Non-exponential Discounting Portfolio Management with Habit Formation

Jingzhen Liu, Liyuan Lin, Ka Fai Cedric Yiu, Jiaqin Wei

Research output: Journal article publicationJournal articleAcademic researchpeer-review

4 Citations (Scopus)


This paper studies the portfolio management problem for an individual with a non-exponential discount function and habit formation in finite time. The investor receives a deterministic income, invests in risky as-sets, buys insurance and consumes continuously. The objective is to maximize the utility of excessive consumption, heritage and terminal wealth. The non-exponential discounting makes the optimal strategy adopted by a naive person time-inconsistent. The equilibrium for a sophisticated person is Nash sub-game perfect equilibrium, and the sophisticated person is time-consistent. We calculate the analytical solution for both the naive strategy and equilibrium strategy in the CRRA case and compare the results of the two strategies. By numerical simulation, we find that the sophisticated individual will spend less on consumption and insurance and save more than the naive person. The difference in the strategies of the naive and sophisticated person decreases over time. Furthermore, if an individual of either type is more patient in the future or has a greater tendency toward habit formation, he/she will consume less and buy less insurance, and the degree of inconsistency will also be increased. The sophisticated person’s consumption and habit level are initially lower than those of a naive person but are higher in later periods.

Original languageEnglish
Pages (from-to)761-783
Number of pages23
JournalMathematical Control and Related Fields
Issue number4
Publication statusPublished - 2020


  • Extended Hamilton-Jacobi-Bellman equation
  • Habit formation
  • Non-exponential discounting
  • Optimal insurance
  • Optimal portfolio

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


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