Science of inexact mathematics. Investment performance measurement. Mortgages and annuities. Computing algorithms. Attribution. Risk valuation
Publisher: AKVY Press
Published: 2009
Format: Hardcover
Number of pages: 591
ISBN-13: 9780980966701
Description: This book presents a coherent and comprehensive study of mathematical methods for investment performance measurement, attribution analysis, mortgages, annuities, and investment risk measurement. It further discusses other advanced topics such as the linking algorithms for rates of return. For the first time, computational algorithms used in these areas of financial mathematics, and the efficiency of their software implementation receive thorough consideration. Overall, this unique work provides a clear conceptual vision of the entire discipline. The high level academic presentation is very well supported by lots of numerical examples, numerous tables and figures.
    The book includes extensive material for a wide range of related undergraduate and graduate courses in finance and computational mathematics. Many of these courses can be built entirely on the book’s content. Academics, researchers and industry specialists, in particular investment analysts and system designers, will find this book an invaluable and comprehensive source of knowledge, reference material, and new ideas.

Preface
I.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 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 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 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 we acquire through years of study, often not knowing exactly what it can be used for, and the loveliness and exuberance of a 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. A 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 an 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, a 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 objective, 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. 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 an author does not think about it, the structure will originate during a creation process. It is better to think of a 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 non-compounding 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 have to make a note about terminology. Originally, this concept has been introduced as a Consistent Linking (CL) operation. However, in the literature, people began referring to this method by the authors’ 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 researches showed that in almost all instances, geometric linking does not produce correct results. On the other hand, Shestopaloff’s Linking (SL) operation allows 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 asset-period 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 more solid 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 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 vary capable 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 in-depth 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 a thickness of the corresponding line. Low priority threads are represented by dashed lines.

I.4. Organization of the book

	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 non-compounding 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 a 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 an 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 computational efficiency of different analytical and numerical methods is the main topic of Chapter 6. First, we introduce numerical methods for solving 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 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 a 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 non-compounding - becomes an additional parameter, or dimension, which has to be associated with investment performance measurement problems. It turns out that IRR equation is the most adequate instrument within the compounding context, while the non-compounding context is best served by a different tool, the generalized Modified Dietz equation. We introduce this equation as a general mathematical instrument to be used within the non-compounding 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 the 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 non-sequential periods and/or different assets. Depending on the task, the input data for these methods are periods’ 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, 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 a 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 an evaluation of attribution methods from this perspective. 
	Chapter 10 studies approaches to risk measurement, and currently 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 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.) 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 appreciation of 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 same 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 problem-solver 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. 

I.5. Acknowledgements

The author is thankful to his family for the support of the entire project. Sons Konstantin and Alexander helped a lot with editing and valuable editorial and style related advices, when the preliminary draft has been completed. Konstantin edited chapters one to five, while Alexander has taken care of chapter eight and the preface.
Table of contents: title: Table of Contents; title: About the Author	15; title: List of graphical notations and symbols	16; title: Preface	18; title: I.1. Who will gain, and what, from this book	18; title: I.2. A goal and general guiding concept of the book	20; title: I.3. Foundations of the study	21; title: I.3.1. Internal rate of return	21; title: I.3.2. Compounding	21; title: I.3.3. Shestopaloff’s Linking	22; title: I.3.4. Other key topics	23; title: I.4. Organization of the book	25; title: I.5. Acknowledgements	29; title: Chapter 1	30; title: 1. Understanding interest rate and rate of return. IRR equation	30; title: 1.1. Interest rate and rate of return	30; title: 1.1.1. Definition of interest rate	30; title: 1.1.2. A general note about mathematical approaches	32; title: 1.2. Introducing compounding	34; title: 1.3. Domain of interest rate. Range of applicability	36; title: 1.3.1. Specifics of negative interest rate	36; title: 1.3.2. The meaning of compounding when the period length is not an integer value	38; title: 1.4. Computing interest rate	40; title: 1.4.1. Relationship between the period length and interest	40; title: rate	40; title: 1.4.2. Computing interest rate for shorter or longer periods. Nominal and effective interest rates	41; title: 1.4.3. Mathematical foundations of interest rate calculations	42; title: 1.4.4. Computing interest rates. Numerical examples	45; title: 1.5. Continuous compounding.	48; title: 1.5.1. Continuous compounding. Definitions	48; title: 1.5.2. Continuous compounding versus discrete compounding. Numerical example	51; title: 1.5.3. Smooth exponential compounding function versus	53; title: the piecewise linear function	53; title: 1.6. IRR equation. An inherent relationship of compounding operation and cash flows	59; title: 1.6.1. How IRR equation accounts for cash transactions	60; title: inside the period	60; title: 1.6.2. Deriving simple form of IRR equation	62; title: 1.6.3. Non-compounding scenario	64; title: 1.7. Deriving a general form of IRR equation	66; title: 1.7.1. IRR Equation with discrete compounding	66; title: 1.7.2. IRR equation with continuous compounding	68; title: 1.7.3. Generalization of continuous compounding.	69; title: Force of interest	69; title: 1.7.4. Getting the “look and feel” of IRR equation	71; title: Chapter 2	77; title: Annuities	77; title: 2.1. Deriving an ordinary annuity from the IRR equation	78; title: 2.1.1. Annuity definition and specifics	78; title: 2.1.4. Annuity’s present value	82; title: 2.2. Annuities and withdrawals. Purchasing an annuity	84; title: 2.2.1. Accumulating required amount	85; title: 2.2.2. Finding the number of annuity payments	86; title: 2.2.3. Price of an annuity to receive regular payments	87; title: 2.3. Discussion of annuity’s features	89; title: 2.3.1. Annuity with a negative interest rate.	89; title: Numerical example	89; title: 2.4. Annuity Due	93; title: 2.4.1. Definition of annuity due	93; title: 2.4.2. Deriving a general formula for an annuity due	94; title: 2.4.3. Finding a lump sum payment	96; title: 2.4.4. Determining the number of payments	97; title: 2.4.5. Calculating a present value	97; title: 2.4.6. Notes with regard to relationship between	98; title: annuities and IRR equation	98; title: 2.5. Annuities with continuous compounding	100; title: 2.5.1. Ordinary annuity	100; title: 2.5.2. Finding a regular payment	102; title: 2.5.3. Calculating a lump sum payment	103; title: 2.5.4. Finding the number of payments	103; title: 2.5.5. Computing unknown interest rate	104; title: 2.5.6. Finding the present value	104; title: 2.6. Relationship between interest rates for periods with different lengths	105; title: 2.6.1. Computing interest rate for an arbitrary period length	108; title: Chapter 3	111; title: Mortgages. Deriving mortgage formulas from IRR equation	111; title: 3.1. Defining mortgage in mathematical terms	113; title: 3.1.1. Deriving a mortgage formula for remaining balance	113; title: 3.1.2. Finding payment amount	115; title: 3.1.3. Calculating a number of periods	116; title: 3.1.4. How much can we afford?	117; title: 3.2. Numerical examples, graphs and discussion	119; title: 3.2.1. Interest and principal amounts	119; title: 3.2.2. Influence of payment frequency	122; title: 3.2.3. Weekly and monthly payments versus annual payments	124; title: 3.3. Choosing the right method to calculate an interest rate for shorter or longer periods	127; title: 3.4. Generalization of mortgage equations	130; title: 3.4.1. Proportionality between the period payment and	130; title: principal amount	130; title: 3.4.2. Relationship between the interest rate and period length	131; title: 3.5. Mortgages with continuous compounding	134; title: 3.5.1. Mortgage equations for continuous compounding	135; title: 3.5.2. Finding the remaining balance	136; title: 3.5.3 Calculating payment amount	137; title: 3.5.4. Finding a number of payments	137; title: 3.5.5. How much can we afford?	138; title: 3.6. Comparing mortgages with discrete and continuous compounding	141; title: 3.7. Chapter conclusion. Merging mortgages and IRR equation	145; title: Chapter 4	146; title: Introducing return on investment.	146; title: Relationship between Modified Dietz formula and IRR equation	146; title: 4.1. Overview of problems and general approaches	146; title: 4.1.1. Relationship of rate of return and compounding context	148; title: 4.1.2. IRR equation as the main conceptual foundation of the study	148; title: 4.2. Dietz’s methods	150; title: 4.3. Taylor series expansion of IRR equation	156; title: 4.3.1. Relationship between the Modified Dietz methods and IRR equation	157; title: 4.3.1.1. Progression of methods for calculating of rate of return	159; title: 4.3.1.2. Sibling of the Modified Dietz method	160; title: 4.4. Accuracy of Modified Dietz method when compared to IRR equation	163; title: 4.5. Geometrical interpretation of Modified Dietz equation in relation to IRR function	167; title: 4.6. Conclusion	170; title: Chapter 5	171; title: Solving IRR equation. New and existing methods. Applications	171; title: 5.1. Newton-Raphson’s method	172; title: 5.2. Number of solutions of IRR equation.	176; title: The Theorem	176; title: 5.2.1. Available approaches. Transformation of IRR Equation	176; title: 5.2.2. Proof of the Theorem’s Corollary	179; title: 5.2.3. A numerical example of IRR equation with three roots	180; title: 5.3. Introducing new methods for solving IRR equation	183; title: 5.3.1. Linear approximation of IRR equation	183; title: 5.3.2. Quadratic approximations of the IRR equation	185; title: 5.3.3. Quadratic solution at point zero	186; title: 5.3.4. Generalization of iterative algorithms for IRR equation	189; title: 5.3.5. Iteration accuracy	191; title: 5.4. Series reversion method	194; title: 5.4.1. Computing series reversion coefficients	195; title: 5.4.2. Discussion of series reversion method	198; title: 5.5. Application of methods for solving IRR equation in the business environment	200; title: 5.5.1. Choosing the first approximate value of rate of return for iterative procedures	200; title: 5.5.2. Factors to be considered when adopting	201; title: computational methods	201; title: 5.5.2.1. Business need	201; title: 5.5.2.2. Compliance with standards. Relevance to existing methods	202; title: 5.5.2.3. Acceptable level of complexity for the average user	202; title: 5.5.2.4. Sufficiency of computational resources	202; title: 5.5.2.5. Data feeds availability	203; title: 5.5.2.6. Flexibility with regard to future business needs	203; title: Chapter 6	204; title: Computational efficiency of algorithms	204; title: for solving IRR equation	204; title: 6.1. Software implementation details	204; title: 6.2. Accuracy of methods for computing rate of return	205; title: 6.2.1. Comparison and hierarchy of computational methods from the accuracy perspective	205; title: 6.2.2. Solutions’ accuracy. Limitations of methods with regard to business scenarios	208; title: 6.4. Comparing computational performance of different methods	216; title: 6.5. New high performance computational methods for mortgages and annuities. Computing algorithms for financial calculators	219; title: 6.5.1. IRR equation for mortgages and annuities	220; title: 6.5.2. Specific properties of mortgage IRR function and its characteristic points	222; title: 6.5.2.1. General form of IRR function. Solution of IRR equation	222; title: 6.5.2.2. Asymptotic behavior of a mortgage IRR function. Characteristic points	224; title: 6.5.2.3. Transforming a mortgage IRR equation	226; title: 6.5.3. Approximating solution of the IRR equation	228; title: 6.5.4. An accuracy of X and Y approximations,	230; title: and their computational performance	230; title: 6.5.5. Generalization of X and Y approximations.	232; title: A-approximation method	232; title: 6.5.6. Adding CP-approximation	235; title: 6.5.7. Practical implementation of A-approximation method. Accuracy evaluation. Numerical examples	236; title: 6.5.7.1. Threshold function	236; title: 6.5.7.2. Power function	238; title: 6.5.7.3. Iterative algorithms	238; title: 6.5.8. Defining an iteration accuracy	240; title: 6.5.9. Conclusion	242; title: 6.6. Series reversion method for mortgages and annuities	243; title: 6.6.1. Mathematical algorithms	243; title: 6.6.2. Computational performance of series reversion algorithms	246; title: Chapter 7	247; title: Climbing to a conceptual level in understanding of rate of return. Mathematical surprises	247; title: 7.1. Compounding context of the IRR equation	250; title: 7.1.1. Distinguishing direct and reverse problems	251; title: 7.2. Non-compounding nature of Modified Dietz equation	254; title: 7.2.1. Calculating non-compounding rate of return	254; title: 7.2.2. Generalized Modified Dietz equation	255; title: 7.2.3. Comparison of methods’ contexts	256; title: 7.3. Numerical examples demonstrating conceptual differences between the Modified Dietz and IRR methods	258; title: 7.3.1. Introducing variable transaction time	258; title: 7.3.2. Influence of transaction time to the IRR and Modified Dietz rates of return. Positive cash flows	258; title: 7.3.3. Converging the time scale	261; title: 7.3.4. Intermediate note about methods’ objectivity	261; title: 7.3.5. Influence of negative cash flows	262; title: 7.3.6. Extreme scenarios with large cash flows	265; title: 7.4. More IRR derivatives.	270; title: NPV (Net Present Value) method	270; title: 7.5. Modified IRR method (MIRR).	273; title: Some common misconceptions	273; title: 7.5.1. Enhancing MIRR formula. Arbitrary period lengths	273; title: 7.5.2. Improving MIRR equation. Advantages and new features	275; title: 7.5.3. Inherent relationships of IRR and MIRR	276; title: 7.5.4. Specifics of compounding context of MIRR. Changing MIRR context	278; title: 7.6. Completing discussion on conceptual differences between methods for calculating rate of return	282; title: Chapter 8	285; title: Shestopaloff’s linking (SL) algorithms for investment performance measurement and trading. Positioning the geometric linking	285; title: 8.1. Introducing SL method	285; title: 8.1.1. A note on terminology	285; title: 8.1.2. Origins and the present status of SL algorithms	285; title: 8.2. Restrictive nature of geometric linking	288; title: 8.2.1. Applicability domain of geometric linking	288; title: 8.2.2. Mathematical formulation of the geometric linking operation	289; title: 8.2.3. Geometric Linking Theorem	290; title: 8.2.4. Importance of the Geometric Linking Theorem	292; title: 8.3. Shestopaloff’s Linking (SL) concept	295; title: 8.3.1. Linking operation. The origin and general features of  SL algorithms	295; title: 8.4. Shestopaloff’s linking for Modified Dietz formula	299; title: 8.4.1. Linking sequential periods	299; title: 8.4.2. Linking non-sequential periods	304; title: 8.4.3. Linking rates of return for different asset classes within the same period	305; title: 8.4.4. Numerical example for Shestopaloff’s linking of rates of return for sequential periods and different assets	306; title: 8.4.5. Discussion of analytical and system design benefits that SL methods provides	307; title: 8.4.6. Linking assets’ rates of return	310; title: 8.4.7. Data mining	310; title: 8.4.8. Data used in numerical examples	311; title: 8.5. Shestopaloff’s Linking for internal rate of return (IRR)	314; title: 8.5.2. Approximating Shestopaloff’s linking for IRR equation	316; title: 8.6. Computational performance of SL algorithms	320; title: 8.7. Cumulative properties of Shestopaloff’s linking algorithms	323; title: 8.8. SL algorithms and multidimensional analytical research of investment portfolio	324; title: 8.9. Business considerations with regard to application of investment performance measurement methods	327; title: 8.9.1. Limitations of existing methods	327; title: 8.9.2. How SL methods can improve the situation	329; title: 8.9.2.1. Diversification of analytical scenarios	330; title: 8.9.2.2. Computational performance	331; title: 8.9.2.3. Enhanced accuracy	331; title: 8.9.2.4. Removing computational redundancy	331; title: 8.9.2.5. Database size	331; title: 8.9.2.6. Optimal data structuring	332; title: 8.9.2.7. Advanced and efficient system design	333; title: 8.9.2.8. Localized changes of data	334; title: 8.9.2.8. Conclusion	334; title: 8.10. Hierarchy of methods for calculating rates of return	336; title: 8.10.1. A brief review of major methods	336; title: 8.10.2. IRR as the most objective method for	338; title: computing rate of return	338; title: 8.10.3. An inherent relationship of the IRR equation and cash flows	340; title: 8.10.4. Relationship of TWRR and Modified Dietz method	341; title: 8.10.5. Summary of IRR method’s features	344; title: 8.10.6. Schema of hierarchy of methods for calculating rate of return.	345; title: 8.10.7. Some considerations regarding industry standards	346; title: 8.11. Trading. Development of trading strategies using SL algorithms	348; title: 8.11.1. Review of trading methods	348; title: 8.11.2. Optimization of investment strategies	350; title: 8.11.3. Market efficiency from a real life perspective	352; title: 8.11.4. Analytical studies and development of trading strategies	353; title: 8.11.5. Trading on relative values instead of prices. Trading on rate of return	355; title: 8.11.6. Programming trading. Thinking realistically	357; title: Chapter 9	358; title: Investment attribution analysis	358; title: 9.1. The scope of attribution analysis	359; title: 9.2. Defining the contribution	361; title: 9.3. Arithmetic attribution	364; title: 9.3.1. Main equations of arithmetic attribution	364; title: 9.3.2. Decomposing contribution to relative return using eigenvalues. Symmetrical arithmetic attribution model (SAA)	371; title: 9.3.3. Removing interaction using symmetry considerations	374; title: 9.3.4. Ideal attribution model	376; title: 9.3.5. Introducing a principle of symmetry	378; title: 9.3.6. Introducing a symmetrical data model.	380; title: Numerical example	380; title: 9.3.7. Summary of SAA’s attribution formulas	383; title: 9.4. Examining Brinson-Fachler’s attribution model	388; title: 9.4.1. Brinson-Fachler’s model and noise factors	388; title: 9.4.2. Highlights of symmetrical attribution models	391; title: 9.4.3. Testing the symmetry of the Brinson-Fachler’s model	392; title: 9.4.4. Introducing symmetry for a referential	393; title: arithmetic attribution model (RAA)	393; title: 9.4. Global attribution. Application of the SAA model	399; title: 9.4.1. Global attribution. Definitions	399; title: 9.4.2. Global attribution basics	402; title: 9.4.3. Choosing the base attribution model	404; title: 9.4.4. Introducing a new global attribution model	404; title: 9.4.5. Currency hedging and multiple currency transactions	409; title: 9.5. Geometric attribution	414; title: 9.5.1. Comparison principles	414; title: 9.5.2. The present geometric attribution model	417; title: 9.5.2.1. Summary of the geometric attribution method’s features	421; title: 9.5.3. A new symmetrical geometric attribution model.	422; title: The Discriminant Theorem	422; title: 9.5.4. Comparing the new symmetrical geometric attribution model with the present geometric attribution method	431; title: 9.5.5. A ratio validation criterion for developing	434; title: attribution models	434; title: 9.5.6. Researching the present geometric attribution model from the perspective of ideal attribution model	435; title: 9.5.7. Reverse transformations	440; title: 9.5.8. Conclusion and general notes about SGA method	440; title: 9.6. Linking attribution parameters for sequential periods	443; title: 9.6.1. Business need to link attribution parameters	443; title: 9.6.2. Input data for linking attribution parameters	444; title: 9.6.3. Linking of geometric attribution parameters	445; title: 9.6.4. Linking of arithmetic attribution parameters	447; title: 9.6.5. Limitations with regard to attribution linking methods. Influence of methods for calculating rate of return	450; title: 9.6.5.1. Restrictions imposed by geometric linking	450; title: of rates of return	450; title: 9.6.5.2. The influence of methods for calculating rate of return to the objectivity of attribution analysis	451; title: 9.6.6. Linking attribution parameters on the basis	452; title: of rates of return	452; title: 9.7. Concluding the study of attribution models and methods	455; title: Chapter 10	457; title: Measuring Risk	457; title: 10.1. Types of risks and available risk measures	460; title: 10.1.1. Interest rate risk	460; title: 10.1.2. Market risk	461; title: 10.1.2.1. Specific risk	462; title: 10.1.3. Measuring market risk	463; title: 10.1.3.1. Value at risk as a probability based measure	463; title: 10.1.3.2. Expected shortfall	466; title: 10.1.3.3. Volatility	467; title: 10.1.4. Credit risk	468; title: 10.1.5. Operational risk	469; title: 10.2. Estimation of Value at Risk (VaR)	472; title: 10.2.1. Time period	472; title: 10.2.2. Confidence level	475; title: 10.2.3. Computing VaR	478; title: 10.2.3.1. Variance – covariance method (VCV)	479; title: 10.2.3.2. Historical simulation	481; title: 10.2.3.3. Monte – Carlo simulation	481; title: 10.2.3.4. General notes	482; title: 10.3. Introducing volatility	484; title: 10.3.1. Defining volatility via a standard deviation.	485; title: Discrete variables	485; title: 10.3.2. Standard deviation for continuous random variables	487; title: 10.3.2.1. Specifics of investment data	489; title: 10.3.2.2. An assumption about data independency	490; title: 10.3.3. Defining volatility via the exponential	491; title: moving average	491; title: 10.4. Sharpe ratio. The role of benchmarks in the risk measurement	494; title: 10.4.1. Importance of Sharpe ratio	497; title: 10.5. Introducing a notion of downside risk	498; title: 10.5.1. The origin and definition of the downside risk	498; title: 10.5.2. Example of downside risk valuation	499; title: 10.6. Implicit embedding of a time period into the risk measures based on volatility	504; title: 10.7. Other volatility based risk measures	506; title: 10.7.1. Sortino ratio	506; title: 10.7.2. Treynor ratio	507; title: 10.7.2.1. Beta coefficient	507; title: 10.7.2.2. Using beta in Treynor ratio	509; title: 10.7.3. Alpha coefficient	510; title: 10.7.4. Tracking error	511; title: 10.7.5. Information ratio	512; title: 10.7.6. M-squared (Modigliani-Modigliani)	513; title: 10.8. Completing the “risky” excursion	517; title: Chapter 11	519; title: Introducing a conceptual framework.	519; title: Exploring the reality	519; title: 11.1. Let us hit the Road. Numbers, formulas and assumptions	523; title: 11.2. Feeling the numbers	526; title: 11.3. Discussing applicability domain	528; title: 11.4. Consequences of separating methods from their applicability domains	531; title: 11.5. Mathematical piety and contempt of mathematics as two extreme approaches	535; title: 11.6. Introducing solutions’ quality	537; title: 11.7. Terms “hypothesis” and “theory”	542; title: 11.8. Solutions’ quality in a three-dimensional space	543; title: 11.9. Improving the adequacy of models to the reality	546; title: 11.10. Specifics of working environments and their influence to the development processes	551; title: 11.11. Openness of business and academic environments as a condition of their optimal development	558; title: 11.11.1. Unbiased acquisition of information as the most important feature of open systems	558; title: 11.11.2. Amateurs and professionals	561; title: 11.11.3. Some specific features of financial industry	563; title: 11.12. Progress as a story about improving adequacy	566; title: References	571; title: Index	575