We analyze two fundamentally different concepts to considering data for planning decisions using the example of a newsvendor problem in which observable features drive variations in demand.
Our work contributes to the extant literature in two ways. First, we develop a novel joint estimation-optimization (JEO) method that adapts the random forest machine learning algorithm to integrate the two steps of traditional separated estimation and optimization (SEO) methods: estimating a model to forecast demand and, given the uncertainty of the forecasting model, determining a safety buffer. Second, we provide an analysis of the factors that drive difference in the performance of the corresponding SEO and JEO implementations. We provide the analytical and empirical results of two studies, one in a controlled simulation setting and one on a real-world data set, for our performance evaluations.
We find that JEO approaches can lead to significantly better results than their SEO counterparts can when feature-dependent uncertainty is present and when the cost structure of overage and underage costs is asymmetric. However, in the examined practical settings the magnitude of these performance differences is limited because of the overlay of opposing effects that entail the properties of the remaining uncertainty and the cost structure.
{"title":"Machine Learning for Inventory Management: Analyzing Two Concepts to Get From Data to Decisions","authors":"J. Meller, Fabian Taigel","doi":"10.2139/ssrn.3256643","DOIUrl":"https://doi.org/10.2139/ssrn.3256643","url":null,"abstract":"We analyze two fundamentally different concepts to considering data for planning decisions using the example of a newsvendor problem in which observable features drive variations in demand.<br><br>Our work contributes to the extant literature in two ways. First, we develop a novel joint estimation-optimization (JEO) method that adapts the random forest machine learning algorithm to integrate the two steps of traditional separated estimation and optimization (SEO) methods: estimating a model to forecast demand and, given the uncertainty of the forecasting model, determining a safety buffer. Second, we provide an analysis of the factors that drive difference in the performance of the corresponding SEO and JEO implementations. We provide the analytical and empirical results of two studies, one in a controlled simulation setting and one on a real-world data set, for our performance evaluations.<br><br>We find that JEO approaches can lead to significantly better results than their SEO counterparts can when feature-dependent uncertainty is present and when the cost structure of overage and underage costs is asymmetric. However, in the examined practical settings the magnitude of these performance differences is limited because of the overlay of opposing effects that entail the properties of the remaining uncertainty and the cost structure.<br>","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123525553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The Wasserstein barycenter is an important notion in the analysis of high dimensional data with a broad range of applications in applied probability, economics, statistics, and in particular to clustering and image processing. In this paper, we state a general version of the equivalence of the Wasserstein barycenter problem to the n -coupling problem. As a consequence, the coupling to the sum principle (characterizing solutions to the n -coupling problem) provides a novel criterion for the explicit characterization of barycenters. Based on this criterion, we provide as a main contribution the simple to implement iterative swapping algorithm (ISA) for computing barycenters. The ISA is a completely non-parametric algorithm which provides a sharp image of the support of the barycenter and has a quadratic time complexity which is comparable to other well established algorithms designed to compute barycenters. The algorithm can also be applied to more complex optimization problems like the k -barycenter problem.
{"title":"On the Computation of Wasserstein Barycenters","authors":"Giovanni Puccetti, L. Rüschendorf, S. Vanduffel","doi":"10.2139/ssrn.3276147","DOIUrl":"https://doi.org/10.2139/ssrn.3276147","url":null,"abstract":"Abstract The Wasserstein barycenter is an important notion in the analysis of high dimensional data with a broad range of applications in applied probability, economics, statistics, and in particular to clustering and image processing. In this paper, we state a general version of the equivalence of the Wasserstein barycenter problem to the n -coupling problem. As a consequence, the coupling to the sum principle (characterizing solutions to the n -coupling problem) provides a novel criterion for the explicit characterization of barycenters. Based on this criterion, we provide as a main contribution the simple to implement iterative swapping algorithm (ISA) for computing barycenters. The ISA is a completely non-parametric algorithm which provides a sharp image of the support of the barycenter and has a quadratic time complexity which is comparable to other well established algorithms designed to compute barycenters. The algorithm can also be applied to more complex optimization problems like the k -barycenter problem.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121913742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In delegated portfolio management, we formulate a central-planned portfolio selection problem by multi-objective programming (utilities of the investor and the manager) to study the Pareto optimal portfolio and find Pareto improvement. First, we solve out two cases of the closed-form Pareto optimal portfolio based on non-smooth and non-concave utility optimization. One case has a special terminal outcome that the manager suffers a loss and the investor loses nothing, resulting that the optimal portfolio has a novel two-peak-three-valley pattern. We originally divide the optimal portfolio into three terms (Merton term, Aggressive term and Conservative term) to explain the pattern and conduct asymptotic analysis to illustrate economic insights. Second, we establish the collection of Pareto points of a single contract and prove that it is a strictly decreasing and strictly concave frontier. Third, we use Pareto frontiers to compare different contracts, showing that among first-loss contracts with long evaluation time, the investor benefits from the one with a smaller incentive rate and a smaller managerial ownership proportion. In addition, when the evaluation time is short, we discover a way of Pareto improvement by simultaneously adding the investor's utility into the manager's investment objective and increasing the manager's incentive rate.
{"title":"Central-planned Portfolio Selection, Pareto Frontier, and Pareto Improvement","authors":"Zongxia Liang, Yang Liu","doi":"10.2139/ssrn.3476392","DOIUrl":"https://doi.org/10.2139/ssrn.3476392","url":null,"abstract":"In delegated portfolio management, we formulate a central-planned portfolio selection problem by multi-objective programming (utilities of the investor and the manager) to study the Pareto optimal portfolio and find Pareto improvement. First, we solve out two cases of the closed-form Pareto optimal portfolio based on non-smooth and non-concave utility optimization. One case has a special terminal outcome that the manager suffers a loss and the investor loses nothing, resulting that the optimal portfolio has a novel two-peak-three-valley pattern. We originally divide the optimal portfolio into three terms (Merton term, Aggressive term and Conservative term) to explain the pattern and conduct asymptotic analysis to illustrate economic insights. Second, we establish the collection of Pareto points of a single contract and prove that it is a strictly decreasing and strictly concave frontier. Third, we use Pareto frontiers to compare different contracts, showing that among first-loss contracts with long evaluation time, the investor benefits from the one with a smaller incentive rate and a smaller managerial ownership proportion. In addition, when the evaluation time is short, we discover a way of Pareto improvement by simultaneously adding the investor's utility into the manager's investment objective and increasing the manager's incentive rate.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130411375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given an investment universe, we consider the vector [Formula: see text] of correlations of all assets to a portfolio with weights [Formula: see text]. This vector offers a representation equivalent to [Formula: see text] and leads to the notion of [Formula: see text]-presentative portfolio, that has a positive correlation, or exposure, to all assets. This class encompasses well-known portfolios, and complements the notion of representative portfolio, that has positive amounts invested in all assets (e.g. the market-cap index). We then introduce the concept of maximally [Formula: see text]-presentative portfolios, that maximize under no particular constraint an aggregate exposure [Formula: see text] to all assets, as measured by some symmetric, increasing and concave real-valued function [Formula: see text]. A basic characterization is established and it is shown that these portfolios are long-only, diversified and form a finite union of polytopes that satisfies a local regularity condition with respect to changes of the covariance matrix of the assets. Despite its small size, this set encompasses many well-known and possibly constrained long-only portfolios, bringing them together in a common framework. This also allowed us characterizing explicitly the impact of maximum weight constraints on the minimum variance portfolio. Finally, several theoretical and numerical applications illustrate our results.
{"title":"Portfolio Rho-Presentativity","authors":"Tristan Froidure, Khalid Jalalzai, Yves Choueifaty","doi":"10.2139/ssrn.2971867","DOIUrl":"https://doi.org/10.2139/ssrn.2971867","url":null,"abstract":"Given an investment universe, we consider the vector [Formula: see text] of correlations of all assets to a portfolio with weights [Formula: see text]. This vector offers a representation equivalent to [Formula: see text] and leads to the notion of [Formula: see text]-presentative portfolio, that has a positive correlation, or exposure, to all assets. This class encompasses well-known portfolios, and complements the notion of representative portfolio, that has positive amounts invested in all assets (e.g. the market-cap index). We then introduce the concept of maximally [Formula: see text]-presentative portfolios, that maximize under no particular constraint an aggregate exposure [Formula: see text] to all assets, as measured by some symmetric, increasing and concave real-valued function [Formula: see text]. A basic characterization is established and it is shown that these portfolios are long-only, diversified and form a finite union of polytopes that satisfies a local regularity condition with respect to changes of the covariance matrix of the assets. Despite its small size, this set encompasses many well-known and possibly constrained long-only portfolios, bringing them together in a common framework. This also allowed us characterizing explicitly the impact of maximum weight constraints on the minimum variance portfolio. Finally, several theoretical and numerical applications illustrate our results.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117048751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Assume asset returns follow a VARMA_MARCH structure, this paper derives the proper multi-horizon mean and covariance matrix estimations that can be used as inputs to mean-variance optimization problem for investors with different horizons. The result is further extended to vector error-correction model with GARCH errors. A simple example is given to show the significant impact of serial correlation to multi-horizon volatility and correlation estimation in asset allocation study. The result can also be applied to calculate multi-horizon volatility estimation for option trading purposes when the underlying model is built upon high frequency data.
{"title":"Multi-Horizon Mean-Covariance Estimation for Serial Correlated Returns","authors":"Zhuanxin Ding","doi":"10.2139/ssrn.3460754","DOIUrl":"https://doi.org/10.2139/ssrn.3460754","url":null,"abstract":"Assume asset returns follow a VARMA_MARCH structure, this paper derives the proper multi-horizon mean and covariance matrix estimations that can be used as inputs to mean-variance optimization problem for investors with different horizons. The result is further extended to vector error-correction model with GARCH errors. A simple example is given to show the significant impact of serial correlation to multi-horizon volatility and correlation estimation in asset allocation study. The result can also be applied to calculate multi-horizon volatility estimation for option trading purposes when the underlying model is built upon high frequency data.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129404396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using constrained optimization, we develop a simple, efficient approach (applicable in both unconstrained and constrained maximum-likelihood estimation problems) to computing profile-likelihood confidence intervals. In contrast to Wald-type or score-based inference, the likelihood ratio confidence intervals use all the information encoded in the likelihood function concerning the parameters, which leads to improved statistical properties. In addition, the method does no suffer from the computational burdens inherent in the bootstrap. In an application to Rust's (1987) bus-engine replacement problem, our approach does better than either the Wald or the bootstrap methods, delivering very accurate estimates of the confidence intervals quickly and efficiently. An extensive Monte Carlo study reveals that in small samples, only likelihood ratio confidence intervals yield reasonable coverage properties, while at the same time discriminating implausible values.
{"title":"Efficient Likelihood Ratio Confidence Intervals using Constrained Optimization","authors":"Gregor Reich, K. Judd","doi":"10.2139/ssrn.3455484","DOIUrl":"https://doi.org/10.2139/ssrn.3455484","url":null,"abstract":"Using constrained optimization, we develop a simple, efficient approach (applicable in both unconstrained and constrained maximum-likelihood estimation problems) to computing profile-likelihood confidence intervals. In contrast to Wald-type or score-based inference, the likelihood ratio confidence intervals use all the information encoded in the likelihood function concerning the parameters, which leads to improved statistical properties. In addition, the method does no suffer from the computational burdens inherent in the bootstrap. In an application to Rust's (1987) bus-engine replacement problem, our approach does better than either the Wald or the bootstrap methods, delivering very accurate estimates of the confidence intervals quickly and efficiently. An extensive Monte Carlo study reveals that in small samples, only likelihood ratio confidence intervals yield reasonable coverage properties, while at the same time discriminating implausible values.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115381603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider indifference pricing of contingent claims consisting of payment flows in a discrete time model with proportional transaction costs and under exponential disutility. This setting covers utility maximisation as a special case. A dual representation is obtained for the associated disutility minimisation problem, together with a dynamic procedure for solving it. This leads to an efficient and convergent numerical procedure for indifference pricing which applies to a wide range of payoffs, a large range of time steps and all magnitudes of transaction costs.
{"title":"Optimal Investment and Contingent Claim Valuation With Exponential Disutility Under Proportional Transaction Costs","authors":"Alet Roux, Zhikang Xu","doi":"10.2139/ssrn.3453265","DOIUrl":"https://doi.org/10.2139/ssrn.3453265","url":null,"abstract":"We consider indifference pricing of contingent claims consisting of payment flows in a discrete time model with proportional transaction costs and under exponential disutility. This setting covers utility maximisation as a special case. A dual representation is obtained for the associated disutility minimisation problem, together with a dynamic procedure for solving it. This leads to an efficient and convergent numerical procedure for indifference pricing which applies to a wide range of payoffs, a large range of time steps and all magnitudes of transaction costs.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"76 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120884673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A simple classical mechanics model is proposed to describe the political dynamics of human society. The model provides a mathematical structure for the philosophy of the state-man paradigm. The proposed model borrows many classical mechanical terms and generalizes them into the political domain. The well-known political terms (like freedom and revolution) are given mathematical definitions. Several abstract but solvable problems are presented to demonstrate the general principles of politiphysics, including the Laffer curve and effects of war, immigration, and random forces on society’s momentum of inertia.
{"title":"Introduction to Politiphysics","authors":"Gary Gindler","doi":"10.2139/ssrn.3454574","DOIUrl":"https://doi.org/10.2139/ssrn.3454574","url":null,"abstract":"A simple classical mechanics model is proposed to describe the political dynamics of human society. The model provides a mathematical structure for the philosophy of the state-man paradigm. The proposed model borrows many classical mechanical terms and generalizes them into the political domain. The well-known political terms (like freedom and revolution) are given mathematical definitions. Several abstract but solvable problems are presented to demonstrate the general principles of politiphysics, including the Laffer curve and effects of war, immigration, and random forces on society’s momentum of inertia.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129287499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give an overview and outlook of the field of reinforcement learning as it applies to solving financial applications of intertemporal choice. In finance, common problems of this kind include pricing and hedging of contingent claims, investment and portfolio allocation, buying and selling a portfolio of securities subject to transaction costs, market making, asset liability management and optimization of tax consequences, to name a few. Reinforcement learning allows us to solve these dynamic optimization problems in an almost model-free way, relaxing the assumptions often needed for classical approaches. A main contribution of this article is the elucidation of the link between these dynamic optimization problem and reinforcement learning, concretely addressing how to formulate expected intertemporal utility maximization problems using modern machine learning techniques.
{"title":"Modern Perspectives on Reinforcement Learning in Finance","authors":"Petter N. Kolm, G. Ritter","doi":"10.2139/ssrn.3449401","DOIUrl":"https://doi.org/10.2139/ssrn.3449401","url":null,"abstract":"We give an overview and outlook of the field of reinforcement learning as it applies to solving financial applications of intertemporal choice. In finance, common problems of this kind include pricing and hedging of contingent claims, investment and portfolio allocation, buying and selling a portfolio of securities subject to transaction costs, market making, asset liability management and optimization of tax consequences, to name a few. Reinforcement learning allows us to solve these dynamic optimization problems in an almost model-free way, relaxing the assumptions often needed for classical approaches. \u0000 \u0000A main contribution of this article is the elucidation of the link between these dynamic optimization problem and reinforcement learning, concretely addressing how to formulate expected intertemporal utility maximization problems using modern machine learning techniques.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125887073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Profit optimization imperative for any business. The businesses that are dealing with lots of stochastic variables he challenges become severe. Almost all of the business situations can be presented through a mathematical model. In this paper, the functioning of a financial institution such as the insurance firm is modelled as a stochastic queue. The cost model for the queue is developed and optimized for different stochastic parameters using pattern search and classical optimization techniques. An algorithm is written in MATLAB for the purpose. The paper can be referred by firms for practical implementation in order to maximize their profit.
{"title":"Optimization of a Multi-Server Stochastic Financial Queue","authors":"Bhupender Kumar Soam, Shweta Bhatia, Kirti Sharma","doi":"10.2139/ssrn.3462986","DOIUrl":"https://doi.org/10.2139/ssrn.3462986","url":null,"abstract":"Profit optimization imperative for any business. The businesses that are dealing with lots of stochastic variables he challenges become severe. Almost all of the business situations can be presented through a mathematical model. In this paper, the functioning of a financial institution such as the insurance firm is modelled as a stochastic queue. The cost model for the queue is developed and optimized for different stochastic parameters using pattern search and classical optimization techniques. An algorithm is written in MATLAB for the purpose. The paper can be referred by firms for practical implementation in order to maximize their profit.<br><br>","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122018250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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