BUSINESS ISSUES, COMPETITION AND ENTREPRENEURSHIP PENSIONS GLOBAL ISSUES, PERSPECTIVES AND CHALLENGES BUSINESS ISSUES, COMPETITION AND ENTREPRENEURSHIP Additional books in this series can be found on Nova’s website under the Series tab Additional e-books in this series can be found on Nova’s website under the eBooks tab No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services BUSINESS ISSUES, COMPETITION AND ENTREPRENEURSHIP PENSIONS GLOBAL ISSUES, PERSPECTIVES AND CHALLENGES ALEXANDRA WEBB EDITOR Copyright © 2017 by Nova Science Publishers, Inc All rights reserved No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description This button is linked directly to the title’s permission page on copyright.com Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: info@copyright.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works Independent verification should be sought for any data, advice or recommendations contained in this book In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services If legal or any other expert assistance is required, the services of a competent person should be sought FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS Additional color graphics may be available in the e-book version of this book Library of Congress Cataloging-in-Publication Data ISBN: H%RRN Published by Nova Science Publishers, Inc † New York CONTENTS Preface vii Chapter Pension Provision in the Robotic Age Qing-Ping Ma Chapter Pension Investment Strategies of Defined Contribution Plan Participants John A Turner and David M Rajnes Chapter Chapter Index Profit-Sharing and Personal Pension Products: A Proposal Valeria D’Amato, Emilia Di Lorenzo, Marilena Sibillo and Roberto Tizzano Searching for Answers to the Maintenance Problem of Insufficiently Financed, Financially Dependent Pension Funds Through Stochastic Diffusion Processes Manuel Alberto M Ferreira 39 97 113 127 PREFACE Understanding the strategies underlying pension investing is important because pension assets are one of the largest pools of investment assets in the world Chapter One reports on how the widespread application of robots and artificial intelligence may dramatically reduce the number of human workers in employment, such that more people will be either unemployed or out of job market altogether Chapter Two discuss both economic theories that predict what rational pension investors would (normative theories) and theories that attempt to explain why individual pension participants hold the portfolios that they actually hold (positive theories) Chapter Three develops a variable annuity pension scheme in which the insureds are entitled to participate in the annual financial results of the invested funds, with an embedded option to withdraw such results during the whole contract duration, that is during both the deferment and the annuitization periods The generic case of a pension fund that it is not sufficiently auto financed and it is thoroughly maintained with an external financing effort is considered in Chapter Four Chapter - Human society is moving toward a new age in which robots and artificial intelligence play key roles in economic activities The widespread application of robots and artificial intelligence may dramatically reduce the number of human workers in employment, such that more people will be either unemployed or out of the job market altogether The employed viii Alexandra Webb workers might receive lower wages, because of the competition from the unemployed workers The lower employment rate and the lower wage income, together with population aging and increased longevity, will present new challenges to the pension provision of the society The lower employment rate implies that a high proportion of the population may reach their retirement without making any meaningful contribution to the state pension systems The defined contribution (DC) pension plans or the notional individual accounts, which have been popular with pension providers who try to offload financial and longevity risks to employees, might not be able to address this challenge Currently, governments and pension researchers are more concerned with the decrease in the potential support ratio (the number of people aged 15–64 per one older person aged 65 or older), because it may lead to insolvency of the pension systems Although automation, robots and artificial intelligence have caused a noticeable loss of employment in some sectors, they are likely to be the solution to the projected pension funding gap as well Their application increases labor productivity, which will offset the effects of the decrease in the potential support ratio When most people are unemployed due to the wide application of robots and artificial intelligence, a guaranteed basic income through general taxation will be a more efficient approach for pension provision than the DC pension plans or the notional individual accounts Chapter - Understanding the strategies underlying pension investing is important because pension assets are one of the largest pools of investment assets in the world Pension investment issues differ between defined benefit plans, defined contribution plans, and hybrid plans, which combine features of both defined benefit and defined contribution plans The shift from defined benefit to defined contribution plans occurring in many countries generally places responsibility on individual participants for making investment decisions Pension investing is different from other types of investing While it is affected by investment principles applicable to investing generally, pension investing differs from non-pension investing due to tax considerations, the life-cycle nature of pension investments, and 118 Manuel Alberto M Ferreira 𝜃𝜑 (𝑟) 𝑣𝑟 (𝑎, 𝜃) = 𝐸[𝑉(𝑟, 𝑎, 𝜃)] = 1−𝜑𝑎 (𝑟) (4.1) 𝜃 It is relevant to note that 𝑢 (𝑎) lim 𝑣𝑟 (𝑎, 𝜃) = −𝑢𝑟´ (0) 𝜃⟶0 (4.2) 𝑟 FINITE TIME MAINTENANCE COST PRESENT VALUE Define the renewal process 𝑁(𝑡), generated by the extended sequence 𝑇0 = 0, 𝑇1 , 𝑇2 , …, by 𝑁(𝑡) = sup{𝑛: 𝑇𝑛 ≤ 𝑡} The present value of the pensions fund maintenance cost up to time t is represented by the stochastic 𝑁(𝑡) process 𝑊(𝑡; 𝑟, 𝑎, 𝜃) = ∑𝑛=1 𝜃𝑒 −𝑟𝑇𝑛 , 𝑊(𝑡; 𝑟, 𝑎, 𝜃) = 𝑖𝑓 𝑁(𝑡) = The important now is the expected value function of the process evaluation: 𝑤𝑟 (𝑡; 𝑎, 𝜃) = 𝐸[𝑊(𝑡; 𝑟, 𝑎, 𝜃)] Begin to note that it may be expressed as a numerical series Indeed, evaluating the expected value function conditioned 1−𝜑𝑛 (𝑟) by 𝑁(𝑡) = 𝑛, it is obtained 𝐸[𝑊(𝑡; 𝑟, 𝑎, 𝜃)|𝑁(𝑡) = 𝑛] = 𝜃𝜑𝑎 (𝑟) 1−𝜑𝜃 (𝑟) 𝜃 Repeating the expectation: 𝑤𝑟 (𝑡; 𝑎, 𝜃) = 𝐸[𝐸[𝑊(𝑡; 𝑟, 𝑎, 𝜃)]|𝑁(𝑡)] = 𝜃𝜑𝑎 (𝑟) − 𝛾(𝑡, 𝜑𝜃 (𝑟)) − 𝜑𝜃 (𝑟) (5.1) where 𝛾(𝑡, 𝜉) is the probability generating function of 𝑁(𝑡) Denote now the 𝑇𝑛 probability distribution function by 𝐺𝑛 (𝑠) and assume 𝐺0 (𝑠) = 1, for 𝑠 ≥ Recalling that 𝑃(𝑁(𝑡) = 𝑛) = 𝐺𝑛 (𝑡) − 𝐺𝑛+1 (𝑡), the above-mentioned probability generating function is 𝑛 𝛾(𝑡, 𝜉) = ∑∞ 𝑛=0 𝜉 𝑛−1 𝑃(𝑁(𝑡) = 𝑛) = − (1 − 𝜉) ∑∞ 𝐺𝑛 (𝑡) 𝑛=1 𝜉 Using the alternative notation, that seems more convenient now (5.2) Searching for Answers to the Maintenance Problem … 119 Entering with (5.2) in (5.1), 𝑤𝑟 (𝑡; 𝑎, 𝜃) is expressed in the form of the series 𝑛−1 𝑤𝑟 (𝑡; 𝑎, 𝜃) = 𝜃𝜑𝑎 (𝑟) ∑∞ 𝑛=1 𝜑𝜃 (𝑟) 𝐺𝑛 (𝑡) (5.3) Then, using (5.3), it will be stated that 𝑤𝑟 (𝑡; 𝑎, 𝜃) satisfies a renewal type integral equation Write for the 𝑤𝑟 (𝑡; 𝑎, 𝜃) ordinary Laplace transform 𝜓(𝜆) = ∞ −𝜆𝑠 ∫0 𝑒 𝑤𝑟 (𝑠; 𝑎, 𝜃)𝑑𝑠 Recalling that the probability distribution function 𝐺𝑛 (𝑠) of 𝑇𝑛 ordinary Laplace transform is given by ∞ ∫0 𝑒 −𝜆𝑠 𝐺𝑛 (𝑠)𝑑𝑠 = 𝜑𝑎 (𝜆) 𝑛−1 (𝜆) 𝜑𝜃 𝜆 and performing the Laplace transforms in 𝜃𝜑 (𝑟)𝜑𝑎 (𝜆) 𝜃 (𝑟)𝜑𝜃 (𝜆)) both sides of (5.3) it is obtained (𝜆) = 𝜆(1−𝜑𝑎 𝜓(𝜆) = 𝜃𝜑𝑎 (𝑟) 𝜑𝑎 (𝜆) + 𝜆 , that is 𝜓(𝜆)𝜑𝜃 (𝑟)𝜑𝜃 (𝜆) (5.4) Inverting the transforms in this equation both sides of (5.4) the following defective renewal equation 𝑡 𝑤𝑟 (𝑡; 𝑎, 𝜃) = 𝜃𝜑𝑎 (𝑟)𝐹𝑎 (𝑡) + ∫0 𝑤𝑟 (𝑡 − 𝑠; 𝑎, 𝜃)𝜑𝜃 (𝑟)𝑓𝜃 (𝑠)𝑑𝑠 (5.5) is achieved Now an asymptotic approximation of 𝑤𝑟 (𝑡; 𝑎, 𝜃) will be obtained through the key renewal theorem, following (Feller, 1971), pg 376 If in (5.5) 𝑡 → ∞ 𝑤𝑟 (∞; 𝑎, 𝜃) = 𝜃𝜑𝑎 (𝑟) + 𝑤𝑟 (∞; 𝑎, 𝜃)𝜑𝜃 (𝑟) (5.6) 𝜃𝜑 (𝑟) or 𝑤𝑟 (∞; 𝑎, 𝜃) = 1−𝜑𝑎 (𝑟) = 𝑣𝑟 (𝑎, 𝜃) 𝜃 That is: the expression (4.1) for 𝑣𝑟 (𝑎, 𝜃)is obtained again Subtracting each side of (5.6) from the corresponding each side of (5.5), and performing some elementary calculations the following, still defective, renewal equation 120 Manuel Alberto M Ferreira 𝑡 𝐽(𝑡) = 𝑗(𝑡) + ∫0 𝐽(𝑡 − 𝑠)𝜑𝜃 (𝑟)𝑓𝜃 (𝑠)𝑑𝑠 (5.7) where 𝐽(𝑡) = 𝑤𝑟 (∞; 𝑎, 𝜃) − 𝑤𝑟 (𝑡; 𝑎, 𝜃) and 𝑗(𝑡) = 𝜃𝜑𝑎 (𝑟)(1 − 𝐹𝑎 (𝑡)) + 𝜃𝜑𝑎 (𝑟)𝜑𝜃 (𝑟) (1 − 𝐹𝜃 (𝑡)) 1−𝜑𝜃 (𝑟) results Now, to obtain a common renewal equation from (5.7), it must be admitted the existence of a value 𝑘>0 such that ∞ 𝑘𝑠 ∫0 𝑒 𝜑𝜃 (𝑟)𝑓𝜃 (𝑠)𝑑𝑠 = 𝜑𝜃 (𝑟)𝜑𝜃 (−𝑘) = This imposes that the transform 𝜑𝜃 (𝜆) is defined in a domain different from the one initially considered, that is a domain that includes a convenient subset of the negative real numbers Multiplying both sides of (5.7) by 𝑒 𝑘𝑡 the common renewal equation 𝑡 is finally obtained: 𝑒 𝑘𝑡 𝐽(𝑡) = 𝑒 𝑘𝑡 𝑗(𝑡) + ∫0 𝑒 𝑘(𝑡−𝑠) 𝐽(𝑡 − 𝑠)𝑒 𝑘𝑠 𝜑𝜃 (𝑟)𝑓𝜃 (𝑠)𝑑𝑠 from which, through the application of the key renewal theorem, it results desired ∞ lim 𝑒 𝑘𝑡 𝐽(𝑡) = 𝑘 ∫0 𝑒 𝑘𝑠 𝑗(𝑠) 𝑑𝑠 𝑡→∞ (5.8) ∞ with 𝑘0 = ∫0 𝑠𝑒 𝑘𝑠 𝜑𝜃 (𝑟)𝑓𝜃 (𝑠)𝑑𝑠 = 𝜑𝜃 (𝑟)𝜑𝜃´ (−𝑘), provided that 𝑒 𝑘𝑡 𝑗(𝑡) is directly Riemann integrable The integral in (5.8) may expressed in terms ∞ of transforms as ∫0 𝑒 𝑘𝑠 𝑗(𝑠) 𝑑𝑠 = 𝜃𝜑𝑎 (𝑟)𝜑𝑎 (−𝑘) 𝑘 So, in this section: An asymptotic approximation, in the sense of (5.8) was obtained: 𝑤𝑟 (𝑡; 𝑎, 𝜃) ≈ 𝑣𝑟 (𝑎, 𝜃) − 𝑐𝑟 (𝑎, 𝜃)𝑒 −𝑘𝑟(𝜃)𝑡 (5.9) where 𝑘𝑟 (𝜃) is the positive value of k, solution of the following equation 𝜑𝜃 (𝑟)𝜑𝜃 (−𝑘) = and (5.10) Searching for Answers to the Maintenance Problem … 𝑐𝑟 (𝑎, 𝜃) = 𝜃𝜑𝑎 (𝑟)𝜑𝑎 (−𝑘𝑟 (𝜃)) 121 (5.11) ´ (−𝑘 (𝜃)) −𝑘𝑟 (𝜃)𝜑𝜃 (𝑟)𝜑𝜃 𝑟 BROWNIAN MOTION EXAMPLE Consider the diffusion process 𝑋(𝑡) underlying the reserves value behavior of the pensions fund is a generalized Brownian motion process, with drift 𝜇(𝑥) = 𝜇, 𝜇 < and diffusion coefficient 𝜎 (𝑥) = 𝜎 , 𝜎 > Observe that the setting satisfies the conditions that were assumed before to the former work, namely 𝜇 < implies condition (2.1) Everything else remaining as previously stated, it will be proceeded to present the consequences of this particularization In general, it will be added a ∗ to the notation used before because it is intended to use these specific results later To get the first passage time 𝑆𝑎 Laplace transform it must be solved (3.1) This is a homogeneous second order differential equation with constant coefficients, which general solution is given by 𝑢𝜆∗ (𝑎) = 𝛽1 𝑒 𝛼1 𝑎 + 𝛽2 𝑒 𝛼2 𝑎 , 𝑤𝑖𝑡ℎ 𝛼1 , 𝛼2 = −𝜇±√𝜇2 +2𝜆𝜎 𝜎2 Condition 𝑢𝜆∗ (∞) = implies 𝛽1 = and 𝑢𝜆∗ (0)=1 implies 𝛽2 =1 so that the following particular solution is achieved: 𝑢𝜆∗ (𝑎) = 𝑒 −𝐾𝜆 𝑎 (= 𝜑𝑎∗ (𝜆)), 𝐾𝜆 = 𝜇+√𝜇2 +2𝜆𝜎 𝜎2 (6.1) For this situation, the perpetual maintenance cost present value of the pensions fund is given by, following (4.1) and using (6.1), 𝑣𝑟∗ (𝑎, 𝜃) = 𝜃𝑒 −𝐾𝑟 𝑎 1−𝑒 −𝐾𝑟 𝜃 (6.2) Note that 𝑣𝑟∗ (𝑎, 𝜃) is a decreasing function of a and an increasing function of 𝜃 Proceeding as before, in particular lim 𝑣𝑟∗ (𝑎, 𝜃) = 𝜃⟶0 𝑒 −𝐾𝑟 𝑎 𝐾𝑟 (6.3) 122 Manuel Alberto M Ferreira To reach an expression for the finite time period maintenance cost present value, start by the computation of 𝑘𝑟∗ (𝜃), solving (5.10) This means finding a positive k satisfying 𝑒 −𝐾𝑟 𝜃 𝑒 −𝐾−𝜆 𝜃 = or 𝐾𝑟 + 𝐾−𝜆 = This identity is verified for the value of k 𝑘𝑟∗ (𝜃) = 𝜇2 −(−2𝜇−√𝜇2 +2𝑟𝜎2 ) , if 𝜇 < −√ 2𝜎 2𝑟𝜎 (6.4) Note that the solution is independent of 𝜃 in these circumstances A simplified solution, independent of a and 𝜃, for 𝑐𝑟∗ (𝑎, 𝜃) was also obtained Using (5.11) the result is 𝑐𝑟∗ (𝑎, 𝜃) = 2𝜎 (−2𝜇−√𝜇2 +2𝑟𝜎2 ) 𝜇2 −(−2𝜇−√𝜇2 +2𝑟𝜎2 ) (6.5) Combining these results as in (5.9) it is observable that the asymptotic approximation for this particularization reduces to 𝑤𝑟∗ (𝑡; 𝑎, 𝜃) ≈ 𝑣𝑟∗ (𝑎, 𝜃) − 𝜋𝑟 (𝑡), where the function 𝜋𝑟 (𝑡) is, considering (6.4) and (6.5), 𝜋𝑟 (𝑡) = 𝜇2 −(−2𝜇−√𝜇2 +2𝑟𝜎2 ) 2𝜎 (−2𝜇−√𝜇2 +2𝑟𝜎 ) 𝜇2 −(−2𝜇−√𝜇2 +2𝑟𝜎 ) 𝑒 − 2𝜎2 𝑡 , (6.6) 2𝑟𝜎 if 𝜇 < −√ ASSETS AND LIABILITY BEHAVIOUR REPRESENTATION It is proposed to consider now an application of the results obtained earlier to an asset-liability management scheme of pensions fund Assume that the assets value process of a pensions fund may be represented by the geometric Brownian motion process 𝐴(𝑡) = 𝑏𝑒 𝑎+(𝜌+𝜇)𝑡+𝜎𝐵(𝑡) with 𝜇 < and 𝑎𝑏𝜌 + 𝜇𝜎 > 0, where 𝐵(𝑡) a standard Brownian motion process is Searching for Answers to the Maintenance Problem … 123 Suppose also that the liabilities value process of the fund performs as the deterministic process 𝐿(𝑡) = 𝑏𝑒 𝜌𝑡 Under these assumptions, consider now the stochastic process 𝑌(𝑡) obtained by the elementary transformation of 𝐴(𝑡): 𝑌(𝑡) = 𝑙𝑛 𝐴(𝑡) 𝐿(𝑡) =𝑎+ 𝜇𝑡 + 𝜎𝐵(𝑡) This is a generalized Brownian motion process exactly as the one studied before, starting at a and with drift 𝜇 and diffusion coefficient 𝜎 Note also that the first passage time of the assets process 𝐴(𝑡) by the mobile barrier 𝑇𝑛 , the liabilities process, is the first passage time of 𝑌(𝑡) by 0-with finite expected time under the condition, stated before, 𝜇 < Consider also the pensions fund management scheme that raises the assets value by some positive constant 𝜃𝑛 when the assets value falls equal to the liabilities process by the 𝑛𝑡ℎ time This corresponds to consider the modification 𝐴̅(𝑡) of the process 𝐴(𝑡) that restarts at times 𝑇𝑛 when 𝐴(𝑡) hits the barrier 𝐿(𝑡) by the 𝑛𝑡ℎ time at the level 𝐿(𝑇𝑛 ) + 𝜃𝑛 For purposes of later computations, it is a convenient choice the management policy where 𝜃𝑛 = 𝐿(𝑇𝑛 )(𝑒 𝜃 − 1), for some 𝜃 > (7.1) The corresponding modification 𝑌̃(𝑡) of 𝑌(𝑡) will behave as a generalized Brownian motion process that restarts at the level 𝑙𝑛 𝐿(𝑇𝑛 )+𝜃𝑛 𝐿(𝑇𝑛 ) = 𝜃 when it hits (at times𝑇𝑛 ) Proceeding this way, it is reproduced via 𝑌̃(𝑡) the situation observed before when the Brownian motion example was treated The Laplace transform in (6.1) is still valid As to former proceedings, the results for the present case will be denoted with the symbol # It is considered the pensions fund perpetual maintenance cost present value, as a consequence of the proposed asset-liability management scheme, given by the random variable: 𝑉 # (𝑟, 𝑎, 𝜃) = −𝑟𝑇𝑛 𝜃 −(𝑟−𝜌)𝑇𝑛 ∑∞ = ∑∞ , 𝑟 > 𝜌 where r represents the 𝑛=1 𝜃𝑛 𝑒 𝑛=1 𝑏(𝑒 − 1)𝑒 appropriate discount interest rate To obtain the above expression it was only 124 Manuel Alberto M Ferreira made use of the 𝐿(𝑡) definition and (7.1) It is possible to express the expected value of the above random variable with the help of (6.2) as 𝑣𝑟# (𝑎, 𝜃) = 𝑏(𝑒 𝜃 −1) 𝜃 ∗ (𝑎, 𝑣𝑟−𝜌 𝜃) = 𝑏(𝑒 𝜃 −1)𝑒 −𝐾𝑟−𝜌 𝑎 1−𝑒 −𝐾𝑟−𝜌𝜃 (7.2) As 𝜃 → lim 𝑣𝑟# (𝑎, 𝜃) = 𝜃→0 𝑏𝑒 −𝐾𝑟−𝜌𝑎 𝐾𝑟−𝜌 (7.3) In a similar way, the maintenance cost up to time t in the abovementioned management scheme, is the stochastic process 𝑁(𝑡) 𝑊 # (𝑡; 𝑟, 𝑎, 𝜃) = ∑𝑛=1 𝑏(𝑒 𝜃 − 1)𝑒 −(𝑟−𝜌)𝑇𝑛 , if 𝑁(𝑡) = 0, with expected value function 𝑤𝑟# (𝑡; 𝑎, 𝜃) = 𝑏(𝑒 𝜃 −1) 𝜃 ∗ (𝑡; 𝑤𝑟−𝜌 𝑎, 𝜃) 𝑊 # (𝑡; 𝑟, 𝑎, 𝜃) = (7.4) CONCLUSION In the general diffusion scenery, the main results are formulae (4.1) and (5.9) The whole work depends on the possibility of solving equation (3.1) to obtain the first passage times Laplacce transforms Unfortunately, the solutions are known only for rare cases An obvious case for which the solution of the equation is available is the one of the Brownian motion diffusion process The main results concerning this particularization are formulae (6.2) and (6.6) Some transformations of the Brownian motion process that allowed to make use of the available Laplace transform may be explored as it was done in section Formulae (7.2) and (7.4) are this application most relevant results Searching for Answers to the Maintenance Problem … 125 REFERENCES Andrade, M., M A M Ferreira, and J A Filipe (2012) Representation of reserves through a Brownian motion model SPMCS 2012, JPCS-Journal of Physics: Conference Series 394 (2012) 012036, IOP Publishing http://dx.doi.org/10.1088/1742-6596/394/1/012036 Andrade, M., M A M Ferreira, J A Filipe and M Coelho (2012) The study of a pensions fund equilibrium through an infinite servers nodes network International Journal of Academic Research, Part A, 4(3), 205-208 Bass, R F (1998) Diffusions and Elliptic Operators Springer-Verlag, New York Bhattacharya, R N., and E Waymire (1990) Stochastic Processes with Applications John Wiley & Sons, New York Feller, W (1971) An Introduction to Probability Theory and its Applications (vol II, 2nd Ed.) John Wiley & Sons, New York Ferreira, M A M (2012) Non-autonomous pensions funds maintenance costs study through a diffusion process International Journal of Academic Research, Part A, (6), 51-56 DOI:10.7813/2075-4124.2012/4-6/A.7 Ferreira, M A M (2016) Results and applications in statistical queuing theory 15th Conference on Applied Mathematics 2016, APLMAT 2016; Bratislava; Slovakia; 362-375 Ferreira, M A M., M Andrade, and J A Filipe (2012) Studying pensions funds through an infinte servers nodes network: a theoretical problem SPMCS 2012, JPCS- Journal of Physics Conference Series 394 (2012) 012035, IOP Publishing http://dx.doi.org/10.1088/1742-6596/394/ 1/012035 Ferreira, M A M., M Andrade, J A Filipe and M Coelho (2011) Statistical queuing theory with some applications International Journal of Latest Trends in Finance and Economic Sciences, 1(4), 190-195 Ferreira, M A M and J A Filipe (2013) Costs study through a diffusion process of pensions funds held with an outside financing effort International Mathematical Forum, (28), 1359-1368 http: dx.doi.org/10.12988/imf.2013.36113 126 Manuel Alberto M Ferreira Figueira, J (2003) Aplicaỗóo dos processos de difusóo e da teoria renovamento num estudo de reservas aleatórias PhD Thesis presented at ISCTE-IUL, Lisboa, Portugal Figueira, J and M A M Ferreira (1999) Representation of a pensions fund by a stochastic network with two nodes: an exercise Portuguese Review of Financial Markets, (1), 75-81 Figueira, J and M A M Ferreira (2000) Financiamento dum fundo sujeito a ruína com base num modelo difusão Boletim Instituto dos Actuários Portugueses, 40, 5-20 Figueira, J and M A M Ferreira (2001) Maintenance cost of non-autonomous pensions funds: an evaluation via a regeneration scheme of diffusion processes Fifth International Congress in Insurance: Mathematics & Economics Pennsylvania State University, USA Figueira, J and M A M Ferreira (2003) Cost of non-autonomous pensions funds via an application of diffusion processes Review of Financial Markets, (1), 39-50 Filipe, J A., M A M Ferreira, and M Andrade (2012) Reserves represented by random walks SPMCS 2012, JPCS- Journal of Physics Conference Series 394 (2012) 012034, IOP Publishing http://dx.doi.org/10.1088/ 17426596/394/1/012034 Gerber, H U and G Parfumi (1998) Stop-loss a tempo continuo e protezione dinamica di un fondo d’investimento Revista di Matematica per le Scienze Economiche e Sociale, 21, 125-146 Karlin, S and H Taylor (1981) A Second Course on Stochastic Processes Academic Press, New York Refait, C (2000) Default risk estimation and stochastic calculus: application to French industrial firms, International Congresso in Insurance and Mathematics, Barcelona, Espanha INDEX # 1/n Investing, 59 20th century, 3, 10, 11, 12, 31 21st century, 3, 11 audit, 102, 109 automate, 19, 24, 25, 27 automated guided vehicles, 20 automatic enrollment, 44 automatic teller machines, 22 automation, viii, 2, 18, 19, 21, 22, 24, 27,32, 33 A access, 51, 52, 99 accounting, 7, 52, 61 accumulation, 42, 53, 99, 104 AlphaGo, 3, 25, 26, 30, 31 Amazon Go, 23 anchoring, 63, 64, 93 annuitize, 78, 94 annuity, 14, 15, 16, 50, 68, 78, 79, 80, 101, 104, 105, 108 arbitrage, 50, 51, 52 artificial intelligence (AI), vii, viii, 1, 2, 3, 4, 6, 17, 18, 19, 21, 25, 26, 28, 29, 31, 35 assets, vii, viii, 10, 15, 35, 39, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 54, 56, 57, 61, 63, 64, 66, 67, 69, 70, 71, 72, 73, 76, 80, 81, 92, 93, 102, 103, 122, 123 asymptotic approximation, 119, 120, 122 B banks, 43, 53 behavioral approaches., 42 behavioral biases, 41 behavioral economics, 58 behavioral finance, 73, 82 behavioral risk, 41 benefits, 5, 6, 9, 10, 14, 28, 30, 50, 54, 57, 72, 80, 81, 99, 101, 105, 108, 109 bequest motive, 78 bias, 52, 57, 58, 65, 73, 74 bond market, 77 bonds, 15, 41, 44, 45, 46, 47, 48, 49, 51, 52, 66, 67, 68, 69, 71, 72, 74, 76, 77 Brownian motion, x, 114, 115, 121, 123, 124, 125 Brownian motion process, 115, 124 128 Index buy and hold, 63 C capital gains, 51, 52, 67 capital markets, 42, 57, 71, 75, 82 capital preservation, 44, 94 cash, 22, 102, 103, 109 challenges, viii, 1, 4, 6, 86 chasing returns, 63, 74 choice overload, 59 chronic diseases, 12 communication technologies, 26 consumer protection, ix, 98, 99, 100 consumers, x, 98, 108 consumption, 14, 27, 28, 66, 76, 78, 80 conveyor belt system, 23 correlation, 49, 56, 57, 60, 77 cost, x, 25, 53, 71, 72, 99, 114, 115, 117, 118, 121, 122, 123, 124, 126 CRRA utility function, 108 customers, 22, 23, 24 D DB pension plans, 7, 14, 16 DC pension plans, viii, 2, 7, 16, 31 decumulation, 42 deep blue, deep learning, 25, 26 default investing, 61 default investment, 44, 45 defective renewal equation, 119 deferred annuity, 79 defined benefit pension, 50 defined benefit plans, viii, 39, 40, 41, 54, 76, 77, 82 defined contribution plans, viii, 39, 40, 41, 44, 47, 48, 49, 51, 54, 55, 56, 58, 64, 70, 71, 73, 76, 77, 79 demographic change, ix, 98, 100 demographic data, demographic dividends, demographic imbalance, 114 diffusion, x, 113, 114, 115, 116, 117, 121, 123, 124, 125, 126 diffusion process, 114, 115, 116, 121, 124, 125, 126 diffusion scale function, 116 distribution, 55, 103, 117 distributive pensions fund, 114 diversification, 45, 47, 48, 49, 54, 55, 56, 57, 58, 59, 73, 83 dollar cost averaging, 72 domestic economy, 57 domestic-only investing, 57 dwindling population, dynamic investment strategies, 42, 65, 71 dynamic strategy, 41 dynasty investing, 54 E earnings, 47, 66 economic development, economic landscape, economic problem, economics, ix, 29, 40, 41, 43, 58, 60, 82 economies of scale, 27 EIOPA, ix, 97, 99, 100, 107, 110, 111 elderly population, emerging markets, 58 empirical studies, 27 employees, viii, 1, 10, 14, 16, 21, 24, 27, 28, 55, 56, 98, 99 employer stock, 55 employers, 56, 98, 99 employment, vii, viii, 1, 2, 6, 19, 21, 28, 31, 50, 56, 79, 99 environment, ix, 46, 53, 96, 98, 100 equity, 15, 48, 51, 53, 58, 59, 60, 67, 68, 76, 77, 78, 80, 81 Index ethically-driven investing, 53 European Commission, ix, 98, 99, 110, 111 European Parliament, 98, 99, 111 evidence, 41, 43, 59, 60, 63, 74, 78 external financing, vii, x, 113, 115 F failing to rebalance, 63 failure to act, 63 federal government, 44, 63, 68 federal reserve, 86, 105 fertility, 2, 5, 6, 7, 8, 9, 10, 17, 26, 29, 31 fidelity, 88 financial, vii, viii, ix, x, 1, 6, 7, 14, 16, 24, 41, 43, 45, 46, 47, 49, 54, 57, 63, 66, 67, 69, 73, 75, 80, 81, 82, 85, 98, 99, 101, 103, 104, 108, 109, 114, 115 financial crisis, 47 financial markets, 57, 82 financial performance, 101 financial planning, 41 financial system, 82 finite time period maintenance cost present value, 122 first passage times, 114, 124 fixed rate, 46 flexibility, 69, 101 framing effects, 59 funded ratio strategy, 76, 77 funding, viii, 2, 35, 77, 85 funds, vii, ix, x, 15, 16, 40, 44, 45, 46, 47, 48, 50, 51, 52, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 73, 74, 77, 78, 80, 83, 85, 98, 99, 101, 105, 110, 113, 114, 116, 125, 126 G G7 countries, 5, 17 GDP, 17, 18, 27 129 General Motors, 20 general taxation, viii, 2, 28, 31 generalized Brownian motion process, x, 114, 121, 123 geometric Brownian motion process, 123 Germany, 5, 7, 8, 10, 12, 17, 44, 86 glide path, 67, 69, 70, 80 government securities, 44 governments, viii, 2, 4, 5, 16, 26 growth, 5, 6, 9, 14, 16, 17, 27, 28, 31, 77, 82 growth rate, 9, 16, 17, 28 guaranteed basic income, viii, 2, 29, 30, 31 H Halal investing, 53 Hawking, Stephen, 4, 35 hedging, 15, 78 heterogeneity, 42, 69, 82 high-end tax sheltering, 52 home-country bias, 57 homogeneous second order differential equation, 121 housing wealth, 36, 81 human, vii, 1, 2, 3, 4, 13, 17, 18, 19, 21, 24, 25, 26, 28, 29, 30, 31, 33, 35, 49, 56, 66, 77, 78, 80, 83 human behavior, 21 human brain, human capital, 49, 56, 66, 77, 78, 80 human workers, vii, 1, 2, 4, 17, 18, 19, 21, 26, 28, 29, 31 hybrid plans, viii, 39, 40 I inattention, 63, 64, 72, 73, 87 income, viii, 1, 2, 5, 10, 12, 29, 30, 31, 35, 36, 42, 44, 45, 46, 49, 50, 51, 52, 57, 62, 68, 70, 72, 76, 77, 78, 80, 98, 99, 101, 102, 111, 114 130 Index income distribution, 52 income hedging, 77 income tax, 29 individual retirement accounts (IRAs), 41, 53 individuals, 15, 40, 41, 49, 50, 51, 63, 64, 66, 69, 70, 79, 99, 100 industrial revolution, 19, 26 industry, 3, 19, 20, 21, 24, 49, 53, 57 inertia, 40, 56, 63, 64, 67, 72, 92 information and communications technology (ICT), 17, 26, 27 information technology, 17, 27 insensitivity to fees, 63 institutions, 14, 24, 41 integrated circuits, 21 intelligence, vii, viii, 1, 2, 35 interest rates, ix, 67, 68, 81, 98, 100, 105, 106 international investment, 58 investment, vii, viii, 15, 26, 27, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 80, 82, 85, 94, 99, 100, 109, 102 investment bank, 15 investment strategies in retirement, 78 investors, vii, ix, 40, 41, 46, 48, 51, 54, 58, 64, 68, 69, 72, 73, 74, 75, 76, 82, 90 issues, viii, ix, 2, 29, 39, 40, 42, 43, 48, 53, 54, 80, 97, 99, 100, 102, 109 K key renewal theorem, 119, 120 L labor force, labor market, 42, 65, 66, 68, 75, 77, 78, 80, 82 labor markets, 42, 65, 75, 82 labor productivity, viii, 2, 4, 6, 9, 16, 17, 26, 27, 31 Laplace transform, 117, 118, 119, 121, 123, 124 Lee Carter, 104 liability driven investing, 76 life assurer, 15 life cycle, 40, 54, 64, 66, 67 life expectancy, 2, 5, 6, 10, 11, 12, 13, 14, 17, 26, 31, 79, 81 life-cycle (target date) investing, 66 lifestyle funds, 48 lifetime, 5, 101, 103, 105 lights-out manufacturing, 19 liquid assets, 55 liquidity, 43, 45, 78, 81 longevity, viii, 1, 5, 6, 13, 14, 15, 16, 30, 34, 35, 46, 79, 101 longevity bonds, 15 longevity risk, viii, 1, 5, 6, 14, 15, 16, 30, 101 long-term investments, 100 long-term savings, 100 low investment risk, 44 M maintenance cost up to time t, 118, 124 management, 69, 102, 103, 109, 115, 122, 123, 124 manufactured goods, 27, 29 manufacturing, 9, 18, 19, 20, 21 market capitalization, 48 market timing, 74 mass, 2, 4, 6, 21, 24, 26, 28, 29 matter, ix, 28, 31, 45, 97 maximum life span, 13 menu-sensitive investing, 59 mobile barrier, 123 models, 81, 100 Index modern portfolio theory, 73 momentum, 48, 71, 73 momentum investing, 72 mortality, 5, 10, 12, 15, 16, 101, 105 mortality risk, 78 mortality swap, 16 myopic investing, 60 N Naïve diversification, 58, 84 national income, 28 negative income tax, 29 O one-child policy, optimization, 47, 48, 50, 86 organization for economic cooperation and development (OECD), 10, 18, 34, 35, 45, 92, 100, 111 P participant loans, 55 participant-directed plans, 82 participants, vii, viii, 39, 40, 41, 42, 43, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 82 participating benefits, 101 pension, vii, viii, ix, 1, 2, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 26, 28, 29, 30, 31, 35, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 85, 97, 98, 99, 100, 101, 103, 104, 108, 110, 111, 114 131 pension plans, viii, 1, 2, 5, 6, 7, 10, 11, 14, 16, 31, 40, 43, 44, 50, 51, 52, 54, 58, 60, 73, 77, 79, 82 pension reforms, 6, 16 pensionable age, 10 pensioners, 14, 98, 114 pensions fund, x, 113, 114, 115, 117, 118, 121, 122, 123, 125, 126 pensions fund perpetual maintenance cost present value, 123 performance analysis, 98 permit, ix, 40, 55 perpetual maintenance cost present value, 117, 121 perpetuity, 114 personal pensions, 99 personality traits, 73 pick-and-place robots, 21 policy, 2, 9, 30, 54, 64, 104, 107, 111, 123 pools, vii, viii, ix, 39, 40, 98, 99 population, viii, 1, 2, 4, 5, 6, 7, 9, 10, 12, 13, 17, 36 population aging, viii, 1, 2, 5, 6, 9, 10 portfolio, 41, 44, 45, 47, 48, 49, 51, 52, 56, 58, 59, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 88, 100, 101 portfolio investment, 70 positive correlation, 60 potential support ratio, viii, 2, 4, 6, 7, 9, 10, 16, 29, 31 present value, 50, 76, 115, 117, 118, 121, 122, 123 preservation, 44, 45, 62, 94 principles, viii, 39, 40 private personal pensions (PPP), 99, 100, 111 private sector, 44, 54, 62 probability density function, 117 probability distribution, 117, 118, 119 probability generating function, 118 product design, ix, 97, 99 132 Index product performance, x, 98 productivity growth, 4, 5, 6, 16, 17, 18, 26, 27, 28, 31 professionals, ix, 25, 40, 85 profit, ix, 98, 101, 102, 103, 105, 108, 109 profitability, x, 98, 101, 109 prospect theory, 74 protection, 50, 107, 110, 111, 115 protection cost present value expectation, 115 public pension, ix, 5, 97, 98 purchasing, 80 Q qualified default investment alternative (QDIA) , 62, 71 queuing theory, 114, 125 quill, 25, 26, 30 R random perpetuity, 117 rate of return, 44, 45, 46, 51, 54, 56, 71, 73 rebalancing, 71, 72 recency bias, 73 regeneration, x, 114, 115, 126 regeneration scheme, 115, 126 regenerative process, 117 regulations, 45, 55, 67 reinsurance, 14, 15 renewal equation, 114, 120 researchers, viii, 2, 4, 5, 13, 16, 17, 19, 25, 26, 30 reserves, x, 113, 115, 116, 121, 125 reserves value process, x, 113, 115, 116 resources, 5, 10, 14, 47, 79, 85, 116 response, 100, 110 restaurants, 21, 22, 23 retail, 22, 23, 83 retirement, viii, ix, 1, 5, 9, 10, 14, 16, 28, 34, 41, 42, 44, 45, 46, 52, 54, 55, 62, 63, 66, 67, 68, 69, 70, 72, 76, 77, 78, 79, 80, 81, 88, 92, 93, 94, 96, 97, 98, 100, 101, 114 retirement age, 5, 9, 10, 14, 28 retirement pension, 114 risk, viii, 1, 5, 6, 14, 15, 16, 19, 24, 30, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 53, 54, 56, 57, 60, 62, 63, 66, 67, 68, 69, 70, 71, 75, 76, 77, 78, 81, 82, 99, 101, 108, 109, 110, 126 risk aversion, 45, 49, 101, 108, 110 risk management, 24, 76 risk preference, 47, 68, 69 risk-return optimization, 47, 49, 50 robotic age, v, 1, 29, 30, 31 robotics, 2, 3, 19, 23, 26 robots, vii, viii, 1, 2, 3, 4, 6, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31 S savings, 4, 45, 46, 51, 66, 78, 88, 98, 100 securities, 15, 44, 52 self-checkout systems, 22 services, 22, 24, 28, 29 social security, 2, 49, 50, 66, 77, 79, 80, 81, 94 social welfare, 28, 29, 30 socially responsible investing (SRI), 53 society, vii, 1, 4, 29, 31 solution, viii, 2, 15, 26, 120, 121, 122, 124 stable value funds, 45 standard Brownian motion process, 123 state, viii, 1, 5, 7, 9, 10, 11, 14, 15, 16, 17, 28, 30, 31, 116 static investment strategies, 42, 43, 65 static strategy, 41 statistics, 18, 52, 111 ...BUSINESS ISSUES, COMPETITION AND ENTREPRENEURSHIP PENSIONS GLOBAL ISSUES, PERSPECTIVES AND CHALLENGES BUSINESS ISSUES, COMPETITION AND ENTREPRENEURSHIP Additional books... understanding that the publisher is not engaged in rendering legal, medical or any other professional services BUSINESS ISSUES, COMPETITION AND ENTREPRENEURSHIP PENSIONS GLOBAL ISSUES, PERSPECTIVES. .. Section looks into how pensions should be provided if robots and other AI systems cause mass unemployment; and Section concludes this article CHALLENGES FROM AN AGING POPULATION AND SLOW PRODUCTIVITY