

PrefaceI.1. Who will gain, and what, from this book
In the preliminary survey of readers who could be interested in this book, we discovered three distinct groups. One large group includes researchers and professors from universities, colleges and business schools. We did not do exhaustive interviewing of the prospective readers in this category (professors are busy people!), however, several of the opinions we collected were not just polite, but expressed real interest. The opinions and recommendations were thoughtful, intelligent and encouraging. So, our impression is that this audience will benefit from the book both from an academic perspective, learning new ideas, methods and concepts, as well as acquiring familiarity with the needs of practitioners working in the industry. In the early stages of the project, we discovered that the subject and the way we presented the material in the first drafts found warm and enthusiastic reception amongst undergraduate students, teaching assistants and instructors. This audience especially praised the idea (which is quite natural in our view) to support the book material with exhaustive numerical examples, tabular and graphical illustrations. The range of courses included different branches of investment business, accounting, underwriting, finance and other disciplines, in which investment and accrual are inherent components. The features which attracted this category of potential readers the most were the logical consistency of the presentation, transparency of mathematical derivations and transformations, abundant numerical examples presented in different forms, and comprehensiveness. Some of these people, let us call them reviewers, told us that they discovered many important details and concepts, of which they were not aware for years, even if they taught the subjects themselves. Their vision and knowledge of the subject, and here we cite their words, “became much more coherent and transparent” (we heard the word “transparent” from several reviewers, so it should not be accidental, and probably describes their genuine impression). As we also learned from reviews and comments, this volume provides a valuable and coherent knowledge base for the third group of readers, who are industry specialists. Besides, we previously received many good comments about our published articles, two of which received Honorable Mention Awards from “The Journal of Performance Measurement”. We also know that some of our results are used in practical applications. This allows us to conclude that financial and investment analysts certainly will be among the beneficiaries. They will find plenty of useful and, in many instances, unique information and approaches which directly relate to their everyday duties and challenges. Much of the material, which includes mathematical and computational algorithms, and analysis of their efficiency, is relevant for software developers and designers of investment performance measurement systems, and, to a lesser extent, trading systems. These specialists will acquire an integral, conceptual vision of the whole discipline from the mathematical and computational perspectives, taken in their entirety as two closely related areas. For the very first time, this book considers the specifics of computational algorithms used in this industry, their efficiency and software implementation issues. We devoted a special section considering computational algorithms for financial calculators. This material is presented in the same academic format as the entire content of this book. However, these results originated from the solution of a real problem, how to advance the computational efficiency of these devices by improving existing and developing new efficient computational algorithms. This is the subject which proved to be of great interest to specialists. On the other hand, through this example, students can learn directly how a real problem can be formulated and efficiently solved from scratch, using acquired knowledge and innovative thinking just a little beyond the traditional boundaries. This real solution of a real problem can give students more confidence in the practicality of their knowledge, and show how to thoughtfully apply it. The author himself had to step over the same threshold, which separates seemingly pure academic knowledge, which we acquire through years of study, often not knowing exactly what it can be used for, and the loveliness and exuberance of real life. In fact, this threshold is neither as big, or as wide as we often think; and with concentrated efforts, desire and stimulus, academic knowledge can quickly become an invaluable and powerful practical instrument, which will help young people move through their lives and, if necessary, to build and pave their own roads.
I.2. The goal and general guiding concept of the book
As it was mentioned already, the author regularly publishes articles on mathematical problems of investment performance measurement in professional journals. These papers mostly consider the core problems of this discipline. At first, some of the author’s results appeared controversial to reviewers, because these results sometimes disagreed with conventional approaches. However, eventually the ideas made their way into the industry. This happened with the author’s idea that all methods for calculating rate of return have a hierarchical structure and, in fact, are related to each other as different approximations of parent methods. The idea of consistent linking was not recognized immediately as well, but eventually the article about this method received the Honorable Mention Award, and so on. Gradually, the content of published and submitted papers emerges as a larger entity with many interrelations and coherence between seemingly different threads. At this point, the scientist realizes that the research paper format confines his perception of the subject. He then realizes the need for a more appropriate form, such as a book, to express his overall vision of the subject to a wider audience (well, if the book is a success story, otherwise the efforts will vanish without a trace, which is always a possible outcome when writing a book!). People write books for other reasons too. For instance, publicity is an important consideration when making such a commitment. It is the hierarchy of reasons that matters, and this is what shapes the conceptual foundations of a manuscript. In this particular case, the number one reason, which is also the goal of the book, is to provide a consistent and comprehensive conceptual platform for the discipline. In order to achieve this goal, the author thoroughly researched all of its principal areas. Although personal preferences towards particular topics influenced the content of the book, the structure and content has been largely dictated by a desire to represent the subject consistently, while exposing its structure and interrelations of constituent topics. Personal development was another reason why this book was written. Writing a book on an intricate academic subject, like investment mathematics, is a challenge. So, why not put in the effort to write a good book, one that will push the limit of your capabilities as a scientist and a writer? This idea became the general guiding principle the author followed in writing this book.
I.3. The foundations of the study
Every phenomenon has a structure. It may be hidden, unknown, it may be not obvious, but it is there. By the same token, every book and every story has a structure. Even if the author does not think about it, the structure will originate during the creation process. It is better to think of structure before beginning a construction project, and not after the third story is completed. The same holds true when planning a book. The deeper and broader our foundation is, the more solid the structure we build will be. Below, you will find a brief outline of the book’s conceptual foundations.
I.3.1. Internal rate of return
We dug deep enough to discover that the internal rate of return (IRR) is a common root of many methods, approaches, and concepts in investment performance measurement. It is for the first time that IRR receives a detailed consideration of its multifaceted nature. It turns out that IRR is a foundation upon which many areas of this discipline are built, though often unknowingly. United by this common foundation, many problems become more understandable and transparent, than they were before, when treated as standalone entities without kith and kin.
I.3.2. Compounding
The notion of compounding, thoroughly researched in this book, obtains a new quality and gains more importance. In fact, it becomes the second main concept that links the pages of this work. For the first time, compounding is embedded into investment performance measurement methods as an inherent and important component. This approach allows us to further distinguish investment performance measurement methods, which brings more clarity into this discipline. For instance, these methods can now be classified into two distinct groups. All investment performance measurement methods that originate within the compounding context are founded upon the notion of IRR, and can be derived mathematically from the IRR equation. The other group unites methods developed within a noncompounding context. This group of methods is based on the generalized Modified Dietz equation, which we introduce in this book. The issue of compounding is also important in applications involving mortgages and annuities. We cover this area as well.
I.3.3. Shestopaloff’s Linking
First, we make a note about terminology. Originally, this concept has been introduced as the Consistent Linking (CL) operation. However, in the literature, people began referring to this method by its author’s name. A possible reason for this change in terminology could be the fact that the term Consistent Linking has been used already in other areas of science and technology. So, we decided to follow this naming convention in this book. The idea of this concept is to provide a mathematically correct linking operation of sequential periods’ rates of return, in order to compute the rate of return of the total period. Presently, geometric linking attempts to link periods’ rates of return, but ours and other research showed that in almost all instances, geometric linking does not produce correct results. On the other hand, Shestopaloff’s Linking (SL) operation allows us to correctly calculate the rate of return for the total period, composed of a set of smaller periods, using the smaller periods’ rates of return. Additionally, we can use Shestopaloff’s Linking to link rates of return across different assets. This is a previously unavailable operation, and it significantly enhances analytical studies related to investing. Combined together, these features of SL allow the computation of total rate of return for any assetperiod combination. There are many other important implications of this concept, which we also look at in this book. For instance, SL can be used for optimizing investment and trading strategies. Some numerical examples show that this is a promising application of SL methods. We see these methods as an important development, which will gradually gain ground in practical applications. The potential of Shestopaloff’s Linking is high, and we think that, as it is used more and more, its real capabilities will be uncovered. SL methods can significantly contribute to further enhancement of existing performance measurement methods, as well as to the introduction of new conceptual approaches. The primary area of application of Shestopaloff’s Linking is in analytical studies. There are also promising implications for system design. Advanced computational performance is another distinguishing feature of this method. In fact, the computing abilities of SL algorithms are so good, that they transform the present time consuming calculations of rates of return from batch jobs to the real time computing. In addition to these, there are certainly other areas that can also benefit from using the Shestopaloff’s Linking concept. We are confident that this is a good concept which provides a good toolset, and we can recommend it to industry academics and practitioners.
I.3.4. Other key topics
Besides the IRR concept, compounding, and Shestopaloff’s Linking, the book includes other key topics related to investment performance measurement. They naturally complement the content of the book in order to provide an indepth coverage of the whole subject. For instance, we analyze the universality of the notion of interest rate. We argue that the notion of interest rate should be used in its single, universal form rather than be unnecessarily split into ambiguous nominal, effective, simple, etc. interest rates. The only thing that needs to be changed in order to remove the present ambiguity is the way of calculating the interest rate for periods of different lengths. The relationships between the discussed topics are presented in Fig. 1. The auxiliary threads are associated with major concepts and support the book’s main content, thus giving the book conceptual integrity.
Fig. 1. Primary and secondary threads of the book. The primacy of a thread is reflected by the thickness of the corresponding line. Low priority threads are represented by dashed lines.
I.4. The book’s organization
The book intentionally starts from scratch. We first introduce basic definitions, such as those of interest rate and rate of return. Then, we build the mathematical apparatus required for analytical studies of investments from the ground up, eventually deriving the internal rate of return (IRR) equation. We explore all possible derivation paths to make sure that we did not leave out any ambiguous issues, or some alternative ways of representing the same phenomenon. Thus, we ensure that the IRR equation is the only possible mathematical representation of lending and investment processes in the compounding context. We also consider noncompounding scenarios to compare these two contextual frameworks. In Chapter 2, we use the IRR equation to derive mathematical formulas required for working with annuities. We thoroughly consider this topic and present many numerical examples, both in graphical and tabular form. This will allow the reader to deeply understand the subject. After completing this chapter, the reader himself will be able to derive all mathematical formulas for similar financial instruments, however exotic and intricate they may be. We pursue a similar goal in Chapter 3, but this time we look at mortgages. Again, our foundation is the IRR equation, from which we derive all of the formulas required to describe the lending business. The mathematical methods we introduce are studied, commented on, and illustrated by abundant numerical examples in graphical and tabular form. Chapter 4 begins the subject of investment performance measurement, to which the four chapters that follow it are devoted. In Chapter 4, we introduce the Modified Dietz method, and describe its relationship with the IRR equation. This is done using a Taylor series expansion. We analyze the domain of applicability of the Modified Dietz equation with regard to the problem of computing rates of return on investments. We consider other interesting developments that accompany the introduction of this formula, and begin to introduce the conceptual foundations of investment performance measurement as they relate to Dietz’s formulas. Chapter 5 continues the application of the IRR equation to practical problems. We introduce and study different methods for computing rate of return from the IRR equation based on various approaches to solving it. Besides the traditional methods, we also consider the series reversion method. This is an approach that was discovered back in the 1950’s; it has been revived recently for computing the interest rate in lending applications. We noticed that many scholars are interested in this subject, probably attracted by its novelty. So, we decided to explore this topic in detail and provide an objective analysis of the method’s features. Given the iterative nature of many numerical methods used for solving the IRR equation, we pay special attention to choosing the initial approximation value. This is an important practical issue which determines how quickly these iterative methods converge to a solution. We look at several approaches for choosing the initial value in order to find ways of improving the computational performance. Evaluation of the computational efficiency of different analytical and numerical methods is the main topic of Chapter 6. First, we introduce numerical methods for solving the IRR equation. Then, we discuss related issues and illustrate them with numerical examples and figures. This is the first time that this important subject is looked at from the point of view of computational efficiency. This chapter can be also be of value to software system designers who develop investment performance measurement systems and applications. In addition, we study computational algorithms for financial calculators, and show how one can significantly improve efficiency of these algorithms by considering the particular problem of computing the interest rate. Chapter 7 generalizes the results of previous chapters. Here, the notion of context of investment performance measurement is researched in depth. This notion has already been introduced in Chapter 1 and successively propagated through the content of the book. We show that the context  compounding or noncompounding  becomes an additional parameter, or dimension, which has to be associated with investment performance measurement problems. It turns out that the IRR equation is the most adequate instrument within the compounding context, while the noncompounding context is best served by a different tool, the generalized Modified Dietz equation. We introduce this equation as a general mathematical tool to be used within the noncompounding context. We also consider two other methods used to evaluate the rate of return, which are the net present value (NPV) and Modified IRR. We show that these methods are closely related to the IRR equation, and thoroughly research them from the mathematical and business perspectives. We improve these methods to enhance their consistency and to widen their range of applicability. Shestopaloff’s Linking is presented in Chapter 8. This is a family of advanced methods to correctly compute rates of return for combinations of sequential and/or nonsequential periods and/or different assets. Depending on the task, the input data for these methods are the period rates of return, a few other integral parameters that are associated with the periods we consider, and information about the assets in the portfolio. The advantages of these methods are numerous and significant. For instance, they introduce unique functionalities, such as the possibility of linking assets’ rates of return. This advanced analytical feature has never been available before. Shestopaloff’s Linking methods enhance analytical studies, the development of investment and trading strategies, and improve computational performance of software systems by hundreds of times. They also allow for the improvement of system design. We think that these methods deserve the close attention of different industry specialists, from analysts to software system designers. Chapter 9 studies investment performance attribution methods. Here, we introduce a conceptual framework and develop attribution models and methods within this framework. We also thoroughly analyze the present methods and compare them to the new ones in terms of validity, accuracy and objectivity. To do this, we introduce validation and verification tools and criteria that allow the evaluation of attribution methods from this perspective. Chapter 10 studies approaches to risk measurement, and the available quantitative instruments. We analyze different investment risk measures developed so far, from a conceptual point of view. Besides detailed mathematical studies, the reader will encounter many original commentaries on this topic. While some of these commentaries are firmly supported by available information, others use more subtle conceptual considerations. We do not provide a wide mathematical coverage of this subject, given the scope of the problem and the amount of different methods available. However, our coverage is sufficient to obtain a solid understanding of the subject, including the interrelations between the numerous available methods. This chapter can be used by scholars and industry professionals who are searching for a consistent and comprehensive presentation of this topic. We think that the readers who need an advanced understanding of investment risk measurement, as well as developers of risk measurement methods who are looking for new ideas and a conceptual vision of the problem, will not be disappointed too. Chapter 11 completes and wraps up our intellectual journey. (The reader can probably immediately recall the infamous Chapter Eleven. Well, knowledge of this material could help many people to avoid becoming closely familiar with this law.) Anyway, this chapter is special in several ways. Throughout the whole book, we make comments of a general nature that are related to the problems we discuss. Sometimes these comments reinforce our assertions and results, but often these considerations represent the conceptual base of our research itself. Here, we elaborate on these notes, presenting them as a sort of conceptual toolset that we applied in our research. Despite its generality, this chapter is not exactly easy reading. It requires critical thinking and a desire to understand various sorts of subtleties. However, we think that understanding the concepts presented in this chapter will be a rewarding experience for readers of any level; having common sense is the only prerequisite. These are concepts that can be applied to a wide range of problems, including ones that are far outside the boundaries of investment performance measurement. Our epoch appreciates specialization, and there are often sound reasons for this being the case. However, overspecialization, in fact, impedes progress; and measure, Aristotle’s “golden mean”, is in many instances trespassed. Real life is a complex and multifactor phenomenon, which in many instances does not respect the artificial boundaries imposed on it by our human passion for classification, structuring, subdivision and, as a consequence, isolation of different factors and aspects of the phenomenon. Such treatment of natural phenomena may make us unable to understand them fully. At the same time, real life problems require real solutions, and this is possible if and only if (the last phrase is a strong mathematical statement!) the problemsolver has an adequate perception of different phenomena in their entirety. This adequacy implies having a vision that reaches across boundaries in all dimensions and directions. In essence, the fundamentals that are necessary for having such a vision are what this chapter is about. 
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