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Article Excerpt
In this paper we consider the optimization problem of an agent who wants to maximize the total expected discounted utility from consumption over an infinite horizon. The agent is under obligation to pay a debt at a fixed rate until he/she declares bankruptcy. At that point, after paying a fixed cost, the agent will be able to keep a certain fraction of the present wealth, and the debt will be forgiven. The selection of the bankruptcy time is taken to be at the discretion of the agent. The novelty of this paper is that at the time of bankruptcy the wealth process has a discontinuity, and that the agent continues to invest and consume after bankruptcy. We show that the solution of a free boundary problem satisfying some additional conditions is the value function of the above optimization problem. Particular examples such as the logarithmic and the power utility functions will be provided, and in these cases explicit forms will be given for the optimal bankruptcy time, investment and consumption processes.
Key words: bankruptcy; consumption and investment; utility maximization; Hamilton-Jacobi-Bellman equation; free boundary problem
MSC2000 subject classification: Primary: 49L05, 60G40, 60G44, 90A09, 90A10, 93E20
OR/MS subject classification: Primary: Dynamic programming/optimal control
History: Received February 14, 2002; revised December 1, 2002, February 23, 2003, and July 9, 2003.
1. Introduction. The Administrative Office of the United States Courts have announced that nonbusiness bankruptcy filings hit a record high of over 1.5 million, while continuing to increase. The last 3 years had 1.27 million, 1.46 million and 1.57 million nonbusiness bankruptcy filings according to the (Administrative Office of the U.S. Courts (15th May 2003)). The rapid increase in personal bankruptcy filings over the last three years may be in part due to the current economic downturn. Sullivan et al. (1989) offer a similar explanation in an earlier study. They find that a large proportion of bankruptcy filers blamed the worsening economy for their plight. Agarwal and Liu (2003) conclude that higher unemployment rates lead to a rise in credit card delinquency rates. However, the increase in the number of filings has also been associated with a steady decline in the social stigma, information search costs, and legal penalties attached to the act of declaring bankruptcy. Gross and Souleles (2002) elaborate on this in their footnotes 1 and 2 and also cite additional relevant references. Concerned lawmakers have reacted to these trends by passing reform legislation that will enact a major overhaul of consumer bankruptcy laws. Their intention is to discourage abuse of the consumer bankruptcy system by making debt forgiveness less attractive and more difficult for the debtor.
At the heart of this reform legislation is a complex "means test" to determine eligibility for filing bankruptcy under Chapter 7 governing liquidations. Li and Sarte (2001) construct a dynamic general equilibrium framework that indicates that "means testing" would at best leave the aggregate bankruptcy filings unchanged. Filing under Chapter 7 offers forgiveness of debt in return for liquidation of nonexempt assets. Once free of debt, the debtor has no obligation to repay from future earnings. She can earn and consume like any other individual ("fresh start"). An alternative mechanism of filing would be to file under Chapter 13, where the debtor keeps her assets but continues to pay from future earnings in exchange for a "reorganization" of debt. Over 70% of consumer bankruptcies are filed under Chapter 7.
The bankruptcy reform underway makes filing for Chapter 7 more difficult, and introduces new categories of debt as nondischargeable. It had not taken effect so far for political reasons (most significant being a veto by the previous President). However, with the mounting number of consumer bankruptcies and the changed political backdrop of the country, it seems highly likely that the bankruptcy reform legislation will come into practice.
The aim of this paper is to give a rigorous mathematical treatment of the issue of complete debt erasure described above, as seen from the debtor's perspective. We provide results that can be used to quantify the benefit of filing for consumer bankruptcy (to the debtor). We do so by first quantifying the value that a risk-averse consumer attaches to future consumption, given an option of debt forgiveness, in return for incurring some fixed and variable costs. From techniques known in literature, the value of consumption without such a facility to go bankrupt can be inferred. One can subtract the present value of all debt from the initial wealth and solve Merton's problem with this lower initial wealth. The assumption, therefore, is that debt is repaid in full and the remainder of wealth determines future consumption strategy. The difference between the two gives the value gain from having the option to declare bankruptcy under Chapter 7. From this value gain, the dollar worth can be inferred using certainty equivalence. To the best of our knowledge, this is the most rigorous analytical characterization in literature of the benefit of a bankruptcy option. The framework and explicit solutions we provide could prove useful in addressing other issues of interest (e.g., changes in consumption patterns of debtors before and after declaring bankruptcy) but we leave these avenues unexplored.
For analytical tractability we assume that all wealth is concentrated in one exempt asset. This is not so damaging as it seems, for two reasons. First, unlimited exemption is a fairly realistic assumption for some of the more generous states (e.g., states like Texas or Florida which have unlimited exemption for wealth held in home equity). There have been several empirical studies documenting effects of exemption levels. Domowitz and Sartain (1999) quantify to what extent lowering exemptions makes the Chapter 7 (debt forgiveness) alternative less attractive, compared to the Chapter 13 (debt reorganization) alternative. Second, it is often the case that the debtor shifts her wealth from nonexempt to exempt assets prior to declaring bankruptcy. Strategy I described in the section on Strategic Behavior of debtors in White (1998) effectively boils down to shifting wealth from nonexempt to exempt assets. Nevertheless, we realize that these simplifying assumptions introduce error and/or limit the scope of application.
We allow for an affine loss function that can accommodate fixed costs of filing bankruptcy as well as real/intangible variable costs such as taxes paid, social status lost or (opportunity cost of) time spent. We do not model the value from the reorganization alternative to bankruptcy. In reality it is possible that a debtor can seek new credit after bankruptcy, and may file for bankruptcy again. We ignore the cost of reduced access to future credit after bankruptcy, Staten (1993) finds that a large proportion of Chapter 7 fliers get postbankruptcy credit without much difficulty and fairly soon after the filing and the possibility of repeated bankruptcies. Pomykala (1999) states "Approximately 8.6 percent fliers have declared bankruptcies once before; 2.5 percent have declared three or more times."
The financial market we consider has a risky asset (stock) with price [S.sub.t] and a fixed interest rate r. We assume that the stock follows a geometric Brownian motion with a constant average rate of return [mu] and dispersion coefficient [sigma]. There is an agent in this market who is under the obligation to pay a debt at a fixed continuous rate d until the time he/she declares bankruptcy. At that point the debt is erased, the agent will pay a fixed cost F and keep a given fraction a of the remaining wealth. The decision variables are the consumption and investment processes {([c.sub.t], [[pi].sub.t]), [less than or equal to] t < [infinity]} and a stopping time [tau] [less than or equal to] to [infinity] representing the time of bankruptcy. We allow [tau] = [infinity] which corresponds to the event that bankruptcy is never declared. We will maximize the quantity E[[[integral].sup.[infinity].sub.0] [e.sub.[gamma]t]U([c.sub.t])dt] where U(*) is a utility function, and [gamma] is a given discount factor. The novelty in this optimization problem is that the agent keeps investing and consuming after the possible bankruptcy, and utility is derived from consumption after time [tau] as well.
Another mixed optimal stopping/control problem has been studied by Karatzas and Wang (2001). In that paper the "duality method" has been used. In order to derive explicit solutions, in particular for the optimal wealth level at bankruptcy, we use the dynamic programming method. The initial impetus for this work came from the above-mentioned paper where in Appendix B it is suggested that an optimization problem over a consumption stream that extends beyond a discretionarily selected stopping time [tau] is an interesting open problem. It is an important feature of our model that the wealth process is discontinuous at time [tau].
We prove that the solution of a particular free boundary problem satisfying some additional conditions is actually the value function of our optimization problem, and that it is optimal to declare bankruptcy if and only if the wealth is less than or equal to the free boundary. We show also that under the optimal policy, bankruptcy will be declared in finite time with probability one. An explicit solution for the free boundary problem will be presented for the logarithmic and the power utility functions.
2. The optimization problem. We consider a financial market that consists of a risky asset whose price [S.sub.t] evolves according to the dynamics
d[S.sub.t]...
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