Optimal Investment, Consumption and Leisure with an Option to File for Bankruptcy

This paper investigates the optimal personal bankruptcy decision of a debtor who participates in the labor market. This paper is based on a mathematical finance model that assumes a Black-Scholes financial market and describes a decision problem as an expected discounted utility maximization problem. Our optimization problem can be cast into a mixed optimal stopping and control problem, and has a symmetry feature with a voluntary retirement decision problem in characterizing the stopping times. To obtain value function and optimal strategies, we use dynamic programming method and transform the relevant nonlinear Bellman equation into a linear equation. Numerical illustrations from our explicit expressions for the optimal strategies reveal how an opportunity to file for bankruptcy affects debtor’s consumption, leisure, and portfolio decisions.


Introduction
Bankruptcy laws in the United States allow overburdened debtors to file for bankruptcy and to get a fresh financial start (Chapter 7) or reorganize debts (Chapter 13). If a debtor files for bankruptcy under Chapter 7, unsecured debts are wiped out in exchange for nonexempt assets and the debtor is allowed to keep future incomes. Chapter 13 petitioners retain all of the assets and should repay their debts over a 3-5 year period.
There is a plenty of economic literature on personal bankruptcy. Reference [1] argued that the dramatic rise of personal bankruptcy in the U.S. was due to the changes in social norms, and [2] attributed it to the expansion of credit card borrowing. How the bankruptcy protection affects on consumers was investigated by [3] (bankruptcy protection increase earnings and employment) and [4] (role of bankruptcy protection as implicit health insurance). Empirical studies [5][6][7] revealed that filers have restrictions on access to credit markets.
On the other hand, personal bankruptcy decision problem, unlike retirement decision problem, receives little attention in mathematical finance literature. Optimal retirement problem attracts enormous interests in mathematical finance literature as well as in economics literature, for examples [8][9][10][11][12][13][14] and others. Our personal bankruptcy decision problem, however, can also be cast into optimal stopping problems, which is the same as an agent's voluntary retirement problem. The symmetry feature between personal bankruptcy and voluntary retirement can be where r > 0 is a constant risk-free interest rate. The risky asset price S := (S t ) {t≥0} evolves according to the following equation where (B t ) t≥0 is a standard Brownian on a probability space (Ω, F , P). Denote by (F t ) t≥0 the P-augmentation of the natural filtration generated by (B t ) t≥0 . The drift µ(> r) and the volatility σ of the risky assets are constants. Let π := (π t ) {t≥0} (call portfolio process) be the amount of money invested in the risky asset, and c := (c t ) {t≥0} (call consumption process) be the nonnegative rate of consumption. The debtor's participation in the labor market is characterized in terms of her choice of leisure. If the sum of labor rate and leisure rate is a constantL, the income rate is given by w(L − l t ), where l := (l t ) {t≥0} (call leisure process) is the leisure rate, and w is the wage rate. The debtor supplies a minimum level of labor rate w(L − L),L > L > 0, and thereby we will have 0 ≤ l t ≤ L. We assume that π, c, and l are F t -progressively measurable and t 0 π 2 s ds < ∞ for all t ≥ 0 a.s., t 0 c s ds < ∞, for all t ≥ 0 a.s., t 0 l s ds < ∞ for all t ≥ 0 a.s. (1)

Utility Function
We employ a Cobb-Douglas utility function as the debtor's utility function of consumption and leisure as follows where γ * is the coefficient of relative risk aversion for consumption and leisure. α measures the contribution of consumption to the debtor's utility. If we define define γ := 1 − α(1 − γ * ), the utility function in Equation (2) can be rewritten as

The Optimization Problems and Solution
Under Chapter 7, unsecured debts are discharged but nonexempt assets should be relinquished to the bankruptcy trustee who will liquidate them to repay creditors. To reconcile this Chapter 7 bankruptcy rule with our model, we postulate that if the debtor files for bankruptcy, the debtor (hence becomes a filer) retains some proportion of wealth less a fixed cost ν. Let X := (X t ) {t≥0} be the wealth level and τ be the time of filing for Chapter 7 bankruptcy. Then we have After filing for Chapter 7 bankruptcy, the debtor is allowed to keep future incomes. We assume that the debtor and the filer have the same labor market parameters. Before filing for bankruptcy, the debtor repays a debt at a constant rate p. Therefore, the wealth level process satisfies the following equation Remark 1. We define θ := (µ − r)/σ and consider two roots m + > 0 and m − < −1 of the equation for solutions to the optimization problems. Assumption 1.

Filer's Optimization Problem
Firstly, we will derive the filer's value function V F and optimal strategies. The filer's wealth level process evolves according to the following equation

Definition 1.
We call (c, π, l) an admissible policy for a filer at x if (a) c, π, and l satisfy Equation (1), Let A F (x) be the set of all admissible policies for a filer at x. Then the value function V F is defined as follows Before we state the filer's value function and optimal strategies, we define the following constants and functions

Theorem 1. The filer's value function V F is given by
where C l F (·) and C L F (·) are the inverse functions of X l F (·) and X L F (·), respectively, andx is given bỹ Moreover, the filer's optimal strategies (c * , π * , l * ) are given as follows Proof. The Bellman equation for the value function V F is given by The maximizer l = l * to Equation (3) is given by and there exists a wealth thresholdx such that the first order conditions (FOCs) for Equation (3) are given by Firstly, we consider the case where −wL/r ≤ x <x. Borrowing the idea from [20], we write the maximizer c to Equation (3) be a function of x such that c = C l F (x) for a function C l F (·).
Combining Equations (4) and (5) and plugging into Equation (3) yields If we differentiate Equation (6) with respect to c, we obtain the following ordinary differential equation which has the general solution of the form We choose A = 0 to discard a rapid growth term c −γ * m + of Equation (7) as c ↓ 0. Substituting Equation (7) into Equation (6), we find For the case where x ≥x, the maximizer l to Equation (3) is L and we write the maximizer c to (3) as c = C L F (x) for a function C L F (·) and denote by X L F (·) the inverse function of Similarly to the way we obtained Equation (7), it is possible to find We take B = 0 due to rapid growth of c −γm − in Equation (8) when c ↑ 0. The value function for this case is given by To find outc, A and B, we apply C 2 condition of V F (x) at x =x, so we have the followings: which is equivalent to Therefore, we havec Combining Equations (9) and (10) yields The optimal strategies (c * , π * , l * ) follow from the first order conditions (FOCs).
With the filer's value function in hand, we are ready to investigate the optimization of the debtor.

Debtor's Optimization Problem
The debtor's wealth level satisfies the following equation

Definition 2.
We call (c, π, l, τ) an admissible policy for a debtor at x if (a) c, π, and l satisfy Equation (1), Let A(x) be the set of all admissible policies for the debtor at x. Then the debtor's optimization problem is to choose optimally consumption, portfolio, leisure processes and time for filing for bankruptcy, which is defined by Similarly to the case of the filer, there exists a wealth thresholdx above which the debtor enjoys the maximum leisure L. Assumption 2. The debtor does not file for bankruptcy while enjoying the maximum leisure L nor enjoys the maximum leisure L immediately after filing for bankruptcy.
Assumption 2 is reasonable since the personal bankruptcy is intended for debt relief for overburdened debtors. The optimal time for filing for bankruptcy τ * is described as the first hitting time of the wealth level on a certain threshold, call the bankruptcy wealth levelx such that τ * = inf{t > 0 | X t ≤x} (See [15]). Therefore, Assumption 2 implies Before we state the debtor's value function and optimal strategies, we define the following constants and functionŝ c :=c,

Theorem 2. The debtor's value function V(x) is given by
where C l D (·) and C L D (·) are the inverse functions of X l D (·) and X L D (·), respectively.x andx are given bŷ

Proof. The Bellman equation of V(x) is given by
Due to the constraint 0 ≤ l ≤ L, there exists a wealth thresholdx with the corresponding optimal consumptionĉ such that the first order conditions (FOCs) of Equation (11) are given by For the case wherex ≤ x <x (resp. x ≥x), write the maximizer c to Equation (11) be a function of x such that c = C l D (x) (resp. c = C L D (x)) for a function C l D (·) (resp. C L D (·)). Denote by X l D (·) (resp. X L D (·)) the inverse function of C l D (·) (resp. C L D (·)). Similar lines to the proof of Theorem 1 leads us to obtain where the rapid growth term c −γm − term of Equation (12) is discarded. Let us denote byc andc l the optimal consumption rate such that In addition, letĉ be the optimal consumption rate that correspondsx. To determine A 1 , A 2 , A 3 ,c,c l , andĉ we apply C 2 condition of V(x) at x =x and the smooth pasting condition of V(x) and V F (x) at x =x. The condition respectively. At x =x, we have the following smooth pasting condition Combining Equations (13) and (14) with Equation (17) we obtain The value matching condition where Equation (17) is used. Combining Equations (15) and (16) results in and Equation (18) implies Plugging Equation (20) into Equation (19) we find From Equation (15), we obtain Theorem 3. The debtor's optimal strategies (c * , π * , l * , τ * ) are given as follows Proof. The optimal strategies c * , π * , l * follow from the first order conditions (FOCs), and the rest of the proof follows similar lines to the proof of Theorem 3.1 of [15].

Numerical Illustrations
We provide some numerical results from our explicit expressions for the optimal strategies of a debtor with an option to file for bankruptcy. Figure 1 shows the optimal consumption, portfolio, and leisure for the debtor and filer. We see that consumption, portfolio, and leisure drop at bankruptcy. From Equation (17), the consumption drop at bankruptcy is given byc regardless of magnitude of the coefficient of relative risk aversion γ * . To take a close look at the debtor's consumption as the wealth level approaches the bankruptcy wealth level, we define the marginal propensity to consume (MPC) out of wealth as follows From the inverse relation of C l D (·) (resp. C L D (·)) and X l D (·) (resp. X L D (·)), the MPC of the debtor is given by The optimal bankruptcy decision problem in this paper is similar to the optimal retirement problem (for example, [8,9,11]) in that both the optimal bankruptcy time and the retirement time are characterized as the first time the wealth level hit a certain threshold from above and below, respectively. From [11], which employ a Cobb-Douglas utility function, consumption jumps (resp. drops) at retirement if the coefficient of relative risk aversion is larger (resp. smaller) than 1. In addition, we see from [11] that the consumption is concave with wealth as the wealth level approaches to the retirement wealth level from below. On the other hand, we see from Figure 2 that when the debtor has an option to file for bankruptcy, consumption is convex with wealth as the wealth level approaches to the retirement wealth level from above.
We compare the optimal strategies of a debtor who are given an opportunity to file for bankruptcy with those of a debtor without such an opportunity. Denote by (c b , π b , l b ) the triple of the optimal consumption, portfolio, and leisure policies of a debtor without an option to file for bankruptcy. It is easy to find the optimal strategies (c b , π b , l b ) which are given by X t ≥x + p r , where C l (·) (resp. C L (·)) is the inverse function of X l (·) := X l F (·) + p/r (resp. X L (·) := X L F (·) + p/r). From Figure 3, we see that an opportunity to file for bankruptcy leads a debtor to consume more and work less than he/she does without an option to file for bankruptcy. Due to the existence of a refuge from financial distress, a debtor with such an option can enjoy consumption and leisure more and reduces savings comparing to a debtor without and option to file for bankruptcy. Also, with an option to file for bankruptcy the debtor tends to invest more in the risky asset than he/she does without the option.

Conclusions
This paper investigates the optimal consumption, portfolio and leisure of a debtor who is eligible for Chapter 7 bankruptcy. Explicit expressions for the optimal strategies can be obtained with a Cobb-Douglas utility function. There exists consumption and leisure drops at bankruptcy regardless of magnitude of the coefficient of relative risk aversion. An option to file for bankruptcy enables the debtor to increase consumption, leisure and to invest more in the risky asset.
Unlike the debtor, we assume that the filer receives a fixed income stream and is not able to control labor supply. Due to the bankruptcy flag, the filer is likely to face restriction to borrow against future income, but we assume that the filer can borrow up the 100% of the present value of the future income stream. These are limitations to this study. Therefore, considering the liquidity constraint of the filer or providing labor supply flexibility to the filer, although methodologically challenging, can increase realism and are possible future research directions.