Dividend Policy and Capital Structure of a Defaultable Firm
aa r X i v : . [ q -f i n . M F ] O c t DIVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM
ALEX S.L. TSE
Abstract.
Default risk significantly affects the corporate policies of a firm. We develop a model inwhich a limited liability entity subject to Poisson default shock jointly sets its dividend policy and capitalstructure to maximize the expected lifetime utility from consumption of risk averse equity investors. Wegive a complete characterization of the solution to the singular stochastic control problem. The optimalpolicy involves paying dividends to keep the ratio of firm’s equity value to investors’ wealth below acritical threshold. Dividend payout acts as a precautionary channel to transfer wealth from the firm toinvestors for mitigation of losses in the event of default. Higher the default risk, more aggressively thefirm leverages and pays dividends. Introduction
Since the capital structure irrelevance principle and dividend irrelevance principle of Modigliani and Miller(1958) and Miller and Modigliani (1961), a vast literature has emerged to explore the factors driving cor-porate policies observed in practice. Historically, some important considerations include tax benefit,asymmetric information, signaling motive, agency costs, financial distress costs, managerial risk aversionand etc. While market frictions and strategic interaction among agents are all realistic concerns, veryfundamental factor such as default risk could indeed also play a crucial role behind corporate financedecisions. In this paper, we examine the impact of default risk of a firm on its joint decision of dividendpolicy and capital structure as well as equity investors’ consumption behavior.Our model features a limited liability firm and risk averse equity investors. At each point of timegiven the amount of equity capital in place, the firm simultaneously decides how much to invest in arisky asset (which implies the amount of debt required and in turn its choice of capital structure) andhow much to pay out to investors with logarithm utility function. Investors can deposit the dividendsreceived in a riskfree retail saving account and consume to derive utility flow. The interests of firmmanagers (who set the capital structure decision and payout policy) and investors (who choose their ownconsumption policy) are perfectly aligned such that their joint economic objective here is to maximizeinvestors’ expected lifetime utility of consumption.
Date : October 9, 2018.
Key words and phrases.
Dividend policy, capital structure, default risk, singular stochastic control, HJB equation.The author would like to thank David Hobson, Bart Lambrecht, Harry Zhang and the seminar participants at ImperialCollege London for the useful discussions and comments.Department of Mathematics, Imperial College London, London SW7 2AZ, UK. [email protected].
We assume wealth can only flow from the firm to investors in form of dividends but capital cannot beinjected into the firm from investors via equity issuance. A reduced form approach is adopted to modelfirm’s default where the equity value of the firm jumps to zero upon the arrival of a Poisson shock. Asa result, our model is perhaps the most suitable to describe the corporate policies of small businessesand start-ups which face a high barrier to equity financing and a high level of operational or financialuncertainty (and hence are prone to default).The incentive to pay out dividends in presence of default risk is intuitive. A unit of wealth retainedin the firm can generate economic value via investment in the risky asset but it comes at the cost of thepotential loss when default arrives. Equity investors, however, can exploit the limited liability structureof the entity by transferring some of the firm’s equity to their private pocket via dividends to earn amediocre but safe retail interest rate, rather than leaving all the money on the table just to be wiped out(for example, to be seized by the creditors) when the firm goes bust.The underlying optimization problem turns out to be a singular stochastic control problem. In contrastto many other models of investment and dividend distribution, the optimal payout policy in our setupis to pay dividends as to keep the ratio of the firm’s equity value to investors’ wealth level below acertain critical threshold, rather than just to pay out the cash to keep the equity reserve below a constantlevel. Moreover, the optimal leverage level is state-dependent with its magnitude being decreasing in theequity-to-wealth ratio instead of being a constant.Analysis of such a singular stochastic control problem is not straightforward in general and the conceptof viscosity solutions sometimes has to be invoked. Nonetheless, based on the transformation techniquesintroduced in the recent work of Hobson and Zhu (2016) and Hobson et al. (2018), we show that theassociated HJB equation can indeed be reduced to a first order crossing problem. The critical dividendpayment boundary can be read from the point at which the solution to a first order differential equationfirst crosses a given analytical function. An important advantage of this approach is that it is relativelyeasy to deduce the comparative statics of the model. We find that a higher default risk leads to a lowercritical equity-to-wealth ratio for dividend payout, a higher leverage level and a lower consumption rateof investors.We close the introduction by relating our work to the existing literature. Both payout policy andcapital structure decision are long-standing research topics in corporate finance and it is impossible togive a full account of the theory development here. Instead, we refer readers to the excellent surveysby Harris and Raviv (1991) and Allen and Michaely (2003). Mainstream finance literature often studiespayout policy and capital structure decision separately but not their joint interaction, as highlighted byLambrecht and Myers (2012). The economic foundation of our model is based on Lambrecht and Myers(2017) and Lambrecht and Tse (2018) where investment and payout policy are examined together in aninter-temporal investment/consumption model. Although our exposition assumes perfect coordination
IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 3 between managers and investors, the same modeling framework can be directly applied to an agencysetup driven by utility maximization of self-interested risk averse managers (see Remark 2 in Section 2).Continuous-time portfolio selection, which is the skeleton of the modeling framework in this paper, isof course another enormous field in the mathematical finance and stochastic control literature. Existingwork usually focuses on optimal consumption/investment models as per the seminal work of Merton (1969,1971) and its many other variants, or optimal risk control/dividend distribution models which are studiedextensively in the field of insurance [e.g. Radner and Shepp (1996), Browne (1997) and Jgaard and Taksar(1999)]. In a certain sense, these two classes of models are equivalent because consumption and dividendare usually viewed interchangeably, as stated by Taksar (2000) that optimal risk control/dividend modelsin most instances are consumption/investment models with linear utility function and arithmetic Brownianmotion return. Implicitly, these models assume that payout from a firm has to be consumed immediatelyand therefore discard the possibility that dividends can at least be deposited for consumption later. Wedisentangle the effect of dividend payout and consumption by introducing a riskfree retail saving accountto investors as an outside option. One novel prediction as a result of this flexibility is that a bad firm willbe voluntarily liquidated because there is no longer the need to keep a bad firm alive just for the purposeof generating dividends over time to match the smooth consumption required by risk averse investors. Tothe best of our knowledge, our current paper is the first one to consider a joint, dynamic model of capitalstructure (i.e. investment/risk control), dividend payout and individuals’ consumption.We also examine the joint impact of default risk on the firms’ corporate policies and consumptionpattern of investors. Incorporation of exogenous default risk is not a new mathematical feature - optimalportfolio choice and consumption problems with random termination time have been considered in lifeinsurance models [see for example Richard (1975) and Pliska and Ye (2007)]. In our current context ofcorporate finance, nonetheless, consideration of default risk leads to some interesting economic phenomenaas revealed by the comparative statics. One important aspect of our model is that equity financing is not possible. Thus our model issomewhat similar to a Merton problem with transaction costs as in Magill and Constantinides (1976),Davis and Norman (1990) and Shreve and Soner (1994). More precisely, the special case studied byHobson and Zhu (2016), where transaction cost is zero on sale and infinite on purchase, is comparable toour model in which equity capital can only be passed to investors as dividends but fresh capital cannot beinjected into the firm. Consequently, our optimal dividend strategy is similar to the investment strategyobtained by Hobson and Zhu (2016). However, our model is inherently a higher dimensional one becauseof the leverage decision involved and thus is not a trivial extension of their model. See the discussionin Section 2. Broadly speaking, our work contributes to the growing literature on solving a singular The interaction among dividend, leverage and firm’s default is also explored in a one-period signaling model of Kucinskas(2018), where high dividend is a bad signal for firms with high leverage because the payout can be driven by the motivationof “cash out” prior to bankruptcy.
IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 4 stochastic control problem via reduction to a first order crossing problem [e.g. Hobson and Zhu (2016),Hobson et al. (2018) and Hobson et al. (2016)]. It showcases the mathematical techniques are amendableoutside the context of portfolio selection under transaction costs and how powerful comparative staticscan be derived based on simple comparison principles.The rest of the paper is organized as follows. Section 2 introduces the modeling setup. Section 3 givesthe main results of the paper and their economic significance. A heuristic derivation of the solution isprovided in Section 4 and we give a full verification argument of the candidate solution in Section 5.Section 6 concludes. Some proofs in the main body of the paper are deferred to the appendix.2.
The setup
Throughout this paper we work with (Ω , F , {F t } , P ) a filtered probability space satisfying the usualconditions which supports a one-dimensional Brownian motion B = ( B t ) t ≥ .A firm can invest in two classes of asset: a bond instrument with interest rate ρ , and a risky assetwhich price process is a geometric Brownian motion with drift µ and volatility σ >
0. For every unit ofequity within the firm at time t , an amount of π t is invested in the risky asset whereas 1 − π t is investedin the bond. A choice of π t > π t − π = ( π t ) t ≥ an investment policy of the firm whichis required to be adapted and satisfy R t π u ( ω ) du < ∞ for all ( t, ω ). Equity within the firm can also bedistributed to risk averse equity investors in form of dividends. Let Φ = (Φ t ) t ≥ with Φ − = 0 be anadapted, non-decreasing process representing the cumulative amount of dividends paid to the investorsup to time t . The equity value of the firm S = ( S t ) t ≥ then evolves as dS t = π t S t ( µdt + σdB t ) + (1 − π t ) S t ρdt − d Φ t = [ ρ + ( µ − ρ ) π t ] S t dt + σπ t S t dB t − d Φ t . (1)The risk averse investors have a logarithm utility function. They possess a private account whichearns a retail riskfree rate of r and they consume at the same time to derive utility flow continuously.A consumption policy c = ( c t ) t ≥ is a non-negative, adapted process with R t c u ( ω ) du < ∞ for all ( t, ω ).The investors’ wealth level X = ( X t ) t ≥ then follows the dynamic dX t = ( rX t − c t ) dt + d Φ t . (2)The firm is exposed to a Poisson shock with intensity λ > IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 5 post-default can be derived by solving the deterministic control problem of F ( x ) := sup c t > Z ∞ e − βt ln c t dt (3)under the dynamic dX t = ( rX t − c t ) dt with X = x . Here β > F ( x ) = 1 β ln x + 1 β (cid:20) rβ + ln β − (cid:21) (4)and the corresponding optimal consumption strategy is given by c ∗ t = βX t .A collection of consumption, investment and dividend policies ( c, π, Φ) is said to be admissible if S t and X t are non-negative with ( S t , X t ) / ∈ (0 ,
0) for all t ≥
0. Denote by A ( s, x ) the class of admissiblestrategies with initial value ( S − = s, X − = x ). Prior to default, equity investors’ expected discountedliftime utility from consumption under a given ( c, π, Φ) is J ( s, x ; c, π, Φ) := E "Z ∞ e − βt ln c t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S − = s, X − = x = E "Z τ e − βt ln c t dt + Z ∞ τ e − βt ln c t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S − = s, X − = x where τ is an exponential random variable with parameter λ defined on the same probability space andit is independent of the underlying Brownian motion B . Firm managers act in the best interest of theinvestors. Their joint objective is to solve V ( s, x ) := sup ( c,π, Φ) ∈A ( s,x ) J ( s, x ; c, π, Φ)which can be rewritten as V ( s, x ) = sup ( c,π, Φ) ∈A ( s,x ) E "Z τ e − βt ln c t dt + e − βτ F ( X τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S − = s, X − = x (5)due to dynamic programming principle [see Jeanblanc et al. (2004)]. Remark . It is straightforward to introduce a dividend tax rate of κ ∈ [0 ,
1) in the model such thatinvestors’ wealth process, prior to firm’s default, satisfies dX t = ( rX t − c t ) dt + (1 − κ ) d Φ t (6)instead. Then under the transformation ˜ X t := X t − κ we can recover a version of the problem without tax. Remark . The optimization problem (5) is a “first best” criteria where firm managers and investors canperfectly coordinate to jointly deduce the optimal corporate policies and consumption strategy to createmaximum value for investors. It is indeed also possible to adopt our mathematical framework withinan agency-based model featuring self-interested, risk averse managers as in Lambrecht and Myers (2012,2017). In this alternative setup, managers capture 1 − κ fraction of the firm’s total payout as a form of rentextraction and pass the remaining κ fraction to investors as dividends. We could interpret X = ( X t ) t ≥ in (6) and c = ( c t ) t ≥ as the private wealth level and consumption rate of the managers respectively.The parameter 1 − κ now reflects the bargaining power of the managers. They simultaneously set thecorporate policies and their consumption strategy to maximize their own lifetime utility as in (5).It is constructive to compare our modeling setup to that of Hobson and Zhu (2016) who consider aMerton consumption and investment problem in which the risky asset can only be sold but not bought.Our problem is similar to theirs in the sense that the transfer of value also occurs in one direction onlyfrom the firm to the investors as dividends but capital cannot be injected into the firm from the investors.In other words, we rule out the possibility of equity issuance within our setup. It is not an unreasonableassumption as equity financing often involves expensive and time consuming procedures especially forsmaller firms.In Hobson and Zhu (2016) wealth is allocated between a risky asset and a riskfree cash account as inthe standard Merton problem. Meanwhile, our model concerns wealth allocation between a risky firmand a riskfree cash account where the value of the former is not an exogenously given process but rathera controlled process based on the capital structure decision π . It is indeed possible to view our setupas a variant of the Merton problem with transaction costs [as considered by Magill and Constantinides(1976), Davis and Norman (1990) and Shreve and Soner (1994)], albeit with a very special transactioncosts structure. The economy features three distinct assets: a risky defaultable asset with drift µ andvolatility σ , a defaultable debt instrument with constant yield ρ and a non-defaultable cash account withinterest rate r . The two defaultable securities can only be sold for cash but not bought with cash, i.e.transaction cost is infinite on purchase and zero on sale. However, these two securities are fully fungiblewhich can be freely converted from one into another at their prevailing value without any friction. Thisunique transaction cost structure makes our problem non-trivial where the solution construction and therelated economic properties do not follow from the existing literature of portfolio optimization undertransaction costs. 3. Main results
We state the key results of this paper where the proof is deferred to Section 5.
Theorem 1.
For the optimization problem (5) : (1) If µ = ρ ≤ λ + r , the optimal strategy is to liquidate the firm immediately by distributing its entireequity to investors in form of dividends and then investors consume their wealth at a rate of β .The corresponding value function is given by V ( s, x ) = F ( s + x ) = 1 β ln( s + x ) + 1 β (cid:20) rβ + ln β − (cid:21) . (7) The sharing rule of the firm’s payout can be justified by solving a repeated bargaining game between managers andinvestors. See Lambrecht and Myers (2012, 2017).
IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 7 (2) If µ = ρ or µ = ρ > λ + r , there exists a constant z ∗ ∈ (0 , ∞ ) such that the optimal strategy(prior to default of the firm) is not to pay any dividend when S t X t ≤ z ∗ . On this region, the optimalconsumption strategy and investment policy are given by the feedback controls c ∗ t = c ∗ ( S t , X t ) and π ∗ t = π ∗ ( S t , X t ) where c ∗ ( s, x ) := 1 V x ( s, x ) , π ∗ ( s, x ) := − ( µ − ρ ) V s ( s, x ) σ sV ss ( s, x ) . (8) V is the value function of the problem to be defined in Proposition 7 in Section 5. When S t X t > z ∗ ,the optimal strategy is to pay a discrete dividend of size S t − z ∗ X t z ∗ to the investors and then thestrategies associated with the region of S t X t ≤ z ∗ are followed thereafter. X t (Investors’ wealth) S t (Equity value of firm) Z t := S t X t = z ∗ no-dividend regiondividend paying region Figure 1.
A graphical illustration of the optimal dividend strategy. When the equityvalue of the firm is too high or investors’ private wealth is too low such that S t /X t > z ∗ ,a discrete amount of dividend is paid out by the firm to bring the ratio S t /X t back to z ∗ . No dividend is paid when the state variables lie within the no-dividend region.In our model, there are two economic motives for the equity investors to invest in the firm. The firstmotive is brought by the investment prospect of the firm which exists for as long as the excess return ofthe risky asset µ − ρ is non-zero. Note that investors are still willing to invest in the firm even if µ < ρ because it is possible for the firm to short sell the poor performing risky asset for value creation. Thesecond motive lies within the funding advantage of the firm due to its access to corporate debt financing.At minimum, the firm can serve as a “risky bank account” with yield ρ and default rate λ . This fundingvehicle is superior to the investors’ private saving account provided that the default-risk-adjusted yield ρ − λ is larger than the retail saving rate r . For the parameter combination in part (1) of Theorem 1, IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 8 there is neither investment nor funding motive to invest in the firm and hence the optimal strategy is toliquidate the firm immediately.This very plausible prediction that a bad firm will be immediately liquidated is a unique featurerelative to other models based on the standard Merton investment/consumption framework (such asLambrecht and Myers (2017)). Under risk aversion, individuals demand a smooth consumption schedule.If dividends are tied with consumption, then risk aversion will force dividends to be smooth as welland hence a bad firm must be run continuously to generate cash flows over time. By distinguishingconsumption and dividend via the possibility of depositing investors’ private wealth in a retail account,individuals can opt to liquidate a bad firm and consume the proceeds over time optimally according totheir own preference.For the more general parameter combination in part (2) of Theorem 1, the optimal dividend strategyresembles the optimal investment strategy of a Merton problem with infinite transaction costs as inHobson and Zhu (2016). Rigorously speaking, the optimal dividend strategy Φ ∗ is a local time policywhich keeps Z t := S t X t ≤ z ∗ , and it can be characterized by the solution to a Skorohod equation withreflecting boundary along Z t = z ∗ . Dividends are paid when the firm value is too high or investors’private wealth level is too low in order to keep ratio of firm value to private wealth below a threshold.See Figure 1. In particular, the payout trigger target is given by S t ≤ z ∗ X t where the right hand sideis not a constant [as commonly seen in standard dividend distribution models such as Radner and Shepp(1996)] but instead it increases with the investors’ private wealth. As investors become more wealthy,consumption can be adequately supported from their private account and hence a larger fraction of equitycan be retained within the firm to further finance its investment activities. It is another unique featureof our model due to the disentanglement of dividend payout and consumption. Remark . In the context of corporate finance, it might be more sensible to impose the constraint π > µ ≤ ρ ≤ λ + r ” and “ µ > ρ , or µ ≤ ρ and ρ > λ + r ” respectively.Although the characterization of the optimal controls in Theorem 1 is somewhat abstract under thenon-degenerate case (2), a lower bound of z ∗ is available and the monotonicity of π ∗ and c ∗ with respectto the state variables can be established. Moreover, in the corner case of µ = ρ > λ + r the closed-formexpressions of the optimal controls can indeed be derived. The results are summarized in the followingtwo propositions which proofs are given in Appendix 7.2. Proposition 1.
Suppose µ = ρ and consider the optimal controls defined in case (2) of Theorem 1: (1) The critical threshold of dividend payment z ∗ satisfies ( ρ − r − λ ) + λ ≤ z ∗ < ∞ . IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 9 (2)
The optimal investment level π ∗ ( s, x ) admits an expression π ∗ ( s, x ) = µ − ρσ θ ( s/x ) where θ ( z ) is abounded, positive and decreasing function on < z ≤ z ∗ with θ ( z ∗ ) > . (3) The optimal consumption rate per unit private wealth ¯ c ( s, x ) := c ∗ ( s,x ) x is a function of z = sx onlywhich is increasing on < z ≤ z ∗ and ¯ c ( z ) ↓ β as z ↓ . Proposition 2.
Suppose µ = ρ > λ + r and consider the optimal controls defined in case (2) of Theorem1. Then z ∗ = ρ − r − λλ , π ∗ ( s, x ) = 0 , c ∗ ( s, x ) = x /β − q ( s/x ) where q = q ( z ) is an implicit function defined as the solution to z = (cid:18) ρ − r − λλ (cid:19) λβ + λ (cid:18) ρ − r − λβ ( ρ − r ) (cid:19) − ρ − rβ + λ (cid:18) βq − βq (cid:19) ββ + λ q ρ − rβ + λ . (9)Analytical expressions of all the important control variables are available in the case of µ = ρ > λ + r where the investment motive vanishes. The firm essentially becomes a pure funding vehicle (subject todefault risk) without any investment in the risky asset as π ∗ = 0. The optimal dividend policy is thenentirely driven by the funding quality of the firm measured by ρ − rλ , which can be interpreted as thecorporate yield spread per unit default risk. Higher this ratio, less often the firm pays dividends since itis more efficient to retain the capital within the firm for value creation.In the more general case of µ = ρ , part (1) of Proposition 1 implies that z ∗ ↑ ∞ as λ ↓ ρ > r . The economic interpretation of this limiting result is the following: as long as the fundingadvantage ρ > r exists, it is in general suboptimal to pay out any dividend because a unit of wealth inthe investors’ private account can only earn the retail rate r whereas a unit of equity within the firm canat least earn a better rate ρ when default of firm is not a concern. Dividends should thus only be paid outafter investors’ private wealth has been entirely depleted. Once X hits zero, the firm pays out dividendcontinuously to match the optimal consumption required by the investors. Although we do not explicitlyconsider λ = 0 in this paper, the optimal strategy of this special case can be characterized rigorouslysimilar to Theorem 10 of Hobson and Zhu (2016).The magnitude of the firm’s investment level is decreasing in equity value S t and increasing in investors’wealth X t . To understand this behavior, it is important to note that investors’ exposure to the risky assetis determined by two factors: the fraction of wealth invested in the firm Z t = S t /X t and the investmentlevel of the firm π t . In general, risk averse investors desire to maintain a target exposure to the riskyasset. A benchmark example is that if there is no friction in the economy such that investors do notneed to rely on the firm as an intermediary to invest in the risky asset, then they will invest a constant Note that the case of µ = ρ ≤ λ + r has been covered by case (1) of Theorem 1 where we can take z ∗ = 0 correspondingto immediate liquidation. IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 10 fraction of their wealth in the risky asset given by the Merton ratio. In our model where equity financingcannot be performed freely, the firm thus has to be leveraged more aggressively when equity value declinesor investors’ wealth increases to provide an efficient risk-return exposure for the investors. Notice that θ ( z ∗ ) > Z t = S t /X t approaches the critical threshold z ∗ becauseinvestors anticipate a dividend payment is due soon which will boost their wealth level and hence a largerconsumption today becomes sustainable.The next proposition highlights the key comparative statics of our model. The proof can be found inAppendix 7.3. Proposition 3.
Suppose we are in the non-degenerate case of either µ = ρ or µ = ρ > λ + r . Then thefollowing comparative statics hold: (1) The critical threshold of dividend payment z ∗ is decreasing in λ , r and σ , and is increasing (resp.decreasing) in µ when µ ≥ ρ (resp. µ ≤ ρ ). (2) The optimal investment level π ∗ ( s, x ) is increasing (resp. decreasing) in λ and r when µ > ρ (resp. µ < ρ ). (3) The optimal consumption level c ∗ ( s, x ) is decreasing in λ and r . We focus on the comparative statics results with respect to λ (see Figure 2 for some numerical illustra-tions). Firstly, with a higher default risk λ the firm pays out dividends more aggressively as reflected bya lower critical threshold z ∗ . This can be understood as a phenomenon of moral hazard. When managersforesee that their firm is more likely to fail, there is a stronger precautionary incentive to transfer valuewithin the firm to the investors’ in form of dividends because the wealth in the private pocket of investorsis left untouched due to limited liability while equity within the firm is seized or wiped out in the eventof default.Among all the results in Proposition 3, perhaps the most surprising one is that the magnitude of π ∗ is increasing in λ . This may seem counter-intuitive in view of the risk averse nature of equity investorsas the result here suggests the (absolute) investment level is higher for a riskier firm (in terms of defaultrisk). Nonetheless, this result should be understood in conjunction with the impact of λ on z ∗ . A higher λ leads to a lower dividend threshold z ∗ and as such a smaller pool of equity (relative to investors’ wealth)is retained in the firm on average. To offset this effect of under-investment, the firm has be leveragedmore aggressively to restore the overall exposure to the risky asset.Finally, the effect of λ on the optimal consumption is intuitive where a higher default risk of the firmencourages precautionary saving of the investors by reducing consumption.The above results could be of interests to the area of corporate finance because they suggest aggressivepayout policy and capital structure could potentially be driven by a high level of default risk where equityholders exploit the limited liability structure of the firm to retain value for themselves, thus a symptom of IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 11 z * (a) Dividend payment boundary. π * ( s , x ) λ = 0.03λ = 0.05λ = 0.07 (b) Optimal investment level. c * ( s , x ) / x λ = 0.03λ = 0.05λ = 0.07 (c) Optimal consumption.
Figure 2.
Comparative statics of the optimal controls with respect to λ . Base parame-ters used are: µ = 0 . σ = 0 . ρ = 0 . r = 0 .
02 and β = 0 . µ and ρ arefixed constants independent of λ . Implicitly, we assume asymmetric information between equity holdersand bondholders. Only the former know the firm is subject to a Poisson default shock of intensity λ whilethe latter do not take this default possibility into account when setting the debt yield. On the other hand,the default risk is assumed to be an idiosyncratic one which does not improve the risky asset return µ . Ina bank lending model of Lambrecht and Tse (2018), both ρ and µ are linked to λ depending on how aninsolvent bank is resolved during an economic downturn. In general, we could incorporate extensions ofthis kind by replacing ρ and µ by some functions of λ . This will not significantly change the mathematicalanalysis of the stochastic control problem. However, the comparative statics with respect to λ will nowdepend on the precise constructions of ρ ( λ ) and µ ( λ ). IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 12 A heuristic derivation of the solution
In this section we use heuristics to derive an equation that the value function should satisfy. Itsoptimality is then verified rigorously in Section 5.It is convenient to reformulate (5) as an infinite horizon control problem. Note that E (cid:20)Z τ e − βt ln c t dt + e − βτ F ( X τ ) (cid:21) = E (cid:20)Z ∞ ( τ>t ) e − βt ln c t dt (cid:21) + E (cid:2) e − βτ F ( X τ ) (cid:3) = E (cid:20)Z ∞ e − ( β + λ ) t ln c t dt (cid:21) + Z ∞ λe − λt E (cid:2) e − βt F ( X t ) (cid:3) dt = E (cid:20)Z ∞ e − ( β + λ ) t (ln c t + λF ( X t )) dt (cid:21) where we have used the fact that τ is an independent exponential random variable. Hence problem (5) isequivalent to V ( s, x ) = sup ( c,π, Φ) ∈A ( s,x ) E "Z ∞ e − ( β + λ ) t (ln c t + λF ( X t )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S − = s, X − = x (10)which is an infinite horizon problem with discount rate β + λ and running reward ln c t + λF ( X t ).Write M t := Z t e − ( β + λ ) u [ln c u + λF ( X u )] du + e − ( β + λ ) t V ( S t , X t ) . Then we expect M is a supermartingale in general, and is a true martingale under the optimal strategy.Suppose V is a C × function, applying Ito’s lemma we find e ( β + λ ) t dM t = (cid:26) ln c t − V x c t + rV x X t + ρV s S t + ( µ − ρ ) V s S t π t + σ V ss S t π t − ( β + λ ) V t + λF ( X t ) (cid:27) dt + ( V x − V s ) d Φ t + σπ t V s S t dB t . Further assume V x > V ss <
0, the drift term can be maximized with respect to c and π . Then weexpect the value function to solve the HJB variational inequalitymin ( −L V, −M V ) = 0 (11)where the operators L and M are defined as L f := sup c> ,π (cid:26) ln c − f x c + rf x x + ρf s s + ( µ − ρ ) f s sπ + σ f ss s π − ( β + λ ) f + λF ( x ) (cid:27) = − ln f x − rf x x + ρf s s − ( µ − ρ ) [ f s ] σ f ss − ( β + λ ) f + λF ( x ) , (12) M f := f x − f s . (13)Inspired by Magill and Constantinides (1976), we conjecture the optimal strategy is to pay dividendsonly when the ratio Z t := S t X t is above a certain threshold z ∗ (refer to Figure 1 again). Following IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 13
Davis and Norman (1990), we postulate the value function has the form V ( s, x ) = 1 β ln x + g (cid:16) sx (cid:17) (14)for some function g to be determined. Write z := sx . Then over z ≤ z ∗ no dividend is paid and we expect L V = 0 which becomes − ln (cid:18) β − zg ′ ( z ) (cid:19) + ( ρ − r ) zg ′ ( z ) − ( µ − ρ ) [ g ′ ( z )] σ g ′′ ( z ) − ( β + λ ) g ( z ) + λβ (cid:18) rβ + ln β − (cid:19) + rβ − . (15)The system can be further simplified by a series of transformation used in Hobson and Zhu (2016) andHobson et al. (2018). Write u := ln z and let h ( u ) = h (ln z ) := g ( z ) − β + λ h λβ (cid:16) rβ + ln β − (cid:17) + rβ − i .Then g ′ ( z ) = ddz g ( z ) = ddz h ( u ) = ddu h ( u ) dudz = h ′ ( u ) z ,g ′′ ( z ) = ddz h ′ ( u ) z = 1 z ddz h ′ ( u ) − h ′ ( u ) z = h ′′ ( u ) z − h ′ ( u ) z . (15) can then be reduced to − ln (cid:18) β − h ′ ( u ) (cid:19) + ( ρ − r ) h ′ ( u ) − ( µ − ρ ) [ h ′ ( u )] σ ( h ′′ ( u ) − h ′ ( u )) − ( β + λ ) h ( u ) = 0 . (16)Set w ( h ) := dhdu such that h ′′ ( u ) = ddu w ( h ) = w ′ ( h ) h ′ ( u ) = w ′ ( h ) w ( h ). (16) then becomes − ln (cid:18) β − w ( h ) (cid:19) + ( ρ − r ) w ( h ) − ( µ − ρ ) w ( h )2 σ ( w ′ ( h ) − − ( β + λ ) h = 0 . (17)Let N be the inverse function of w , i.e. N := w − , and write q := w ( h ). (17) can then be written as − ln (cid:18) β − q (cid:19) + ( ρ − r ) q − ( µ − ρ ) q σ (1 /N ′ ( q ) − − ( β + λ ) N ( q ) = 0 . (18)Suppose for now µ = ρ . The special case of µ = ρ is discussed at the end of this section. Set n ( q ) := ln (cid:16) β − q (cid:17) + βN ( q ). After some algebra of substituting N and N ′ away by n and n ′ in (18), wecan obtain a first order ODE n ′ ( q ) = O ( q, n ( q )) where O ( q, n ) := β q − βq m ( q ) − nn − ℓ ( q ) (19)and m ( q ) := ββ + λ (cid:26) ( ρ − r ) q + λβ ln (cid:18) β − q (cid:19) + ( µ − ρ ) σ β (cid:27) ,ℓ ( q ) := ββ + λ (cid:26)(cid:20) ρ − r + ( µ − ρ ) σ (cid:21) q + λβ ln (cid:18) β − q (cid:19)(cid:27) = m ( q ) − ββ + λ ( µ − ρ ) σ (cid:18) β − q (cid:19) . We now derive the form of the value function on the dividend paying regime sx = z > z ∗ . Underthe conjectured strategy, a lump sum dividend D is paid out by the firm to restore the equity value toinvestors’ private wealth ratio back to z ∗ . D should then solve s − Dx + D = z ∗ which gives D = s − z ∗ x z ∗ . The IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 14 value function does not change on this corporate action and thus V ( s − D, x + D ) = V ( s, x ) which isequivalent to 1 β ln( x + D ) + g ( z ∗ ) = 1 β ln x + g ( z ) . From this we obtain g ( z ) = 1 β ln (cid:18) Dx (cid:19) + g ( z ∗ ) = 1 β ln (cid:18) z z ∗ (cid:19) + g ( z ∗ ) = 1 β ln(1 + z ) + A ∗ (20)on z > z ∗ where A ∗ is some constant.Now we apply the same set of transformations to (20). We have h ( u ) = g ( e u ) − β + λ (cid:20) λβ (cid:18) rβ + ln β − (cid:19) + rβ − (cid:21) = 1 β ln(1 + e u ) + A ∗ − β + λ (cid:20) λβ (cid:18) rβ + ln β − (cid:19) + rβ − (cid:21) =: 1 β ln(1 + e u ) + ¯ A. Then w ( h ) = h ′ ( u ) = 1 β e u e u = 1 β e β ( h − ¯ A ) − e β ( h − ¯ A ) = 1 β (1 − e β ( ¯ A − h ) )and the inverse function of w is found to be N ( q ) = w − ( q ) = ¯ A − β (cid:18) ln β + ln (cid:18) β − q (cid:19)(cid:19) . Finally, n ( q ) = ln (cid:18) β − q (cid:19) + βN ( q ) = β ¯ A − ln β = βA ∗ − (cid:18) rβ + ln β − (cid:19) + λλ + β ln 1 β = βA ∗ − (cid:18) rβ + ln β − (cid:19) + ℓ (0) (21)which is a constant. The above relationships hold as long as z > z ∗ , on which q = w ( h ) = 1 β e u e u = 1 β z z . Hence the equivalent range in the q -coordinate is q > q ∗ := β z ∗ z ∗ .We expect the transformed value function n to solve n ′ = O ( q, n ) on q ≤ q ∗ and to be a constant on q > q ∗ . To solve such a free boundary value problem, we further require an initial value associated withthe system. Along the boundary s = 0 the equity value of the firm is zero and hence cannot invest inthe risky asset nor the bond (if a firm with zero net equity attempts to borrow to invest or short sell theasset to support purchase of the bond, the Brownian nature of the asset price will make it impossible forthe firm to maintain non-negative equity value). Then essentially the firm ceases to exist and the onlyfeasible strategy is for the investors to consume their existing private wealth optimally. Hence V (0 , x ) = F ( x ) = β ln x + β h rβ + ln β − i . This boundary condition translates into g (0) = β h rβ + ln β − i , h ( −∞ ) = g (0) − β + λ h λβ (cid:16) rβ + ln β − (cid:17) + rβ − i = ln ββ + λ and we expect h ′ ( −∞ ) = 0. Then w (cid:16) ln ββ + λ (cid:17) = w ( h ( −∞ )) = h ′ ( −∞ ) = 0, N (0) = ln ββ + λ . and finally n (0) = ln β + ββ + λ ln β = λβ + λ ln β = ℓ (0). IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 15
In summary, we are looking for a solution n with initial value n (0) = ℓ (0) which solves n ′ = O ( q, n ) on0 ≤ q ≤ q ∗ and n ( q ) being a constant on q > q ∗ , where q ∗ is an unknown boundary to be identified. Theconjectured second order smoothness of the original value function V now translates into the first ordersmoothness of n and as such we expect n ′ ( q ∗ ) = 0. But the form of (19) suggests that, away from q = 0, n ′ = O ( q, n ) = 0 if and only if n = m . Hence the boundary point q ∗ must be given by the q -coordinatewhere the solution n intersects the function m ( q ).The next proposition confirms that a solution n and the corresponding free boundary point q ∗ indeedexist. Proposition 4.
Suppose µ = ρ . Consider an initial value problem n ′ ( q ) = O ( q, n ( q )) := β q − βq m ( q ) − n ( q ) n ( q ) − ℓ ( q ) , n (0) = ℓ (0) and n ′ (0) > ℓ ′ (0) where m ( q ) := ββ + λ (cid:26) ( ρ − r ) q + λβ ln (cid:18) β − q (cid:19) + ( µ − ρ ) σ β (cid:27) ,ℓ ( q ) := ββ + λ (cid:26)(cid:20) ρ − r + ( µ − ρ ) σ (cid:21) q + λβ ln (cid:18) β − q (cid:19)(cid:27) . A unique solution to the above problem exists at least up to ≤ q ≤ q ∗ with n ′ (0) > and q ∗ := inf { q > n ( q ) ≥ m ( q ) } ∈ ( β ( ρ − r − λ ) + ( ρ − r − λ ) + + λ , β ) . The solution n is strictly increasing and lies between ℓ ( q ) and m ( q ) on (0 , q ∗ ) .Proof. We first show that a unique solution exists in the neighborhood of q = 0. Let χ ( q ) := n ( q ) − ℓ ( q ).Then the initial value problem is equivalent to χ ′ ( q ) = n ′ ( q ) − ℓ ′ ( q ) = O ( q, χ ( q ) + ℓ ( q )) − ℓ ′ ( q )= β − βq (cid:20) ββ + λ ( µ − ρ ) σ (cid:18) β − q (cid:19) − χ ( q ) (cid:21) qχ ( q ) − ββ + λ (cid:18) ρ − r + ( µ − ρ ) σ − λ − βq (cid:19) =: A ( q, χ ( q )) qχ ( q ) + B ( q ) :=: ˆ O ( q, χ ( q )) (22)subject to χ (0) = 0 and χ ′ (0) > χ ′ ( q ) = A qχ ( q ) + B, χ (0) = 0 and χ ′ (0) > A, B are constants with
A >
0. Making use of the substitution y = χq , we can obtain an ODE interms of y = y ( q ) as y + q dydq = Ay + B and in turn yA + By − y dy = 1 q dq. (23) IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 16
Since
A >
0, the denominator on the left hand side of (23) admits an expression of ( α − y )( γ + y ) for some α, γ >
0. (23) can then be solved via partial fraction which leads to the solution of | q | α + γ | α − y | α | γ + y | γ = C for some arbitrary constant C , or equivalently | αq − χ | α | γq + χ | γ = C. The initial condition χ (0) = 0 forces C = 0. Hence the solution is either χ ( q ) = αq or χ ( q ) = − γq wherethe former satisfies the initial condition χ ′ (0) > A (0 ,
0) = β β + λ ( µ − ρ ) σ > q ↓ χ ( q ) q = α where α is thepositive root to the quadratic equation A (0 ,
0) + B (0) y − y = 0 . (24)The conclusion that n ′ (0) > n ′ ( q ) = O ( q, n ( q ))around q = 0 gives n ′ (0) = β [ m (0) − ℓ (0)] n ′ (0) − ℓ ′ (0) = β β + λ ( µ − ρ ) σ n ′ (0) − ℓ ′ (0)and thus n ′ (0) > ℓ ′ (0) implies n ′ (0) > , n ′ (0) > ℓ ′ (0) and n (0) = ℓ (0) < m (0), n is initially lying between ℓ ( q ) and m ( q ). It is trivial from the form of O ( q, n ) that the solution n cannot cross ℓ , and that the solutionis increasing for as long as q < β and n ( q ) stays between m ( q ) and ℓ ( q ). As m ( q ) → −∞ when q → β , n must cross m somewhere on q < β which guarantees the existence of q ∗ := inf { q > n ( q ) ≥ m ( q ) } .Moreover, as n ( q ) < m ( q ) on q < q ∗ and the derivative of n has to be zero when n crosses m , we musthave m ′ ( q ∗ ) ≤
0. A simple calculus exercise shows that m ( q ) has an inverted U-shape when ρ − r − λ > q = ρ − r − λβ ( ρ − r ) and thus q ∗ ≥ ρ − r − λβ ( ρ − r ) . Otherwise if ρ − r − λ ≤ m ( q ) isdecreasing for all 0 ≤ q < /β and in this case we can only conclude q ∗ >
0. Combining these two casesleads to q ∗ ∈ ( β ( ρ − r − λ ) + ( ρ − r − λ ) + + λ , β ). (cid:3) Remark . Note that we have imposed an additional constraint of n ′ (0) > ℓ ′ (0) in Proposition 4. If weinstead pick the solution with n ′ (0) < ℓ ′ (0), then n ( q ) will be initially below ℓ ( q ). The form of O suggeststhat n is decreasing and does not cross ℓ , and in turn m , for all q . Then there does not exist a boundarypoint q ∗ at which smooth pasting holds. The resulting n therefore is not a sensible candidate solution.Finally, we consider the special case of µ = ρ . We can indeed obtain an explicit solution for n where(18) gives a closed-form expression of N as N ( q ) = 1 β + λ (cid:20) ( ρ − r ) q − ln (cid:18) β − q (cid:19)(cid:21) IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 17 and thus n ( q ) = ln (cid:18) β − q (cid:19) + βN ( q ) = ββ + λ (cid:20) ( ρ − r ) q + λβ ln (cid:18) β − q (cid:19)(cid:21) = m ( q ) = ℓ ( q ) . This result should not be surprising. From Proposition 4, ℓ ( q ) < n ( q ) < m ( q ) on 0 < q < q ∗ and m ( q ) − ℓ ( q ) → µ → ρ , we must have n ( q ) → m ( q ) = ℓ ( q ) when µ approaches ρ .What should be the correct value of q ∗ when µ = ρ ? Again, we expect n is a constant on q > q ∗ andfirst order smoothness suggests q ∗ should satisfy n ′ ( q ∗ ) = m ′ ( q ∗ ) = 0. Thus q ∗ should be the q -coordinateof the turning point of m ( q ) if it exists. The existence condition is given by m ′ (0) > ρ − r − λ > q ∗ is ρ − r − λβ ( ρ − r ) .If ρ − r − λ ≤
0, then one cannot locate any positive boundary point at which the first order smoothnessholds. Our conjecture in this case is that the no-dividend region vanishes which is economically equivalentto q ∗ = 0. The firm is liquidated immediately at time zero and investors receive the entire equity of thefirm as dividends. The value function should thus satisfy V ( s, x ) = F ( s + x ) = 1 β ln( s + x ) + 1 β (cid:20) rβ + ln β − (cid:21) = 1 β ln x + 1 β ln(1 + z ) + 1 β (cid:20) rβ + ln β − (cid:21) and thus g ( z ) = β ln(1 + z ) + β h rβ + ln β − i . If we apply the transformation which takes (20) to (21),we can obtain n ( q ) = λλ + β ln β = m (0) = ℓ (0) for all q ≥ n , m and ℓ under different parameter combinationsare shown in Figure 2.5. Construction of the candidate value function and verification
The heuristics in Section 4 guide us to write down a first order system that the transformed valuefunction should satisfy (with closed-form expressions available in some special cases). Conversely, giventhe solution to the first order system we can reverse the transformation to construct a second order smoothcandidate value function and prove its optimality via a formal verification argument.We first construct the candidate value function in the special case of µ = ρ ≤ r + λ in which case weexpect the optimal strategy is to liquidate the firm immediately by transferring all equity to investors viadividends payment. Proposition 5.
Suppose µ = ρ ≤ r + λ . On ( s, x ) ∈ R \ { (0 , } define V C ( s, x ) = 1 β ln( s + x ) + 1 β (cid:20) rβ + ln β − (cid:21) . (25) Then V C ( s, x ) is a concave function increasing in both s and x . Moreover, L V C ≤ and M V C = 0 ,where L and M are the operators defined in (12) and (13) . IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 18 q n(q)m(q)l(q) (a) µ = ρ and ρ > r + λ . q n(q)m(q)l(q) (b) µ = ρ and ρ ≤ r + λ . q n(q)m(q)=l(q) (c) µ = ρ > r + λ . q n(q)m(q)=l(q) (d) µ = ρ ≤ r + λ . Figure 3.
The possible shapes of the transformed value function n under different pa-rameter combinations. When µ = ρ , n is first increasing and then becomes flat when itcrosses m at q = q ∗ . When µ = ρ , n , m and ℓ coincide and q ∗ is the turning point of m if exists. If µ = ρ ≤ r + λ such that m is decreasing for all q ≥
0, we take q ∗ = 0 and thetransformed value function is a flat horizontal line n ( q ) = m (0) = ℓ (0). Proof.
It is trivial that V C is concave and is increasing in both of its arguments. Direct computationgives V Cs = V Cx = β ( s + x ) and in turn M V C = 0. Finally, L V C = − ln 1 β ( s + x ) − rβ xs + x + ρβ ss + x − ( β + λ ) (cid:20) β ln( s + x ) + 1 β (cid:18) rβ + ln β − (cid:19)(cid:21) + λ (cid:20) β ln x + 1 β (cid:18) rβ + ln β − (cid:19)(cid:21) = λβ ln xs + x + ρ − rβ ss + x ≤ λβ ln xs + x + λβ ss + x = λβ (cid:18) ln xs + x + 1 − xs + x (cid:19) . Simple calculus exercise shows that f ( α ) := ln α + 1 − α ≤ α ∈ (0 ,
1) and hence L V C ≤ (cid:3) Away from the special case of µ = ρ ≤ r + λ , we cannot write down the candidate value functionexplicitly. In the following two propositions, we describe how the transformation introduced in Section 4can be reversed and several important analytical properties of the constructed candidate value functionare provided. IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 19
Proposition 6. If µ = ρ , let n = ( n ( q )) ≤ q ≤ q ∗ be the solution to the initial value problem in Proposition4 where q ∗ := inf { q > n ( q ) ≥ m ( q ) } . Otherwise if µ = ρ > r + λ , define n ( q ) := m ( q ) on ≤ q ≤ q ∗ := ρ − r − λβ ( ρ − r ) .In both case, let z ∗ := βq ∗ − βq ∗ and N ( q ) := β h n ( q ) − ln (cid:16) β − q (cid:17)i . Let w be the inverse function of N .For −∞ < u ≤ ln z ∗ , define h = h ( u ) as the solution to Z N ( q ∗ ) h dvw ( v ) = ln z ∗ − u (26) which is equivalent to Z q ∗ w ( h ) N ′ ( v ) v dv = ln z ∗ − u. (27) u → h ( u ) is then a strictly increasing bijection from ( −∞ , ln z ∗ ] → ( N (0) , N ( q ∗ )] .Finally, set g C ( z ) := h (ln z ) + β + λ h λβ (cid:16) rβ + ln β − (cid:17) + rβ − i , < z ≤ z ∗ ; β ln(1 + z ) + β h n ( q ∗ ) + (cid:16) rβ + ln β − (cid:17) − ℓ (0) i , z > z ∗ . (28) Then g C : (0 , ∞ ) → R is a C function.Proof. For ease of notation we suppress the superscript C in g C throughout the proof. We give the proofin the case of µ = ρ . It is much easier to establish the results under µ = ρ since n and q ∗ are then availablein closed-form.We first show that h ( u ) is an increasing bijection. Recall from Proposition 4 that n is increasing. Then N ′ ( v ) = n ′ ( v ) β + − βv > v < β and in turn the integrand on the left hand side of (27) is strictlypositive such that w ( h ( u )) is strictly increasing in u . Moreover, Z · N ′ ( v ) v dv > Z · dvv (1 − βv ) = + ∞ . Hence u → w ( h ( u )) is a bijection from ( −∞ , ln z ∗ ] to (0 , q ∗ ]. Since N = w − is strictly increasing on(0 , q ∗ ], u → h ( u ) is a bijection from ( −∞ , ln z ∗ ] → ( N (0) , N ( q ∗ )].Now we proceed to show that g is a C function. On z > z ∗ , g is trivially a C function. On 0 < z < z ∗ , n is a C function and the continuity property is inherited by ( N, N ′ ) and then on integration by ( h, h ′ , h ′′ )and finally ( g, g ′ , g ′′ ). It is thus sufficient to check the continuity of g , g ′ and g ′′ at z = z ∗ > h (ln z ∗ ) = N ( q ∗ ) = β h n ( q ∗ ) − ln (cid:16) β − q ∗ (cid:17)i = n ( q ∗ ) β − β ln (cid:16) β (1+ z ∗ ) (cid:17) . Hence g ( z ∗ ) = h (ln z ∗ ) + 1 β + λ (cid:20) λβ (cid:18) rβ + ln β − (cid:19) + rβ − (cid:21) = 1 β ln(1 + z ∗ ) + 1 β (cid:20) n ( q ∗ ) + (cid:18) rβ + ln β − (cid:19) − ℓ (0) (cid:21) = g ( z ∗ +) . IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 20
We now check the continuity of zg ′ ( z ) at z = z ∗ . Let u ∗ := ln z ∗ and h ∗ := h ( u ∗ ). Then by construction, z ∗ g ′ ( z ∗ ) = h ′ ( u ∗ ) = w ( h ∗ ) = q ∗ . Meanwhile, ( z ∗ +) g ′ ( z ∗ +) = z ∗ β (1+ z ∗ ) = q ∗ . This implies the continuityof g ′ at z ∗ .Similarly we check the continuity of z g ′′ ( z ) at z = z ∗ . From construction we can deduce( z ∗ ) g ′′ ( z ∗ ) = h ′′ ( u ∗ ) − h ′ ( u ∗ ) = w ( h ∗ )[ w ′ ( h ∗ ) −
1] = q ∗ (cid:18) N ′ ( q ∗ ) − (cid:19) = − β ( q ∗ ) where we have used the fact N ′ ( q ) = β (cid:16) n ′ ( q ) + β − βq (cid:17) and n ′ ( q ∗ ) = 0. On the other hand,( z ∗ +) g ′′ ( z ∗ +) = − ( z ∗ ) β (1 + z ∗ ) = − β ( q ∗ ) . This completes the proof. (cid:3)
When we transform the original HJB equation in Section 4, a crucial step is a change of the independentvariable via q := w ( h ) which leads to the transformed value function n = ( n ( q )) ≤ q ≤ q ∗ . Proposition 6is about the reversal of the transformation. While q is a dummy independent variable associated withthe candidate value function n in the transformed system, one should keep in mind that q is relatedto the original coordinate system through q := w [ h ( u )] = w [ h (ln z )]. The following lemma provides animportant link between the two coordinate systems which will be utilized extensively in many of thesubsequent proofs in this paper. Lemma 1.
Recall the notations introduced in Proposition 6. Write q := w [ h ( u )] = w [ h (ln z )] . Then z and q are linked via z = z ( q ) = βq − βq exp − Z q ∗ q n ′ ( v ) βv dv ! . (29) In particular, z : [0 , q ∗ ] → [0 , z ∗ ] is an strictly increasing function, z ( q ) ↓ as q ↓ and z ( q ) ↑ z ∗ as q ↑ q ∗ .Proof. Starting from (27),ln z ∗ − ln z = Z q ∗ q N ′ ( v ) v dv = Z q ∗ q (cid:18) n ′ ( v ) βv + 1 v (1 − βv ) (cid:19) dv = Z q ∗ q (cid:18) n ′ ( v ) βv (cid:19) dv + ln q ∗ − βq ∗ − ln q − βq = Z q ∗ q (cid:18) n ′ ( v ) βv (cid:19) dv + ln z ∗ − ln βq − βq and we can arrive at (29) after a slight rearrangement of the terms. Since n ′ ( v ) > v ∈ (0 , q ∗ ], z = z ( q ) is increasing in q . IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 21
From the form of (29) it is trivial that z ↑ z ∗ as q ↑ q ∗ because n ′ ( v ) /v is bounded near v = q ∗ . Toestablish z ( q ) ↓ q ↓
0, observe that0 ≤ z ( q ) = βq − βq exp − Z q ∗ q n ′ ( v ) βv dv ! ≤ βq − βq . Taking limit gives the desired result. (cid:3)
Now we formally define the candidate value function in the non-degenerate case and provide a fewuseful properties.
Proposition 7.
For µ = ρ or µ = ρ > r + λ , define V C ( s, x ) = 1 β ln x + g C (cid:16) sx (cid:17) , s > , x > where g C is defined in Proposition 6. Then: (1) V C can be extended to s = 0 and x = 0 by continuity leading to V C (0 , x ) = 1 β ln x + 1 β (cid:20) rβ + ln β − (cid:21) , x > ,V C ( s,
0) = 1 β ln s + 1 β (cid:20) m ( q ∗ ) + (cid:18) rβ + ln β − (cid:19) − ℓ (0) (cid:21) , s > . (2) V C ( s, x ) is a concave function and is increasing in both s and x . (3) On { ( s, x ) : s > xz ∗ , s ≥ , x ≥ } , M V C = 0 and L V C ≤ . (4) On { ( s, x ) : 0 ≤ s ≤ xz ∗ , x ≥ , sx = 0 } , L V C = 0 ; On { ( s, x ) : 0 < s ≤ xz ∗ , x ≥ } , M V C ≤ .Proof. Again we will suppress the superscript C in g C throughout the proof for brevity.(1) Recall that u → h ( u ) is a bijection from ( −∞ , ln z ∗ ] to ( N (0) , N ( q ∗ )]. Thenlim z ↓ g ( z ) = lim u ↓−∞ h ( u ) + 1 β + λ (cid:20) λβ (cid:18) rβ + ln β − (cid:19) + rβ − (cid:21) = N (0) + 1 β + λ (cid:20) λβ (cid:18) rβ + ln β − (cid:19) + rβ − (cid:21) = 1 β (cid:20) ℓ (0) − ln 1 β (cid:21) + 1 β + λ (cid:20) λβ (cid:18) rβ + ln β − (cid:19) + rβ − (cid:21) = 1 β (cid:20) rβ + ln β − (cid:21) . Thus V C (0 , x ) := lim s ↓ V C ( s, x ) = 1 β ln x + lim z ↓ g ( z ) = 1 β ln x + 1 β (cid:20) rβ + ln β − (cid:21) . On the other hand, for all x = 0 and sx = z > z ∗ we have V C ( s, x ) = 1 β ln x + 1 β ln(1 + s/x ) + 1 β (cid:20) n ( q ∗ ) + (cid:18) rβ + ln β − (cid:19) − ℓ (0) (cid:21) = 1 β ln( x + s ) + 1 β (cid:20) n ( q ∗ ) + (cid:18) rβ + ln β − (cid:19) − ℓ (0) (cid:21) IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 22 and hence V C ( s,
0) := lim x ↓ V C ( s, x ) = 1 β ln s + 1 β (cid:20) n ( q ∗ ) + (cid:18) rβ + ln β − (cid:19) − ℓ (0) (cid:21) since z ∗ < ∞ .(2) On s > xz ∗ , V C ( s, x ) = 1 β ln x + 1 β ln(1 + s/x ) + 1 β (cid:20) n ( q ∗ ) + (cid:18) rβ + ln β − (cid:19) − ℓ (0) (cid:21) = 1 β ln( s + x ) + 1 β (cid:20) n ( q ∗ ) + (cid:18) rβ + ln β − (cid:19) − ℓ (0) (cid:21) which is obviously a concave function increasing in both s and x . On s ≤ xz ∗ or equivalently q ≤ q ∗ , V Cx = 1 x (cid:18) β − zg ′ ( z ) (cid:19) = 1 x (cid:18) β − h ′ ( u ) (cid:19) = 1 x (cid:18) β − w ( h ) (cid:19) ≥ x (cid:18) β − q ∗ (cid:19) > q ∗ < /β from Proposition 4, and V Cs = 1 x g ′ ( z ) = 1 s zg ′ ( z ) = 1 s h ′ ( u ) = 1 s w ( h ) > . Since g is second-order smooth at z = z ∗ by Proposition 6, to show that V C concave it is sufficientto check that the Hessian matrix H := V Css V Csx V Cxs V Cxx is semi-negative definite on s ≤ xz ∗ . From the transformation adopted, V ss = g ′′ ( z ) x = 1 s z g ′′ ( z ) = 1 s [ h ′′ − h ′ ] = w ( h ) s [ w ′ ( h ) −
1] = qs [1 /N ′ ( q ) − < N ′ ( q ) = n ′ ( q ) β + 11 − βq ≥ − βq > n is increasing. Meanwhile, the determinant of H can be evaluated as V Css V Cxx − [ V Cxs ] = 1 x (cid:20) g ′′ ( z ) (cid:18) z g ′′ ( z ) + 2 zg ′ ( z ) − β (cid:19) − ( g ′ ( z ) + zg ′′ ( z )) (cid:21) = − x (cid:20) g ′′ ( z ) β + ( g ′ ( z )) (cid:21) = − z x (cid:20) z g ′′ ( z ) β + ( zg ′ ( z )) (cid:21) = − z x (cid:20) h ′′ − h ′ β + ( h ′ ) (cid:21) = − w ( h ) z x (cid:20) w ′ ( h ) − β + w ( h ) (cid:21) = − qz x (cid:20) /N ′ ( q ) − β + q (cid:21) . Hence det( H ) ≥ /N ′ ( q ) − β + q ≤ N ′ ( q ) ≥ − βq . But againthe latter holds due to (31). Thus V C is concave. IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 23 (3) From construction of V C on s > xz ∗ it is trivial that and V Cs = V Cx = β ( s + x ) and hence M V C = 0.Then it remains to show L V C ≤
0. Suppose x = 0. A direct evaluation of L V C on sx = z > z ∗ gives L V C = − ln 1 βx (1 + z ) − rβ (1 + z ) + ρzβ (1 + z ) + ( µ − ρ ) σ β − ( β + λ ) (cid:20) β ln x + 1 β ln(1 + z ) + 1 β (cid:20) n ( q ∗ ) + (cid:18) rβ + ln β − (cid:19) − ℓ (0) (cid:21)(cid:21) + λβ ln x + λβ (cid:20) rβ + ln β − (cid:21) = ( ρ − r ) (cid:20) β z z (cid:21) + λβ ln (cid:20) β − β z z (cid:21) + ( µ − ρ ) σ β − β + λβ n ( q ∗ )= β + λβ (cid:20) m (cid:18) β z z (cid:19) − n ( q ∗ ) (cid:21) ≤ β + λβ (cid:20) m (cid:18) β z ∗ z ∗ (cid:19) − n ( q ∗ ) (cid:21) = β + λβ [ m ( q ∗ ) − n ( q ∗ )] = 0since m ( q ) is decreasing on q ≥ q ∗ = z ∗ β (1+ z ∗ ) and n ( q ∗ ) = m ( q ∗ ) by the definition of q ∗ . Theinequality can be extended to x = 0 by continuity on observing that m (cid:16) β z z (cid:17) = m (cid:16) β ss + x (cid:17) .(4) On z ≤ z ∗ the candidate value function V C is constructed from n = ( n ( q )) ≤ q ≤ q ∗ which bydefinition solves L V C = 0. Thus we only have to verify that M V C = V Cx − V Cs ≤ < z ≤ z ∗ .Under the transformation adopted the desired inequality is x (cid:16) β − (1 + z ) g ′ ( z ) (cid:17) ≤ zg ′ ( z ) ≥ zβ (1+ z ) and in turn q = w ( h ) = h ′ = zg ′ ( z ) ≥ zβ (1+ z ) or equivalently z ≤ βq − βq . But this immediately follows from (29). (cid:3) Remark . While we can extend the definition of L V C to x = 0 by continuity, we do not require M V C at s = 0. The rationale is that along s = 0 the net equity value of the firm is zero and hence no dividendcan be paid out, i.e. d Φ t = 0 is the only admissible strategy whenever S t = 0. The marginal utilitycontributed by the dividend term M V C d Φ t is thus zero.The following lemma provides some useful results which will facilitate the proof of the verificationtheorem. Lemma 2. (1)
For V C defined in Proposition 5 or 7, sV Cs and xV Cx are bounded everywhere. (2) Suppose µ = ρ or µ = ρ > λ + r such that z ∗ > . Then V Cs sV Css and ( V Cs ) V Css are bounded on ≤ s ≤ xz ∗ .Proof. (1) In the case of µ = ρ ≤ r + λ we have a closed-form expression of V C as in Proposition 5where the desired results can be established easily.For the more general case where V C is defined in Proposition 7, on 0 ≤ s ≤ xz ∗ we have sV Cs = zg ′ ( z ) = h ′ ( u ) = w ( h ) ∈ [0 , q ∗ ] IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 24 and 1 xV Cx = 11 /β − zg ′ ( z ) = 11 /β − h ′ ( u ) = 11 /β − w ( h ) (32)which is bounded as w ( h ) = q ∈ [0 , q ∗ ] ⊂ [0 , /β ). Meanwhile, on s > xz ∗ V C equals β ln( s + x )plus a constant such that sV Cs and xV Cx are trivially bounded.(2) Similarly, V Cs sV Css = zg ′ ( z ) z g ′′ ( z ) = h ′ ( u ) h ′′ ( u ) − h ′ ( u ) = 1 w ′ ( h ) − /N ′ ( q ) − < q ≤ q ∗ as N ′ ( q ) is continuous, N ′ (0) = n ′ (0) β + 1 and n ′ (0) is non-zerofrom Proposition 4. Finally,[ V Cs ] V Css = [ zg ′ ( z )] z g ′′ ( z ) = [ h ′ ( u )] h ′′ ( u ) − h ′ ( u ) = w ( h ) w ′ ( h ) − q /N ′ ( q ) − < q ≤ q ∗ . (cid:3) We are now ready to prove Theorem 1.
Proof of Theorem 1.
To show that V C is indeed the value function, it is sufficient to show that V C issimultaneously an upper bound and a lower bound of V defined in (10).(1) In this case the candidate value function is given by (25). We first show that V ≤ V C whichrelies on a perturbation argument based on Davis and Norman (1990). Fix ǫ > V C ( s, x ) := V C ( s, x + ǫ ) such that ˜ V C is bounded below by V C (0 , ǫ ) = β ln ǫ + β h rβ + ln β − i .For an arbitrary admissible strategy ( c, π, Φ), let˜ M t := Z t h e − ( λ + β ) u ln c u + λF ( X u ) i du + e − ( λ + β ) t ˜ V C ( S t , X t ) . Since ˜ V C is C × , generalized Ito’s lemma gives˜ M t = ˜ M + Z t e − ( β + λ ) u ( ln c u − ˜ V Cx c u + r ˜ V Cx X u + ρ ˜ V Cs S u + ( µ − ρ ) ˜ V Cs S u π u + σ V Css S u π u − ( β + λ ) ˜ V C + λF ( X u ) ) du + Z t e − ( β + λ ) u ( ˜ V Cs − ˜ V Cx ) d Φ u + X υ ≤ t e − ( β + λ ) υ h ˜ V C ( S υ , X υ ) − ˜ V C ( S υ − , X υ − ) − ˜ V Cs ∆ S υ − ˜ V Cx ∆ X υ i + Z t e − ( β + λ ) u σπ u S u ˜ V Cs dB u =: ˜ M + N t + N t + N t + N t . (35)Consider a sequence of stopping times T n := inf { t > R t ( π u S u ˜ V Cs ) du ≥ n } under which thestopped local martingale N t ∧ T n = R t ∧ T n e − ( β + λ ) u σπ u S u ˜ V Cs dB u is a true martingale for each n . IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 25
Since R t π u du < ∞ and sV Cs (and in turn s ˜ V Cs ) is bounded as shown in part (1) of Lemma 2, wehave T n ↑ ∞ almost surely.Using Proposition 5, M ˜ V C ( s, x ) = V Cx ( s, x + ǫ ) − V Cs ( s, x + ǫ ) = M V C ( s, x + ǫ ) = 0and L ˜ V C = − ln V Cx ( s, x + ǫ ) − rxV Cx ( s, x + ǫ ) + ρsV Cs ( s, x + ǫ ) − ( β + λ ) f + λF ( x )= " − ln V Cx ( s, x + ǫ ) − r ( x + ǫ ) V Cx ( s, x + ǫ ) + ρsV Cs ( s, x + ǫ ) − ( β + λ ) f + λF ( x + ǫ ) − rǫV Cx ( s, x + ǫ ) − λ ( F ( x + ǫ ) − F ( x )) ≤ − rǫV Cx ( s, x + ǫ ) − λ ( F ( x + ǫ ) − F ( x )) ≤ V C solves L V C = 0 and the last inequalityis due to V C and F being both increasing in x . Thus N t ≤ Z t e − ( β + λ ) u L ˜ V C du ≤ , N t = Z t e − ( β + λ ) u M ˜ V C d Φ u = 0 . Moreover, the concavity property of V C is inherited by ˜ V C and as such N t ≤ t ∧ T n leads to E "Z t ∧ T n h e − ( λ + β ) u ln c u + λF ( X u ) i du + e − ( λ + β )( t ∧ T n ) ˜ V C ( S t ∧ T n , X t ∧ T n ) = E ( ˜ M t ∧ T n ) ≤ ˜ M = ˜ V C ( s, x ) . Sending n ↑ ∞ , monotone convergence and bounded convergence theorem give respectivelylim n ↑∞ E "Z t ∧ T n h e − ( λ + β ) u ln c u + λF ( X u ) i du = lim n ↑∞ E "Z t ∧ T n h e − ( λ + β ) u ln c u + λF ( X u ) i + du − lim n ↑∞ E "Z t ∧ T n h e − ( λ + β ) u ln c u + λF ( X u ) i − du = E (cid:20)Z t h e − ( λ + β ) u ln c u + λF ( X u ) i du (cid:21) andlim n ↑∞ E h e − ( λ + β )( t ∧ T n ) ˜ V C ( S t ∧ T n , X t ∧ T n ) i ≥ lim n ↑∞ E h e − ( λ + β )( t ∧ T n ) min( ˜ V C ( S t ∧ T n , X t ∧ T n ) , i = E h e − ( λ + β ) t min( ˜ V C ( S t , X t ) , i . Hence we obtain E (cid:20)Z t h e − ( λ + β ) u ln c u + λF ( X u ) i du + e − ( λ + β ) t min( ˜ V C ( S t , X t ) , (cid:21) ≤ ˜ V C ( s, x ) = V C ( s, x + ǫ ) . On letting t ↑ ∞ and then ǫ ↓
0, we can conclude E (cid:20)Z ∞ h e − ( λ + β ) u ln c u + λF ( X u ) i du (cid:21) ≤ V C ( s, x ) IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 26 and for any admissible ( c, π,
Φ) and thus V ( s, x ) ≤ V C ( s, x ).To show that V C ≤ V it is sufficient to demonstrate an admissible strategy which attains thecandidate value function. Consider the strategy of liquidating the firm by distributing the entireequity to investors in form of dividends, and then investors consume their private wealth at a rateof β . In other words, the candidate optimal dividend and consumption policy are given by Φ ∗ t = s and c ∗ t = βX ∗ t respectively for t ≥
0. The resulting wealth process X ∗ is thus the solution to dX ∗ t = ( r − β ) X ∗ t dt, X ∗ = s + x and hence X ∗ t = ( s + x ) e ( r − β ) t . The candidate optimal consumption policy can be written as c ∗ t = βX ∗ t = β ( s + x ) e ( r − β ) t . The resulting expected lifetime utility is E (cid:20)Z ∞ e − βt ln (cid:16) β ( s + x ) e ( r − β ) t (cid:17) dt (cid:21) = 1 β ln( s + x ) + 1 β (cid:20) rβ + ln β − (cid:21) = V C ( s, x ) . Thus V C ≤ V .(2) The proof of V ≤ V C is omitted since it is almost identical to part (1), except that N t ≤ N t ≤ V ≥ V C , we again want to demonstrate there exists an admissible strategy under which V C ( s, x ) = E (cid:2)R ∞ e − ( β + λ ) t [ln c t + λF ( X t )] dt (cid:3) . Suppose the initial value ( s, x ) is such that sx ≤ z ∗ . Define thefeedback controls c ∗ = ( c ∗ t ) t ≥ , π ∗ = ( π ∗ t ) t ≥ as in (8), and Φ ∗ := (Φ ∗ t ) t ≥ an adapted, local timestrategy which keeps S t X t ≤ z ∗ . By part (2) of Lemma 2, π ∗ ( s, x ) and c ∗ ( s, x ) /x are bounded andthus ( c ∗ , π ∗ ) is a pair of valid consumption/investment policy. Denote by ( S ∗ , X ∗ ) = ( S ∗ t , X ∗ t ) t ≥ the state variable processes evolving under these controls.Let M ∗ t := R t (cid:2) e − ( λ + β ) u ln c ∗ u + λF ( X ∗ u ) (cid:3) du + e − ( λ + β ) t V C ( S ∗ t , X ∗ t ) be the value process under( c ∗ , π ∗ , Φ ∗ ). Using part (4) of Proposition 7 and the fact that d Φ ∗ t = 0 on Z ∗ t := S ∗ t /X ∗ t ≤ z ∗ ,Ito’s lemma gives M ∗ t = M ∗ + R t σπ ∗ u S ∗ u V Cs dB u . By part (2) of Lemma 2 the integrand of thestochastic integral is bounded and thus it is a true martingale such that E (cid:20)Z t e − ( λ + β ) u (ln c ∗ u + λF ( X ∗ u )) du (cid:21) + E h e − ( λ + β ) t V C ( S ∗ t , X ∗ t ) i = E ( M ∗ t ) = M ∗ = V C ( s, x ) . (36)To show that ( c ∗ , π ∗ , Φ ∗ ) is admissible, we want to demonstrate that S ∗ t ≥ X ∗ t ≥ t and also T := inf { t ≥ S ∗ t , X ∗ t ) ∈ (0 , } = ∞ . The design of Φ ∗ immediately implies S ∗ t ≥ X ∗ t ≥
0. Applying Ito’s lemma directly to V C ( S ∗ t , X ∗ t ) gives V C ( S ∗ t , X ∗ t ) = V C ( s, x ) + Z t (cid:20) rV Cx X ∗ u + ρV Cs S ∗ u + ( µ − ρ ) V Cs S ∗ u π ∗ u + σ V ss ( π ∗ u S ∗ u ) − V x c ∗ u (cid:21) du + Z t ( V Cx − V Cs ) d Φ ∗ u + σ Z t π ∗ u V Cs S ∗ u dB u IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 27 = V C ( s, x ) + Z t (cid:2) − ln c ∗ u + ( β + λ ) V C − λF ( X ∗ u ) (cid:3) du + σ Z t π ∗ u V Cs S ∗ u dB u = V C ( s, x ) + Z t (cid:2) ln V Cx + ( β + λ ) V C − λF ( X ∗ u ) (cid:3) du − µ − ρσ Z t ( V Cs ) V Css dB u =: V C ( s, x ) + Z t f ( S ∗ u , X ∗ u ) du + Z t f ( S ∗ u , X ∗ u ) dB u (37)where we have used the fact that L V C = 0 on z ≤ z ∗ . But f ( s, x ) = ln V Cx + ( β + λ ) V C − λF ( x )= ln (cid:18) β − zg ′ ( z ) (cid:19) + ( β + λ ) g ( z ) − λβ (cid:18) rβ + ln β − (cid:19) = ln (cid:18) β − h ′ ( u ) (cid:19) + ( β + λ ) h ( u ) + rβ −
1= ln (cid:18) β − q (cid:19) + ( β + λ ) N ( q ) + rβ − n ( q )which is bounded on q ≤ q ∗ and f ( s, x ) is bounded as well by part (2) of Lemma 2. Now V C ( S ∗ t ∧ T , X ∗ t ∧ T ) = V C ( s, x ) + Z t ∧ T f ( S ∗ u , X ∗ u ) du + Z t ∧ T f ( S ∗ u , X ∗ u ) dB u and thus we must have T = ∞ because V C ( S ∗ T , X ∗ T ) = V C (0 ,
0) = −∞ but the integrands on theright hand side are bounded.(37) and the boundedness of f and f also allow us to deduce the transversality conditionlim t →∞ E (cid:2) e − ( β + λ ) t V C ( S ∗ t , X ∗ t ) (cid:3) = 0. Upon taking limit t → ∞ in (36) we can conclude V ( s, x ) ≥ E (cid:2)R ∞ e − ( λ + β ) u (ln c ∗ u + λF ( X ∗ u )) du (cid:3) = V C ( s, x ) on sx ≤ z ∗ .Finally, if the initial value ( s, x ) is such that sx > z ∗ , then consider a strategy of paying adiscrete dividend D ∗ = s − z ∗ x z ∗ at time zero such that the ex-dividend equity to private wealthratio is restored to z ∗ , and then follow the candidate optimal strategies ( c ∗ , π ∗ , Φ ∗ ) described inthe regime of sx ≤ z ∗ thereafter. By construction of V C on z > z ∗ , V C ( s, x ) = V C ( s − D ∗ , x + D ∗ ).Then (36) gives E (cid:20)Z t e − ( λ + β ) u (ln c ∗ u + λF ( X ∗ u )) du (cid:21) + E h e − ( λ + β ) t V C ( S ∗ t , X ∗ t ) i = V C ( s − D ∗ , x + D ∗ ) = V C ( s, x ) . and again we can conclude V ( s, x ) ≥ V C ( s, x ). (cid:3) Concluding remarks
We develop a continuous-time stochastic control model which jointly determines the optimal dividendpolicy and capital structure of a defaultable firm as well as the consumption strategy of its risk averseequity investors. We give a complete characterization of the solution to the problem. The optimal dividendpolicy is a local time strategy which keeps the ratio of the firm’s equity value to investors’ wealth below
IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 28 a target threshold. Comparative statics of economic importance are derived where the impact of defaultrisk on the corporate policies is highlighted. A firm facing a higher default risk has stronger incentive topay out dividends aggressively as a precautionary move to preserve value for investors against potentialdefault. To offset the negative effect on investment due to the shrunk equity basis, the firm adopts ahigher leverage level to boost its return. This feature can potentially be interesting to mainstream financeliterature as dividends and leverage decision of a firm now reflect its riskiness (default probability), andhence they could have important asset pricing implications.We have exclusively focused on the equity value evolution without considering the payoff to the bond-holders. In particular, the corporate yield ρ is a given constant. A possible variant of the current modelmay involve bondholders who understand the default probability of the firm and adjust the cost of debtaccordingly. An example of the modeling strategy can be found in Lambrecht and Tse (2018), whererisk-neutral bondholders charge a fair corporate yield as a function of the leverage level π and default risk λ . Our analysis can be extended in a similar fashion and this can potentially shed colors on the issuesof agency cost of debt and their impacts on the corporate policies although the analysis might then relymore heavily on numerical studies.Investors have logarithm utility function in the current model. A natural and tempting extensionof the model is to consider a more general power utility function such that the effect of risk aversioncan be investigated. Unfortunately, it appears difficult to apply the same set of transformation schemeto facilitate the analysis of the HJB system since the “bequest” term λF ( x ) now has a multiplicative(rather than additive) form and the resulting first order system n ′ = O ( q, n ) is much more complicated.Moreover, we also expect that under power utility function the issue of well-posedness will lead to extracomplications of the problem. Identification of the exact conditions under which well-posedness holdsfor stochastic control problems of this type has historically been a very difficult task. For example,since the rigorous formulation of the Merton consumption/investment problem under transaction costs byDavis and Norman (1990), it has taken more than two decades for the precise well-posedness conditionsto be established by Choi et al. (2013). A full generalization of the model in the current paper to powerutility function should prove to be a challenging open problem for future research.In our framework, the only outside investment option available for the investors is the retail savingaccount. Another possible direction of future research is to allow equity investors to also invest in anotherrisky market asset which can potentially be correlated to the risky asset of the firm. While it is expectedthat the extra dimension introduced will bring significant challenges to the analysis of the underlyingHJB equation, it is perhaps not an impossible task in view of the recent progress by Hobson et al. (2016) For utility function in form of u ( c ) = c − R / (1 − R ) where R ∈ (0 , ∞ ) \ { } is the risk aversion level, the parametercombination of R < β ≤ (1 − R ) r leads to an ill-posed problem since the deterministic optimal consumption problemunder such parameters is ill-posed and thus a version of the problem with the defaultable firm is also ill-posed. IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 29 who completely solve a multi-asset Merton problem with transaction costs (albeit the special case wheretransaction cost is only payable for one of the assets).
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Appendix
A bound of n . The following lemma will be useful when establishing the proofs related to thedependence of the optimal controls on the state variables.
Lemma 3.
Suppose µ = ρ and recall the notations introduced in Proposition 4. Let χ ( q ) := n ( q ) − ℓ ( q ) .Then χ ( q ) ≤ αq for ≤ q ≤ q ∗ where α is defined to be the positive root to the quadratic equation (24) .Proof. On differentiating both side of (22), setting q → χ ′′ (0+) = − β ( µ − ρ ) β + λ ) σ α χ ′′ (0+) − β β + λ which give χ ′′ (0+) = − β / ( β + λ )1 + β ( µ − ρ ) β + λ ) σ α < . Hence χ is concave near q = 0 and then χ must be initially lying below L ( q ) := χ ′ (0) q = αq .Write the ODE (22) as χ ′ ( q ) = ˆ O ( q, χ ( q )). Thenˆ O ( q, L ( q )) = ˆ O ( q, αq ) = β β + λ ( µ − ρ ) σ α − ββ + λ (cid:18) ρ − r + ( µ − ρ ) σ (cid:19) − β − βq (cid:18) βq − λβ + λ (cid:19) = β β + λ ( µ − ρ ) σ α − ββ + λ (cid:18) ρ − r + ( µ − ρ ) σ (cid:19) − λ + β β q − βq + λβλ + β IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 31 ≤ β β + λ ( µ − ρ ) σ α − ββ + λ (cid:18) ρ − r + ( µ − ρ ) σ (cid:19) + λβλ + β = A (0 , α + B (0) = α = L ′ ( q ) . Thus χ ( q ) can only downcross L ( q ) and from this we conclude χ ( q ) ≤ L ( q ) = αq . (cid:3) Dependence of the optimal controls on the state variables.
Proof of Proposition 1. (1) Recall from Proposition 4 that q ∗ ∈ (cid:16) β ( ρ − r − λ ) + ( ρ − r − λ ) + + λ , β (cid:17) . The result followsimmediately under the relationship z ∗ = βq ∗ − βq ∗ .(2) The result mainly follows from part (2) of Lemma 2 that π ∗ ( s, x ) = µ − ρσ (cid:20) − V s sV ss (cid:21) = µ − ρσ (cid:20) − zg ′ ( z ) z g ′′ ( z ) (cid:21) =: µ − ρσ θ ( z ) (38)such that θ ( z ) = − zg ′ ( z ) z g ′′ ( z ) = − /N ′ ( q ) which is positive, strictly larger than one and boundedsince N ′ is a positive function bounded away from 1 on q ∈ (0 , q ∗ ].It remains to show that θ ( z ) is decreasing in z which is equivalent to showing that N ′ ( q ) isincreasing in q . By definition of N , we can obtain N ′′ ( q ) = n ′′ ( q ) β + β (1 − βq ) . We work out thesecond derivative of n as n ′′ ( q ) = ddq O ( q, n ( q )) = ddq (cid:20) β q − βq m ( q ) − n ( q ) n ( q ) − ℓ ( q ) (cid:21) = ddq (cid:26) β q − βq (cid:20) m ( q ) − ℓ ( q ) n − ℓ ( q ) − (cid:21)(cid:27) = β β + λ ( µ − ρ ) σ [ n ( q ) − ℓ ( q )] − q [ n ′ ( q ) − ℓ ′ ( q )][ n ( q ) − ℓ ( q )] − β (1 − βq ) and hence N ′′ ( q ) = ββ + λ ( µ − ρ ) σ [ n ( q ) − ℓ ( q )] − q [ n ′ ( q ) − ℓ ′ ( q )][ n ( q ) − ℓ ( q )] = ββ + λ ( µ − ρ ) σ χ ( q ) − qχ ′ ( q )[ χ ( q )] where χ ( q ) := n ( q ) − ℓ ( q ) as introduced in the proof of Proposition 4. Hence to show that N ′ isincreasing it is necessary and sufficient to show that χ ′ ( q ) ≤ χ ( q ) q for all 0 < q ≤ q ∗ .Suppose on contrary that there exists ¯ q ∈ (0 , q ∗ ] such that χ ′ (¯ q ) > χ (¯ q )¯ q =: ¯ α . From Lemma 3, χ ( q ) ≤ αq for q ∈ (0 , ¯ q ] where α := χ ′ (0) and thus we must have ¯ α ≤ α . Let L ( q ) := ¯ αq . Thenwe have L ′ (¯ q ) = ¯ α < χ ′ (¯ q ) and hence χ ( q ) upcrosses L ( q ) at q = ¯ q .Since χ ′ (0) = α > ¯ α , χ ( q ) must initially be large than L ( q ) for q near zero. Hence there mustexist some ˜ q < ¯ q where χ downcrosses L at q = ˜ q . But, recall the definition of ˆ O in (22),¯ α < χ ′ (¯ q ) = ˆ O (¯ q, χ (¯ q )) = ˆ O (¯ q, ¯ α ¯ q )= β β + λ ( µ − ρ ) σ ¯ α − ββ + λ (cid:18) ρ − r + ( µ − ρ ) σ (cid:19) − β − β ¯ q (cid:18) β ¯ q − λβ + λ (cid:19) = β β + λ ( µ − ρ ) σ ¯ α − ββ + λ (cid:18) ρ − r + ( µ − ρ ) σ (cid:19) − λ + β β ¯ q − β ¯ q + λβλ + β IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 32 < β β + λ ( µ − ρ ) σ ¯ α − ββ + λ (cid:18) ρ − r + ( µ − ρ ) σ (cid:19) − λ + β β ˜ q − β ˜ q + λβλ + β = ˆ O (˜ q, ¯ α ˜ q ) = ˆ O (˜ q, χ (˜ q )) = χ ′ (˜ q ) . But this contradicts the fact that χ downcrosses L at q = ˜ q .(3) The result is immediate from (32) where c ∗ ( s, x ) x = 11 /β − zg ′ ( z ) = 11 /β − q . (39)The above is increasing in q and in turn z and it tends to β as q ↓ z ↓ (cid:3) Proof of Proposition 2. q ∗ = ρ − r − λβ ( ρ − r ) under µ = ρ > λ + r and hence z ∗ = βq ∗ − βq ∗ = ρ − r − λλ . π ∗ = 0 is trivialwhen µ = ρ . Finally, the expression of c ∗ ( s, x ) can be established using (39) and the fact that n = m when µ = ρ > λ + r . Thus (29) in Lemma 1 can be further simplified to z = z ( q ) = βq − βq exp − Z ρ − r − λβ ( ρ − r ) q m ′ ( v ) βv dv ! = βq − βq exp " − Z ρ − r − λβ ( ρ − r ) q β + λ (cid:18) ρ − rv − λv (1 − βv ) (cid:19) dv (40)and we can arrive at (9). (cid:3) Comparative statics.
Proof of Proposition 3.
In the case of µ = ρ > λ + r , z ∗ = ρ − r − λλ and π ∗ = 0 and hence their comparativestatics are trivial. While it may be less trivial to deduce the comparative statics of c ∗ ( s, x ) directly from(9), one trick is to observe its integral form as in (40) to deduce that z = z ( q ) is increasing in λ and r . As z = z ( q ) is an increasing bijection, its inverse function q = q ( z ) is decreasing in λ and r . Thismonotonicity is then inherited by c ∗ ( s, x ) = x /β − q ( s/x ; λ,r ) .Now we proceed to give the proof in the general case of µ = ρ :(1) Recall that the transformed value function n is the solution to the ODE n ′ = O ( q, n ) where O is defined in Proposition 4, and the transformed dividend payment boundary is given by q ∗ :=inf { q > n ( q ) ≥ m ( q ) } .Let b ( q ) := β + λβ ( m ( q ) − n ( q )). Then the ODE becomes m ′ ( q ) − ββ + λ b ′ ( q ) = O (cid:16) q, m ( q ) − ββ + λ b ( q ) (cid:17) which can be written as b ′ ( q ) = ρ − r − λ − βq − ( β + λ ) βq − βq b ( q ) ( µ − ρ ) σ (cid:16) β − q (cid:17) − b ( q ) =: P ( q, b ( q )) (41)subject to initial condition b (0) = β + λβ ( m (0) − ℓ (0)) = ( µ − ρ ) βσ . The dividend payment boundarycan now be expressed as q ∗ := inf { q > b ( q ) ≤ } . Note that P ( q, b ) is decreasing in λ for as IVIDEND POLICY AND CAPITAL STRUCTURE OF A DEFAULTABLE FIRM 33 long as 0 ≤ ( µ − ρ ) σ (cid:16) β − q (cid:17) which must be satisfied along the solution trajectory b = ( b ( q )) ≤ q ≤ q ∗ since the transformed value function n = n ( q ) always lies between m ( q ) and ℓ ( q ) on [0 , q ∗ ].Consider λ hi > λ lo and denote by b hi (resp. b lo ) the solution to the ODE b ′ = P ( q, b ( q ); λ hi )(resp. b ′ = P ( q, b ( q ); λ lo )) with initial condition b (0) = ( µ − ρ ) βσ . Since P ( q, b ; λ hi ) < P ( q, b ; λ lo ), b hi can only downcross b lo . Thus b hi is dominated by b lo at least up to min( q ∗ hi , q ∗ lo ), where q ∗ hi := inf { q > b hi ( q ) ≤ } (and q ∗ lo is defined similarly). Hence we must have q ∗ hi < q ∗ lo fromwhich we conclude q ∗ and in turn z ∗ = βq ∗ − βq ∗ are both decreasing in λ .The exact same argument can be used to establish the comparative statics of z ∗ with respectto µ , σ and r . From (41) it is easy to see that P ( q, b ) is increasing in ( µ − ρ ) σ (while keeping allthe other parameters fixed) and is decreasing in r . The result follows immediately.(2) From (38), ( µ − ρ ) π ∗ ( s, x ) ∝ − /N ′ ( q ) and hence to show that ( µ − ρ ) π ∗ is increasing in λ itis sufficient to show that N ′ ( q ) = N ′ ( q ( z ; λ ); λ ) is decreasing in λ . Using the substitution of b ( q ) := β + λβ ( m ( q ) − n ( q )) again, we have N ′ ( q ; λ ) = n ′ ( q ; λ ) β + 11 − βq = m ′ ( q ; λ ) − ββ + λ b ′ ( q ; λ ) β + 11 − βq = βq − βq b ( q ; λ ) ( µ − ρ ) σ (cid:16) β − q (cid:17) − b ( q ; λ ) + 11 − βq and thus N ′ ( q ; λ ) is decreasing in λ under a fixed q as b ( q ; λ ) is decreasing in λ as shown in part(1) of this proof. Then ddλ N ′ ( q ( λ ); λ ) = ∂∂λ N ′ ( q ( λ ); λ ) + ∂∂q N ′ ( q ( λ ); λ ) ∂∂λ q ( λ ) < q ( λ ) is decreasing in λ and N ′ ( q ) is increasing in q as already shown in part (1) of this proofand part (2) of the proof of Proposition 1 respectively. Similarly, we can show that ( µ − ρ ) π ∗ isincreasing in r .(3) From (39) the optimal consumption rate per unit wealth is given by c ∗ ( s,x ) x = /β − q . We firstshow that the expression is decreasing in λ which is equivalent to showing that q = q ( z ; λ ) isdecreasing in λ .Using (29) and the substitution of b ( q ) := β + λβ ( m ( q ) − n ( q )) again, we have z = z ( q ; λ ) = βq − βq exp − Z q ∗ ( λ ) q β − βv b ( v ; λ ) ( µ − ρ ) σ (cid:16) β − v (cid:17) − b ( v ; λ ) dv . We have shown in part (1) of the proof that both q ∗ ( λ ) and b ( · ; λ ) are decreasing in λ . Hence z = z ( q ; λ ) is increasing in λ . As z = z ( q ; λ ) is increasing in q , q = q ( z ; λ ) is decreasing in λ .Hence the result follows. Using the exact same argument, it can be shown that q = q ( z ; r ) and inturn c ∗ ( s,x ) x are decreasing in r ..