arXiv subjects

Takashi Shinzato

Publications and source records attributed to Takashi Shinzato.

At least 19 recordsLinked to original sources

arXiv2025

Trigonometric Ratios Can Prove the Pythagorean Theorem

Recent interest in noncircular trigonometric proofs has underscored the need for alternative methodologies. Jackson and Johnson's 2024 study addresses a longstanding gap in the foundations of trigonometric proofs. Inspired by the work of Jackson and Johnson [JJ24], we present three noncircular proofs of the Pythagorean theorem based on trigonometric identities. First, we establish the Pythagorean theorem via an isosceles triangle construction and the tangent double-angle formula. Second, we present an alternative proof utilizing an isosceles-triangle and the angle-bisector theorem. Third, we derive a novel trigonometric relation from the angle-bisector theorem, thereby unifying and extending the two preceding approaches. These approaches collectively demonstrate that the principal contribution of Jackson and Johnson lies in their strategic use of the double-angle formula. These proofs clarify the role of trigonometric identities independent of infinite series.

math.HO↗

arXiv2019

Relationship between optimal portfolios which can maximize and minimize the expected return

In recent years, the evaluation of the minimal investment risk of the quenched disordered system of a portfolio optimization problem and the investment concentration of the optimal portfolio has been actively investigated using the analysis methods of statistical mechanical informatics. However, the work to date has not sufficiently compared the optimal portfolios of different portfolio optimization problems. Therefore, in this paper, we use the Lagrange undetermined multiplier method and replica analysis to examine the relationship between the optimal portfolios of the expected return maximization problem and the expected return minimization problem with constraints of budget and investment risk. In particular, we derive the mean square error and the correlation coefficient of the optimal portfolios of these maximization and minimization problems as functions of a variable (the degree of risk tolerance) that can characterize the feasible subspace defined by the two constraints.

q-fin.PM↗

arXiv2019

Macroscopic theorem of the portfolio optimization problem with a risk-free asset

The investment risk minimization problem with budget and return constraints has been the subject of research using replica analysis but there are shortcomings in the extant literature. With respect to Tobin's separation theorem and the capital asset pricing model, it is necessary to investigate the implications of a risk-free asset and examine its influence on the optimal portfolio. Accordingly, in this work, we explore the investment risk minimization problem in the presence of a risk-free asset with budget and return constraints. Moreover, we discuss opportunity loss, the Pythagorean theorem of the Sharpe ratio, and Tobin's separation theorem.

q-fin.PM↗

arXiv2019

Minimal Investment Risk with Cost and Return Constraints: A Replica Analysis

Previous studies into the budget constraint of portfolio optimization problems based on statistical mechanical informatics have not considered that the purchase cost per unit of each asset is distinct. Moreover, the fact that the optimal investment allocation differs depending on the size of investable funds has also been neglected. In this paper, we approach the problem of investment risk minimization using replica analysis. This problem imposes cost and return constraints. We also derive the macroscopic theory indicated by the optimal solution and confirm the validity of our proposed method through numerical experiments.

q-fin.PM↗

arXiv2018

Replica Analysis for Maximization of Net Present Value

In this paper, we use replica analysis to determine the investment strategy that can maximize the net present value for portfolios containing multiple development projects. Replica analysis was developed in statistical mechanical informatics and econophysics to evaluate disordered systems, and here we use it to formulate the maximization of the net present value as an optimization problem under budget and investment concentration constraints. Furthermore, we confirm that a common approach from operations research underestimates the true maximal net present value as the maximal expected net present value by comparing our results with the maximal expected net present value as derived in operations research. Moreover, it is shown that the conventional method for estimating the net present value does not consider variance in the cash flow.

q-fin.PM↗

arXiv2018

Replica Approach for Minimal Investment Risk with Cost

In the present work, the optimal portfolio minimizing the investment risk with cost is discussed analytically, where this objective function is constructed in terms of two negative aspects of investment, the risk and cost. We note the mathematical similarity between the Hamiltonian in the mean-variance model and the Hamiltonians in the Hopfield model and the Sherrington{Kirkpatrick model and show that we can analyze this portfolio optimization problem by using replica analysis, and derive the minimal investment risk with cost and the investment concentration of the optimal portfolio. Furthermore, we validate our proposed method through numerical simulations.

q-fin.PM↗

arXiv2017

Random matrix approach for primal-dual portfolio optimization problems

In this paper, we revisit the portfolio optimization problems of the minimization/maximization of investment risk under constraints of budget and investment concentration (primal problem) and the maximization/minimization of investment concentration under constraints of budget and investment risk (dual problem) for the case that the variances of the return rates of the assets are identical. We analyze both optimization problems by using the Lagrange multiplier method and the random matrix approach. Thereafter, we compare the results obtained from our proposed approach with the results obtained in previous work. Moreover, we use numerical experiments to validate the results obtained from the replica approach and the random matrix approach as methods for analyzing both the primal and dual portfolio optimization problems.

q-fin.PM↗

arXiv2017

Replica Analysis for Portfolio Optimization with Single-Factor Model

In this paper, we use replica analysis to investigate the influence of correlation among the return rates of assets on the solution of the portfolio optimization problem. We consider the behavior of the optimal solution for the case where the return rate is described with a single-factor model and compare the findings obtained from our proposed methods with correlated return rates with those obtained with independent return rates. We then analytically assess the increase in the investment risk when correlation is included. Furthermore, we also compare our approach with analytical procedures for minimizing the investment risk from operations research.

q-fin.PM↗

arXiv2017

Pythagorean theorem of Sharpe ratio

In the present paper, using a replica analysis, we examine the portfolio optimization problem handled in previous work and discuss the minimization of investment risk under constraints of budget and expected return for the case that the distribution of the hyperparameters of the mean and variance of the return rate of each asset are not limited to a specific probability family. Findings derived using our proposed method are compared with those in previous work to verify the effectiveness of our proposed method. Further, we derive a Pythagorean theorem of the Sharpe ratio and macroscopic relations of opportunity loss. Using numerical experiments, the effectiveness of our proposed method is demonstrated for a specific situation.

q-fin.PM↗

arXiv2017

Property Safety Stock Policy for Correlated Commodities Based on Probability Inequality

Deriving the optimal safety stock quantity with which to meet customer satisfaction is one of the most important topics in stock management. However, it is difficult to control the stock management of correlated marketable merchandise when using an inventory control method that was developed under the assumption that the demands are not correlated. For this, we propose a deterministic approach that uses a probability inequality to derive a reasonable safety stock for the case in which we know the correlation between various commodities. Moreover, over a given lead time, the relation between the appropriate safety stock and the allowable stockout rate is analytically derived, and the potential of our proposed procedure is validated by numerical experiments.

math.OC↗

arXiv2016

Replica Analysis for the Duality of the Portfolio Optimization Problem

In the present paper, the primal-dual problem consisting of the investment risk minimization problem and the expected return maximization problem in the mean-variance model is discussed using replica analysis. As a natural extension of the investment risk minimization problem under only a budget constraint that we analyzed in a previous study, we herein consider a primal-dual problem in which the investment risk minimization problem with budget and expected return constraints is regarded as the primal problem, and the expected return maximization problem with budget and investment risk constraints is regarded as the dual problem. With respect to these optimal problems, we analyze a quenched disordered system involving both of these optimization problems using the approach developed in statistical mechanical informatics, and confirm that both optimal portfolios can possess the primal-dual structure. Finally, the results of numerical simulations are shown to validate the effectiveness of the proposed method.

q-fin.PM↗

arXiv2016

Maximizing and Minimizing Investment Concentration with Constraints of Budget and Investment Risk

In this paper, as a first step in examining the properties of a feasible portfolio subset that is characterized by budget and risk constraints, we assess the maximum and minimum of the investment concentration using replica analysis. To do this, we apply an analytical approach of statistical mechanics. We note that the optimization problem considered in this paper is the dual problem of the portfolio optimization problem discussed in the literature, and we verify that these optimal solutions are also dual. We also present numerical experiments, in which we use the method of steepest descent that is based on Lagrange's method of undetermined multipliers, and we compare the numerical results to those obtained by replica analysis in order to assess the effectiveness of our proposed approach.

q-fin.PM↗

arXiv2016

Universality of Makespan in Flowshop Scheduling Problem

Makespan, which is defined as the time difference between the starting time and the terminate time of a sequence of jobs or tasks, as the time to traverse a belt conveyor system, is well known as one of the most important criteria in scheduling problems. It is often used by manufacturing firms in practice in order to improve the operational efficiency with respect to the order of job processing to be performed. It is known that the performance of a machine depends on the particular timing of the job processing even if the job processing order is fixed. That is, the performance of a system with respect to flowshop processing depends on the procedure of scheduling. In this present work, we first discuss the relationship between makespan and several scheduling procedures in detail by using a small example and provide an algorithm for deriving the makespan. Using our proposed algorithm, several numerical experiments are examined so as to reveal the relationship between the typical behavior of makespan and the position of the fiducial machine, with respect to several distinguished distributions of the processing time. We also discuss the behavior of makespan by using the properties of the shape functions used in the context of percolation theory. Our contributions are firstly giving a detail discussion on the universality of makespan in flowshop problems and obtaining several novel properties of makespan, as follows: (1) makespan possesses universality in the sense of being little affected by a change in the probability distribution of the processing time, (2) makespan can be decomposed into the sum of two shape functions, and (3) makespan is less affected by the dispatching rule than by the scheduling procedure.

math.OC↗

arXiv2016

Validation of the Replica Trick for Simple Models

We discuss replica analytic continuation using several simple models in order to prove mathematically the validity of replica analysis, which is used in a wide range of fields related to large scale complex systems. While replica analysis consists of two analytical techniques, the replica trick (or replica analytic continuation) and the thermodynamical limit (and/or order parameter expansion), we focus our study on replica analytic continuation, which is the mathematical basis of the replica trick. We apply replica analysis to solve a variety of analytical models, and examine the properties of replica analytic continuation. Based on the positive results for these models we propose that replica analytic continuation is a robust procedure in replica analysis.

cond-mat.dis-nn↗

arXiv2016

Asymptotic Eigenvalue Distribution of Wishart Matrices whose Components are not Independently and Identically Distributed

In the present work, eigenvalue distributions defined by a random rectangular matrix whose components are neither independently nor identically distributed are analyzed using replica analysis and belief propagation. In particular, we consider the case in which the components are independently but not identically distributed; for example, only the components in each row or in each column may be {identically distributed}. We also consider the more general case in which the components are correlated with one another. We use the replica approach while making only weak assumptions in order to determine the asymptotic eigenvalue distribution and to derive an algorithm for doing so, based on belief propagation. One of our findings supports the results obtained from Feynman diagrams. We present the results of several numerical experiments that validate our proposed methods.

q-fin.PM↗

arXiv2016

Minimal Investment Risk of Portfolio Optimization Problem with Budget and Investment Concentration Constraints

In the present paper, the minimal investment risk for a portfolio optimization problem with imposed budget and investment concentration constraints is considered using replica analysis. Since the minimal investment risk is influenced by the investment concentration constraint (as well as the budget constraint), it is intuitive that the minimal investment risk for the problem with an investment concentration constraint be larger than that without the constraint (that is, with only the budget constraint). Moreover, a numerical experiment shows the effectiveness of our proposed analysis.

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arXiv2016

Portfolio Optimization Problem with Non-identical Variances of Asset Returns using Statistical Mechanical Informatics

The portfolio optimization problem in which the variances of the return rates of assets are not identical is analyzed in this paper using the methodology of statistical mechanical informatics, specifically, replica analysis. We define two characteristic quantities of an optimal portfolio, namely, minimal investment risk and concentrated investment level, in order to solve the portfolio optimization problem and analytically determine their asymptotical behaviors using replica analysis. Moreover, numerical experiments were performed, and a comparison between the results of our simulation and those obtained via replica analysis validated our proposed method.

q-fin.PM↗

arXiv2014

Self-Averaging Property of Minimal Investment Risk of Mean-Variance Model

In portfolio optimization problems, the minimum expected investment risk is not always smaller than the expected minimal investment risk. That is, using a well-known approach from operations research, it is possible to derive a strategy that minimizes the expected investment risk, but this strategy does not always result in the best rate of return on assets. Prior to making investment decisions, it is important to an investor to know the potential minimal investment risk (or the expected minimal investment risk) and to determine the strategy that will maximize the return on assets. We use the self-averaging property to analyze the potential minimal investment risk and the concentrated investment level for the strategy that gives the best rate of return. We compare the results from our method with the results obtained by the operations research approach and with those obtained by a numerical simulation using the optimal portfolio. The results of our method and the numerical simulation are in agreement, but they differ from that of the operations research approach.

q-fin.PM↗