Capital Investment and Liquidity Management with collateralized debt.


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1 TSE 54 Novembe 14 Capital Investment and Liquidity Management with collatealized debt. Ewan Piee, Stéphane Villeneuve and Xavie Wain 7
2 Capital Investment and Liquidity Management with collatealized debt. Ewan Piee Stéphane Villeneuve Xavie Wain Abstact: This pape examines the dividend and investment policies of a cash constained fim that has access to costly extenal funding. We depat fom the liteatue by allowing the fim to issue collatealized debt to incease its investment in poductive assets esulting in a pefomance sensitive inteest ate on debt. We fomulate this poblem as a bidimensional singula contol poblem and use both a viscosity solution appoch and a veification technique to get qualitative popeties of the value function. We futhe solve quasiexplicitly the contol poblem in two special cases. Keywods: Investment, dividend policy, singula contol, viscosity solution JEL Classification numbes: C61; G35. MSC Classification numbes: 6G4; 91G5; 91G8. 1 Intoduction In a wold of pefect capital maket, fims could finance thei opeating costs and investments by issuing shaes at no cost. As long as the net pesent value of a poject is positive, it will find investos eady to supply funds. This is the cental assumption of the Modigliani and Mille theoem [17. On the othe hand, when fims face extenal financing costs, these costs geneate a pecautionay demand fo holding liquid assets and etaining eanings. This depatue fom the ModiglianiMille famewok has eceived a lot of attention in ecent yeas and has given bith to a seie of papes explaining why fims hold liquid assets. Pioneeing papes ae Jeanblanc and Shiyaev [13, Radne and Shepp [1 while moe ecent studies include Bolton, Chen and Wang [4, Décamps, Maiotti, Rochet and Villeneuve [7 and Hugonnie, Malamud and Moellec [1. In all of these papes, it is assumed that fims have only access to extenal equity capital. Should it un out of liquidity, the fim eithe liquidates o aises new funds in ode to continue opeations by issuing equity. This binay decision only depends on the seveity of issuance costs. Even if ou wok is elated to these ecent papes that incopoate financing fictions into dynamic models of copoate finance to explain why fims hold lowyielding cash eseves, the model we develop hee depats fom the liteatue above in the following way. In this EDF R&D OSIRIS. Toulouse School of Economics (CRMIDEI, Manufactue des Tabacs, 1, Allée de Bienne, 31 Toulouse, Fance. EDF R&D & FiME, Laboatoie de Finance des Machés de l Enegie ( 1
3 pape, we assume that the fim can aise new funds at any time only by issuing collatealized debt. The collatealized debt continuously pays a vaiable coupon indexed on the fim s outstanding debt that chages a highe inteest ate as the fim s debt inceases with the covenant that debt value cannot exceed the value of total assets: liquid and poductive. The collatealized debt poposed in this pape is somehow simila to the pefomancesensitive debt studied in [16 except that the shaeholdes ae hee foced to go bankupt when they ae no moe able to collatealize a loan with thei assets. Theefoe, the liability side of the balance sheet of the fim consists in two diffeent types of ownes: shaeholdes and debtholdes. Should the fim be liquidated, the debtholdes have senioity ove shaeholdes on the total assets. Many models initiated by Black and Cox [3 and Leland [14 that conside the taditional tadeoff between tax and bankuptcy costs as an explanation fo debt issuance study fims liabilities as contingent claims on its undelying assets, and bankuptcy as an endogenous decision of the fim management. On the othe hand, these models assume costless equity issuance and thus put aside liquidity poblems. As a consequence, the fim s decision to boow on the cedit maket is independent fom liquidity needs and investment decisions. Ou model belongs to the class of models that conside endogenous bankuptcy but takes the opposite viewpoint by examining debt issuance of a cashconstained fim in a model without tax. Fom a mathematical point of view, poblems of cash management have been fomulated as singula stochastic optimal contol poblems. As efeences fo the theoy of singula stochastic contol, we may mention the pioneeing woks of Haussman and Suo [9 and [1 and fo application to cash management poblems Højgaad and Taksa [11, Asmussen, Højgaad and Taksa [1, Choulli, Taksa and Zhou [5, Paulsen [19 among othes. To mege copoate liquidity, capital investment and debt financing in a tactable model is challenging because it involves a athe difficult theedimensional singula contol poblem with stopping whee the state vaiables ae the book value of equity, the size of poductive asset and the outstanding debt while the stopping time is the decision to default. The liteatue on multidimensional contol poblems elies mainly on the study of leading examples. A seminal example is the socalled finitefuel poblem intoduced by Benes, Shepp and Witsenhausen [. This pape povides a ae example of a bidimensional optimization poblem that combines singula contol and stopping that can be solved explicitly by analytical means. Moe ecently, Fedeico and Pham [8 have solved a degeneate bidimensional singula contol poblem to study a evesible investment poblem whee a social planne aims to contol its capacity poduction in ode to fit optimally the andom demand of a good. Ou pape complements the pape by Fedeico and Pham [8 by intoducing fims that ae cashconstained and indebted 1. To ou knowledge, this is the fist time that such a combined appoach is used. This makes the poblem much moe complicated and we do not petend solving it with full geneality, but athe, we pave the way fo futue developments of these multidimensional singula contol models. In paticula, we lose the global convexity popety of the value function that leads to the necessay smoothfit popety in [8 (see Lemma 8. Instead, we will give popeties of the value function (see Poposition 6 and chaacteize it by means of viscosity solution (see Theoem 1. Futhemoe, we will solve explicitly by a standad veification agument the peculia case of costless evesible 1 Ly Vath, Pham and Villeneuve [15 have also studied a evesible investment poblem in two altenative technologies fo a cashconstained fim that has no access to extenal funding
4 investment. A new esult is ou chaacteization of the endogenous bankuptcy in tems of the pofitability of the fim and the coupon ate. The emainde of the pape is oganized as follows. Section intoduces the model with a poductive asset of fixed size, fomalizes the notion of collatealized debt and defines the shaeholdes value function. Futhemoe, it descibes the optimal debt issuance and gives the analytical chaacteization in tems of fee bounday poblems. Section 3 built the value function by solving explicitly the fee bounday poblem associated to the contol poblem. Section 4 extends the analysis to the case of evesible investment on poductive assets. The Model We conside a fim with a poductive asset of fixed size K that is chaacteized at each date t by the following balance sheet: K X t M t L t K epesents the fim s poductive assets, assumed to be constant and nomalized to one. M t epesents the amount of cash eseves o liquid assets. L t epesents the volume of outstanding debt. Finally, X t epesents the book value of equity. The poductive asset continuously geneates cashflows ove time. The cumulative cashflows pocess R = (R t t is modeled as an aithmetic Bownian motion with dift µ and volatility σ which is defined ove a complete pobability space (Ω, F, P equipped with a filtation (F t t. Specifically, the cumulative cashflows evolve as dr t = µ dt + σ db t whee (B t t is a standad onedimensionnal Bownian motion with espect to the filtation (F t t. We allow the fim to incease its cash eseves o cove losses by aising funds in the capital maket. In this pape, we assume that fim can issue collatealized loans which enable it to boow against the value of its assets. Moeove, we assume that the inteest payments ae vaiable and modelled by a function α depending only on the volume of debt the fim has issued. The extension to the case of vaiable size will be studied in Section 4 3
5 Assumption 1 α is a stictly inceasing, continuously diffeentiable convex function. Futhemoe, it is assumed that the collatealized debt is isky, that is In this famewok, the cash eseves evolve as x, α (x and α( = (1 dm t = (µ α(l t dt + σdb t dz t + dl t ( whee (Z t t is an inceasing ightcontinuous (F t t adapted pocess epesenting the cumulative dividend payment up to time t and (L t t is a positive ightcontinuous (F t t adapted pocess epesenting the outstanding debt at time t. Using the accounting elation 1 + M t = X t + L t, we deduce the dynamics fo the book value of equity dx t = (µ α(l t dt + σdb t dz t. (3 Assumption The cash eseves must be non negative and the fim management is foced to liquidate when the book value of equity hits zeo. Using the accounting elation, this is equivalent to assume the debtholdes get back all the assets afte bankuptcy. The goal of the management is to maximize shaeholdes value which is defined as the expected discounted value of all futue dividend payout. Shaeholdes ae assumed to be iskneutal and futue cashflows ae discounted at the iskfee ate. The fim can stop its activity at any time by distibuting all of its assets to stakeholdes. Thus, the objective is to maximize ove the admissible contol π = (L, Z the functional ( τ V (x, l; π = E x,l e t dz t whee τ = inf{t, X π t } accoding to Assumption. Hee x (esp. l is the initial value of equity capital (esp. debt. We denote by Π the set of admissible contol vaiables and define the shaeholdes value function by V (x, l = sup V (x, l; π (4 π Π.1 Optimal debt issuance The shaeholdes optimization poblem (4 involves two state vaiables, the value of equity capital X t and the value of debt L t, making its esolution difficult. Fotunately, the next poposition will enable us to educe the dimension and make it tactable the computation of V. Poposition 1 shows that debt issuance is only optimal when the cash eseves ae depleted. Poposition 1 A necessay and sufficient condition to hold debt is that the cash eseves ae depleted, that is t R +, L t M t = o equivalently L t = (1 X t + 4
6 Poof: Fist, by Assumption, it is clea that the fim management must issue debt when cash eseves ae nonpositive. Convesely, assume that the level of cash eseves m is stictly positive. We will show that it is always bette off to educe the level of outstanding debt by using the cash eseves. We will assume that the initial value of debt is l > and intoduce π t = (L t, Z t any admissible stategy. We fist assume that L = l. Because m >, we will built a stategy fom π as follows: { L = l fo < < min(m, l and L t L t, Z t = Z t + t (α(l s α(l s ds Note that the debt issuance stategy L consists in always having less debt that unde the debt issuance stategy L and because α is inceasing, the dividend stategy Zt pays moe than the dividend stategy Z t. Futhemoe, denoting by π = (L t, Zt, the bankuptcy time unde π stating fom (x, l and the bankuptcy time unde π stating fom (x, l ae the same stopping time. Theefoe, we will show that it is bette off to follow π than π. ( τ π V (x, l; π = E (x,l e s dzs ( τ π > E (x,l = E (x,l ( τ π = V (x, l; π e s dz s e s dz s So if m > l, it is optimal to set l = by using m l units of cash eseves while if m < l, it is optimal to educe the debt to l m. In any case, at any time L t = (1 X t +. Now, if we have L, two cases have to be consideed. L = which is possible only if m > l. In that case, we set L t = L t and Z t = Z t fo t >. L >. In that case, we take the same stategy π with < < min(m, l + L. Poposition 1 implies the following dynamics fo the book value of equity dx t = (µ α((1 X t + dt + σdb t dz t, X( = x (5 We thus define the value function as v (x = V (x, (1 x +. (6 5
7 . Analytical Chaacteization of the optimal policy of the fim Because the level of capital is assumed to be constant, Poposition 1 makes ou contol poblem onedimensional. Thus, we will follow a veification pocedue to chaacteize the value function in tems of a fee bounday poblem. We denote by L the diffeential opeato: LΦ = (µ α((1 x + Φ (x + σ Φ (x Φ (7 We stat by poviding a standad esult which establishes that a smooth solution to a fee bounday poblem coincides with the value function v. Poposition Assume thee exists a C 1 and piecewise twice diffeentiable function w on (, + togethe with a pai of constants (a, b R + R + such that, x a, Lw and w(x = x a x b, Lw = and w (x 1 x > b, Lw and w (x = 1 (8 then w = v. with w (b = (9 Poof: Fix a policy π = (Z Π. Let : dx t = (µ α((1 X t + dt + σdb t dz t, X( = x be the dynamic of the book value of equity unde the policy π. Let us decompose Z t = Zt c + Z t fo all t whee Zt c is the continuous pat of Z. Let τ the fist time when X t =. Using the genealized Itô s fomula, we have : e (t τ w(x t τ = w(x + t τ t τ e s Lw(X s ds + t τ e s w (X s dzs c + e s [w(x s w(x s s t τ σe s w (X s db s Because w is bounded away fom zeo, the thid tem is a squae integable matingale. Taking expectation, we obtain [ t τ w(x = E x [e (t τ w(x t τ E x e s Lw(X s ds [ t τ + E x e s w (X s dzs c [ E x e s [w(x s w(x s s t τ 6
8 Because w 1, we have w(x s w(x s X s = Z s theefoe the thid and the fouth tems ae bounded below by ( t τ E x e s w (X s dz s. Futhemoe w is positive because w is inceasing with w( = and Lw thus the fist two tems ae positive. Finally, ( t τ ( t τ w(x E x e s w (X s dz s E x e s dz s Letting t + and we obtain w(x v (x. To show the evese inequality, we will pove that thee exists an admissible stategy π such that w(x = v(x, π. Let (X t, Z t be the solution of whee, with X t = t Z t = (x 1 {x a} + (x b + 1 {t= } + (µ α((1 X s + ds + σb t Z t (1 t τ a τ a = inf{t, X t a} 1 {X s =b}dz s + a1 {t τa} (11 whose existence is guaanteed by standad esults on the Skookhod poblem (see fo example Revuz and Yo [. The stategy π = (Zt is admissible. Note also that Xt is continuous on [, τa. It is obvious that v(x, π = x = w(x fo x a. Now suppose x > a. Along the policy π, the liquidation time τ coincides with τ a because Xτ a =. Poceeding analogously as in the fist pat of the poof, we obtain [ w(x = E x e (t τ w(xt τ [ t τ + E x [ = E x e (t τ w(xt τ [ t τ + E x e s w (X s dz s e s w (bdz s [ = E x e (t τ w(xt τ [ t τ + E x e s dzs, + E x [ 1t>τ e τ (w(x τ w(x τ + E x [ 1t>τ e τ a whee the last two equalities uses, w(a = a w (b = 1 and ( Z τ = a. Now, because w( =, [ E x e (t τ w(xt τ [ = E x e t w(xt 1 t τ. Futhemoe, because w has at most linea gowth and π is admissible, we have lim E [ x e t w(xt 1 t τ =. t 7
9 Theefoe, we have by letting t tend to +, [ τ w(x = E x e s dzs = v(x, π which concludes the poof. Remak 1 We notice that the poof emains valid when a = and w ( is infinite which will be the case in section 4. The veification theoem allows us to chaacteize the value function when the fim pofitability is lowe than the inteest ate. Coollay 1 If µ then it is optimal to liquidate the fim, v (x = x. Poof: We will show that the function w(x = x satisfies Poposition. To see this, we have to show that Lw(x is nonpositive fo any x. A staightfowad computation gives fo x > 1, Lw(x = µ x < µ, fo x 1, Lw(x = µ α(1 x x, Using Equation (1 of Assumption 1, we obseve that Lw(x is nondeceasing fo x 1 and nonpositive at x = 1 when µ. Heeafte, we will assume that µ >. 3 Solving the fee bounday poblem We will now focus on the existence of a function w and a pai of constants (a, b satisfying Poposition. We will poceed in two steps. Fist we ae going to establish some popeties of the solutions of the diffeential equation Lw =. Second, we will conside two diffeent cases one whee the poductivity of the fim is always highe than the maximal inteest payment α(1 µ, the othe whee the inteest payment of the loan may exceed the poductivity of the fim α(1 > µ. Standad existence and uniqueness esults fo linea secondode diffeential equations imply that, fo each b, the Cauchy poblem : w(x = (µ α((1 x + w (x + σ w (x w (b = 1 (1 w (b = has a unique solution w b ove [, b. By constuction, this solution satifies w b (b = µα((1b+. Extending w b linealy to [b, [ as w b (x = x b + µα((1b+, fo x b yields a twice continuously diffeentiable function ove [, [, which is still denoted by w b. 8
10 3.1 Popeties of the solution to the Cauchy Poblem We will establish a seie of peliminay esults of the smooth solution w b of (1. Lemma 1 Assume b > 1. If w b ( = then w b is inceasing and thus positive. Poof: Because w b ( =, w b (b = µ and Lw b =, the maximum pinciple implies w b > on (, +. Let us define c = inf{x >, w b(x = } If c = then w b ( = w b ( = w b ( =. By unicity of the Cauchy poblem, this would imply w b = which contadicts w b (b = µ. Thus, c >. If c < b, we would have w b(c >, w b (c = and w b (c and thus Lw b(c < which is a contadiction. Theefoe w b is always positive. Lemma Assume b > 1. We have w b > 1 and w b < on [1, b[. Poof: Because w b is smooth on 1, b, we diffeentiate Equation (1 to obtain, w b (b = σ > As w b (b = and w b (b = 1, it follows that w b <, and thus w b > 1 ove some inteval b, b[, whee >. Now suppose by way of contadiction that w b (x 1 fo some x [1, b and let x = sup{x [1, b, w b (x 1}. Then w b ( x = 1 and w b (x > 1 fo x x, b[, so that w b (b w b (x > b x fo all x x, b[. Because w b (b = µ, this implies that fo all x x, b[, w b (x = σ [w b(x µw b(x < σ [(x b + w b(b µ = (x b < σ which contadicts w b (b = w b ( x = 1. Theefoe w b Lemma 1, > 1 ove [1, b[. Futhemoe, using w b (x = σ [w b(x µw b(x < σ [w b(x µ < σ [w b(b µ =. The next esult gives a sufficient condition on b to ensue the concavity of w b on (, b. Coollay Assume b α(1 and µ α(1, we have w b > 1 and w b < ove, b[. 9
11 Poof: Poceeding analogously as in the poof of Lemma, we define x = sup{x [, b, w b (x 1} such that w b ( x = 1 and w b (x > 1 fo x x, b[, so that w b(b w b (x > b x fo all x x, b[. Because b > α(1 > 1, w b (b = µ, we have Denote by g the function w b (x = σ [w b(x (µ α((1 x + w b(x < σ [(x b + w b(b (µ α((1 x + < σ [(x b + α((1 x+ g(x = [(x b + α(1 x, x [, 1[. σ We have g (x = [ α (1 x < by Assumption 1. Because g( = [b + α(1 σ σ if b α(1, we have w b (x < fo x, 1 which contadicts w b ( x = 1 and w b (1 > 1 by Lemma. Theefoe w b > 1 ove [, 1[, fom which it follows w b < and w b is concave on, 1[. Because Lemma gives the concavity of w b on [1, b[, we conclude. The next poposition establishes some esults about the egulaity of the function b w b (y fo a fixed y [, 1[. Lemma 3 Fo any y [, 1[, b w b (y is an inceasing function of b ove [y, 1 and stictly deceasing ove 1, + [. Poof: Conside the solutions H y and H y 1 to the linea secondode diffeential equation LH = ove [y, [ chaacteized by the initial conditions H y (y = 1, (H y (y =, H y 1 (y =, (H y 1 (y = 1. We fist show that (H y and (H y 1 ae stictly positive on y, [. Because H y (y = 1 and (H y (y =, one has (H y (y = >, such that σ (H y (x > ove some inteval y, y + [ whee >. Now suppose by way of contadiction that x = inf{x y +, (H y (x } <. Then (H y ( x = and (H y ( x. Because LH y =, it follows that H y ( x, which is impossible because H y (y = 1 and H y is stictly inceasing ove [y, x. Thus (H y > ove y; [, as claimed. The poof fo H y 1 is simila, and is theefoe omitted. Next, let W H y,hy 1 (y = 1 and W H y,hy 1 = H y (H y 1 H y 1 (H y be the Wonskian of H y and H y 1. One has x y, W H y,hy 1 (x =H y (x(h y 1 (x H y 1 (x(h y (x = σ [Hy (x(h y 1 (x (µ α((1 x + (H y 1 (x H y 1 (x(h y (x (µ α((1 x + (H y (x = [µ α((1 x+ W σ H y (x,hy 1 1
12 Because α is integable, the Abel s identity follows by integation: [ ( x x y, W H y (x = exp µ(x y + α((1 u + du,hy 1 σ Because W H y,hy 1 >, Hy and H y 1 ae linealy independent. As a esult of this, (H y, H y 1 is a basis of the twodimensional space of solutions to the equation LH =. It follows in paticula that fo each b >, on can epesent w b as : x [y, b, w b (x = w b (yh y (x + w b(yh y 1 (x Using the bounday conditions w b (b = µα((1b+ and w b (b = 1, on can solve fo w b(y as follows: w b (y = (Hy 1 (b µα((1b+ H y 1 (b W H y (b,hy 1 Using the deivative of the Wonskian along with the fact that H y 1 is solution to LH =, it is easy to veify that: b [y, 1[, dw b(y db b 1, [, dw b(y db = ( (H y 1 (b α (1b + 1 W H y (b,hy 1 = (Hy 1 (b W H y (b,hy 1 So w b (y is an inceasing function of b ove [y, 1 and stictly deceasing ove 1, [. Coollay 3 If b > b 1 > 1, then w b < w b1. y Poof: Let us define W = w b1 w b. Clealy, W > on [b, + [. Moeove, we have LW = on [, b 1 and W ( > by Lemma 3. Moeove, w b1 (b 1 = w b (b and w b (b > w b (b 1 by Lemma. Theefoe, the maximum pinciple implies w b < w b1 on [, b 1. Finally, w b is concave and w b (b = 1 theefoe fo b 1 x b, w b (x w b (b + x b = µ + x b < µ + x b 1 = w b1 (x. 3. Existence of a solution to the fee bounday poblem We ae now in a position to chaacteize the value function and detemine the optimal dividend policy. Two cases have to be consideed: when the pofitability of the fim is always highe than the maximal inteest payment (µ α(1 and when the inteest payment exceeds the pofitability of the fim (µ < α(1. 11
13 3..1 Case: µ α(1 The next lemma establishes the existence of a solution w b to the Cauchy poblem (1 such that w b ( =. Lemma 4 Thee exists b 1, µ [ such that the solution to (1 satisfies w b ( =. Poof: Because µ α(1, we know fom Coollay that w µ is a concave function on [, µ. Moeove, because µ >, w µ ( µ = µ. Because w µ is stictly concave ove, µ [ with w µ ( µ = µ and w µ = 1, w µ (x x fo all x, µ [. In paticula, w µ ( <. Moeove, we have : w ( = µ α(1. Theefoe, Lemma 3 implies w 1 ( >. Finally by continuity thee is some b 1, µ [ such that w b ( = which concludes the poof. The next lemma establishes the concavity of w b. Lemma 5 The function w b is concave on [, b Poof: Because b > 1, Lemma implies that w b is concave on [1, b thus w b (1. Fo x < 1, we diffeentiate the diffeential equation satisfied by w b to get, σ w b (x + (µ α(1 xw b (x + (α (1 x w b (x = (13 Because w b ( = we have w b ( = σ (µ α(1w b (. Now, suppose by a way of contadiction that w b > on some subinteval of [, 1. Because is continuous and nonpositive at the boundaies of [, 1, thee is some c such that w b w b (c = and w (c >. But, this implies b w b which is a contadiction with Lemma 1. α(1 cw b (c = (µ (c < α (1 c Poposition 3 If µ α(1, w b is the solution of the contol poblem (9. Poof: Because w b is concave on [, b and w (b = 1, w 1 on [, b. Theefoe we have a twice continuously diffeentiable concave function w b and a pai of constants (a, b = (, b satisfying the assumptions of Poposition and thus w b = v. Figue 1 plots some value functions, when µ α(1, using a linea function fo α, α(x = λx with diffeent values of λ. Remak When the maximal inteest payment is lowe than the fim pofitability, the value function is concave. This illustates the shaeholdes fea to liquidate a pofitable fim. In paticula, the shaeholdes value is a deceasing function of the volatility. 1
14 Figue 1: Compaing shaeholdes value functions with µ =, =.5, σ = 1 and µ α(1 fo diffeent values of λ whee α(x = λx. 3.. Case: µ < α(1 We fist show that, fo all y [, 1[, thee exists b y such that w by Cauchy Poblem (1 with w by (y = y. is the solution of the Lemma 6 Fo all y [, 1[, we have w 1 (y > y. Poof: Because ( α is continuous with α( = and µ >, thee exists such that w 1 (1 = µα( > 1. Diffeentiating Equation (1, we obseve w 1(1 = σ ( α ( < using Equation (1. Theefoe w 1 is convex in a left neighbohood of 1. If w 1 is convex on (, 1 then w 1 (x x (1 + µα( > x fo small enough and the esult is poved. 13
15 If w 1 is not convex on (, 1 then it will exist some x < 1 such that w 1( x =, w 1( x > and w 1 convex on x, 1. Diffeentiating Equation (1 at x gives w 1( x <. Theefoe w 1 is noninceasing in a neighbohood of x. Assume by a way of contadiction that w 1 is inceasing at some point ˆx [, x[. This would imply the existence of x < x such that w 1( x =, w 1( x < and w 1 ( x > which contadicts Equation (1. Theefoe w 1 is deceasing on (, x and convex on ( x, 1 which implies that w 1 (x > x fo all x 1. To conclude, fo any y < 1, we can find small enough to have w 1 (y > y which can be extended to w 1 (y > y by Lemma 3. Coollay 4 Fo all y [, 1[, thee is an unique b y 1, 1 + µ [ such that w b y (y = y. Poof: By Lemma 1, w 1+ µ is concave on 1, 1 + µ [, thus w 1+ µ (1 < µ + (1 (1 + µ =. Suppose that thee exists c in [, 1[ such that w 1+ µ (c >, then thee exists x c, 1[ such that w 1+ µ ( x <, w 1+ ( x =, w µ 1+ ( x > yielding to the standad contadiction with µ the maximum pinciple. We thus have w 1+ µ (y < y fo all y [, 1 + µ. Using Lemma 6 and the continuity of the function b w b (y, it exists fo all y < 1 a theshold b y 1, 1 + µ [ such that w by (y = y. The uniqueness of b y comes fom Coollay 3. We will now study the behavio of the fist deivative of w by. Lemma 7 Thee exists > such that w b 1 (1 1 and b 1 < µ. Poof: Moeove w µ Because α( = and µ >, it exists η > such that x [1 η, 1, α(1 x + x µ <. (14 is stictly concave on [1, µ [ by Lemma and thus w µ (1 w µ (µ + (1 µ w µ ( µ = 1. Because by Lemma, we have w µ > 1 on [1, µ [, thee exists ν > such that x [1 ν, 1, w µ (x < x. Let = min(η, ν. By Coollay 4, it exists b 1 1, 1 + µ [ such that w b1 (1 = 1. We have w b1 (1 > w µ (1 and then b 1 < µ by Coollay 3. Let us conside the function W (x = w b1 (x x, we have W (1 =, W (b 1 = µ b 1 >. Moeove, W is solution (µ α((1 x + W (x + σ W (x W (x = α((1 x + + x µ. (15 On [1, 1, the second membe of Equation (15 is negative due to Equation (14. On [1, b 1, it is equal to x µ which is negative because b 1 < µ. Assume by a way of contadiction that thee is some x [1, b 1 such that W (x <, then it would exist x [1, b 1 such that W ( x <, W ( x = and W ( x > which is in contadiction with Equation (15. Hence, W is a positive function on [1, b 1 with W (1 = which implies w b 1 (
16 Lemma 8 When µ < α(1, w b is a convexconcave function. Poof: Accoding to Coollay 4, thee exists b 1, 1 + µ [ such that w b ( = and by Lemma 1, w b > on (, b. Using Equation (1, we thus have w b ( > implying that w b is stictly convex on a ight neighbohood of. Because b > 1, Lemma implies w b (x < on [1, b [. If thee is moe than one change in the concavity of w b, it will exist x [, 1[ such that w b ( x >, w b ( x = and w b ( x yielding the standad contadiction. Poposition 4 If µ < α(1 and w b ( 1, w b is the shaeholdes value function (4 Poof: It is staightfowad to see that the function w b satisfies Poposition when w b ( 1. Now, we will conside the case w b ( < 1. Lemma 9 If w b ( < 1, it exists a, 1[ such that w ba (a = a and w b a (a = 1. Poof: Let φ(x = w b x (x. By assumption, we have φ( < 1 and by Lemma 7, φ(1 > 1. By continuity of φ, thee exists a, 1[ such that w b a (a = 1. By definition, the function w ba satisfies w ba (a = a. Lemma 1 w ba is a convexconcave function on [a, b a. Poof: Fist, we show that w ba is inceasing on [a, b a. Because w b a (a = 1, we can define x = min{x > a, w b a (x }. If x b a, we will have w b a ( x =, w ba ( x > and w b a ( x yielding the standad contadiction. Accoding to Lemma 1, we have w b a (x < ove [1, b a [ because b a > 1. Poceeding analogously as in the poof of Lemma 8, we pove that w ba is a convexconcave function because it cannot change of concavity twice. Lemma 11 We have w ba > 1 on (a, b a with b a < µ. Poof: Accoding to Lemma 1, w ba is convexconcave with w b a (a = 1 and w b a (b a = 1, theefoe x a, b a [, w b a (x > 1. As a consequence, w ba (x > x on a, b a and in paticula w ba (1 > 1. Remembeing that w µ (1 < 1 and using Coollay 3, we have b a < µ. Poposition 5 If w b ( < 1, the function x w(x = w ba (x x b a + µ fo x a fo a x b a fo x b a is the shaeholdes value function (4. 15
17 Poof: it is staightfowad to check that w satisfies Poposition. Figue plots some value functions, when α(1 > µ, using a linea function fo α, α(x = λx fo diffeent values of λ. Figue : Compaing shaeholdes value functions with µ =, =.5, σ = 1 and α(1 > µ fo diffeent values of λ whee α(x = λx. Remak 3 When the cost of debt exceeds the fim pofitability, the shaeholdes value is convex fo low value of equity. This illustates the shaeholdes option to abandon a highly indebted fim. This option value, measued by the theshold a, inceases with the cost of debt while the value function deceases with the cost of debt. 4 Capital investment In this section, we will extend ou model to allow vaiable investment in the poductive assets. We will assume deceasing etun to scale by intoducing an inceasing concave function β with lim x β(x = β that impacts the dynamic of the book value of equity as 16
18 follows: { dxt = β(k t (µdt + σdw t α((k t X t + dt γ di t dz t dk t = di t = di + t di t whee I t + (esp. It is the cumulative capital invested (esp. disinvested in the poductive assets up to time t, γ > is an exogenous popotional cost of investment. We assume that the fim is foced to liquidate when the level of outstanding debt eaches the sum of the liquidation value of the poductive assets and the liquid assets, (1 γk t + M t. The goal of the management is to maximize ove the admissible stategies π = (Z t, I t t the iskneutal shaeholdes value ( τ V (x, k = sup E x,k e t dz t (17 π whee By definition, we have τ = inf{t, X t γk t }. (16 k, V (γk, k = ( Dynamic pogamming and fee bounday poblem In ode to deive a classical analytic chaacteization of V in tems of a fee bounday poblem, we shall appeal to the dynaming pogamming pinciple as follows Dynamic Pogamming Pinciple: Fo any (x, k S whee S = {(x, k R +, x γk}, we have ( θ V (x, k = sup E e t dz t + e θ V (X θ, K θ (19 π whee θ is any stopping time. Let us conside the second ode diffeential opeato Lw = ( β(kµ α((k x + w x + σ β(k w w. ( x The aim of this section is to chaacteize via the dynamic pogamming pinciple the shaeholdes value as the unique continuous viscosity solution to the fee bounday poblem whee F (x, k, V, DV, D V = (1 ( F (x, k, w, Dw, D w = min Lw, w w 1, γ x x w k, γ w x + u. k We will fist establish the continuity of the shaeholdes value function which elies on some peliminay wellknown esults about hitting times we pove below fo sake of completeness. Lemma 1 Let a < b and (x n n a sequence of eal numbes such that lim n + x n = b and min n x n > a. Let (X n t n the solution of the stochastic diffeential equation { dx n t = µ n (X n t dt + σ n dw t X n = x n 17
19 whee µ n and σ n satisfy the standad global Lipschitz and linea gowth conditions. Moeove, (σ n n ae stictly positive eal numbes conveging to σ > and (µ n n is a sequence of bounded functions conveging unifomly to µ. Let us define T n = inf{t, Xt n = a} and θ n = inf{t, Xt n = b}. We have lim P(θ n < T n = 1 n + Poof: (a, b as Let us define the functions U n, F n : I R, on some bounded inteval I containing U n (y = y µ n (z + x n dz, F n (y = y e Un(z σn dz. Because (µ n n conveges unifomly to µ, we note that (F n, U n n conveges unifomly to (F, U whee and F (y = U(y = y y e U(z σ dz. µ(z + bdz Let Yt n = Xt n x n, Mt n = F n (Yt n and τ n = inf{t, Yt n / a n, b n [} with a n = a x n et b n = b x n. We fist show that τ n is integable. Because F n is the scale function of the pocess Yt n, Mt n is a local matingale with quadatic vaiation Because < M n > t = t σne 4Un(Y σn ( E(< M n > t τn σnt exp 4 min U σn n (y < + y [a n,b n the pocesses (M n t τ n t and ((M n t τ n < M n > t τn t ae both matingales. By Optional sampling theoem E[(M n t τ n < M n > t τn = which implies [ t E 1 [,τn(sσne 4Un(Y σn and ( σn exp 4 σn thus thee is a constant K n > such that n s n s max U n(y E[t τ n y [a n,b n t, E[t τ n K n ds ds = E[Fn(Y t τ n n max F n(y. y [a n,b n We conclude by dominated convegence that τ n is integable. The matingale popety implies E[F n (Yt τ n n = 18
20 which yields by dominated convegence because E[F n (Y n τ n =, t, F n (Yt τ n n max F n(y. y [a n,b n This is equivalent to with p(a n, b n = P(Y n τ n = b n. Hence, Moeove, F n (a n (1 p(a n, b n + F n (b n p(a n, b n = p(a n, b n = F n (a n F n (b n F n (a n P(θ n < T n = P(X n τ n = b Using the unifom convegence of F n, we have = P(Y n τ n = b x n = p(a n, b n lim P(θ n < T n = lim p(a n, b n n + n + F (a b = F ( F (a b =1 Lemma 13 Let a < b and (x n n a sequence of eal numbes such that lim n + x n = b and min n x n > a. Let (X n t n the solution to { dx n t = µ n (X n t dt + σ n dw t X n = x n with the same assumptions as in Lemma 1. Thee exist constants A n and B n such that ( exp b x n ( ( A σ n n + σn A n E[e θn exp b x n ( B σ n n + σn B n ( Poof: Because µ n ae bounded functions, thee ae two constants A n and B n such that A n µ n (x B n fo all a < x < b. We define X t n = x n + A n t + σ n W t. By compaison, we 19
21 have X t n Xt n and θ n θ n, with θ n = inf{t, X t n = b}. But the Laplace tansfom of θ n is explicit and given by ( E[e θ n = exp b x n ( A σ n n + σn A n which gives the left inequality of (. The poof is simila fo the ight inequality intoducing X n t = x n + B n t + σ n W t. Poposition 6 The shaeholdes value function is jointly continuous. Poof: Let (x, k S and let us conside (x n, k n a sequence in S conveging to (x, k. Theefoe, {(x n γ k k n, k, (x γ k k n, k n } S fo n lage enough. We conside the following two stategies that ae admissible fo n lage enough: Stategy πn: 1 stat fom (x, k, invest if k n k > (o disinvest if k n k < and do nothing up to the minimum between the liquidation time and the hitting time of (x n, k n. Denote (X π1 n t, K π1 n t t the contol pocess associated to stategy πn. 1 Stategy πn: stat fom (x n, k n, invest if k n k < (o disinvest if k n k > and do nothing up to the minimum between the liquidation time and the hitting time of (x, k. Denote (X π n t, K π n t t the contol pocess associated to stategy πn. To fix the idea, assume k n > k. The stategy π 1 makes the pocess (X, K jump fom (x, k to (x γ(k n k, k n. Define θn 1 = inf{t, (X π1 n t, K π1 n t = (x n, k n }, θ n = inf{t, (X π n t T 1 n = inf{t, X π1 n,x t, K π n t = (x, k}, γk π1 n,k t }
22 and T n = inf{t, X π n,xn t γk π n,kn t } Dynamic pogamming pinciple and V (X T 1 n, K T 1 n = on T 1 n θ 1 n yield V (x, k E [ θ 1 n T 1 n e t dz π1 n t + e (θ1 n T n 1 1 {θ 1 n <Tn 1} V (X θ 1 n, K θ 1 n [ E e θ1 n 1{θ 1 n <Tn 1} V (x n, k n ( E ( ( e n θ1 E e θn 1 1{θ 1 n Tn} 1 V (x n, k n ( E ( ( e n θ1 P θ 1 n Tn 1 V (x n, k n (3 On the othe hand, using V (X T n, K T n = on Tn θn [ θ n T V n (x n, k n E e t dz π n t + e (θ n T n 1 {θ n <Tn}V (X θ n, K θ n [ E e θ n 1{θ n <Tn } V (x, k ( E ( ( e n θ E e θn 1{θ n Tn} V (x, k ( E ( ( e n θ P θ n Tn V (x, k (4 The convegence of (x n, k n implies fom which we deduce using Lemma 1 that and lim (x n γ k k n, k = (x, k n + lim n + P(θ1 n Tn 1 = (5 lim n + P(θ n Tn = (6 Let µ n (X n t = β(k n µ α((k n X n t + and σ n = β(k n σ. The function µ n is bounded by thus, accoding to Lemma 13 exp ( κn ( A σ n n + σn A n with κ n = x x n + γ k n k. Letting n tend to + and using A n = β(k n µ α(k n B n = β(k n µ E[e θ1n exp ( κn 1 ( B σ n n + σn B n
23 we obtain Finally, we have fom (3 and (4, lim n + A n lim n + B n lim n + σ n = β(kµ α(k = β(kµ = β(kσ lim n + E(eθ1 n = lim n + E(eθ n = 1 (7 V (x, k lim sup V (x n, k n lim inf V (x n, k n V (x, k, n n which poves the continuity of V. We ae now in a position to chaacteize the shaeholdes value in tems of viscosity solution of the fee bounday poblem (1. Theoem 1 The shaeholdes value V is the unique continuous viscosity solution to (1 on S with linea gowth. Poof: The poof is postponed to the Appendix 4. Absence of Investment cost In this section, we will assume that thee is no cost of investment/disinvestment, that is γ =. Let us define H(kw(x = (β(kµ α((k x + w (x + σ β(k w (x w(x. We will constuct a C solution (w, b to the fee bounday poblem and max k H(kw(x = and w (x 1 fo x b (8 max k H(kw(x and w (x = 1 fo x b (9 which will chaacteize the shaeholdes value using a veification theoem simila to Poposition whose poof is omitted. Fist, poceeding analogously as in the poof of Coollay 1, it is staighfowad to pove Coollay 5 If µβ ( then it is optimal to liquidate the fim thus v (x = x. Heeafte, we will assume that µβ ( >.
24 4..1 High cost of debt: α ( > µβ ( Let us define δ = and a the unique nonzeo solution (if it exists of the equation σ µ + σ (3 σ (1 δβ(a = µa. (31 Because β is concave and β goes to, the existence of a is equivalent to assume σ β ( µ (1 δ (3 Let us define the function w A fo A > as the unique solution on (a, + of the Cauchy poblem µβ(xw A(x + σ β(x w A(x w A (x = with w A (x = Ax δ fo x a and w A diffeentiable at a. Remak 4 The Cauchy poblem is well defined with the condition w A diffeentiable at a. Moeove, it is easy to check, using the definition of a, that the function w A is also C. Because the cost of debt α is high, the shaeholdes optimally choose not to issue debt but athe adjust costlessly thei level of investment. Lemma 14 Fo evey A > the function w A is inceasing. Poof: Clealy, w A is inceasing and thus positive on [, a. Let c = min{x > a, w A (c = }. w A (c > because w A is inceasing and positive in a left neighbohood of c. Thus, accoding to the diffeential equation, we have w A (c which implies that w A is also inceasing in a ight neighbohood of c. Theefoe, w A cannot become negative. Lemma 15 Fo evey A >, thee is some b A such that w A (b A = and w A is a concave function on a, b A [. Poof: Assume by a way of contadiction that w A does not vanish. Using Equations (3 and (31, we have σ β (a w A(a = Aa δ. Theefoe, we equivalently assume that w A <. This implies that w A is sticly deceasing and bounded below by by lemma 14 theefoe w A is an inceasing concave function. Theefoe, lim x + w A(x exists and is denoted by l. Letting x + in the diffeential equation, we obtain, because β has a finite limit, σ β lim x w A(x = lim w A (x µ βl. x 3
25 Theefoe, eithe lim x w A (x is + fom which we get a contadiction o finite fom which we get lim x + w A(x = by mean value theoem. In the second case, diffeentiating the diffeential equation, we have µβ (xw A(x + µβ(xw A(x + σ β (xβ(xw A(x + σ β(x Poceeding analogously, we obtain that the diffeential equation, we get w A(x w A(x = (33 lim x + w A(x = and thus l =. Coming back to = lim x w A (x which contadicts that w A is inceasing. Now, define b A = inf{x a, w A (x = } to conclude. Lemma 16 Thee exists A such that w A (b A = 1. Poof: Fo evey A >, we have Let A 1 = µ β a δ. Lemma 14 yields µβ(b A w A(b A = w A (b A. (34 w A1 (b A1 w A1 (a = µ β µβ(b A 1 Theefoe, Equation (34 yields w A 1 (b A1 1. On the othe hand, let A = a1δ. By constuction, w δ A (a = 1 and thus w A (b A 1 by concavity of w A on (, b A. Thus, thee is some A [min(a 1, A, max(a 1, A such that w A = 1. Heeafte, we denote b = b A. Lemma 17 We have µβ (b. Poof: Diffeentiating the diffeential equation and plugging x = b, we get σ β(b w A(b + µβ (b = Because w A is inceasing in a left neighbohood of b, we have w A (b implying the esult. Let us define { w A (x x b v = x b + µβ(b x b We ae in a position to pove the main esult of this section. 4
26 Poposition 7 The shaeholdes value is v. Poof: We have to check that (v, b satisfies the fee bounday poblem (8 and (9. By constuction, v is a C concave function on (, + satisfying v 1. It emains to check max k H(kv(x. Fo x > b, we have H(kv(x = µβ(k α((k x + µβ(b (x b. If k x, concavity of β and Lemma 17 implies H(kv(x = µ(β(x β(b (x b (µβ (b (x b. If k x, we diffeentiate H(kv(x with espect to k and obtain using again concavity of β and convexity of α, H(kv(x k = µβ (k α (k x µβ ( α (. Theefoe, H(kv(x H(xv(x. Let x < b, because v is concave, the same agument as in the pevious lines shows that and theefoe H(kv(x k Fist ode condition gives fo k < x Thus fo < x < a, we have fo k x max H(kv(x = max H(kv(x. k k x k (H(kv = µβ (kv (x + σ β (kβ(kv (x = β (k[µv (x + σ β(kv (x k (H(kv = β (ka x δ δ[µx + σ β(k(δ 1 which gives, k (H(v > k (H(xv < Theefoe the maximum k (x of H(kv(x lies in the inteio of the inteval [, x and satisfies: < x < a, β(k (x = 5 µx σ (1 δ
27 Hence, fo x a, we have by constuction max {H(kv} = µ x k x σ (1 δ A δx δ1 + σ µ x σ 4 (1 δ A δ(δ 1x δ A x δ = Now, fix x (a, b. We note that k (H(kv has the same sign as µv (x + σ β(kv (x because β is stictly inceasing. Moeove, because v is concave and β inceasing, we have min k x µv (x + σ β(kv (x = µv (x + σ β(xv (x. Thus, it suffice to pove µv (x + σ β(xv (x fo x (a, b o equivalently because β is a positive function that the function φ defined as φ(x = µβ(xv (x + σ β(x v (x is positive. We make a poof by contadiction assuming thee is some x such that φ(x <. As φ(a = by Equation 31 and φ(b > then thee is some x 1 [a, b such that { φ(x1 < φ (x 1 = Using the diffeential equation (33 satisfied by v, we obtain fom we deduce φ (x 1 = ( µβ (x 1 v (x 1 µβ(x 1 v (x 1 = φ(x 1 = µβ(x 1 v (x 1 + σ β(x 1 v (x 1 = µβ(x 1 v (x 1 + σ β(x 1 ( µβ (x 1 v (x 1 µ = β(x 1 v (x 1 (µ + σ µ σ β (x 1 But x 1 a and thus β (x 1 β (a. Moeove, by definition of a, we have σ β (a µ. (1δ Theefoe, Equation (3 yields φ(x 1 β(x 1 v (x 1 (µ + σ µ µ 1 δ ( σ β(x 1 v (x 1 µ µ δ 1 δ ( σ β(x 1 v (x 1 µ µ σ µ + σ µ + σ µ which is a contadiction. = Figue 3 plots some shaeholdes value functions whith α ( > µβ ( and σ β ( µ fo diffeent values of (1δ β ( and using : 6
28 a linea function fo α, α(x = λx. an exponential function fo β, β(x = β max ( 1 e β ( βmax. x Figue 3: Compaing shaeholdes value functions with µ =, =.5, σ = 1, λ = 1, β max = 8, fo diffeent values of β (. Remak 5 This esult gives us the best stategy in tems of investment/disinvestment. When thee is a high cost of debt and when thee is no cost of investment/disinvestment, the optimal level of poductive assets is given, when (3 is fulfilled, by [ x a, k(x = β 1 µx σ (1 δ x a, k(x = x It shows that the manage shouldn t invest all the cash eseve in poductive assets when the book value of equity is low, even if thee is no cost of investment/disinvestment. The manage optimally disinvests to lowe the volatility of the book value of equity. 7
29 Figue 4 plots the optimal level of poductive assets fo diffeent values of σ. It shows that, fo a given level of the book value of equity, the investment level in poductive assets is a deceasing function of the volatility. Figue 4: Compaing optimal level of poductive assets with µ =, =.5, λ = 1, β max = 8, β ( = 5 fo diffeent values of σ. To complete the chaacteization of the shaeholdes value when the cost of debt is high, we have to study the optimal policy when (3 is not fulfilled. We expect that a = in that case which means that fo all x, the manage should invest all the cash in poductive assets. Thus we ae inteested in the solutions to such that w( =. µβ(xw (x + σ β(x w (x w(x = (35 Poposition 8 Suppose that the functions x x and x x ae analytics in with β(x β(x a adius of convegence R. The solutions w to Equation (35 such that w( = ae given 8
30 by with k 1, A k = whee the functions p and q ae the function I is given by and y 1 is the positive oot of I y 1 = w(x = 1 k1 I(k + y 1 j= A k x k+y 1 k= p(x = µx σ β(x (j + y 1 p (kj ( + q (kj ( A j (k j! q(x = x σ β(x I(y = µβ (y + σ β ( y(y 1 µ + σ β ( + (µ σ The adius of convegence of w is at least equal to R. β ( + σ. σ β ( Poof: This esult is given by the Fuchs theoem [18. Note that the solutions of Equation (35 vanishing at zeo can be witten w A (x = A w 1 (x. If the adius of convegence of the Fobenius seie is finite, then the peviously defined function w 1 can be extended by use of the Cauchy theoem. Because µβ (, we have y 1 < 1. As a consequence, we have lim x w 1(x = + and lim w x 1(x = Thus, poceeding analogously as in Lemma 15, we pove the existence of b such that w 1(b =. Because w A is linea in A, we choose A = A = 1 to get a concave solution w 1 (b w to (35 with w ( =, (w (b = 1 and (w (b =. We extend w linealy on (b, + as usual to obtain a C function on [, + [. Poposition 9 The shaeholdes value is w. 9
31 Poof: It suffices to check that w satisfies the fee bounday poblem (8 and (9. By constuction w is a C concave function on R +. Because (w (b = 1, we have and x, b, (w (x 1 x b, (w (x = 1 On [b, + [, we have [ max {H x,kw } = max µβ(k α((k x + µβ(b + (b x k k [ = max max µβ(k µβ(b + (b x, k x max µβ(k α(k x µβ(b + (b x k x Using β concave inceasing, α convexe, α ( + > µβ ( +, we have Then using the concavity of β, It emains to show that fo evey x < b max k {H x,kw } = µβ(x µβ(b + (b x x b, max k {H x,kw } max k {H x,kw } = Using β concave, α convexe, α ( > µβ ( and w concave inceasing, we have thus, Moeove, k > x, k (H x,kw = (µβ (k α (k x(w (x + σ β (kβ(k(w (x max k {H x,k(w } = max k x {H x,k(w } < k < x, k (H x,k(w = µβ (k(w (x + σ β (kβ(k(w (x = β (k[µ(w (x + σ β(k(w (x We expect Notice that β (k and x, b, k x, k (H x,k(w min k x µ(w (x + σ β(k(w (x = µ(w (x + σ β(x(w (x 3
32 because (w (x and β is inceasing. Thus it is enough to pove fo evey x < b, o equivalently, using β, µ(w (x + σ β(x(w (x φ(x = µβ(x(w (x + σ β(x (w (x fo x < b. We make a poof by contadiction assuming the existence of x such that φ(x <. In a neighbohood of, we have and (w (x A y 1 x y 11 (w (x A y 1 (y 1 1x y 1 Fom which we deduce because β(xx y 11 β (x y 1, lim x β(x(w (x = lim x β(x (w (x = yielding lim φ(x = x But φ(b > thus thee is x 1, b[ such that { φ(x1 < Using the deivative of Equation (35 fom which we deduce : φ (x 1 = φ (x 1 = ( µβ (x 1 (w (x 1 µβ(x 1 (w (x 1 = φ(x 1 = µβ(x 1 (w (x 1 + σ β(x 1 (w (x 1 = µβ(x 1 (w (x 1 + σ β(x 1 ( µβ (x 1 (w (x 1 µ = β(x 1 (w (x 1 (µ + σ µ σ β (x 1 Now, emembe that x 1 > and thus using the concavity of β, we have β (x 1 β (. Futhemoe, β ( µ +σ when Equation (31 is not fulfilled. Hence, σ µ φ(x 1 β(x 1 (w (x 1 (µ + σ µ µ + σ µ which yields to a contadiction and ends the poof. Figue 5 plots some shaeholdes value functions whith α ( > µβ ( and σ β ( µ fo diffeent values of (1δ β ( and using : 31
33 a linea function fo α, α(x = λx. an exponential function fo β, β(x = β max ( 1 e β ( βmax. x Figue 5: Compaing shaeholdes value functions with µ =, =.5, σ = 1, λ = 1, β max = 8, fo diffeent values of β (. 4.. Low cost of debt: α ( µβ ( When the cost of debt is low, it can be optimal to issue debt in ode to incease the investment level. Thee is a tadeoff between the cost of debt α and the investment gain function β. The following lemma gives us a bounday fo the cost of the debt when (3 is not fulfilled. Lemma 18 If α ( + < µβ ( σ β ( (1 and σ β ( < µ such that x, d[, max {H x,k(w } = max {H x,k(w } k k>x (1δ then it exists d > 3
34 Poof: We can show as in Poposition 9 that so, k < x, k H x,kw max k {H x,k(w } = max k x {H x,k(w } We make a poof by contadiction assuming that in the neighbohood of, we have thus we have and max k {H x,k(w } = H x,x w (w (x A y 1 x y 11 (w (x A y 1 (y 1 1x y 1 fom which we deduce that in a neighbohood of k H x,x +(w (β (x α ( + A y 1 x y 11 + β (xβ(xa y 1 (y 1 1x y 1 A y 1 x y 11 [µβ ( σ β ( (1 α ( + > and thee is a contadiction. Figue 6 plots the optimal level of poductive assets fo diffeent values of the cost of debt α. 5 Appendix 5.1 Poof of Theoem 1 Supesolution popety. Let ( x, k S and ϕ C (R + s.t. ( x, k is a minimum of V ϕ in a neighbouhood B ( x, k of ( x, k with small enough to ensue B S and V ( x, k = ϕ( x, k. Fist, let us conside the admissible contol ˆπ = (Ẑ, Î whee the shaeholdes decide to neve invest o disinvest, while the dividend policy is defined by Ẑt = η fo t, with η. Define the exit time τ = inf{t, (Xt x, K k t / B ( x, k}. We notice that τ < τ fo small enough. Fom the dynamic pogamming pinciple, we have [ τ h ϕ( x, k = V ( x, k E e t dẑt + e (τ h V (Xτ x h, K k τ h [ τ h E e t dẑt + e (τ h ϕ(xτ x h, K k τ h. (36 33
35 Figue 6: Compaing optimal level of poductive assets with µ =, =.5, σ = 1, λ = 1, β max = 8, β ( =, fo diffeent values of α. Applying Itô s fomula to the pocess e t ϕ(x x t, K k t between and τ h, and taking the expectation, we obtain [ E e (τ h ϕ(xτ x h, K k τ h [ τ h = ϕ( x, k + E e t Lϕ(Xt x, K k t dt [ + E e t [ϕ(xt x, K k t ϕ(x x t, K k t. (37 t τ h Combining elations (36 and (37, we have [ τ h E e t (Lϕ(Xt x, K k t dt [ E t τ h [ τ h E e t dẑt e t [ϕ(x x t, K k t ϕ(x x t, K k t. (38 34
36 Take fist η =. We then obseve that X is continuous on [, τ h and only the fist tem of the elation (38 is non zeo. By dividing the above inequality by h with h, we conclude that Lϕ( x, k. Take now η > in (38. We see that Ẑ jumps only at t = with size η, so that [ τ h E e t (Lϕ(Xt x, K k t dt η (ϕ( x η, k ϕ( x, k. By sending h, and then dividing by η and letting η, we obtain ϕ x ( x, k 1. Second, let us conside the admissible contol π = ( Z, Ī whee the shaeholdes decide to neve payout dividends, while the investment/disinvestment policy is defined by Īt = η R fo t, with < η. Define again the exit time τ = inf{t, (Xt x, K k t / B ( x, k}. Poceeding analogously as in the fist pat and obseving that Ī jumps only at t =, thus [ τ h E e t (Lϕ(Xt x, K k t dt (ϕ( x γ η, k + η ϕ( x, k. Assuming fist η >, by sending h, and then dividing by η and letting η, we obtain γ ϕ x ( x, k ϕ k ( x, k. When η <, we get in the same manne γ ϕ x ( x, k + ϕ k ( x, k. This poves the equied supesolution popety. Subsolution Popety: We pove the subsolution popety by contadiction. Suppose that the claim is not tue. Then, thee exists ( x, k S and a neighbouhood B ( x, k of x, k, included in S fo small enough, a C function ϕ with (ϕ V ( x, k = and ϕ V on B ( x, k, and η >, s.t. fo all (x, k B ( x, k we have Lϕ(x, k > η, (39 ϕ (x, k 1 x > η, (4 (γ ϕ x ϕ (x, k k > η. (41 (γ ϕ x + ϕ (x, k k > η. (4 Fo any admissible contol π, conside the exit time τ = inf{t, (X x t, K k t / B ( x, k} and notice again that τ < τ. Applying Itô s fomula to the pocess e t ϕ(x x t, K k t between and τ, we have 35
37 E[e τ ϕ(x τ, K τ = ϕ( x, k E + E + E E + E [ τ [ τ [ τ [ [ τ e u Lϕdu e u (γ ϕ x + ϕ k dic,+ u e u (γ ϕ x ϕ u ϕ e x dzc u k dic, u <s<τ e s [ϕ(x s, K s ϕ(x s, K s (43 (44 (45 (46 (47 Using elations (39,(4,(41,(4, we obtain V ( x, k = ϕ( x, [ k τ ηe e u du + E[e τ ϕ(x τ, K τ (48 + ηe + ηe [ τ [ τ + (1 + ηe E [ e u di c,+ u e u di c, u [ τ e u dz c u <s<τ e s [ϕ(x s, K s ϕ(x s, K s (49 (5 (51 (5 36
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