Chapter 6 dynamic asset and liability management

CHAPTER Dynamic Asset and Liability Management Ricardo Matos Chaim CONTENTS 6.1 I ntroduction 6.2 Modeling of Pension Fund Dynamics 6.3 A sset/Liability Management 6.4 Pension Fund Governance 6.5 P opulational Dynamics 6.6 Multi-Paradigm Approach to Get a Dynamic ALM Model 6.6.1 Prospective and Retrospective Scenario Analyses 6.6.2 Multi-Criteria Analysis for a Decision-Making Process 6.6.3 Analytic Hierarchy Process 6.6.4 Measuring Attractiveness by a Categorical-Based Evaluation Technique (Macbeth) 6.6.5 Agent-Based Combined System Dynamics Models 6.6.6 Agent-Based Modeling and Fuzzy Logic 6.6.7 B ayes Theo rem 6.6.8 Monte Carlo Simulation 6.6.9 M arkov Chains 6.7 Mathematical Provision Simulation 6.8 C onclusion Acknowledgment References 130 131 132 135 137 141 141 142 142 143 143 144 147 148 149 149 150 51 54 129 © 2010 by Taylor and Francis Group, LLC 130 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling G ov er na nc e is a sy ste m composed of a great number of interdependent entities, with different degrees of relationships between many actors The g overnance o f a soc ioeconomic a nd po litical en vironment under a pension fund’s perspective is considered as a complex system in which t he interactions a mong t he actors influence t he governance a nd the governance influence their interactions, in a recursive way In order to be risk oriented and to cope with the peculiarities of complex systems, asset and liability management (ALM) models of pension funds problems incorporate, among others, stochasticity, liquidity control, populational dynamics, a nd dec ision dela ys to be tter f orecast a nd f oresee so lvency in t he long ter m Once A LM wa s established a s a fac tor-based model, a research consisting of 25 Brazilian pension funds chosen from among 313 that together had almost 70% of total segment assets of such entities was made so as to identify risk factors and to represent their cause-and-effect relationships Then, t o m odel u ncertainties o r t o enab le m ulti-criteria analysis, m ethods such a s a nalysis o f p rospective a nd r etrospective scenarios, s ystem dy namics, agent-based mo delling, Bayesian a nalysis, Markov chains, and measuring attractiviness by a categorical-based evaluation technique (Macbeth) were presented and discussed In conclusion, t his cha pter e vidences t he po wer o f a m ulti-paradigm m odel to study complex environments and offers a wa y to manage a dy namic ALM problem 6.1 INTRODUCTION Pension funds are nonfinancial institutions with a nonspeculative nature, and thus, assets and liabilities management is different from those of financial institutions, and so is risk management Due to the long-term nature of financial assets and because most pa rts of liabilities must be linked to pensions released on the occasion of workers’ retirement, pension funds are exposed to many risks that are difficult to be dealt with Those i nherent r isks t o ben efit p lans dema nd a st ructure o f ma ny investment policies that looks for an optimal allocation strategy and acts to seek sustained growth along with a socially responsible behavior Thei r complex goal is to offer benefit plans and to get an adequate income return in a way to assure an actuarial equilibrium in the long term The main functionalities of dynamic ALM models help to support decisions about asset allocations, pension costs, populational dynamics, mathematical p rovisions, ac tuarial eq uilibrium, a nd s o o n A s a fac tor-based model, ALM studies usually address many factors mostly qualitatively and © 2010 by Taylor and Francis Group, LLC Dynamic Asset and Liability Management ◾ 131 subjectively The co rporate g overnance o f a p ension f und inc ludes a s et of practices that may optimize its performance and protect all eco nomic agents involved, such as in vestors, employees, sponsors, and other interested parties Once ALM facili tates decisio n-making p rocesses in a p ension f und, methods and techniques such as p rospective and retrospective scenarios, multi-criteria analysis for decision-making processes, agent-based models combined with system dynamics (SD) models, fuzzy logic, Bayes theorems, Monte Carlo simulation, and Markov chains help to estimate and calibrate many parameters and insert stochasticity in the model 6.2 MODELING OF PENSION FUND DYNAMICS Essentially, a pens ion f und needs to dec ide per iodically how to a llocate investments over different asset classes and what would be t he contribution rate in order to fund its liabilities Modeling is an important part of a decision-making process Beginning with the key relationships among variables, models try to capture the best understanding of the current situation and to project available data so as to forecast the future, considering our interference or the absence of it The modeling of a pension fund benefit plan is generally concerned in achieving at least one of the following goals: Identify and estimate risk factors Imitate populational dynamics Characterize prices evolution to get a prospective cash flow Control solvency and liquidity Model changeable factors to get their systemic complexity Design stochastic scenarios and maintain stochastic control Control and charge shortfalls Calculate asset and liability durations Analyze hedging strategies 10 Produce assets allocation based on estimated liabilities 11 Produce pension payment over time on a long-term basis © 2010 by Taylor and Francis Group, LLC 132 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 12 Produce quantitative information for the board of trustees 13 Produce insurance pricing 14 Optimize social insurance policies Therefore, u ncertainties a nd r isks a re f requently ma naged by combined qualitative and quantitative methods and techniques that ensure adequate data for the analysis process 6.3 ASSET/LIABILITY MANAGEMENT Once e ach ben efit p lan s d ifferent l iabilities a nd o bjectives, pens ion funds need to produce a h igh-income return to correspond to long-term actuarial expectations and to pay different k inds of benefits ALM more usually refers to the projections of assets and liabilities over a l ong-term period, co mmonly o ver 30 y ears, co nsidering d ifferent s cenarios a nd under a stochastic modeling ALM ma y b e s een as a met hod t o ma nage r isks a nd uncer tainties in pension funds so as t o ensure adequate solvency and liquidity over time Considering their long-term obligations, pension funds’ planning horizon is large and generally address solvency and liquidity by many policies Acceptable and prudential investments, clear contribution criteria, and actuarial assessments over time are practices that aim to assure annually the equilibrium between accumulation and future payments to the participants of the plan If a shortfall or surplus occurs, many actions to assure actuarial equilibrium take place The process requires a g reat amount of information about the organization, its operations, and the performance of the market It comprises The analysis of an organization’s balance sheet Many actions and controls to manage credit, liquidity, market, and operational risks The use of statistical and mathematical methods to predict or forecast the future or to define a finite number of scenarios so as to model uncertainty Many b iometric, demogra phic, eco nomic, a nd administra tive fac tors in ALM mo dels treat s ome degree of uncer tainty Actuaries, directors, and economists in pension funds must interact with each other to decide over © 2010 by Taylor and Francis Group, LLC Dynamic Asset and Liability Management ◾ 133 allocation processes based on subjective information and variable liabilities Thus, there are basic principles that an ALM model must adhere to: • Deterministic modeling involves using a single “best guess” estimation of each variable within a model to determine its outcomes • Sensitivities determine how much that outcome might vary by structuring what-if scenarios Every possible value that each variable can take is weighted by the probability of its occurrence to achieve this • Within a risk analysis model, available data and expert opinions are important so urces o f i nformation a s a wa y t o cha racterize a nd t o quantify uncertainty In ALM studies, the term “risk management” can be somewhat misleading as “management” tends to imply some ability to influence or “control” events, a nd t his is not a lways t he c ase As a f ormal process, r isk fac tors in a pa rticular co ntext a re s ystematically i dentified, a nalyzed, a ssessed, ranked, and provided for As shown in Figure 6.1, Chaim (2006) related drivers of decision, their inherent risk factors, and many typical actions to better comprehend a pension fund’s dynamics over time ALM st udies a re g enerally co mbined t o o ne o r m ore m ean–variance models or techniques to quantify financial risks: Markowitz portfolio theory, capital asset pricing model (CAPM), asset pricing theory (APT), value at risk—V@r, Sharpe, duration, and many others Generally, attempting to predict the future based on past behavior or to take the present value of a future position, they try to know more about time series, and thus mitigate risks and reduce uncertainties A stochastic programming model for ALM is dynamic since the information on the actual value of uncertain parameters is revealed in stages From Drijver et al (2002), it is assumed that Because of the risks of underfunding, decisions on asset mix, contribution rate, and remedial contributions are made once a year Uncertainty is modeled through a finite number of scenarios given by a scenario tree Each scenario demands a complete set of decision variables at each time period because like t he total asset va lue, t he portfolio market value is given by the value of investments in each asset class and by the total value of liabilities © 2010 by Taylor and Francis Group, LLC Accumulation Maturity All stages Drivers of Decision ↑ Strategic asset allocation Inherent Risk Factors Typical Actions ↑ High-income (market risks) A portfolio with more risky assets is structured due to the need of credibility and to correspond to participants expectations ↓ Low-solvency (liquidity risks) Interest on new adhesions to reduce costs and to get more income ↑ Higher returns over investments Loans and other facilities to add value to participants and attract more of them ↓ Strategic asset allocation ↓ Low-income (market risks); A portfolio with less risky assets is structured to assure liquid yields to pay liabilities ↑ Assure punctual payments ↑ High-solvency (liquidity risks) The adhesions are generally closed ↓ Lower returns The loans follow a historical behaviour to maintain credibility and participant satisfaction Assure prudential investments Authorize new benefits plan Better manage the assets Low costs Good solvency Higher yields Legal risks: out of the limits fixed by the regulation Compliance Legal obligations and schedule Bad corporative governance Reducing transaction costs Market monitoring Actuarial assessments Emphasis on actuarial constraints and the plan’s equilibrium A program to maintain good internal controls is desirable in order to assure better corporate governance Economies of scale through volume of transactions and controlling the information flow to better decide and act accordingly the needs FIGURE 6.1 Inherent risks by maturity stage of a b enefit plan (Adapted f rom Chaim, R M., Combining ALM a nd system dy namics in pension f unds, in 24th International Conference of System Dynamics Society, Proceedings of th e 24th International Conference, Nijmegen, the Netherlands, 2006, Wiley Inter Science Available at: http://www.systemdynamics.org/conferences/2006/proceed/papers/ CHAIM315.pdf, accessed April 15, 2009.) © 2010 by Taylor and Francis Group, LLC 134 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling PF Phase Dynamic Asset and Liability Management ◾ 135 In a changing and complex environment, pension funds wealth management needs a m ore r obust i nvestment a llocation a pproach t han a st atic mean–variance a nalysis C ariño et a l (1994) proposed a m ultistage stochastic dynamic ALM model that includes stochastic controls and shortfall penalties Also, techniques like Brownian motion have been used in the search for better results (Kaufmann, 2005) Boulier et al (1996) consider that “stock returns are uncertain in efficient ma rkets, so st ochastic co ntrol w ould h elp i n finding t he o ptimal investment policy, as well as t he adequate level of contribution” (Cariño et al., 1994) Kaufmann (2005) used stochastic volatility models with jumps to estimate quartiles of financial risks for a 2-week period Due t o u ncertainty, it is d ifficult to quantify risk, especially in some special c ases I n t his way, Aderbi e t a l (2006) st udied t he properties of expected shortfall from a fi nancial risk management’s point of view “As a measure for assessing the financial risks of a portfolio,” they conclude that “expected shortfall appears a s a na tural choice to resort to when v@r i s unable to distinguish between portfolios with different riskiness” (Aderbi et al., 2006) Expected shortfall may be defined as “the average loss when value-at-risk is exceeded” giving “information about frequency and size of large losses” (Kaufmann, 2005) Therefore, ALM is a proactive, systematic analysis of possible events and responses to risks of different nature rather than a mere reaction mechanism t o t hose l imited e vents de tected I t i s abo ut ma naging t he f uture rather than administering past events, and it must be d irectly connected to actions that aim to assure good governance of a pension fund 6.4 PENSION FUND GOVERNANCE Governance is a system composed of a great number of interdependent entities, with different degrees of relationships As complex systems, there would be many interactions among diverse actors that may cause relevant discrepancies on their institutional performances Liability ma nagement decisions must consider u ncertain outcomes of events relevant to a pension fund business environment such as regulation, multiple accounts, multiple horizons for different goals, provisions for end effects, the uncertainty of future assets, and liabilities Figure 6.2 uses the SD’s feedback loop method to represent the dynamic of a t ypical pension fund and some organizational processes involved The symbol “+” means that factors are directly related, and “−” when they are inversely related It is possible to verify that many cause relations between variables, expressed by fe edback lo ops R1, R2, B1, B2, and B3, wh ich may ex plain © 2010 by Taylor and Francis Group, LLC 136 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling New benefit plan Fractional growth rate – + New participants + – + Expect liability Credibility + – R1 + B3 + + Contributions + Transaction and risk management costs + (2) R2 Pension payments B1 + Capital gains + Maturity delay – + Asset allocation + Shortfall costs + B2 FIGURE 6.2 Pension funds—typical dynamics some pens ion f und’s dy namics a nd c an a id t o p lan ac tions t o a ssure a better governance: R1—Good solvency (new participants → contributions → asset allocation → capital gains → credibility → new participants) More attractive plans could enhance contributions by means of new participants or more sponsors that may generate m ore acc umulation This way, m ore c redibility could a ttract more participants and, once there is more money, the solvency tends to get better by, for example, the share and reduction of estimated costs of the plan R2—Accumulation (a sset a llocation → c apital gains → asset a llocation) is a s ituation where money produces money A c apital gain means more money to invest This could lead to an exponential growth or decay phenomena in the future, depending on capital gains or losses B1—Credibility (new participants → contributions → asset allocation → transaction and risk management costs → new participants): More attractive p lans ma y a ttract m ore pa rticipants a nd t hen t he cost s ten d t o g et higher This dynamic may reduce adhesions © 2010 by Taylor and Francis Group, LLC Dynamic Asset and Liability Management ◾ 137 B2—Good wealth (transaction and risk management costs → asset allocation → shortfall costs → transaction and risk management costs): If there were costs, there would be less money to invest and, consequently, lower shortfall costs that may reduce costs If they reduce, there would be more money, much shortfall costs, and more costs This ba lancing dy namic could explain actions that aim to reduce costs B3—Benefit payments (ne w pa rticipants → exp ect lia bility → p ension payments → transaction and risk management costs → new participants) describe the process of accumulating funds and paying benefits It includes actions and practices to control the pension fund’s liquidity and solvency After co nsidering t he dy namics o f a s ystem, “t o u se co mputational based m odels i t i s n ecessary t o define t he w orld i n ter ms o f va riables,” “to imagine the world in terms of variables, to understand rates of change, to t hink at a s ystem level a nd to u nderstand c ausation i n a s ystem” (de Santos, 1992) Macroeconomics, biometrics, and actuarial classes of variables must be h olistically considered Ma ny r andom va riables identified by Rocha (2001) were returns over investment, interest rates, administrative taxes, capacity factor of salaries and benefits, and the rates of increase in salaries, all of them being economical factors Appendix 6.A.1 shows many factors identified by conducting research with financial ma nagers a nd ac tuaries a nd t heir i nterrelation o n a motricity-dependence ba sis ( Godet, 004) a s a wa y t o cla ssify t hem t o explain result variables On t he basis of this, the causation between variables on a pension fund is shown in Figure 6.3 It i s pos sible to verify ma ny c ause relations a mong va riables a nd t he influence of populational rates on a pension fund system 6.5 POPULATIONAL DYNAMICS The difference b etween a defined contribution and defined benefit plans is the lower cost of the former due to loss sharing among participants In both cases, there are many risk factors There are many particular risk factors that explain the system behavior once a defined benefit plan reaches maturity As the number of active participants decrease and pension payments increase, it becomes more important to hedge against liquidity risks ALM served an important role in elici ting requirements to b etter elaborate benefit and investment plans and to review the predictions underlying choice preferences It had a significant impact on the structure and parameterization of the final simulation model © 2010 by Taylor and Francis Group, LLC 138 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Salary Administrative taxes Withdrawal/ termination rate – – + Plan’s – – attractivity – + New participants – + + Present value of future contributions Present value of – future benefits – Assets + – + + Mortality rate – Withdrawal/ termination rate – + Mathematical provisions + Actuarial goals + Actuarial goals + Salary increases + Time of contribution + + + Average age of participants and relatives + Plan´s estimated costs Expected return – Actuarial interest rate Contributions + – + Pension costs Disability rate Long-term inflation Retirement rate – – Liabilities + – Perfomance of the plan + Liquidity Investment return Plan’s maturity FIGURE 6.3 Actuarial factors and their interrelationship in an ALM Model According to Winklevoss (1977, p 56), a population of a pension plan members consists of many subpopulations, for example, active employees, retired employees, vested terminated employees, disabled employees, and beneficiaries Figure shows t he populational dy namics a nd its i nfluences over the costs of a plan The word “S” means that factors are directly related a nd beha ve i n t he s ame s ide a nd “ O” wh en t hey a re i nversely related, behaving in the opposite side As pens ion f unds a re t ypically a m ulti-decrement en vironment (Winklevoss, 1977, pp 10–22), the causal loop diagram in Figure 6.4 shows the dynamics of a benefit plan Credibility reinforce new adhesions made by word of mouth and ad campaigns (R1); many decrements like mortality (B1), withdrawal (B2), disability (B3), and retirement (B4) are the balancing ways to reduce this population, and thus the costs of the benefit plan (Winklevoss, 1977, pp 10–22) Some details about each decrement follows: R1—Credibility me ans t hat p eople b ecome more a nd more i nterested i n adhering to a benefit plan of a pension fund This means more assets come from the participants and the organizations that are sponsors of the benefit © 2010 by Taylor and Francis Group, LLC Dynamic Asset and Liability Management ◾ 141 problem To cope with the complexity of pension systems, one can use the combination of many simulation approaches like agent-based models, system dynamics, and discrete event simulation (econometric model) They all require a co mbination of ma ny qualitative a nd quantitative methods and techniques in order to organize and represent information necessary to control risks and uncertainties 6.6 MULTI-PARADIGM APPROACH TO GET A DYNAMIC ALM MODEL Uncertainties sometimes mean risks and their associated probabilities of occurrence, which must be defined in tangible operational terms Many methods and techniques try to fit a theoretical distribution to observed data and give ways to a d ynamic model to forecast the possible results by estimators and probabilities A multi-paradigm approach based on a combination of subjective methods and techniques that, by nature, are scenarios st udies inc lude AHP, M acbeth, a nd a pproaches lik e syst em dynamics, ag ent-based mo deling, f uzzy logic, M arkov c hains, M onte Carlo simulations, and Bayesian nets may enhance ALM capability so as to be dynamically risk oriented 6.6.1 Prospective and Retrospective Scenario Analyses According to t he Centre for Tax Policy and Administration (CTPA, 2001), risk management implies abilities to influence or control future events Future studies focus on t he context in which t he problem under analysis is introduced and try to identify events that may frustrate organizational goals It can be seen as a formal process in which risk factors of a pa rticular context are systematically identified, analyzed, assessed, and prioritized Among o thers, t he p rocess a ims t o en hance t he co mprehension o f t he system’s environment a nd t heir interconnections, treat efficiently uncertainties, fac ilitate i nformation flows, a nd i ntegrate ma ny o rganizational sectors According t o Ma rshall ( 2002, p p 8–80), m odels ba sed o n sc enario analysis try to comprehend impacts of events over the organization They basically are stories that describe and determine the combination of risk events (Marshall, 2002, p 78) The author refers to multiple and prospective scenarios as a way to cope with uncertainties and market turbulences In such environments, it is difficult to define strategies and it is necessary to use tools capable of minimizing their effects over the organization © 2010 by Taylor and Francis Group, LLC 142 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling By m ultiple sc enarios, o ne o rganization ma y ex plore s ystematically uncertainties and their consequences over defined strategies It results in many t rajectories, r espective r isks, a ssociated a ctions, a nd st rategies t o follow in the event of the occurrence of risk events After quantitative or qualitative classification, the management should determine what risks will be t reated by the plan of action and with what resources At a conceptual level, four actions can occur with regard to the risk: (1) avoid, for example, moving away the system so that the risk cannot occur for a longer period of time; (2) reduce, for example, mitigating itself the probability and/or the consequences; (3) transfer, for example, by contract, by insurance, or by t he legislation; or (4) contain, for example, when the relation cost/benefit will become unfavorable in the handling of the risk Those o ptions a re e valuated f rom t he p reliminary f orm t o t he m ore efficient a nd a ppropriate dec ision ch osen f or t reating t he r isk The first three preventive measures should produce plans of contingency to t reat the effects of significant risks, if those are likely to occur 6.6.2 Multi-Criteria Analysis for a Decision-Making Process Different decision alternatives mostly demand methodologies to a nalyze diverse criteria that are presented in complex scenarios Such methodologies seek a so lution of consensus, since t he ideal one can be o utside t he adequate pos sibilities A mong d iverse ex isting tech niques, t here a re t he analytic hierarchy process (AHP) and Macbeth 6.6.3 Analytic Hierarchy Process As stated by Gomes et al (2004, p 67), the AHP method focuses on imprecise and subjective information to address model subjectivity When giving his preference or when relating to imprecise information, a ma nager uses a j udging sc ale t hat n ormally i s ba sed o n sen tences t o ex press h is preferences over any action or point of view, particularly when there is a comparison a mong va rious po ints o f v iew This p rocess co nsiders va riables t hat a re not de termined yet, t hat c annot be m easured, or t hat a re vague, fuzzy, or imprecise by nature AHP has been used to define priorities, to assess costs and benefits, to allocate resources, to measure performances, to make market researches, to elicitate requirements, to research the market, to perform forward and backward planning, to predict scenarios thus giving forecasts and insights over © 2010 by Taylor and Francis Group, LLC Dynamic Asset and Liability Management ◾ 143 future possibilities, to negotiate and solve conflicts, to anticipate decisions, and to make political or social predictions (Shimizu, 2001, p 278) The method starts on a de scription of t he d iverse criteria required to construct ma ny co mparative ma trices A fter esta blishing p riorities f or each c riteria a nd t heir co nsistency te sts, t he pa rity ma trice a ggregates each criteria to decision alternatives Then, plural priorities are composed in order to produce the information that could help managers to choose the better decision alternative 6.6.4 Measuring Attractiveness by a Categorical-Based Evaluation Technique (Macbeth) Asset allocations and liabilities’ estimation frequently require new methods that could help to numerically represent the judgments of managers on the global activity or actions over criteria used by an assessment model for the efficacy and the efficiency of the organization In this way, Macbeth is an interactive approach that helps to rationalize resources and to systematize procedures in troubled and complex contexts In order to structure a model and to develop a set of alternatives, judgments about t he degree of attractivity between t he elements of a finite group of potential ac tions ba sed o n sema ntical j udgments o ver t he d ifference between many attractivities noticed by managers based on their values 6.6.5 Agent-Based Combined System Dynamics Models The use of agent-based models to represent the population behavior of a pension fund participants and other socioeconomic and political environments is a method to provide deeper insights by simulation experiments The main stage is the definition of rules for the model agents’ behavior This approach a llows modeling of t he complexity a nd helps to clarify agents’ interactions a nd beha viors, f or ex ample, t he n onlinear beha viors o f t he system that are difficult to be captured with mathematical formalisms This m odeling tech nique d oes n ot a ssume a u nique co mponent t hat takes dec isions for t he s ystem a s a wh ole Agents a re i ndependent entities that establish their own goals, and have rules for the decision-making process and for the interactions with other agents The agents’ rules can be sufficiently simple, but the behavior of the system can become extremely complex (Gilbert, 2002), pa rticularly due to ma ny u ncertainties a nd t he risks to be managed © 2010 by Taylor and Francis Group, LLC 144 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Since decisions under uncertainty become complex especially because of the low comprehension of the system’s long-term interests as a wh ole, SD methods may provide a h olistic overview on t he u ncertainties of a n ALM analysis result The combination may improve managers’ abilities to set out tacit k nowledge, understand complexity, plan under uncertainty, and to establish policies enhancing, thus, the discussions and the training of businesses strategies in pension funds Figure 6 represents a dy namic A LM i n a pens ion f und b y a n S D method, in this case, a stock and flow diagram A stock and flow diagram is a system dynamic technique that shows the structure of a s ystem where correlations of factors automatically emerge from the system (Sterman, 2000) SD tools provide automatically stochastic d ifferential equations t hat could ex plain t he i nteractions of d ifferent factors in a pension fund Some key features of the dynamic’s mechanism are the ability to perform multiple simulations on a m odel under different conditions and to test the impact of different policies and predict side effects and the reactions provoked by many decisions over the system It is possible to consider a m odel enhancement that would permit the generation of an efficient set of alternate balance sheets It will be possible to explore t he price of risk associated with t he trade-off between investment a nd u nderwriting o pportunities Reg ulators, o n t he o ther nd, would be ab le t o o bserve u seful i nformation abo ut t he firm’s ab ility t o mediate risky managerial decisions and economic environments 6.6.6 Agent-Based Modeling and Fuzzy Logic As shown in Figure 6.7, an agent-based model’s goal is to improve the u nderstanding of t he behavior of a gents who ma ke dec isions t hat impact pension f und systems Th is model focuses on t he populational dynamics of a pension fund and its effects on a benefit plan cost Using table AT-2000, a g roup of 1000 participants is simulated for 100 years All participants start at different ages The outcomes of the system were as follows Every year, living participants are exposed to risks of death and disability, or retirement Retired participants are only exposed to risks of death Credibility is distributed randomly throughout the group by a triangular distribution S ome o utcomes o f t he m odel a re p robability d istributions that co uld ex plain t he beha vior o f t he s ystem, cost e stimates, a nd t he results of simulation events © 2010 by Taylor and Francis Group, LLC Rentability rate Assets Net growth of pension funds segment Capital gains Benefits net growth rate Contributions Credibility Non-participants Participants Plan’s adhesion Relation between participants and nonparticipants Expected losses Net growing rate Active participants Retirees Cessation rate Pensioners Cessation rate Relation between participants and liabilities Selic rate oscillation Market growth Stock market oscillation Retirement for age rate Historical volatilities Liabilities net present value Total risk INPC oscillation Influence of external market FIGURE 6.6 Stock and flow diagram including risks restrictions © 2010 by Taylor and Francis Group, LLC Plan’s equilibrium Assets Dynamic Asset and Liability Management ◾ 145 Survival probability Expected contributions by participant Benefits to pay 146 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling FIGURE 6.7 An agent-based simulation for a population of 1000 participants Once the agents in t he model and their behaviors over time a re identified, fuzzy rules are one of many possibilities to manage subjective factors or outside agents from a pension fund’s internal perspective As shown in Figur e 6.8, it is bas ed o n f uzzy r ules o f t he IF -THEN typ e t hat has Pressure of the agent financial market on the agent monetary authority If (interest rate quite high) AND (inflation not quite high or not high) THEN (high pressure for interest rate reduction) Expectations of the agent financial market in relation to decisions on interest rates If (inflation increases very much) THEN (expectation:aggressive increase interest rate) If (inflation decreases very little) THEN (expectation: reasonable increase interest rate OR expectation maintenance of interest rate) Statements of agent monetary authority to influence expectations on interest rate decisions If (inflation quite high OR high) AND (pressure reduction interest rates) THEN (statement difficulties) Perception of agent financial market in relation to the credibility of agent monetary authority If (inflation decreases) THEN (monetary authority credibility increases) If (inflation does not decrease) AND (decision interest rate reduction) THEN (monetary authority credibility decreases) If (high interest rate) AND (decision interest rate unchanged OR small increase) THEN (monetary authority credibility is maintained) Subset o f f uzzy r ules o f a n a gent o ver o thers ( From St reit, R and B orenstein, D., Appl Ar tif I ntell., 23 , 316, 009, a vailable a t http://dx.doi org/10.1080/08839510902804796.) FIGURE 6.8 © 2010 by Taylor and Francis Group, LLC Dynamic Asset and Liability Management ◾ 147 encapsulated agent beliefs, desires, or objectives This kind of knowledge influences the response to important external events relevant to decisionmaking processes Since pension f und ALM studies are related to probabilistic, complex human systems, “it is impractical or even impossible to rely only on mathematical model” (Streit and Boresnteins, 2009) The basic idea behind the use of the fuzzy extension for modeling multiagent systems is the specification and description of the agent behavior by means of fuzzy rules The inference of these rules can be understood as the mapping between a se t of inputs and a se t of outputs The practical reasoning of the agent consists of two principal activities (Schut et al., 2004; Shen et al., 2004): (1) deliberation, in which the agent decides what to (which intention to carry out) and (2) planning, which is the decision of how to carry out the intentions Thus, the inference of these rules during simulation establishes the dynamic behavior of each agent in the system, and a s a co nsequence, t he behavior of t he system a s a wh ole (Streit a nd Borenstein, 2009) 6.6.7 Bayes Theorem Thomas B ayes p roposed a met hod t o vie w p henomena, co nsidering probabilities in t erms o f b eliefs a nd in degr ees o f uncer tainty B ayes’ theorem relates the conditional and marginal probabilities of two random e vents It is o ften us ed t o co mpute p osterior p robabilities, gi ven observations, so that beliefs can be altered according to new data or new evidence (Figure 6.9) Bayes theorem is based on past information (a priori probabilities) and conditional probabilities that represent knowledge about phenomena, mostly subjective and based on specialist opinions Some actual information serves to e stimate c onditional pro babilities For e xample, w hen d iscussing l ife A priori probability P(B/A) P(A) P(A/B) = [P(B/A) P(A)] + [P(B/Ac) P(Ac)] A posteriori probability FIGURE 6.9 Bayes theorem © 2010 by Taylor and Francis Group, LLC Conditional probability 148 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling expectancy or population increases in a populational dynamics model, such a theorem can be used to compute the correct probability of the proposed parameter, g iven t hat o bservation B ecause i n a dy namic m odel, t here i s not only one response to uncertainty but new information is used to revise inferences over past information, giving more credibility to the model 6.6.8 Monte Carlo Simulation Many decision problems comprehend a dynamic sequence of decision problems and multiple stages (Shimizu, 2001, p 315) Monte Carlo simulation uses information of a historical series to estimate probability distributions that explain the random behavior of variables in a m odel that could interact with each other and that is used to visualize test parameters and thus assess simulation parameters As shown by the financial activity on a pension fund in Figure 6.10, the idea is to construct one empirical distribution of random variables that is desirable to estimate available values and to produce percentiles that may indicate extreme values of the distribution The simulation produces a great quantity of information and calculates average values, variance, and minimum and maximum values of the variables of the simulated problem in order to get results that can be verified with statistical tests when reproducing simulated phenomena FIGURE 6.10 SD simulation with factors based on probability distribution © 2010 by Taylor and Francis Group, LLC Dynamic Asset and Liability Management ◾ 149 6.6.9 Markov Chains Complex p roblems f ormulated o ver inf ormation ob tained f rom ma rket surveys or opinion researches are difficult to assimilate by managers at once; for them, it is difficult to comprehend new information because of their incapacity to visualize al ternative actions that may be presented A st ochastic p rocess is us ed t o mo del p henomena in w hich ma ny ndom va riables o r distr ibutions a re enco untered Markov c hains allo w t o model systems using a set of states and transitions over them, represented in scales of discrete time or continuous time, i.e., that the transitions over states may occur, for the former, in discrete points of time and, in the latter, at any instant of time According to Markov chains, t he st ates of t he mo del are dis crete and countable and the transition of states of the model may be of continuous or discrete means The transitions of a st ate to another does not consider past states, nor future states E ach transition uses a te or a p robability So, Markov chains require structuring of a probability tree to each event of the model and to evidence different alternatives and respective associated probabilities by means of a transition tree, which could reflect the desired situation against the original situation The process reaches its absorbent state when there is a n ull probability that it gets over this state and, thus, stays in such a state indefinitely 6.7 MATHEMATICAL PROVISION SIMULATION Each event (death, disability, retirement) has an associated event cost and is expressed in a mathematical provision formula: aa MPx = FCS [S x (1 + CS)r − x r − x Px a r v r − x − (Sx →r CN(%) ax :r − x ¬]{x < r} where MP = mathematical provisions FCS = capacity factor of salary; it reflects inflation CS = salary enhancement Sx · (1 + CS) r−x = s alary of one participant, projected to t he retirement age r r − x = for a participant of age x, the time remaining between the assessment date and the retirement date (r) © 2010 by Taylor and Francis Group, LLC 150 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling FIGURE 6.11 A simulation model of mathematical provision factors’ relationship pxaa = the probability of a pa rticipant of age x to be a live and active when reaching the age x of retirement ar = fac tor o f a nticipated ac tuarial i ncome r elated t o t he pa rticipant when initiating the retirement vr−x = discount factor considering the interval between ages r and x Sx→r = All salaries between ages x and r CN (%) = Taxes that represents the cost of the plan ax:r−x¬ = factor of an anticipated actuarial income, temporary, related to the activity period of the participant r −x Figure 6.11 shows outcomes of such a simulation model The model imitates reality representing t he acc umulation where a g reat a mount of money will finance retirements and pensioners in the future 6.8 CONCLUSION All methods described have a multidisciplinary and interdisciplinary characteristic as they draw from various fields of mathematics, statistics, software engineering, and administration in o rder to obtain the dynamics of the syst emic interrelationship of many r isk fac tors and t o quantify t heir probabilities of occurrence, so as to define complex scenarios and to perform a multi-criteria analysis that may aid organize available information This inf ormation ca n als o b e us ed in co mputational sim ulation mo dels © 2010 by Taylor and Francis Group, LLC Dynamic Asset and Liability Management ◾ 151 that ca n imi tate r eality, r epresent i ts co mplexity, a nd t hose t hat ma nage stochastically risks and uncertainties The need to anticipate the regulatory environment and factors interactions lead to dynamic models that may show stochastically their risk characteristics, a nd may a nticipate s ome issues t hat a re lik ely t o e volve The portfolio has to be managed against relevant benchmarks that must reflect yield targets, spreads, convexity, duration, quality, and liquidity A d ynamic ALM mo del r equires ma ny multi-paradigm met hods a nd techniques to mo del p ension f und r isks and uncer tainties Multiple s cenarios a nd m ulti-criteria a pproaches a re us eful t o ob tain sub jective a nd qualitative estimates based on specialists’ opinions combined with a historical series that aid in predicting the future based on projections or forecasts and, thus, proceed to better diagnosis over ALM problems With t he m ethods a nd tech niques p resented, A LM c an a pply o ther approaches a nd m ethodologies t hat ma y en hance t he ma nagement o f a pension f und i n order to ach ieve better governance, to better cope w ith risks, and to reduce uncertainties To model the dynamics of asset and liabilities’ relationship on a pension fund, i t i s n ecessary t o m odel ma ny r isk fac tors a nd t heir st ructural, complex, a nd dy namic interrelationship in a wa y to produce action a nd contingency plans to mitigate, reduce, or even eliminate risks Many ben efits came from this possibility The first benefit i s t o p ut together k nowledge t hat c ame f rom ac tuaries on one side a nd financial managers on the other side so as to test pension fund politics and to estimate t heir co nsequence o ver t he s ystem wh en o ne r isk fac tor va ries o r when some uncertainty is added to the model The s econd b enefit is t o p rovide simulation mo dels as co mputational games to train managers and to enhance the technical skills of technicians Some assum ptions a nd p ractices m ust b e w ell cumented, q uantified, and under stood in o rder t o b etter ma nage t he co mmunication b etween the board of directors, the administrative council, actuaries, and financial managers In this way, politics may be better managed and decision-making processes may be facilitated ACKNOWLEDGMENT Work pa rtially supported by Fundaỗóo de A poio P esquisa Di strito Federal (FAP-DF) © 2010 by Taylor and Francis Group, LLC 1(+) 1(+) 1(−) 1(−) 1(+) 1(+) 1(−) 1(−) 0 1(+) 0 1(+) 1(+) 1(+) 1(−) Total Liquidity Investment Return Plan’s Attractivity Withdrawal /Termination Rate Time of Contribution Average Age of Participants and Relatives New Participants Maturity of the Plan Expected Return Salary Rates of Mortality, Retirement, and Disability Longevity Indexes Long-Term Inflation Administrative Taxes Salary Increases 1(−) Contributions Motricity × Dependencies of Pension Variables 2 152 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling © 2010 by Taylor and Francis Group, LLC Mathematical Provisions Plan’s Cost Performance of the Plan Actuarial Goals Interest rates Actuarial goals Performance of the plan Plan’s cost Mathematical provisions Contributions Salary increases Administrative taxes Interest Rates APPENDIX 6.A.1 1(−) 1(+) 1(+) 1(−) 1(−) 1(+) 1(+) 1(+) 1(−) 1(+) 0 0 0 1(+) 1(+) 1(+) 1(−) 1(−) 1(+) 1(+) 1(+) 1(−) 1(+) 1(+) 1(+) 1(+) 11 © 2010 by Taylor and Francis Group, LLC 0 0 0 0 4 1(+) 1(+) 1(−) Dynamic Asset and Liability Management ◾ 153 Long-term inflation Longevity indexes Rates of mortality, retirement, and disability Salary Expected return Maturity of the plan New participants Average age of participants and relatives Time of contribution Withdrawal/ termination rate Plan’s attractivity Investment return Liquidity Total 154 ◾ 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Taylor and Francis Group, LLC Dynamic Asset and Liability Management ◾ 137 B2—Good wealth (transaction and risk management costs → asset allocation → shortfall costs → transaction and risk management. .. external market FIGURE 6. 6 Stock and flow diagram including risks restrictions © 2010 by Taylor and Francis Group, LLC Plan’s equilibrium Assets Dynamic Asset and Liability Management ◾ 145 Survival... Mortality decrements FIGURE 6. 5 Population dynamics—stock and flow diagram © 2010 by Taylor and Francis Group, LLC Cessation Pensioners mortality ratio Dynamic Asset and Liability Management ◾ 141 problem