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Articles on finance and financial mathematics are below |
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Articles by Yuri K. Shestopaloff (PDF format) |
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Color 3-D diagrams of rates of return for period and asset combinations. This is an illustration of how to use the consistent linking algorithms. |
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NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES 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. The Journal of Performance Measurement, 2011, Vol. 15, No. 2, p. 41-54 |
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A SIMPLIFIED METHOD FOR CALCULATING THE MONEY-WEIGHTED RATE OF RETURN 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. The Journal of Performance Measurement (JPM), 2004, volume 9, No. 2, pp. 14-23. |
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(presently the method is referred to in the literature as Shestopaloff's Linking operation) This article received "Honourable Mention Award" from The Journal of Performance Measurement. 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), 2005, Volume 10, No. 1, pp. 50-63. |
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A HIERARCHY OF METHODS FOR CALCULATING RATES OF RETURN This article received "Honourable Mention Award" from The Journal of Performance Measurement. 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 , 2007, Fall, Vol. 12, pp. 39-52. |
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THE ROLE OF CONCEPTUAL CONTEXT IN THE PROBLEM OF FINDING RATE OF RETURN 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. The Journal of Performance Measurement , 2008, Winter, Vol. 13, pp. 37-50. |
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GEOMETRIC ATTRIBUTION MODEL AND A SYMMETRY PRINCIPLE 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. The Journal of Performance Measurement, 2008, No. 4, pp.29-39. |
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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, 2009, V. 3, No. 13, pp.42-49. |