Pension fund risk management financial and actuarial modeling 1

Preface INTEGRATED RISK MANAGEMENT IN PENSION FUNDS Marco Micocci, Greg N Gregoriou, and Giovanni B Masala The world of pension funds is facing a period of extreme changes Countries around t he w orld ve ex perienced u nexpected i ncreases i n l ife ex pectancy and fertility rates, changing accounting rules, contribution reductions, low financial returns, and abnormal volatility of markets All these elements have led to a fall in funded systems and to an increase in the dependency ratios in many countries U.K and U.S pension funds, which have traditionally had relatively high equity allocations, have been hit hard Many public pay-as-you-go (PAYGO) systems in Europe are reducing t heir “ generosity” w ith n ew c alculation r ules po inting t oward t he reduction of the substitution ratios of workers Europe is moving toward a risk-based approach also for the regulation and the control of the technical risk of funded pension schemes Risk management is becoming highly complex both in public pension funds and in private pension plans, requiring the expertise of different specialists who are not frequently disposable in the professional market The world is quite rich with skilled investment managers but their comprehension of the demographic and of the actuarial face of pension risk is often inadequate On the other hand, you have many specialized actuaries who are able to perform very sophisticated calculations and forecasts of pension liabilities but who are not able to fully understand the coexistence (or integration) of financial and actuarial risks Also, the international accounting standards i ntroduce new ac tuarial a nd financial elements i n t he ba lance sheet of the firms that may affect the corporate dividend and its investment ix © 2010 by Taylor and Francis Group, LLC x ◾ Preface policy In other words, little is being said about the integration of actuarial and financial risks in the risk management of pension funds We believe t he chapters in t his book highlight and shed new light on the current state of pension fund risk management and provide the reader new technical tools to face pension risk from an integrated point of view The exclusive new research for t his book can assist pension f und executives, r isk m anagement d epartments, c onsultancy firms, a nd ac ademic researchers to hopefully get a clearer picture of the integration of risks in the pension world The chapters in this book are written by well-known academics a nd p rofessionals w orldwide wh o ve p ublished n umerous journal articles and book chapters The book is divided into four parts— Part I: Financial Risk Management; Part II: Technical Risk Management; Part I II: Reg ulation a nd S olvency T opics; a nd P art I V: I nternational Experience in Pension Fund Risk Management In P art I , C hapter f ocuses o n t he co rrect m easurement o f r isk i n pension funds The author formalizes an intuitive concept of investment risk i n p roviding f or pens ions, t aking i t a s a m easure o f t he financial impact when the actual investment experience differs from the expected Investment risk can be explicitly measured and, through a series of case studies, the author estimates the investment risk associated with different investment strategies in different markets over the twentieth century He shows t hat w ithin a b road r ange, t he relative i nvestment r isk a ssociated w ith d ifferent st rategies i s n ot pa rticularly sens itive t o h ow t he pension objective is framed The investment risk associated with equity investment can be o f t he same order of magnitude as bond i nvestment if the bond duration mismatches those of the targeted pension He suggests that failure to explicitly measure investment risk entails that pension portfolios might not be optimally structured, holding the possibility that i nvestment r isks co uld be r educed w ithout r educing t he ex pected pension proceeds In Chapter , t he authors scrutinize t he f und dy namics u nder a per formance-oriented a rrangement (i.e., bonus fees a nd downside pena lty), whereby a st ochastic co ntrol i s f ormulated t o f urther cha racterize t he defined contribution (DC) pension schemes A five-fund separation theorem i s der ived to cha racterize its optimal st rategy W hen per formanceoriented arrangement is taken into account, the fund managers tend to increase the holdings in risky assets Hence, an incentive program has to be carefully implemented in order to balance the risk and the reward in DC pension fund management © 2010 by Taylor and Francis Group, LLC Preface ◾ xi Chapter p roposes a n a ttribution m odel f or m onitoring t he per formance a nd t he r isk o f a defined benefit ( DB) pens ion f und The model is ba sed on a l iability benchmark t hat reflects t he r isk a nd return cha racteristics of the liabilities As a result, the attribution model focuses the attention of the portfolio managers on creating a portfolio that replicates liabilities The attribution model a llocates d ifferences i n return between the actual portfolio and the benchmark portfolio to decisions relative to the benchmark portfolio In addition, the model decomposes risks according to the same structure by using a measure of downside risk Chapter i nvestigates a n optimal investment problem faced by a DC pension f und ma nager u nder inflationary risk It is assumed t hat a r epresentative member of a DC pension plan contributes a fi xed share of his salary to the pension fund during the time horizon The pension contributions are invested continuously in a risk-free bond, an index bond, and a stock The objective is to maximize the expected utility of terminal value of the pension fund By solving this investment problem, the author presents a way to deal with the optimization problem, in case of an (positive) endowment (or contribution), using the martingale method Chapter deals with the study of a pension plan from the point of view of dy namic o ptimization This sub ject i s c urrently w idely d iscussed i n the literature The optimal management of an aggregated type of DB pension fund, which is common in the employment system, is analyzed by a mean–variance portfolio selection problem The main novelty is that the risk-free ma rket interest rate is a t ime-dependent f unction a nd t he benefits are stochastic In C hapter , t he a uthor h ighlights t he fac t t hat a pens ion f und i s a complex system Asset a nd liability ma nagement (ALM) models of pension f und p roblems i ncorporate, a mong o thers, st ochasticity, l iquidity control, population dynamics, and decision delays to better forecast and foresee solvency i n t he long ter m I n order to model u ncertainties or to enable multicriteria analyses, many methods are considered and analyzed to obtain a dynamic asset and liability management approach In Chapter 7, the authors investigate the optimal asset allocation of U.S pension f unds b y t aking i nto acco unt t he f unds’ l iabilities B esides t he traditional i nputs, such a s ex pected r eturns a nd t he covariance ma trix, the uncertainty of expected returns plays a crucial role in creating robust portfolios t hat a re less sensitive to small cha nges i n i nputs The authors illustrate this with an example of a pension fund that decides on investing in emerging market equities © 2010 by Taylor and Francis Group, LLC xii ◾ Preface Chapter explains that most pension funds already manage the different risks they face, but usually from a “single stakeholder” pension fund perspective, typically expressed in, e.g., the risk of funding shortfall The many d ifferent st akeholders i n pens ion f unds, such a s t he em ployees, retirees, and sponsors, all bear different risks, but there is often hardly any insight i n t he objective ma rket va lue of t hese r isks In add ition, t here is usually no explicit compensation agreement for those who bear the risks Therefore, a tech nique that identifies and values these stakeholders’ risks has many useful applications in pension fund management Chapter focuses on value-at-risk (VaR) VaR has become a popular risk measure of financial risk and is a lso used for regulatory capital requirement purposes in banking and insurance sectors The VaR methodology has be en de veloped ma inly f or ba nks t o co ntrol t heir sh ort-term ma rket risk A lthough, VaR is a lready w idespread in financial industry, t his method has yet to become a standard tool for pension funds However, just as any other financial institution, pension funds recognize the importance of m easuring t heir financial r isks The a im o f t his cha pter i s t o spec ify conditions u nder wh ich V aR co uld be a g ood m easure o f l ong-term market risk Chapter 10 examines the effects of taxation, risk sharing between t he employer and employees, and default insurance on the asset allocation of DB pension schemes These three factors can have a powerful effect on the optimal asset allocation of a fund The authors show that the three factors have the potential to create conflict between the employer and the employees, particularly when the employer is not subject to taxation In Part II, Chapter 11 is devoted to examining how uncertainty regarding f uture m ortality a nd l ife ex pectancy o utcomes, i e., l ongevity r isk, affects employer-provided DB private pension plan liabilities The author argues t hat to a ssess u ncertainty a nd a ssociated r isks adequately, a st ochastic a pproach t o m odel m ortality a nd l ife ex pectancy i s p referable because it allows one to attach probabilities to different forecasts In this regard, t he cha pter p rovides t he r esults o f e stimating t he L ee–Carter model f or se veral OECD co untries F urthermore, i t co nveys t he u ncertainty su rrounding f uture m ortality a nd l ife ex pectancy o utcomes b y means o f M onte-Carlo s imulations o f t he L ee–Carter m odel I n o rder to assess the impact of longevity risk on employer-provided DB pension plans, the author examines the different approaches that private pension plans f ollow i n p ractice wh en i ncorporating l ongevity r isks i n t heir actuarial calculations © 2010 by Taylor and Francis Group, LLC Preface ◾ xiii Chapter 12 analyzes the pension plan of a firm that offers wa ge-based lump sum payments by death, retirement, or dismissal by the employer, but no payment is made by the employer when the employee resigns An actuarial risk model for funding severance payment liabilities is formulated and studied The yearly aggregate lump sum payments are supposed to follow a classical collective model of risk theory with compound distributions The final wealth at an arbitrary time is described explicitly including formulas for the mean and the variance Annual initial level premiums required for “dismissal f unding” a re de termined a nd u seful g amma approximations for confidence intervals of the wealth are proposed A specific numerical example illustrates the non-negligible probability of a bankruptcy in case the employee structure of a “dismissal plan” is not well balanced Chapter 13 starts from the fact that retirement is being remade owing to the confluence of demographic, economic, and policy factors The authors empirically i nvestigate ma jor i nfluences on t he re tirement b ehavior of older U.S workers f rom 1992 t hrough 004 u sing su rvey d ata f rom t he Health and Retirement Study Their analysis builds on the large empirical literature on retirement, in particular, by examining how market booms and busts affect the likelihood and timing of retirement, an issue that will be o f g rowing i mportance g iven t he o ngoing sh ift f rom t raditional D B pensions t o 01(k)s They co mprehensively m odel a ll ma jor so urces o f health i nsurance co verage a nd i dentify t heir va rying i mpacts, a nd a lso reveal t he significant policy-driven retirement differences across cohorts that are attributable to the changes in social security full-retirement age These f undamental r etirement cha nges n eed t o be t aken i nto acco unt when we design corporate and public retirement programs Chapter 14 deals w ith a st udy on occ upational pension i nsurance for Germany—a country where Pillar II pension schemes are (still) widely based on a book reserve system The insurance of occupational pension schemes is prov ided for by t he P ensions-Sicherungs-Verein Versicherungsverein auf Gegenseitigkeit (PSVaG), which is the German counterpart to the U.S PBGC This study investigates potential adverse selection and moral hazard problems originating from the introduction of reduced premiums for funded pens ions a nd a ssesses whether t he r isk-adjusted r isk premiums, as i ntroduced b y t he U.K Pension Pr otection F und, c an be a m eans t o mitigate these problems Chapter de scribes t he l ongevity r isk sec uritization i n pens ion schemes, f ocusing ma inly o n l ongevity bo nds a nd su rvivor s waps The authors analyze the evaluation of these mortality-linked securities in an © 2010 by Taylor and Francis Group, LLC xiv ◾ Preface incomplete market using a risk-neutral pricing approach A Poisson Lee– Carter model is adopted to represent the mortality trend The chapter concludes with an empirical application on Italian annuity market data In Part III, Chapter 16 h ighlights t hat t he i nternational t rend toward adopting a “ fair va lue” a pproach t o pens ion acco unting s t ranspired the r isks i nvolved i n promises of DB pensions The hunt i s on for ways to remove or l imit t he employers’ r isk ex posures to financial statements volatility This chapter examines the U.K firms’ risk management of their pension f und a sset a llocation over a per iod when t he new U.K pension GAAP (FRS 17) became effective The findings suggest that firms manage their pension risk exposure in order to minimize cash contribution risks associated w ith t he ad option o f “ fair va lue”–based pens ion acco unting rules, consistent with a risk offsetting explanation Chapter 17 develops and tests a theory of competition among pressure groups over political influence in the context of conflicting U.K standards concerning the factors affecting the recent development of pension fund accountability rules The chapter models both sources of pressure affecting the accountability relationship as well as how those factors combined to i nfluence U.K pens ion f und ma nagers’ d iscretion o ver t he ad option and retention of disclosure regulations The author finds that auditors and pension management groups exerted most political pressure, which translated to political influence during the extended adoption period The findings are mostly consistent with a capture or private interest perspective on pension accounting regulation Chapter 18 r eviews t hree u seful i nstruments—notional defined-contribution acco unts (N DCs), t he ac tuarial ba lance ( AB), a nd a utomatic balance mechanisms (ABMs)—derived from actuarial analysis methodology that can be applied to the public management of PAYGO systems to improve t heir fa irness, t ransparency, a nd so lvency The a uthors su ggest that these tools are not simply theoretical concepts but, in some countries, an already legislated response to the growing social demand for transparency in the area of public finance management as well as the desire to set the pension system firmly on the road to long-term financial solvency In C hapter 19, t he authors review t he r isk-based solvency regime for pension funds in the Netherlands The supervision of pension funds aims to ensure that institutions are always able to meet their commitments to t he beneficiaries I n add ition, t he pension f und must be l egally separated from the employer offering the pension arrangement Furthermore, the ma rked-to-market va lue o f t he a ssets m ust be a t l east eq ual t o t he © 2010 by Taylor and Francis Group, LLC Preface ◾ xv marked-to-market value of the liabilities at all times (full funding prerequisite) Risk-based solvency requirements are intended as a buffer to absorb the r isks f rom u nexpected cha nges i n t he va lue of a ssets a nd l iabilities Finally, a ke y element of the Dutch regulatory approach is the continuity analysis for assessing the pension fund’s solvency in the long run In Chapter 20, the author addresses the fact that the global financial crisis of 2008 highlighted the importance of shielding pension participants from market volatility This policy concern is of general relevance due to the global shift from DB to DC as main mechanisms for financing retirement income Policy options being debated in the aftermath of the crisis include, but are not necessarily limited to, the following: (1) the introduction o f l ifetime m inimum r eturn g uarantees, ( 2) t he r eview o f defa ult investment options, and (3) the outright reversal to PAYGO earning–related pensions This chapter reviews the performance during the crisis of countries that a lready rely on mandatory DC plans The author suggests that important welfare gains can be ach ieved by requiring the introduction of liability-driven default i nvestment products based on a m odified version of the target date funds commonly available in the retail industry for retirement wealth Such products would reconnect the accumulation with the decumulation phase, improve the hedging of annuitization risk, but avoid the introduction of liabilities for plan managers In P art I V, C hapter h ighlights t he D B pens ion f reezes i n la rge healthy firms such as Verizon and IBM, as well as terminations of plans in the struggling steel and airline industries that cannot be v iewed as riskfree from t he employee’s perspective The a uthors de velop a n em pirical dynamic programming framework to investigate household saving decisions in a simple life cycle model with DB pensions subject to the risk of being f rozen The model i ncorporates i mportant sources of u ncertainty facing households, including asset returns, employment, wages, and mortality, as well as pension freezes Chapter 22 is referred to as the Italian experience In Italy, social security contributions of Italian employees finance a two-pillar system: public and p rivate pens ions t hat a re bo th c alculated i n a DC sch eme (funded for the private pension and unfunded for the public one) In addition to this, a la rge number of workers have also termination indemnities at the end of their active service The authors aim to answer the following questions A re t he d ifferent flows o f contributions co herent w ith t he a im o f minimizing the pension risk of the workers? Given the actual percentages of contributions, is the asset allocation of private pension funds optimal? © 2010 by Taylor and Francis Group, LLC xvi ◾ Preface What per centages w ould o ptimize t he pens ion r isk ma nagement o f t he workers ( considering pu blic p ension, pr ivate p ension, a nd t ermination indemnities)? Chapter 24 examines the Greek experience in limiting the opportunity of investments of pension funds in foreign assets In fact, suffering from inefficient funding, the current imbalance of the Greek social security system, to some extent, was the result of the restrictive investment constraints in the period 1958–2000 that directed reserves to low-yielding deposits with the Bank of Greece with little or no exposure to market yields or the stock market As shown in the 43 year analysis, these investment restrictions i ncurred a s ignificant economic opportunity loss both in terms of inferior returns as well as lower risks Chapter 25 examines the effect of a company’s unfunded pension liabilities on its stock market valuation Using a sample of UK FTSE350 firms with DB pension schemes, the authors find that although unfunded pension liabilities reduce the market value of the firm, the coefficient estimates indicate a less t han one-for-one effect Moreover, there is no evidence of significantly negative subsequent abnormal returns for highly underfunded schemes These results suggest that shareholders take into consideration the unfunded pension liabilities when valuing the firm, but not fully incorporate all available information Chapter 26 f ocuses on t he sel ection o f a n a ppropriate st yle m odel t o explain t he returns of Spanish ba lanced pension plans as well as on t he analysis of the relevance of these strategic allocations on portfolio performance Results suggest similar findings than those obtained in previous studies, providing e vidence t hat a sset a llocations ex plains about 0% of portfolio r eturns o ver t ime, m ore t han 0% o f t he va riation o f r eturns among plans, and about 100% of total returns MATLAB® i s a r egistered t rademark o f The M athworks, In c F or p roduct information, please contact: The Mathworks, Inc Apple Hill Drive Natick, MA 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: info@mathworks.com Web: www.mathworks.com © 2010 by Taylor and Francis Group, LLC Editors Marco Micocci is a f ull professor of financial mathematics and actuarial science in the Faculty of Economics, University of Cagliari, Italy He has received deg rees i n eco nomics, ac tuarial st atistics, a nd t he finance of financial i nstitutions H is r esearch i nterests i nclude financial a nd ac tuarial risk management of pension funds and insurance companies, enterprise risk management, and operational and reputational risk va luation He has published nearly 90 books, chapters of books, journal articles, and papers He also works as a consultant actuary Greg N Gregoriou has published 34 books, over 50 refereed publications in peer-reviewed journals, and 22 book chapters since his arrival at SUNY (Plattsburgh, New York) in August 2003 Professor Gregoriou’s books have been published by John Wiley & Sons, McGraw-Hill, Elsevier Butterworth/ Heinemann, T aylor & F rancis/Chapman-Hall/CRC P ress, P algraveMacMillan, a nd R isk/Euromoney boo ks H is a rticles ve a ppeared i n the Journal o f P ortfolio M anagement, t he Journal o f F utures M arkets, the European Journal of O perational Re search, t he Annals of O perations Research, and C omputers and O perations Re search Professor Gregoriou is a coed itor a nd ed itorial boa rd m ember f or t he Journal o f D erivatives and Hedge Funds, a s well a s a n ed itorial boa rd member for t he Journal of W ealth Man agement, t he Journal o f Ri sk M anagement i n F inancial Institutions, and the Brazilian Business Review A na tive of Montreal, he received his joint PhD at the University of Quebec at Montreal, Quebec, Canada, in finance, which merges the resources of Montreal’s major universities (McGill University, Concordia University, and École des Hautes Études C ommerciales, M ontreal) H is i nterests f ocus o n h edge f unds, funds of hedge funds, and managed futures He is also a m ember of the Curriculum Committee of the Chartered Alternative Investment Analyst Association xvii © 2010 by Taylor and Francis Group, LLC xviii ◾ Editors Giovanni B Ma sala is a r esearcher i n mathematical methods for economy and finance at the Faculty of Economics, University of Cagliari, Italy He re ceived h is PhD i n pu re m athematics (differential geometry) at t he University o f M ulhouse, F rance H is c urrent r esearch i nterest i ncludes mathematical risk modeling for financial a nd ac tuarial applications He attended n umerous i nternational c ongresses to le arn more a bout t hese topics His results have been published in refereed national and international journals © 2010 by Taylor and Francis Group, LLC 60 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling However, the distribution of market return is not symmetrical; thus, the performance co ntract s t o se t d ifferent ben chmark r ates a nd pa rticipated rates of bonus fees as well as downside penalty 2.5.1 Optimal Investment Decision The optimal problem is reset as follows: dS dS dB ⎤ ⎡ dWN = WN ⎢(1 − w S − w B ) + w S + w B K ⎥ + γ dL S0 S BK ⎦ ⎣ ⎛⎡ dS dS dB ⎤ ⎞ − e1WN max ⎜ ⎢(1 − w S − w B ) + w S + w B K ⎥ , p1 ⎟ S0 S BK ⎦ ⎠ ⎝⎣ ⎛⎡ dS dS dB ⎤ ⎞ − e2WN ⎜ ⎢(1 − w S − w B ) + w S + w B K ⎥ , p2 ⎟ S0 S BK ⎦ ⎠ ⎝⎣ (2.15) Note t hat it i s a ssumed t hat t here ex ist i n t he i nvestment ma ndate t wo benchmarks p1 an d p2 i n t riggering t he bo nus f ees a nd d ownside pen alty This kind of structural setup is intended to be similar to the contract requirement, that is, according to the Taiwan labor pension fund management regulation, the return rate cannot be less than the interest rate of year fi xed deposit In this regard, when the fund performance is better than p1, the fund manager is entitled to get the bonus fee; otherwise, the manager is required to reduce the management fee due to downside penalty In our performance mechanism, the optimal problem contains two kinds of financial options and the explicit solution is sometimes hard to find Thus, in this section, the optimization method is employed to solve the problem In case (2.15), we find t hat t he f und manager takes greater risks in seeking higher return since the fund manager is required to guarantee t he m inimum r eturn r ate p2, wh ich sh ows t hat t he f und spo nsor need not worry about the downside risk of the investment Thus, the optimal a sset a llocation i s i nvestigated f rom t he v iew o f t he pens ion f und manager The optimal investment problem becomes ⎛⎡ dS dS dB ⎤ ⎞ dV = e1WN max ⎜ ⎢(1 − w S − w B ) + w S + w B K ⎥ , p1 ⎟ S0 S BK ⎦ ⎠ ⎝⎣ ⎛⎡ dS dS dB ⎤ ⎞ + e2WN ⎜ ⎢(1 − w S − w B ) + w S + w B K ⎥ , p2 ⎟ S0 S BK ⎦ ⎠ ⎝⎣ © 2010 by Taylor and Francis Group, LLC (2.16) Investment Decision in Defined Contribution Pension Schemes ◾ 61 where V denotes the fund surplus, that is, the fund manager is rewarded through obtaining t he bonus fee when t he f und per formance i s be tter than p1 On the other hand, the management fee of the fund manager has to be reduced due to the downside penalty when his or her performance is worse than the investment benchmark p2 This is a combined structure of a sset-based a nd t arget-based i ncentive m echanisms I n co mputation, the MATLAB® p rogram i s w ritten t o a pply t he p roposed o ptimization method i n computing t he optimal i nvestment weights i n Equation 16 In each scenario, 50,000 realizations are simulated and t he short-selling restriction is a lso employed The trade-off parameters e2 and the performance b enchmark p2 a re a ssumed t o be % a nd %, r espectively The investment time horizon in our illustration is set to be 10 years In Figure 2.6, the optimal multi-period investment strategies are illustrated given different e1 and p1 As seen in Figure 2.6a through c, the optimal p1 = 3% e1 = 0.7% 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 10 (a) Proportion of wealth 0.8 0.6 0.4 0.2 0 (d) (e) 10 (c) 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 10 10 (b) p1 = 5% 0.8 e1 = 0.5% p1 = 4% 10 Investment horizon (year) Stock Bond Cash 10 (f ) Optimal p ortfolio h oldings o f c ash, s tocks, a nd n ominal b onds given tim e h orizon T = y ears u nder i ncentive p rograms ( bold s olid l ine, weights of stock index fund; gray dot line, weights of rolling bond fund; gray dashed line, weights of cash (a) e1 = 0.7% and p1 = 3%, (b) e1 = 0.7% and p1 = 4%, (c) e1 = 0.7% and p1 = 5%, (d) e1 = 0.5% and p1 = 3%, (e) e1 = 0.5% and p1 = 4%, and (f) e1 = 0.5% and p1 = 5%) FIGURE 2.6 © 2010 by Taylor and Francis Group, LLC 62 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling weights with increasing p1 under e1 is 0.7% In Figure 2.6d through f, e1 is 0.5% and p1 is rising from 3% to 5% The bo ld so lid l ine r epresents t he optimal investment weights of t he stock index f und The red dashed line denotes the proportion of fund in cash The third line illustrates the weights of rolling bond fund First, Figure shows t hat t he f und ma nager w ill i ncrease t he holding of stock index f und as t he investment horizon approaches maturity A p robable ex planation i s t hat u nder t he o ptimal i nvestment dec ision, the performance of fund will be better than the benchmark; hence, the manager increases the risk of portfolio in order to seek higher bonus fees Comparing Figure 2.1 with Figure 2.4 shows that the optimal investment strategy i s t o h old m ore r isky a ssets s ince t he w eights i n t hese figures exceed 100% However, in Figure 2.6, we set t he short-selling constrain; thus the holding weight of stock index fund is close to at maturity date Second, the weights of cash are small because its return rate is low Fund managers w ould n ot p refer t o h old c ash bec ause t hey ve t o m eet t he requirement of minimum guarantees p2 However, Figure 2.6 shows certain diverse characteristics of the investment behaviors In Figure 2.6a through c, the weights of stock index fund (bold solid line) in the beginning are decreasing from 0.88 to 0.58 when p1 increases On the other hand, the allocation in Figure 2.6d through f show that when e1 decreases to 0.5%, the fund manager would hold less stock index fund from 0.48 to 0.62 in the beginning with increasing p1 This is interesting that the settlement of bonus fees would affect the investment behaviors o f f und ma nagers I n o ther w ords, t his p henomenon i mplies that the settlement of incentive mechanism contract is important on the delegated management contract because the parameters would affect the investment behaviors of fund managers 2.5.2 Financial Implication In Section 2.4.3, the optimal investment decisions are simulated under certain constrains (e1 = e2 = e and p1 = p = 0) The f und ma nager w ill increase t he h olding i n r isky a sset U nder t he d ownside p rotection arrangement, a fund holder will tend to increase the risk profi le of fund portfolio I n Section 5.1, m ore r ealistic per formance m echanisms including e1 ≠ e2 an d p1 ≠ p2 a re i nvestigated M oreover, R aghu e t a l (2003) conclude that there exists the agency problem between the fund managers and the plan participants In Section 2.5.1, we try to investigate the financial influence of performance mechanisms on the optimal © 2010 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution Pension Schemes ◾ 63 investment d ecisions The a sset a llocation p roblem i n Section 4.3 i s simplified t o d erive t he e xplicit s olution The o ptimal so lution va ries according to t he va rious scenarios of t he fi nancial ma rket In order to explore t he realistic impact of t he incentive mechanism, t he optimization p rogram i s i mplemented t o a pproximate t he o ptimal i nvestment weights through simulations For t he DC pens ion f und ma nagement, t he se tup i n Section 5.1 i s more p ractical a nd v ital The o ptimal i nvestment dec isions o f f und managers u nder per formance m echanisms a re i nvestigated O ur m odel extends the previous research through implementing the unlimited liability downside protection The unlimited liability is incorporated since the limited mechanisms would motivate the fund manager to increase the risk profile of portfolio after a period of poor performance (Edwin et al 2003) Moreover, t he labo r pens ion p lan i mplemented i n Taiwan i ncludes a lso this kind of guarantee arrangement We find that under this performance contract, the benchmark rate (p1) and the participated rate (e1) of bonus fees would change the investment behavior of the fund manager This is consistent with the conclusion made by Raghu et al (2003), that the performance would be influenced by the commission rate That is, the participated rate of bonus fees would affect the investment behaviors However, Raghu et al (2003) propose that the efficacy of limited incentive is better than unlimited contract Moreover, t he optimal i nvestment dec ision i s a nalyzed a nnually The performance of fund is measured and the fund managers are also rebalancing their asset class every year Mark (1987) discovers that the length of the time horizon is very crucial During the shorter period, the performance contract would not identify whether t he success of t he f und performance is the true investment ability or pure luck of the fund manager Lawrence a nd S tephen (1987) a lso su ggest t hat t he proper per formance index should employ the moving year time period 2.6 CONCLUSION In this study, we investigate the asset allocation issue for DC labor pension f und t hat considers not only t he ma rket r isk a nd i nterest r isk but also t he u ncertainties f rom labo r i ncomes, t he i nflation r isk, a nd t he incentive scheme We fi nd that if the fund manager would like to maximize t he expected exponential utility of his or her terminal wealth, he can adopt the mutual fund separation theorem through five components in its optimal asset allocation Hence, the optimal investment behaviors © 2010 by Taylor and Francis Group, LLC 64 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling of the pension fund managers are characterized by the relative weights among t he sepa rated m utual f unds acco rding t o t heir p reference, t he fi nancial market, and the influential factors With both the financial a nd background risks incorporated, pension fund ma nagers a re recommended to consider t he short-term f und performance a nd t he h edge r equirements s imultaneously B ecause back ground risks cannot be controlled by fund managers, a comprehensive dynamic framework is formulated to describe the decision-making process A s t he r esults sh ow, t he dy namic po rtfolio o f t he r estricted f orm that ma ximizes t he ex pected u tility o f t he p lan pa rticipant co nsists o f five components: the market portfolio, the state variables hedge portfolio, the inflation hedge portfolio, the salary uncertainty hedge portfolio, and cash By solving explicitly the optimal portfolio problem, the numerical results i ndicate t hat t he i nflation h edge po rtfolio co nstitutes t he o verwhelming proportion of stocks in the optimal portfolios In addition, the inflation hedge portfolio and the state variable hedge portfolio constitute the overwhelming proportions of bond holdings This shows t hat longterm investors should hedge inflation rate risk by holding the stock index In addition, these investors should respond to the inter-temporal hedging demands in the financial markets by increasing the average allocation to their bond fund To understand the roles of these components, it is necessary to explore the economic interpretations by solving the dynamic optimization problems With respect to the most common approach used in the literature, the i ncorporation o f t he labo r i ncome a nd i nflation r isks a llows u s t o characterize the general pattern of the optimal strategy The results indicate that the inflation hedge portfolio constitutes the main proportion of the optimal stock portfolios, while in the earlier stage the market portfolio makes up t he larger part of t he stock index f und However, in t he labor income h edge po rtfolio, t he i nvestor sh ould sh ort-sell h is o r h er st ock index and the bond portfolio in order to preserve the salary uncertainty over his or her investment horizon Finally, t he optimal a sset a llocation st rategy i s solved for t he general incentive mechanism The optimal behaviors of the fund managers alter according to various parameter settings within the incentive mechanism Our results are also consistent with the findings of Richard and Andrew (1987) and Lawrence and Stephen (1987), who confirm that the incentive setting is essential in the delegated management contract (Lawrence and Stephen 1987, Richard and Andrew 1987) © 2010 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution Pension Schemes ◾ 65 APPENDIX A H* is as follows: H * = µ′ν JW + JW [WN (r − re − µ π ) + γLµ L − ( A′ + B′)Γ ′(ΓΓ ′)−1 M] + tr(Ω′ΩJ νν ) − J ( J W )2 M ′(ΓΓ ′)−1 M − W M ′(ΓΓ ′)−1 ΓΩJ νW JWW JWW + ( A′ + B′)( I − Γ ′(ΓΓ ′)−1 Γ)ΩJ vW − 1 J ν′ W ΩΓ ′(ΓΓ ′)−1 ΓΩJ νW + JWW ( A′ + B′)( I − Γ ′(ΓΓ ′)−1 Γ)( A + B), JWW (2.17) where we denote A = FNΦ, B = γLΛ, and that I is the identity matrix Then, substituting Equation 2.13 into Equation 2.17, we obtain J(t;W, v) ht + H* = 0,h(T, v(T) ) = a nd after dividing by J, we can write Equation 2.17 in the following way: = ht + µ ν′ hν + UW ⎡WN (r − re − µ π ) + γLµ L − ( A ′ + B ′)Γ ′(ΓΓ ′)−1 M ⎤ ⎦ U ⎣ 1 (UW )2 (U )2 M ′(ΓΓ ′)−1 M − W M ′(ΓΓ ′)−1 ΓΩhν + tr(Ω ′Ω(hνν + hνhν′ )) − UWWU 2 UWWU 1 (UW )2 hν′ Ω ′Γ ′(ΓΓ ′)−1 ΓΩhν + (A ′ + B ′)(I − Γ ′(ΓΓ ′)−1 Γ )Ωhν − 2 UWW U + UWW ( A′ + B ′ )(I − Γ ′(ΓΓ ′ )−1 Γ )( A + B) U We h ave t hat UW U is β2 and (UW )2 UWWU is 1, and then substitute these two values into the above equation Therefore, the HJB equation can be written as follows: = ht + [µ′ν − M ′(ΓΓ ′)−1 ΓΩ + β2 ( A′ + B′)(I − Γ ′(ΓΓ ′)−1 Γ )Ω]hν + tr(Ω′Ωhνν ) − hν′ Ω′Γ ′(ΓΓ ′)−1 ΓΩhν + β2[WN (r − re − µ π ) + γLµ L − ( A′ + B′)Γ ′(ΓΓ ′)−1 M ] 1 − M ′(ΓΓ ′)−1 M + β22 ( A′ + B′)(I − Γ ′(ΓΓ ′)−1 Γ )( A + B) 2 © 2010 by Taylor and Francis Group, LLC 66 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling This kind of partial differential equation can be solved using the Feynman– Kac t heorem, a nd so w e can find t he f unctional for m of h(v;t), which is ⎡ T ⎤ given by h(ν; t ) = Ε t ⎢ g (v(s), s)ds ⎥ , where ⎣ t ⎦ ∫ dν(s) = [µ ′ν − M ′(ΓΓ ′)−1 ΓΩ + β2 ( A ′ + B ′)(I − Γ ′(ΓΓ ′)−1 Γ )Ω]′ ds + Ω(v(s), s)′ dZ ν(s) = ν(s), g (ν(t ), t ) = [WN (r − re − µ π ) + γLµ L − ( A′ + B ′)Γ ′(ΓΓ ′)−1 M ] − 1 M ′(ΓΓ ′)−1 M ] + β2 ( A′ + B ′)(I − Γ ′(ΓΓ ′)−1 Γ )( A + B) β2 Finally, the optimal portfolio is written as follows: 1 1 (ΓΓ ′)−1 M − (ΓΓ ′)−1 ΓΩ ⋅ β2 WN (1 − e) β2 WN (1 − e) wG* = − − ∫ T t ∂ Ε t [ g (ν(s), s)]ds ∂ν 1 (ΓΓ ′)−1 ΓΦ − γL(ΓΓ ′)−1 ΓΛ WN (1 − e) 1−e APPENDIX B Now we are interested in the second component wG*(2) of the optimal portfolio, which is the state variable hedge portfolio We follow Battocchio and Menoncin (2004) t o d erive wG*(2) S ince t he ter m M′(ΓΓ′)−1 M es not rely on the state variables, this term is deleted We rearrange wG*(2) as the following equation: wG*(2) = (ΓΓ ′)−1 ΓΩ⋅ WN (1 − e) ∫ T t ∂ Et [Q1 + Q2 ]ds, ∂ν where Q1 = [WN (r − re − µ π )] − WN Φ′Γ′(ΓΓ′)−1 M + β2WN2 σ2π Q2 = γLµ L − γLΛ′Γ′(ΓΓ′)−1 M + β2[−2γLWN σL σ π + γ L2 σ2L ] © 2010 by Taylor and Francis Group, LLC (2.18) Investment Decision in Defined Contribution Pension Schemes ◾ 67 Next, we derive the derivative of the last term in Equation 2.18: ⎡ ∂ ⎤ ⎢ ∂r(t ) Et [Q1 + Q2 ]⎥ ⎡ (1 − e)WN (t ) ⎤ ⎢ ⎥=⎢ ⎥ 2 −1 ⎢ ∂ E [Q + Q ]⎥ ⎢⎣γ µ L − γΛ ′Γ ′(ΓΓ ′) M − β2 γσ L σ πWN (t ) + β2 γ σ L Et [L]⎥⎦ t ⎢⎣ ∂L(t ) ⎥⎦ In t he above equation, we have to compute t he ex pected va lue of t he ∼ modified process of labor incomes, Et[L], which is called the modified real contribution First, we need to compute the following matrix product: −1 −1 ⎡ ⎤ ⎣ − M ′(ΓΓ ′) ΓΩ + β2 (WN Φ ′ + γLΛ ′)(I − Γ ′(ΓΓ ′) Γ )Ω ⎦ ′ For simplicity, we assume that Γ′(ΓΓ′)−1Γ = I Then, the above equation is equal to the first term, and we can write it as ′ ⎡ w1 ⎤ ⎡⎣ − M ′(ΓΓ′)−1 ΓΩ ⎤⎦ = ⎢ ⎥ ⎣ Lw2 ⎦ where w1 and w2 are given by w1 ≡ σr λ r , w2 ≡ −2σ Lm σSr λ r σSm − σ Lm λ m + σLr λ r Thus, we can get the modified differential of the state variables ∼ v (s) as follows: ⎡ dr ⎤ ⎡a(b − r ) − w1 ⎤ ⎡ σr dt + ⎢ ⎢ ⎥=⎢ ⎥ w ⎣ σLr ⎣⎢dL L ⎦⎥ ⎣ µL − ⎦ σLm ⎡ dz ⎤ ⎤⎢ r ⎥ dz σL ⎦⎥ ⎢ m ⎥ ⎢ dz L ⎥ ⎣ ⎦ In particular, for s < t, the solutions of the interest rate process and the modified labor income process are r (s) = r (t )ea(t − s ) + ab − w1 (1 − ea(t − s ) ) + σr e −as a s ∫e aτ t dzr (τ) ⎡⎛ 1 ⎞ L(s) = L(t )exp ⎢⎜ µ L − w2 − σ2Lr − σ2Lm − σ2L ⎟ (s − t ) ⎝ 2 ⎠ ⎣ ⎤ + σLr (zr (s) − zr (t )) + σLm (z m (s) − z m (t )) + σL (z L (s) − z L (t ))⎥ ⎦ © 2010 by Taylor and Francis Group, LLC 68 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Then, according to the boundary equation (v∼(s) = v(s) ), we can obtain ∼ the expected value Et[L (s)] = L(t)eR(s−t), where 1 ⎞ ⎛ R(s − t ) = ⎜ µ L − w2 − σ2Lr − σ2m − σ2L ⎟ (s − t ) ⎝ 2 ⎠ + σ Lr (z r (s) − z r (t )) + σ Lm (z m (s) − z m (t )) + σ L (z L (s) − z L (t )) Thus, the integral term of wG*(2) in the optimal portfolio becomes ∫ T t (1 − e)WN (t ) ⎡ ⎤ ⎢ ⎥ −1 ∂ Et [Q1 + Q2] ds = ⎢ γ µ L − γΛ ′Γ ′(ΓΓ ′) M − β2 γσ L σ πWN (t )⎥ ∂ν ⎢ ⎥ 2 R ( s −t ) ⎢⎣ + β2 γ σ L L(t )e ⎥⎦ In the end, we could get the solution of wG*(2) as wG*(2) (1 − e)WN (t ) ⎡ ⎤ ⎢ ⎥ − (ΓΓ ′)−1 ΓΩ⋅ ⎢ γ µ L − γΛ ′Γ ′(ΓΓ ′) M − β2 γσ L σ πWN (t )⎥ = WN (1 − e) ⎢ ⎥ 2 R ( s −t ) ⎢⎣ + β2 γ σ L L(t )e ⎥⎦ REFERENCES Battocchio, P and Menoncin, F 2002 Optimal portfolio strategies with stochastic wage income and inflation: The case of a defined contribution pension plan Working Paper CeRP, No 19-02 Torino, Italy Battocchio, P and Menoncin, F 2004 Optimal pension management in a stochastic framework Insurance: Mathematics and Economics 34: 79–95 Bodie, Z 1990 P ensions 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Financial Analysts Journal 43: 27–38 Ritter, J R and Warr, R S 2002 The decline of inflation and the bull market of 1982–1999 Journal of Financial and Quantitative Analysis 37: 29–61 Roy, K and William, T Z 2007 Incentives and risk taking in hedge funds Journal of Banking and Finance 31: 3291–3310 Rutkowski, M 1999 Self-financing trading strategies for sliding, rolling-horizon, and consol bonds Mathematical Finance 9: 361–385 Shapiro, A F 1977 The relevance of expected persistency rates when projecting pension costs Journal of Risk and Insurance 44: 623–638 Shapiro, A F 1985 Contributions to the evolution of pension cost analysis Journal of Risk and Insurance 52: 81–99 Vasicek, O E 1977 An equilibrium characterization of the term structure Journal of Financial Economics 5: 177–188 Viceira, L M 2001 Optimal portfolio choice for long-horizon investors with nontradable labor income Journal of Finance 56: 433–470 © 2010 by Taylor and Francis Group, LLC CHAPTER Performance and Risk Measurement for Pension Funds Auke Plantinga CONTENTS 3.1 I ntroduction 3.2 Liability-Driven Investing and Pension Liabilities 3.3 Liability-Driven Risk and Performance Attribution 3.3.1 P erformance Measure 3.3.2 R isk Measures 3.3.3 B enchmark 3.3.4 P erformance Attribution 3.4 An Application of the Model 3.5 Conclusion and Discussion References 72 72 76 76 76 77 78 80 83 T his ch a pter p r opos es a n a ttribution m odel f or m onitoring t he performance and risk of a defined benefit pension fund In order to facilitate easy interpretation of the results, the return is expressed as a percentage of the value of the liabilities The model is based on a liability benchmark t hat re flects t he r isk a nd return cha racteristics of l iabilities As a r esult, t he a ttribution m odel f ocuses t he a ttention o f t he po rtfolio managers on creating a portfolio that replicates liabilities The attribution model allocates differences in return between the actual and benchmark portfolio to decisions relative to the benchmark portfolio In addition, the model decomposes risks according to the same structure by using a measure of downside risk 71 © 2010 by Taylor and Francis Group, LLC 72 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 3.1 INTRODUCTION Pension funds around the world suffered dramatically from falling stock markets in the period 1999–2003 as well as during the current 2008–2009 financial crisis Many pension funds have become underfunded, and face funding ratios well below 100% In both periods, stock markets performed badly a nd i nterest r ates d ropped considerably These a re conditions t hat have a ser ious impact on pension f und ba lance sheets, by decreasing t he market value of the assets and increasing the market or fair value of the liabilities Ryan and Fabozzi (2002) provide evidence of the existence of large mismatches between assets and liabilities in the 200 largest U.S definedbenefit plans prior to the first pension crisis For example, in 1995 the assets of these 200 pension funds yielded an average positive return of 28.70%, which was insufficient to match with the liability return of 41.16% Liability-driven in vesting i s a n in vestment p hilosophy t hat im plies managing a ssets r elative t o l iabilities.* W ith l iability-driven i nvesting, pension f unds c an reduce t heir ex posure to financial crises by reducing the mismatches between assets and liabilities In order to encourage pension funds to adopt liability-driven investment strategies, i t i s n ecessary t o r edesign per formance a nd r isk m easurement systems The best way to accomplish this is by adopting a so-called liabilitydriven benchmark for performance and risk management In this study, we discuss a performance and risk measurement system that f ocuses o n l iability-driven i nvesting W ithout a ppropriate per formance and risk measurement, liability-driven investing will be difficult to manage as portfolio managers will be tempted to focus on traditional asset-only benchmarks We propose a number of new elements in measuring per formance a nd r isk for pension f unds The first new element is by measuring the performance of a pension fund relative to the value of the liabilities in order to focus on the societal objective of a pens ion fund as the provider of reliable pensions The second new element is that we propose to use a decomposition of downside risk consistent with the decomposition of performance 3.2 LIABILITY-DRIVEN INVESTING AND PENSION LIABILITIES Liability-driven investing is a strategy aimed at reducing a pension plan’s risks by choosing assets that serve as a hedge for the risks implicit in the liabilities This i s o nly pos sible f or a l imited se t o f r isk fac tors t hat a re * See, for example, Leibowitz (1986), Siegel and Waring (2004), and Waring (2004) © 2010 by Taylor and Francis Group, LLC Performance and Risk Measurement for Pension Funds ◾ 73 present bo th i n t he a sset po rtfolio a nd t he l iability po rtfolio The most important of these risks are the interest rate risk and the inflation risk.* Some other risks in the liability portfolio are difficult to hedge with traditional a sset cla sses, such a s t he mortality r isk or t he operational r isk Those types of risks can be reduced by, for example, reinsurance, or simply by creating an a dditional s olvency buffer f or u nexpected l osses I n t his study, we implement liability-driven investing using cash flow matching Cash flow matching is a robust strategy for dealing with any type of term structure movement Practical considerations may result in choosing for a less robust solution, such as duration matching Pension claims in a defined-benefit plan are largely determined by the wages earned by the beneficiaries and the number of years of employment With a final pay system, the pension claims are based on the wage earned in the last year However, it is also common to have claims as a function of past wages earned, such a s t he average wage earned during t he years of em ployment A co rrect va luation o f t he l iabilities i s c rucial i n ma naging pens ion f unds The va lue o f l iabilities ser ves a s a ben chmark f or the level of assets needed to ser vice the future cash flows of the pension fund The valuation starts with an appropriate estimate of the future cash flows resulting from the current promises made to beneficiaries Next, the present value of these cash flows is calculated using a d iscount rate This discount rate is a c rucial element i n t he l iability-driven benchmark We propose to use the risk-free rate on government bonds as the discount rate In combination with a matching strategy and sufficient assets, this means that the liabilities will be met with certainty.† From the perspective of the beneficiary, a va luation ba sed on t he r isk-free government bond r ates is the preferred alternative since it provides the opportunity to create riskless pension claims It is important to see that the resulting value is not the market value of the liabilities, but the value of a risk-free cash flow desired by the beneficiaries If the present value of the liabilities based on government bond r ates exceeds t he va lue of t he a ssets, it i s clear t hat t he t rue market value of the liabilities is less, unless the sponsor provides a credible guarantee to make up for any deficits The return on the cash flow matching strategy is the minimum acceptable return (MAR) to be achieved on the assets (see Sortino et al., 1999) If the assets not yield this return, the future pension benefits cannot be * Another factor is currency risk, which we ignore in this chapter by assuming that the pension fund has single currency pension liabilities † The conditions for creating an immunized portfolio are specified by Redington (1952) © 2010 by Taylor and Francis Group, LLC 74 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling realized without further sponsor contributions The actual realization of reliable pensions depends on the investment strategy In order to keep the focus of portfolio managers on the objective of the pension beneficiaries, the cash flow matching strategy, which gives the best chance of realizing the objective, has to be embedded in the asset benchmark Elton and Gruber (1992) provide a t heoretical justification for the use of matching strategies in the context of a mean–variance framework For a pension fund with assets A and present value of liabilities L, the return on surplus is rs = St +1 A(1 + ) − L(1 + rl ) −1 = −1 St A−L (3.1) The r iskless st rategy, F, is to invest an amount, L, i n a po rtfolio of c ash flow–matched a ssets, a nd t o i nvest t he r emainder i n t reasury b ills The return of this strategy is rF = ( A − L)(1 + rf ) + (L − L)(1 + rl ) − = rf A−L (3.2) Similarly, the return on a risky strategy R is rR = A(1 + rR ) − L(1 + rl ) L − = rR + (rR − rl ) A−L S (3.3) Any co mbination o f t hese t wo po rtfolios r esults i n a po rtfolio o n a straight l ine connecting F and R i n t he mean st andard de viation space, similar to the capital market line in the CAPM However, the underlying portfolios are different and depend on the leverage and liability returns Consequently, different investors have opportunity sets The actual choice for an asset portfolio depends on the risk aversion of the pension fund Pension funds have a multitude of participants that also have different interests Therefore, t he q uestion h ow t o ch oose a l evel o f r isk a version for the pension fund that is representative for all participants is not easy and perhaps even impossible to answer In many of the surplus optimization models, the surplus is owned by the plan sponsor, and the criterion for choosing an asset portfolio is to maximize the utility of the surplus.* As stated by Erza (1991): “A defined benefit pension plan is effectively an operating d ivision o f t he spo nsor.” W ith pens ion f unds bei ng o perating divisions of sponsors, conflicts of interest with the beneficiaries may arise * See, for example, Elton and Gruber (1992), Erza (1991), or Sharpe and Tint (1992) © 2010 by Taylor and Francis Group, LLC ... integration of actuarial and financial risks in the risk management of pension funds We believe t he chapters in t his book highlight and shed new light on the current state of pension fund risk management. .. of financial i nstitutions H is r esearch i nterests i nclude financial a nd ac tuarial risk management of pension funds and insurance companies, enterprise risk management, and operational and. .. parts— Part I: Financial Risk Management; Part II: Technical Risk Management; Part I II: Reg ulation a nd S olvency T opics; a nd P art I V: I nternational Experience in Pension Fund Risk Management