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Value Maximization under Stochastic Quasi-Hyperbolic Discounting

Kaixin Yan, Wenyuan Wang, Jinxia Zhu

Abstract

We investigate a value-maximizing problem incorporating a human behavior pattern: present-biased-ness, for a firm which navigates strategic decisions encompassing earning retention/payout and capital injection policies, within the framework of Lévy processes. We employ the concept of stochastic quasi-hyperbolic discounting to capture the present-biased inclinations and model decision making as an intra-personal game with sophisticated decision-makers. Our analysis yields closed-form solutions, revealing that double-barrier strategies constitute Markov equilibrium strategies. Our findings reveal that firms, influenced by present-biased-ness, initiate dividend payments sooner, diminishing overall value compared to scenarios without present-biased-ness (under exponential discounting). We also discuss bailout optimality, providing necessary and sufficient conditions. The impact of behavioral issues is examined in the Brownian motion and jump diffusion cases.

Value Maximization under Stochastic Quasi-Hyperbolic Discounting

Abstract

We investigate a value-maximizing problem incorporating a human behavior pattern: present-biased-ness, for a firm which navigates strategic decisions encompassing earning retention/payout and capital injection policies, within the framework of Lévy processes. We employ the concept of stochastic quasi-hyperbolic discounting to capture the present-biased inclinations and model decision making as an intra-personal game with sophisticated decision-makers. Our analysis yields closed-form solutions, revealing that double-barrier strategies constitute Markov equilibrium strategies. Our findings reveal that firms, influenced by present-biased-ness, initiate dividend payments sooner, diminishing overall value compared to scenarios without present-biased-ness (under exponential discounting). We also discuss bailout optimality, providing necessary and sufficient conditions. The impact of behavioral issues is examined in the Brownian motion and jump diffusion cases.
Paper Structure (11 sections, 6 theorems, 112 equations, 12 figures)

This paper contains 11 sections, 6 theorems, 112 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Equilibrium results
  4. A class of double-barrier strategies
  5. Markov equilibrium strategies
  6. Impact of present-bias
  7. "Optimality" of bailout/capital injections
  8. The Brownian Motion Case
  9. The Jump-diffusion Case
  10. Conclusion
  11. Appendix

Key Result

Lemma 3.1

(i) For any $b\ge 0$, the function $V_{0,b}^E(x)$ has the following representation (ii) For any $b\in(0,\infty)$, the function $V_{0,b}^E(x)$ is strictly increasing and continuously differentiable on $(-\infty,\infty)\setminus\{b\}$. And, if $X$ has paths of unbounded variation, $V_{0,b}^E(x)$ is continuously differentiable on $(-\infty,\infty)$ and twice continuously differentiab (ii) The functi

Figures (12)

  • Figure 1: Example 4.1: The optimal dividend barriers when $\beta$ varies
  • Figure 2: Example 4.1: The optimal dividend barriers when $\lambda$ varies
  • Figure 3: Example 4.1: The optimal dividend barriers when $\phi$ varies
  • Figure 4: Example 4.1: Losses
  • Figure 5: Example 4.1: The uncontrolled process and the controlled process $X_t$ with capital injection barrier $0$ and dividend payment barrier $b^E$ and $b^*$ when $x=2,\,\mu=-0.2,\,\sigma=2,\,\delta=0.05,\,\beta=0.9,\,\lambda=1,\,\phi=1.2,$ and $V_{0,b^*}(0)<0$
  • ...and 7 more figures

Theorems & Definitions (8)

  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Definition 3.2
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 3.1