خطر طول عمر، پس اندازهای بازنشستگی، و نوآوری های مالی

Over the last couple of decades unprecedented increases in life expectancy have raised important concerns for retirement savings. We solve a life-cycle model with longevity risk, which can be hedged through endogenous saving and retirement decisions. We investigate the benefits of financial assets designed to hedge the shocks to survival probabilities. When longevity risk is calibrated to match forward-looking projections, those benefits are substantial. This lends support to the idea that such hedging should be pursued by defined benefit pension plans on behalf of their beneficiaries. Finally, we draw implications for optimal security design.

Over the last few decades, an unprecedented increase in life expectancy has occurred. For example, a 65-year-old United States male in 1970 had a life expectancy of 78 years.1 Roughly three and a half decades later, in 2007, a 65-year-old male had a life expectancy of 82.5 years. This represents an increase of 1.2 years per decade. To understand what such an increase implies in terms of the savings needed to finance a given stream of retirement consumption, consider a fairly priced annuity that pays $1 real per year and assume that the real interest rate is 2%. The price of such annuity for a 65-year-old male would have increased from $10.5 in 1970 to $13.5 by 2007. This is an increase of roughly 29%. In other words, to finance a given stream of real retirement consumption, a defined benefit (DB) pension plan (or a 65-year-old male) would have needed 29% more wealth in 2007 than in 1970. These large increases in life expectancy were, to a large extent, unexpected and as a result they have often been underestimated by pension plans, Actuary's, and insurers. This is hardly surprising given the historical evidence on life expectancy. From 1970 to 2007 the average increase in the life expectancy of a 65-year-old male was 1.2 years per decade, but, from 1933 to 1970, the corresponding increase had been only 0.2 years per decade. This pattern of increases in life expectancy has not been confined to the US. In the United Kingdom, a country for which a longer time series of data on mortality is available, the average increase in the life expectancy of a 65-year-old male was 1.5 years per decade from 1970 to 2009, but only 0.1 years per decade from 1849 to 1970. These unprecedented longevity increases are to a large extent responsible for the underfunding of pay-as-you-go state pensions,2 and defined benefit company and state-sponsored pension plans, and they show either their benefits must be lowered or contribution rates increased.3 The response of governments has been to decrease the benefits of state pensions and to give tax and other incentives for individuals to save privately, through defined contribution pension schemes. Likewise, many companies have closed company sponsored defined benefit plans to new members, while others have reduced their benefits, or increased contribution rates, or both. For instance, in their 2010 review of employer rates, the members of the benefits and program administration committee of Calpers note that what was causing the biggest increase in employer contribution rates was the proposed change to more realistic post-retirement mortality assumptions.4 This also raises a question as to how DB pension plans should address the issue of longevity risk or further changes in mortality rates going forward. The risk that plan members live longer could be reduced by the pension plan by the purchase of annuities at retirement age. However, considerable uncertainty exists for younger pension plan members with respect to the level of aggregate life expectancy, and consequently the annuity prices, that they will face when they retire. How should pension plans address such risk? Should they try to hedge it to the extent that it is possible to do so? Naturally, the answer to these questions depends on the extent to which the pension plan beneficiaries might benefit from such insurance. This paper studies the extent to which individuals are affected by longevity risk and the role that different instruments, including financial assets, play in hedging it. Thus, we focus on the demand for such assets. We first present the existing empirical evidence on longevity, focusing on its historical evolution, on forward-looking estimates of mortality rates, and on the uncertainty surrounding these estimates. For this purpose we use current long-term projections made by the US Social Security Administration (SSA) and by the UK Government Actuary's Department (GAD). We use this evidence to parameterize a life-cycle model of consumption and saving choices. The main distinctive feature of the model is that the survival probabilities are stochastic and evolve according to the specification proposed in Renshaw and Haberman (2006). This generalizes the one proposed in Lee and Carter (1992), which is the leading statistical model of mortality in the demographic literature, by allowing for cohort effects. For part of the analysis we focus on the simpler Lee and Carter formulation. In our model, the individual receives a stochastic labor income each period and decides how much to consume and save. She knows the current survival probabilities, but she does not know the future survival probabilities, because those are stochastic. Naturally the individual forms an expectation of such probabilities when making her decisions. We allow for endogenous retirement so that, in addition to adjusting her savings in response to changes in life expectancy, she can revise her retirement decision. Traditionally markets were incomplete in that agents did not have at their disposal the financial assets that would allow them to hedge longevity risk. We say “traditionally” because recent attempts have been made to address this market incompleteness. In December 2003, Swiss Reinsurance Company Ltd. (Swiss Re.) issued a $400 million three-year life catastrophe bond.5 Swiss Re. tried to insure itself against a catastrophic mortality deterioration (e.g., a pandemic). More recently, there has been a growing interest in longevity swaps. These assets allow pension funds and other annuity providers to hedge the longevity risk to which they are exposed. In recent years several pension funds have actively started hedging their longevity exposure using financial products such as these, provided by some of the major financial institutions. Business Week reported a volume of $15 billion in new life settlement backed securities issued in the US in 2006, and this number was expected to double in 2007.6 According to Risk.net the volume of new issuances in the UK exceeded £7 billion in 2009, with comparable numbers reported by the Financial Times.7 It is with this process in mind that we allow the agent in our model to invest in financial assets whose returns are correlated with the shocks to the survival probabilities, which we call longevity bonds. We study the portfolio allocation between these bonds and risk-free assets, namely how the demand for them changes over the life-cycle and with individual characteristics. Therefore, our model allows us to identify, in a microsetting, which individuals benefit most from longevity bonds and which benefit less, that is, those who might be the counterparty for such bonds. We find that agents in our model respond to longevity improvements by increasing their savings, and in this way they are able to at least partially self insure against longevity shocks. Because longevity risk is realized slowly over the life-cycle, agents have time to react to the shocks. This requires that agents are well informed of the improvements in life expectancy, and the implications of such improvements for the retirement savings needed. In addition, our model shows that retiring later is a typical response to improvements in life expectancy, even though such a decision carries a utility cost in terms of the foregone utility of (additional) leisure. Thus our model lends support to the argument that it is important to allow for flexible retirement arrangements as a mechanism for individuals to react to increases in life expectancy. However, and importantly, our results show that even when agents are allowed to respond to shocks to life expectancy by saving more and by retiring later, longevity risk can have significant utility implications. More precisely, we find that individuals would benefit from being able to invest in longevity bonds or financial assets whose returns are correlated with longevity shocks. This is particularly the case when the extent of longevity risk in our model is calibrated to match the forward-looking projections of the US Social Security Administration and, particularly, those of the UK Government Actuary's Department. Furthermore, the benefits from investing in longevity bonds are substantially higher when it is taken into account that the payouts of defined benefit pension plans are likely to decrease and that this decrease is likely to be correlated with aggregate survival rates. This scenario is motivated by recent events, which suggest that as survival rates increase, retirement benefits are progressively decreased (or contributions increased). In this case, when longevity increases and households need more wealth to finance their retirement consumption, they are also more likely to receive a lower pension. Currently, there is not a liquid market for the longevity bonds that we model, but the partial-equilibrium utility gains associated with them do provide a meaningful metric of the extent to which individuals are affected by longevity risk under these different scenarios. This metric is also useful for DB pension plans deciding on whether to hedge longevity risk on behalf of members. Evidence exists of a link between income and life expectancy, with the larger mortality improvements occurring for individuals in the top half of the earnings distribution. We study the effects of such correlation, to find out that in our model individuals with higher income and life expectancy tend to accumulate more wealth. Finally, we use the model to study the optimal design of longevity bonds. Because investors face short-selling constraints, an increase in the volatility of the payoffs to the longevity bond allows them to achieve levels of hedging that were previously unfeasible. However, excessive levels of volatility might deter young households from buying these assets, because they do not allow households to hedge labor income risk, which is their primary concern. This is an important trade-off to consider, when designing these securities. A vast and rich literature is available on annuities, to which our paper is related. This literature has studied the welfare benefits that individuals derive from purchasing annuities and why, in spite of the large theoretical benefits of such purchases, in practice individuals do not seem to annuitize a significant part of their savings. Recent contributions to this literature include Mitchell, Poterba, Warshawasky, and Brown (1999), Brown, Davidoff, and Diamond (2005), Brown and Poterba (2006), Inkmann, Lopes, and Michaelides (2011), Hari, Melenberg, Nijman, and Waegenaere (2008), Horneff, Maurer, Mitchell, and Stamos (2009), Chai, Horneff, Maurer, and Mitchell (2009), Yogo (2009), and Post (2011). In spite of being related to these papers, important differences exist between the financial asset that we study and annuities. First, in our model, longevity bonds are available for individuals to purchase even before retirement age, allowing individuals to hedge increases in life expectancy from early in life. In this respect alone this asset resembles more the purchase of forward annuities. Second, longevity bonds provide a hedge against aggregate life expectancy, not individual life expectancy. While this reduces the benefits for individuals wishing to hedge increases in their own life expectancy, it also means that, relative to annuities, adverse selection problems are reduced or eliminated. Therefore, a market for longevity bonds could be more easily developed than a market for forward annuities, with individuals being able to buy and sell longevity bonds in each period. A growing literature studies the optimal pricing of longevity bonds and related instruments (see, for example, Dahl, 2004; Carins, Blake, and Dowd; 2006). In our paper, we take bond prices as given and investigate their role in household portfolios, in the context of an empirically parameterized life-cycle model with labor income risk and endogenous retirement (see Farhi and Panageas, 2007 for a model with endogenous retirement but without longevity bonds). We are interested in evaluating in a realistic setting the potential demand for longevity bonds by households saving for retirement and, consequently, by pension providers acting on their behalf.8 The paper is organized as follows. In Section 2 we use long-term data for a cross section of countries to show the existing empirical evidence on longevity. In Section 3 we set up and parameterize a life-cycle model of the optimal consumption and saving choices of an individual who faces longevity risk. The results of the model are discussed in Section 4. Section 5 explores the relation between life expectancy and income. Section 6 addresses security design implications. The final section concludes.

The large somewhat unexpected increases in life expectancy that have occurred over the last few decades have contributed to the current underfunding of defined benefit pension plans. As a result, many of them have had to either reduce promised benefits to employees or increase contributions. Looking forward, pension plans must decide how to address the issue of longevity risk. Ultimately, they should try to do so if that would be in the interest of their beneficiaries. In this paper we quantify, in the context of a life-cycle model, the impact of longevity risk on individual saving and retirement decisions, and we determine the extent to which individuals would benefit from the hedging of such risk. We start by studying the historical evidence and the current projections for life expectancy. The latter show considerable uncertainty with respect to future improvements in mortality rates. We use this evidence to parameterize a life-cycle model with longevity risk, in the context of which we assess how much longevity risk affects the consumption, saving, retirement and portfolio decisions of an individual saving for both buffer stock and retirement motives. We show several ways in which the agents in our model react to shocks to life expectancy. First, because longevity risk is realized slowly over the life cycle, agents optimally save more throughout the life cycle in response to an improvement in longevity. Second, when faced with large improvements in life expectancy individuals decide to retire later, even though this entails a utility cost. We show that even when agents optimally decide how much to consume and save, and when to retire in response to shocks to life expectancy, they still benefit from being able to invest in financial assets that allow them to insure against longevity risk. We find that when we parameterize longevity risk to match the current projections of the US SSA and particularly when we try to match those of the UK GAD, the benefits of investing in assets whose returns are correlated with shocks to life expectancy can be economically very significant. They are particularly large in the current context of declining benefits in defined benefit pension plans and if such declining benefits are a direct response to improvements in life expectancy. These findings lend support to the idea that pension plans should, through their investment decisions, try to hedge the longevity risk that arises from their liabilities, because this would be in the interest of pension plan members. Evidence seems to exist of a link between mortality rates and income, with higher mortality improvements occurring for individuals in the top half of the earnings distribution (e.g., Waldron, 2007). We use our model to investigate the effects of such links. We find that individuals who expect to live longer and who have higher lifetime earnings save more. The benefits of hedging longevity risk depend on the source of the correlation. If individuals are identical ex ante, but a positive correlation exists between permanent income shocks and mortality improvements, then labor income acts as an hedge for longevity risk. However, we also show that if individuals are ex ante heterogeneous, then those who expect to live longer and to have higher lifetime earnings would benefit more from the investment in longevity bonds. Finally, even though our model reflects the demand for insurance against longevity risk, we use it to shed some light on the optimal design of longevity bonds, and we also discussed who might be a counterparty for such bonds. This is an area that deserves further research.