Paper: arXiv 2401.08323

Abstract

This paper addresses the continuous-time portfolio selection problem under generalized disappointment aversion (GDA). The implicit definition of the certainty equivalent within GDA preferences introduces time inconsistency to this problem. We provide the sufficient and necessary condition for a strategy to be an equilibrium by a fully nonlinear integral equation. Investigating the existence and uniqueness of the solution to the integral equation, we establish the existence and uniqueness of the equilibrium. Our findings indicate that under disappointment aversion preferences, non-participation in the stock market is the unique equilibrium. The semi-analytical equilibrium strategies obtained under the constant relative risk aversion utility functions reveal that, under GDA preferences, the investment proportion in the stock market consistently remains smaller than the investment proportion under classical expected utility theory. The numerical analysis shows that the equilibrium strategy’s monotonicity concerning the two parameters of GDA preference aligns with the monotonicity of the degree of risk aversion.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is highly theoretical, featuring a fully nonlinear integral equation and proofs of existence/uniqueness, warranting a high math complexity score. However, it contains no backtesting, datasets, or statistical metrics, and is purely analytical, resulting in a very low empirical rigor score.

Research Flowchart

  flowchart TD
  subgraph A ["Research Goal"]
      A1["Continuous-time portfolio selection<br>under Generalized Disappointment Aversion<br>to address time inconsistency"]
  end

  subgraph B ["Methodology & Analysis"]
      B1["Derive fully nonlinear integral equation<br>for equilibrium condition"]
      B2["Analyze existence & uniqueness<br>of solution"]
  end

  subgraph C ["Key Findings"]
      C1["Unique Equilibrium: Non-participation<br>in stock market"]
      C2["Investment proportion in stocks<br>consistently lower than EUT"]
      C3["Monotonicity aligns with<br>risk aversion degree"]
  end

  A --> B
  B --> C

Cited References (87)

  1. Exotic Preferences for Macroeconomists (2004) — link
  2. Growth uncertainty, generalized disappointment aversion and production-based asset pricing (2015)
  3. Behavioral Portfolio Choice and Disappointment Aversion: An Analytical Solution with ``Small’’ Risks (2010)
  4. Portfolio choices and hedge funds: a disappointment aversion analysis (2021)
  5. Estimation of the Mean of a Multivariate Normal Distribution (1981)
  6. Generalized Disappointment Aversion and the Variance Term Structure (2023) — link
  7. Time-Inconsistent Stochastic Linear-Quadratic Control: Characterization and Uniqueness of Equilibrium (2017)
  8. A theory of Markovian time-inconsistent stochastic control in discrete time (2014)
and 79 more
  1. Optimal Dynamic Portfolio Selection: Multiperiod Mean-Variance Formulation (2000)
  2. Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework (2000)
  3. Time-Consistent Portfolio Management (2012)
  4. The golden rule when preferences are time inconsistent (2010)
  5. Optimum consumption and portfolio rules in a continuous-time model (1971)
  6. Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case (1969)
  7. Portfolio Selection (1952)
  8. Myopia and Inconsistency in Dynamic Utility Maximization (1955)
  9. Existence and Uniqueness for Non-Markovian Triangular Quadratic BSDEs (2022)
  10. Multi-dimensional backward stochastic differential equations of diagonally quadratic generators: The general result (2023)
  11. Multi-dimensional backward stochastic differential equations of diagonally quadratic generators (2016)
  12. On time-inconsistent stochastic control in continuous time (2017)
  13. Generalized Expected Utility Theory: The Rank-Dependent Model (1993)
  14. A theory of anticipated utility (1982)
  15. Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory (2017)
  16. Multidimensional quadratic BSDEs with separated generators (2017) — link
  17. Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games (2018)
  18. A rank-dependent generalization of zero utility principle (2003)
  19. Equilibrium investment with random risk aversion (2023)
  20. Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l’Ecole Americaine (1953)
  21. Axiomatic utility theories with the betweenness property (1989)
  22. An Operational Measure of Riskiness (2009)
  23. An Economic Index of Riskiness (2008)
  24. A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox (1983)
  25. Asset Pricing and Portfolio Choice Theory (2017)
  26. Learning equilibrium mean-variance strategy (2023)
  27. A dynamic mean-variance analysis for log returns (2021)
  28. An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom (1986)
  29. Splitting multidimensional BSDEs and finding local equilibria (2014)
  30. A system of quadratic BSDEs arising in a price impact model (2016) — link
  31. Solvability of backward stochastic differential equations with quadratic growth (2008)
  32. A Theory of Disappointment Aversion (1991)
  33. Generalized Disappointment Aversion and Asset Prices (2010)
  34. Me, Myself and I: A general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents (2023)
  35. Extended backward stochastic volterra integral equations and their applications to time-inconsistent stochastic recursive control problems (2021) — link
  36. Equilibrium asset pricing with transaction costs (2021) — link
  37. Utility maximization in incomplete markets (2005)
  38. Continuous Exponential Martingales and BMO (1994)
  39. Brownian Motion and Stochastic Calculus (1998)
  40. Mean field portfolio games (2023)
  41. Completeness of security markets and solvability of linear backward stochastic differential equations (2006)
  42. Time-inconsistent consumption-investment problems in incomplete markets under general discount functions (2021) — link
  43. A General Chain Rule for Derivatives and the Change of Variables Formula for the Lebesgue Integral (1969)
  44. Stochastic Controls (1999)
  45. Multidimensional Diffusion Processes (1997)
  46. Equilibrium master equations for time-inconsistent problems with distribution dependent rewards (2021) — link
  47. Consistent investment of sophisticated rank-dependent utility agents in continuous time (2021) — link
  48. Why stocks may disappoint (2005)
  49. Optimal Portfolio Selection under Disappointment Averse Utility (2009)
  50. Dynamic mean-variance asset allocation (2010)
  51. Long-term dynamic asset allocation under asymmetric risk preferences (2024)
  52. Myopic loss aversion, disappointment aversion, and the equity premium puzzle (2007)
  53. Time-inconsistent consumption-investment problems in incomplete markets under general discount functions (2021)
  54. Equilibrium stochastic control with implicitly defined objective functions — → 2312.15173
  55. Dynamic portfolio selection for nonlinear law-dependent preferences — → 2311.06745
  56. Macroeconomic tail risks and asset prices (2020)
  57. Disappointment aversion, term structure, and predictability puzzles in bond markets (2021)
  58. Optimal portfolio choice under loss aversion (2004)
  59. Asymmetries and portfolio choice (2017)
  60. Consistent plans (1980)
  61. Multiple solutions under quasi-exponential discounting (2009)
  62. Being serious about non-commitment: subgame perfect equilibrium in continuous time (2006)
  63. Mean–variance portfolio optimization with state-dependent risk aversion (2014)
  64. Generalized disappointment aversion, long-run volatility risk, and asset prices (2011)
  65. First order versus second order risk aversion (1990)
  66. Time-inconsistent stochastic linear–quadratic control (2012)
  67. Investment and consumption without commitment (2008)
  68. Portfolios of value and momentum: disappointment aversion and non-normalities (2022)
  69. On the equilibrium strategies for time-inconsistent problems in continuous time (2021)
  70. Anxiety in the face of risk (2016)
  71. Horizon-dependent risk aversion and the timing and pricing of uncertainty (revised 2023) (2014)
  72. The value of a statistical life and the coefficient of relative risk aversion (2005)
  73. Ordinary Differential Equations (1980)
  74. A random walk down Wall Street: including a life-cycle guide to personal investing (1999)
  75. Optimal portfolio selection with transaction costs and finite horizons (2002)
  76. Prospect theory: An analysis of decision under risk (2013)
  77. Investing for the long run when returns are predictable (2000)
  78. On the risk of stocks in the long run (1995)
  79. The long-term case for equities (1994)

2 of 87 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.