubles which might occur when using this algorithm.

Throughout this entire chapter we will study a full example of valuing a closely-held company, ―Trick Track‖. All the data for this company is imaginary but plausible. We do not

46 use a real company data to avoid the excessive complexity of the example and concentrate on the concept rather than the calculations. We will introduce different pieces of data when they will be required for a corresponding step of the algorithm.

A private company called ―Trick Track‖ produces navigating devices for the cars. The factory works on its full capacity and the net income for the previous five years has been equal to ̌ a month (the difference from year to year is not statistically significant, net income after the normalization). The analysis has detected three possible options: an advertising campaign, expanding the factory and accepting to produce the navigation devices on the government order. If the last option is chosen the net income will be constant for five years ( a month), then the contract expires. The government order can be accepted on one single date – at the very end of month 4 (assuming today month 1 starts); at this time some specific requirements should be satisfied among which some adjustment of the production lines is. If the company chooses to expand its factory we assume for simplicity that the capacity is increased twice and the amount of additional sales is 60% from the initial, so the new factory will work on 80% of its capacity. At any moment the additional belt can be removed and all the equipment sold, if required. There is a possibility for further expansion but the analysis has shown that there is no positive effect from this action, so this option is not considered. The advertising campaign will cost some money but the effect is unpredictable – the additional amount of sales is characterized by the random value . All the amounts not stated in the text are described under the automaton.

Figure 8. Trick Track – Automaton.

Trick Track – Automaton

횯 푝 (36 5푡 3) S

푝

(

4 4

)

휀

푡

(

푡

)

푝

)

푚 ( 4 ),훼- B A

The automaton for ―Trick Track‖ is shown above. We will now explain the process of its creation and at the same time have a closer look on the internal structure of the algorithm.

We start from two states, and , and put down all three options we have (we say ―three options‖ here missing the quitting option, it exists in most cases and we always put it at the very top of the list with number 0):

0. Shutting down

1. Government order

2. Advertising

3. Expanding

To make the following notes more compact let us use the following notation. If we need to say, for example, that option 2 leads from state S to state P we will write and if option 2 is not associated with state S we will write it as . These notations will help us to structure the information better. We start the analysis with determining the options associated with S.

The description of the current state says that the factory is working on 100% of its capacity, therefore the advertising campaign is just a waste of money because we cannot fulfill the demand if it increases, .

We are able to shut down at any moment. We assume that the assets today are worth

$3.6M and the depreciation is $15k per month, so if we quit at some point of time the amount of money we can sell all assets for is equal to 36 5 . The process of selling all the assets lasts three months; though it does not affect the total value of the company we incorporate this number into our calculations for uniformity of notations.

Therefore we draw an arrow from state S to state and sign it as (36 5 3) All the transitions with the key of 0, if they do exist always end in , so we write . As we have already noted is the final state and does not have any options associated with it so we do not need to bother about this transition any longer.

Being in the initial state we can use the option to expand. Here we should use the algorithm presented in this chapter to determine whether this transition is a loop or it leads to a new state. Though in theory we should look at the whole set of the options associated with the initial state and the auxiliary state X here the question is much simpler – we have already found out that , but definitely if we build a new production belt there are no more problems with the advertising campaign, so and we see that X differs from S in at least one position, which is enough to clarify that these are two different states. We rename X as B and draw an arrow 3 We suppose that all the expenditures on the new factory will cost a total of $5M and it will take 8 months from the actual decision to extend to the new conveyor starting production, so the arrow from S to B is signed with (5 ). The numeration of the transition functions does not matter – the main thing about them is that they should be always connected with the corresponding states.

In the initial state we can accept the government order. For simplicity we will assume that the government order requires full capacity of our production lines and one of the terms of the contract is that we do not expand the production until the order expires, so the end state for the transition is distinguished from the beginning state by the following: 3 but 3 ; therefore X is a new state which will be denoted as A. It will cost us $240k to adjust our machinery so that it satisfies the government requirements and these adjustments will take 1 month, so the arrow from S to A is denoted as ( 4 ). State A is unstable, which means that it is not always available for us; the only moment in time which we can use for transition is – four months from now, so if we decide to go this way we must start readjusting our belt at the beginning of month 4 so that when the work actually starts everything is ready. For simplicity we ignore different procedures connected with the government orders and assume that there is no competition and this contract is only for

―Trick Track‖.

We have considered all the options with respect to the initial state and discovered two more states, A and B. Step three of the algorithm prescribes us to repeat the same procedure for each of these. Let us start with state A.

If the company is in state A it means that the work on government order has been accepted. We have already mentioned that there is a condition of no expansion so 3 , we are not capable of performing an advertisement campaign because we are not producing anything for the customers, , we cannot take more than one government order, ; finally, we cannot shut down while performing the government order because our contract does not allow us to stop the production, . This situation is called a trap, it is very rare and in most cases temporary. Here, for example, even though we have detected no outgoing arrows from A we know that the contract will expire in five years. This is another kind of a transition because it is mandatory; we use the dotted line to show this difference. We assume

50 that the return to the initial state S requires neither time nor money, so it is a blank transition

(denoted as ). After this example is finished we will look at the classification of transitions in detail.

State B has much more possibilities associated with it. In addition to the options we have detected when we started this analysis there is one more option, which is sometimes hidden behind other ones. If there is a way from S to B the first thing to do is always checking the availability of a backwards transition. We have not written it down initially on purpose: almost never in real analysis all the possible options will be uncovered, but in most cases the process of creating an automaton itself will reveal the missing ones and the final system will be closed in a sense that all the possible actions were considered. If such a situation happens, the missed option is called dual to the one which generated the initial transition, we will write it as 3 and by definition if 3 then 3 . Note, however, that in some complicated situation the same option might have several different dual options and they are not related to each other; it is a researcher‘s concern to do all the process carefully and study all possibilities, because if the input has at least one flaw the result might have no connection to the actual situation. We can sell the equipment for its initial price less depreciation ($25k per month) accumulated during the period of usage, which is equal to the time the company has been in state B (we denote it here as ).̅ The process will take three months. The arrow from B to S is signed with 4 (5 5 ̅ 3).

Nothing can prevent us from shutting down the business after the expansion, so there is a path , which is denoted as . We assume that there are no cross effects and we can sell all the assets we would have in state B for the sum of the price of new equipment and old assets, of course the algorithm will not change if the price would be different. Three months is the amount of time required for this transaction and the arrow from B to S is signed with ( 6 5 5 ̅ 3).

After one expansion we can have another one as well, but the research of the market shows that there is no demand and another expansion will be deeply detrimental; therefore

3 .After finishing with this example we will see what could have happened if we did not make this assumption and supposed that another expansion was not unprofitable.

We have already discussed the relationship between the expansion opportunity and the government order and found out that if the government order is taken no expansion can be performed until it matures. However, this requirement alone does not prevent us to accept the government order after the expansion. This opportunity is blocked for another reason: the government order can be accepted in one single moment of time, 4 months from now, and the expansion lasts eight months, so even if we start it right now we would not be able to finish until the required time. Therefore . This analysis can be sometimes performed automatically but in more complicated cases it is the researcher‘s duty to check the consistency of the transition and their actual feasibility. In this case it would not be a problem to include the transition 3 because it would be blocked anyway due to the constraints

(we will talk about these in the next section), but in other cases such transitions can increase significantly the amount of calculations and therefore make the whole algorithm‘s performance much worse.

Finally, we now have enough room to increase the production in case of the demand growth and the advertising campaign is a real opportunity. By simple reasoning we can conclude that the advertising campaign does not affect the possibility of any other transitions, so this option generates a loop: . Let us assume that the advertising campaign costs

4 and lasts three months, after it is finished another one can be started at once so the amount of the advertising campaigns is virtually unlimited. We denote this transition as

( 4 ), - (m here stands for modification meaning that the transition is modifying the state but there is no change of states). One additional symbol which we have not used before

52 is a letter in square brackets (, -) after the cost and length; this is the random variable associated with this particular transition and determining its effect. For this situation we will assume that the effect of the campaign (measured as a percentage of additional net income) is distributed as ( 5 3) – the mean increase in the net income is 5% and the deviation is

3%.

We have revised all possible transitions and created an automaton for our model company. For the purpose of clear understanding we have described each transition in detail, but once a researcher has enough experience it would take him about five minutes to create the whole automaton for such a small company. If we drop all the figures the final automaton can be written down as follows:

( ) ( ) 3 ( 4) ( ) ( ) 3

For the computer this line is coding an automaton we have just created.

3.2 Decision Rules and Constraints

After creating an automaton we need to define all the necessary information for its work. Everything we need has already been collected, now we need to rearrange the information in such a way that it will be recognized by the computer.

A constraint is a specific condition which prevents something from happening. It can be put upon any transition or state, in some cases it can be a global constraint affecting several states or even entire process of calculation. Constraints are fixed conditions and are used in those cases when the origin of the constraint is a regulation or the outer condition.

A decision rule is a much more flexible and complicated unit of an algorithm. They are usually representing the desires, wishes, any additional requirements which cannot be transformed into a logical construction or for which this transformation requires an extremely complicated analysis. One of the most important applications of the decision rules is the

53 outcome structure; because the outcome is flexible using the proper decision rules will result in having additional information.

The algorithm of creating the decision rules is following one for creating an automaton step by step. The first constraint is always the horizon, the first rule is always how the residual value should be dealt with. Both of these points are extremely significant and are associated with the final state . After defining these we continue with each of the states and then with each of the transitions.

To understand the concept of the decision rules and constraints better let us continue the example we used in the previous section. The first global constraint is the observer horizon – we will assume that for ―Trick Track‖ it is equal to 9 years. The first decision rule is dealing with the residual value, by default we use the mandatory transition principle but in some cases there are much better ways of estimating the residual value. Here we will use the default option.

We start with the state S. Does it have any constraints or rules associated with it?

From the text we see no boundaries whatsoever, so the constraint section is blank, nether are there any rules (we assume that there are no problems with the social or ecologic responsibility and the only ultimate goal of the company is creating value). For state A the constraint lies upon its existence – the order expires in five years and we should take this into account. For this purpose we will need to create a new variable, called , and use it as a counter. Once we transit to state A this variable starts decrementing itself from the initial value of 60, as soon as it gets equal to 0 the trigger is initialized which automatically brings the automaton back to state S (this is how the mandatory -transition is dealt with).

For state B there are no constraints, but there is a decision rule which determines whether any more advertisement campaigns should be conducted. It can be simple or not, for our purposes let us make is as plain as possible. We have supposed that the effect of one campaign is

54 normally distributed as ( 5 3) and the production lines in state B are working on

80% of their capacity. Then the rule will be as follows: after each advertising campaign we calculate the probability that the total demand will increase the utilization of the production lines to 100% or more, this is a finite value between 0 and 1 stored in the variable called

; if it is above, say, 0.25 we stop conducting these campaigns, while it is lower than 0.25 we start another campaign once the previous one ends. This is, as we have said, the easiest version of the decision rule, the more complicated ones will include comparing the increase in net income with the cost of one campaign and base the decision to stop on the marginal utility principles. Here we see the difference between the constraints and the rules: the constraints could be dealt with one single way only and there is no freedom of choice while the decision rules can be written in lots of different ways for the same situations.

After finishing with the constraints for the states we have to deal with the transitions.

In our case transition ( ) requires a constraint which will bind its availability, it will state that this transition can be committed at one point in time only, 4. The pack of rules constraints is tightly limited with the speed of the algorithm and its productivity; we will discuss these in the last section of this chapter.

Let us now look at the constraints and rules defining the final aim. After the computer builds all possible routes it calculated the distribution of the present value for each of them and stores in the memory. The problem here is that we cannot compare two functions directly, when we say that ( ) ( ) we implicitly mean that for any value of the argument the first function provides higher value than the second one; however, two functions ( ) and ( ) are incomparable on , - because for the first function is bigger than the second one and for the situation is the opposite. As there should be some precise output we need to transform all the functions into the quantities by using some functional on them (we will denote the functional over the probability distribution

55 which transforms it into a single quantity for the purposes of comparison as ( )). After this operation the heaviest path is the one for which ( ) has the greatest value. The examples of this functional can be:

 The simple mean, ( ) 〈 〉, is used almost always and serves as the

default.

 The quantiles, which are defined as such values of argument which distinguish

the distribution into two parts. For example, the first quartile is the value

which the random value with some propability distribution ( ) will exceed

with the probability of exactly 75% (or not exceed with the probability of

exactly 25%, which is the same); these values are calculated using the

cumulative distribution functions. The median is the 50%-quantile by

definition. We will denote these as ( ) ⟦ ⟧ for -quantile. If we use the

high quantile (higher than 0.5, for example 65), the distributions which

are right-skewed and ones with higher standard deviation will be preferred to

the normal ones and with low deviation, so this can be used if the management

is ready to take high risk; on the contrary, the pessimistic 3 will result

in choosing the distributions with lowest deviation.

 The linear combinations, for example, ( ) 〈 〉 ⟦ ⟧ ⟦ ⟧ . This is

an advanced technique and should be used in some very specific cases.

 The external joins. These can be used in case the ultimate goal of the company

is not only maximizing its value but also something else, like environmental

protection. In this case one or several variables are used throughout the whole

algorithm and store some value which will be taken into account when time

comes to choose the best path. If the company wants to lower its carbon

dioxide emissions to the atmosphere we can use the variable E to store the

amount of these emissions; then a final function can look like this: ( )

〈 〉 , where is the value and is the factor showing our amount of

concern about the emissions, the higher it is the more we are going to pay for

reducing them. The external join can include as many factors as seem

important to the decision maker.

3.3 Computations

After two previous steps are finished it is the time for the algorithm to do its job. In this section we will describe the algorithm in general: its idea, formal description, input, output and interpretation of the output.

Input:

1. Service data: the observer horizon , duration of one period or amount of periods in one year , the final aim, the constraints , decision rules

2. External data: the discount index ( ), additional external variables and forecasts for them * +

3. Internal data: the forecast for cash flows ( )

Output:

The distribution of the value and the heaviest path.

Interpretation of the output:

Under all the constraints, conditions and with all the estimates which were put in, the algorithm has determined the route with the highest decision value and the distribution of the value for this route. Based on these data the analyst should make his final valuation.

Idea:

The main idea can be reduced to the analog of the travelling salesman problem. We have the automaton as a map and the states representing the cities, we have the finite amount of time and we need to travel in such a way that maximizes the overall profit. Fallen trees on the roads are the constraints, desire to visit some particular city is a decision rule. Of course, there are dozens of additional factors in this model and each of them affects the path and must be taken into account, but globally this type of tasks has already been studied for quite a while and some of the results should be used in designing the program.

Formal Description:

At the very beginning the automaton is in the initial state, all the service variables are equal to zero, the current state is checked for satisfying the constraints. Then for each time step all possible actions are analyzed (including the null-action which is just waiting and all the possible loops and transitions), for each of these actions the effect (the total present value distribution of all the cash flows from the start till the current step) is calculated and stored in the memory. After the last step is performed the algorithm is compulsorily transited to the final state (if no other rule for dealing with the residual value has been used), the cash flow from this operation is added to the previous sum and then the highest of all the values is treated as the value of the company/project and the path to it is along with the distribution of this amount is the outcome. On each step one or more service variables can change their values depending on the structure of the constraints, decision rules and internal conditions of the algorithm, ultimately the only thing being affected by these changes is the distribution of the cash flow on the current step and the following ones.

3.4 Stability Analysis

So far we have been dealing with the algorithm somewhat loosely, especially when we treated the input and the rules as if they were precisely correct. Some important actions were performed to deal with the probabilities, but still there are lots of variables which cannot be treated that way. Let us look, for example, at the role of the discount index in the calculations.

The discount index ( ) does not influence the cash flows unless some rules or constraints have been prescribed which create the direct dependency; its role is huge on the last step when the final distribution is calculated. Predicting the discount rate is not an easy problem anyway, so the probability of making an error is in fact very high; therefore we need to know to what extent an error in this value will influence the result. The theory provides us with the following terms and definitions.

Let us consider the initial estimate, ( ), and the real discount index ̿( ) (we have mentioned previously that the discount index in unobservable but for now let us imagine that we know for sure what its value has been for the past). We will say that the algorithm is stable in strong sense if the final outcome for both cases has the same path and the following inequality is true: ‖ ̿‖ ‖ ( ) ̿( )‖, where is the distribution of the value (the outcome function), is the coefficient determined beforehand and ‖ ‖ is the norm for the corresponding functional space. The algorithm is called stable in weak sense if the path is the same for both cases. As we see from here, the definitions provide useful information but it is only actual when we already know the true value of the discount index; nevertheless there is a method of conducting the stability analysis at the very beginning. From the financial point of view we are interested in the weak form of the stability because it means that regardless of the actual events the course of actions remains the same and therefore we do not need to take the differences between the actual and the predicted value into account.

Let us use an example to show this idea. Suppose the discount index is ( ) , which holds true for a constant discount rate of per year. If the horizon is small this assumption is acceptable. We are not completely sure that the discount rate will actually remain on this level so we perform the stability analysis using the following concept. Along

with the initial discount index, ( ), two other indices, ( ) and ( ) are considered, . Then we calculate the area of weak stability, the minimum value of

and the maximum value of , for which the algorithm provides the same path. It is done using the result of the computations so no recalculation of the entire algorithm is required.

The graph below shows the outcome of the stability analysis.

Figure 9a. Stability Area

The red line represents our initial forecast of the discount index, the green line is for

( ) and the blue one is for ( ). The area between these lines is the weak stability space, if the actual discount index never crosses any of these we can be quite sure that the path will remain the same (this situation is shown on the graph below).

Figure 9b. A Stable Result

The stability analysis is performed over each variable which has a significant probability of being predicted wrongfully. This analysis is made on request of the researcher automatically but requires some skill for its outcome to be treated in a proper way.

2.5 Computation Time and Optimization

Apart from the complicated structure of the algorithm itself, which requires deep knowledge of both finance and mathematics, there is also a huge problem with the computations. If we denote the average amount of options associated with each of the states by and the number of periods for which the result is calculated by the algorithm will require about ( ) operations to finish. If we have annual data, our horizon is 10 years and the average amount of options per state is equal to 3 then the algorithm will provide the result in no time: 4 operations will take less than a second if the computer frequency is one billion operations per second (1 GHz). At the same time, if we have the quarterly data for the same 10 years and the amount of options per state is 5 we have a problem: 64 operations will take more time than the current age of the Universe is. On the other hand, as we have already mentioned increasing the length of a period results in significant decrease of the quality of final result. The aim of the researcher is to maintain the equilibrium between

61 the quality and the speed; some of the optimization techniques will reduce this dramatic difference and make the calculations be performed faster.

The first way to deal with never-ending calculations is the so-called method of ballast.

To use it we need to rank the states as ―best‖ and ―worst‖ depending on the cash flows they produce; in most cases it is actually clearly seen after the automaton is created that some of the states will never be reached because they provide much less net income that the others, these are labeled the ―worst‖ states and placed at the bottom of the rating. If the algorithm requires unacceptable time to finish the researcher can start dropping these states and all transitions leading to and from them, one by one, until he believes that all the states left are important and deleting them might cause the mistake in the result. The outcome of this method is very significant: in the example we had with 5 transitions per state and 40 steps, if we assume that there were 8 states in total, deleting one state will on average decrease the amount of transitions per state by 0.5 and the total estimate of time will be 5 54 – we have decreased the amount of calculations approximately 100 times.

The second way of solving this problem is called the weakest link method. It is much more complicated than the first one and its idea could be reduced to the following: the system analyzes the loops containing two or more transitions (something like ) and breaks the most of the links. If we consider the ―Trick Track‖ example we remember that there is a loop , which represents the possibility of expanding the production line and shrinking it back again; even though it is possible it does not make sense unless for some very specific circumstances (like new laws which limit the amount of production lines), so the system basing on the conditions and decision rules can delete one of these transitions which is worse in effect. If the calculations lead to a result that the transition is worse than there will be no way back from state S, so the amount of the possibilities will become a little less, but in case is worse state B can no longer be achieved and should

62 be dropped with all income and outcome transitions; if this happens it means in fact that state

B is the ―worst‖ using the terminology of the previous method, so these techniques are connected to each other and share the same general idea.

If all the possible states and transitions have been dropped and the result is still unacceptable the final optimization method is used which is called ―now or never‖. If we look at most of the transitions we will see that the overall effect of their application is either an increasing or a decreasing function during the entire horizon. For example, the earlier we quit the production, the more our assets will cost because the less will their depreciation be but the more will be the opportunity cost; we need to compare these two quantities and in most cases we will find out that the alternative of quitting comparing to continuing production generates a stable negative net income (if we continue production we will earn much more than is we sell all assets and quit). This will be a ―never‖ situation – this transition destroys value and is deleted from the automaton (notice here that this does not hold true for the mandatory transition to the -state at the very end; even if the quitting alternative is a ―never‖ we still make this transition regardless of any parameters). The ―never‖ transitions are colored red.

On the contrary, the expansion should be made as soon as possible if the market is growing rapidly because there will soon be the uncovered demand which must be taken advantage of; these are ―now‖ transitions and they are marked green, the meaning of it is that we transit to some state and there is a green arrow leading out of it this action is chosen automatically at the next step of the algorithm. After all the arrows are colored red, green or black the automaton acts as follows: it ignores red arrows, interprets green ones as mandatory transitions and the black ones are treated the same way as usually, the optimization is achieved because both green and red arrows are not the branch nodes and the amount of calculations in some cases can drop by half of the current power – from 4 to , for instance.

CONCLUSION

A great scientist Enrique Fermi once said: ―Ignorance is never better than knowledge‖, and from my point of view it is an appropriate time to repeat this statement once more. The financial engineering era has begun about two decades ago, when the personal computer became affordable for companies and people. The possibilities it brought are almost limitless, starting with absolute precision of calculations and ending with such software products as SAP and Oracle databases, but in the valuation sphere the techniques remained the same. They are passed down from generation to generation and even the most principal books more that thousand pages in length (like Pratt‘s ―Valuation‖) do not mention any of the new approaches. The businesses become more and more complicated, the frauds are amazing us with the virtue they are committed with and in such a situation new approaches should be used to maintain the quality of final decision at the proper level. The approach described in this paper is no more than a draft, it requires lots of additional improvements and adjustments to fit most of the real-world situations, but this is one of the ways to make a huge step ahead in the process of evaluation.

In this study we have reviewed some of the most common methods used for the purpose of evaluating the closely-held companies and investment projects and created a new method being significantly different from all the rest both in methodology and in requirements. It consists of several steps and requires much more calculations and initial information, what we gain is a very thorough analysis which is both detailed and adjustable for each particular case. A specialist has to a professional in lots of areas to efficiently use this algorithm, including the probability theory, analytic skills and deep knowledge of the industry, but if the ultimate aim of the whole process is estimating the value of a company or a project with the maximum precision available the method we introduced here allows to achieve much higher results than all the rest. The algorithm we have created has several advantages, which makes it preferable to any other methods. The first main point is that all the calculations are adjustable: the method can cover the overwhelming majority of situations, except the most unusual and non-trivial.

The methodological essence is structured in such a way that there should be much less flaws due to the estimates – as we have seen, we use all the information available without reducing its quantity on any stage and the result is representing the situation with the maximum precision possible. This is the great advantage comparing to the income approach. The outcome of the algorithm is not only the value, but also the course of actions which leads to it, therefore it is a new level of analysis, which is unavailable to any of the methods currently existing. Finally, in spite of looking extremely complicated at a glance, one can find two out of three stages being performed automatically once the data is collected and the automaton with all correspondent items (constraints, decision rules, etc.) are put inside. For a skilled analyst the whole process will take about six to twelve hours.

Let us also point out the implications of the method we have constructed. Of course, it should be used for its initial purpose of evaluation of closely-held companies and investment projects, but these aims do not exhaust the possibilities of its application. If we imagine the situation that the manager of some company has a dilemma of whether to imply some decision or not he can just run the algorithm two times – with and without this decision and see whether it affects the result and how. As the outcome includes the path leading to the highest NPV (or whatever quantity to be maximized) this is a powerful tool for all decision- makers to make a draft of the strategy. Finally, is a researcher wants to investigate some aspect of business model in practice this tool will help him in detecting the most crucial information for the model he is examining.

So far the whole work done is only a concept without any real practice implications.

Before this algorithm could be tested on some real-world situation a big work needs to be

65 performed in the area of classification of the options, methodological principles of estimation and forecasting and programming the whole thing. This is not something which can be done within a month or even a year, so lots of time will pass before the actual testing can be started. Nevertheless, I strongly believe that one way or another the concept introduced here is useful and in case it will be turned into the complete valuation system the valuation process will be more precise, convenient and justified. Therefore the process of valuation will become up-to-date and progressive and might even give a new boost to the overall progress of financial engineering.

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