Investment Performance Measurement. Mortgages and Annuities. Computing Algorithms. Attribution. Risk Valuation


Midwest Book Reviews (Oregon, WI, USA)

For dedicated mathematicians, there is as much art and beauty as there is science in their calculations, formulas, precepts, concepts, and expositions. There is also utility, practicality, insight, and value in the application of mathematical principals to financial systems and the economy which are complex compilations of factors that mathematicians develop models to explain otherwise inexplicable and seemingly random phenomena. That's why Yuri Shestopaloff's "Science of Inexact Mathematics: Investment Performance Measurement, Mortgages and Annuities, Computing Algorithms, Attribution, Risk Valuation" is such a seminal work in the field of applied mathematics to financial issues and economic performances with respect to investment strategies and interpretations. Offering detailed computing algorithms (including software implementation), the informed and informative text is enhanced with numerical examples, graphical and tabular illustrations throughout. A work of impressive scholarship, Yuri Shestopaloff's "Science of Inexact Mathematics" is especially recommended for academic, governmental, and professional library collections and is a valued contribution as a graduate level mathematics curriculum supplemental resource.

ForeWord Magazine

Business & Economics. SCIENCE OF INEXACT MATHEMATICS: INVESTMENT PERFORMANCE MEASUREMENT, MORTGAGES AND ANNUITIES, COMPUTING ALGORITHMS, ATTRIBUTION, RISK VALUATION by Yuri K. Shestopaloff (AKVY Press, 592 pages, hardcover, $89.95, 978-0-9809667-0-1): Doctor of Sciences and author of Sums of Exponential Functions and their New Fundamental Properties and more than eighty academic articles on mathematical modeling presents a comprehensive monograph on investment analysis, introducing new methods and unifying existing ones within a single conceptual framework; the book ranges from its theoretical underpinnings to the software implementation of particular algorithms, e.g. for the fast computation of the Internal Rate of Return.

Kirkus Discoveries, Nielsen Business Media, 770 Broadway, New York, NY 10003,


Mathematician and consultant Shestopaloff thoroughly explores the world of financial mathematics in

a volume that will be valuable to anyone in the field.

Beginning with interest and considering annuities, mortgages, and investment and risk measurement methods, Shestopaloff uncovers the complexities of investment mathematics with clear, understandable text accompanied by numerous derivations, examples, graphs and tables. Topics studied include the internal rate

of return—which the author considers in a lengthy discussion that includes its relationship with similar calculations—and nominal and effective interest rates. He also considers compounding using various computational methods and linking—a more accurate alternative to geometric linking, which is applied to

financial trading. Shestopaloff discusses measurement of risk with details of the various risks and quantifying methods that are involved in investing, such as risks in interest rate, volatility, operational risk, downside risk and more. He briefly explains the probabilistic calculations involved. The introductory text includes definitions of all terms and rapidly advances through equations to allow mathematicians of different skill levels to follow the explanations. An associated software package is available, and the author briefly reviews computation methods, as well as the accuracy obtained by different methods. Shestopaloff ends with a caution that—although software

may make many of these calculations invisibly and easily—it is still imperative to understand the mathematics behind the software. His

explanations are thorough without excessive wordiness and the text smoothly accompanies equations and derivations. The author helpfully

analyzes business consequences alongside the mathematics. The detailed index and table of contents, with paged references to subtopics, make

this a very convenient reference book. Although additional editing could have corrected minor linguistic issues, readers will find the text easy to

comprehend. Shestopaloff has presented many of these topics in previous peer-reviewed journal papers, but academics, students and

professionals—from programmers to financial mathematicians—will find this a convenient one-volume guide, well-written and seamless.

A valuable addition to the financial mathematician’s library.

Zentralblatt MATH Database 1931 – 2009, c 2009 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag (Berlin, Germany)

Zbl pre05526839

Shestopaloff, Yuri K.

Science of inexact mathematics. Investment, performance, measurement.

Mortgages and annuities, computing algorithms, attribution, risk valuation.


Toronto: AKVY Press. 591 p. EUR 69.95; $ 89.95; £ 69.95 (2009). ISBN 978-0-9809667-0-1/hbk

The book presents a coherent and comprehensive study of mathematical methods for investment performance measurement, attribution analysis, mortgages, annuities and

investment risk measurement. Mathematical backgrounds are comparatively simple, so the book can be useful both for academic studies and for practitioners. It consists of 11 chapters.

In Chapter 1 the well-known Internal Rate of Return (IRR) equation for cash flows with fixed compound interest rate is derived and discussed. The applications of this equation

to annuities and mortgages are considered in the Chapters 2 and 3. Chapters 4 and 5 are devoted to the calculation of rate of return and solving IRR equation. Computational

efficiency of algorithms for solving IRR equation is discussed on Chapter 6. Chapter 7 proceeds with a conceptual level in understanding rate of return. Influence of negative

cash flows, extreme scenarios with large cash flows, modified IRR method are studied. The new linking algorithms for investment performance measurement and trading are

introduced in Chapter 8. Chapter 9 and 10 are devoted to investment attribution analysis and measuring risk, correspondingly. Chapter 11 contributes to the problems

of “real” calculations: solutions’ quality, applicability domains, some specific features of financial industry.

Keywords : internal rate of return; lending and investment; annuities; investment performance

measurement; attribution models; modified Dietz equation; risk measurement

Classification :

91-01 Instructional exposition (Social and behavioral sciences)

91B28 Finance etc.

Library of Congress, USA

Science, Tech and Business recommender wrote about books by Yuri K. Shestopaloff the following (“Sums of exponential function and their new fundamental properties” and “Science of inexact mathematics”): “Both titles are very impressive academic works and the Library is very pleased to acquire these publications for inclusion to the collections”

Dr. Wolfgang Marty, Vice President of CREDIT SUISSE, Zurich, Switzerland

As I have worked in numerical analysis and in the banking industry, I think that the book adds great value to the area of performance measurement. The concepts are introduced with mathematical precision accompanied by thorough descriptions. I found many issues addressed for the first time. As an example, the role of the Newton-Raphson iteration and similar methods for computing the internal rate of rate are illustrated in detail. Generally, the importance and power of mathematical methods in performance measurement is greatly emphasized. The book adds value to in-depth portfolio analysis, because return and risk considerations are in the centre of the material presented. This book is a must for every professional in the finance industry.

Sergei Beliaev, Former Director of Department developing Investment Performance Measurement Systems at Royal Bank of Canada

I knew the author as a system architect, when he was designing an investment performance measurement system for the Royal Bank of Canada. (That time, I was a Director of this department.) In this review, I am looking at the book from the perspective of a system designer, one who has to understand all related mathematics to implement the system. I wish we had this book when we were developing our investment performance measurement systems. All the answers we were looking for are in this book, and even more. At the outset, we lacked an understanding of relationships between the different types of rates of return. The book completely covered this subject. We also did not know which computational algorithms are better and how they compared to each other in terms of performance, accuracy, etc. This book covers this information in-depth and is in itself a treasure trove for system designers. There is an interesting chapter on attribution analysis. The author covers all available methods and in addition, introduces a framework and develops new methods on his own, with impressive numerical results. I would also pay attention to new linking methods named in the literature for author. These are very valuable methods from a systems development perspective. Asset and period “slicing and dicing”, “asset schemas”, etc. are always a headache for system designers complicating things immensely, and this is something that these methods solve. Overall this book is a must have for any serious systems designer or performance measurement analyst.

5.0 out of 5 stars Useful for beginners and professionals, August 23, 2009, Review from

  A. Sharikov (LA, USA)  The author calls this manuscript a reference book. This is true, because beginners who specialize in the given area will find accurate definitions, necessary formulas for compound and non-compound use cases, many illustrations and practical examples of calculations of internal rate of return.
For the wider public, the book will be useful as good reading about the pitfalls of calculating internal rates of return when simple non-compound formulas are used to simplify calculation instead of more accurate compounding approaches.
On the other hand, professional practitioners will find analyses and examples on the implementation of numeric methods and computer algorithms, including a comprehensive first-hand explanation of Shestopaloff's linking (SL) method from its author.
SL allows one to combine internal rates information about different investment periods to find total rate of return. The method can be used to link sequential and non-sequential periods.
The author shows the relationship between SL and well-known geometric linking and how SL extends the geometric linking approach.
The author compares the results of all algorithms available today to prove SL effectiveness.
I found interesting the discussion of the important role of modified Dietz formula and its usage in numeric calculation.
The book describes different mathematical aspects of annuities, mortgages, the internal rate of return equation, investment attribution analysis, and risk assessments, and can probably be used for the development of new trading techniques.

5.0 out of 5 stars, Review from, by Angel

In my view, the pros are the comprehensiveness and depth of the subject coverage; a consistently scrupulous level of detail; flawless clarity of presentation. What appeals is that the reader can start from scratch and get to an advanced level in all areas of investment performance measurement just by reading this one book, in parts or as a whole. The volume combines the qualities of a reference book, well thought through and conceptually seamless monograph, and a handy manual, when it comes to practical application of investment performance methods and computational algorithms. All in one book! As a note, the computational and system implementation parts are unique, both with regards to the subject itself and, again, in the comprehensiveness of coverage. System designers should be happy to have this book when designing financial systems. Financial analysts, like myself, will benefit in all aspects of business knowledge. The good thing is that the book will be up-to-date for a long time, the depth of conceptual thinking that it presents will unlikely change for decades. I am not sure that all people need the small occasional insertions of general considerations, some of them almost of a philosophical level. Personally, I found some of them interesting. However, this is a minor thing. This is not that these insertions should not be there. They are just not for everybody. Otherwise, the book is an excellent buy.






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