# Guide to Recursive Models ## With applications from the _®_ ## Rapid Recursive toolbox ### June 2025 Dan Lipsy _President and COO,_ _Supported Intelligence, LLC_ #### Supported Intelligence, LLC 1555 Watertower Place, Suite 300 East Lansing, Michigan 48823 Tel: (973) 800-2024 [http://www.supportedintelligence.com](/content/site-root.html) ISBN:0989083306 _© 2025 Supported Intelligence LLC_ _All rights reserved, except those granted to licensed users_ _under a valid agreement with Supported Intelligence LLC._ * * * # Note on Intellectual Property The Rapid Recursive® toolbox is patent-pending. It is licensed by Supported Intelligence LLC to users under an agreement that protects the intellectual prop- erty of both the supplier of the software and the user. Licensed users have access to a wide variety of information, maintenance and support services, and contrib- uted files. If you are not already a licensed user, please visit the Supported Intelligence LLC website and learn about your options for licensing it. This Guide to Recursive Models is copyrighted. You may not reproduce it except as allowed by the license agreement, which grants licensed users of the Rapid Recursive ® toolbox the right to use the contents of this Guide and distrib- ute it (with copyright notices intact) to other licensed users, but prohibits the posting or public distribution of its contents and the creation of derivative works. Readers who wish to excerpt portions of the Guide in their research or writing should properly cite the excerpt by title, date, and author or editor. “Rapid Recursive” and “Supported Intelligence” are registered trademarks of Supported Intelligence LLC. Other marks (such as Matlab, Microsoft Word and Adobe Acrobat) are registered trademarks of other parties. # Note on the Use and Limitations of Analytical Models This Guide is intended to assist interested people in the general power, use, and features of recursive models; and to provide instructive examples using the Rapid Recursive® toolbox. It is not intended as a comprehensive report on val- uation, accounting, finance or business administration; neither it is intended to convey an estimate of value for any specific business or investment. Readers are reminded that all models represent simplifications of the real world, and involve assumptions and forecasts about the future that cannot be expected to be correct. Supported Intelligence LLC, as well as the editors and contributors of this _Guide, do not possess knowledge of any specific investment or business deci-_ sion contemplated by a reader, and have no financial interests in any such deci- sions. For these and other reasons, responsibility for any such decisions rests solely with the reader. _Guide to Recursive Models_ **ii** * * * # Table of Contents ## Part I: Introduction to Sequential Decision Problems 1 ## Chapter 1: Introduction ... 2 Purpose of this Guide ... 2 Limitations of this Guide ... 2 User Guide and Help Documents ... 3 Special Assistance to Scholars and Authors ... 3 Accredited Consultants; Customized Models ... 3 Respecting Intellectual Property ... 4 Outline of this Guide ... 4 Let us know how you use the Rapid Recursive® Toolbox! ---------------- 5 ## Chapter 2: Sequential Decision Problems ... 6 Sequential Decision Problems ... 6 Examples of SDPs ... 7 Investment problems ------------------------------------------------------------ 7 Entrepreneurial Problems ------------------------------------------------------- 7 Managerial Decisions ----------------------------------------------------------- 7 Real Options Problems---------------------------------------------------------- 7 Life Decisions -------------------------------------------------------------------- 7 Examples of Other Decision Problems ... 8 Advice on Recognizing and Structuring Sequential Decision Problems ... 8 ## Chapter 3: Approaches to Decision Problems ... 10 Possible Approaches ... 10 Four Specific Decision Rules ... 10 The Net Present Value Rule and DCF ... 12 Assumptions in these Methods ... 12 Motivation for the Recursive Method ... 14 Difficulties in Using a Recursive Model ... 14 ## Part II: Guidance for the Use of Recursive Models 16 ## Chapter 4: Problems with the DCF Model: Ignoring Management ## Flexibility ... 17 Real Options ... 17 Failure of DCF in Capital Budgeting ... 17 Prevalence of “Adjustments” and Subjective Methods ... 18 Alternatives to DCF ... 19 ## Chapter 5: The Recursive Model ... 20 Recursive Thinking ... 20 The Bellman Equation --------------------------------------------------------- 20 * * * Table 1. Functions in Discrete-Time, Discrete Variable Case and Continuous Variable Cases------------------------------------------------------------ 21 The Principle of Optimality --------------------------------------------------- 21 Essential Elements of a Recursive Model of Business................................. 22 Non Business Models ---------------------------------------------------------- 22 Questions to Answer ----------------------------------------------------------- 23 Chapter 6: Using the Rapid Recursive® Toolbox............................ 24 First Steps in Composing the Problem......................................................... 24 State Variables for Businesses..................................................................... 24 State Variables for Other Subject Persons................................................... 25 “Importance of Importance” Principle --------------------------------------- 25 Difference Between Econometric and Recursive Models ----------------- 25 Use of Index and Transformed Variables.................................................... 26 Chapter 7: Discounting and Discount Rates..................................... 27 Rationale and Data on Discounting.............................................................. 27 Theories of Discounting ------------------------------------------------------- 27 Data on Rates of Return-------------------------------------------------------- 27 Two Critical Assumptions: Utility, and Time Preference............................ 28 Assumptions in Recursive models: Utility and Time Preferences............... 30 Practical Advice on Discount Rates............................................................. 31 Practical Advice by Subject of Decision Problem -------------------------- 32 Table 2. Suggestions for Discount and Growth Rates............................ 33 Chapter 8: Additional Topics in Discount Rates and State Variable Diffusions.......................................................................................... 35 Timing of Rewards and actions................................................................... 35 Time Index Starts at t=0 ------------------------------------------------------- 35 Cash Flows and Discounting........................................................................ 36 Discount and Growth Rates......................................................................... 37 Note on Continuous Compounding------------------------------------------- 38 Decomposition of Total Return and the HJB Formula................................. 39 Chapter 9: Setting Up the Reward Matrix........................................ 41 What does the “R” Matrix Represent?......................................................... 41 What is the Reward?.................................................................................... 41 Estimating Elements of the Reward Matrix................................................. 41 Chapter 10: General Advice: Getting Data into the Workspace..... 43 Advice on Getting Tabular Data into the Matlab workspace....................... 43 Importance of Checking and Sourcing......................................................... 44 * * * Second Checks: Data Conformance ... 47 Third Checks: Model Composition ... 48 Fourth Checks: Model Calibration ... 49 ## Part III: Selected Solution Templates 51 ## Chapter 12: Selected Solution Templates ... 52 Introduction ... 52 PUrpose and Limitations ... 52 Automaker Market Share ... 54 A. Problem Description -------------------------------------------------------- 54 B. Suggested Uses -------------------------------------------------------------- 54 C. Interpreting the Results ----------------------------------------------------- 54 Beverage Wholesaler Value ... 57 A. Description of Problem----------------------------------------------------- 57 B. Interpreting the Results ----------------------------------------------------- 58 Basic “Black Swan” Analysis ... 60 A. Problem Description -------------------------------------------------------- 60 _Table 4. Reward Matrix for Black Swan Template ------------------------- 61_ B. Results for the Example ---------------------------------------------------- 62 _Table 5. Results for Solution Template “Black Swan” -------------------- 62_ C. Other Suggested Uses------------------------------------------------------- 62 Classic Reinvestment Problem ... 63 A. Problem Description -------------------------------------------------------- 63 Results --------------------------------------------------------------------------- 63 _Table 6. Results for Classic Reinvestment Problem ------------------------ 64_ Machine Replacement Problem ... 64 A. Problem Description -------------------------------------------------------- 64 B. Results ------------------------------------------------------------------------ 65 _Table 7. Results for Machine Replacement Problem ----------------------- 65_ ## Part IV: Appendices 1 ## Appendix A. ## Contacting Supported Intelligence LLC ... 2 Authors, Scholars, Researchers ... 2 Training for Accredited Consultants ... 2 Accredited Consultants ... 2 General Help and Troubleshooting ... 2 Planned Improvements; Warranty --------------------------------------------- 3 Help from Accredited Consultants -------------------------------------------- 3 ## Appendix B. ## Bibliography and References ... 1 An Annotated Bibliography ... 1 * * * Control Theory and Mathematics ---------------------------------------------- 1 Economic Applications: Investment Behavior ------------------------------- 1 Mathematics of Recursive Models -------------------------------------------- 2 Macroeconomics and Agricultural Economics------------------------------- 2 Business Valuation and Forensic Economics--------------------------------- 2 Real Options and Failure of Discounted Cash Flow------------------------- 3 References.......................................................................................................3 Appendix C. Sample Worksheets................................................................................ 1 List of Worksheets......................................................................................... 1 Usage Rights Granted to Licensed Users....................................................... 1 Worksheet A: Subject of Decision Problem-- - - - - - - - - - - - -2 Worksheet B: States and Rewards - - - - - -- - - - - - - - - - - - -3 Worksheet C: Actions and Effect on Reward -- - - - - - - - - - - - -4 Worksheet D: Transitions - - - - - - - - - -- - - - - - - - - - - - -5 * * * **List of Tables, Figures, and Equations** # List of Tables, Figures, and Equations Table 1. Functions in Discrete-Time, Discrete Variable Case and Continuous Variable Cases ... 21 Equation 1. Functional Equation Variables --------------------------------------21 Equation 2. Value Functional Equation-------------------------------------------21 Table 2. Suggestions for Discount and Growth Rates ... 33 Equation 3. NPV Summation ------------------------------------------------------36 Equation 4. Stochastic Discount Factor -------------------------------------------37 Equation 5. Basic Pricing Equation -----------------------------------------------37 Equation 6. Conventions for Discounting ----------------------------------------38 Equation 7. Diffusions, Differential and Integral Forms -----------------------40 Table 3. Importing Tabular (Matrix) Data from a Spreadsheets and Other File Formats ... 45 Figure 1. Input Structure for Automaker Market Share Decision Problem55 Figure 2. Automaker Market Share Code Fragments -----------------------56 Figure 3. Beverage Wholesaler Code Fragments -----------------------------58 Figure 4. Input and Output Structures for Wholesaler Valuation Problem 59 Table 4. Reward Matrix for Black Swan Template ... 61 Table 5. Results for Solution Template “Black Swan” ... 62 Table 6. Results for Classic Reinvestment Problem ... 64 Figure 5. Possible Inputs for Machine Replacement Problem --------------65 Table 7. Results for Machine Replacement Problem ... 65 Figure 6. Graphical Output from Machine Replacement Problem ---------66 **vii** **© 2025 Supported Intelligence LLC** * * * # Part I. Introduction to Sequential Decision Problems **© 2025 Supported Intelligence, LLC** * * * Chapter 1. Introduction PURPOSE OF THIS GUIDE The purpose of this Guide is threefold: 1. To introduce the concepts of sequential decision problems, and the recursive approach to such problems; To describe a step-by-step process to organize the information necessary to 2. To describe a step-by-step process to organize the information necessary to compose a sequential decision problem in a form that can be solved by the Rapid Recursive® toolbox, error-check the model, solve the model, and report the results; and To present a set of such models that represent common decision problems, 3. To present a set of such models that represent common decision problems, and which have already been composed into models that can be immediately solved by the Rapid Recursive® toolbox. Some of these are Solution Templates that are included as part of the software licensed to toolbox users. In addition, this Guide can assist readers to: 4. Identify a set of practical issues—such as the selection of discount factors and state variables—for which there is currently no standard convention. Provide the reader a set of references on the theory of value functional mod- 5. Provide the reader a set of references on the theory of value functional models, real options, sequential decision problems, related valuation concepts, and the underlying mathematics. LIMITATIONS OF THIS GUIDE This Guide is a valuable resource. However, it is not intended to be provide basic instruction in the use of the toolbox, complete historical discussion of the underlying theory, or an accurate assessment of any specific practical decision problem. We suggest the following resources for these purposes: • The User’s Guide is a program reference for the Rapid Recursive® toolbox, and should be consulted for information on the proper command syntax and usage, as well as troubleshooting, licensing, and installation. • The Rapid Recursive® toolbox, version 1, requires MATLAB®, a product of • The published works cited here include derivations of the theory behind sequential decision problems, recursive models, and related concepts such as control theory, MDPs, dynamic programming, finance, economics, and valuation. See “Appendix B. Bibliography and References” on page B-1. * * * **Introduction** ## USER GUIDE AND **HELP** **© 2025 Supported Intelligence, LLC** * * * RESPECTING INTELLECTUAL PROPERTY Bringing the power of “recursive thinking” to our customers required many years of work, and major financial investments by our founders and investors. We ask that you respect this investment and the effort we have placed into developing this product by limiting the distribution of our software to licensed users. We also respect the work that you do, and the importance you place on being able to share the results of your analyses with your co-workers, colleagues, and customers. For that reason, we acknowledge your rights under our licensing agreement, and specifically allow our licensed users to use, adapt, and report the results of Rapid Recursive® toolbox models in a manner consistent with that agreement. OUTLINE OF THIS GUIDE While we ask that you cite the Rapid Recursive® toolbox (and any solution tem- plate you adapted) in any report or document distributed to those who are not 1 1 licensed users. You may also excerpt short passages from this document in your work, provided you properly quote the passage and cite it by title, author, and date. However, any proprietary data you use in such an analysis, and the resulting results and report, remain your property until you distribute it to others. Before setting up a model in the Rapid Recursive® toolbox, it is important to understand the recursive approach to sequential decision problems. Thus, this Guide is laid out as follows: 1. Part I introduces the software; and then discusses sequential decision problems. These are discussed in “Sequential Decision Problems” on page 6. It provide examples of problems in a variety of fields that can be attacked with recursive methods. We then discuss a set of possible decision rules for solving such problems, only a few of which are commonly mentioned in financial curriculum. These are discussed in “Approaches to Decision Problems” on page 10. Part II provides guidance on a variety of topics important to those actually 2. Part II provides guidance on a variety of topics important to those actually using recursive models. It includes a discussion of the serious problems with using traditional discounted cash flow (“DCF”) analysis to evaluate decisions under uncertainty in “Problems with the DCF Model: Ignoring Management Flexibility” on page 17. Practical steps for composing a recursive model and identifying the informa- 3. An acceptable citation would be something like the following: We used the Rapid Recursive® toolbox, a software program from Supported Intelligence LLC, to compose and solve the decision problems described in this research project. The web site for Supported Intelligence LLC is: [http://www.supportedintelligence.com](/content/site-root.html). * * * **Introduction** and growth rates, and certain mathematical conventions is contained in the following chapters, including “Discounting and Discount Rates” on page 27. **3.** Part III introduces a set of selected Solution Templates that are provided to licensed users of the Rapid Recursive® toolbox. These cover various decision problems in valuation, management, finance, risk analysis, and economic policy. Excerpts from these templates as well as output from running them are excerpted here. Licensed users have access to thousands of lines of unen- crypted code in these Solution Templates, that they can adapt and use pro- fessionally and in their academic research in a manner consistent with the License Agreement. See “Selected Solution Templates” on page 52. **4.** A set of appendices, including: - \\*\\*\\*\\* Appendix A. Contacting Supported Intelligence LLC - \\*\\*\\*\\* Appendix B. Bibliography and References - \\*\\*\\*\\* Appendix C. Sample Worksheets _Let us know how you use the Rapid Recursive® toolbox!_ We hope you will find this Guide useful. If you have suggestions for inclusion in future versions, or think you have found a mistake, mis-statement, or paragraph that needs clarification, please contact by mail or email using the addresses listed in the appendix. Also, if you find the toolbox particularly useful, or use it for something particu- larly interesting, or have some of your work published, please let us know. I am sure we will be pleased, and may be able to share any published work with oth- ers. **© 2025 Supported Intelligence, LLC** * * * Chapter 2. Sequential Decision Problems SEQUENTIAL DECISION PROBLEMS Sequential decision problems (“SDPs”) are problems of the following type: • There is a time sequence of events. Information on market conditions, the amount of resources the subject controls, and other relevant facts arrive as events unfold in this time sequence. • The subject must make a decision at one or more points in that time sequence. These decisions are represented by actions of the subject. These actions could affect the current reward or cost, and also market and other conditions that will affect the subject in the future. • At each point in time, the subject receives some type of reward (which might be a payoff, benefit, earnings of some type) or incurs some type of cost. • The subject faces the problem of choosing the best decision, according to some decision rule, based on the information available to him or her at that time. 2 was formalized by the statistician Abraham Wald in the 1940s. Wald’s statistical approach to decision problems spawned many areas of fruitful research in fields such as economics, statistics, natural sciences, finance, control theory, and operations research. 1. Puterman (1994, page 1) describes the model as follows: The key ingredients of this sequential decision model are the following: 2. In particular, Wald (1947) and Wald (1950) formalized sequential decision problems and described approaches to solving them. Many of the subject of Wald’s analyses were statistical decision problems, and these are sometimes called Statistical Decision Problems in the literature. On the intellectual history of decision theory, see Brown (2000). The sequential decision problem is to choose, prior to the first decision epoch, a policy to maximize a function of this reward sequence. 2\. In particular, Wald (1947) and Wald (1950) formalized sequential decision problems A set of decision epochs A set of available actions * * * EXAMPLES OF SDPS Many of the most difficult, important, and common decision problems faced by people around the world can be structured as sequential decision problems. For example, all of the following are sequential decision problems: Investment problems • You have a chance to invest in a company developing a promising technology, which may or may not result in significant future earnings. Alternatively, you can invest in very safe short term bonds that pay a predictable, but very low, return. Which do you do? Entrepreneurial Problems • You are one of the founders of a new technology firm. You and your partners own 100% of the company, which is burning through its capital at an alarming rate. A “angel” investor offers to purchase 50% of the stock in return for an infusion of funds, but also wants a liquidation preference. Do you take the badly-needed funds in return for a loss of control, and a much lower potential payout in the future? Managerial Decisions • You are managing a division of a very large manufacturer of consumer goods in a competitive industry. You know that significant investments in new product is necessary to maintain your current market share, but that size of investment would cost more than your entire current profits. Without the investment, however, your market share will decline. Should you make the investment? Real Options Problems • You are managing a natural resource company (a logging, mining, or similar enterprise) that has the rights to continue to exploit the raw materials in a certain area. However, the size of those reserves are steadily declining. You can sell the land (and lose the rights) for a predictable, and stable, price anytime during the next five years. When, if ever, do you sell? You can observe that all of the decision problems listed above have a time sequence, where information about relevant conditions are revealed over time, and where the manager, owner, or other subject person is capable of taking actions that affect the future course of events and future value of the business or other asset. Life Decisions * * * **Sequential Decision Problems** **EXAMPLES OF OTHER** Sequential decision problems are qualitatively different from many other deci- **DECISION PROBLEMS** sions. In particular, they are different from what are known as “one-off” deci- sions, in which a person makes a one-time decision with no possibility of changing course in the future. 1 Throwing a coin into a fountain is one example (assuming, of course, that you are not going to climb into the fountain to retrieve it!) This is a single event: you throw the coin into the fountain, and it is gone. The sequence stops there. Sequential decision problems are also different from whimsical actions that result in no consequences. An example is choosing between chocolate and vanilla ice cream on a spur-of-the-moment visit to an ice cream parlor, once you have already arrived at the parlor and decided to enjoy some ice cream. This is a decision, of course, but it is difficult to call it a decision problem. One can conceptually broaden these examples to arrive at a sequential decision problem (for example, by visiting the fountain or ice cream parlor every day, and having financial or physical health depend on the decisions made there). However, in this book we will consider only sequential decision problems with non-trivial elements including a time sequence, set of rewards, and decision opportunities that carry with them consequences for the future. **ADVICE ON** The examples above provide the basis for some important practical advice about **RECOGNIZING AND** using analytical tools in general, and decision analytical tools in particular. **STRUCTURING** **1.** Use an analytical tool only when it makes sense to use it. **SEQUENTIAL** The corollary is also sound advice: if a person doesn’t think they have a **DECISION PROBLEMS** problem, face a decision, or have the ability to affect future conditions that are important to him; then it probably doesn’t make sense to model a sequential decision problem for that person. It may be possible for another person, such as a skilled analyst, to identify the necessary components for a formal decision model within the person’s situation. However, one should be very cautious about assuming that other people have the same motivations, information, and objectives as you have. **2.** Be sure that the conditions, rewards, and costs you put in your model are recognized by the subject of the problem Good models capture the major incentives and beliefs of the subject of those models. Therefore, be sure that the rewards, costs, probabilities, and other mathematical elements of your model comport in broad terms with the state of the world as seen by the subject of the problem. If it does not, you may have a great analytical model—but you can’t expect it will explain the behavior of the original subject! 1. The term “one-off” appears to originate in the artificial intelligence literature. See, e.g., Poole & Mackworth (2010, section 9.2). **© 2025 Supported Intelligence, LLC** * * * **Sequential Decision Problems** If the purpose of the model is to analyze a decision problem, then the model and its inputs should reflect the perspective of the subject, not the analyst. 1 **3.** Ensure your model captures the major motivation and opportunities of the subject of the decision problem Recursive models allow the analyst to capture much more complexity in incentives, information, possible actions, and motivations than traditional models. However, it is possible in any model to ignore the true motivations, information, and possible actions of the subject of a problem and substitute those described in a textbook or an academic exercise. Don’t fall prey to this error. Useful models reflect practical conditions and constraints faced by actual people. Reality is often somewhat messy. While you can never capture everything in a model, strive to capture what the sub- ject feels is important. 1. This is another illustration of the “first maxim of business economics:” Remember, it _is a social science._ See Anderson (2004). **© 2025 Supported Intelligence, LLC** * * * **Approaches to Decision Problems** # Chapter 3. Approaches to Decision Problems **POSSIBLE** People with training in accounting, finance, business management or economics **APPROACHES** have been introduced to at least one approach to solving decision problems that involve time, risk, and money. In fact, it may appear from reading the literature in these fields—as well as popular investment advice columns and websites— that there is one universally accepted decision rule in finance. However, before discussing that common approach, we should note that there are many possible methods to address decision problems where risk and uncer- tainty are present. These include random selection, appeals to moral authority, listening to experts, “follow the crowd” and “don’t follow the crowd” advice; as well as selecting choices on the basis of avoidance of risk, seeking of risk, diversification of risk, highest possible outcome, best typical outcome, and least- worst outcome. We will introduce a novel approach that can be used to solve many sequential decision problems. However, first we consider the most common approach. **FOUR SPECIFIC** In this Guide, we will focus on sequential decision problems that involve impor- **DECISION RULES** tant financial and economic considerations, where we assume that the courses of action available are consistent with the laws of society and the philosophy of the subject person. For these, it is worth discussing the following possible approaches to financial decisions, which have been proven to be powerful enough to be applied in a wide variety of situations. Each is very briefly described below: **1.** The DCF method, with the net present value rule This method requires an analyst to identify on scenario for the future, usu- ally involving forecasting future cash revenue and expenses. From this fore- cast, the analyst discounts the future earnings and expenses back to the current day, using a discount factor that (at least in principle) incorporates the risk in the investment. DCF analysis is probably the most widely-used tool for analyzing financial decisions, as well as many managerial, entrepreneurial, and life decisions. Countless loan applications, business plans, valuation schedules, and cost- benefit analyses include variations on the basic discounted cash flow model, and (implicitly or explicitly) rely upon the net present value decision rule. It is discussed further below. **2.** Monte Carlo analysis One response to the defects of the standard DCF model when faced with real options, as well as to the possibility of responses to random events that are likely to occur in the future, has been the development of Monte Carlo meth- ods. Rather than project one scenario and evaluate it, Monte Carlo analysis **© 2025 Supported Intelligence, LLC** * * * **Approaches to Decision Problems** begins with one scenario and then evaluates a large number of randomly- generated paths that are based on the one scenario. These are not truly independent scenarios, but instead are random “trials” that are explicitly based on a single scenario that allows for a different ran- dom outcome in each trial. The decision rules that are followed after Monte Carlo analysis can be based on selecting the best likely outcome, the best outcome on average, the least-worst case. **3.** Financial option valuation methods Since the 1970s, an analytical method for pricing certain pure financial options in complete markets has been widely available. As in the classic Black-Scholes-Merton formula, option pricing methods provide a price that is determined by an “intrinsic value” based on net present value, plus an “option premium” based on the likelihood that a randomly varying price will end up “in the money.” These methods can be used only for traded securities where option contracts are widely available, prices are quoted on a regular basis, investors can borrow and lend easily at known rates, and counter-party risk can be largely ignored. However, in these cases, option pricing methods are very powerful, and avoid the failings of DCF models in valuing these assets. Option valuation methods imply a decision rule to exploit any arbitrage opportunities, meaning selling or buying whenever market prices deviate substantially from the analytical price. **4.** The market method. The basis of the market method is the principle that similar traded assets should trade in the same market at a similar price. This method is only avail- able where a specific security or asset is similar to others that are traded on a market sufficiently frequently to establish a buying and selling price. When data on comparable asset sales are available, and the asset is actively traded, this is a powerful tool. As with option pricing methods, the market valuation method implies the exploiting of arbitrage possibilities. **5.** The “value functional” or “recursive” method In the 1950s, a theoretical approach to evaluating sequential decision prob- lems was first outlined. This approach involves the specification of a func- _tional equation that incorporates both state and action variables, as well as a_ time index and transition probabilities. The functional equation often has a _recursive structure, meaning that a multi-period problem is formulated as a_ series of two-period problems that form a mathematical recursion. This method requires an analyst to identify and forecast future cash earnings or other benefits for a set of possible actions and market conditions, as well as a discount factor and a transition function or set of transition probabilities that specify how market conditions change over time. The value functional approach is also distinguished by its decision rule: the maximization of value, rather than current-period earnings or the highest possible discounted earnings from any one scenario. **© 2025 Supported Intelligence, LLC** * * * **Approaches to Decision Problems** As stated in these very brief descriptions, certain types of investments, securi- ties, characteristics of markets, and institutions are required for some approaches. Furthermore, all of these approaches place demands on the analyst to estimate, forecast, identify, and calculate various factors. **THE NET PRESENT** The most widely-cited decision rule in economics, finance, and business man- **VALUE RULE AND** agement is known as the “net present value rule” in finance and business, or (in **DCF** economics) the “neoclassical investment rule.” That rule is to maximize the expected net present value of any investment opportunity. Versions of this “rule” appear in almost every standard corporate finance textbook, an entire stream of literature in economics, as well as in finance literature dating back at least to the Modigliani-Miller articles of the late 1950s. 1 Following that rule can appear to be deceptively easy: simply project out the future cash flows for any investment (using your best estimate given the infor- mation at the time), and then discount it back to the present date. If the net pres- ent value of those expected cash flows exceeds the cost of the investment, then undertake the investment. If not, choose another use for your funds. The prac- tice of discounting cash flows in a manner consistent with this decision rule is so common that it has its own acronym: DCF. **ASSUMPTIONS IN** While there are many wrinkles and complications involved in each of these **THESE METHODS** tasks involved in all the methods listed above. Furthermore, these and all analyt- ical decision models involve some assumptions. The following critical assump- tions are worth noting: **1.** The DCF method assumes that the investor has the capital necessary to make the investment, and has a known discount rate. The former is often true, although the latter is often a suspect assumption. To some extent, these assumptions are shared across many methods, although vary in importance among them. **2.** Furthermore, DCF relies on the assumption that analyzing one scenario (or, perhaps, a handful of scenarios, each with its own cash flow schedule) is adequate as a basis for evaluating the decision. As is apparent by testing the DCF model with any investment involving real or financial options, this is often not true. 1. See Anderson (2013) for citations on the ubiquity of the NPV rule in business and in the academic literature, and the intellectual origins of the “rule.” Anderson (2013) also cites studies showing the failure of business investors to actu- ally follow it. See Schwartz & Trigeorgis (2001) for an assembly of classical papers on the failure of the NPV rule in the face of “real options.” Complete citations to these works are in “Appendix B. Bibliography and Refer- ences” on page B-1. **© 2025 Supported Intelligence, LLC** * * * **Approaches to Decision Problems** The failure of the DCF method to consider management flexibility—the ability to change course in the future—is probably the most damaging to its use in critical situations. We discuss this further in “Guidance for the Use of Recursive Models” on page 16. **3.** The DCF method also relies on the assumption that the investor is just as happy with receiving income 2 and 3 years from now as tomorrow, as long as the amount received is bigger (by an amount represented by his or her dis- count rate) in the future. This is known as “time separable utility.” However, very few investors (or business managers) are actually indifferent to the timing of future expenses or earnings! (See the discussion in “Discounting and Discount Rates” on page 27, and in particular “Two Critical Assumptions: Utility, and Time Preference” on page 28.) **4.** Similarly, the market method (and the option pricing method) relies on the key assumption that there are enough trades of similar assets to establish a market price for the subject investment. This is often true for commodities such as oil and gold, and close enough to being true to make the method very useful for pricing many used cars and trucks, houses, and commercial real estate, as well as some kinds of collect- ible goods and certain well-identified businesses. It is not true for most other types of assets, notably including most entrepreneurial and closely-held businesses. **5.** The Monte Carlo method assumes that a relatively simple structure of deci- sion nodes and rules can be identified in advance, and that the underlying risk distributions can be well-defined by a standard statistical formula. Under these conditions, Monte Carlo by itself provides a richer analysis of one scenario; Monte Carlo combined with a decision tree (or “event tree”) provides a richer analysis of a handful of scenarios. The assumption about well-behaved risks, which can be modeled using stan- dard statistical distributions, might be adequate in certain situations. How- ever, it is clearly not true for most intellectual property, almost all start-up businesses, and many closely-held firms, not to mention investments in new technology, many natural resource leases and rights to segments of the com- munication spectrum, and companies with significant real options to grow or facing substantial risks that may imperil their future viability. Furthermore, the limitations of an event tree become apparent as soon as you consider more than a few possible states and actions that could be undertaken, or more than a few time periods. Again, no analytical method avoids relying on at least some assumptions. How- ever, some methods require assumptions that are often heroic and frequently are clearly incorrect. Others rely upon much milder assumptions with smaller pen- alties involved for minor violations of assumptions. These are worth noting when selecting a model. **© 2025 Supported Intelligence, LLC** * * * MOTIVATION FOR THE RECURSIVE METHOD One of the methods listed above is at least as powerful as the others, and arguably relies on fewer unrealistic assumptions. It is the value functional or recursive method, and it is the focus of the majority of this Guide. The major reasons why a recursive method is worth pursuing are: 1. As noted above, it does not require certain assumptions that are infrequently true in actual practice. For example, it does not require an assumption of time-separable utility, nor of low transaction costs, perfect information about the market today, or that the subject is widely-traded securities. It does not require the assumption of symmetric distributions of possible 2. It does not require the assumption of symmetric distributions of possible outcomes, such as the normal distribution, or even a “bell curve” shape of possible outcomes. Indeed, it can explicitly model asymmetric risks, including the risk of an unusual and potentially catastrophic “black swan” event. It can recognize explicitly the existence of managerial choices, including 3. It can recognize explicitly the existence of managerial choices, including real options, and natively incorporate them in the decision model. It can incorporate information in a formal, objective manner that is not con- 4. It can incorporate information in a formal, objective manner that is not considered in other models. For example, an analyst may be aware that a real estate parcel could be sold for a known price, but cannot include that informa- tion in a DCF analysis unless the analyst assumes it is sold. DIFFICULTIES IN USING A RECURSIVE MODEL While the recursive model is powerful and has significant advantages over other methods, it also suffers from some disadvantages that have formed significant barriers to entry. In particular: It is novel. Until very recently, the entire concept was obscure except in a 1. It is novel. Until very recently, the entire concept was obscure except in a fairly narrow set of research and academic settings. Because it is novel, there are far fewer guidelines to usage and examples, not 2. Because it is novel, there are far fewer guidelines to usage and examples, not to mention well-established teachings, than exist for all the other methods. The underlying logical paradigm is different from that of the standard DCF 3. The underlying logical paradigm is different from that of the standard DCF model, and involves concepts that are often not highlighted in other approaches. Usage of a recursive model therefore requires an analyst to learn a bit of a new “recursive language” as well as understand a new model. Even though there are typically fewer assumptions about markets, assets, 4. Even though there are typically fewer assumptions about markets, assets, and investors required by for a recursive model, one still must make some assumptions. 5. Until very recently, no commercial software existed that allowed interested Fortunately, much progress has been made over the past decade on the first ® three barriers. With the public release of the Rapid Recursive Toolbox by Supported Intelligence LLC in December 2012, the latter burden was finally overcome. This Guide is intended to describe how the Rapid Recursive® software can be used in practical situations, and to briefly explain the underlying logical paradigm of a recursive model and introduce some of the elements of the recur- * * * **Approaches to Decision Problems** sive language. It addresses the remaining barrier, although it does not com- pletely overcome it. In addition to the advice in the main part of this Guide, the references included in an Appendix provide extensive theoretical and mathematical background on this and related topics. **© 2025 Supported Intelligence, LLC** * * * # Part II. Guidance for the Use of Recursive Models **© 2025 Supported Intelligence, LLC** * * * **Problems with the DCF Model: Ignoring Management Flexibility** # Chapter 4. Problems with the DCF Model: Ignoring Management Flexibility In addition to the inherent problems of forecasting future events, the DCF meth- odology has a serious flaw in cases where the investor or manager has the power to change course in the future. This is most easily seen with real or financial options. An “option” is an oppor- tunity, but not the obligation, to do something. A financial option (such as a call option) is the right, but not the obligation, to purchase or sell a traded security for a certain price during a certain time period. Such financial options typically have some value, even when they are “out of the money,” because they provide insurance against other events, act as a hedge against other risks, or provide the chance to earn money (or avoid losses) under certain circumstances. **REAL OPTIONS** “Real options” are opportunities to buy, sell, wait, reinvest, scrap, or take other actions involving real assets. The term “real options” was coined by Stewart Myers (1977), and the existence and value of real options in business situations has been extensively studied by Kester (1984), Dixit & Pindyck (1994), and in the papers compiled in Schwartz & Trigeorgis (2001). (See complete references in the Appendix.) Both real options and financial options derive some of their value from the pos- _sibility that events will produce a valuable outcome for the investor in the_ future, even if such events do not occur. Because the discounted cash flow methodology relies on projecting the most likely scenario, it inherently ignores other scenarios. If other scenarios are very similar to the one forecasted, there may be little loss in ignoring them. However, if they are markedly different—as is often the case with a real option to buy or sell—the single-scenario DCF model can provide very bad advice. **FAILURE OF DCF IN** DCF methods are the standard approach to evaluating investment opportunities **CAPITAL BUDGETING** in large businesses. (See the survey in Anderson \[2013\]). However, the failure of the net present value criteria in the “capital budgeting” decision of companies facing real options or asymmetric risks has been apparent for some time. (In portfolio analysis, the failure of a pure NPV rule was noted at least as far back as Harry Markowitz \[1952\], who proposed adjusting the expected NPV for risk, creating the basis for the mean-variance analysis.) The failure of DCF was noted in the economics literature at least as far back as the 1970s (including by Myers \[1977\]). By the 1990s, Dixit & Pindyck (1994) concluded that the DCF model and NPV rule were inherently flawed: Hence the simple NPV rule is not just wrong, it is often very wrong. **© 2025 Supported Intelligence, LLC** * * * By the 2000s, the causes of the failure of the NPV rule had been well-identified: • DCF requires the creation of a single-decision scenario, which plays itself out over multiple time periods, and for which the net present value of the benefits is the key decision criteria. A single discount rate is used to capture all elements of uncertainty and time preference. However, both real and financial options imply a multiple-decision sce- • However, both real and financial options imply a multiple-decision scenario. Management has flexibility that is not captured in a DCF model. Financial option pricing formulas, even in simple cases, imply value that • Financial option pricing formulas, even in simple cases, imply value that 1 is not the same as implied by DCF or the NPV rule. Risks are asymmetrical for business operations when real or financial • Risks are asymmetrical for business operations when real or financial options are present. Typical DCF analyses are based on an assumption of symmetric risks. (This assumption is often based on a presumption that their is a distribution of returns that is similar to that of a bell-shaped normal distribution.) For entrepreneurial firms, the standard DCF models are often completely • For entrepreneurial firms, the standard DCF models are often completely 2 incapable of providing a good valuation assessment. Thus, standard DCF analysis will often produce, as Dixit & Pindyck put it in 1994, “very wrong” answers. PREVALENCE OF “ADJUSTMENTS” AND SUBJECTIVE METHODS Of course, the workhorse DCF model is used by sophisticated investors and analysts. They, presumably, are aware of asymmetric risks and at least some of the real options that are present in business settings. How do they deal with the problems with DCF? Among the common strategies are the following: • Use DCF as a primary method only when the embedded assumptions are approximately fulfilled, at least for major risks. • Create an “expanded net present value” (or XNPV) measure, which is the • Create an “expanded net present value” (or XNPV) measure, which is the sum of the expected NPV of current operations, plus the value of the embedded real options. • Create a DCF schedule on the basis of objective assumptions. Then, “adjust” the assumptions until the result matches a subjective assessment of value, or the value implied by another method. By altering a growth rate and discount rate assumptions, analysts can frequently produce vastly different net present value calculations on multi-year DCF schedules. • Ignore the flaws and plow ahead anyway. • Ignore the flaws and plow ahead anyway. 1. See, for example, an option pricing reference such as Hull (2008). The companion critique involving real options is amply covered in Schwartz & Trigeorgis (2001). 2. See Smith, Smith, & Bliss (2011), as well as Anderson (2013). 3. See Smith, Smith, & Bliss (2011), as well as Anderson (2013). * * * **Problems with the DCF Model: Ignoring Management Flexibility** - **Make decisions on “gut instincts” or other reasons, and then instruct an ana-** lyst to create a DCF schedule that supports the decision. In some cases, the analyst is forced to use one of these methods due to lack of resources or data. Hopefully, the analyst discloses the limitations and major assumptions underlying his or her estimate in such cases. Unfortunately, how- ever, it is all too common to see naive (and even reckless) use of DCF analyses, as well as undisclosed use of significant subjective adjustments. **ALTERNATIVES TO** In the past, there were often no alternatives to DCF methods in practical work. **DCF** Managers, regulators, and investors concerned about risk were often forced to use blunt instruments to evaluation asymmetric risks, use Monte Carlo simula- tion to at least describe the possible outcomes, or ignore such risks entirely. However, with the discussion of recursive methods in the literature in econom- ics, and the introduction of commercial software designed to make use of these methods practical, they should now be considered by analysts, managers and investors that face problems involving real options or asymmetric risks. **© 2025 Supported Intelligence, LLC** * * * Chapter 5. The Recursive Model RECURSIVE THINKING The essence of a recursive model is to separate a multi-period decision into a series of two-period decisions. The theory behind this originated with Richard Bellman’s 1957 book Dynamic Programming. Bellman argued that many multi-period decision problems could be separated into sets of the following problems: maximize, across all available actions, the sum of these two quantities: 1. The current-period earnings or other reward; and 2. The discounted value expected in the next period, given the decision made in the current period. This gives rise to a value functional equation, where the value in the current state of affairs is the maximization of the sum described above. This equation is, in mathematical terms, a functional, rather than a simple function. A functional is a “function of functions,” which was described by Dreyfus (1963) as a function that takes curves as its arguments. The Bellman Equation The value functional equation is sometimes called a Bellman equation. Because such an equation leads to a recursive set of calculations, the method of creating a value functional equation and solving it is often called a “recursive method.” In various forms, such methods may also be called dynamic programming, stochastic control, impulse control, the value functional method, or a Markov Decision Process. The concepts and variables in a value functional equation, and then the value 3 functional (“Bellman”) equation itself, are shown below. These equations are general enough to apply to both continuous and discrete cases, although the common terminology differs slightly. The following table provides a quick comparison between the two forms. 3. There is a slight difference in notation between these two involving the time index. In one, time is shown as an argument; in the other, it is an index on the state variable. Both are possible in value functional equations. In cases with explicit, time-sensitive real options, time is an argument to the value function. * * * TABLE 1. Functions in Discrete-Time, Discrete Variable Case and Continuous Variable Cases | | Continuous Variables | Discrete-Time, Discrete Variables Case | Notes on Usage in Rapid Recursive toolbox | | --- | --- | --- | --- | | Reward | reward function f(s,x) | reward matrix R | R matrix is size SxA with each row a state and each column an action | | Transition | transition function g(s,x,e) | probability transition matrix P | P matrix is size SxSxA with each frame of the matrix corresponding to a certain action.Because these are probabilities,each row of the matrix must sum to 1 | Functional Equation Variables (EQ 1) V (s, t) = value at time t given state s f (s, x) ≡ reward function given state and action g (s, x) ≡ transition function t = 0,...,T time index st= s0,..., sTstate variables xi= x0,..., xMaction or control variables β = discount factor ε = random error term Value Functional Equation (EQ 2) V (s ) = maxt{f (s, x) + β E\[V (st +1)\]}; x∈Γ where : s = state; x = action; Γ = feasible set of actions. S\_{t}=s\_{0},...,s\_{T} * * * **The Recursive Model** **ESSENTIAL** **© 2025 Supported Intelligence, LLC** * * * **The Recursive Model** In many cases, personal and organizational decisions can be modeled in a recur- sive manner. _Questions to Answer_ The following clarifying questions are helpful in setting up a decision problem, especially if the decision problem is cast in a recursive fashion: - **What person (or what organization) is the subject of the decision problem?** - **What is the reward (or penalty) that the subject person receives, dependent** _on the state of affairs and decisions that are made?_ - **How does the subject person discount future rewards (and costs) compared** to current rewards and costs? - **In what time periods does the subject typically recognize the state of affairs,** receive rewards, and take actions? - **What actions can the subject person take that affects the current reward, or** the future state of affairs? Worksheets that include these questions are included in “Appendix C. Sample Worksheets” on page C-1. **© 2025 Supported Intelligence, LLC** * * * Chapter 6. Using the Rapid Recursive® Toolbox The RR tool allows a person to: Compose a decision problem (or “model”), using information a. Compose a decision problem (or “model”), using information gained from the user and other sources of data. Error check and correct the model. b. Error check and correct the model. Formulate the underlying problem into a mathematical value c. Formulate the underlying problem into a mathematical value functional equation. Solve the equation, yielding a value in each state and a value- d. Solve the equation, yielding a value in each state and a valuemaximizing action (“policy”) in each state. Report the results. e. Report the results. Practical guidance for using the toolbox is included in this chapter. FIRST STEPS IN COMPOSING THE PROBLEM STATE VARIABLES FOR BUSINESSES The first steps in using the toolbox to analyze a decision problem often involve thinking about the problem, rather than using the software. The essential elements of a recursive decision problem must be identified. This is discussed in “The Recursive Model” on page 20. Also, we have created a set of worksheets for users that helps outlining a decision problem; and identify states. These are included in “Appendix C. Sample Worksheets” on page C-1 Anderson (2012) provides a prototype state vector for an operating firm, with elements from the following four categories: The human capital of the entrepreneur, which he describes as “a critical vari- 1. The human capital of the entrepreneur, which he describes as “a critical variable and frequently watched by other employees, customers, investors, and (of course) the entrepreneur herself.” Economic and competitive conditions in the relevant marketplace. 2. Economic and competitive conditions in the relevant marketplace. Balance sheet information, including invested capital. 3. Employees, technology or other intellectual property. 4. Balance sheet information, including invested capital. Employees, technology or other intellectual property. Some variation of these items are likely to be good state variables for an operating firm. * * * STATE VARIABLES FOR OTHER SUBJECT PERSONS Possible state variables for individuals, families, nonprofit organizations, schools, countries, and other organizations include: • Enrollment or attendance • The number of customers, dues-paying members, or affiliates • The weather (represented by temperature, wind speed, number of storms, rainfall, or other variables) • Crop size, type of crops planted, growth of trees or other crops • Number of predators or prey • Number of predators or prey • Number of people, families, churches, schools, streets, or cars • Income, jobs, after-tax earnings, savings or investment balances • Years of schooling, degree attainment, or other human capital measures • Years to retirement or graduation In effect, any variable that is observed and considered to be important to the subject of a decision problem is a possible state variable. “Importance of Importance” Principle The last sentence should not be read over lightly. Unless the variable is observed and important to the subject, it is not a good state variable. One often sees poor analyses in a variety of fields that suffer from the (often fatal) logical flaw of analyzing variables that the subjects do not consider important, and as a result “finding” unusual results. One common variant of this error is to compare the response of individual residents of a country or area to a change in government policy, when that policy is unobserved or measured in a 6 way that is foreign, unknown, or unimportant to the subjects. The policy may be important to the government agency or other organization that conducted or commissioned the research, but if it isn’t observed and considered important by the subjects, it is a poor choice for a state variable in a decision model. Remember the “importance of importance” principle: if a variable is not observed and important to the subject, it is not important to the decision model. Difference Between Econometric and Recursive Models 6. For example, consider the use (in economics) of Gini coefficients, top marginal tax rates, “effective” tax rates, and total government program expenditures as explanatory variables for a variety of models. Although these variables, in some cases, change over time in a way that provides some information about conditions affecting taxpayers and employers, very few of them actually face the top marginal tax rate, pay the average “effective” rate, or know what the Gini coefficient is. * * * **Using the Rapid Recursive® Toolbox** Continuing on this line of logic, it is worth noting that econometric models that seek to explain aggregate behavior over time are not the same as recursive mod- els, in the following way: - **Data on individual behavior that is measured by aggregates suffers from the** well-known and obvious problem of aggregation and inference. Recursive models can more easily focus on the individual decision maker. - **Econometric models of time-series data usually involve a model that does** not change over time, preferences that remain the same over time, and an underlying model that assumes that responses to changes in state variables are additive. 7 Recursive models, on the other hand, can involve transition probabilities that are different in different states or when actions are differ- ent. Although the recursive models considered in the examples presented in this Guide rely on transition probabilities that do not change over time, they do not assume that the conditions or consequences of actions do not change in a way that affects the subject’s decisions. Of course, econometric methods are sometimes the best available given the data and problem at hand. Furthermore, one may find results from econometric anal- yses useful in estimating parameters in a recursive model. However, always consider the fundamental difference in the logic between a recursive (two- period, decision-maker-centered) model and a multi-period, aggregate-data econometric model. **USE OF INDEX AND** Often the available data is messy, or involves multiple possible metrics. In some **TRANSFORMED** such cases, the data can be transformed into an index or representative indicator. **VARIABLES** Although this procedure is sometimes necessary, it should be approached with care. Just as noted above, it is quite possible to transform data that the subjects see and find important to an index that does not reflect conditions important to the subject. 7. Experienced econometricians will note that the underlying model can be multiplica- tive and then transformed (such as by logarithms) into an additive model, or can involve a more complicated nonlinear regression. However, this does not change the main point of this observation. **© 2025 Supported Intelligence, LLC** * * * RATIONALE AND DATA ON DISCOUNTING Chapter 7. Discounting and Discount Rates Theories of Discounting Economists as long ago as Adam Smith conjured up reasons why consumers preferred to receive money now rather than later, and why lenders and investors insisted on what we would now call a rate of return. Smith, in Wealth of Nations (1776, book I, chapter 8) described how the potential for loss, the difficulty in ensuring repayment, and the possible alternative earnings all affected the market rate of interest or “ordinary rate of profit.” The early 20th-century economist Irving Fisher catalogued the reasons for “impatience” among consumers in his monumental Theory of Interest (1930, 8 chapter IV). Fisher even described why consumers with relatively little income had much stronger reasons to prefer income immediately than those whose income afforded them the ability to patiently invest. In the latter half of the 20th century, a large number of theories and variations of those theories arose regarding expected rates of return for investors. The most well-known of these (such as the Mean-Variance Framework, the Capital Asset Pricing Model, the Fama-French model, the APT, and many variants of these) have been employed to create models of expected rates or return for certain investors. For the purposes of this Guide, we will merely introduce the theories and authors listed above, and suggest that one or more of these could be applicable to specific decision problems. At least as important, however, is likely to be the availability of data for that problem. We discuss this next. Data on Rates of Return The availability of data on rates of return vary tremendously by the type of investment or security. 2. Its expected distribution in time, or its time shape—that is, whether it is constant, or increasing, or decreasing, or sometimes one and sometimes the other. 3. The size (measured in dollars) of his expected real income stream. 4. Its composition—to what extent it consists of nourishment, of shelter, of amusement, of education, and so on. 5. Its probability, or degree of risk or uncertainty. \[Theory of Interest, IV, sec. 5\] * * * **Discounting and Discount Rates** **1.** Over the past few decades, price and dividend data from U.S. stock and bond markets have been compiled into data on actual rates of return (and, sometimes, defaults) on traded securities representing many classes of equity and debt in these markets. These data focus only a small fraction of companies (those that are publicly-traded), are often subject to “survivor bias,” and typically ignore transaction costs. However, these data provide a very rich source of information to those that can observe both the limitations and the benefits. **2.** Unfortunately, nothing remotely comparable exists for investments in pri- vately-held firms. This is a huge area of missing information, as investments in private firms are clearly not the same (in terms of control, liquidity, mar- ketability, and tax implications) as investments in publicly-traded firms through the stock market. **3.** The burgeoning field of entrepreneurial finance—and the increasing visibil- ity of venture capital—has provided some public awareness of investment terms that are common for start-up and early-stage firms. However, compa- rable data here for venture capital investments across industries is very diffi- cult to obtain. **4.** Some data exist on the value over time of investments in houses (for many households, their most valuable asset), and on what households report as their retirement savings. **5.** Data on consumer loan rates are available. However, these are subject to the considerable limitations, in that many consumers avoid taking out loans (including those represented by credit card balances), and others use the strong differences in terms among these loans to shift funds from one pur- pose to another. **6.** Finally, there are enormous data available on interest rates and yields on Treasury securities for the United States and many other nations. These, of course, do not reflect the borrowing terms of most private sector lenders or borrowers, but do provide an indication of the base rate of interest and infla- tion expectations. It is important to note that the wide availability of data for two of the above cat- egories does not mean that such data can be readily applied to investments or loans made in the other categories. **TWO CRITICAL** The literature of finance, economics, and professional valuation can differ sub- **ASSUMPTIONS:** stantially in terms of their embedded assumptions, conventions, and nomencla- **UTILITY, AND TIME** ture.9Unfortunately, readers familiar with one literature can easily overlook **PREFERENCE** differences in assumptions in texts from another field. 9. For a pointed discussion, see Anderson (2013, chapter 1). **© 2025 Supported Intelligence, LLC** * * * **Discounting and Discount Rates** To avoid this problem, and to further explain the intellectual basis for the recur- sive approach, it is worth discussing the following two critical assumptions and how they often differ depending on the field of use: **1.** “Utility” versus cash flows Since the emergence of the neoclassical school over a century ago, econo- mists have typically rested their analyses on the bedrock of a set of prefer- ences of a consumer. These preferences are embodied by a “utility function” that maps the receipt of income and other benefits to subjective benefits as seen by the consumer. Much effort within the profession has focused on the form of this utility function, the implied view of risk, and (recently) the apparent deviations from ideal models of utility that are implied by actual consumer behavior. 10 Depending on the utility function used in an economic model, the first dollar earned in a time period is often “worth” much more to a consumer than the thousandth or ten thousandth. 11 On the other hand, most finance and valuation texts largely ignore this entire concept, using the simplifying assumption that money earned is money earned, and that twice as much money is worth exactly twice as much. Of course, for many tasks (such as managing a large investment fund), this is quite accurate. The common analytical tasks in each field underline this difference: econo- mists often “maximize utility” in their models, while finance professionals “discount cash flows” in theirs. **2.** “Time separable” versus “recursive” utility Recursive models incorporate the series of two-period decisions that inves- tors, managers, and consumers make. Each of these involve a trade-off between benefits (or costs) today, and expected value tomorrow. 10\. The latter has spawned an entire “behaviorist” school of finance and economics, including the notion of “bounded rationality” and “loss aversion.” Kahneman & Tversky (1979) is one of the seminal contemporary articles establishing this school of thought. It interesting to note the recent re-discovery of the “behaviorist” tendencies of classi- cal economists. For example, Adam Smith clearly outlined “loss aversion” 200 years before it was studied by contemporary economists. In Theory of Moral Sentiments (1759), he wrote: Pain...is, in almost all cases, a more pungent sensation than the opposite and corre- spondent pleasure. The one almost always depresses us much more below the ordi- nary, or what may be called the natural state of our happiness, than the other ever raises us above it. \[Theory of Moral Sentiments, III, i\] On the classical behaviorists, see Ashraf, et al (2005). 11\. This is the famous “diminishing marginal returns” argument that often surfaces in introductory economics classes. It is most easily expressed in terms of the desire of a person to eat when very hungry: the first several bites can be the difference between life and death; those that follow are beneficial and enjoyable; those still afterwards can result in sickness. **© 2025 Supported Intelligence, LLC** * * * In contrast, the standard finance model involves a stream of future cash flows discounted to the present. The implied trade-off here is between an 12 investment today, and a sum of future expected earnings. This implied attitude is known as “time separable” or “additive” utility. In fact, very few investors (or consumers, or managers) are truly indifferent In fact, very few investors (or consumers, or managers) are truly indifferent to the timing of their payments. Instead, they behave as if they have recursive utility, meaning that they attach meaning to the timing of income and 13 expenses as well as the net present value of their sum. Here, only a sliver of the current economics literature observes the differ- Here, only a sliver of the current economics literature observes the difference between time-separable and recursive utility, with the large majority typically assuming time-separable (additive) utility. ASSUMPTIONS IN RECURSIVE MODELS: UTILITY AND TIME PREFERENCES With the above discussion in mind, it is important to observe that recursive models embed the following assumptions: 1. Utility in a recursive model can be measured by cash income, cash expenses, or any utility measure that can be defined. The analyst can define the measure used for current-period benefits. This The analyst can define the measure used for current-period benefits. This can be—and often will be in practical models—a simple expression of income or expenses. It can also be something that incorporates a utility function that incorporates risk aversion, diminishing marginal benefits, loss avoidance, and other commonly-observed human preferences. In the Rapid Recursive® toolbox, this can be accomplished by using some In the Rapid Recursive® toolbox, this can be accomplished by using some type of mathematical construct that implements the utility function, and putting the result into the reward matrix. For example, very strong avoidance behavior can be implemented by using an extremely low number to represent the subjective reaction of the subject to a particularly bad situation that he or she wishes to avoid. Time preferences in a recursive model are inherently recursive, and can 2. Time preferences in a recursive model are inherently recursive, and can mimic time-separable preferences if appropriate. The discounting factor in a recursive model is always recursive. This means The discounting factor in a recursive model is always recursive. This means the appropriate discount factor is the one the subject uses to compare current-period benefits and the expected value in the next time period. Combined with the wide flexibility available for describing benefits, a very large 12. The logic of net present value is inescapable: When the discount rate is 10%, $1 today is worth the same as $1.10 next year, $1.21 the year after that, and $2 some years after that. The net present value of each of these possibilities (and of many combinations of them) is the same. This use of a NPV measure or decision criteria implies that an investor is indifferent This use of a NPV measure or decision criteria implies that an investor is indifferent to the timing of payments, as long as the net present value of the eventual sum of payments is the same. 13. Systematic treatment of recursive utility is found in Stokey and Lucas (1989, chapter 5), and Duffie (2001, chapters 1 and 2 including notes); Anderson (2013, chapter 8) contrasts time separable and recursive utility in decision models such as those discussed here. * * * **Discounting and Discount Rates** range of preferences (including strong loss aversion, asymmetrical attitudes about earnings over different time periods, and desire to rationally evaluate time-sensitive real options) can be modeled. This being said, it is quite possible—and sometimes desirable—to use a sim- ple discount factor, straightforward cash earnings for a reward matrix, and structure a model so that the subject is indeed largely indifferent about the timing of cash flows as long as they have identical net present value. **PRACTICAL ADVICE** Given the discussion of theory, and then data, and then critical embedded **ON DISCOUNT RATES** assumptions above, we can offer the following practical advice on using dis- count rates in a recursive model: **1.** Think hard about the underlying problem before selecting a discount rate. The correct discount rate will always depend on the subject of the decision problem, and only sometimes on stock market or other widely-available data. Always think hard about the problem itself before selecting a discount rate. **2.** The correct discount rate will be based on factors that are apparent and important to the subject of the decision problem. Remember that the discount rate reflects the competing incentives for the subject of the problem, given the other factors surrounding him or her. Take those factors into account Avoid relying on a theory of discounting, or a data source on discount rates, that does not match the conditions of the problem. **3.** Consider transaction costs and liquidity constraints in any problem. One of the strengths of the recursive approach is the ability to explicitly include such factors in the model. When selecting a discount rate, be sure that you don’t ignore these factors. **4.** The recursive structure of the problem implies the use of a two-period dis- count rate. This may or may not be represented by a “short term” interest rate. In the recursive model, the subject makes a series of two-period decisions. Therefore, they are always choosing between this period and the next. In cases where the subject can repeatedly use short-term borrowing, and faces no liquidity constraints, this suggests the use of a short-term borrowing rate as a discount rate. However, the more likely case for both businesses and households will be that repeated short-term borrowing could only involve costly unsecured credit. The alternative sources of funds are equivalent to equity financing, such as home-equity loans, refinancing of existing debt, implicit use of proceeds from car loans or student loans, and reducing house- hold expenditures. Some of these sources have high transaction costs, and all are limited in some way. **5.** Take into account inflation and growth trends consistently throughout the model. **© 2025 Supported Intelligence, LLC** * * * **Discounting and Discount Rates** In times of low inflation, low interest rates, and relatively stable economic growth, one might get away with subsuming inflation expectations into a discount and growth projections without much additional thought. However, such times are the exception, not the rule. Therefore, always consider the growth and inflation trends in the problem, and recognize them consistently throughout the model. _Practical Advice by Subject of Decision Problem_ Table 2, “Suggestions for Discount and Growth Rates,” on page 33, contains a set of recommendations for growth and discount rates for use in recursive mod- els. The recommendations are separated by type of subject. Note that the advice in the table primarily reflects important considerations for the subjects of the problem, rather than extensive data or econometric analysis. This is consistent with the recommendations under “Practical Advice on Dis- count Rates” on page 31. **© 2025 Supported Intelligence, LLC** * * * TABLE 2. Suggestions for Discount and Growth Rates | Subject and Problem | Possible d | Possible g | Notes | | --- | --- | --- | --- | | Publicly-traded corporation with access to short-term and long-term borrowingSubject considers whether to invest or re-invest in current business | commercial prime rate; commercial paper rate;“cost of capital” estimate | zero; long-term real growth | This is a common subject in corporate finance, but is the exception rather than the rule in the economy.“Cost of capital” could be estimated in a variety of ways,taking into account debt and equity mix for corporation. | | Classic micro-economic household unitSubject considers whether to save or consume | borrowing rate for household assets,such as a mortgage | long-term productivity or wage growth trends | This is a common teaching and analytical exampleTransaction costs,short-term uncertainty,and even mortality are often ignored in these examples. | | Realistic household unit or individual workerSubject considers whether to save or consume | available borrowing rates,which could include high-cost credit card debt,mortgage rates,home equity loans,and family borrowing | human capital influenced wage growth rate,which could unemployment and risks related to health,industry,and skills | The actual constraints and market conditions imposed on typical families and workers are often severe.For realism,ensure that you have taken into account transaction costs and liquidity constraints,which can dramatically increase the costs of borrowing,as well as access (or lack thereof)to financial markets and institutions. | | Closely-held business in operation for at least several years and with good financial and managerial resourcesSubject considers whether to re-invest in current industry,perhaps to expand or acquire a competitor,or distribute income to owners | | growth trend for that company in that industry,adjusted for“key man”or similar risksinflation trends in industry | Note that even a successful privately-held firm cannot borrow using commercial paper;cannot issue stock on major exchanges;and may or may not be able to obtain conventional bank loans.Note also that the owners may have the benefits of controlling the entity in a manner that stockholders or large corporations do not. | | Start-up or entrepreneurial firm;entrepreneur starting firm or creating inventionEntrepreneur may be considering whether to continue re-investing in unprofitable company(or invention without current market),or stop and return to regular employment.Firm may be struggling to develop a product and get it to market;or to become profitable after a product is first introduced | Cost of entrepreneur's time and money;cost of“angel”capital;total cost of venture funding | zero;high-growth potential(if properly offset by high discount rate);modest growth rate for productivity of entrepreneur | Liquidity constraints and transaction costs are very significant in this sector.Often,the only financing available to entrepreneurs is“3-F”financing.Terms for VC and angel financing are often harsh.Bootstrapping"for some period of time is often the only possible path.Risks, and possible rewards,are extremely high. | Source: Author’s advice * * * TABLE 2. Suggestions for Discount and Growth Rates | Subject and Problem | Possible d | Possible g | Notes | | --- | --- | --- | --- | | General treatment of inflation in all these problems | | | Price inflation trends should be considered consistently across all elements in problem, including reward function, available actions, and both discount and growth assumptions embedded in discount factor. | Source: Author’s advice * * * Chapter 8. Additional Topics in Discount Rates and State Variable Diffusions There are a variety of conventions and practical considerations that develop over time in the use of any method. Because recursive methods are new, these conventions and practical considerations are not well established. By convention, we mean a certain manner of usage that is assumed to be known and followed. Conventions are often the result of tradition, precedence, and reasoning. However, the use of a convention does not mean that there is no other way to do something, or that only one way is correct. For example, people in many parts of the world drive on the right side of the road. Others drive on the left. It is very, very important to have a convention about what side of the road should be driven. However, it is difficult to make a strong case that one is logically superior to another. One purpose of this Guide is to help users of the Rapid Recursive® toolbox quickly become proficient in the use of recursive techniques. In this chapter, we discuss a set of conventions and practical considerations that can be very helpful in actual use. TIMING OF REWARDS AND ACTIONS The Rapid Recursive® toolbox uses the following conventions regarding the tim- ing of rewards and actions. Rewards are assumed to be received at the beginning of each period. 1. Rewards are assumed to be received at the beginning of each period. Actions take place simultaneously with the reward. 2. Actions take place simultaneously with the reward. The results of the action (and any random factors that are incorporated into 3. The results of the action (and any random factors that are incorporated into the transition probabilities) occur with the arrival of the state at the beginning of the next period. At that point, the process begins again. As noted below, practitioners in other fields may use this convention for timing of rewards that are cash earnings, or they may use other conventions. Observing this convention is particularly important when the discounting is also captured in the summation. For example, the following summation includes an exponent on the discount factor beta that is multiplied by a series of cash flows Ct: Time Index Starts at t=0 The Rapid Recursive® toolbox uses the convention that time starts at t=0, and that the first reward is received at that point in time. Note that some summation formulas in finance textbooks and articles either omit the time subscripts, or start with t=1. C\_{t}. * * * NPV Summation N!P P V=\\sum\_{t=0}^{T}\\beta^{\\prime}C\_{t} where : t = time index, β =discount factor; 0 < β < 1; β =discount factor; 0 < β < 1; C = cash flow at time t. 0!