JSTOR

A. The Economy

We consider a finite‐horizon, , economy with a single consumption good (the numeraire). Uncertainty is represented by a filtered probability space [ ], on which is defined an N‐dimensional Brownian motion . All stochastic processes are assumed adapted to , the augmented filtration generated by w. All stated (in‐)equalities involving random variables are understood to hold P almost surely. In what follows, given that our focus is on characterization, we assume all stated processes to be well‐defined, without explicitly listing the regularity conditions (Karatzas and Shreve 1998) ensuring this.

There are N+1 investment opportunities: one instantaneously riskless and the remainder risky. The vector of instantaneous net returns on the investment opportunities follows the dynamics where the interest rate r, the drift coefficients , and the volatility matrix might be path dependent.

Dynamic market completeness (under no arbitrage) implies the existence of a unique state price density process, ξ, given by where is the market price of risk process, and . The quantity is interpreted as the Arrow‐Debreu price per unit probability P of 1 unit of consumption good in state at time T . Without loss of generality, we set .

The borrower (bound by a zero‐coupon debt contract described in Section II.B) is endowed with an initial wealth of W(0), net of borrowing proceeds. The borrower chooses a nonnegative, planning‐horizon wealth, , representing terminal net worth, and an investment policy, θ, where denotes the vector of fractions of wealth invested in each risky investment opportunity.4 The wealth process W before (and, when relevant, after) the debt‐maturity date then follows: Prior to debt maturity, the total value of the assets, managed by the borrower, V, is endogenously determined, given by , where D is the value of the debt.

The borrower maximizes the expected value of . The function v(·) is assumed twice continuously differentiable, strictly increasing, strictly concave, and to satisfy the Inada conditions, and . A concave objective function renders our analysis widely applicable as it allows us to represent the objective function of any utility‐maximizing agent (as in, e.g., Zame 1993; Alvarez and Jermann 2000; Chang and Sundaresan 2005); to incorporate, in a reduced form, the presence of managerial self‐interest (as argued, e.g., by Stulz 1984; Allen and Santomero 1998) or concave compensation structures (as advocated, e.g., by John and John 1993; John, Saunders, and Senbet 2000); or to capture risk‐neutral managers/shareholders facing a concave nonstochastic investment opportunity beyond the modeled horizon (investment opportunities as in, e.g., Froot, Scharfstein, and Stein 1993; Froot and Stein 1998). In the sequel, for expositional convenience, we sometimes emphasize results by adopting for the borrower the interpretation of a levered firm, but our results are equally valid for an individual borrower or household.5

B. Modeling the Debt Contract and the Costs of Default

Our objective is to examine, in as simple a setting as possible, how the possibility of costly default affects optimal policies of the borrower (who controls the dynamics of the assets V) and to study the implication of this optimal behavior for aggregate quantities in the economy of Section II.A. We would like to capture two observed phenomena associated with defaultable debt contracts. First, upon default, the lender is able to seize only a fraction of the borrower's assets, which is reflected in deviations from the absolute priority rule. Second, a borrower may default despite managing enough assets to service the debt. To this end, we assume a given debt contract in place between the borrower and the lender, where the contract structure is specified as follows.

Assumption 1 (Debt Contract). The payoff of a zero‐coupon debt contract with face value F, maturity date , and retaining rate β is , where .

The contract asserts that default occurs at the debt‐maturity date whenever the face value is not repaid in full, implying solvency for , and we refer to as the default boundary. The retaining rate β captures, in reduced form, the two aforementioned observed phenomena (see also [iii] of Remark 1), where βV is the value retained by the borrower upon default. One natural way to interpret our debt‐contract formulation is to note that any borrower's assets are made up of tangible and intangible components. The fraction of assets seizable by the lender, (1−β)V, represents then the tangible, collateralizable part, which is the liquidation value of the assets (for any value of V). The intangible, noncollateralizable part, βV, represents borrower‐specific intangible assets, such as human capital and organizational knowledge base. Prior to debt maturity, the intangible assets, , are an integral part of total assets and fully capitalized in our complete markets setting.6 But, according to this interpretation, once (1 − β)V(T) is seized, the intangible part can no longer be capitalized (yet it is valuable to the borrower, i.e., can be used by the borrower to generate future cash flows). Note that our formulation conforms to the traditional approach (as in Merton 1974) to model defaultable (discount) debt. In particular, default may occur only at a deterministic date, T, when the debt matures, where this date is fixed to precede, or coincide with, the planning horizon, T . Section III examines the case of , and Section IV the case of (Section V analyzes the case with two debt payments, where default may occur at T, T , or both, thereby introducing endogeneity in the timing of default).

It is undisputed that, in general, corporate or personal default is costly (due to impaired business reputation and stigmatization, unfavorable asset‐liquidation terms, or other direct expenses). We focus on the following structure of the borrower's costs associated with default.

Assumption 2 (Borrower's Default Costs). Upon default , the borrower incurs fixed costs and proportional costs with the total costs of . Otherwise,

To capture essential components of default costs, our cost structure combines the two primary types of costs discussed in the literature: we allow for a fixed‐costs component φ, and for a component proportional to the amount of default , where λ is the proportional cost per unit of default.7 Both cost components may include the commonly empirically documented direct out‐of‐pocket expenses and the remaining indirect expenses (e.g., Warner 1977b; Altman 1984). The costs in our model may be interpreted more generally as financial distress costs (that include, e.g., lost business and wasted managerial resources) incurred when the borrower is in danger of defaulting (Bodie and Merton 2000, pp. 429–30) and are not necessarily limited to bankruptcy costs. The financial‐distress region (as a function of V) can be defined explicitly by financial ratios within debt covenants (in our setting ) or implicitly by the market's perception of distressed financial ratios. From assumptions 1 and 2, it is clear that the borrower may default and incur costs while having . This is consistent, for example, with our tangibles‐plus‐intangibles interpretation of total assets, where the borrower cannot liquidate the intangibles and, hence, costs cannot be avoided. More generally, when costs are interpreted as costs of financial distress, these may be incurred (due to third party) regardless of debt service (even if face value is repaid). Our debt‐contract formulation then captures, in reduced form, scenarios where lenders are able to seize only assets valued less than F (perhaps due to intangibility but possibly due to other reasons, such as bargaining between different stakeholders).8

Our formulation nests the benchmark investment model (henceforth B) with no debt (Merton 1971; Cox and Huang 1989). Specifically, when , there is no debt ( ) and the optimal solution is the B‐model wealth, . Moreover, when , to satisfy the Inada conditions, the borrower never defaults, guaranteeing , and again is optimal. In a third extreme, φ = λ = 0, default is costless, and although it may occur, it does not affect the borrower, who therefore can still finance the optimal policy . Therefore, in the latter case, the face value F and the retaining rate β affect only the value of the debt, hence V but not W, and leverage (V/W) has no impact on how the borrower's net worth is invested. Given our interest in focusing on borrowers' wealth, we collectively refer to these three cases (although differing in V) as the benchmark.9 Our results in the sequel are straightforwardly mapped into results for total assets V, which inherit the main qualitative features of borrower's wealth W (see the discussion of Proposition 2 and note 12) and thus are not presented to avoid repetition.

Remark 1 (Alternative Modeling of Debt and Costs). Consider the following:

(i)	Debt payoff is given by ;
(ii)	Larger default costs accrue for a larger asset base: , when ;
(iii)	The borrower's retaining rate is given by β1, while the default region is parameterized by β2, , if ; otherwise, .

We adopt the formulation in assumption 1, instead of (i), to clarify that represents the costs born by the borrower (which is still the case in (i)) and these costs do not necessarily represent immediate expenses; default affects future business and financing opportunities, so that although is retained upon default, it cannot be “consumed” entirely. Moreover, our formulation lends itself to more convenient comparisons with the benchmark. Our focus here is on the borrower, and any costs incurred by the lender must subsequently be deducted from or, equivalently, affect the lender's budget constraint, analogously to the borrower's budget constraint in equation (4). We will demonstrate (see Remark 2) that specifying the debt as in (i), employing alternative cost structures, such as in (ii), as well as capturing by two separate parameters the two aforementioned default‐related observed phenomena, as in (iii), does not qualitatively change the insights gained from our setting.

A. Borrower's Optimization

We solve the dynamic optimization problem of the borrower using the martingale representation approach (Cox and Huang 1989; Karatzas, Lehoczky, and Shreve 1987), which allows the problem to be restated as the following static variational problem: where the costs of default , the terminal net worth , and the associated total‐assets value satisfy assumptions 1 and 2. The budget constraint states that initial wealth, net of borrowing proceeds, must be sufficient to cover the value of terminal wealth plus potential costs.10 We note that the optimization problem in (4) is nonstandard, as it is complicated by the nonlinearity and discontinuity in the cost structure, introducing not only nonconcavity into the objective, but also nonconvexity into the budget constraint. Proposition 1 characterizes the optimal solution.11

Proposition 1. When debt maturity coincides with the borrower's planning horizon ( ), the borrower's optimal terminal net worth is where I(·) is the inverse function of and solve the following system: with . The benchmark borrower's optimal terminal net worth is , where solves .

Consequently,

(i)	If , then ;
(ii)	under no default, and for under default; However, under default for , we may have ;
(iii)	For , .

We describe the solution of Proposition 1 via figure 1, which depicts the optimal terminal net worth of the borrower (equation [5]) and illustrates how it may relate to the B‐policy.

Figure 1 reveals the borrower to exhibit three distinct patterns of economic behavior, mapped into three regions of the state space: no default, default, and in between, an extended region of default resistance (resistance for brevity). In the last, the borrower resists default and the target wealth does not change upon maturity in response to deteriorating economic conditions (represented by increasing values of ). In the no‐default “good” states (low ), the borrower behaves as in the B‐case, while not defaulting on the debt obligation. However, unfavorable states ( above ) are endogenously classified into two subsets: the default “bad” states , in which the borrower defaults, and the default‐resistance “intermediate” states , in which the wealth level is maintained at the default boundary.12 Hence, the probability of default is endogenously set by the choice of to equal the probability mass of the states where .

The optimal behavior is driven by the undesirability of costly default. The default‐resistance region then arises due to the asymmetry of the cost structure across the state space.13 Specifically, striving to not default in states where, without default costs it would have been optimal to default, the borrower attempts to maintain, over some of these states, the minimum wealth level that avoids triggering default costs. This level must then correspond to the value of the default boundary, and the flat, constant‐wealth shape arises because, in our setting, the default boundary is state‐independent. However, when the default‐resistance value is too costly to maintain, recognizing that default is allowed, the borrower chooses to default. Default is chosen in the worst states, as in these states it is most expensive to finance the state‐independent default‐boundary wealth. To compensate for the wealth level in the default‐resistance states and for the costs incurred upon default, the wealth across the no‐default region must be decreased (property (ii) in Proposition 1), although it maintains the B‐like structure.

Fixed costs, φ, contribute to the borrower's incentive to extend the resistance region and are the sole cause for the discontinuity of the net worth in the transition into the default region. However, once default occurs, fixed costs are incurred regardless of the amount of default and it is optimal to revert to the B‐like policy. Consequently, only the proportional costs parameter, λ, affects the shape of in the default region. The lower is , and hence , in the default region, the lower is the debt payment, leading to larger proportional costs.14 To counteract this, the borrower aims at a higher wealth upon default. Therefore, the optimal policy differs from the B‐policy by being positively related to the proportional‐costs parameter λ. This explains the somewhat unexpected feature of the optimal policy stated in property (ii) of Proposition 1 and explicitly depicted in figure 1, where the solid line (denoting borrower's net worth with costly default) is above the dotted line (denoting the B‐policy net worth) over the default region [ ). That is, could be such that a levered firm defaulting at a time of economic downturn, despite suffering default costs, fairs better than an otherwise equal unlevered firm or a firm facing costless default. A distinct feature of the solution is that the discontinuity in figure 1 is larger than the fixed costs, φ. This is due to two effects of fixed costs when the debt maturity coincides with the planning horizon. The first is the direct effect of fixed costs on wealth. The second is the indirect effect arising from the concave objective over wealth at debt maturity. This latter effect overextends the resistance region, introducing upon default an additional discontinuity over and above φ.

Inspection of Figure 1 allows us to summarize the dependence of the solution on the parameters F, β, λ, and φ, driving the borrower's optimal behavior presented in Proposition 1. As the face value, F, or the retaining rate, β, increase, so does the default boundary. Then, region boundaries, and , decrease, but so that the resistance region shrinks. (Indeed, in the limit of β =1, the default region extends over all states and maximal costs are incurred even though .) This, along with decreasing in the default and no‐default regions, allows the borrower to meet the higher default‐resistance level. For high enough F or β, the wealth in the default region falls below the benchmark . As the proportional costs parameter, λ, increases, the borrower acts to decrease the probability of default and, at the same time, to raise the wealth in the default region to minimize the burden of proportional costs. Accordingly, the resistance region expands in both directions and the wealth in the shrinking no‐default region is decreased, thereby financing the increased level at the bad states. A higher λ also increases the curvature of the policy upon default, rendering it more variable across states. An increase in fixed costs, φ, similarly extends the resistance region, achieving the goal of lowering the default probability and hence decreasing the deadweight of fixed default costs. However, being insensitive to the magnitude of default, increased φ induces a decreased level of wealth in the shrinking no‐default and default regions. When φ increases high enough relative to λ, the wealth in the default region falls below the benchmark value. At the other extreme, when φ vanishes, so does the discontinuity in .

Figure 2 depicts the shape of the probability density function corresponding to the terminal wealth policies in figure 1. There is a probability mass build up in the borrower's terminal wealth, at the default boundary . The borrower then has a discontinuity, with no states having wealth between and . Note that, relative to the benchmark, the depicted distribution in the default region is shifted to the right, meaning more wealth with higher probability, as in figure 1. On the other hand, when fixed costs dominate, the default region tail shrinks while shifting to the left relative to the benchmark, similarly to the left‐shifted no‐default‐region tail, whereas the probability mass build up at the default boundary increases.

Corollary 1 elaborates upon the borrower's optimal capital structure, when debt maturity coincides with the planning horizon, describing the equity component (cum default costs) and the debt liability.

Corollary 1. When debt maturity coincides with the borrower's planning horizon ( ),

(i)	The borrower's optimal terminal wealth cum costs is given by
(ii)	The optimal debt payout policy is given by where , and y, x, , are as in proposition 1. Moreover, as . The default‐region boundary, , may lie above or below the benchmark costless‐default‐region boundary, .

In Corollary l(i), the equity component, cum costs, takes the form of the B‐wealth plus a put option thereon, plus a short position in a package that includes a put and a “binary” option. The long put position guarantees the default‐boundary value in the resistance region, while the short package is structured to guarantee the funds necessary to cover the default costs. The binary‐option component arises because of the aforementioned additional discontinuity in the wealth upon transition into the default region, arising due to the indirect effect of fixed cost.

Figure 3 describes the payoff of the debt contract across the state space, following Corollary 1 (ii). In the presence of fixed default costs incurred at the planning horizon, Corollary 1 (ii) illustrates that the debt credit‐risk component, although being a put option when expressed in terms of , it is in fact a portfolio of options when analyzed across the state space; the credit‐risk component combines a put option and a binary option (the latter accounting for the discontinuity at in figure 3). This portfolio of options enters into the debt contract due to the debt's structural dependence on the assets value, , and hence on the terminal net worth. Note that, since must include funds to cover default costs (unlike in the B‐case), is higher than the B‐value for a large enough , even though the default‐region boundary in the costless‐default benchmark, , may be higher than for some parameter values. Therefore, at the most adverse states, in our setting, lenders recover a larger fraction of the face value from borrowers that incur default costs than from borrowers that default costlessly.

Remark 2 (Solution with Alternative Modeling of Debt and Costs). The optimal policies corresponding to the formulations (i), (ii), and (iii) in Remark 1 are:

(i)	For debt payoff given by ,
(ii)	for defaults costs given by , when
(iii)	When if , otherwise where , where region boundaries are specified analogously to the specification in Proposition 1 and are omitted here for brevity. Consistent with the economic rationale underlying inherit the shape depicted in figure 1. However, the default boundary in (i) is cost dependent, the wealth in (ii) always falls below the B‐wealth upon default (as the lower is the net worth, the lower are the costs), and the default boundary in (iii) does not depend on the retaining rate β1. Also note that the debt payoff in (i) vanishes at the most adverse states.

B. Further Properties of the Borrower's Optimal Policy

To perform a detailed analysis of the optimal behavior of a borrower, we specialize the setting to an isoelastic objective function, , and to lognormal state prices with constant interest rate and the market price of risk. Under this setting, we can derive explicit expressions for the borrower's optimal wealth and investment policy before the planning horizon/debt maturity, as reported in proposition 2.

Proposition 2. When debt maturity coincides with the borrower's planning horizon we assume , and r and κ are constant. Then,

(i)	The borrower's optimal wealth before the debt‐maturity date is given by where 𝒩(·) is the standard‐normal cumulative distribution function, ln solve and .
(ii)	The fraction of wealth invested in the risky investment opportunities is where the B‐value, , and the exposure to risky investments relative to the benchmark, , are respectively, and φ(·) is the standard‐normal probability distribution function.
(iii)	The exposure to risky investments relative to the benchmark is bounded below: . Under no fixed costs, , . However, for , we may have .

The explicit expression for the optimal wealth , in equation (6) of Proposition 2, reveals that it, hence the value of the assets (see note 14 for the mapping between W and V), inherit stochastic mean return and volatility in their dynamics. The importance of this observation is that it is in contrast to the widely accepted practice to model asset value dynamics as a geometric Brownian motion with exogenously specified constant mean return and volatility. The option‐based interpretation in Corollary 1 (i) clarifies the expression of the time‐t optimal wealth in equation (6). The first term takes the form of the optimal B‐wealth, the second and third terms represent the cost of a Black and Scholes (1973)‐type put option on the B‐wealth with strike price , the fourth and fifth terms are the proceeds from shorting a portfolio of a put plus a binary option. Consequently, when the fraction invested in the risky investments is expressed as a multiple of the B‐policy, the three curly‐bracketed terms in equation (7) correspond, respectively, to the positions replicating the long put and the short options portfolio. Similarly, since the value of the assets is the sum of the items in Corollary 1 (i) and (ii), and since the option package in (ii) already appears in (i), the implications we discuss here for the borrower's wealth and its dynamics are also inherited by the assets value and its dynamics.

Following proposition 2, figure 4 plots the borrower's optimal time‐t wealth (equation [6]) and risk exposure (equation [7]), and compares these with the B‐case. Figure 4(a) reveals that the prehorizon borrower's wealth behaves similarly to the benchmark in all states, while lower in the good states and higher in the bad. In the intermediate region, borrower's wealth exhibits concavity in and it is easy to visualize how this concavity increases as time approaches the horizon and tends to the discontinuous shape in figure 1. In these intermediate states, the borrower begins to accumulate wealth to guarantee the resistance level, whereas in the bad states, the borrower starts to allocate funds to cover the almost imminent default costs, rendering bounded away from zero. The shift of wealth into the intermediate and bad states is feasible due to the decreased wealth in the good states.

Figure 4(b) illustrates the typical shape of the borrower's optimal investment policy, characterized by four segments in the ξ(t) space. First, in the good states, with default being unlikely, the benchmark behavior prevails. Second, in the relatively cheap unfavorable states, the borrower increases the fraction of wealth in the riskless investment, aiming to secure the default‐resistance level. Third, as ξ(t) rises further, the borrower's risk exposure begins to rise as well, tending back toward the B‐policy but, in the case of figure 4(b), not surpassing it. The fourth segment occurs when ξ(t) is high enough to deter the borrower from further risk taking, and the optimal policy gradually shifts toward a totally riskless position. It is straightforward to verify, using equations (6) and (7), that the humped shape in Figure 4(b) survives for all parameter values. Formally, this nonmonotonic behavior across the state‐space is linked to the replication of the options described in (i). Intuitively, the borrower's investment policy is driven by the combined need to finance the default‐resistance region, as well as the funds required to cover default‐costs in the default region. The hump in Figure 4(b) then arises when ξ(t) is in the proximity of , because only a risky position, sensitive to economic fluctuations, can facilitate the financing of the two distinct wealth levels over nearby states. Clearly, when ξ(t) is already very high, default is very likely, it is too costly to bet on a favorable realization of a large risky investment, and a borrower favors riskless investments that, although unlikely to lead to solvency, nevertheless cover the costs of default.

Figure 5 displays a sensitivity analysis of and time. Increasing F or β raises the default boundary and has the qualitatively similar effect of increasing the likelihood of default and shrinking the resistance region. Therefore in figures 5(a) and 5(b), the humped deviation from the benchmark is reduced to a smaller region of states. When default costs (φ or λ) increase, the threat of costly default exerts more influence, extending the resistance region, and hence in figures 5(c) and 5(d), the humped deviation from the benchmark spreads to a larger region of states.

Figures 5(d) and 5(e) illustrate the somewhat surprising result, stated in Proposition 2 (iii), that indeed, for some parameter values and in particular as the time‐to‐horizon decreases, the exposure of a borrower to risky investments increases in some states compared to the benchmark. The increase occurs across the region of states that straddles . Then, conditions are such that, in these states, large investment in risky projects is the only strategy allowing avoidance of the penalties of default, by reaching the resistance level, should economic conditions turn favorable, although leading to default if the state of the economy deteriorates slightly. Interestingly, there is justification for such an aggressive behavior only when the presence of the fixed‐costs wedge is coupled with the debt maturing at the planning horizon. Otherwise, φ=0 eliminates the sharp disparity of wealth around ; and , which we study next, removes the urgency of the highly levered bets, even if .

The analysis therefore illustrates that, overall, as depicted in figures 4(b) and 5, following Proposition 2 (ii), in many scenarios of interest (and under most of the examined parameter space), the optimal policy of a levered firm is in fact the one of a perennial lower risk exposure relative to the benchmark. This is in contrast to the commonly made “asset‐substitution” (increased risk exposure) arguments in the literature for a risk‐neutral borrower with a net worth truncated at zero due to limited liability (e.g., Jensen and Meckling 1976).

C. Economic Significance of the Borrower's Behavior

To quantify the economic significance of some of the model's major implications, in this section, we further examine the borrower's behavior in light of the imperfection of costly default. Since costly default is a critical feature of our model, we first discuss the empirical evidence for the existence and magnitude of default costs. For direct out‐of‐pocket bankruptcy costs (legal fees and professional services), Warner (1977b) suggests that “there are substantial fixed costs.” Weiss (1990) estimates direct costs to average about 3% of the book value of debt plus the market value of equity, while Lawless and Ferris (2000) report average out‐of‐pocket fees of about 18% of total assets. Mapping into our model parameters, it is not unreasonable to assume that the borrower incurs 0.5%–2% fixed costs, in W(0) units. Indirect default costs account for all costs on top of direct out‐of‐pocket expenses (ranging from the obvious costs of deteriorating business relations to the more subtle costs of a “locked‐in” suboptimal capital structure; Gilson 1997). Altman (1984) notes that “in many cases [bankruptcy costs] exceed 20% of the value of the firm.” Andrade and Kaplan (1998) estimate financial distress costs to be 10%–20% of firm value (including a fixed component). In terms of costs as a fraction of , assuming that unpaid debts are on the order of magnitude of distressed firm value, it is therefore of interest to examine the impact of λ being of up to 20%. Although some argue that indirect costs may imply market irrationality (Haugen and Senbet 1978), we merely take the existing empirical evidence as stylized facts. Moreover, most values used for default costs in this section are chosen conservatively.

In our model with no default costs, upon maturity, the state space is separated into two regions: no default , and default . In the presence of default costs, an intermediate region of default resistance arises, over which the borrower strives to not default, as discussed in Sections III.A and III.B. Toward assessing the economic significance of this effect, within economic environments with reasonable costs of default, the extent of default resistance can be captured by the difference in default probabilities: Another major implication is that the borrower may emerge wealthier upon default, despite incurring default costs. Comparing to the case of costless default, we refer to the wealth in the region as distressed wealth, because in the presence of default costs, the borrower either resists default or defaults in this region. To measure the effect of the borrower's behavior on the distressed wealth, we can examine by how much the present value of distressed wealth changes when default is costly:

Table 1 reports that, within empirically reasonable economic environments, the model‐obtained values for the decrease in the probability of default can be as high as 21.8%, while distressed wealth may be increased by up to 23.5%. Overall, the results in the table indicate that borrower's optimal behavior indeed carries economically significant effects on quantities of interest, such as default probabilities and distressed wealth, even when default costs are small by empirical standards. The effects are only amplified for longer debt maturity (e.g., for , , , , horizon of 5 years instead of 1 year, decreases the default probability by 1.1%, and increases distressed wealth by 32.3% instead of 0.1% and 10.4%, respectively, in the table). Clearly, if regulators are concerned about externalities affected by low‐net‐worth firms or individuals in distress, then supporting an economic environment with an appropriate level of default costs could be socially desirable.

A. Borrower's Optimization

Using the martingale representation approach, the dynamic optimization problem of the borrower is restated as the following static variational problem: where default costs, , and net worth upon debt maturity, , satisfy assumptions 1 and 2. The budget constraint is broken into two components to clarify the impact of possible default: the first component states that initial wealth must be sufficient to cover potential default costs upon maturity and the second component is as in the B‐case.

While we retain all the main features of default coinciding with the planning horizon, to highlight the implications unique to the model with prehorizon default, we henceforth assume isoelastic objective and lognormal state prices.

Proposition 3. When debt maturity is prior to the borrower's planning horizon , we assume , and r and κ are constant. Then, the borrower's optimal planning‐horizon wealth is The borrower's optimal wealth upon debt maturity and the Lagrange multiplier are where , , X(s) is as in proposition 2, and solves the budget constraint .

Consequently,

(i)	If , then
(ii)	under no default and for under default. However, under default for , we may have .
(iii)	The optimal planning‐horizon policies, post‐no‐default, post‐default‐resistance, and post‐default, after the realization of either or , respectively, satisfy   (a). for , holding with equality for .   (b). for or .   (c). for and , with the inequality reversed for and . When and , , with the inequality reversed for .

Proposition 3 (and properties (i) and (ii) reveals that upon maturity the three‐region structure, and the emerging behaviors within each region resemble those in proposition 1 and figure 1, with the wealth upon maturity now being the present value of the planning‐horizon wealth in the no‐default and default regions. However, we now clearly see how the three regions are formed. The no‐default region is set by the choice of , then to set the other two regions, the proportional costs parameter, λ, and the fixed costs, φ, enter separately into two multiplicative terms that determine in relation to . So, the structure of the default costs explicitly determines how aggressive the borrower is in avoiding default by extending the default‐resistance region. The larger is λ, or φ, the larger is the default‐resistance region relative to the no‐default and default regions. Moreover, due to the adverse impact of fixed costs, hitting the borrower for the slightest amount of default, increases the more concave is the borrower's objective function.

Focusing on the optimal behavior in the no‐default region vs. the default region, property (iii) (a) states that, despite paying proportional default costs, the planning‐horizon wealth, , is higher post default compared to post no default.15 This is true regardless of whether, under default, the deflated wealth, , is above or below the deflated wealth under no default. The wealth upon debt maturity, , deviates from the B‐structure in the default region only when , and, as with , is bumped up to reduce the proportional default costs. Hence, the path independence of the B‐solution no longer holds, and for a given the post‐default wealth exceeds the post‐no‐default wealth. The value of post default is the same as post no default in the case of fixed costs only ( , ), because there are no proportional costs to modify the B‐like structure of the optimal policy upon maturity. Therefore, the resulting inherits the B‐like path independence in both the no‐default and default regions.

Examining the optimal planning‐horizon wealth post default resistance, property (iii)(b) shows it to be higher than the post‐no‐default wealth, and in the case of fixed costs only ( , ), property (iii)(c) shows it to be higher than the post‐default wealth. This is because the extra funds allocated to reach the default boundary at maturity are subsequently used to finance the post‐default‐resistance wealth. However, post default resistance does not exceed the post‐default wealth in the case of proportional costs only ( ), because of the bumped‐up across the default region. When and , the default‐resistance region is further stretched to the right (via increased ). Then, there are resistance states with close enough to in which the deflated wealth at T is exceptionally high, resulting in a higher than the value achieved post default, for a given .

Corollary 2 presents the borrower's optimal capital structure upon debt maturity.

Corollary 2. When debt maturity is prior to the borrower's planning horizon ( ), we assume , and r and κ are constant. Then,

(i)	The borrower's optimal wealth cum costs upon debt maturity is given by
(ii)	The optimal debt payout policy is given by where is as in proposition 2, and , , y are as in proposition 3, x is as in proposition 1.

The expressions in corollary 2(i) and 2(ii) are similar to those in corollary 1 and arise due to the same arguments as in Section III. The major difference here is that put options are the only instruments embedded within the optimal policies. The reason is that, when debt maturity precedes the planning horizon ( ), the fixed default costs upon debt maturity do not immediately affect the concave objective over the planning‐horizon wealth. The ability to spread the impact of fixed costs over the planning‐horizon states removes the urgency to avoid fixed costs upon maturity and, hence, undermines the rationale for overextending the default‐resistance region. So, when , the fixed costs have only a direct effect on wealth, reducing it by φ. Therefore, unlike in the case of , there is no need for investment strategies (implemented by binary options) designed to finance upon maturity a larger than φ net‐worth discontinuity. As a result, when , the debt credit‐risk component, analyzed across the time‐T state space, is entirely captured by a put option, which is in the money when .

B. Further Properties of the Borrower's Optimal Policy

Proposition 4 presents explicit expressions for the borrower's optimal wealth and investment policy before the debt‐maturity date.

Proposition 4. When debt maturity is prior to the borrower's planning horizon ( ), we assume , and r and κ are constant. For ,

(i)	The borrower's optimal wealth before the debt‐maturity date is given by where , , and y are as in proposition 3.
(ii)	The fraction of wealth invested in the risky investment opportunities is where the exposure to risky investments relative to the benchmark, , is
(iii)	The exposure to risky investments is bounded below and above: 0≤ (t)≤1.

Since corollary 2 illustrated that the separation of the maturity and planning‐horizon dates eliminated the need for aggressive risky betting prior to T, in (10), although similar to (6), does not include a binary component. In (10), and are set so that is composed from a first term in the form of the B‐wealth, plus a put option thereon with strike , and units of a short put thereon with strike . This portfolio of options guarantees the default‐resistance wealth as well as the funds needed to cover default costs; hence, it increases the fraction of wealth invested in the riskless investment. This results in a typical shape for as in figure 4(b), for all t < T and all parameter values, meaning that levered firms, facing prehorizon costly default, unambiguously reduce their risk exposure relative to unlevered firms or firms facing no default costs.

A. Equilibrium in the Presence of Credit Risk

Given the prevalence of defaultable debt in the economy, it is of interest to evaluate its impact on asset prices at an aggregate level. In this section, we provide a simple general‐equilibrium production model in which the partial‐equilibrium behavior of the borrower persists and affects market value and dynamics. It is not our intention to provide the most general setting where most pertinent quantities are endogenously determined.

B. Extension to Defaultable Coupon Debt or Repeated Borrowing

Our analysis so far focused, for clarity, on a single debt contract. However, our setting readily lends itself to dealing with multiple debt contracts, cross‐sectionally or intertemporally. To highlight the intertemporal dimension, in this section, we consider a defaultable coupon debt contract with maturity T and payments F at time T, F at time T (>T), where the borrower may default on any of the two payments. In our setting, this multiple‐payment defaultable debt is equivalent to the case of repeated borrowing, where as an initial debt contract matures at time T, the borrower enters into a new contract with face value F , maturity T . Default in this setting is allowed both before and at the planning horizon T . Moreover, under the multipayment debt interpretation, prematurity default is also allowed, since the borrower may optimally default on the first payment at time T (as described next in proposition 6).17 The payoff and the associated default costs of the first payment are as previously specified in Section II.B, while those of the second debt payment are (following the structure in Section II.B): , respectively, where . The borrower's optimization problem is reduced then to solving the following problem: Proposition 6 characterizes the optimal solution (where we use the hat superscript, ^, to distinguish the endogenous quantities here from those in the previous sections).

Proposition 6. Consider a defaultable coupon debt contract with maturity T and payment F at time T, F at time T , and assume , and r and κ are constant. Then, the time‐T optimal wealth of the borrower is where . The time‐T optimal wealth and the Lagrange multiplier z(T) are where the constants , , , and the functions are given in the appendix. And, y ≥ 0 solves the budget constraint .

Consequently,

(i)	If , then .
(ii)	for either or , where is as in proposition 1 (with T replacing T).
(iii)	for either F = 0, or , where is as in proposition 3.

Proposition 6 (properties (ii) and (iii)) asserts that the optimal planning‐horizon wealth, , coincides with the optimal policy of proposition 1, or that prehorizon time‐T wealth, , coincides with the optimal policy of, when some parameter values vanish so that only one of the contract payments affects the borrower. However, for general parameter values, although both and are structured similarly to their counteparts in the single‐payment contract cases, each features a notable difference.18

The optimal policy upon maturity of the first payment, , is modified to account for the default costs that may be incurred at time T , on top of those as of time T considered in proposition 3. The reason is that, from time 0, the borrower is conscious of future borrowing. Therefore, the choice of region boundaries in the ξ(t) space and the wealth within each region are affected not only by the desire for a balance between the costs of resisting default and the costs incurred upon default at time T, but also by the desire for a similar balance with respect to time T . Consequently, even at the most adverse states at time is maintained above a floor (H > 0), thereby enabling the borrower to finance the default‐resistance region and the costs of default at the planning horizon T , irrespective of the magnitude of default at time T. The borrower's wealth at the most adverse states at time T is therefore always higher than in the cases of no borrowing or costless‐planning‐horizon default.

Examining the planning‐horizon optimal policy, , reveals that, while retaining the three region structure of propositon 1, the boundaries of the regions in the ξ(T) space, and , are now path dependent and identified by whether the outcome at time T is no default, default resistance, or default. The behavior after T is driven by similar arguments to those outlined in the context of propositon 3. In particular, the optimal policy post default or post no default is not sensitive to the particular realization of ξ(T). Moreover, the fact that, conditional on ξ(T ), the borrower is never worse off post default than post no‐default is translated with repeated borrowing not only to the time‐T wealth level but also to the time‐T region boundaries—as, for example, is illustrated by the larger no‐default region post time‐T default, , compared to post time‐T no default, .

C. Managing Credit Risk

The credit risk associated with a debt issue can be managed by the appropriate choice of the debt‐contract parameters. Specifically, focusing for illustration on a parameter easily adjustable in practice, a firm, or its creditors, can choose the face value of the debt, F, all else being equal, so that to fix a prespecified probability of default α, which may be necessary, for example, to maintain a desirable credit rating. Moreover, from a rating agency's perspective, our model helps identify those levered firms (as characterized by F, β, φ, λ) that can meet a given default probability required for a target rating (for more on credit ratings migration see, e.g., Hull 2003, chap. 26).

We now return to the framework of Section IV, where a single debt contract matures prior to the planning horizon. Propositon 7 characterizes the debt face value required to maintain a desired probability of default under the optimal policy:

Proposition 7. With a single debt contract maturing prior to the borrower's planning horizon ( ), assume , and r and κ are constant. Then, the borrower's optimal policy results in a default probability α, when the face value of the debt contract is set to where is the value of for which is as in corollary 1, (·) is the inverse of the standard normal cumulative distribution function, y solves the time‐0 budget constraint [0; y, (·) is as in (10), and , with X(s) as in proposition 2.

Figure 6(a) summarizes the comparative‐statics analysis of the probability of default with respect to the face value. By depicting the correspondence between the face value and the probability of default, we can clearly see which debt contract complies with a required range of default probabilities. It is interesting to note that, at the relatively low levels of leverage associated with the lower end of default probabilities, a firm facing default costs is less likely to default on a given debt contract than a firm facing costless default; and vice versa at the higher levels of leverage. Clearly, bearing the costs of default disciplines the levered firm to better service its debt, striving to avoid costly default, and the . This behavior is illustrated by figure 6(b) (as well as by figure 1).19 However, as shown in Figure 6(c), with a higher debt face value, resisting default becomes much more costly. This extends the default region, which in turn increases the burden of default costs, further weakening the firm's ability to support the default‐resistance wealth. Overall, for a given debt contract associated with the higher end of default probabilities, a levered firm facing default costs is more likely to default, despite the disciplinary impact of default costs, than a firm facing no default costs, and then .

Proposition 7 may be also useful to borrowers in the context of more formal risk‐management practices. In particular, it is evident from proposition 3 and figure 1 that a borrower's time‐T value at risk (VaR) at the α × 100% significance level, VaR(α), is given by (see, e.g., Jorion 2000 for more an VaR). Therefore, those who track their VaR, for example, at the α=0.01 level (possibly due to regulatory requirements), can enter into a debt contract with a face value F(0.01), using (17), thereby guaranteeing to lose over[0, T] not more than , with probability 0.99.