Paper: arXiv 2409.15325

Authors: John Armstrong, Cristin Buescu, James Dalby

Abstract

We study the optimal investment problem for a homogeneous collective of $n$ individuals investing in a Black-Scholes model subject to longevity risk with Epstein–Zin preferences. %and with preferences given by power utility. We compute analytic formulae for the optimal investment strategy, consumption is in discrete-time and there is no systematic longevity risk. We develop a stylised model of systematic longevity risk in continuous time which allows us to also obtain an analytic solution to the optimal investment problem in this case. We numerically solve the same problem using a continuous-time version of the Cairns–Blake–Dowd model. We apply our results to estimate the potential benefits of pooling longevity risk over purchasing an insurance product such as an annuity, and to estimate the benefits of optimal longevity risk pooling in a small heterogeneous fund.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents rigorous mathematical derivations for optimal investment and consumption under Epstein-Zin preferences and longevity risk, achieving elegant analytic solutions. However, it lacks deep empirical validation or empirical backtesting on real historical market datasets, relying instead on stylized models and numerical illustrations.

Research Flowchart

  flowchart TD
    A[Goal: Optimize Post-Retirement Collective Investment under Longevity Risk & Epstein-Zin Preferences] --> B[Inputs: Black-Scholes Financial Model & Cairns-Blake-Dowd Longevity Model]
    B --> C[Analytical Modeling: Solve Optimal Investment & Discrete Consumption without Systematic Risk]
    B --> D[Stylized & Continuous Modeling: Formulate Systematic Longevity Risk Framework]
    C --> E[Numerical Analysis: Solve Continuous-Time CBD Model for Small/Heterogeneous Funds]
    D --> E
    E --> F[Outcomes: Analytic Investment Formulas & Quantified Benefits of Pooling Risk vs. Annuities]

Cited References (94)

  1. Rates of weak convergence and asymptotic expansions for classical central limit theorems (1971)
  2. Continuous-time stochastic control and optimization with financial applications (2009)
  3. Lifetime portfolio selection under uncertainty: The continuous-time case (1969)
  4. Complex analysis (2010)
  5. Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework (1989) — link
  6. Temporal resolution of uncertainty and dynamic choice theory (1978)
  7. An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom (1986)
  8. Axiomatic utility theories with the betweenness property (1989)
and 86 more
  1. Risk aversion with many commodities (1974)
  2. Stationary ordinal utility and impatience (1960)
  3. The structure of intertemporal preferences under uncertainty and time consistent plans (1985)
  4. Nonexpected utility preferences in a temporal framework with an application to consumption-savings behaviour (1990)
  5. Health and (Other) Asset Holdings (2012)
  6. Chapter 21 - Household Finance: An Emerging Field (2013) — link
  7. Household Finance and the Value of Life (2017) — link
  8. Asset Pricing with Limited Risk Sharing and Heterogeneous Agents (2008)
  9. Optimal Life-Cycle Asset Allocation: Understanding the Empirical Evidence (2005)
  10. A Rank-Dependent Utility Model of Uncertain Lifetime (2015)
  11. Stochastic Finance: an Introduction in Discrete Time (2011)
  12. Theory of Games and Economic Behavior (1953)
  13. Classifying Markets up to Isomorphism (2018) — link
  14. A Theory of Justice (1971)
  15. Convex Analysis (2015)
  16. Numerical methods for stochastic control problems in continuous time (2013)
  17. Numerical methods for controlled H amilton- J acobi- B ellman PDE s in finance (2007)
  18. Intertemporal substitution in consumption (1988)
  19. Pensions: challenges and choices: the first report of the pensions commission (2004)
  20. Government Response to the Consultation: Delivering Collective Defined Contribution Pension Schemes (2019)
  21. Cross-country heterogeneity in intertemporal substitution (2015)
  22. Disentangling intertemporal substitution and risk aversion under the expected utility theorem (2019)
  23. Risk aversion and the elasticity of substitution in general dynamic portfolio theory: Consistent planning by forward looking, expected utility maximizing investors (2009) — link
  24. What level of pension contribution is needed to obtain an adequate retirement income? (2013)
  25. CDC: International insights (2022)
  26. What is CDC and how might it work in the UK ? (2018)
  27. Risk Sharing Pension Plans: The Canadian Experience (2014)
  28. Risk Sharing Pension Plans: The Dutch Experience (2014)
  29. Supplementary forecast information release: Long-term economic determinants – March 2019 (2019)
  30. Personal Incomes Statistics 2016-17 (2019)
  31. The DC Future Book: 2018 Edition (2018)
  32. Working Paper 119. CMI Mortality Projections Model: CMI_2018 (2019)
  33. Optimization by vector space methods (1997)
  34. Collectivised post-retirement investment (2019) — link
  35. Collectivised Pension Investment with Exponential K ihlstrom– M irman Preferences (2019) — link
  36. Consumption–investment optimization with E pstein– Z in utility in incomplete markets (2017)
  37. E pstein- Z in Utility Maximization on Random Horizons (2019) — link
  38. Risks for the long run: A potential resolution of asset pricing puzzles (2004)
  39. Long-run risks and financial markets (2007)
  40. Explaining asset pricing puzzles associated with the 1987 market crash (2011)
  41. The levered equity risk premium and credit spreads: A unified framework (2009)
  42. Stochastic differential utility (1992)
  43. Optimal lifetime consumption-portfolio strategies under trading constraints and generalized recursive preferences (2003)
  44. Consumption and portfolio decisions when expected returns are time varying (1999)
  45. New Dutch pension contracts and lessons for other countries (2019)
  46. Can collective pension schemes work in the United Kingdom? (2012)
  47. Backward stochastic differential equations and integral-partial differential equations (1997)
  48. BSDE s with default jump (2016)
  49. Mathematical methods for financial markets (2009)
  50. Sharing risk: the N etherlands’ new approach to pensions (2009)
  51. Saving and investing over the life cycle and the role of collective pension funds (2007)
  52. King William’s Tontine: why the retirement annuity of the future should resemble its past (2015)
  53. Tontine pensions (2014)
  54. The analytics of a single-period tontine (2016)
  55. Fair tontine annuity (2010)
  56. Intergenerational risk-sharing and risk-taking of a pension fund (2008)
  57. Measuring time preferences (2016)
  58. Theory of interest: as determined by impatience to spend income and opportunity to invest it (1930)
  59. Uncertain lifetime, life insurance, and the theory of the consumer (1965)
  60. Uncertain lifetime and intertemporal choice: risk aversion as a rationale for time discounting (2006)
  61. Consumption and portfolio choice over the life cycle (2005)
  62. Optimal consumption and portfolio choice for pooled annuity funds (2008)
  63. The simple analytics of a pooled annuity fund (2005)
  64. Optimal risk-sharing in pension funds when stock and labor markets are co-integrated (2018)
  65. Intergenerational risk sharing within funded pension schemes (2011)
  66. Equitable retirement income tontines: Mixing cohorts without discriminating (2016)
  67. Optimal retirement income tontines (2015)
  68. Options on tontines: An innovative way of combining tontines and annuities (2019)
  69. On the optimal combination of annuities and tontines (2019)
  70. A new approach for satisfactory pensions with no guarantees (2020)
  71. A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration (2006)
  72. Optimal consumption and investment with Epstein–Zin recursive utility (2017)
  73. The infinite-horizon investment–consumption problem for Epstein–Zin stochastic differential utility. I: Foundations (2023)
  74. An Introduction to Computational Stochastic PDEs (2014)
  75. Systematic longevity risk: to bear or to insure? (2019)
  76. Asset pricing with stochastic differential utility (1992)
  77. Collectivised post-retirement investment (2019) — link
  78. Inter-generational crosssubsidies in the UK’s first CDC pension scheme
  79. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework (1989)
  80. Numerical Methods for Controlled Hamilton-Jacobi-Bellman PDEs in Finance (2007)
  81. Risk sharing pension plans: The dutch experience (2014)
  82. What is CDC and how might it work in the UK? (2018)
  83. How might CDC develop in the UK? (2024)
  84. Optimal Investment (2013)
  85. Supplementary forecast information release: Long-term economic determinants (2019)
  86. The DC Future Book: 2018 Edition (2018)

0 of 94 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.