Books on financial mathematics can be viewed on the publisher's website


Published articles on financial mathematics (.pdf format) are below

Financial Mathematics



(The Journal of Performance Measurement, 2013, V. 18, No. 1, p. 33-48).

Article received Dietz Award for the year 2014.

We consider the problem of multiple mathematical solutions of IRR equation, which is a mathematical base for numerous applications in financial industry, such as investment performance measurement, all yields’ related valuations for bonds, spot interest rates, forward rates, and lots of other applications. Previously, this problem has been studied mostly from the mathematical perspective and no satisfactory resolution has been found. Our research takes into account both mathematical and business aspects of the problem. We discovered and convincingly proved that the largest root of the IRR equation, which accordingly produces the largest rate of return, is the most adequate solution of the IRR equation, both from the mathematical and business perspectives, which should be used in practical computations of rate of return based on the IRR equation. Based on our study, we introduced the "Rule of the largest root" for choosing the right solution of the IRR equation, which effectively solves the problem of multiple roots of IRR equation. Solving this long standing problem, which is of very high practical and also theoretical importance in finance, opens lots of new opportunities for developing new robust financial instruments and advanced analytical methods.



(The Journal of Performance Measurement, 2012 Fall issue, p. 41-52). Note, this is a preprint version of the article.

Article considers the area of investment attribution analysis. It introduces mathematically grounded, objective principles and verification criteria for developing attribution models, which together create a conceptual attribution framework. Existing models have been scrutinized using these new concepts; the known inadequacies of these models with regard to objective representation of attribution parameters, such the interaction effect, have been confirmed. New attribution models have been developed based on introduced principles, thoroughly researched, verified, and compared with existing methods in terms of objectivity. The results proved that the new attribution models provide more objective and meaningful results than existing methods.


NEW High performance computational methods for mortgages and annuities

The Journal of Performance Measurement, 2011, Vol. 15, No. 2,  p. 41-54.

Generally, mortgage and annuity equations do not have analytical solutions for unknown interest rate, which has to be found using numerical methods. Another issue is that these equations have multiple solutions. We discovered an interesting property of mortgage and annuity functions that can be beneficially used for computing the interest rate. Namely, these functions have a single minimum and one or no inflection point, while the value of interest rate that corresponds to the minimum is equal to approximately one-half of the value of the correct solution. The appropriate values of interest rates for the minimum and inflection point have a very simple analytical presentation. Based on this discovered property, we introduce new computational methods that allow unambiguously choosing the correct solution and finding a very accurate value of unknown interest rate without solving the mortgage or annuity equation at all. This value can be used directly or, if its accuracy is not sufficient, as the first initial value in iterative procedures. Such an initial value guarantees convergence to the correct solution. We also propose some additional computational enhancements, which, taken together, significantly improve the overall computational performance of mortgage and annuity related calculations.


Properties of IRR Equation with Regard to Ambiguity of Calculating of Rate of Return and a Maximum Number of Solutions

Yu. K. Shestopaloff, W. Marty, The Journal of Performance Measurement, 2011, Vol. 15, No. 3,  p. 8-22.

 IRR equation is widely used in financial mathematics for different purposes, such as computing rate of return on investment, calculation of implied volatility, yield to maturity, calculation of interest rates for mortgages and annuities, etc. However, in general, IRR equation may have several solutions, which restricts its application. Thus, the knowledge of how to find these solutions and choose the right one is important. Our understanding of properties of IRR equation is not complete. This article studies when solutions of IRR equation exist and what factors define the number of solutions. It also explores other properties of IRR equation with regard to the problem of calculating rate of return in general, and interest rate for mortgages and annuities.


A MODEL FOR A global INVESTMENT attribution ANALYSIS, BASED ON a symmetricAL arithmetic ATTRIBUTION model

The Journal of Performance Measurement, 2009, Vol. 13, No. 3,  pp.42-49

This article proposes a method for a global investment attribution. This method can be applied to equity portfolios, when portfolios are composed of assets traded in different currencies; active currency management strategies, and other investment policies. We consider the problem of computing and decomposition of rates of return for such investments portfolios with multiple currencies, and provide a detailed and complete study of mathematical aspects, illustrated by a numerical example confirming the validity of analytical results. In the attribution analysis, we used a symmetrical arithmetic attribution model to find the global attribution parameters. The advantage of this attribution model is that it does not have the ‘noise’ terms, such as an interaction; thus, it provides more objective analysis. In addition, the article considers currency hedging and multiple currency conversions with regard to the global attribution analysis and computing rates of return.



The Journal of Performance Measurement, 2008, No. 4,  pp.29-39.

This article considers an existing geometric attribution model and introduces a new one. Analysis is based on a conceptual framework grounded on a symmetry principle. This principle closely relates to a more general principle of invariance to permutations, well developed in mathematics. The framework consists of a few fundamental principles, or concepts, validation tools and verification criteria. Geometric attribution methods are analyzed within this framework. Some inadequacies of the present attribution method have been discovered, and a new model has been developed and verified by the introduced validation criteria. Numerical examples are presented, and they confirm the analytical results.



This article received "Honourable Mention Award" from The Journal of Performance Measurement.

The Journal of Performance Measurement, 2007, Fall, Vol. 12, pp. 39-52.

This article explores two topics. The first is the relationships between mathematical algorithms for calculating rates of return on investment portfolios. Article introduces and describes a hierarchy between different methods. It is shown that mathematically the Internal Rate of Return (IRR) is the most adequate method among the all presently used approaches. It should be used as a standard or a true value all other methods have to refer to. Secondly the article describes new mathematical algorithms that overcome many drawbacks of the present approaches. These new methods are collectively called Shestopaloff’s linking (SL) methods. The following advantageous aspects of SL methods are considered - analytical research, system performance and system design.



The Journal of Performance Measurement, 2008, Winter, Vol. 13, pp. 37-50.

This article uncovers and explores fairly subtle issue at a first glance, namely the compounding and non-compounding contexts in the problem of finding rate of return for investment portfolios. In reality this is a very important factor that should be considered as an essential characteristic of any method for calculating rate of return. IRR equation represents purely compounding context of application. The appropriate equation for non-compounding context has been derived. It turned out to be a flavor of Modified Dietz equation. Despite the mathematical similarity this equation was derived independently based only on assumption of non-compounding context of the problem finding rate of return. It also has some additional features so that we decided to name it as a generalized Modified Dietz equation in order to emphasize its non-compounding origin.


CONSISTENT LINKING CONCEPT FOR FAST CALCULATION OF RATE OF RETURN AND RESEARCH OF INVESTMENT STRATEGIES (presently it is referred to in the literature as Shestopaloff's Linking operation)

This article received "Honourable Mention Award" from The Journal of Performance Measurement.

The Journal of Performance Measurement (JPM), 2005, Volume 10, No. 1, pp. 50-63.

This article introduces the concept of consistent linking and algorithms for implementing this concept. Consistent linking is functionally similar to geometric linking. Geometric linking is used for calculating rate of return for the overall period based on sub-period returns. However, geometric linking generally produces a different result from rate of return calculated for the overall period considered as a single period. This makes impossible the usage of geometric linking for precise calculation of rate of return based on sub-period returns. Consistent linking, unlike geometric linking, always produces the same rate of return as if it was calculated for entire period as a single one.



The Journal of Performance Measurement (JPM), 2004, volume 9, No. 2, pp. 14-23.

This study explores internal rate of return. High accuracy approximation method for calculating internal rate of return is suggested. It is also demonstrated that dollar weighted and time weighted methods closely relate to each other with the time weighted method being just a particular case of dollar weighted rate of return when considering linear approximation of internal rate of return equation.