Measuring the Benefits of Dynamic Asset Allocation Strategies in the Presence of Liability Constraints

Transcription

1 An EDHEC Risk and Asset Management Research Centre Publication Measuring the Benefits of Dynamic Asset Allocation Strategies in the Presence of Liability Constraints March 2009 in partnership with

2 Table of Contents Abstract...5 Acknowledgements Introduction A Formal Model of Asset-Liability Management Measuring the Cost of Regulatory Short-Termism Introducing Funding Ratio Constraints Introducing Minimum and Maximum Funding Ratio Constraints The (Ir)relevance of Risk Management Robustness Checks Conclusion and Extensions Appendix References...71 About the EDHEC Risk and Asset Management Research Centre EDHEC Position Papers and Publications About BNP Paribas Investment Partners Printed in France, March Copyright EDHEC The opinions expressed in this survey are those of the authors and do not necessarily reflect those of EDHEC Business School and BNP Paribas Investment Partners. This paper is a shortened version of a paper entitled "How Costly is Regulatory Short-Termism for Defined-Benefit Pension Funds?". This research has benefited from the support of the "Asset-Liability Management and Institutional Investment Management Chair", BNP Paribas Investment Partners.

3 Foreword The present publication represents the results of the first-year research work within the EDHEC/BNP Paribas Investment Partners research chair on Asset-Liability Management and Institutional Investment Management. This chair, under the responsibility of Lionel Martellini, the Scientific Director of the EDHEC Risk and Asset Management Research Centre, is examining dynamic allocation strategies in asset-liability management in order to formulate an integrated ALM model. Of critical importance, especially in the current context, is the cost of shorttermism (namely short-term funding constraints) for pension funds and more globally institutional investors. In the present study we observe ultimately that, contrary to commonly held beliefs, this cost is quite low, but what does turn out to be costly is the lack of dynamic risk management (which of course is the only genuine form of risk management). I would like to thank the co-authors, Lionel Martellini and Vincent Milhau, for the quality of their extensive research in this area. We hope that you will find their analysis and conclusions valuable and will continue to monitor and contribute to our research in this area. Finally, we would like to extend our sincere thanks to our partners at BNP Paribas Investment Partners for their commitment to this research chair and their close involvement with our research. Wishing you an enjoyable and informative read, Noël Amenc Professor of Finance Director of the EDHEC Risk and Asset Management Research Centre An EDHEC Risk and Asset Management Research Centre Publication 3

4 About the authors Lionel Martellini is professor of finance at EDHEC Business School and scientific director of the EDHEC Risk and Asset Management Research Centre. Lionel has consulted on risk management, alternative investment strategies, and performance benchmarks for various institutional investors, investment banks, and asset management firms, both in Europe and in the United States. His research has been published in leading academic and practitioner journals, including the Journal of Mathematical Economics, Management Science, Review of Financial Studies, the Journal of Portfolio Management, Financial Analysts Journal, and Risk. He sits on the editorial board of the Journal of Portfolio Management and the Journal of Alternative Investments. Lionel has co-authored and co-edited reference texts on fixed-income management and alternative investment such as the much-praised Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies (Wiley Finance) and is regularly invited to deliver presentations at leading academic and industry conferences. He holds graduate degrees in business administration, economics, statistics, and mathematics, as well as a PhD in finance from the Haas School of Business at UC Berkeley. Vincent Milhau holds master s degrees in statistics (ENSAE) and financial mathematics (Université Paris VII). He joined the EDHEC Risk and Asset Management Research Centre in 2006 and is currently a research engineer and a PhD candidate in finance at the Université de Nice-Sophia Antipolis. His research focuses on quantitative asset allocation and asset and liability management. 4 An EDHEC Risk and Asset Management Research Centre Publication

5 Abstract An EDHEC Risk and Asset Management Research Centre Publication 5

6 Abstract The recent pension crisis has triggered a fierce debate in most developed countries between advocates of a tighter regulation designed to provide explicit incentives for pension funds to increase their focus on risk management, and those arguing that imposing short-term funding constraints and solvency requirements on such long-term investors would only increase the cost of pension financing. We analyse this question in the context of a formal continuous-time dynamic asset allocation model for an investor facing liability commitments subject to inflation and interest rate risks. In an empirical exercise, we find that the presence of short-term funding ratio constraints indeed involves a positive welfare cost, but that cost is not found to be prohibitive for reasonable parameter values. Recognising that the presence of minimum funding ratio constraints, whether desirable or not, should affect the optimal allocation policy, we then provide the formal solution to the asset allocation problem in the presence of such constraints. We compare these risk-controlled strategies to unconstrained allocation strategies coupled with additional contributions, and find that the latter involve severe welfare costs in the presence of irreversible contributions and regulatory short-termism, especially when marginal utility decreases sharply beyond a given threshold. In essence, we show that risk-management strategies can turn reversible contributions and short-term constraints into irreversible contributions and long-term constraints. Overall, our results suggest that it is not so much the presence of short-term funding ratio constraints that is in itself costly for pension funds as their reluctance to implement risk-management strategies that are optimal given such regulatory constraints. 6 An EDHEC Risk and Asset Management Research Centre Publication

7 Acknowledgements An EDHEC Risk and Asset Management Research Centre Publication 7

8 Acknowledgements This work has benefitted from fruitful exchanges with the financial engineering team at BNP Paribas Investment Partners. We would also like to thank Noël Amenc, Peter Carr, Nicole El Karoui, Samuel Sender, Volker Ziemann as well as participants at the Bachelier mathematical finance seminar, the Bloomberg finance seminar, the EDHEC- Risk finance seminar and the University Paris-Dauphine finance seminar for very useful comments 8 An EDHEC Risk and Asset Management Research Centre Publication

9 1. Introduction An EDHEC Risk and Asset Management Research Centre Publication 9

10 1. Introduction 1 - See Standard Life Investments (2003) and Watson Wyatt (2003). 2 - See Standard Life Investments (2003). 3 - See Lane, Clark & Peacock Actuaries and Consultants (2003). 4 - "Valuation & Accounting: No Respite from Pension Risk", Morgan Stanley Research Europe, October Market difficulties at the turn of the millennium have drawn attention to risk management practices of institutional investors in general and defined benefit pension plans in particular. What has been labelled as a perfect storm of adverse market conditions has devastated many corporate pension plans, with negative equity market returns that have eroded plan assets at the same time as declining interest rates have increased the marked-to-market value of benefit obligations. In 2003, the defined-benefit pension plans for the companies included in the S&P 500 and the FTSE 100 index faced a cumulative deficit estimated at $225 billion and $55 billion, respectively, while the worldwide deficit reached an estimated $1,500 to $2,000 billion. 1 To better understand the magnitude of the crisis, and the scale and rapidity of the deterioration in pension funding status, it is perhaps worth noting that in the United States defined-benefit plans from S&P500 companies were enjoying a total surplus of $239 billion at the end of 1999, a mere three years earlier! 2 That pension funds have been so dramatically affected by market downturns has raised with heightened scrutiny the question of risk management for pension funds, and many have argued that their asset allocation policies have not been entirely consistent with a sound liability risk hedging process. In particular, it has been argued that asset allocation decisions of defined-benefit pension schemes were often too heavily skewed towards equities in the absence of any protection with respect to their downside risk relative to liabilities. According to an annual survey, it turns out that by 1992, percentage holdings in equities by pension funds were 75% in the UK, 47% in the US, 18% in the Netherlands and 13% in Switzerland. In 2001, midway through the bear market, pension funds had 64% of their total assets in equities in the UK, 60% in the US, 50% in the Netherlands, and 39% in Switzerland. 3 As a result of such a domination of equities, the increase in liability value that followed the decrease in interest rates was only partially offset by the parallel increase in the value of the bond portfolio. While it is too early to assess the exact impact of the current credit crisis on pension fund financial health, a recent report has provided evidence that deficits are again widening for European pension plans. 4 In October 2008, MSCI Europe pension plan assets have fallen by an estimated 19% on average since the start of the year, due to the fall in equities and other assets, while obligations have fallen only by 10% year to date based on a 97 basis point rise in the AA bond rates. From an academic perspective, a number of papers have focused on extending Merton's intertemporal selection analysis (see Merton 1971) to account for the presence of liability constraints in the asset allocation policy. Broadly speaking, asset-liability management (ALM) distinguishes itself from pure asset management by the fact that what matters from an ALM perspective is not total terminal wealth, but how terminal wealth compares to terminal value of future liabilities. More precisely, the presence of future liability commitments is accounted for by a focus on terminal wealth net of the value at the horizon of future liability payments, a quantity known as the pension fund surplus (or as the pension fund deficit when it 10 An EDHEC Risk and Asset Management Research Centre Publication

11 1. Introduction 5 - In fact, defined benefit pension commitments represent a short position in collateralised defaultable bonds issued by the sponsor company and privately held by employees, where the assets of the pension plans are the collateral, in exchange for which the company receives the present value of lower wage demands. 6 - We first consider minimum funding ratio constraints, and then introduce in the next section maximum funding ratio targets, which allow for a decrease in the cost of downside protection. is negative). 5 A first step towards the application of optimal portfolio selection theory to the problem of pension funds was taken by Merton (1993), who studies the allocation decision of a university that manages an endowment fund. In a similar spirit, Rudolf and Ziemba (2004) have formulated a continuoustime dynamic programming model of pension fund management in the presence of a time-varying opportunity set, where state variables are interpreted as currency rates that affect the value of the pension's asset portfolio. Also related is a paper by Sundaresan and Zapatero (1996), which is specically aimed at asset allocation and retirement decisions in the case of a pension fund. Van Binsbergen and Brandt (2007) complement this early work by analysing how various regulatory rules with respect to the valuation of liabilities impact optimal investment decisions, while Detemple and Rindisbacher (2008) consider an ALM problem with a limited tolerance for a shortfall in the funding ratio at the terminal date. In a nutshell, the main insight from this strand of the literature is that the presence of liability risk induces the introduction of a specific hedging demand component in the optimal allocation strategy, as typical in intertemporal allocation decisions in the presence of stochastic state variables. These results suggest that very conservative pension funds would optimally invest a significant fraction of their assets in the liabilityhedging portfolio, mostly invested in nominal or inflation-indexed bonds, so as to minimise as much as possible the probability and magnitude of a shortfall. As a consequence, they also suggest that high levels of investment in equities before the start of the bear market might indeed have been the primary cause of the pension crisis. In an attempt to provide direct incentives for pension funds to increase the focus on risk management, and in view of protecting the interests of beneficiaries, formal funding ratio constraints have been introduced by the regulator in most developed countries (see section 4 for a quick overview of regulatory constraints). Roughly speaking, the regulation can be looser or stricter when it comes to funding constraints (see Pugh 2003 for more a detailed analysis). A strict regulatory requirement over a minimum funding ratio rule is for example present in some European countries, as in Germany, where Pensionskassen and Pensionsfonds must be fully funded at all times to the extent of the guarantees they have given, or in the Netherlands, where the minimum funding level is 105% plus additional buffers for investment risks. In the United States, on the other hand, there is no formal minimum funding ratio requirement, but the Pension Benefit Guaranty Corporation (PBGC), which provides a partial insurance of pensions, charges a higher premium to funds reporting a funding level of under 90% of current liabilities, thus providing strong incentives for maintaining the funding ratio over that minimum 90% threshold. In the UK, there was a formal general minimum funding requirement (MFR) that came into effect in 1995, which eventually was replaced in the 2004 Pensions Bill with a schemespecific statutory funding objective to be determined by the sponsoring firm and fund trustees. 6 An EDHEC Risk and Asset Management Research Centre Publication 11

12 1. Introduction 7 - In Europe it has even been proposed that the Solvency II regulatory framework, originally designed for insurance companies, should also be applied to pension funds. This issue has given rise to a fierce debate between advocates of a tighter regulation, not only in the US but also in Europe, and those arguing that it would result only in a severe welfare loss. 7 The introduction of funding ratio constraints has been particularly criticised by a number of experts, who find that imposing such short-term constraints to long-term investors could be counter-productive. For example, in a report prepared on pension funding rules for the OECD, Pugh (2003) makes the following argument. "Minimum funding standards in many countries are designed around (...) current market yields on longterm bonds. In order to avoid problems, especially in jurisdictions that require immediate correction of the (perceived) underfunding, a plan sponsor is tempted to over-invest in such long-term bonds. (...) However, pension plans in the long term (...) need substantial investments in equities. Otherwise, the investments may be inefficient, and the cost of the pension plan to the plan sponsor will therefore increase. (...) In the area of minimum funding, as with other areas of legislation, there is a fine line between (over)protecting the interests of DB plan members and destroying the incentives for employers to sponsor such plans". In fact, there are two main (related) arguments that are put forward by advocates of looser regulation of pension funds. The first argument is related to the cost of short-termism. In the presence of short-term funding ratio constraints, it is required that the sponsor company make an additional contribution so as to bring the funding ratio back to minimum required value when needed. Consider, for example, a very tight regulation that enforces funding ratio constraints every year, and requires immediate funding of any deficit with respect to the liabilities. Such "short-termism" of the regulation prevents pension funds from benefiting from possible recoveries due to market performance. In other words, when a pension fund is under-funded at a given date, there is a non-zero probability that it will get back to a fully funding status (or better) at a later horizon date without any additional contribution, simply because of solid market performance. When the regulation imposes that funding ratio constraints be satisfied every year, however, the sponsor company has to make such additional contributions, even though they eventually would prove to be unnecessary in those states of the world, for which the pension fund will as a consequence end up with unnecessarily high funding levels. The second argument is related, as it happens to contribution irreversibility. While sponsor companies are required to make additional contributions, when needed, to bring asset value back to minimum regulatory levels, it is typically not possible for them to obtain refunds in those states of the world where funding ratios are very high. In the end, excess fund levels typically result in improved benefits to pensioners, and it is only in exceptional circumstances that shareholders of sponsor companies can benefit from some of the pension fund's excess assets. These two arguments have been summarised by Pugh (2003) in the following statement: "In many countries, the minimum funding standards focus on the pension fund assets exceeding the pension plan's accrued liabilities on every measurement date. 12 An EDHEC Risk and Asset Management Research Centre Publication

13 1. Introduction (...) If an asset/liability type of minimum funding measure is to be introduced or retained, then legislation should not require the immediate and complete correction of any underfunding that the test purports to reveal. Asset values fluctuate, and funding shortfalls may disappear as quickly as they had appeared. It is counterproductive for a plan sponsor to make high additional contributions and then find, one or two years later, that the markets have recovered and the plan now has an embarrassing funding excess". Short-term constraints on pension funds have in fact been criticised not only for being prohibitively costly, but also for being mostly irrelevant for longterm investors that do not face shortterm solvency concerns. This stands in stark contrast to banks and insurance companies, where the risk of client runs justifies the short-term focus. Because banking corporations borrow short and lend long, transforming savings into longer-term investments, they are subject to liquidity risk when clients exercise the implicit put option on their deposits. In the savings contracts provided by insurance companies, financial risk and losses can be passed through to employees in the form of reduced profit-sharing, as long as clients remain until the contract maturity. However, insurance policy holders may also exercise their put option on the asset value of the insurance and surrender their contracts in the event of short-term losses. Pension funds, on the other hand, have the unique ability to behave as very long-term investors, not only because the liabilities they face typically have a very long horizon, but also, and more importantly perhaps, because long-term ties bind employers and employees. After all, pension fund benefits are a by-product of the employment contract, and not a competitive financial service, and this prevents the risk of client runs: employees may be able to surrender their pension contracts only by breaking their employment contracts, and option that is rarely exercised in a massive collective fashion. While these arguments seem to make compelling intuitive sense, a formal analysis of the welfare loss, if any, related to regulatory short-termism and contribution irreversibility, has yet to be performed. This paper extends the aforementioned literature on asset allocation decisions with liability commitments by analysing the impact of funding ratio constraints in the context of a continuoustime model for intertemporal allocation decisions. Given that interest rate and inflation uncertainty are the two main risk factors impacting pension liability values, we cast the problem in a setting with stochastic interest and inflation rates. Using the martingale approach to portfolio optimisation problems, we obtain analytical solutions that allow us to confirm that the optimal strategy involves a fund separation theorem that legitimates investing in a liabilityhedging portfolio, in addition to the standard performance-seeking portfolio (speculative demand). In an empirical exercise, we find that the presence of short-term funding ratio constraints involves a positive welfare cost, but that cost is not found to be prohibitive for reasonable parameter values. Recognising that the presence of minimum funding ratio constraints, An EDHEC Risk and Asset Management Research Centre Publication 13

14 1. Introduction 8 - CPPI strategies were originally introduced by Black and Jones (1987) and Black and Perold (1992). whether desirable or not, should affect the optimal allocation policy, we then provide the formal solution to the asset allocation problem in the presence of such constraints. When funding ratio constraints are explicitly accounted for, optimal policies, for which we obtain analytical expressions, are shown to involve a dynamic allocation to the performance-seeking portfolio that is a function of the margin for error measured in terms of the distance between the current asset value and the minimum level consistent with the funding ratio constraints. These strategies, which fall within the category of risk-controlled dynamic asset allocation strategies, are reminiscent of constant proportion portfolio insurance (CPPI), which they extend to a relative (with respect to liabilities) risk context. 8 We also show that the introduction of maximum funding ratio targets would allow pension funds to decrease the cost of downside liability risk protection while giving up part of the upside potential beyond levels where marginal utility of wealth (relative to liabilities) is low or almost zero. Finally, we introduce an irrelevance principle (see subsection 6.1), which states that following an unconstrained strategy and adding contributions to make up for the deficit, if any, with respect to the minimum funding ratio requirement leads to the same terminal net wealth as following a constrained strategy when contributions are reversible and funding ratio constraints are imposed only at the terminal date. As a corollary, this result suggests that risk management matters in the presence of contribution irreversibility and regulatory short-termism. To provide a numerical assessment of the welfare gain induced by risk management, we implement various optimal risk-controlled strategies in the context of a typical liability payment schedule of an actual pension fund. For reasonable parameter values, these strategies are found to allow significant access to the upside potential of the performance-seeking portfolio, coupled with downside protection relative to the liability-based benchmark. We finally confirm that risk management adds value in the presence of contribution irreversibility and regulatory shorttermism, and we find that the induced welfare gains are very significant. These results have a number of important implications in the perspective of the current debate regarding pension fund regulation. They suggest that tight regulations, with short-term funding ratio constraints, do not involve significant welfare losses and additional burdens in terms of additional contributions, especially when marginal utility decreases sharply beyond a given threshold. They also suggest that what is actually costly for long-term investors is not so much the presence of funding ratio constraints per se as the reluctance of such investors to follow risk-controlled strategies that are optimal given these regulatory funding ratio constraints. The rest of the paper is organised as follows. In section 2, we introduce a formal continuous-time model of dynamic asset allocations decisions in the presence of liability commitments. In section 3, we provide a numerical analysis of the cost of regulatory short-termism. In sections 4 and 5, we analyse the impact of formal minimum and/or maximum 14 An EDHEC Risk and Asset Management Research Centre Publication

15 1. Introduction funding ratio constraints on optimal allocation strategies. In section 6, we provide a numerical measure of the welfare cost induced by the reluctance of following risk-controlled strategies that are optimal given regulatory constraints. In section 7, we test for the impact of various forms of market imperfections or incompleteness on the behaviour of optimal strategies introduced in section 4. In section 8, we present a conclusion as well as suggestions for further research. Technical details and proofs of the main results are relegated to an appendix. An EDHEC Risk and Asset Management Research Centre Publication 15

16 1. Introduction 16 An EDHEC Risk and Asset Management Research Centre Publication

17 2. A Formal Model of Asset-Liability Management An EDHEC Risk and Asset Management Research Centre Publication 17

18 2. A Formal Model of Asset-Liability Management 9 - It is well known that a limitation of this model is that it allows for negative interest rates with positive probability, although this probability will be small for a proper choice of parameter values Brennan and Xia (2002) and Munk et al. (2004) assume that the expected inflation rate follows an Ornstein-Uhlenbeck process. In this section, we introduce a general continuous-time asset allocation model for a pension fund facing liability constraints. This continuous-time stochastic control approach to assetliability management (ALM) is appealing in spite of its highly stylised nature because it leads to tractable solutions, allowing one to explicitly understand the various mechanisms affecting the optimal allocation strategy. We let [0; T 0 ] denote the (finite) time span of the economy, where uncertainty is described through a standard probability space (Ω,, ). We assume that financial markets are frictionless, and we denote by r t the nominal shortterm interest rate at time t. 2.1 State Variables Regarding the liability side, inflation risk and interest rate risk appear as the two most relevant risk factors. This is because pension benefits are typically indexed to inflation, and on the other hand the typically long duration of liability payments make their current value highly sensitive to changes in interest rates. As a result, it is critical for our model to incorporate both stochastic interest and inflation rates. In what follows, we model the nominal short-term interest rate as an Ornstein-Uhlenbeck process (see Vasicek 1977): (2.1) where follows a standard Wiener process under. 9 On the asset side, we assume that the menu of asset classes includes a unit zero-coupon bond with payoff 1 at maturity τ 1, where the maturity τ 1 [0; T 0 ] and will be specified later, and a price B(t; τ 1 ) at time t given by a deterministic function of (t; r t ; τ 1 ), which is given in appendix A. Our model also accounts for stochastic inflation, by assuming that the price index follows a Geometric Brownian motion: (2.2) where is a -Brownian motion correlated to. represents the instantaneous expected inflation rate, which we assume to be constant for simplicity. 10 So as to stay within a complete market environment, we assume on the asset side that there exists an inflation-indexed unit zero-coupon bond of maturity τ 2, i.e., a bond with payoff given by Φ τ2. The price at time t of the inflation-linked bond is denoted by I(t,τ 2 ), a deterministic function of (t, r t, Φ t, τ 2 ) which is also given in appendix A. Moreover, we assume that there exists one stock index with value S t at time t, where S evolves as: (2.3) where is a -Brownian motion correlated to and. Because the number of stocks is identifical to the number of risk factors impacting stock returns, the stock market is complete, as was the bond market. In this complete market setting, assuming away arbitrage opportunities implies that there exists a unique market price of risk vector λ, which we will assume to be constant, and 18 An EDHEC Risk and Asset Management Research Centre Publication

19 2. A Formal Model of Asset-Liability Management a unique equivalent probability measure under which discounted prices are martingales. (2.1), (2.2) and (2.3) can be rewritten using a n+2-dimensional Brownian motion z under (see e.g. Shreve 2005): of the nominal bond and weight of the indexed bond. We denote by A t the asset value of the pension fund at time t, which is given as the market value of its asset portfolio after taking into account pension payments before t. Hence, the change in A over a small period [t; t + dt] is equal to the return on the financial portfolio, from which is subtracted the pension payment over the period. Mathematically, this is written as: 11 - In a more general setting, one should allow for an endogenous contribution policy from a sponsor company perspective. This would require a formal dynamic capital structure model in the presence of a pension fund (see suggestions for further research in the conclusion). Note that we consider in section 6 a specific kind of contribution strategy aiming at covering the funding gap when needed. where and are respectively the long-term mean of r and the drift of Φ under the new probability measure. σ r, σ Φ and σ s are the volatility vectors of r, Φ and S. In appendix A, we derive the expressions for the volatility vectors of both types of bonds, σ B (.τ 1 ) and σ I (.,τ 2 ). We denote with σ t the square matrix with columns σ S, σ B (.,τ 1 ) and σ I (.,τ 2 ). 2.2 Liability and Net Wealth of the Pension Fund We consider a pension fund managing financial assets and making a stream of pension payments. For simplicity, we do not model the stream of contributions from the sponsor company, and assume instead that it can be summarised by an initial endowment A 0 to the pension fund. 11 This initial wealth can be invested in the n stocks, the two zero-coupon bonds, and a locally risk-free asset, whose value S 0 is the continuously compounded risk-free nominal interest rate. We denote with ω t the following vector of weights at time t: weight of the stock index, weight (2.4) where V is an increasing right-continuous stochastic process with left limits, so that dv t represents the pension payment over the interval [t, t+ dt]. This model of pension payments is inspired by models of lump-sum dividend payments for stocks (see section 6.M of Duffie 2001). In fact, this general formulation can encompass various situations with either continuous or discrete pension payments. Hence we can assume that dv t = l t dt, where l is an -adapted process, a model that captures a stream of continuous payments. Alternatively, we can take, where 0 t 1 < < t n = T 0. Here H ti is the Heaviside function,, l ti is an -measurable random variable. This model captures a stream of discrete payments from date 0 to date T 0. In what follows, we focus on a specific case of the discrete payment model, where we allow for a single payment at date T 0, a situation we shall refer to as the zerocoupon case. An EDHEC Risk and Asset Management Research Centre Publication 19

20 2. A Formal Model of Asset-Liability Management 12 - In this paper, we assume away the presence of default risk on liability payments. See the conclusion for a discussion of this important aspect The base case empirical exercises of subsections 3.1, 4.2 and 5.2, however, will assume that several payments take place before date T 0. Since the financial market is complete, the stream of future payments can be valued as the dividend flow of a financial asset. Hence the quantity (2.5) is the price that an agent would have to pay at time t to receive the payment stream dv from date t excluded to date T 0 included. 12 We will take throughout the paper l t = n t Φ t, where n is a non-negative deterministic function of time representing the size of the population to which benefits will be provided for. Hence, the pension fund is pre-committed to pay n t in real terms at time t, which amounts to a nominal payment of n t Φ t. In the zero-coupon case, a single payment takes place at time T 0, so (2.5) can be simplified into: (2.6) In this particular case, the volatility vector of L is σ I (., T 0 ). In the theoretical results of this paper, we restrict the problem of optimal allocation decisions to this zerocoupon case, that is we will assume that L t is given by (2.6). This assumption does not really simplify the computations, but it leads to allocation policies that are more easily interpreted. The general case, in which several payments take place, is addressed in Martellini and Milhau (2008) (referred to as MM08 from now on) Objectives and Optimal Asset Allocation Decisions In terms of objectives, it is customary to assume that the preferences of the investor are captured by an expected utility framework: (2.7) where u is a growing and concave utility function and T is some horizon date lying between 0 (the initial date) and T 0 (the date of the latest pension payment, beyond which the pension fund has no longer any reason to exist). In what follows we take u to be the constant relative risk aversion (CRRA) utility function, defined as: where γ lies in [1, [. If γ = 1 we obtain the logarithmic utility function. One key problem with this objective in an asset-liability management context is that it fails to recognise that further liability payments are scheduled beyond the horizon T for any choice of T < T 0. Hence, while past liability payments have been accounted for in the dynamics of asset value, the presence of future liability payments will not be incorporated in the pension fund objective. One natural approach to tackle this problem consists of recognising that terminal wealth at date T is made of a long position in the asset portfolio with value A T, but also involves a short position in the liability portfolio with value L T. Hence, preferences of the pension fund manager are expressed over the terminal surplus, as opposed to 20 An EDHEC Risk and Asset Management Research Centre Publication

21 2. A Formal Model of Asset-Liability Management 14 - Solving the optimal allocation problem when preferences are expressed in terms of the surplus A T - L T orin terms of the funding ratio generates in fact, and quite unsurprisingly, very similar results. In particular, we obtain the exact same expression for the relationship between the optimal wealth conditional upon solving the unconstrained problem and the optimal wealth conditional upon solving the constrained problem when using either the surplus of funding ratio as objectives. These results can be found in MM08. over the terminal asset value, so that the optimisation program reads: Another related approach to account for the presence of liability payments beyond the horizon consists of introducing an additional state variable, the funding ratio, defined as the ratio of assets to liabilities: (2.8) which is well-defined as long as L t is not zero. This quantity is commonly used in industry practice, where a pension trust is said to be overfunded when the funding ratio is greater than 100%, to be fully funded when the funding ratio equals 100%, and to be underfunded when the funding ratio is lower than 100%. One may therefore capture the pension fund objective as: (2.9) if pension payments are not zero after the horizon T. This objective, which is perhaps the most natural since it recognises that what really matters in pension fund management is not the value of the assets per se, but how asset value compares to liability value at each point in time, has been used by Van Binsbergen and Brandt (2007). Maximisation of expected utility of the funding ratio accounts for the presence of future liability payments since by definition (see 2.8) it is the asset value net of past liability payment expressed in terms of number of units of the current value of future liability payments. From an interpretation standpoint, this amounts to using the liability value process (L t ) t 0, as opposed to the bank account, as the numeraire, a natural choice given that what matters for the pension fund is not asset value per se, but how asset value compares to liability value. If liability commitments consist of a single inflation-indexed payment, then this objective is equivalent to maximising real wealth, as opposed to nominal wealth, a problem considered Brennan and Xia (2002). We choose in what follows to present detailed results for the case of the funding ratio, which amounts to using the current value of all future liability payments as the natural numeraire portfolio. 14 The following proposition presents the expression for the optimal policy and optimal asset value process for this program in the zero-coupon case under the assumption of complete markets. In section 7, we perform a series of robustness checks with respect to the introduction of various forms of market incompleteness. To obtain this solution, we use the martingale approach in complete markets developed by Karatzas et al. (1987) and Cox and Huang (1989), which involves two main steps. In a first step, the dynamic programme is rewritten in a static form that allows for the derivation of the optimal terminal net wealth, A* T and of the optimal net wealth process (A* t ). In a second step, the optimal dynamic portfolio strategy is obtained as the replicating portfolio for the optimal terminal wealth. Details of derivation are relegated to appendices. An EDHEC Risk and Asset Management Research Centre Publication 21

22 2. A Formal Model of Asset-Liability Management Proposition 1 The solution to (2.9) in the zero-coupon case is given by the following asset strategy: which can also be written as: where: The optimal terminal net wealth reads: Proof. This proposition is obtained as a particular case of proposition 2 below for k = 0. We find that the solution involves the standard performance-seeking portfolio (PSP) and a liability-hedging or liabilitymatching portfolio (LMP). This portfolio has the following property, which is typical of intertemporal hedging demand terms in dynamic asset allocation models (see Merton 1973): maximises the correlation between the returns on the asset portfolio and the return on the present value of future pension payments. In fact, in this complete market setting, the maximum correlation achieved is equal to 1. In case the maturity of the inflationlinked bond coincides with the date of the unique payment T 0, the liabilitymatching portfolio is fully invested in this inflation-linked bond; otherwise, it involves the combination of cash and the inflation-linked bond needed for reaching the target duration. It should be noted that the optimal portfolio strategy does not involve a separate interest rate hedging component. While interest rate risk impacts the asset value, it also impacts liability value in such a way that the net impact at the funding ratio level is trivial. Overall this portfolio strategy represents an example of a fund separation theorem, transposed from a pure asset management context to an asset-liability management context, which intuitively states that risk and performance objectives are two conflicting objectives that are best managed when managed separately. We confirm that the allocation to the performance-seeking portfolio is a decreasing function of the risk-aversion parameter γ. In the limit of an infinite risk-aversion, 100% of the assets are invested in the liability-matching portfolio which represents the true risk-free asset in an ALM context 22 An EDHEC Risk and Asset Management Research Centre Publication

23 3. Measuring the Cost of Regulatory Short-Termism An EDHEC Risk and Asset Management Research Centre Publication 23

24 3. Measuring the Cost of Regulatory Short-Termism 15 - In practice, inflation indexation is sometimes conditional, with indexation conditions that can be complex and typically depend on the funding ratio of the pension fund and the inflation rate, combined with a minimum and maximum level of indexation. We shall assume away this additional complexity in the empirical exercise that follows, and consider for simplicity a full indexation payment The main difference between their model and ours is that they also assume an Ornstein-Uhlenbeck process for the expected inflation rate, whereas we take it as a constant, which we assume is given by the long-term mean level used by Munk et al. (2004). In addition, Munk et al. (2004) provide an estimate for the market price of interest rate risk, but neither for the market price for inflation risk nor for the market price for equity risk. We set the former at 0, and we set the latter to the value found used in Brennan and Xia (2002), namely We now turn to an empirical testing of the optimal strategy discussed in the previous section. To this end, we use a schedule of liability payments provided for by a Dutch pension fund, whose name shall not be revealed. The stream of payments is displayed in table 1, from which we obtain that the date of the last scheduled payment is seventy-five years. These cash flows represent real expected pension payments, to which a cumulative inflation factor should be added so as to obtain the actual liability payment. 15 The duration of the pension fund liability is the maturity of the indexed zero-coupon bond that has the same sensitivity to interest rates as the coupon bond that models the liability. The present value at time 0 of all future payments is given by: where nti is the real payment to be made at time t i. The duration τ 0 is defined by: (3.1) Numerically, we get that τ 0 = years. Our goal in this section is not only to illustrate the empirical properties of the dynamic allocation strategy in the presence of liability constraints, but also to provide a formal measure of the cost of regulatory short-termism. 3.1 Base Case Empirical Exercise This subsection serves the purpose of providing an illustration and empirical testing of the strategy presented in the previous section, based on actual pension fund data. We emphasise that in this subsection 3.1, as well as in the corresponding table 3, we are not in the zero-coupon case: we take the horizon T to be strictly less than the date of the last payment, and greater than one, so that liability payments take place before T, according to table 1. With no loss of generality, we assume that the investment opportunity set includes a single stock index, regarded as an efficient combination of individual stocks, in addition to a zero-coupon bond and an inflation-indexed bond with maturity corresponding to the duration of pension payments. Our base case parameters are taken from Munk et al. (2004), who also model the nominal interest rate as an Ornstein-Uhlenbeck process and the price index as a Geometric Brownian motion. 16 table 2 summarises our base case set of parameter values. For each set of parameters values, N = 5000 optimal terminal asset values are obtained. These terminal values can be achieved only through the implementation in continuous-time of the corresponding dynamic allocation strategy. In all cases, we have assumed that the pension fund was initially fully funded, that is A 0 = L 0. In table 3 and figure 1, we provide information regarding the distribution of the final funding ratio. Parameters are fixed at their base case values (see table 2), the risk-aversion parameter γ takes on the values 2, 5 and 10, and the investment horizon T takes on the values 1, 10 and 20. As expected, we find that the dispersion of the final funding ratio increases with 24 An EDHEC Risk and Asset Management Research Centre Publication

25 3. Measuring the Cost of Regulatory Short-Termism Table 1: Schedule of annual liability payments expressed in real terms. Year Payment Year Payment Year Payment Year Payment This table presents the schedule of annual pension payments expressed in real terms. The data has been provided for by a Dutch pension fund. The duration of this payment stream, as computed in (3.1), is τ 0 = years. Table 2: Base case parameters. Parameter Interest rate process Estimate a b σ r Price index process ϕ σ Φ Stock value process σ s Correlation parameters ρ rφ ρ Sr ρ SΦ Risk premium parameters λ r λ Φ 0 λ S This table contains parameter values for interest rate, price index and stock return processes. These parameter values, as well as the price for interest rate risk, are borrowed from from Munk et al. (2004), while the equity risk premium parameter is taken from Brennan and Xia (2002) and the inflation risk premium is set to zero. T and decreases with γ. Indeed, a lower risk-aversion parameter implies a higher investment in the performance-seeking portfolio, and hence a higher performance potential coupled with a higher funding risk. On the other hand, for a given risk-aversion parameter value, we find that the range of funding ratio values increases with the time-horizon T as more time is allowed for uncertainty to play a role. Even for γ = 10, we find that the minimum funding ratio obtained is lower than 90% for a one-year horizon, and lower than 70% for a fifty-year horizon. This provides justication for the introduction of funding ratio constraints aiming at imposing a left truncation of the final funding ratio distribution. More importantly perhaps, we find that the average relative deficit can be rather sizable. It is equal to 28% when T = 20 and γ = 2. Similarly, we find that maximum An EDHEC Risk and Asset Management Research Centre Publication 25

26 3. Measuring the Cost of Regulatory Short-Termism Table 3: Distribution of the nal funding ratio when utility is from terminal funding ratio and no lower bound is imposed. γ = 2 T Min % % % % % Max Mean St. Dev γ = 5 T Min % % % % % Max Mean St. Dev γ = 10 T Min % % % % % Max Mean St. Dev An EDHEC Risk and Asset Management Research Centre Publication

27 3. Measuring the Cost of Regulatory Short-Termism This table reports the minimum and the maximum of the distribution of the terminal funding ratio, the 2.5%, 25%, 50%, 75% and 97.5% quantiles, the mean and the standard deviation. Also reported are the the shortfall probability, the expected shortfall and the conditional mean of the funding ratio given it lies above k = 0.9, or between 0.9 and 1.1, or between 0.9 and 1.3. Parameters are xed at their base case values (see Table 2). Several liability payments take place and we have T 0 = 75 years. Figure 1: Distribution of the final funding ratio when utility is from terminal funding ratio and no lower bound is imposed. This figure plots the distribution of the optimal terminal funding ratio there is no lower bound on this ratio. when utility is from terminal funding ratio and funding ratio values can be exceedingly high, especially for long horizons and low risk aversion. 3.2 Additional Contributions Induced by Regulatory Short-Termism In this subsection, we test for the impact of short-term funding ratio constraints in terms of additional contributions. As outlined in the introduction, there are in fact two aspects in the shorttermism of regulatory constraints. The first aspect relates to the frequency of regulatory actuarial valuations, which typically take place every year (Belgium, Brazil, Netherlands, US) or every three years (Canada, UK, Portugal, Switzerland or Ireland). In fact, in some countries, such as Portugal and Switzerland, funding ratio constraints are re-assessed every year in practice, especially for large pension plans, even though the regulation only imposes a frequency every three years (see Pugh 2003 for more details). An EDHEC Risk and Asset Management Research Centre Publication 27

28 3. Measuring the Cost of Regulatory Short-Termism 17 - Note that a distinction should be drawn at this stage between economic versus accounting impact. For example, the Financial Reporting Standards in the UK (FRS 17) imply that funding deficits have an immediate impact on sponsor balance sheets, even though pension funds have a number of years to close the funding gap. This is different from the US, where temporal smoothing of pension deficits (and surpluses) is allowed by International Accounting Standards (IAS 19) As a consequence, when δt = T, m is necessarily equal to 1. The second aspect relates to how much time is allowed for recovery plans, that is how many years the sponsor company has to fund the deficit measured at time t. This recovery time also varies across countries and regulatory environments. In rare instances, immediate funding is required, as in Germany when the funding ratio is below 100%. Most often, some recovery time is allowed. This recovery time is equal to three years in Ireland and the Netherlands, versus five years in the US and in Canada (where an additional margin is allowed). In other countries, the regulation does not impose any general recovery time, so recovery time becomes plan specific; this is the case for Belgium, the UK (where it reaches 7.5 years on average), or Portugal (between three years and ten years), where sponsor companies are expected to provide the regulator with a reasonable recovery plan indicating how they expect to manage to close the funding gap, and how much time they expect to take to do so. 17 Given that an analysis of the impact of different regulatory environments is obviously of high practical importance, we formalise this question as follows. We consider a pension fund with horizon T = 10 years and a single liability payment at date T 0 =τ 0 = years. For simplicity, we assume away sponsor default risk. The pension fund is assumed to follow the optimal unconstrained strategy, as written in proposition 1. The regulator requires that the funding ratio be checked every δt years, where typical values of δt are 1 or 3, as recalled earlier. For ; we denote by s i = iδt the date of the i th actuarial valuation. At every measurement date, the sponsor is called to contribute if the funding ratio lies below the required minimum level k. The net asset of the pension fund just after the contribution is thus equal to the net asset just before, plus the contribution. The level of the contribution is equal to the gap between the net asset and the minimum asset imposed by regulatory constraint, but the sponsor company may split its contributions into m parts, consistent with an m year duration allowed for recovery. In other words, we take the actual contribution at time s i < T is: (3.2) with m = 1 describing a stringent regulation where the sponsor is required to close the whole funding gap immediately. For all m values, we assume that the sponsor has to finance any remaining deficit immediately at the horizon date T, so that: 18 (3.3) The present value of the stream of contributions reads: where we have emphasised the dependence upon δt, m and A A Formal Measure of the Cost of Regulatory Short-Termism Our goal here is to compare two regulatory environments, and related unconstrained allocation strategies (a) and (b), with (b) being a long-termist environment, and (a) being a more or less short-termist environment. In the environment (a), the pension fund invests A 0 in the unconstrained strategy and the funding 28 An EDHEC Risk and Asset Management Research Centre Publication

29 3. Measuring the Cost of Regulatory Short-Termism ratio is checked after δt years. If the minimum funding constraint is not satisfied at this date, checking takes place every year until the funding ratio is back above the limit. As soon as a sufficiently funded status is recovered, checking takes place every δt years, and so on. Moreover, if at some checking date the constraint is not met, the sponsor is called to a contribution whose level is 1/m times the current deficit. A last check is made at terminal date T: if the constraint is not satisfied, the sponsor has to fill the gap. In environment (b), the pension fund invests A 0 + x in the unconstrained strategy, the funding ratio is checked only at terminal date T (and the sponsor has to fill the gap immediately if necessary, according to our assumption). The total cost of strategy (a) is A 0 + C 0 (δt, m, A 0 ), and the total cost of (b) is A 0 + x + C 0 (1, T, A 0 + x). We compute the value of x that makes expected utilities (with respect to utility function u) from terminal funding ratios in (a) and (b) equal. We denote this particular value with a eq (δt, m, A 0 ). We then compare the cost of (a) with the cost of (b) when x = a eq (δt, m, A 0 ) by measuring the welfare gain/loss of the short-termist regulatory environment versus the looser regulatory environment through the following factor (relative to initial asset value): Results are reported in table 4, where they are all expressed as a percentage of initial asset value. We find that the present value of future contribution decreases when δt and m increase, which was to be expected and confirms that shorttermism is costly since it induces more contributions. On the other hand, one should also recognise that early contributions might lead to higher terminal funding ratios. For example, the very tight regulatory environment δt = 1 (funding ratios assessed every year) and m = 1 (immediate funding of the deficit) is costly (for γ = 2, the cost of additional contributions is 28.18%, to be compared to 15.71% when δt = 10), but leads to a better (in terms of expected utility) funding ratio compared to the loose regulation that consists of measuring the funding ratio only at horizon (δt = T), as can be seen from the fact that the certainty equivalent amount a eq (1, 1, A 0 ) is positive, and relatively high (11.18%). More generally, we find that a eq (δ, m, A 0 ) decreases when δt and m increase, which also is intuitive. Beside, C 0 (T, 1, A 0 + a eq (δt, m, A 0 )) increases when a eq (δt, m, A 0 ) decreases, which is natural since the initial wealth is lower, and therefore there are more states of the world that require a contribution at the final date. We also see that Δ is a decreasing function of δt and m. This has a very natural interpretation: the cost of shorttermism decreases when regulation is less short-termist, whether it controls funding status less frequently or it leaves more time for refunding of deficits. Overall, the impact in terms of Δ, our measure of the cost of short-termism, shows that short-termism is costly but that the induced cost is limited. Focusing on the case γ = 2, we find that in the worst case, that is when the regulation is highly short-termist (δt = 1 and m = 1), the welfare loss is limited to %. Realistic regulatory environments, where funding ratios are measured every year and recovery times are at the minimum An EDHEC Risk and Asset Management Research Centre Publication 29

30 3. Measuring the Cost of Regulatory Short-Termism equal to three years, are in fact more accurately captured by δt = 1 and m = 3, with a cost of short-termism that goes from % for γ = 2 to % for γ = 10. Overall, absolute values of Δ are relatively small, albeit economically significant. These results provide some interesting perspective on the current debate regarding pension fund regulation, as they suggest that a short-termist regulation is costly in terms of welfare for the pension fund but that the welfare loss is relatively small. Table 4: Present value of contributions and certainty equivalents for different values of δt and m when the pension fund has utility function u. δt m This table reports the present value of contributions, C 0 (δt, m, A 0 ), when the initial asset is A 0, the funding ratio is checked every δt years and the sponsor can split contributions into m parts, and the present value of contributions, C 0 (T, 1, A 0 + a eq (δt, m, A 0 )), when the initial asset is A 0 + a eq (δt, m, A 0 ), the funding ratio is checked only at time T and the sponsor has to fill the gap immediately. a eq (δt, m, A 0 ) denotes the certainty equivalent, such that strategies (a) and (b) lead to the same expected utilities. Δ is defined as a eq (δt, m, A 0 ) + C 0 (T, 1, A 0 + a eq (δt, m, A 0 )) - C 0 (δt, m, A 0 )). Unless otherwise stated, parameters are fixed at their base case values (see Table 2), and the initial funding of the pension fund is 1, i.e., A 0 = L 0 = The minimum legal funding k is set at 0.9. The date of the single payment is T 0 = years and the time horizon of the fund is T = 10 years. γ = δt m γ = γ = 10 δt m An EDHEC Risk and Asset Management Research Centre Publication

31 4. Introducing Funding Ratio Constraints An EDHEC Risk and Asset Management Research Centre Publication 31

32 4. Introducing Funding Ratio Constraints As discussed before, funding ratio constraints, whether desirable or not, are dominant in pension fund environment. The allocation strategy presented in proposition 1 is in fact not optimal in the presence of liability constraints. As opposed to following an unconstrained strategy, and eventually requesting additional contributions, we now turn to the analysis of the optimal allocation strategy when funding ratio constraints are explicitly accounted for. To this end, we now consider the following optimisation program with explicit constraints: 4.1 The Constrained Solution We now solve the optimisation program when the minimum funding ratio is explicitly introduced in the investor's objective. So as to better analyse the results, we focus on the zero-coupon case, where we can have an explicit expression for the optimal dynamic allocation strategy. Proposition 2 The solution to (4.1) in the zero-coupon case is given by: (4.1) such that A T kl T almost surely. Imposing an explicit lower bound intuitively means that the pension fund has infinitely low utility from funding ratios below k. This leads us to consider the following equivalent program: where (4.2) where is a (non-smooth) utility function defined as follows: 19 and where the optimal net wealth process is given by: (4.3) It should be noted that because of the budget constraint, we need to have A o kl o in order to get F T k almost surely. Note also that the complete market assumption is critical here, since the presence of a non-hedgeable source of risk would make it impossible for the constraint to hold almost surely. while constant ξ is chosen so that the budget constraint A* 0 = A 0 holds. The optimal portfolio can also be expressed as: 32 An EDHEC Risk and Asset Management Research Centre Publication

33 4. Introducing Funding Ratio Constraints 20 - CPPI strategies were introduced by Black and Jones (1987) and Black and Perold (1992) One might be surprised by the fact that when the formal constraint is reached, i.e., when A t = kl t, there is a strictly positive remaining risk budget equal to (in relative terms). In fact, one can show that is the probability that the (limited) exposure to the unconstrained strategy eventually generates higher wealth than the liability benchmark level, starting from a situation where the funding ratio constraint is just met. In fact, it is only when = 0 that the risk budget is entirely spent and the investment in the performance-seeking portfolio vanishes. Proof. See appendix B. This expression is strongly reminiscent of CPPI (constant proportion portfolio insurance) strategies, which the present setup extends to an asset-liability relative risk management context. 20 Indeed, the dollar investment in the performance portfolio is given by a multiplier equal to equal to, multiplied by a measure of the margin for error (also known as the cushion in the standard CPPI strategies), defined by the difference between the asset value and a floor given by the minimum value given funding ratio constraints kl t weighted by the term. 21 While CPPI strategies are designed to prevent final terminal wealth from falling below a specific threshold, risk-controlled strategies introduced in this proposition are designed to protect asset value from falling below a pre-specified fraction of liability benchmark value. To the best of our knowledge, this result is novel and rationalises/extends the so-called contingent optimisation technique, a concept introduced by Leibowitz and Weinberger (1982) and Leibowitz and Weinberger (1983) with no theoretical justification in a simple setting where the opportunity set only includes bonds with various durations. Note, of course, that when k is zero, we recover the unconstrained optimal terminal net wealth. A comparison between the optimal terminal wealth under the unconstrained strategy and the constrained strategy can be found in the following proposition, which shows that conditional upon the funding ratio being greater than the lower bound k, the terminal wealth under the constrained strategy is lower than the terminal wealth under the unconstrained strategy. This result formalises the intuition according to which insurance of downside risk (relative to liabilities) has a cost in terms of performance potential. Proposition 3 For the states of the world ω such that, or equivalently such that we have that. Proof. To see this, first note that for k > 0, the price of the exchange option lies between the no-arbitrage bounds given on the one hand by the intrinsic value of the option (ξa 0 - kl 0 ) + and on the other hand by the underlying price ξa 0 : which implies that ξ < 1 and also that. Of course, when k = 0, we have that ξ = Empirical Analysis In this subsection 4.2, we return to the case where several liability payments take place before T. We take T 0 = 75 years. In table 5 and figure 2, we test for the introduction of explicit funding ratio constraints, with a minimum set at k = 90%. In this case, we find that the minimum funding ratio is indeed limited to k, a value that is reached with a relatively high probability, suggesting that the margin for error is fully utilised with these strategies. In fact, the dispersion of the funding ratio distribution is narrower on both sides ; An EDHEC Risk and Asset Management Research Centre Publication 33

34 4. Introducing Funding Ratio Constraints when the strategy with explicit constraints is implemented compared to the unconstrained counterpart, confirming that downside protection has a cost in terms of performance potential. Hence, in the case T = 20 and γ = 5, the maximum funding ratio is 3.1 in the explicit constraint case while it reaches 4.01 in the unconstrained case. That downside protection comes at the cost of a more limited access to the upside potential can also be seen from the fact that the expected terminal funding ratio conditional upon being larger than the constraint k;, is always strictly lower in the constrained case compared to the unconstrained case. For example, when T = 20 and γ = 5, the conditional mean reaches 1.16 in the (explicitly) constrained case, while it is 1.51 in the unconstrained case. On the other hand, the average deficit is also significantly higher in the latter case (13%, as opposed to 8% in the constrained case). Table 5: Distribution of the final funding ratio when utility is from terminal funding ratio and an explicit lower bound is imposed. γ = 2 T Min % % % % % Max Mean St. Dev γ = 5 T Min % % % % % Max Mean St. Dev An EDHEC Risk and Asset Management Research Centre Publication

35 4. Introducing Funding Ratio Constraints γ = 10 T Min % % % % % Max Mean St. Dev This table reports the minimum and the maximum of the distribution of the terminal funding ratio, the 2.5%, 25%, 50%, 75% and 97.5% quantiles, the mean and the standard deviation. Also reported are the the shortfall probability, the expected shortfall and the conditional mean of the funding ratio given that it lies between k = 0.9 and 1.1 or 1.3. Parameters are fixed at their base case values (see table 2). The explicit lower bound k is set to 0.9. Several liability payments take place and we have T 0 = 75 years. Figure 2: Distribution of the final funding ratio when utility is from terminal funding ratio and an explicit lower bound is imposed. This figure plots the distribution of the optimal terminal funding ratio and there is an explicit lower bound on this ratio. when utility is from terminal funding ratio An EDHEC Risk and Asset Management Research Centre Publication 35

36 4. Introducing Funding Ratio Constraints 36 An EDHEC Risk and Asset Management Research Centre Publication

37 5. Introducing Minimum and Maximum Funding Ratio Constraints An EDHEC Risk and Asset Management Research Centre Publication 37

38 5. Introducing Minimum and Maximum Funding Ratio Constraints 22 - Introducing implicit minimum and maximum funding ratio constraints could also be done, and would require the introduction of preferences defined in a piece-wise manner: v (x) = u (x - k) for x τ and v (x) = u (k' - x) for x τ, where τ is some threshold value, which should for example be taken equal to if one wishes to obtain a continuous utility function. Given the complexity of surplus sharing rules, it is unclear whether pension funds have any utility over exceedingly large surpluses. This is emphasised by Pugh (2003), who notes: "In practical terms, funding shortfalls are the employer's problem and funding excesses belong to the members. (...) Given that it is impossible to return any funding excess to the plan sponsor, overfunding is not a sound philosophy for most employers. (...)". It should also be noted that in some regulatory environments, maximum funding constraints are imposed by tax authorities to prevent the deliberate or accidental build-up of excessive assets within the pension fund. More generally, maximum funding ratios, when they are not a constraint, can be a target. In the Netherlands, for example, there exists a target funding ratio that is a function of investment risk, and reaches 130% on average. In this context, and given that pension funds have preferences that likely reach satiation beyond a given funding ratio level, it seems reasonable to analyse how the introduction of maximum funding ratio targets would impact the optimal strategy. The idea is that by giving up part of the upside potential beyond levels where marginal utility of wealth (relative to liabilities) is low or almost zero, the investor can decrease the cost of downside protection. 5.1 The Solution with Upper and Lower Constraints To this end, we consider the simultaneous introduction of a minimum and a maximum funding ratio constraints, so that the optimisation program is given by (2.9) subject to the additional constraints F T k and F T k': 22 (5.1) In order to have the constraint k F T k' satisfied almost surely, the initial asset A 0 should satisfy: kl 0 A 0 k'l 0 (5.2) A related programme reads: (5.3) ~ where u is a non-smooth utility function defined as follows: (5.4) In appendix E, we present an argument suggesting that the solution to (5.3) is identical to the solution to (5.1). In other words, this argument indicates that in program (5.3), one optimally implements a strategy that will not lead to funding ratios above k', even though this is not strictly forbidden as in program (5.1). Intuitively, this is because asset allocation decisions leading to funding ratios beyond k' involve an additional risk while they do not involve any marginal utility gain. The equivalence between programs (5.1) and (5.3) will be used in section 6, where we provide an analytical and numerical comparison of the constrained strategies with respect to unconstrained strategies coupled with additional contributions. The following proposition presents the solution to these optimisation programs in the presence of minimum and maximum 38 An EDHEC Risk and Asset Management Research Centre Publication

39 5. Introducing Minimum and Maximum Funding Ratio Constraints funding ratio constraints (or targets). As before, we focus on the zero-coupon case. Proposition 4 The solution to the program (2.9) with constraint k F T k' in the zero-coupon case is given by: where: The optimal net wealth process is given by: The constant ξ' is adjusted to make the budget constraint hold. Proof. We explain how to derive the optimal terminal net wealth. The first-order optimality condition for the optimisation program reads: where kl T A* T k'l T, v 2 and v 3 are non-negative and This implies that: The end of the proof involves the pricing of two exchange options and application of Ito's lemma. It is very similar to the proof of the single-constrained case, which is presented in appendix B. Comparing this solution to the case of the solution with lower constraint only, it can be seen that the imposition of an explicit upper bound involves a reduction of the risk budget. Intuitively, the idea is to reduce the cost of downside protection by giving up some access to the upside potential beyond the funding ratio threshold k'. This reduction of the cost of downside protection obtained through the introduction of the upper bound on the funding ratio implies that the optimal terminal asset value conditional upon the funding ratio lying between k and k' is higher when both constraints are imposed compared to the case when the lower constraint only is imposed. This is the content of the following proposition. Proposition 5 Let denote the optimal terminal funding ratio when the lower bound k and the upper bound k' are imposed. Similarly, let denote the optimal terminal funding ratio when only the lower bound k is imposed. For the states of the world ω such that, we have that. Proof. See appendix D. An EDHEC Risk and Asset Management Research Centre Publication 39

40 5. Introducing Minimum and Maximum Funding Ratio Constraints 5.2 Empirical Analysis In this subsection 5.2, several liability payments take place, and we take T 0 = 75 years. In table 6 and figure 3, we introduce an additional upper bound constraint, with a maximum funding ratio value set at k' = 110%. Giving up access to the upside potential above 110% allows one to decrease the cost of downside protection, as can be seen from the fact that the average of terminal funding ratio values conditional upon being in the range between 90% and 110% are higher when the upper bound is introduced than when it is not. In fact, focusing again on T = 20 and γ = 5, we have that the conditional expected funding ratio when both constraints are imposed, while it merely reached 0.95 when only the lower constraint was imposed. Comparing the solution with both constraints to the unconstrained case from table 3, we find in the unconstrained case that Hence, the addition of the short option position allows for an increase in the mean funding ratio on the range of values between 0.9 and 1.1, not only with respect to the case with minimum funding requirement only, but also with respect to the unconstrained case. for the strategy constrained at 130% as opposed to 110%. On the other hand, we find that the conditional mean of the funding ratio is higher when the upper bound is set at 110% than when it is set at 130%. Of course, the case with minimum funding ratio constraints only is recovered for k' =. Overall, this analysis suggests that introducing an upper limit allows one to decrease the cost of downside protection. Finally, in table 7 and figure 4, we test for a higher value of the maximum funding ratio that is now set at k' = 130%. Of course, the maximum value obtained is now 130%, higher than the value obtained in table 6, which was 110%. Comparing those two tables, we also find that the conditional mean of the funding ratio is higher 40 An EDHEC Risk and Asset Management Research Centre Publication

41 5. Introducing Minimum and Maximum Funding Ratio Constraints Figure 3: Distribution of the final funding ratio when utility is from terminal funding ratio and explicit lower and upper bounds are imposed. This figure plots the distribution of the optimal terminal funding ratio and there are explicit lower and upper bounds on this ratio. when utility is from terminal funding ratio Figure 4: Distribution of the final funding ratio when utility is from terminal funding ratio and explicit lower and upper bounds are imposed. This figure plots the distribution of the optimal terminal funding ratio and there are explicit lower and upper bounds on this ratio. when utility is from terminal funding ratio An EDHEC Risk and Asset Management Research Centre Publication 41

42 5. Introducing Minimum and Maximum Funding Ratio Constraints Table 6: Distribution of the final funding ratio when utility is from terminal funding ratio and explicit lower and upper bounds are imposed (k' = 1.1). T Min % % % % % Max Mean St. Dev γ = T Min % % % % % Max Mean St. Dev γ = γ = 10 T Min % % % % % Max Mean St. Dev An EDHEC Risk and Asset Management Research Centre Publication

43 5. Introducing Minimum and Maximum Funding Ratio Constraints This table reports the minimum and the maximum of the distribution of the terminal funding ratio, the 2.5%, 25%, 50%, 75% and 97.5% quantiles, the mean and the standard deviation. Also reported are the shortfall probability and the expected shortfall and the conditional mean of the funding ratio given that it lies between k = 0.9 and 1.1 or 1.3. Parameters are fixed at their base case values (see table 2). The explicit lower bound k is set to 0.9. Several liability payments take place and we have T 0 = 75 years. The explicit lower bound k is set to 0.9 and the explicit upper bound k 0 at1.1. Table 7: Distribution of the final funding ratio when utility is from terminal funding ratio and explicit lower and upper bounds are imposed (k' = 1.3). T Min % % % % % Max Mean St. Dev γ = T Min % % % % % Max Mean St. Dev γ = γ = 10 T Min % An EDHEC Risk and Asset Management Research Centre Publication 43

44 5. Introducing Minimum and Maximum Funding Ratio Constraints 25% % % % Max Mean St. Dev This table reports the minimum and the maximum of the distribution of the terminal funding ratio, the 2.5%, 25%, 50%, 75% and 97.5% quantiles, the mean and the standard deviation. Also reported are the shortfall probability and the expected shortfall and the conditional mean of the funding ratio given that it lies between k = 0.9 and 1.1 or 1.3. Parameters are fixed at their base case values (see Table 2). The explicit lower bound k is set to 0.9. Several liability payments take place and we have T 0 = 75 years. The explicit lower bound k is set to 0.9 and the explicit upper bound k' at An EDHEC Risk and Asset Management Research Centre Publication

45 6. The (Ir)relevance of Risk Management An EDHEC Risk and Asset Management Research Centre Publication 45

46 6. The (Ir)relevance of Risk Management 23 - Note that nothing guarantees that the unconstrained strategy, which is optimal in the absence of funding ratio constraints, remains optimal when the presence of additional contributions from the sponsor company is accounted for. In practice, pension funds do not usually implement the risk-controlled strategies introduced in section 4, even though these were shown to be optimal in the presence of funding ratio constraints. The alternative to these risk-controlled strategies, which is not consistent with risk management principles but is broadly applied in current industry practice, consists of following unconstrained allocation strategies, and requiring additional contributions from the sponsor company when and if needed for bringing the funding ratio back to the minimum required level. A formal analysis of the welfare cost induced by this alternative is of great practical interest and relevance, and is the subject of this section. In other words, we try to understand whether following an unconstrained strategy, which is sub-optimal in the presence of funding ratio constraints, and eventually making additional contributions if needed, leads to a severe welfare loss compared to following the optimal constrained strategy Two Irrelevance Principles In what follows, we are able to isolate a number of conditions under which risk management is irrelevant, in the sense that both approaches lead to the exact same terminal net wealthes. This irrelevancy principle is, however, subject to a number of conditions and simplifying assumptions, and we obtain as a corollary that any deviation from these conditions will induce a utility cost, for which we will provide a quantitative measure in subsection The Case of Minimum Funding Ratio Constraints The dynamic allocation strategies introduced in the previous sections allow investors to benefit from downside protection in the sense that they guarantee that the pension fund asset value stays above the level consistent with the minimum funding requirement. Obviously, they are superior to the unconstrained strategy when a minimum funding ratio requirement is introduced, since they have been shown to be optimal under such circumstances. While we have so far considered a single initial contribution A 0 and assumed away the possibility of additional contributions, it should be noted that there exists an alternative to implementing a risk-controlled dynamic allocation strategy, which consists of following the unconstrained strategy, and requiring that the sponsor company eventually make a final contribution so as to bring the funding ratio back to the minimum required value (that is, kl T ) when needed. Hence the final contribution for the sponsor company is. (Note that in this expression, and in the analysis that follows, we make explicit the dependence upon A 0 in the notation for the optimal asset value process that was introduced in proposition 1. This is because it will prove necessary to analyse various strategies with different initial wealth levels.) The cost of downside protection for the sponsor company can therefore be measured as the initial price, denoted by C 0, of the additional contribution. In other words, the cost of the strategy that consists of investing A 0 in the unconstrained strategy and holding on the side an option with payoff 46 An EDHEC Risk and Asset Management Research Centre Publication

47 6. The (Ir)relevance of Risk Management, and the terminal net wealth associated to this strategy is max. In what follows, we would like to compare the (direct) cost of downside risk protection enforced through the aforementioned contribution strategy versus the (opportunity) cost of downside risk protection enforced through a suitable dynamic allocation strategy. To do so, one should compare the terminal net wealth max to the terminal net wealth generated by following the optimal constrained strategy (with explicit constraint), starting from the initial asset level A 0 +C 0. In fact, it can be shown (see MM08) that the two strategies are equivalent in the sense that they yield the same terminal net wealth in each state of the world. This result is related to a put-call parity relation between the prices of two exchange options, and it is rather intuitive: it shows that imposing an explicit funding ratio constraint is equivalent to buying a guarantee from the sponsor company, and has the same initial cost. This result seems to suggest that risk management is irrelevant; as opposed to implementing risk-controlled dynamic allocation strategies aiming at protecting a minimum value of assets relative to liabilities, the pension fund can simply follow an unconstrained strategy, and eventually require an additional contribution if and when needed. A number of comments should, however, be made about this conclusion. First, in a complete market environment, the approach aiming at enforcing downside protection through dynamic allocation strategies allows a respect of the minimum funding ratio requirement not only at terminal date, but also at all possible dates. This stands in sharp contrast to the approach aiming at requiring a final contribution from the sponsor company, which can involve large deficits at intermediate dates. While an analytical expression for the present value of the stream of additional contributions is not easily obtained in closed-form, it can be numerically computed. In subsection 6.2, as it happens, we do a numerical analysis of the additional cost (in terms of present value of contributions) induced by implementing downside protection through such contribution strategies. The key difference between enforcing minimum funding ratio constraints at the terminal date versus imposing them at all dates is that in the latter situation, the pension fund cannot expect that good performance on the risky asset classes will help reabsorb the deficit. The difference will be interpreted as the cost of short-termism. Rather surprisingly, and in contrast to a widespread belief, we find that short-termism induces only a very mild cost, and sometimes in fact leads to a welfare gain, especially when utility reaches a plateau beyond a given funding ratio threshold. Furthermore, it should be noted, as an additional advantage of allocation versus contribution strategies, that the efficiency of not managing risk, and requiring additional contributions, is seriously impaired by the fact that contributions are irreversible. In other words, if the pension fund ends up with very large surplus, it will typically prove very complex for the sponsor to enjoy a "negative contribution", that is to withdraw money from the pension fund. In practice, the best it can hope for is a contribution holiday. On the other An EDHEC Risk and Asset Management Research Centre Publication 47

48 6. The (Ir)relevance of Risk Management 24 - If default risk is introduced, one should also note that the process giving the value of the real assets of the firm, a key element in how and when default will be triggered, will appear as an additional state variable, which shall justify the introduction of another "fund" in the optimal unconstrained strategy. hand, dynamic allocation strategies can be designed to achieve a terminal net wealth that will never exceed a given fraction of the liability value. Analysing how optimal dynamic allocation strategies consistent with the introduction of maximum funding ratio targets would allow pension funds to decrease the cost of downside protection while giving up part of the upside potential beyond levels where marginal utility of wealth (relative to liabilities) is low or almost zero is precisely the purpose of the next subsection. As a final note, it should be emphasised that this analysis has abstracted away from default risk from the sponsor company perspective. If the required additional contribution needed to make up for the deficit with respect to the minimum funding requirement happens to be too large compared to the value of the assets of the sponsor company, the sponsor can default on its obligations with respect to employees and pensioners. 24 Regarding default on pension obligations, it should be noted that in some countries like the US, there exists a pension insurance system (called the Pension Benefit Guaranty Corporation or PBGC), which is in charge of compensating for the deficit, if any, in pension payment in case of default from the sponsor company (see Bodie 1996 for a discussion of the pension put, and Bodie et al for empirical evidence that PBCG creates an incentive for distressed companies to underfund their pension plan and invest in risky assets.) Analysing the impact of these additional sources of complexity is beyond the scope of the present paper, and is left for further research The Case of Minimum and Maximum Funding Ratio Constraints In the spirit of subsection 6.1.1, we wish to compare the terminal net wealth obtained by imposing the double constraint to that achieved through an unconstrained strategy coupled with a state-dependent contribution from the sponsor at the terminal date. To do this, we first consider the very hypothetical situation of a pension fund that can receive a negative contribution from the sponsor. In other words, we now assume not only that if the net asset at time T is insufficient to meet the minimum funding requirement, the sponsor contributes so that the constraint is satisfied but also that the sponsor company can obtain a "refund" if the net asset value at time T is higher than what is required in terms of maximum (or target) funding ratio k'. This, of course, is highly unrealistic, since contributions are essentially irreversible. It can be shown that the dynamic asset allocation strategy with double constraints is formally equivalent to the unconstrained strategy with reversible contribution at the terminal date (see MM08). One may define the cost of irreversibility as the difference between C 0, which is the present value of a single contribution at time T when irreversibility prevents the sponsor from recovering the assets in excess of kl T, and C' 0, which is the present value of a single contribution at time T when reversibility allows the sponsor to recover the excess of assets over minimum threshold. In other words, we set: 48 An EDHEC Risk and Asset Management Research Centre Publication

49 6. The (Ir)relevance of Risk Management As outlined in the introduction, a second reason why implementing risk management through dynamic allocation strategies differs from implementing risk management through additional contribution strategies is related to the presence of short-term minimum funding requirements, which impose that additional contributions occur every year if the minimum funding ratio is not met, thus preventing the sponsor company from enjoying the possibility for the fund to get back to solvency levels following good financial market performance. In the next subsection, we perform a separate numerical assessment of the impact of contribution irreversibility and short-termism when comparing unconstrained strategies to constrained strategies. 6.2 Measuring the Costs and Benefits of Risk Management We now wish to compare the following two strategies: (a) the pension fund invests A 0 in the unconstrained strategy of proposition 1, the funding ratio is checked every δt years as long as the funding ratio stays above the limit, and every year while it lies below, and the sponsor has to finance 1/m of the current deficit each year; (c) the pension fund invests A 0 + x in the constrained strategy and the sponsor does not (need to) contribute. The constrained strategy in (c) can be either with a minimum funding ratio constraint only or with both minimum and maximum funding ratio constraints. As before, we compute the value, which we denote by, of x that makes expected utilities from terminal funding ratios in (a) and (c) equal. The choice between (a) and (c) is a choice between not implementing a risk management process versus implementing an asset allocation strategy that offers as a builtin property downside risk (relative to liabilities) protection. We then compare the cost of (a) with the cost of (c) when by measuring the welfare loss of non-implementing a risk-controlled strategy as: As a consequence of lemma 1 in appendix C, when is unique if it exists. Moreover, it follows from the discussion in subsection that, in the case of a single constraint, is equal to C 0 (T, 1, A 0 ). Indeed, starting from the initial asset value A 0 + C 0 (T, 1, A 0 ) and following the constrained strategy with minimum funding ratio constraint leads to the same terminal net wealth, and hence the same expected utility, as starting from A 0, following the unconstrained strategy and requiring additional contributions from the sponsor at terminal date T. Note that the costs of additional contributions, denoted by C c 0, are always 0 for the constrained strategies since they precisely have been designed to avoid with probability 1 (in complete market environments) funding ratios below the lower limit k. In fact, it can be argued that following a risk-controlled strategy precisely amounts to turning short-term constraints into long-term constraints (i.e., transforming δt < T into δt = T). As a result, the cost of not following a risk-controlled strategy is identical to the cost of short-termism, i.e., Δ c = Δ, and has therefore already been estimated in table 4. To see this, first note that (again as a consequence of a result from MM08) An EDHEC Risk and Asset Management Research Centre Publication 49

50 6. The (Ir)relevance of Risk Management investing A 0 + a eq (δt, m, A 0 ) + C 0 (T, 1,A 0 + a eq (δt, m, A 0 )) in the constrained strategy yields the same terminal net wealth, hence the same expected utility, as investing A 0 + a eq (δt, m, A 0 ) in an unconstrained strategy and requiring a unique final contribution from the sponsor at time T. Then, by definition of a eq (δt, m, A 0 ), investing A 0 +a eq (δt, m, A 0 )+C 0 (T, 1, A 0 + a eq (δt, m, A 0 )) in the constrained strategy with a funding ratio checked only at date T leads to the same expected utility as investing A 0 in an unconstrained strategy with funding ratio checked every δt and a recovery time of m years (and subsequently adding contributions with present value C 0 (δt, m, A 0 )). Hence, the value that makes expected utilities from (a) and (c) equal is simply the sum of and C 0 (T, 1, A 0 + a eq (δt, m, A 0 )), two quantities that are displayed in table 5, and we have that Δ c = Δ. In table 8, we thus provide estimates for and C 0 (δt, m, A 0 ), expressed as a percentage of the initial asset value A 0, for different values of γ, δt and m. Hence, we can conclude that the cost of not following a risk-controlled strategy, which is identical to the cost of short-termism, is relatively low but economically significant, for example equal to % of the initial asset value for γ = 2 and δt = m = 1. As before, we now implement the same exercise, but we consider the imposition of a double explicit constraint rather than a single one, thus recognising that there is little marginal utility in reaching very high funding ratios. Since imposing a double constraint amounts to assuming that the pension fund has utility function, expected utilities from both strategies (a) and (c) are computed using this function. Again, we compute the amount that needs to be added to A 0 in order to get the same expected utility from strategy (a) as from strategy (c), a quantity that is expressed in percentage terms relative to the initial wealth A 0. In table 9, we provide information about our measure Δ c of welfare gains/losses induced by implementing, versus not implementing, risk management strategies. These results are obtained for different values of the risk-aversion parameter, when the pension fund horizon is assumed to be T = 10, k = 90%, and k' is taken to be equal to be 100%, 110% or 130%. The values we obtain now provide a very different picture, and show a very significant cost associated with the absence of risk management, a cost that reaches % (as a percentage of the initial asset value) when γ = 2, t = 1, m = 1 and k' = 1.1. This suggests that risk-controlled strategies with minimum and maximum constraints have the appealing property of allowing significant upside benefits at a relatively low cost. From an intuitive standpoint, these results can be explained by the fact that the cost of contribution irreversibility is now added to the cost of regulatory short-termism. In fact, MM08 show that by following the optimal constrained dynamic strategy, the pension fund allows the sponsor company to save an amount which is exactly equal to the present value of negative contributions that would be enjoyed by the sponsor company if it could withdraw the excess value of assets with respect to a target funding level. In other words, the welfare gains associated with risk management are large because the risk-controlled strategies now not only avoid additional contributions by respecting the minimum funding requirement, but also avoid the unnecessarily 50 An EDHEC Risk and Asset Management Research Centre Publication

51 6. The (Ir)relevance of Risk Management high funding ratios associated with no utility improvement. Overall, and taken together, the results we obtain strongly suggest that risk-controlled strategies can add significant value when compared to unconstrained strategies that result in additional short-term and irreversible contributions from the sponsor company. In the end, it appears that what is costly is not so much the presence of funding ratio constraints per se, as the reluctance to use the dynamic risk-controlled allocation strategies that are optimal given these regulatory constraints. Table 8: Comparison of the costs of a strategy involving contributions and of a strategy with explicit lower bound. γ = 2 δt m γ = 5 δt m γ = 10 δt m This table reports the present value of contributions, C 0 (δt, m, A 0 ), when the initial asset is A 0, the funding ratio is checked every δt years and the sponsor can split contributions into m parts, and the present value of contributions,, when the pension fund implements the constrained strategy (c). denotes the certainty equivalent, such that strategies (a) and (c) lead to the same expected utilities. Δ c is defined as. Unless otherwise stated, parameters are fixed at their base case values (see Table 2), and the initial funding of the pension fund is 1, i.e., A 0 = L 0 = The explicit lower bound k, which is also the minimum legal funding, is set at 0.9. The date of the single payment is T 0 = years and the time horizon of the fund is T = 10 years. An EDHEC Risk and Asset Management Research Centre Publication 51

52 6. The (Ir)relevance of Risk Management Table 9: Comparison of the costs of a strategy involving contributions and of a strategy with explicit lower and upper bounds. k' = 1.1 δt m k' = 1.3 δt m k' = 1.5 δt m k' = 1.7 δt m k' = 1.9 δt m This table reports the present value of contributions, C 0 (δt, m, A 0 ), when the initial asset is A 0, the funding ratio is checked every t years and the sponsor can split contributions into m parts, and the present value of contributions,, when the pension fund implements the constrained strategy (c). denotes the certainty equivalent, such that strategies (a) and (c) lead to the same expected utilities. c is defined as. Unless otherwise stated, parameters are fixed at their base case values (see table 2), and the initial funding of the pension fund is 1, i.e., A 0 = L 0 = The explicit lower bound k, which is also the minimum legal funding, is set at 0.9 and the risk aversion parameter at 2. The date of the single payment is T 0 = years and the time horizon of the fund is T = 10 years. 52 An EDHEC Risk and Asset Management Research Centre Publication

53 7. Robustness Checks An EDHEC Risk and Asset Management Research Centre Publication 53

54 7. Robustness Checks 25 - We have also tested the introduction of other forms of market incompleteness on the asset side, including stochastic equity volatility and stochastic equity risk premiums, and found that the risk-controlled strategies performed extremely well. In particular, violations of the minimum funding ratio requirements were extremely limited even for unreasonably large values of volatility or risk premium uncertainty. This section will serve the purpose of assessing the robustness of the results we have obtained in the context of a stylised model when the strategies are transported in more general contexts. In particular, we test for the introduction of specific liability risk, which induces a particular form of market incompleteness. In fact, our goal is to test for the robustness of the strategies derived in a simple completemarket environment in such incomplete market settings. We also test for the impact of dynamic incompleteness induced by the inability of pension funds to implement continuous trading. 7.1 Introducing Dynamic Incompleteness The optimal terminal net wealth obtained in section 4 under various kinds of maximisation objectives can only be generated through dynamic trading in continuous time. In this subsection, we analyse the impact of implementing the dynamic portfolio strategies in a discrete-time, as opposed to a continuoustime setting. With no real loss of generality, we consider the zero-coupon case, which leads to analytical expressions for the optimal strategies. For the sake of brevity, we focus solely on the comparison of terminal funding ratio distributions with an explicit minimum funding requirement, when trading takes place at various frequencies, ranging from annual to bimonthly, monthly and weekly. The results are compared to the case when the strategies are implemented in continuous time. Results are presented in table 10. As expected, we find that the introduction of dynamic incompleteness leads to a deterioration of the performance of the strategy as far as respect of minimum funding ratio constraints are concerned. Hence the minimum value for the funding ratio happens to be lower than the minimum funding value k in the yearly and bimonthly situations for various values of the risk-aversion parameters. On the other hand, these violations are very limited in size and probability for reasonable trading frequencies. Assuming a monthly trading frequency, we find a violation of the minimum value only in the case γ = 10, with no violations at the 2.5% probability. 7.2 Introducing Jump Risk on the Asset Side So as to check the robustness of the terminal net wealth distribution with respect to various forms of market incompleteness, we also introduce the presence of jumps in the process describing the dynamics of stock returns. 25 Following Merton (1976), we model the stock price as the mixture of a pure diffusion process and a Poisson process: In this equation, N denotes a Poisson process with constant intensity under denoted by ι, a process which we assume to be independent of. x represents the jump size, which we assume to be constant for simplicity. The stock price can be expressed in terms of the Brownian motion and the Poisson process N, based on an extended version of Ito's formula that can be applied to the right-continuous process log S (see for example Protter 2004 or Duffie 2001): 54 An EDHEC Risk and Asset Management Research Centre Publication

55 7. Robustness Checks 26 - Liu et al. (2003) consider a model in which a -25% jump takes place at the average frequency of twenty-five years. The instantaneous expected rate of return on the stock is denoted by μ s. Following Liu et al. (2003), we assume that it is given as the sum of the risk-free interest rate, r t, and a risk premium that rewards the owner of the stock for both diffusion and jump risks. Consistent with the base case Geometric Brownian motion model, we take to be constant. For the numerical application, we take again the horizon of the pension fund to be T = 10, and we assume a monthly implementation of the strategy with explicit lower bound set at 90% over the period. Results are presented in table 11. We find that jumps of exceedingly large size or frequency are needed for the strategy to exhibit severe and frequent violations of the minimum funding ratio requirements. For example, assuming a jump that takes place on average every five years, with a -20% intensity, we find for γ = 2 that the minimum funding ratio is 75%, with the 2.5% bottom percentile already very close to the minimum funding requirement, with the value 87%. If a jump of the same size were to occur on average every year, the strategy would not perform as well: for γ = 2, the probability of not satisfying the minimum funding requirement would be more than 50%. However, a -20% jump on average every year appears to be an extreme assumption. 26 For γ = 2, in each but this particular case, the probability of not breaching the floor is greater than 75%. For greater values of the relative risk aversion, it is easier to meet the constraint since a greater γ implies a lower allocation to the performance-seeking portfolio. Since stocks receive a lower allocation, the terminal net wealth is less impacted by jump risk. For instance, when γ = 10, the constraint is satisfied with a probability of 2.5% in all cases, unless a -20% jump occurs on average every year. 7.3 Introducing Specific Liability Risk So far we have assumed that the pension payments are of the form lti = ntiφti where nti is deterministic. Hence the only sources of uncertainty driving the liability value L t are interest rate and inflation risks. Since nominal and indexed bonds are assumed to be traded on the financial market, liability risk is therefore entirely spanned by existing securities. In practice, however, the presence of specific liability risk related in particular to actuarial uncertainty induces a particular form of market incompleteness. We do not attempt here to present a realistic model for the dynamics of the work force demography, but instead to introduce a stylised model that will allow us to test the robustness of the dynamic allocation strategies in the presence of actuarial risk that is not spanned by traded securities. In order to do so, we assume that the amount to be paid at time T 0 is equal, where and is a -Brownian motion independent of. Hence, we assume that actuarial risk is independent from financial risk, arguably a reasonable approximation of reality. Note that we have: Hence, the expected level of the payment to be made at time T 0 given all financial information available at time T 0 is equal to so that the complete market case An EDHEC Risk and Asset Management Research Centre Publication 55

56 7. Robustness Checks with no actuarial uncertainty is recovered for. When, the financial market is no longer complete since traded assets only span the uncertainty arising from Brownian motion. More precisely, the market price of risk vector associated with is completely determined by the returns on existing securities, and hence is identical to the one introduced in section 2, but the market price of risk associated with is a degree of freedom, which we take to be zero. This amounts to assuming that the pricing measure is the same as that introduced in section 2. In particular, still follows a standard Wiener process under, and we have: We take again the horizon of the pension fund to be T = 10, and we assume a monthly implementation of the strategy with an explicit lower bound set at 90% over the period. Results are presented in table 12 for different values of, which is the parameter driving the uncertainty on through the equality: As expected, we find that the introduction of liability risk also leads to a deterioration of the performance of the risk-controlled strategy. For reasonable parameter values leading to 5% standard-deviation in, we find that minimum funding requirement is respected with a 2.5% probability when γ = 10, and close to being respected at the 2.5% confidence level for γ = 2 and γ = 5. Hence, the 2.5% percentile of the funding ratio distribution for liability risk with 5% standard deviation is 0.87 for γ = 2, 0.88 for γ = 5; and 0.9 for γ = 10. Overall, it seems that the benefits of the risk-controlled strategies are relatively robust with respect to the introduction of various forms of market incompleteness. Table 10: Distribution of the terminal funding ratio when utility is from terminal funding ratio, there is an explicit lower bound, and trading takes place either at discrete instants or in continuous time. γ = 2 Trading frequency Yearly Bi-monthly Monthly Weekly Continuous Δt Min % % % % % Max Mean St. Dev γ = 5 Trading frequency Yearly Bi-monthly Monthly Weekly Continuous Δt Min % An EDHEC Risk and Asset Management Research Centre Publication

57 7. Robustness Checks 25% % % % Max Mean St. Dev γ = 10 Trading frequency Yearly Bi-monthly Monthly Weekly Continuous Δt Min % % % % % Max Mean St. Dev This table reports the minimum and the maximum of the distribution of the terminal funding ratio, the 2.5%, 25%, 50%, 75% and 97.5% quantiles, the mean and the standard deviation. Also reported are the shortfall probability and the expected shortfall of the funding ratio. t denotes the time interval between two consecutive trading instants, expressed in fractions of years. Parameters are set at their base case values (see table 2), except for the risk aversion parameter γ. The explicit lower bound is set to k = 0.9. The date of the single payment is T 0 = years and the time horizon of the fund is T = 10 years. Table 11: Distribution of the terminal funding ratio when utility is from terminal funding ratio, and the stock price process exhibits jumps. ι x Min % % % % % Max Mean St. Dev γ = An EDHEC Risk and Asset Management Research Centre Publication 57

58 7. Robustness Checks ι x Min % % % % % Max Mean St. Dev This table reports the minimum and the maximum of the distribution of the terminal funding ratio, the 2.5%, 25%, 50%, 75% and 97.5% quantiles, the mean and the standard deviation. Also reported are the shortfall probability and the expected shortfall of the funding ratio. ι denotes the intensity of the Poisson process, that is 1/ ι represents the average time between two consecutive jumps. x stands for the size of jumps. Other parameters are set at their base case values (see Table 2). The date of the single payment is T 0 = years and the time horizon of the fund is T = 10 years. The explicit lower bound is set to k = 0.9. γ = γ = 10 ι x Min % % % % % Max Mean St. Dev Table 12: Distribution of the terminal funding ratio when utility is from terminal funding ratio, and there are both an explicit lower bound and a specific liability risk. γ = Min % % % % % Max Mean An EDHEC Risk and Asset Management Research Centre Publication

59 7. Robustness Checks St. Dev This table reports the minimum and the maximum of the distribution of the terminal funding ratio, the 2.5%, 25%, 50%, 75% and 97.5% quantiles, the mean and the standard deviation. Also reported are the shortfall probability and the expected shortfall of the funding ratio. is the standard deviation of. Parameters are set at their base case values (see table 2). The date of the single payment is T 0 = years and the time horizon of the fund is T = 10 years. Trading takes place every month, and the explicit lower bound k is set to 0.9. γ = Min % % % % % Max Mean St. Dev γ = Min % % % % % Max Mean St. Dev An EDHEC Risk and Asset Management Research Centre Publication 59

60 7. Robustness Checks 60 An EDHEC Risk and Asset Management Research Centre Publication

61 8. Conclusion and Extensions An EDHEC Risk and Asset Management Research Centre Publication 61

62 8. Conclusion and Extensions Defined-benefit pension funds are currently facing a serious challenge and dilemma. On the one hand, the desire to alleviate the burden of contributions leads them to invest significantly in equity markets and other classes poorly correlated with liabilities but offering superior long-term performance potential. On the other hand, stricter regulatory environments and accounting standards give them greater incentives to invest a dominant fraction of their portfolios in assets highly correlated to liabilities. While there is general agreement that some regulatory constraints are needed, there is a fierce debate regarding whether it makes sense to impose short-term constraints on long-term investors. The introduction of funding ratio constraints has been particularly criticised by a number of experts, who find that imposing such shortterm constraints on long-term investors could be counter-productive. The point of our paper is not to determine whether minimum funding ratio constraints are desirable, but to try to provide a formal measure of the cost of regulatory short-termism. We analyse this question in the context of a continuoustime dynamic asset allocation model for an investor facing liability commitments subject to inflation and interest rate risks. Perhaps surprisingly, we first find that short-term funding ratio constraints do not involve significant welfare losses, especially when marginal utility decreases sharply beyond a given threshold. Recognising that the presence of minimum funding ratio constraints, whether desirable or not, should affect the optimal allocation policy, we provide the formal solution to the asset allocation problem in the presence of such constraints. We then compare these risk-controlled strategies to unconstrained allocation strategies coupled with additional contributions, and find that the latter involve severe welfare costs in the presence of irreversible contributions and regulatory short-termism. In an empirical analysis, we find that the benefits of risk-controlled strategies are relatively robust with respect to the introduction of various forms of market incompleteness. Overall, our results suggest that it is not so much the presence of short-term funding ratio constraints per se that is costly for pension funds as their reluctance to implement risk-management strategies that are optimal given such regulatory constraints. In essence, our results show that risk-management strategies can turn reversible contributions and short-term constraints into irreversible contributions and long-term constraints, hence the severe opportunity cost for pension funds that do not follow them. The benefits of risk-controlled strategies are illustrated in figure 5: in such strategies, the exposures to the performance-seeking portfolio and the liability-hedging portfolio are contingent on the funding level so as to make any additional contribution from the sponsor plan unnecessary (see figure 5(b)). If the unconstrained strategy is instead implemented, these additional contributions are required in situations where the funding ratio falls below the minimum regulatory level (see figure 5(a)). Making such additional contributions is all the more unfortunate that they are irreversible, and can not be recovered 62 An EDHEC Risk and Asset Management Research Centre Publication

63 8. Conclusion and Extensions Figure 5: Impact of a risk-controlled strategy on the funding ratio. (a) Unconstrained approach Funding ratio constraints: Contributions may be necessary in case of underfunding and are irreversible (C1, C2). Surplus of the pension fund is unavailable to the sponsor company since contributions are irreversible (C3). (b) Risk-controlled approach Dynamic LDI strategy depending on the current funding ratio: Contributions are no longer necessary, which involves a lower cost. The management of the portfolio is based on a risk-controlled strategy, involving dynamic adjustments in the exposures to PSP and LHP. An EDHEC Risk and Asset Management Research Centre Publication 63