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Optimal consumption with loss aversion and reference to past spending maximum

Xun Li, Xiang Yu, Qinyi Zhang

Abstract

This paper studies an optimal consumption problem for a loss-averse agent with reference to past consumption maximum. To account for loss aversion on relative consumption, an S-shaped utility is adopted that measures the difference between the non-negative consumption rate and a fraction of the historical spending peak. We consider the concave envelope of the utility with respect to consumption, allowing us to focus on an auxiliary HJB variational inequality on the strength of concavification principle and dynamic programming arguments. By applying the dual transform and smooth-fit conditions, the auxiliary HJB variational inequality is solved in piecewise closed-form and some thresholds of the wealth variable are obtained. The optimal consumption and investment control can be derived in the piecewise feedback form. The rigorous verification proofs on optimality and concavification principle are provided. Some numerical sensitivity analysis and financial implications are also presented.

Optimal consumption with loss aversion and reference to past spending maximum

Abstract

This paper studies an optimal consumption problem for a loss-averse agent with reference to past consumption maximum. To account for loss aversion on relative consumption, an S-shaped utility is adopted that measures the difference between the non-negative consumption rate and a fraction of the historical spending peak. We consider the concave envelope of the utility with respect to consumption, allowing us to focus on an auxiliary HJB variational inequality on the strength of concavification principle and dynamic programming arguments. By applying the dual transform and smooth-fit conditions, the auxiliary HJB variational inequality is solved in piecewise closed-form and some thresholds of the wealth variable are obtained. The optimal consumption and investment control can be derived in the piecewise feedback form. The rigorous verification proofs on optimality and concavification principle are provided. Some numerical sensitivity analysis and financial implications are also presented.

Paper Structure

This paper contains 15 sections, 17 theorems, 145 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model Setup and Problem Formulation
  3. Market Model and Preference
  4. Concave envelope of the realization utility
  5. Equivalent problem
  6. Derivation of the Solution
  7. Properties of Optimal Controls
  8. Proofs
  9. Proof of Proposition \ref{['prop: dual_solution_beta<1']}
  10. Proof of Theorem \ref{['thm: beta<1']} (Verification Theorem)
  11. Proof of Corollary \ref{['cor: beta<1']}
  12. Proof of Proposition \ref{['prop: samevalue']} (Concavification Principle)
  13. Proof of Lemma \ref{['lemma: beta<1']}
  14. Proof of Corollary \ref{['cor: asy_infty_wealth']}
  15. Proof of Corollary \ref{['thm: long_run']}

Key Result

Proposition 2.1

Two problems eq: primalvalue and eq: concavevalue admit the same optimal control $(\pi_t^*, c_t^*)$ so that two value functions coincide, i.e., $u(x,h) = \tilde{u}(x,h)$ for any $(x,h)\in\mathbb{R}_+\times\mathbb{R}_+$.

Figures (4)

  • Figure 1: Concave envelopes when $0<\beta_2<1$: (left panel) the subcase (i) when $z(h)\neq h$; (right panel) the subcase (ii) when $z(h)= h$.
  • Figure 2: Four cases of boundary curves caused by different parameters
  • Figure 3: Sensitivity analysis on the reference degree $\lambda$.
  • Figure 4: Sensitivity analysis on the expected return $\mu$.

Theorems & Definitions (34)

  • Remark 2.1
  • Proposition 2.1: Concavification Principle
  • Proposition 3.1
  • Remark 3.1
  • Theorem 3.1: Verification Theorem
  • Remark 3.2
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • proof
  • ...and 24 more