An Optimal Dividend and Investment Control Problem under Debt ...
An Optimal Dividend and Investment Control Problem under Debt Constraints Etienne CHEVALIER∗ †
Vathana LY VATH‡
†
Simone SCOTTI§
December 12, 2011
Abstract
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This paper concerns with the problem of determining an optimal control on the dividend and investment policy of a firm. We allow the company to make an investment by increasing its outstanding indebtedness, which would impact its capital structure and risk profile, thus resulting in higher interest rate debts. We formulate this problem as a mixed singular and switching control problem and use a viscosity solution approach combined with the smooth-fit property to get qualitative descriptions of the solution. We further enrich our studies with a complete resolution of the problem in the tworegime case. Keywords: stochastic control, optimal singular / switching problem, viscosity solution, smooth-fit property, system of variational inequalities, debt constraints. MSC2000 subject classification: 60G40, 91B70, 93E20.
∗
Université d’Evry, Laboratoire d’Analyse et Probabilités, France , [email protected]. This research benefitted from the support of the “Chaire Risque de Crédit”, Fédération Bancaire Française. ‡ ENSIIE and Université d’Evry, Laboratoire d’Analyse et Probabilités, France, [email protected]. § Université Paris Diderot, LPMA, France, [email protected]. †
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1
Introduction
The theory of optimal stochastic control problem, developed in the seventies, has over the recent years once again drawn a significance of interest, especially from the applied mathematics community with the main focus on its applications in a variety of fields including economics and finance. For instance, the use of powerful tools developed in stochastic control theory has provided new approaches and sometime the first mathematical approaches in solving problems arising from corporate finance. It is mainly about finding the best optimal decision strategy for managers whose firms operate under uncertain environment whether it is financial or operational, see [3] and [9]. A number of corporate finance problems have been studied, or at least revisited, with this optimal stochastic control approach. In this paper, we consider the problem of determining the optimal control on the dividend and investment policy of a firm. There are a number of research on this corporate finance problem. In [6], Décamps and Villeneuve study the interactions between dividend policy and irreversible investment decision in a growth opportunity and under uncertainty. We may equally refer to [20] for an extension of this study, where the authors relax the irreversible feature of the growth opportunity. In other words, they consider a firm with a technology in place that has the opportunity to invest in a new technology that increases its profitability. The firm self-finances the opportunity cost on its cash reserve. Once installed, the manager can decide to return back to the old technology by receiving some cash compensation. As in a large part of the literature in corporate finance, the above papers assume that the firm cash reserve follows a drifted Brownian motion. They also assume that the firm does not have the ability to raise any debt for its investment as it holds no debt in its balance sheet. In our study, as in the Merton model, we consider that firm value follows a geometric Brownian process and more importantly we consider that the firm carries a debt obligation in its balance sheet. However, as in most studies, we still assume that the firm assets is highly liquid and may be assimilated to cash equivalents or cash reserve. We allow the company to make investment and finance it through debt issuance/raising, which would impact its capital structure and risk profile. This debt financing results therefore in higher interest rate on the firm’s outstanding debts. Furthermore, we consider that the manager of the firm works in the interest of the shareholders, but only to a certain extent. Indeed, in the objective function, we introduce a penalty cost P and assume that the manager does not completely try to maximize the shareholders’ value since it applies a penalty cost in the case of bankruptcy. This penalty cost could represent, for instance, an estimated cost of the negative image upon his/her own reputation due to the bankruptcy under his management leadership. Mathematically, we formulate this problem as a combined singular and multiple-regime switching control problem. Each regime corresponds to a level of debt obligation held by the firm. In terms of literature, there are many research papers on singular control problems as well as on optimal switching control problems. One of the first corporate finance problems using singular stochastic control theory was the study of the optimal dividend strategy, see for instance [5] and [15]. These two papers focus on the study of a singular stochastic control problem arising from the research on optimal dividend policy for a firm whose cash
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reserve follows a diffusion model. Amongst singular stochastic control problems, we may equally refer to problems arising from mathematical biology, in particular studies on optimal harvesting strategies, see [18] and [24]. In the study of optimal switching control problems, a variety of problems are investigated, including problems on management of power station [4], [13], resource extraction [2], firm investment [10], marketing strategy [17], and optimal trading strategies [7], [25]. Other related works on optimal control switching problems include [1] and [19], where the authors employ respectively optimal stopping theory and viscosity techniques to explicitly solve their optimal two-regime switching problem on infinite horizon for one-dimensional diffusions. We may equally refer to [22], for an interesting overview of the area. In the above studies, only problems involving the two-regime case are investigated. There are still too few studies on the multi-regime switching problems. The main additional feature in the multiple regime problems consists not only in determining the switching region as opposed to the continuation region, but also in identifying the optimal regime to where to switch. This additional feature sharply increases the complexity of the multi-regime switching problems. Recently, Djehiche, Hamadène and Popier [8], and Hu and Tang [14] have studied optimal multiple switching problems for general adapted processes by means of reflected BSDEs, and they are mainly concerned with the existence and uniqueness of solution to these reflected BSDEs. In [23], the authors investigated an optimal multiple switching problem on infinite horizon for a general one-dimensional diffusion and used the viscosity techniques to provide an explicit characterization of the switching regions showing when and where it is optimal to change the regime. However, the studies that are most relevant to our problem are the one investigating combined singular and switching control problems. Recently an interesting connection between the singular and the switching problems was given by Guo and Tomecek [12]. In [20], the authors studied a problem which combines features of both optimal switching and singular control. They proved that the mixed problem can be decoupled in two pure optimal stopping and singular control problems and provided results which are of quasi-explicit nature. However, the switching part of this problem is limited to a two-regime problem. In this paper, we combine the difficulties of the multiple switching problem [23] with those of the mixed singular and two-regime switching problem [20]. In other words, not only do we have to determine the three regions comprising the continuation, dividend and switching regions, but also, within the switching region, we have to identify the regime to where to switch. The latter feature considerably increases the complexity of our problem. In terms of mathematical approach, we characterize our value functions as unique viscosity solution to an associated system of variational inequalities. Furthermore, we use viscosity and uniqueness results combined with smooth-fit properties to determine the solutions of our HJB system. The results of our analysis take qualitatively different forms depending on the parameters values. The plan of the paper is organized as follows. We define the model and formulate our stochastic control problem in the second section. In section 3, we characterize our problem as the unique viscosity solution to the associated HJB system and obtain some regularity properties, while, in section 4, we obtain qualitative description of our problem. Finally, in
3
section 5, we further enrich our studies with a complete resolution of the problem in the two-regime case.
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The model
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We consider a firm whose value follows a process X. The firm also has the possibility to raise its debt level in order to satisfy its financial requirement such as investing in growth opportunities. It may equally pay down its debt. We consider an admissible control strategy α = (Zt , (τn )n≥0 , (kn )n≥0 ), where Z represents the dividend policy, the nondecreasing sequence of stopping times (τn ) the switching regime time decisions, and (kn ), which are Fτn -measurable valued in {1, ..., N }, the new value of debt regime at time t = τn . Let denote the process X x,i,α as the enterprise value of the company with initial value of x and initially operating with a debt level Di and which follow the control strategy α. However, in order to reflect the fact that a firm also holds a significant amount of debt obligation either to financial institutions, inland revenues, suppliers or through corporate bonds, we assume that the firm debt level may never get to zero. We assume that the firm assets is cash-like, i.e. the manager may dispose of some part of the company assets and obtain its equivalent in cash. In other words, the process X could be seen as a cash-reserve process used in most papers on optimal dividend policy, see for instance [6], [15]. We assume that the cash-reserve process X x,i,α , denoted by X when there is no ambiguity and associated to a strategy α = (Zt , (τn )n≥0 , (kn )n≥0 ), is governed by the following stochastic differential equation: dXt = bXt dt − rIt DIt dt + σXt dWt − dZt + dKt
(2.1)
where It =
X
kn 1τn ≤t 0, in the case of a holding company looking to liquidate one of its own affiliate or activity. In the case of the penalty, it mainly assumes that the manager does not completely try to maximize the shareholders’ value since it applies a penalty cost in the case of bankruptcy. We therefore define the value functions which the manager actually optimizes as follows "Z
#
T−
−ρt
vi (x) = sup E α∈A
e
−ρT
dZt − P e
,
x ∈ R, i ∈ {1, ..., N },
(2.7)
0
where A represents the set of admissible control strategies, and ρ the discount rate. The next step would be to compute the real value function ui of the shareholders. Indeed, once we obtain the optimal strategy, α∗ = (Zt∗ , (τn∗ )n≥0 , (kn∗ )n≥0 ) of the above problem (2.7), then we may compute the real shareholders’ value by following the strategy α∗ , as numerically illustrated in Figure 3 and 4: "Z ui (x) = Eα∗
#
T−
e
−ρt
dZt∗
,
x ∈ R, i ∈ {1, ..., N }
(2.8)
0
Remark 2.1 If b > ρ, the value functions is infinite for any initial value of x > Di and any regime i. The proof is quite straightforward. We simply consider a sequence of strategy controls which consists in doing nothing up to time tk , where (tk )k≥1 is strictly nondecreasing and goes to infinity when k goes to infinity, and then at tk , distribute Xtk − DItk in dividend and allow the company to become bankrupt. We then need to notice that lim E e−ρtk Xtxk = +∞. k→∞
For the rest of the paper, we now consider that the discount rate ρ is always bigger than the growth rate b.
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3
Viscosity Characterization of the value functions
We first introduce some notations. We denote by Rx,i the firm value in the absence of dividend distribution and the ability to change the level of debt, fixed at Di . dRtx,i = [bRtx,i − ri Di ]dt + σRtx,i dWt , R0x,i = x
(3.1)
The associated second order differential operator is denoted Li : 1 Li ϕ = [bx − ri Di ]ϕ0 (x) + σ 2 x2 ϕ00 (x) 2
(3.2)
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Using the dynamic programming principle, we obtain the associated system of variational inequalities satisfied by the value functions: min −Ai vi (x) ,
vi0 (x)
− 1 , vi (x) − max vj (x + Dj − Di − g) = 0, x > Di , i ∈ IN j6=i
vi (Di ) = −P,
where the operator Ai is defined by Ai φ = Li φ − ρφ. We now state a standard first result for this system of PDE. Proposition 3.1 Let (ϕi )i∈IN smooth enough on (Di , ∞) such that ϕi (Di+ ) := lim ϕi (x) ≥ x↓Di
−P , and 0 min −Ai ϕi (x) , ϕi (x) − 1, ϕi (x) − max ϕj (x + Dj − Di − g) ≥ 0, for x > Di , i ∈ IN j6=i
where we set by convention ϕi (x) = −P for x < Di , then we have vi ≤ ϕi , for all i ∈ IN . Proof: Given an initial state-regime value (x, i) ∈ (Di , ∞) × IN , take an arbitrary control α = (Z, (τn ), (kn )) ∈ A, and set for m > 0, θm,n = inf{t ≥ T ∧ τn : Xtx,i,α ≥ m or Xtx,i,α ≤ DIt + 1/m} % ∞ a.s. when m goes to infinity. Apply then Itô’s formula to e−ρt ϕkn (Xtx,i,α ) between the stopping times T ∧ τn and τm,n+1 : = T ∧ τn+1 ∧ θm,n . Notice that for T ∧ τn ≤ t < τm,n+1 , Xtx,i stays in regime kn . Then, we have Z τm,n+1 x,i x,i −ρτm,n+1 −ρ(T ∧τn ) e ϕkn (Xτ − ) = e ϕkn (XT ∧τn ) + e−ρt (−ρϕkn + Lkn ϕkn )(Xtx,i )dt m,n+1 T ∧τn Z τm,n+1 Z τm,n+1 + e−ρt σϕ0kn (Xtx,i )dWt − e−ρt ϕ0kn (Xtx,i )dZtc T ∧τn T ∧τn h i X x,i −ρt + e ϕkn (Xt ) − ϕkn (Xtx,i ) , (3.3) − T ∧τn ≤t τn a.s., so that equation (3.3) holds true a.s. for all n, m (recall that ϕkn (XTx,i ) = −P ). 6
x,i Since ϕ0kn ≥ 1, we have by the mean-value theorem ϕkn (Xtx,i ) − ϕkn (Xtx,i − Xtx,i − ) ≤ Xt − = −(Zt − Zt− ) for T ∧ τn ≤ t < τm,n+1 .
By using also the supersolution inequality of ϕkn , taking expectation in the above Itô’s formula, and noting that the integrand in the stochastic integral term is bounded by a constant (depending on m), we have Z τm,n+1 h i x,i −ρ(T ∧τn ) −ρt c E e−ρτm,n+1 ϕkn (Xτx,i ) ≤ E e ϕ (X ) − E e dZ kn − t T ∧τn m,n+1 T ∧τn X − E e−ρt (Zt − Zt− ) , T ∧τn ≤t θr,i,0 ] → 0,
as x ↓ Di .
Notice that θr,i,0 < θr,i and combined with (3.8), we obtain P [θi > θr,i ] ≤ P [θi,0 > θr,i,0 ]. As such,
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P [θi > θr,i ] → 0,
as x ↓ Di .
(3.9)
Let α = (Z, (τn )n≥1 , kn≥1 ) be an arbitrary policy in A, and denote η = T ∧θr,i = T x,i,α ∧θr,i . For t ≤ η, from the definition of an admissible control, there is no regime shift. As such, for t ≤ η, we have Xtx,i ≤ Rtx,i ≤ Rtx,0 . We also have T x,i ≤ θi . We then write : "Z − # "Z − # " # Z T− T η −ρt −ρt −ρt E e dZt = E e dZt + E 1T >η e dZt 0
0
η
## e−ρt dZt Fθ− ≤ E E 1T >η r,i η "Z − ## " T e−ρt dZt Fθ− ≤ E 1T >θr,i E r,i θr,i ≤ E 1T >θr,i e−ρθr,i vi Xθx,i , − " "
Z
T−
(3.10)
r,i
where we also used in the second inequality the fact that on {T > η}, η = θr,i , and θr,i is a predictable stopping time. Now, since vi is nondecreasing, we have vi (Xθx,i − ) ≤ vi (Di + r). r,i
Moreover, recalling that T ≤ θi , inequalities (3.10) and (3.9) yield "Z − # T −ρt 0 ≤ E e dZt ≤ vi (Di + r)P[θi > θr,i ] −→ 0,
as x ↓ Di .
(3.11)
0
Furthermore, using the fact that T x,i ≤ θi ≤ θi,0 , we have h i E −P e−ρT ≤ −P E e−ρθi,0 Noticing that θi,0 is the hitting time of a drifted Brownian, it is straightforward that E e−ρθi,0 −→ 1, as x ↓ Di and recalling (3.11), we obtain "Z − # T −ρt −P ≤ vi (x) ≤ E e dZt − E P e−ρT 0
i h ≤ vi (r)P[θi > θr ] − P E e−ρθi,0 −→ −P, 9
as x ↓ Di .
We may therefore conclude that vi (Di+ ) = −P . b) We now turn to the continuity of the value functions vi . Let γ > 0 and x ∈ (Di , ∞). We set T γ = inf{t ≥ 0; Rtx,i ≥ x + γ}. We now consider a control strategy α = (Z, (τn ), kn ), where τ1 > T γ and Zt = 0, ∀ t < τ1 . Notice that ∀ t < τ1 , Xtx,i = Rtx,i . Applying the programming dynamic principle (DP), we obtain vi (x) ≥ E e−ρXT γ ∧T x,i vi (XT γ ∧T x,i ) ,
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therefore γ vi (x + γ) − vi (x) ≤ E (1 − e−ρT )vi (x + γ)1T γ Di . (3.13) j6=i
Actually, we obtain some more regularity results on the value functions. Proposition 3.3 The value functions vi , i ∈ IN , are C 1 on (Di , ∞). Moreover, if we set for i ∈ IN : Si = x ≥ Di , vi (x) = max vj (x + Dj − Di − g), (3.14) j6=i
Di = int ({x ≥ Di , vi0 (x) = 1}),
(3.15)
Ci = (Di , ∞) \ (Si ∪ Di ),
(3.16)
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then vi is C 2 on the open set Ci ∪ int(Di ) ∪ int(Si ) of (Di , ∞), and we have in the classical sense ρvi (x) − Li vi (x) = 0,
x ∈ Ci .
Si , Di , and Ci respectively represent the switching, dividend, and continuation regions when the outstanding debt is at regime i. The proofs of Theorem 3.1 and Proposition 3.3 are omitted as they follow essentially arguments from [11], [21] and in particular [20].
4
Qualitative results on the switching regions
For i, j ∈ IN and x ∈ [Di , +∞), we introduce some notations:
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δi,j = Dj − Di ,
∆i,j = (b − rj )Dj − (b − ri )Di
and xi,j = x + δi,j − g.
We set x∗i = sup{x ∈ [Di , +∞) : vi0 (x) > 1} for all i ∈ IN We equally define Si,j as the switching region from debt level i to j. Si,j = {x ∈ (Di + g, +∞), vi (x) = vj (xi,j )}. From the definition (3.14) of the switching regions, we have the following elementary decomposition property : Si = ∪j6=i Si,j , i ∈ IN . We now begin with two obvious results. Since there is a fixed switching cost g > 0, it is not optimal to continuously change your debt structure. Moreover, if it is optimal to distribute dividends and to switch to another regime, it is still optimal to distribute dividend after the regime switch. Lemma 4.1 Let i, j ∈ IN such that i 6= j. Assume that there exists x ∈ Si,j then we have i) xi,j := x + Dj − Di − g 6∈ Sj . ii) vi0 (x) = vj0 (xi,j ). Especially, if x ∈ Si,j ∩ Di then xi,j ∈ Dj \ Sj . Proof: For k ∈ IN \ {j}, we have vj (xi,j ) = vi (x) ≥ vk (xk,i ). As vk is strictly non-decreasing, we get vj (xi,j ) > vk (xk,i − g). Let h ∈ R. For h going to 0, we have vj (xi,j + h) = vj (xi,j ) + hvj0 (xi,j ) + o(h) = vi (x) + hvj0 (xi,j ) + o(h) = vi (x + h) + h(vj0 (xi,j ) − vi0 (x)) + o(h). 11
As vi (x + h) ≥ vj (xi,j + h), we obtain h(vj0 (xi,j ) − vi0 (x)) ≤ o(h). 2
Hence, we have vj0 (xi,j ) = vi0 (x).
In the following Lemma, we state that there exists a finite level of cash such that it is optimal to distribute dividends up to this level. Lemma 4.2 For all i ∈ IN , we have x∗i := sup{x ∈ [Di , +∞) : vi0 (x) > 1} < +∞.
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Proof: Assume that there exists k ∈ {0, ..., N − 1} such that x∗i < +∞ for all i ∈ Ik := {i1 , ..., ik } ⊂ IN . Notice that Ik = ∅ if k = 0. We will show that there exists j ∈ IN \ Ik such that x∗j < +∞. From Corollary 3.2 and Proposition 3.3, we deduce that, for all i ∈ IN , the function x → vi (x) − x is continuous, non decreasing and bounded. We set ai := lim (vi (x) − x). x→+∞
Moreover, for all (i, j) ∈ IN such that i 6= j, we have aj − (ai + δj,i − g) = lim (vj (x) − vi (x + δj,i − g)) ≥ 0. x→+∞
Let j0 ∈ IN \ Ik such that aj0 + Dj0 = maxj∈IN \Ik (aj + Dj ). For all j ∈ IN \ Ik , we have aj + δj0 ,j − g < aj0 . It is easy to see that there exists x ¯ ∈ [Dj , +∞) satisfying the following conditions: vj0 (¯ x) > x ¯+
max
j∈IN \Ik ;j0 6=j
(aj + δj0 ,j − g),
ρvj0 (¯ x) > b¯ x − rj0 Dj0 , x ¯ > x∗i − (δj0 ,i − g),
∀i ∈ Ik .
At this point, we introduce a continuous function defined on [Di , +∞): ( vj0 (x) if x < x ¯ Vˆ (x) = x−x ¯ + vj0 (¯ x) if x ≥ x ¯ Let x ≥ x ¯. We have −Aj0 Vˆ (x) = (ρ − b)(x − x ¯) + [ρvj0 (¯ x) − (b¯ x − rj0 Dj0 )] > 0. Moreover, for j ∈ IN \ Ik such that j 6= j0 , we have Vˆ (x) ≥ x + aj + δj0 ,j − g ≥ vj (x + δj0 ,j − g). For i ∈ Ik , we have vi (x + δj0 ,i − g) − Vˆ (x) = x + δj0 ,i − g − x∗i + vj (x∗i ) − (x − x ¯ + vj0 (¯ x)) = vi (x∗i ) − x∗i + (¯ x + δj0 ,i − g) − vj0 (¯ x) ≤ vi (¯ x + δj0 ,i − g) − vj0 (¯ x) ≤ 0. 12
Finally, for all j ∈ IN \ {j0 }, Vˆ ((x) ≤ vj0 (x) ≤ vj (x − δj,j0 + g). As Vˆ 0 (x) = 1, Vˆ is a continuous solution of equation (3.13). From Theorem 3.1, we deduce ¯. 2 that vj0 = Vˆ and x∗j0 ≤ x Now, we shall study properties of x∗i and more generally, properties of left-boundaries of Di in the sense as detailed in the following definition. Definition 4.1 Let i ∈ IN and x ∈ (Di , +∞). x is a left-boundary of Di if there exists ε > 0 and a sequence (yn )n∈N with values in (Di , x) \ Di such that [x, x + ε] ∈ Di
lim yn = x.
and
n→+∞
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Notice that, if x∗i > Di then x∗i is a left-boundary of Di . In order to compute the dividend regions, we establish the following lemma. Lemma 4.3 Let i, j ∈ IN such that j 6= i. We assume that there exists x ˆi a left-boundary of Di . i) Assume that x ˆi 6∈ Si , then we have (b − ri )Di > −ρP and ρvi (ˆ xi ) = bˆ xi − ri Di . As x → ρvi (x) − bx + ri Di is increasing, it implies that ρvi (x) < bx − ri Di on (Di , x ˆi )
and
ρvi (x) > bx − ri Di on (ˆ xi , +∞).
ii) Assume that x ˆi ∈ Si,j then we have ii.a) [ˆ xi , x ˆi + ε] ⊂ Si,j and x ˆi + δi,j − g is a left-boundary of Dj . ii.b) ρvi (ˆ xi ) = bˆ xi − ri Di + ∆i,j − bg and ∆i,j > 0. ii.c) ∀k ∈ IN − {i, j}, x ˆi 6∈ Si,k . Notice that the last equality implies that −ρP + bg < (b − rj )Dj . Remark 4.1 We have ρvi (ˆ xi ) ≥ bˆ xi − ri Di , ∀i ∈ In . Proof: i). We assume that x ˆi 6∈ Si . As Si is closed and x ˆi > Di , we can choose ε > 0 such that (ˆ xi − ε, x ˆi + ε) ∩ Si = ∅. Moreover, vi0 ≥ 1 and vi0 (ˆ xi ) = 1 so there exists a sequence (yn )n∈N ∈ (ˆ xi − ε, x ˆi ) ∩ Ci such that lim yn = x ˆi and vi00 (yn ) ≤ 0. We have n→+∞
0 ≥ vi00 (yn ) =
2 σ 2 yn2
ρvi (yn ) − (byn − ri Di )vi0 (yn )
and letting n going to infinity, we get 0 ≥ ρvi (ˆ xi ) − (bˆ xi − ri Di ) = leading to the desired equality.
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lim
y→ˆ xi ;y>ˆ xi
−Ai vi (y) ≥ 0,
Now let us show that (b − ri )Di > −ρP . Assume that (b − ri )Di ≤ −ρP , we then obtain the following inequality: vi (ˆ xi ) ≤
b (ˆ xi − Di ) − P, ρ
leading to a contradiction as ρ > b and vi (ˆ xi ) ≥ x ˆ i − Di − P . ii). We assume that x ˆi ∈ Si,j . ii.a). We first prove that [ˆ xi , x ˆi + ε] ⊂ Si,j and that x ˆj := x ˆi + δi,j − g is a left-boundary of Dj . Let y ∈ [ˆ xi , x ˆi + ε]. We have
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vj (y + δi,j − g) ≤ vi (y) = y − x ˆi + vi (ˆ xi ) = y − x ˆi + vj (ˆ xj ). On the other hand, vj0 ≥ 1 so y − x ˆi +vj (ˆ xj ) ≤ vj (y +δi,j −g). It follows that vj (y +δi,j −g) = vi (y) and [ˆ xi , x ˆi ] ⊂ Si,j . Moreover, we have proved that [ˆ xj , x ˆj + ε] ∈ Dj . We assume that there exists η > 0 such that (ˆ xj − η, x ˆj ) ⊂ Dj and show that it leads to a contradiction. Let x ∈ (ˆ xj − η, x ˆj ). We have
vj (x) = x − x ˆj + vj (ˆ xj ) = (x − δi,j + g) − x ˆi + vi (ˆ xi ) > vi (x − δi,j + g). The last inequality follows from the fact that x ˆi is a left-boundary of Di and contradicts the fact that vi is solution of equation (3.13). Hence, to show that x ˆj is a left-boundary of Dj it remains to prove that x ˆj > Dj . However, if it was not the case, we would have, vi (ˆ xi ) = vj (ˆ xi + δi,j − g), since x ˆi ∈ Si,j , = vj (Dj ) = −P. But x ˆi = x ˆj −δi,j +g = Di +g, leading to the contradiction −P = vi (Di ) < vi (Di +g) = −P . ii.b). We now prove that ρvi (ˆ xi ) = bˆ xi −ri Di +∆i,j −bg and ∆i,j := (b−rj )Dj −(b−ri )Di > 0. From Lemma 4.1, we know that x ˆj 6∈ Sj . Therefore it follows from step i) that (b − rj )Dj > −ρP and ρvj (ˆ xj ) = bˆ xj − rj Dj . We obtain ρvi (ˆ xi ) = ρvj (ˆ xi + δi,j − g) = ρvj (ˆ xj ) = bˆ xj − rj Dj = bˆ xi + b(δi,j − g) − rj Dj = bˆ xi − ri Di + ∆i,j − bg.
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(4.1)
As ρvi (ˆ xi ) − (bˆ xi − ri Di ) =
lim
y→ˆ xi ;y>ˆ xi
ρvi (y) − Li vi (y) ≥ 0, we have ρvi (ˆ xi ) ≥ bˆ xi − ri Di and
then ∆i,j ≥ bg > 0. ii.c). It remains to show that ∀k ∈ IN − {i, j}, x ˆi 6∈ Si,k . This fact is an elementary result as highlighted earlier because if there exists k ∈ IN − {i, j} such that x ˆi ∈ Si,k ∩ Si,j , it would implies that ∆i,k = ∆i,j . Relation (4.1) gives us the last equality. 2 Corollary 4.1 Let i ∈ IN . We have the following results: i) Assume that x∗i 6∈ Si . If (b − ri )Di > −ρP then ρvi (x∗i ) = bx∗i − ri Di else x∗i = Di .
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ii) Assume that x∗i ∈ Si,j where j ∈ IN − {i}. We have ∆i,j > 0 and (b − rj )Dj > −ρP + bg and ρvi (x∗i ) = bx∗i − ri Di + ∆i,j − bg. Proof: Most results follow directly from Lemma 4.3. We just have to prove that when we assume that x∗i 6∈ Si and (b − ri )Di ≤ −ρP then x∗i = Di . If it was not the case, we would have, following the proof of Lemma 4.3, ρvi (x∗i ) − (bx∗i − ri Di ) = 0. We then obtain vi (x∗i ) ≤
b ∗ (x − Di ) − P, ρ i
leading to a contradiction as ρ > b and vi (x∗i ) ≥ x∗i − Di − P . Hence if (b − ri )Di ≤ −ρP , we have x∗i = Di . 2 We now turn to the following result which basically states that when it is optimal to distribute dividend and/or to switch regime, then it is still optimal when the firm is richer. Lemma 4.4 Let (i, j) ∈ I2N such that i 6= j. If (x∗i , +∞) ∩ Si,j 6= ∅ then there exists ∗ ∈ [x∗ , +∞) such that yi,j i ∗ ∗ ∗ [x∗i , +∞) ∩ Si,j = [yi,j , +∞) and ρvi (yi,j ) = byi,j − ri Di + ∆i,j − bg. ∗ = inf [x∗ , +∞) ∩ S ˚i,j . If x∗ ∈ Si,j , the result has been proved in Proof: We set yi,j i i ∗ . Let y > y ∗ . Using the same argument as in Corollary 4.1. We now assume that x∗i < yi,j i,j ∗ , +∞) ⊂ S . ii) of Corollary 4.1, we may get vj (y + δi,j − g) = vi (y) and [yi,j i,j ∗ Moreover, we know that yi,j + δi,j − g 6∈ Sj and Sj is a closed set, so there exists ε > 0 such ∗ − ε, y ∗ ] ∩ S = ∅ where we set y ∗ = y∗ + δ ∗ ˚ that [¯ yi,j ¯i,j ¯i,j j i,j − g. As yi,j ∈ Di , we can find a i,j ∗ ∗ sequence (yk )k∈N with values in [¯ yi,j − ε, y¯i,j ], such that yk 6∈ Dj (if not we may obtain a ˚i,j .), i.e. contradiction by straightforwardly showing that y ∗ > inf [x∗ , +∞) ∩ S i,j
∀k ∈ N, yk ∈ Cj
and
i
∗ lim yk = y¯i,j .
k→+∞
We finally obtain σ 2 yk2 00 v (yk ) 2 j ∗ − rj Dj )vj0 (¯ yi,j ).
0 = ρvj (yk ) − (byk − rj Dj )vj0 (yk ) − ∗ ∗ = ρvj (¯ yi,j ) − (b¯ yi,j
15
∗ ) = v (y ∗ ) and v 0 (¯ ∗ 0 ∗ ∗ Using vj (¯ yi,j i i,j j yi,j ) = vi (yi,j ) = 1 (from Lemma 4.1 and yi,j ∈ Di ), we may obtain the desired results and conclude the proof. 2
We now establish an important result in determining the description of the switching regions. The following Theorem states that it is never optimal to expand its operation, i.e. to make investment, through debt financing, should it result in a lower “drift” ((b − ri )Di ) regime. However, when the firm’s value is low, i.e. with a relatively high bankruptcy risk, it may be optimal to make some divestment, i.e. sell parts of the company, and use the proceedings to lower its debt outstanding, even if it results in a regime with lower “drift”. In other words, to lower the firm’s bankruptcy risk, one should try to decrease its volatility, i.e. the diffusion coefficient. In our model, this clearly means making some debt repayment in order to lower the firm’s volatility, i.e. σXt .
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Theorem 4.1 Let i, j ∈ IN such that (b−rj )Dj > (b−ri )Di . We have the following results: ˚j = (x∗ , +∞). 1) x∗j 6∈ Sj,i and D j ˚j,i ⊂ (Dj + g, x∗ ). Furthermore, if Dj < Di , then S ˚j,i = ∅. 2) S j Proof: 1). Since (b − rj )Dj > (b − ri )Di , we have ∆j,i < 0. It follows from part ii) of Corollary ˚j . There exists ε > 0 such that (y − ε, y + ε) ⊂ D ˚j . For 4.1 that x∗j 6∈ Sj,i . Let y ∈ D x ∈ (y − ε, y + ε), we have 0 ≤ −Aj vj (x) = ρvj (x) − (bx − rj Dj ). Hence, ρvj (x) ≥ bx − rj Dj and y ≥ x∗j . ˚j,i . we first need to prove that y < x∗ . 2). Let us assume that there exists y ∈ S j Let’s assume that y ≥ x∗j . From Lemma 4.4, we know that [y, +∞) ⊂ Sj,i . We set ˚j,i ∩ Dj . As x∗ 6∈ Sj , we have x∗ ≤ s∗ . On the other hand, it is easy to see that s∗j,i = inf S j j j,i ∗ sj,i + δj,i − g = x∗i and x∗i 6∈ Si . We obtain ρvj (s∗j,i ) = ρvi (x∗i ) = bx∗i − ri Di = bs∗j,i − rj Dj − (∆i,j + bg) < bs∗j,i − rj Dj < ρvj (x∗j ). We may deduce that x∗j > s∗j,i ,, which contradicts the fact that x∗j ≤ s∗j,i , so y < x∗j . ˚j,i = ∅. We now prove that if Dj < Di , S ˚j,i . From the first step, we know that x 6∈ Dj . We deduce Assume that there exists x ∈ S from Lemma 4.1 that x ¯ := x + δj,i − g ∈ Ci then we have
16
1 2 2 00 σ x vj (x) + (bx − rj Dj ) vj0 (x) ≤ ρvj (x) 2 = ρvi (¯ x) 1 2 2 00 = σ x ¯ vi (¯ x) + (b¯ x − ri Di ) vi0 (¯ x) 2 1 2 2 00 = σ x ¯ vj (x) + (b¯ x − ri Di ) vj0 (x). 2 Combining these equations, we get 0≤
σ2 2 (¯ x − x2 )vj00 (x) − (∆i,j + bg)vj0 (x). 2
As x ¯2 − x2 ≥ 0 (using Dj < Di and Assumption (2.5)), (∆i,j + bg)vj0 (x) > 0 and vj00 < 0 on
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r D
r D
[ j b j , x∗j ), we necessarily have x ∈ (Dj + g, j b j ∧ x∗j ). Therefore, if b ≥ rj , Sj,i = ∅. ˚j,i 6= ∅. Let Sj,i = sup S ˚j,i and if we set S¯j,i := Sj,i +δj,i −g, Now, we assume that b < rj and S it follows that σ 2 ¯2 − 2 (S − Sj,i )vj00 (Sj,i ) − (∆i,j + bg)vj0 (Sj,i ). 0≤ 2 j,i Hence, we have ! 2 S¯j,i 0 ≤ ρvj (Sj,i ) − bS¯j,i − ri Di + ¯2 (∆i,j + bg) vj0 (Sj,i ) 2 Sj,i − Sj,i ! Sj,i 2 (4.2) (∆i,j + bg) vj0 (Sj,i ). ≤ ρvj (Sj,i ) − bSj,i − rj Dj + ¯2 2 Sj,i − Sj,i On the other hand, we have (Sj,i , x∗j ) ⊂ Cj so 2 σ 2 Sj,i Sj,i 2 + (∆i,j + bg)vj0 (Sj,i ). ) − ¯2 0≤ vj00 (Sj,i 2 2 Sj,i − Sj,i + Especially, we have vj00 (Sj,i ) > 0. Moreover, vj is a C 2 function and vj0 > 1 on (Sj,i , x∗j ), it follows that there exists y ∈ (Sj,i , x∗j ) such that vj00 (y) = 0 since v 0 (x∗j ) = 1. We set r D
yj = inf{y ∈ (Sj,i , x∗j ) : vj00 (y) ≤ 0}. As vj00 ≤ 0 on [ j b j , +∞), we know that yj ≤ We have vj00 (yj ) = 0 and yj ∈ Cj , so we can assert that h(yj ) = 0 where we have set
rj Dj b .
h(x) = (bx − rj Dj )vj0 (x) − ρvj (x). On (Sj,i , yj ), we have vj00 > 0 so h is decreasing. Indeed, we have h0 (x) = (bx − rj Dj )vj00 (x) − (ρ − b)vj0 (x) ≤ 0. Finally, this proves that ρvj (Sj,i ) ≤ (bSj,i − rj Dj )vj0 (Sj,i ). Reporting this in the inequality (4.2), we get Sj,i 2 (∆i,j + bg)vj0 (Sj,i ). 0 ≤ − ¯2 2 Sj,i − Sj,i 2 > S2 , ∆ 0 ˚ This is impossible as S¯j,i i,j + bg > 0 and vj (Sj,i ) ≥ 1. In conclusion, Sj,i = ∅. 2 j,i We now turn to an important corollary.
17
Corollary 4.2 Let m ∈ IN such that (b − rm )Dm = maxi∈IN (b − ri )Di . ˚m = (x∗ , +∞). 1) x∗m 6∈ Sm and D m 2) For all i ∈ IN − {m}, we have: ˚m,i = ∅. i) If Dm < Di , S ˚m,i ⊂ (Dm + g, x∗m ). Furthermore, if b ≥ ri , then S ˚m,i ⊂ ii) If Di < Dm , S (Dm + g, (a∗i + δi,m + g) ∧ x∗m ), where a∗i is the unique solution of the equation ρvi (x) = (bx − ri Di )vi0 (x). We further have a∗i 6= x∗i .
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Proof: The only point left to show is 2.ii). We now assume that there exists i ∈ IN − {m} such ˚m,i 6= ∅ that Di < Dm , b ≥ ri and S We prove that the equation ρvi (x) = (bx − ri Di )vi0 (x) admits a unique solution a∗i and ˚m,i ⊂ (Dm + g, a∗ + δm,i − g). prove that S i ˚m,i . It follows from the first step that S ˚m,i ∩ Dm = ∅. Hence, from Lemma 4.1, Let x ∈ S we have x := x + δm,i − g ∈ Ci . We obtain σ 2 x2 00 v (x) + (bx − rm Dm )vi0 (x) − ρvi (x) 2 i σ2 = Ai vi (x) + (x2 − x2 )vi00 (x) + (∆i,m + bg)vi0 (x) 2 x2 − x2 = − Hi (x), x2
0 ≥
where we have set Hi (x) = bx − ri Di −
x2 (∆i,m + bg) vi0 (x) − ρvi (x). (x + δi,m + g)2 − x2
Hence, we have ˚m,i ⊂ {x ∈ (Dm + g, +∞) : Hi (x) ≥ 0} ⊂ {x ∈ (Dm + g, +∞) : Gi (x + δm,i − g) ≤ 0}, S where we set Gi (y) = ρvi (y) − (by − ri Di )vi0 (y). We notice that, for all y ∈ (Di , +∞), Gi (y) ≥ b ≥ ri , it follows that G0i (y) ≥ (ρ − b)vi0 (y) −
σ 2 x2 00 2 vi (y).
Recalling our assumption that
2(by − ri Di ) 2(by − ri Di ) Gi (y) > − Gi (y). 2 2 σ y σ2y2
As Gi is continuous on (Di , +∞) and Gi (Di ) < 0, it implies that the equation Gi (y) = 0 ˚m,i ⊂ (Dm + admits a unique solution which will be denoted by a∗i . Therefore, we have S g, a∗i + δi,m + g). Furthermore, from Corollary 4.1, we either have G(x∗i ) = 0 or G(x∗i ) > 0. As such, we deduced that a∗i ∈ (Di , x∗i ).
We now turn to the following results ordering the left-boundaries regions (Di )i∈IN . 18
(x∗i )i∈IN
2 of the dividend
Proposition 4.1 Consider i, j ∈ IN , such that (b − ri )Di < (b − rj )Dj . We always have x∗i + δi,j − g ≤ x∗j unless there exists a regime k such that (b − rj )Dj < (b − rk )Dk and x∗i ∈ Si,k , then we have x∗j − δi,j + g < x∗i < x∗k − δi,k + g. Proof: First, we assume that x∗i 6∈ Si . From Lemma 4.3, we know that ρvi (x∗i ) = bx∗i −ri Di . On the other hand, we have ρvi (x∗j − (δi,j − g)) ≥ ρvj (x∗j ) ≥ bx∗j − rj Dj ≥ b(x∗j − (δi,j − g)) − ri Di + ∆i,j − bg
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> b(x∗j − (δi,j − g)) − ri Di . Hence, we have x∗i + δi,j − g < x∗j . Now, we assume that there exists k ∈ IN − {i} such that x∗i ∈ Si,k . If k = j, we have x∗i + δi,j − g = x∗j . If k 6= j, we have x∗i + δi,k − g = x∗k and ρvi (x∗i ) = bx∗i − ri Di + ∆i,k − bg = b(x∗i + δi,k − g) − rk Dk . On the other hand, we have ρvi (x∗j − (δi,j − g)) ≥ ρvj (x∗j ) = bx∗j − rj Dj = b(x∗j − (δi,j − g)) − ri Di + ∆i,j − bg = b x∗j − (δi,j − g) + (δi,k − g) − rk Dk + ∆k,j . If (b − rj )Dj > (b − rk )Dk , i.e. ∆k,j > 0, then we have ρvi (x∗j − (δi,j − g)) > b x∗j − (δi,j − g) + (δi,k − g) − rk Dk . Hence, we have x∗i + δi,j − g < x∗j . However, in the case that (b − rj )Dj < (b − rk )Dk , then ρvi (x∗j − (δi,j − g)) < b x∗j − (δi,j − g) + (δi,k − g) − rk Dk . 2
Hence, we have x∗i + δi,j − g > x∗j .
5
The two regime-case
Before investigating the two-regime case, for the sake of completeness, we give the results in the case where there is no regime change, i.e. the firm’s debt level remains constant. Proposition 5.1 The value function Vˆ is C 2 on (D1 , ∞). There exists x ˆ ≥ D such that ˆ = (ˆ Cˆ = (D, x ˆ), and D x, ∞). Furthermore, If (b − rD) > −ρP , then, on Cˆ = (D, x ˆ), Vˆ is the unique solution (in the classical sense) to ρv − Lv = 0 19
and Vˆ (x)
=
where Vˆ (ˆ x)
=
x−x ˆ + Vˆ (ˆ x), bˆ x − rD . ρ
x≥x ˆ
If (b − rD) ≤ −ρP , then x ˆ = D and the optimal value function V (x) = x − D − P . 2
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This result directly derives from Corollary 4.1.
Throughout this section, we now assume that N = 2, in which case, we will get a complete description of the different regions. We will see that the most important parameter to consider is the so-called “drifts” ((b − ri )Di )i=1,2 and in particular their relative positions. To avoid cases with trivial solution, i.e. immediate consumption, we will assume that −ρP < (b − ri )Di , i = 1, 2. We now distinguish the two following cases: (b − r2 )D2 < (b − r1 )D1 and (b − r1 )D1 < (b − r2 )D2 . Throughout Theorem 5.1 and Theorem 5.2, we provide a complete resolution to our problem in each case. Theorem 5.1 We assume that (b − r2 )D2 < (b − r1 )D1 . We have ˚1 = ∅ where ρv1 (x∗ ) = bx∗ − r1 D1 . C1 = [D1 , x∗1 ), D1 = [x∗1 , +∞), and S 1 1 1) If S2 = ∅ then we have C2 = [D2 , x∗2 ), and D2 = [x∗2 , +∞) where ρv2 (x∗2 ) = bx∗2 − r2 D2 . 2) If S2 6= ∅ then there exists y2∗ such that S2 = [y2∗ , +∞) and we distinguish two cases a) If x∗2 + δ2,1 − g < x∗1 , then y2∗ > x∗2 , y2∗ = x∗1 + δ1,2 + g and C2 = [D2 , x∗2 ), and D2 = [x∗2 , +∞) where ρv2 (x∗2 ) = bx∗2 − r2 D2 . b) If x∗2 + δ2,1 − g = x∗1 then y2∗ ≤ x∗2 , ρv2 (x∗2 ) = bx∗2 − r2 D2 + ∆2,1 − bg. We define a∗2 as the solution of ρv2 (a∗2 ) = ba∗2 − r2 D2 and have two cases i) If a∗2 6∈ D2 , we have D2 = [x∗2 , +∞)
and
C2 = [D2 , y2∗ ).
ii) If a∗2 ∈ D2 , there exists z2∗ ∈ (a∗2 , y2∗ ) such that D2 = [a∗2 , z2∗ ] ∪ [x∗2 , +∞)
and
C2 = [D2 , a∗2 ) ∪ (z2∗ , y2∗ ).
Remark 5.1 Theorem 5.1 clearly states that it is never optimal to make growth investment through debt financing when it results in lower “drift” (b − ri )Di . However, when the firm value process exceeds the threshold, yi∗ , it may be optimal to switch to a lower debt regime should it result in a higher “drift” (b − ri )Di . 20
Figure 1: Switching regions: case (b − r1 )D1 > (b − r2 )D2 . Div2
C2
Regime 2
x
D2
D1
D2
D1
C ase 1
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S2
x2*
z 2*
y2* Div2
Regime 1
x
x1*
a
* 2
C1
Div1
C2
Regime 1
C2 Div2 C2
Regime 2
* 2
x1*
Div1
C ase 2
Proof: From Theorem 4.2, we have D1 = [x∗1 , +∞) where ρv1 (x∗1 ) = bx∗1 − r1 D1 and S1 = ∅. ˚2 = ∅. From Corollary 4.1, we know that ρv2 (x∗ ) = bx∗ − r2 D2 . If 1) We assume that S 2 2 ˚2 ∩ (D2 , x∗ ), we would have 0 ≤ ρv2 (x) − (bx − r2 D2 ) but this is there exists x ∈ D 2 impossible for x < x∗2 . Hence we have C2 = [D2 , x∗2 ), and D2 = [x∗2 , +∞). ˚2 6= ∅ and set y ∗ = inf S ˚2 . We first prove that S2 = [y ∗ , +∞). 2) Now, we assume that S 2 2 We define the following function ( v2 (x) if D2 ≤ x < y2∗ V2 (x) = v1 (x + δ2,1 − g) if y2∗ ≤ x. V2 is a C 1 function on [D2 , +∞). We prove that V2 is the solution of the variational inequality satisfied by v2 . We obviously have V20 (x) ≥ 1 and V2 (x) ≥ v1 (x + δ2,1 − g). Moreover, we have ( v2 (x + δ1,2 − g) ≤ v1 (x) if D2 ≤ x < y2∗ + δ2,1 + g V2 (x + δ1,2 − g) = v1 (x − 2g) ≤ v1 (x) if y2∗ + δ2,1 + g ≤ x. It remains to prove that A2 V2 (x) ≤ 0 on [y2∗ , +∞). For x ≥ y2∗ , we set x = x + δ2,1 − g and we have σ 2 x2 00 v (x) + (bx − r2 D2 )v10 (x) − ρv1 (x) 2 1 σ2 = A1 v1 (x) + (x2 − x2 )v100 (x) − (∆2,1 + bg)v10 (x) 2 σ2 2 ≤ (x − x2 )v100 (x). 2
A2 V2 (x) =
21
x
As D1 < D2 , we have x2 > x2 . On the other hand, we have seen that v1 is concave so we can assert that A2 V2 (x) ≤ 0 on [y2∗ , +∞). This proves that v2 = V2 and especially ˚2 = (y ∗ , +∞). that S 2 a) If x∗2 + δ2,1 − g < x∗1 , then using Proposition 4.1, we have y2∗ > x∗2 and x∗2 6∈ S2 , so it follows from Corollary 4.1 that ρv2 (x∗2 ) = bx∗2 − r2 D2 . Moreover, we have ˚2 = (x∗ , +∞) and from Lemma 4.4, we have y ∗ = x∗ + δ1,2 + g. D 2 2 1 b) If x∗2 + δ2,1 − g = x∗1 , then using Proposition 4.1, we have y2∗ ≤ x∗2 . In this case, x∗2 ∈ S2 and it follows from Corollary 4.1 that ρv2 (x∗2 ) = bx∗2 − r2 D2 + ∆2,1 − bg. We define a∗2 as the solution of ρv2 (a∗2 ) = ba∗2 − r2 D2 and distinguish two cases: i) If a∗2 6∈ D2 , it follows from Lemma 4.3 that
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D2 = [x∗2 , +∞) and C2 = [D2 , x∗2 ). ii) Finally, we assume that a∗2 ∈ D2 . We set z2∗ = inf{x ≥ a∗2 : v20 > 1} and have D2 = [a∗2 , z2∗ ] ∪ [x∗2 , +∞) and C2 = [D2 , a∗2 ) ∪ (z2∗ , x∗2 ). 2 We now turn to the case where (b − r1 )D1 < (b − r2 )D2 . Theorem 5.2 We assume that (b − r1 )D1 < (b − r2 )D2 , 1) we have D2 = [x∗2 , +∞) where ρv2 (x∗2 ) = bx∗2 − r2 D2 ˚2 = ∅ or there exist s∗ , S ∗ ∈ (D2 + g, x∗ ) such that S ˚2 = (s∗ , S ∗ ). S 2 2 2 2 2 ˚1 = ∅ then we have 2) If S C1 = [D1 , x∗1 ), and D1 = [x∗1 , +∞) where ρv1 (x∗1 ) = bx∗1 − r1 D1 . ˚1 = ˚1 = (y ∗ , +∞) 3) If S 6 ∅ there exists y1∗ such that S 1 a) If x∗1 + δ1,2 − g < x∗2 , then y1∗ > x∗1 , y1∗ = x∗2 + δ2,1 + g and C1 = [D1 , x∗1 ), and D1 = [x∗1 , +∞) where ρv1 (x∗1 ) = bx∗1 − r1 D1 . b) If x∗2 + δ2,1 − g = x∗1 , then y1∗ ≤ x∗1 , ρv1 (x∗1 ) = bx∗1 − r1 D1 + ∆1,2 − bg. We define a∗1 as the solution of ρv1 (a∗1 ) = ba∗1 − r1 D1 and have two cases. i) If a∗1 6∈ D1 , we have D1 = [x∗1 , +∞) 22
and
C1 = [D1 , y1∗ ).
ii) If a∗1 ∈ D1 , there exists z1∗ ∈ (a∗1 , y1∗ ) such that D1 = [a∗1 , z1∗ ] ∪ [x∗1 , +∞)
C1 = [D1 , a∗1 ) ∪ (z1∗ , y1∗ ).
and
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Remark 5.2 Theorem 5.2 states that when the firm’s value is sufficiently high (above y1∗ threshold), it’s optimal to switch to a higher-debt regime which operates at a higher drift (b−ri )Di , see figure 2, case 2. However, when the firm is too small, it may be optimal not to postpone dividend payment and to operate under as a medium size company (cash-reserve lower than the threshold a∗1 , as in figure 2, case 3), i.e. to distribute dividend, whenever the cash-reserve exceed the threshold a∗1 . However, one should not switch to a lower drift regime unless it lowers the firm’s bankruptcy risk. It may happen, when the value firm dangerously approaches bankruptcy threshold, i.e. when its cash reserve stands between s∗2 and S2∗ .
Figure 2: Switching regions: case (b − r1 )D1 < (b − r2 )D2 . Regime 2
D2
x
* 2
Div2
C2
Regime 2
x
D2
S2
Div1
C2 S 2*
s2*
x2*
Div2
Div1
Regime 1
C1
Regime 1 * 1
x
D1
D1
Div1
C ase 1
C2
Regime 2
D2
S2
Div1
s
* 2
S1 * 1
x
* 1
y
Div1
C ase 2
C2 S
* 2
x2*
Div2
x
Div1
Div1
C1
Regime 1
D1
* 1
a
S1
C1 z1*
y1* x1*
Div1
C ase 3
Proof: Throughout the proof, for x ∈ R, we set x ¯ = x + δ1,2 − g and x = x + δ2,1 − g. Notice that we have x < x < x ¯.
23
x
1.) From Theorem 4.2 we have D2 = [x∗2 , +∞) where ρv2 (x∗2 ) = bx∗2 − r2 D2 ˚2 ⊂ (D2 + g, (a∗1 + δ2,1 − g) ∧ x∗2 ), S
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where a∗1 is the unique solution of the equation ρv1 (x) = (bx − r1 D1 )v10 (x). ˚2 6= ∅. We set s∗ = inf S ˚2 and S ∗ = sup S ˚2 . Now we prove that S ˚2 = (s∗ , S ∗ ). Assume that S 2 2 2 2 On [D2 , +∞), we define the following function: if x < s∗2 v2 (x) V2 (x) = v1 (x + δ2,1 − g) if s∗2 ≤ x ≤ S2∗ v (x) if x > S2∗ . 2 V2 is a continuous function on [D2 , +∞) and it is easy to see that V20 ≥ 1. For all x ∈ [D2 , +∞), we have V2 (x) ≥ v1 (x+δ2,1 −g) and for x+δ1,2 −g ∈ [s∗2 , S2∗ ], V2 (x+δ1,2 −g) = v1 (x − 2g) < v1 (x). We now prove that A2 V2 ≤ 0. Let x ∈ [s∗2 , S2∗ ], we have x2 − x2 σ 2 x2 00 x2 0 A2 V2 (x) = A1 v1 (x) + v (x) + 2 (∆1,2 + bg)v1 (x) x2 2 1 x − x2 x2 − x2 = − H1 (x). x2 We recall that H1 (x) = bx − r1 D1 −
x2 (∆1,2 + bg) v10 (x) − ρv1 (x). (x + δ1,2 + g)2 − x2
˚2 ⊂ {x ∈ (Dm , +∞) : H1 (x) ≥ 0}. We have seen in the proof of Theorem 4.2 that S Especially, we have H1 (S2∗ ) ≥ 0, with S2∗ ≤ a∗1 . Now, we prove that H1 is decreasing on (D1 , a∗1 ). As H1 is continuous, this will lead to A2 V2 ≤ 0 and allows us to assert that ˚2 = (s∗ , S ∗ ). v2 = V2 and especially that S 2 2 We may rewrite H1 : H1 (x) = U1 (x) − G1 (x), where G1 (x) = ρv1 (x) − (bx − r1 D1 )V10 (x) x2 U1 (x) = − (∆1,2 + bg)v10 (x). (x + δ1,2 + g)2 − x2 From the proof of Theorem 4.2, we have G1 is strictly non-decreasing on (D1 , a∗1 ). Furthermore, a straight study of the function U1 and recalling that on (D1 , a∗1 ], v100 (x) ≤ 0, we may obtain that U1 is non-increasing. As such, H1 is strictly non-increasing. ˚1 = ∅, then x∗ 6∈ S1 . Using the arguments from 1.) of Theorem 4.2, we may obtain 2.) If S 1 C1 = [D1 , x∗1 ), and D1 = [x∗1 , +∞) where ρv1 (x∗1 ) = bx∗1 − r1 D1 .
24
˚1 6= ∅. We set y ∗ = inf S˚1 and prove that S˚1 = (y ∗ , +∞). 3.) We now assume that S 1 1 On [D1 , +∞), we define the following function: ( v1 (x) V1 (x) = v2 (x + δ1,2 − g)
if if
x < y1∗ y1∗ ≤ x.
V1 is a C 1 function on [D1 , +∞) and it is easy to see that V10 ≥ 1. For all x ∈ [D1 , +∞), we have V1 (x) ≥ v2 (x + δ1,2 − g) and for x ≥ y1∗ , V1 (x + δ2,1 − g) = v2 (x − 2g) < v2 (x). We now prove that A1 V1 ≤ 0. Let x ∈ [y1∗ , +∞), we have (¯ x2 − x2 )σ 2 00 0 v2 (¯ x) − ∆− x) 1,2 v2 (¯ 2 (¯ x2 − x2 )σ 2 00 0 ≤ − v2 (¯ x) − ∆− x). 1,2 v2 (¯ 2
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A1 V1 (x) = A2 v2 (¯ x) −
If x ¯ ∈ D2 , we obviously have A1 V1 (x) ≤ 0. Assume that x ¯ ∈ C2 , then we have A1 V1 (x) =
x ¯ 2 − x2 H2 (¯ x). x ¯2
As H2 is decreasing, we have H2 (¯ x) ≤ H2 (y1∗ ) ≤ 0 so A1 V1 (x) ≤ 0. Finally, we assume that ˚2 . In this case, we have x ¯∈S (x − g)2 σ 2 (x − g)2 00 (x − g)2 − 0 A V (x) ≤ − v (x − g) − ∆ v (x − g) 1 1 1 x ¯ 2 − x2 2 x ¯2 − x2 1,2 1 (x − g)2 − = −ρv1 (x − g) + b(x − g) − r1 D1 − 2 ∆ v 0 (x − g) x ¯ − x2 1,2 1 (x − g)2 − v 0 (¯ x) = −ρv2 (¯ x) + b¯ x − r2 D2 − ∆1,2 − 2 ∆ x ¯ − x2 1,2 2 2 x ¯ − (x − g)2 − = H2 (¯ x) + ∆1,2 − ∆1,2 v20 (¯ x) x ¯ 2 − x2 2 x ¯ − (x − g)2 − ≤ ∆1,2 − ∆1,2 v20 (¯ x). x ¯2 − x2 However, we have (¯ x2 − (x − g)2 )∆− x2 − x2 )∆1,2 = −bg(¯ x2 − x2 ) + g(2x − g))(∆1,2 − bg) 1,2 − (¯ 2 = g 2(r1 D1 − r2 D2 )x − g∆− − b(δ − g) 1,2 1,2 ≤ 0. Therefore, A1 V1 ≤ 0 on (D1 , +∞). This allows us to assert that v1 = V1 and especially ˚1 = (y ∗ , +∞). that S 1 a) If x∗1 + δ1,2 − g < x∗2 , then using Proposition 4.1, we have y1∗ > x∗1 and x∗1 6∈ S1 . So ˚1 = it follows from Corollary 4.1 that ρv1 (x∗1 ) = bx∗1 − r1 D1 . Moreover, we have D ∗ ∗ ∗ (x1 , +∞) and from Lemma 4.4, we have y1 + δ1,2 − g = x2 . 25
b) If x∗1 + δ1,2 − g = x∗2 , then using Proposition 4.1, we have y1∗ ≤ x∗1 . In this case, x∗1 ∈ S1 and it follows from Corollary 4.1 that ρv1 (x∗1 ) = bx∗1 − r1 D1 + ∆1,2 − bg. We define a∗1 as the solution of ρv1 (a∗1 ) = ba∗1 − r1 D1 and distinguish two cases: i) If a∗1 6∈ D1 , it follows from Lemma 4.3 that D1 = [x∗1 , +∞) and C1 = [D1 , x∗1 ). ii) Finally, we assume that a∗1 ∈ D1 . We set z1∗ = inf{x ≥ a∗1 : v10 > 1} and have D1 = [a∗1 , z1∗ ] ∪ [x∗1 , +∞) and C1 = [D1 , a∗1 ) ∪ (z1∗ , x∗1 ).
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2 Remark 5.3 The arguments used to obtain the above results in Theorems 5.1 and 5.2 in the two-regime problem may also apply to higher regime problems although the required analysis involved would be much lengthier and depends on many more parameters. It is particularly the case when we reconsider our initial multi-switching problem with a slight but realistic change to our initial model: we only allow the firm to change, i.e. increase or repay, its debt level to the one immediately above or below. This latter case may be subject to further studies in the future, but we may already obtain: • The elementary decomposition of the switching regions becomes Si = Si,i−1 ∪ Si,i+1 . The system of variational inequalities becomes : 0 min −Ai vi (x) , vi (x) − 1 , vi (x) − max vj (x + (j − i)D − g) ≥ 0, x > Di (5.1) j=i−1,i+1
• With the exception of part i) of Lemma 4.1, all the other results still hold. For results obtained by using part i) of Lemma 4.1, it suffices to slightly modify the existing proofs. • The complete solution to our modified problem may be obtained by applying iteratively the results from Theorem 5.1 and 5.2. Some numerical illustrations: Below are some numerical analysis on value functions as defined in equation (2.7) versus the real value for shareholders as defined in equation (2.8) for different values of P , see Figure 3. Finally, Figure 4 shows the contribution of the management team in creating value for shareholders for different values of P .
26
Figure 3: Optimal values for managers (vi ) Vs shareholders’ value (ui ) for increasing penalty P .
R eal equity values u i (x) for different ↓ P
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V alue functions vi (x ) for different ↓ P
Figure 4: Excess shareholders’ values Vs immediate consumption.
W i ( x) = ui ( x) − ( x − Di ) for different ↓ P
27
References [1] Bayraktar E. and M. Egami (2010) : “On the One-Dimensional Optimal Switching Problem", Mathematics of Operations Research, 2010, 35 (1), 140-159. [2] Brekke K. and B. Oksendal (1994) : “Optimal switching in an economic activity under uncertainty", SIAM J. Cont. Optim., 32, 1021-1036. [3] Brennan M. and E. Schwartz (1985) : “Evaluating natural resource extraction", J. Business, 58, 135-137.
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[4] Carmona R. and M. Ludkovski (2010) : “Valuation of Energy Storage: An Optimal Switching Approach”, Quantitative Finance, 10(4), 359-374, 2010. [5] Choulli T. , Taksar M. and X.Y. Zhou (2003) : “A diffusion model for optimal dividend distribution for a company with constraints on risk control", SIAM J. Cont. Optim., 41, 1946-1979. [6] Décamps J.P. and S. Villeneuve (2007) : “Optimal dividend policy and growth option", Finance and Stochastics, 11, 3-27. [7] Dai,M., Q. Zhang, and Q. Zhu (2010): “Trend Following Trading under a Regime Switching Model", SIAM Journal on Financial Mathematics 1, 780-810. [8] Djehiche B., S. Hamadène and A. Popier (2009) : “A finite horizon optimal switching problem", SIAM J. Control and Optimization, Volume 48, Issue 4, pp. 2751-2770. [9] Dixit A. and R. Pindick (1994) : Investment under uncertainty, Princeton University Press. [10] Duckworth K. and M. Zervos (2001) : “A model for investment decisions with switching costs", Annals of Applied Probability, 11, 239-250. [11] Guo X. and H. Pham (2005) : “Optimal partially reversible investment with entry decision and general production function", Stochastic Processes and their Applications, 115, 705-736. [12] Guo, X. and Tomecek, P. (2008) : “Connections between singular control and optimal switching", SIAM J. Control Optim., 47(1): 421-443. [13] Hamadène S. and M. Jeanblanc (2007) : “On the Starting and Stopping Problem: Application in reversible investments”, Mathematics of Operation Research, vol. 32 no. 1 182-192. [14] Hu Y. and S. Tang (2010) : “Multi-dimensional BSDE with oblique reflection and optimal switching", Prob. Theory and Related Fields, 147(1-2) 89-121. . [15] Jeanblanc M. and A. Shiryaev (1995) : “Optimization of the flow of dividends", Russian Math. Survey, 50, 257-277.
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[16] I. Karatzas, D. Ocone, H. Wang and M. Zervos (2000) : “Finite-Fuel Singular Control with Discretionary Stopping", Stochastics and Stochastics Reports, 71, pp.1-50 [17] Lon P. C. and M. Zervos (2011) :‘A model for optimally advertising and launching a product”, Mathematics of Operations Research, to appear. [18] Lungu E. M. and Oksendal B. (1997) : “Optimal harvesting from a population in a stochastic crowded environment",Mathematical Biosciences Volume 145, Issue 1, Pages 47-75 [19] Ly Vath V. and H. Pham (2007) : “Explicit solution to an optimal switching problem in the two-regime case", SIAM J. Cont. Optim., 46, 395-426.
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[20] Ly Vath V., H. Pham and S. Villeneuve (2008) : “A mixed singular/switching control problem for a dividend policy with reversible technology investment", Annals of Applied Probability, 2008, 18, pp. 1164-1200. [21] Pham H. (2007) : “On the smooth-fit property for one-dimensional optimal switching problem", Séminaire de Probabilités, vol XL, 187-201. [22] Pham H. (2009) : “Continuous-time Stochastic Control and Optimization with Financial Applications", Stochastic Modelling and Applied Probability, Vol. 61, 232p. [23] Pham H., V. Ly Vath, and X.Y. Zhou (2009) : “Optimal switching over multiple regimes", SIAM Journal on Control and Optimization, 48, 2217-2253. [24] Yang R. and Liu K. (2004) : “Optimal singular stochastic problem on harvesting system", Applied Mathematics E-Notes, 133-141. [25] Zervos M., T. Johnson, and F. Alazemi (2011) :“Buy-low and sell-high investment strategies", to appear in Mathematical Finance.
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Vathana LY VATH‡
†
Simone SCOTTI§
December 12, 2011
Abstract
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This paper concerns with the problem of determining an optimal control on the dividend and investment policy of a firm. We allow the company to make an investment by increasing its outstanding indebtedness, which would impact its capital structure and risk profile, thus resulting in higher interest rate debts. We formulate this problem as a mixed singular and switching control problem and use a viscosity solution approach combined with the smooth-fit property to get qualitative descriptions of the solution. We further enrich our studies with a complete resolution of the problem in the tworegime case. Keywords: stochastic control, optimal singular / switching problem, viscosity solution, smooth-fit property, system of variational inequalities, debt constraints. MSC2000 subject classification: 60G40, 91B70, 93E20.
∗
Université d’Evry, Laboratoire d’Analyse et Probabilités, France , [email protected]. This research benefitted from the support of the “Chaire Risque de Crédit”, Fédération Bancaire Française. ‡ ENSIIE and Université d’Evry, Laboratoire d’Analyse et Probabilités, France, [email protected]. § Université Paris Diderot, LPMA, France, [email protected]. †
1
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1
Introduction
The theory of optimal stochastic control problem, developed in the seventies, has over the recent years once again drawn a significance of interest, especially from the applied mathematics community with the main focus on its applications in a variety of fields including economics and finance. For instance, the use of powerful tools developed in stochastic control theory has provided new approaches and sometime the first mathematical approaches in solving problems arising from corporate finance. It is mainly about finding the best optimal decision strategy for managers whose firms operate under uncertain environment whether it is financial or operational, see [3] and [9]. A number of corporate finance problems have been studied, or at least revisited, with this optimal stochastic control approach. In this paper, we consider the problem of determining the optimal control on the dividend and investment policy of a firm. There are a number of research on this corporate finance problem. In [6], Décamps and Villeneuve study the interactions between dividend policy and irreversible investment decision in a growth opportunity and under uncertainty. We may equally refer to [20] for an extension of this study, where the authors relax the irreversible feature of the growth opportunity. In other words, they consider a firm with a technology in place that has the opportunity to invest in a new technology that increases its profitability. The firm self-finances the opportunity cost on its cash reserve. Once installed, the manager can decide to return back to the old technology by receiving some cash compensation. As in a large part of the literature in corporate finance, the above papers assume that the firm cash reserve follows a drifted Brownian motion. They also assume that the firm does not have the ability to raise any debt for its investment as it holds no debt in its balance sheet. In our study, as in the Merton model, we consider that firm value follows a geometric Brownian process and more importantly we consider that the firm carries a debt obligation in its balance sheet. However, as in most studies, we still assume that the firm assets is highly liquid and may be assimilated to cash equivalents or cash reserve. We allow the company to make investment and finance it through debt issuance/raising, which would impact its capital structure and risk profile. This debt financing results therefore in higher interest rate on the firm’s outstanding debts. Furthermore, we consider that the manager of the firm works in the interest of the shareholders, but only to a certain extent. Indeed, in the objective function, we introduce a penalty cost P and assume that the manager does not completely try to maximize the shareholders’ value since it applies a penalty cost in the case of bankruptcy. This penalty cost could represent, for instance, an estimated cost of the negative image upon his/her own reputation due to the bankruptcy under his management leadership. Mathematically, we formulate this problem as a combined singular and multiple-regime switching control problem. Each regime corresponds to a level of debt obligation held by the firm. In terms of literature, there are many research papers on singular control problems as well as on optimal switching control problems. One of the first corporate finance problems using singular stochastic control theory was the study of the optimal dividend strategy, see for instance [5] and [15]. These two papers focus on the study of a singular stochastic control problem arising from the research on optimal dividend policy for a firm whose cash
2
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reserve follows a diffusion model. Amongst singular stochastic control problems, we may equally refer to problems arising from mathematical biology, in particular studies on optimal harvesting strategies, see [18] and [24]. In the study of optimal switching control problems, a variety of problems are investigated, including problems on management of power station [4], [13], resource extraction [2], firm investment [10], marketing strategy [17], and optimal trading strategies [7], [25]. Other related works on optimal control switching problems include [1] and [19], where the authors employ respectively optimal stopping theory and viscosity techniques to explicitly solve their optimal two-regime switching problem on infinite horizon for one-dimensional diffusions. We may equally refer to [22], for an interesting overview of the area. In the above studies, only problems involving the two-regime case are investigated. There are still too few studies on the multi-regime switching problems. The main additional feature in the multiple regime problems consists not only in determining the switching region as opposed to the continuation region, but also in identifying the optimal regime to where to switch. This additional feature sharply increases the complexity of the multi-regime switching problems. Recently, Djehiche, Hamadène and Popier [8], and Hu and Tang [14] have studied optimal multiple switching problems for general adapted processes by means of reflected BSDEs, and they are mainly concerned with the existence and uniqueness of solution to these reflected BSDEs. In [23], the authors investigated an optimal multiple switching problem on infinite horizon for a general one-dimensional diffusion and used the viscosity techniques to provide an explicit characterization of the switching regions showing when and where it is optimal to change the regime. However, the studies that are most relevant to our problem are the one investigating combined singular and switching control problems. Recently an interesting connection between the singular and the switching problems was given by Guo and Tomecek [12]. In [20], the authors studied a problem which combines features of both optimal switching and singular control. They proved that the mixed problem can be decoupled in two pure optimal stopping and singular control problems and provided results which are of quasi-explicit nature. However, the switching part of this problem is limited to a two-regime problem. In this paper, we combine the difficulties of the multiple switching problem [23] with those of the mixed singular and two-regime switching problem [20]. In other words, not only do we have to determine the three regions comprising the continuation, dividend and switching regions, but also, within the switching region, we have to identify the regime to where to switch. The latter feature considerably increases the complexity of our problem. In terms of mathematical approach, we characterize our value functions as unique viscosity solution to an associated system of variational inequalities. Furthermore, we use viscosity and uniqueness results combined with smooth-fit properties to determine the solutions of our HJB system. The results of our analysis take qualitatively different forms depending on the parameters values. The plan of the paper is organized as follows. We define the model and formulate our stochastic control problem in the second section. In section 3, we characterize our problem as the unique viscosity solution to the associated HJB system and obtain some regularity properties, while, in section 4, we obtain qualitative description of our problem. Finally, in
3
section 5, we further enrich our studies with a complete resolution of the problem in the two-regime case.
2
The model
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We consider a firm whose value follows a process X. The firm also has the possibility to raise its debt level in order to satisfy its financial requirement such as investing in growth opportunities. It may equally pay down its debt. We consider an admissible control strategy α = (Zt , (τn )n≥0 , (kn )n≥0 ), where Z represents the dividend policy, the nondecreasing sequence of stopping times (τn ) the switching regime time decisions, and (kn ), which are Fτn -measurable valued in {1, ..., N }, the new value of debt regime at time t = τn . Let denote the process X x,i,α as the enterprise value of the company with initial value of x and initially operating with a debt level Di and which follow the control strategy α. However, in order to reflect the fact that a firm also holds a significant amount of debt obligation either to financial institutions, inland revenues, suppliers or through corporate bonds, we assume that the firm debt level may never get to zero. We assume that the firm assets is cash-like, i.e. the manager may dispose of some part of the company assets and obtain its equivalent in cash. In other words, the process X could be seen as a cash-reserve process used in most papers on optimal dividend policy, see for instance [6], [15]. We assume that the cash-reserve process X x,i,α , denoted by X when there is no ambiguity and associated to a strategy α = (Zt , (τn )n≥0 , (kn )n≥0 ), is governed by the following stochastic differential equation: dXt = bXt dt − rIt DIt dt + σXt dWt − dZt + dKt
(2.1)
where It =
X
kn 1τn ≤t 0, in the case of a holding company looking to liquidate one of its own affiliate or activity. In the case of the penalty, it mainly assumes that the manager does not completely try to maximize the shareholders’ value since it applies a penalty cost in the case of bankruptcy. We therefore define the value functions which the manager actually optimizes as follows "Z
#
T−
−ρt
vi (x) = sup E α∈A
e
−ρT
dZt − P e
,
x ∈ R, i ∈ {1, ..., N },
(2.7)
0
where A represents the set of admissible control strategies, and ρ the discount rate. The next step would be to compute the real value function ui of the shareholders. Indeed, once we obtain the optimal strategy, α∗ = (Zt∗ , (τn∗ )n≥0 , (kn∗ )n≥0 ) of the above problem (2.7), then we may compute the real shareholders’ value by following the strategy α∗ , as numerically illustrated in Figure 3 and 4: "Z ui (x) = Eα∗
#
T−
e
−ρt
dZt∗
,
x ∈ R, i ∈ {1, ..., N }
(2.8)
0
Remark 2.1 If b > ρ, the value functions is infinite for any initial value of x > Di and any regime i. The proof is quite straightforward. We simply consider a sequence of strategy controls which consists in doing nothing up to time tk , where (tk )k≥1 is strictly nondecreasing and goes to infinity when k goes to infinity, and then at tk , distribute Xtk − DItk in dividend and allow the company to become bankrupt. We then need to notice that lim E e−ρtk Xtxk = +∞. k→∞
For the rest of the paper, we now consider that the discount rate ρ is always bigger than the growth rate b.
5
3
Viscosity Characterization of the value functions
We first introduce some notations. We denote by Rx,i the firm value in the absence of dividend distribution and the ability to change the level of debt, fixed at Di . dRtx,i = [bRtx,i − ri Di ]dt + σRtx,i dWt , R0x,i = x
(3.1)
The associated second order differential operator is denoted Li : 1 Li ϕ = [bx − ri Di ]ϕ0 (x) + σ 2 x2 ϕ00 (x) 2
(3.2)
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Using the dynamic programming principle, we obtain the associated system of variational inequalities satisfied by the value functions: min −Ai vi (x) ,
vi0 (x)
− 1 , vi (x) − max vj (x + Dj − Di − g) = 0, x > Di , i ∈ IN j6=i
vi (Di ) = −P,
where the operator Ai is defined by Ai φ = Li φ − ρφ. We now state a standard first result for this system of PDE. Proposition 3.1 Let (ϕi )i∈IN smooth enough on (Di , ∞) such that ϕi (Di+ ) := lim ϕi (x) ≥ x↓Di
−P , and 0 min −Ai ϕi (x) , ϕi (x) − 1, ϕi (x) − max ϕj (x + Dj − Di − g) ≥ 0, for x > Di , i ∈ IN j6=i
where we set by convention ϕi (x) = −P for x < Di , then we have vi ≤ ϕi , for all i ∈ IN . Proof: Given an initial state-regime value (x, i) ∈ (Di , ∞) × IN , take an arbitrary control α = (Z, (τn ), (kn )) ∈ A, and set for m > 0, θm,n = inf{t ≥ T ∧ τn : Xtx,i,α ≥ m or Xtx,i,α ≤ DIt + 1/m} % ∞ a.s. when m goes to infinity. Apply then Itô’s formula to e−ρt ϕkn (Xtx,i,α ) between the stopping times T ∧ τn and τm,n+1 : = T ∧ τn+1 ∧ θm,n . Notice that for T ∧ τn ≤ t < τm,n+1 , Xtx,i stays in regime kn . Then, we have Z τm,n+1 x,i x,i −ρτm,n+1 −ρ(T ∧τn ) e ϕkn (Xτ − ) = e ϕkn (XT ∧τn ) + e−ρt (−ρϕkn + Lkn ϕkn )(Xtx,i )dt m,n+1 T ∧τn Z τm,n+1 Z τm,n+1 + e−ρt σϕ0kn (Xtx,i )dWt − e−ρt ϕ0kn (Xtx,i )dZtc T ∧τn T ∧τn h i X x,i −ρt + e ϕkn (Xt ) − ϕkn (Xtx,i ) , (3.3) − T ∧τn ≤t τn a.s., so that equation (3.3) holds true a.s. for all n, m (recall that ϕkn (XTx,i ) = −P ). 6
x,i Since ϕ0kn ≥ 1, we have by the mean-value theorem ϕkn (Xtx,i ) − ϕkn (Xtx,i − Xtx,i − ) ≤ Xt − = −(Zt − Zt− ) for T ∧ τn ≤ t < τm,n+1 .
By using also the supersolution inequality of ϕkn , taking expectation in the above Itô’s formula, and noting that the integrand in the stochastic integral term is bounded by a constant (depending on m), we have Z τm,n+1 h i x,i −ρ(T ∧τn ) −ρt c E e−ρτm,n+1 ϕkn (Xτx,i ) ≤ E e ϕ (X ) − E e dZ kn − t T ∧τn m,n+1 T ∧τn X − E e−ρt (Zt − Zt− ) , T ∧τn ≤t θr,i,0 ] → 0,
as x ↓ Di .
Notice that θr,i,0 < θr,i and combined with (3.8), we obtain P [θi > θr,i ] ≤ P [θi,0 > θr,i,0 ]. As such,
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P [θi > θr,i ] → 0,
as x ↓ Di .
(3.9)
Let α = (Z, (τn )n≥1 , kn≥1 ) be an arbitrary policy in A, and denote η = T ∧θr,i = T x,i,α ∧θr,i . For t ≤ η, from the definition of an admissible control, there is no regime shift. As such, for t ≤ η, we have Xtx,i ≤ Rtx,i ≤ Rtx,0 . We also have T x,i ≤ θi . We then write : "Z − # "Z − # " # Z T− T η −ρt −ρt −ρt E e dZt = E e dZt + E 1T >η e dZt 0
0
η
## e−ρt dZt Fθ− ≤ E E 1T >η r,i η "Z − ## " T e−ρt dZt Fθ− ≤ E 1T >θr,i E r,i θr,i ≤ E 1T >θr,i e−ρθr,i vi Xθx,i , − " "
Z
T−
(3.10)
r,i
where we also used in the second inequality the fact that on {T > η}, η = θr,i , and θr,i is a predictable stopping time. Now, since vi is nondecreasing, we have vi (Xθx,i − ) ≤ vi (Di + r). r,i
Moreover, recalling that T ≤ θi , inequalities (3.10) and (3.9) yield "Z − # T −ρt 0 ≤ E e dZt ≤ vi (Di + r)P[θi > θr,i ] −→ 0,
as x ↓ Di .
(3.11)
0
Furthermore, using the fact that T x,i ≤ θi ≤ θi,0 , we have h i E −P e−ρT ≤ −P E e−ρθi,0 Noticing that θi,0 is the hitting time of a drifted Brownian, it is straightforward that E e−ρθi,0 −→ 1, as x ↓ Di and recalling (3.11), we obtain "Z − # T −ρt −P ≤ vi (x) ≤ E e dZt − E P e−ρT 0
i h ≤ vi (r)P[θi > θr ] − P E e−ρθi,0 −→ −P, 9
as x ↓ Di .
We may therefore conclude that vi (Di+ ) = −P . b) We now turn to the continuity of the value functions vi . Let γ > 0 and x ∈ (Di , ∞). We set T γ = inf{t ≥ 0; Rtx,i ≥ x + γ}. We now consider a control strategy α = (Z, (τn ), kn ), where τ1 > T γ and Zt = 0, ∀ t < τ1 . Notice that ∀ t < τ1 , Xtx,i = Rtx,i . Applying the programming dynamic principle (DP), we obtain vi (x) ≥ E e−ρXT γ ∧T x,i vi (XT γ ∧T x,i ) ,
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therefore γ vi (x + γ) − vi (x) ≤ E (1 − e−ρT )vi (x + γ)1T γ Di . (3.13) j6=i
Actually, we obtain some more regularity results on the value functions. Proposition 3.3 The value functions vi , i ∈ IN , are C 1 on (Di , ∞). Moreover, if we set for i ∈ IN : Si = x ≥ Di , vi (x) = max vj (x + Dj − Di − g), (3.14) j6=i
Di = int ({x ≥ Di , vi0 (x) = 1}),
(3.15)
Ci = (Di , ∞) \ (Si ∪ Di ),
(3.16)
10
then vi is C 2 on the open set Ci ∪ int(Di ) ∪ int(Si ) of (Di , ∞), and we have in the classical sense ρvi (x) − Li vi (x) = 0,
x ∈ Ci .
Si , Di , and Ci respectively represent the switching, dividend, and continuation regions when the outstanding debt is at regime i. The proofs of Theorem 3.1 and Proposition 3.3 are omitted as they follow essentially arguments from [11], [21] and in particular [20].
4
Qualitative results on the switching regions
For i, j ∈ IN and x ∈ [Di , +∞), we introduce some notations:
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δi,j = Dj − Di ,
∆i,j = (b − rj )Dj − (b − ri )Di
and xi,j = x + δi,j − g.
We set x∗i = sup{x ∈ [Di , +∞) : vi0 (x) > 1} for all i ∈ IN We equally define Si,j as the switching region from debt level i to j. Si,j = {x ∈ (Di + g, +∞), vi (x) = vj (xi,j )}. From the definition (3.14) of the switching regions, we have the following elementary decomposition property : Si = ∪j6=i Si,j , i ∈ IN . We now begin with two obvious results. Since there is a fixed switching cost g > 0, it is not optimal to continuously change your debt structure. Moreover, if it is optimal to distribute dividends and to switch to another regime, it is still optimal to distribute dividend after the regime switch. Lemma 4.1 Let i, j ∈ IN such that i 6= j. Assume that there exists x ∈ Si,j then we have i) xi,j := x + Dj − Di − g 6∈ Sj . ii) vi0 (x) = vj0 (xi,j ). Especially, if x ∈ Si,j ∩ Di then xi,j ∈ Dj \ Sj . Proof: For k ∈ IN \ {j}, we have vj (xi,j ) = vi (x) ≥ vk (xk,i ). As vk is strictly non-decreasing, we get vj (xi,j ) > vk (xk,i − g). Let h ∈ R. For h going to 0, we have vj (xi,j + h) = vj (xi,j ) + hvj0 (xi,j ) + o(h) = vi (x) + hvj0 (xi,j ) + o(h) = vi (x + h) + h(vj0 (xi,j ) − vi0 (x)) + o(h). 11
As vi (x + h) ≥ vj (xi,j + h), we obtain h(vj0 (xi,j ) − vi0 (x)) ≤ o(h). 2
Hence, we have vj0 (xi,j ) = vi0 (x).
In the following Lemma, we state that there exists a finite level of cash such that it is optimal to distribute dividends up to this level. Lemma 4.2 For all i ∈ IN , we have x∗i := sup{x ∈ [Di , +∞) : vi0 (x) > 1} < +∞.
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Proof: Assume that there exists k ∈ {0, ..., N − 1} such that x∗i < +∞ for all i ∈ Ik := {i1 , ..., ik } ⊂ IN . Notice that Ik = ∅ if k = 0. We will show that there exists j ∈ IN \ Ik such that x∗j < +∞. From Corollary 3.2 and Proposition 3.3, we deduce that, for all i ∈ IN , the function x → vi (x) − x is continuous, non decreasing and bounded. We set ai := lim (vi (x) − x). x→+∞
Moreover, for all (i, j) ∈ IN such that i 6= j, we have aj − (ai + δj,i − g) = lim (vj (x) − vi (x + δj,i − g)) ≥ 0. x→+∞
Let j0 ∈ IN \ Ik such that aj0 + Dj0 = maxj∈IN \Ik (aj + Dj ). For all j ∈ IN \ Ik , we have aj + δj0 ,j − g < aj0 . It is easy to see that there exists x ¯ ∈ [Dj , +∞) satisfying the following conditions: vj0 (¯ x) > x ¯+
max
j∈IN \Ik ;j0 6=j
(aj + δj0 ,j − g),
ρvj0 (¯ x) > b¯ x − rj0 Dj0 , x ¯ > x∗i − (δj0 ,i − g),
∀i ∈ Ik .
At this point, we introduce a continuous function defined on [Di , +∞): ( vj0 (x) if x < x ¯ Vˆ (x) = x−x ¯ + vj0 (¯ x) if x ≥ x ¯ Let x ≥ x ¯. We have −Aj0 Vˆ (x) = (ρ − b)(x − x ¯) + [ρvj0 (¯ x) − (b¯ x − rj0 Dj0 )] > 0. Moreover, for j ∈ IN \ Ik such that j 6= j0 , we have Vˆ (x) ≥ x + aj + δj0 ,j − g ≥ vj (x + δj0 ,j − g). For i ∈ Ik , we have vi (x + δj0 ,i − g) − Vˆ (x) = x + δj0 ,i − g − x∗i + vj (x∗i ) − (x − x ¯ + vj0 (¯ x)) = vi (x∗i ) − x∗i + (¯ x + δj0 ,i − g) − vj0 (¯ x) ≤ vi (¯ x + δj0 ,i − g) − vj0 (¯ x) ≤ 0. 12
Finally, for all j ∈ IN \ {j0 }, Vˆ ((x) ≤ vj0 (x) ≤ vj (x − δj,j0 + g). As Vˆ 0 (x) = 1, Vˆ is a continuous solution of equation (3.13). From Theorem 3.1, we deduce ¯. 2 that vj0 = Vˆ and x∗j0 ≤ x Now, we shall study properties of x∗i and more generally, properties of left-boundaries of Di in the sense as detailed in the following definition. Definition 4.1 Let i ∈ IN and x ∈ (Di , +∞). x is a left-boundary of Di if there exists ε > 0 and a sequence (yn )n∈N with values in (Di , x) \ Di such that [x, x + ε] ∈ Di
lim yn = x.
and
n→+∞
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Notice that, if x∗i > Di then x∗i is a left-boundary of Di . In order to compute the dividend regions, we establish the following lemma. Lemma 4.3 Let i, j ∈ IN such that j 6= i. We assume that there exists x ˆi a left-boundary of Di . i) Assume that x ˆi 6∈ Si , then we have (b − ri )Di > −ρP and ρvi (ˆ xi ) = bˆ xi − ri Di . As x → ρvi (x) − bx + ri Di is increasing, it implies that ρvi (x) < bx − ri Di on (Di , x ˆi )
and
ρvi (x) > bx − ri Di on (ˆ xi , +∞).
ii) Assume that x ˆi ∈ Si,j then we have ii.a) [ˆ xi , x ˆi + ε] ⊂ Si,j and x ˆi + δi,j − g is a left-boundary of Dj . ii.b) ρvi (ˆ xi ) = bˆ xi − ri Di + ∆i,j − bg and ∆i,j > 0. ii.c) ∀k ∈ IN − {i, j}, x ˆi 6∈ Si,k . Notice that the last equality implies that −ρP + bg < (b − rj )Dj . Remark 4.1 We have ρvi (ˆ xi ) ≥ bˆ xi − ri Di , ∀i ∈ In . Proof: i). We assume that x ˆi 6∈ Si . As Si is closed and x ˆi > Di , we can choose ε > 0 such that (ˆ xi − ε, x ˆi + ε) ∩ Si = ∅. Moreover, vi0 ≥ 1 and vi0 (ˆ xi ) = 1 so there exists a sequence (yn )n∈N ∈ (ˆ xi − ε, x ˆi ) ∩ Ci such that lim yn = x ˆi and vi00 (yn ) ≤ 0. We have n→+∞
0 ≥ vi00 (yn ) =
2 σ 2 yn2
ρvi (yn ) − (byn − ri Di )vi0 (yn )
and letting n going to infinity, we get 0 ≥ ρvi (ˆ xi ) − (bˆ xi − ri Di ) = leading to the desired equality.
13
lim
y→ˆ xi ;y>ˆ xi
−Ai vi (y) ≥ 0,
Now let us show that (b − ri )Di > −ρP . Assume that (b − ri )Di ≤ −ρP , we then obtain the following inequality: vi (ˆ xi ) ≤
b (ˆ xi − Di ) − P, ρ
leading to a contradiction as ρ > b and vi (ˆ xi ) ≥ x ˆ i − Di − P . ii). We assume that x ˆi ∈ Si,j . ii.a). We first prove that [ˆ xi , x ˆi + ε] ⊂ Si,j and that x ˆj := x ˆi + δi,j − g is a left-boundary of Dj . Let y ∈ [ˆ xi , x ˆi + ε]. We have
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vj (y + δi,j − g) ≤ vi (y) = y − x ˆi + vi (ˆ xi ) = y − x ˆi + vj (ˆ xj ). On the other hand, vj0 ≥ 1 so y − x ˆi +vj (ˆ xj ) ≤ vj (y +δi,j −g). It follows that vj (y +δi,j −g) = vi (y) and [ˆ xi , x ˆi ] ⊂ Si,j . Moreover, we have proved that [ˆ xj , x ˆj + ε] ∈ Dj . We assume that there exists η > 0 such that (ˆ xj − η, x ˆj ) ⊂ Dj and show that it leads to a contradiction. Let x ∈ (ˆ xj − η, x ˆj ). We have
vj (x) = x − x ˆj + vj (ˆ xj ) = (x − δi,j + g) − x ˆi + vi (ˆ xi ) > vi (x − δi,j + g). The last inequality follows from the fact that x ˆi is a left-boundary of Di and contradicts the fact that vi is solution of equation (3.13). Hence, to show that x ˆj is a left-boundary of Dj it remains to prove that x ˆj > Dj . However, if it was not the case, we would have, vi (ˆ xi ) = vj (ˆ xi + δi,j − g), since x ˆi ∈ Si,j , = vj (Dj ) = −P. But x ˆi = x ˆj −δi,j +g = Di +g, leading to the contradiction −P = vi (Di ) < vi (Di +g) = −P . ii.b). We now prove that ρvi (ˆ xi ) = bˆ xi −ri Di +∆i,j −bg and ∆i,j := (b−rj )Dj −(b−ri )Di > 0. From Lemma 4.1, we know that x ˆj 6∈ Sj . Therefore it follows from step i) that (b − rj )Dj > −ρP and ρvj (ˆ xj ) = bˆ xj − rj Dj . We obtain ρvi (ˆ xi ) = ρvj (ˆ xi + δi,j − g) = ρvj (ˆ xj ) = bˆ xj − rj Dj = bˆ xi + b(δi,j − g) − rj Dj = bˆ xi − ri Di + ∆i,j − bg.
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(4.1)
As ρvi (ˆ xi ) − (bˆ xi − ri Di ) =
lim
y→ˆ xi ;y>ˆ xi
ρvi (y) − Li vi (y) ≥ 0, we have ρvi (ˆ xi ) ≥ bˆ xi − ri Di and
then ∆i,j ≥ bg > 0. ii.c). It remains to show that ∀k ∈ IN − {i, j}, x ˆi 6∈ Si,k . This fact is an elementary result as highlighted earlier because if there exists k ∈ IN − {i, j} such that x ˆi ∈ Si,k ∩ Si,j , it would implies that ∆i,k = ∆i,j . Relation (4.1) gives us the last equality. 2 Corollary 4.1 Let i ∈ IN . We have the following results: i) Assume that x∗i 6∈ Si . If (b − ri )Di > −ρP then ρvi (x∗i ) = bx∗i − ri Di else x∗i = Di .
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ii) Assume that x∗i ∈ Si,j where j ∈ IN − {i}. We have ∆i,j > 0 and (b − rj )Dj > −ρP + bg and ρvi (x∗i ) = bx∗i − ri Di + ∆i,j − bg. Proof: Most results follow directly from Lemma 4.3. We just have to prove that when we assume that x∗i 6∈ Si and (b − ri )Di ≤ −ρP then x∗i = Di . If it was not the case, we would have, following the proof of Lemma 4.3, ρvi (x∗i ) − (bx∗i − ri Di ) = 0. We then obtain vi (x∗i ) ≤
b ∗ (x − Di ) − P, ρ i
leading to a contradiction as ρ > b and vi (x∗i ) ≥ x∗i − Di − P . Hence if (b − ri )Di ≤ −ρP , we have x∗i = Di . 2 We now turn to the following result which basically states that when it is optimal to distribute dividend and/or to switch regime, then it is still optimal when the firm is richer. Lemma 4.4 Let (i, j) ∈ I2N such that i 6= j. If (x∗i , +∞) ∩ Si,j 6= ∅ then there exists ∗ ∈ [x∗ , +∞) such that yi,j i ∗ ∗ ∗ [x∗i , +∞) ∩ Si,j = [yi,j , +∞) and ρvi (yi,j ) = byi,j − ri Di + ∆i,j − bg. ∗ = inf [x∗ , +∞) ∩ S ˚i,j . If x∗ ∈ Si,j , the result has been proved in Proof: We set yi,j i i ∗ . Let y > y ∗ . Using the same argument as in Corollary 4.1. We now assume that x∗i < yi,j i,j ∗ , +∞) ⊂ S . ii) of Corollary 4.1, we may get vj (y + δi,j − g) = vi (y) and [yi,j i,j ∗ Moreover, we know that yi,j + δi,j − g 6∈ Sj and Sj is a closed set, so there exists ε > 0 such ∗ − ε, y ∗ ] ∩ S = ∅ where we set y ∗ = y∗ + δ ∗ ˚ that [¯ yi,j ¯i,j ¯i,j j i,j − g. As yi,j ∈ Di , we can find a i,j ∗ ∗ sequence (yk )k∈N with values in [¯ yi,j − ε, y¯i,j ], such that yk 6∈ Dj (if not we may obtain a ˚i,j .), i.e. contradiction by straightforwardly showing that y ∗ > inf [x∗ , +∞) ∩ S i,j
∀k ∈ N, yk ∈ Cj
and
i
∗ lim yk = y¯i,j .
k→+∞
We finally obtain σ 2 yk2 00 v (yk ) 2 j ∗ − rj Dj )vj0 (¯ yi,j ).
0 = ρvj (yk ) − (byk − rj Dj )vj0 (yk ) − ∗ ∗ = ρvj (¯ yi,j ) − (b¯ yi,j
15
∗ ) = v (y ∗ ) and v 0 (¯ ∗ 0 ∗ ∗ Using vj (¯ yi,j i i,j j yi,j ) = vi (yi,j ) = 1 (from Lemma 4.1 and yi,j ∈ Di ), we may obtain the desired results and conclude the proof. 2
We now establish an important result in determining the description of the switching regions. The following Theorem states that it is never optimal to expand its operation, i.e. to make investment, through debt financing, should it result in a lower “drift” ((b − ri )Di ) regime. However, when the firm’s value is low, i.e. with a relatively high bankruptcy risk, it may be optimal to make some divestment, i.e. sell parts of the company, and use the proceedings to lower its debt outstanding, even if it results in a regime with lower “drift”. In other words, to lower the firm’s bankruptcy risk, one should try to decrease its volatility, i.e. the diffusion coefficient. In our model, this clearly means making some debt repayment in order to lower the firm’s volatility, i.e. σXt .
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Theorem 4.1 Let i, j ∈ IN such that (b−rj )Dj > (b−ri )Di . We have the following results: ˚j = (x∗ , +∞). 1) x∗j 6∈ Sj,i and D j ˚j,i ⊂ (Dj + g, x∗ ). Furthermore, if Dj < Di , then S ˚j,i = ∅. 2) S j Proof: 1). Since (b − rj )Dj > (b − ri )Di , we have ∆j,i < 0. It follows from part ii) of Corollary ˚j . There exists ε > 0 such that (y − ε, y + ε) ⊂ D ˚j . For 4.1 that x∗j 6∈ Sj,i . Let y ∈ D x ∈ (y − ε, y + ε), we have 0 ≤ −Aj vj (x) = ρvj (x) − (bx − rj Dj ). Hence, ρvj (x) ≥ bx − rj Dj and y ≥ x∗j . ˚j,i . we first need to prove that y < x∗ . 2). Let us assume that there exists y ∈ S j Let’s assume that y ≥ x∗j . From Lemma 4.4, we know that [y, +∞) ⊂ Sj,i . We set ˚j,i ∩ Dj . As x∗ 6∈ Sj , we have x∗ ≤ s∗ . On the other hand, it is easy to see that s∗j,i = inf S j j j,i ∗ sj,i + δj,i − g = x∗i and x∗i 6∈ Si . We obtain ρvj (s∗j,i ) = ρvi (x∗i ) = bx∗i − ri Di = bs∗j,i − rj Dj − (∆i,j + bg) < bs∗j,i − rj Dj < ρvj (x∗j ). We may deduce that x∗j > s∗j,i ,, which contradicts the fact that x∗j ≤ s∗j,i , so y < x∗j . ˚j,i = ∅. We now prove that if Dj < Di , S ˚j,i . From the first step, we know that x 6∈ Dj . We deduce Assume that there exists x ∈ S from Lemma 4.1 that x ¯ := x + δj,i − g ∈ Ci then we have
16
1 2 2 00 σ x vj (x) + (bx − rj Dj ) vj0 (x) ≤ ρvj (x) 2 = ρvi (¯ x) 1 2 2 00 = σ x ¯ vi (¯ x) + (b¯ x − ri Di ) vi0 (¯ x) 2 1 2 2 00 = σ x ¯ vj (x) + (b¯ x − ri Di ) vj0 (x). 2 Combining these equations, we get 0≤
σ2 2 (¯ x − x2 )vj00 (x) − (∆i,j + bg)vj0 (x). 2
As x ¯2 − x2 ≥ 0 (using Dj < Di and Assumption (2.5)), (∆i,j + bg)vj0 (x) > 0 and vj00 < 0 on
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r D
r D
[ j b j , x∗j ), we necessarily have x ∈ (Dj + g, j b j ∧ x∗j ). Therefore, if b ≥ rj , Sj,i = ∅. ˚j,i 6= ∅. Let Sj,i = sup S ˚j,i and if we set S¯j,i := Sj,i +δj,i −g, Now, we assume that b < rj and S it follows that σ 2 ¯2 − 2 (S − Sj,i )vj00 (Sj,i ) − (∆i,j + bg)vj0 (Sj,i ). 0≤ 2 j,i Hence, we have ! 2 S¯j,i 0 ≤ ρvj (Sj,i ) − bS¯j,i − ri Di + ¯2 (∆i,j + bg) vj0 (Sj,i ) 2 Sj,i − Sj,i ! Sj,i 2 (4.2) (∆i,j + bg) vj0 (Sj,i ). ≤ ρvj (Sj,i ) − bSj,i − rj Dj + ¯2 2 Sj,i − Sj,i On the other hand, we have (Sj,i , x∗j ) ⊂ Cj so 2 σ 2 Sj,i Sj,i 2 + (∆i,j + bg)vj0 (Sj,i ). ) − ¯2 0≤ vj00 (Sj,i 2 2 Sj,i − Sj,i + Especially, we have vj00 (Sj,i ) > 0. Moreover, vj is a C 2 function and vj0 > 1 on (Sj,i , x∗j ), it follows that there exists y ∈ (Sj,i , x∗j ) such that vj00 (y) = 0 since v 0 (x∗j ) = 1. We set r D
yj = inf{y ∈ (Sj,i , x∗j ) : vj00 (y) ≤ 0}. As vj00 ≤ 0 on [ j b j , +∞), we know that yj ≤ We have vj00 (yj ) = 0 and yj ∈ Cj , so we can assert that h(yj ) = 0 where we have set
rj Dj b .
h(x) = (bx − rj Dj )vj0 (x) − ρvj (x). On (Sj,i , yj ), we have vj00 > 0 so h is decreasing. Indeed, we have h0 (x) = (bx − rj Dj )vj00 (x) − (ρ − b)vj0 (x) ≤ 0. Finally, this proves that ρvj (Sj,i ) ≤ (bSj,i − rj Dj )vj0 (Sj,i ). Reporting this in the inequality (4.2), we get Sj,i 2 (∆i,j + bg)vj0 (Sj,i ). 0 ≤ − ¯2 2 Sj,i − Sj,i 2 > S2 , ∆ 0 ˚ This is impossible as S¯j,i i,j + bg > 0 and vj (Sj,i ) ≥ 1. In conclusion, Sj,i = ∅. 2 j,i We now turn to an important corollary.
17
Corollary 4.2 Let m ∈ IN such that (b − rm )Dm = maxi∈IN (b − ri )Di . ˚m = (x∗ , +∞). 1) x∗m 6∈ Sm and D m 2) For all i ∈ IN − {m}, we have: ˚m,i = ∅. i) If Dm < Di , S ˚m,i ⊂ (Dm + g, x∗m ). Furthermore, if b ≥ ri , then S ˚m,i ⊂ ii) If Di < Dm , S (Dm + g, (a∗i + δi,m + g) ∧ x∗m ), where a∗i is the unique solution of the equation ρvi (x) = (bx − ri Di )vi0 (x). We further have a∗i 6= x∗i .
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Proof: The only point left to show is 2.ii). We now assume that there exists i ∈ IN − {m} such ˚m,i 6= ∅ that Di < Dm , b ≥ ri and S We prove that the equation ρvi (x) = (bx − ri Di )vi0 (x) admits a unique solution a∗i and ˚m,i ⊂ (Dm + g, a∗ + δm,i − g). prove that S i ˚m,i . It follows from the first step that S ˚m,i ∩ Dm = ∅. Hence, from Lemma 4.1, Let x ∈ S we have x := x + δm,i − g ∈ Ci . We obtain σ 2 x2 00 v (x) + (bx − rm Dm )vi0 (x) − ρvi (x) 2 i σ2 = Ai vi (x) + (x2 − x2 )vi00 (x) + (∆i,m + bg)vi0 (x) 2 x2 − x2 = − Hi (x), x2
0 ≥
where we have set Hi (x) = bx − ri Di −
x2 (∆i,m + bg) vi0 (x) − ρvi (x). (x + δi,m + g)2 − x2
Hence, we have ˚m,i ⊂ {x ∈ (Dm + g, +∞) : Hi (x) ≥ 0} ⊂ {x ∈ (Dm + g, +∞) : Gi (x + δm,i − g) ≤ 0}, S where we set Gi (y) = ρvi (y) − (by − ri Di )vi0 (y). We notice that, for all y ∈ (Di , +∞), Gi (y) ≥ b ≥ ri , it follows that G0i (y) ≥ (ρ − b)vi0 (y) −
σ 2 x2 00 2 vi (y).
Recalling our assumption that
2(by − ri Di ) 2(by − ri Di ) Gi (y) > − Gi (y). 2 2 σ y σ2y2
As Gi is continuous on (Di , +∞) and Gi (Di ) < 0, it implies that the equation Gi (y) = 0 ˚m,i ⊂ (Dm + admits a unique solution which will be denoted by a∗i . Therefore, we have S g, a∗i + δi,m + g). Furthermore, from Corollary 4.1, we either have G(x∗i ) = 0 or G(x∗i ) > 0. As such, we deduced that a∗i ∈ (Di , x∗i ).
We now turn to the following results ordering the left-boundaries regions (Di )i∈IN . 18
(x∗i )i∈IN
2 of the dividend
Proposition 4.1 Consider i, j ∈ IN , such that (b − ri )Di < (b − rj )Dj . We always have x∗i + δi,j − g ≤ x∗j unless there exists a regime k such that (b − rj )Dj < (b − rk )Dk and x∗i ∈ Si,k , then we have x∗j − δi,j + g < x∗i < x∗k − δi,k + g. Proof: First, we assume that x∗i 6∈ Si . From Lemma 4.3, we know that ρvi (x∗i ) = bx∗i −ri Di . On the other hand, we have ρvi (x∗j − (δi,j − g)) ≥ ρvj (x∗j ) ≥ bx∗j − rj Dj ≥ b(x∗j − (δi,j − g)) − ri Di + ∆i,j − bg
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> b(x∗j − (δi,j − g)) − ri Di . Hence, we have x∗i + δi,j − g < x∗j . Now, we assume that there exists k ∈ IN − {i} such that x∗i ∈ Si,k . If k = j, we have x∗i + δi,j − g = x∗j . If k 6= j, we have x∗i + δi,k − g = x∗k and ρvi (x∗i ) = bx∗i − ri Di + ∆i,k − bg = b(x∗i + δi,k − g) − rk Dk . On the other hand, we have ρvi (x∗j − (δi,j − g)) ≥ ρvj (x∗j ) = bx∗j − rj Dj = b(x∗j − (δi,j − g)) − ri Di + ∆i,j − bg = b x∗j − (δi,j − g) + (δi,k − g) − rk Dk + ∆k,j . If (b − rj )Dj > (b − rk )Dk , i.e. ∆k,j > 0, then we have ρvi (x∗j − (δi,j − g)) > b x∗j − (δi,j − g) + (δi,k − g) − rk Dk . Hence, we have x∗i + δi,j − g < x∗j . However, in the case that (b − rj )Dj < (b − rk )Dk , then ρvi (x∗j − (δi,j − g)) < b x∗j − (δi,j − g) + (δi,k − g) − rk Dk . 2
Hence, we have x∗i + δi,j − g > x∗j .
5
The two regime-case
Before investigating the two-regime case, for the sake of completeness, we give the results in the case where there is no regime change, i.e. the firm’s debt level remains constant. Proposition 5.1 The value function Vˆ is C 2 on (D1 , ∞). There exists x ˆ ≥ D such that ˆ = (ˆ Cˆ = (D, x ˆ), and D x, ∞). Furthermore, If (b − rD) > −ρP , then, on Cˆ = (D, x ˆ), Vˆ is the unique solution (in the classical sense) to ρv − Lv = 0 19
and Vˆ (x)
=
where Vˆ (ˆ x)
=
x−x ˆ + Vˆ (ˆ x), bˆ x − rD . ρ
x≥x ˆ
If (b − rD) ≤ −ρP , then x ˆ = D and the optimal value function V (x) = x − D − P . 2
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This result directly derives from Corollary 4.1.
Throughout this section, we now assume that N = 2, in which case, we will get a complete description of the different regions. We will see that the most important parameter to consider is the so-called “drifts” ((b − ri )Di )i=1,2 and in particular their relative positions. To avoid cases with trivial solution, i.e. immediate consumption, we will assume that −ρP < (b − ri )Di , i = 1, 2. We now distinguish the two following cases: (b − r2 )D2 < (b − r1 )D1 and (b − r1 )D1 < (b − r2 )D2 . Throughout Theorem 5.1 and Theorem 5.2, we provide a complete resolution to our problem in each case. Theorem 5.1 We assume that (b − r2 )D2 < (b − r1 )D1 . We have ˚1 = ∅ where ρv1 (x∗ ) = bx∗ − r1 D1 . C1 = [D1 , x∗1 ), D1 = [x∗1 , +∞), and S 1 1 1) If S2 = ∅ then we have C2 = [D2 , x∗2 ), and D2 = [x∗2 , +∞) where ρv2 (x∗2 ) = bx∗2 − r2 D2 . 2) If S2 6= ∅ then there exists y2∗ such that S2 = [y2∗ , +∞) and we distinguish two cases a) If x∗2 + δ2,1 − g < x∗1 , then y2∗ > x∗2 , y2∗ = x∗1 + δ1,2 + g and C2 = [D2 , x∗2 ), and D2 = [x∗2 , +∞) where ρv2 (x∗2 ) = bx∗2 − r2 D2 . b) If x∗2 + δ2,1 − g = x∗1 then y2∗ ≤ x∗2 , ρv2 (x∗2 ) = bx∗2 − r2 D2 + ∆2,1 − bg. We define a∗2 as the solution of ρv2 (a∗2 ) = ba∗2 − r2 D2 and have two cases i) If a∗2 6∈ D2 , we have D2 = [x∗2 , +∞)
and
C2 = [D2 , y2∗ ).
ii) If a∗2 ∈ D2 , there exists z2∗ ∈ (a∗2 , y2∗ ) such that D2 = [a∗2 , z2∗ ] ∪ [x∗2 , +∞)
and
C2 = [D2 , a∗2 ) ∪ (z2∗ , y2∗ ).
Remark 5.1 Theorem 5.1 clearly states that it is never optimal to make growth investment through debt financing when it results in lower “drift” (b − ri )Di . However, when the firm value process exceeds the threshold, yi∗ , it may be optimal to switch to a lower debt regime should it result in a higher “drift” (b − ri )Di . 20
Figure 1: Switching regions: case (b − r1 )D1 > (b − r2 )D2 . Div2
C2
Regime 2
x
D2
D1
D2
D1
C ase 1
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S2
x2*
z 2*
y2* Div2
Regime 1
x
x1*
a
* 2
C1
Div1
C2
Regime 1
C2 Div2 C2
Regime 2
* 2
x1*
Div1
C ase 2
Proof: From Theorem 4.2, we have D1 = [x∗1 , +∞) where ρv1 (x∗1 ) = bx∗1 − r1 D1 and S1 = ∅. ˚2 = ∅. From Corollary 4.1, we know that ρv2 (x∗ ) = bx∗ − r2 D2 . If 1) We assume that S 2 2 ˚2 ∩ (D2 , x∗ ), we would have 0 ≤ ρv2 (x) − (bx − r2 D2 ) but this is there exists x ∈ D 2 impossible for x < x∗2 . Hence we have C2 = [D2 , x∗2 ), and D2 = [x∗2 , +∞). ˚2 6= ∅ and set y ∗ = inf S ˚2 . We first prove that S2 = [y ∗ , +∞). 2) Now, we assume that S 2 2 We define the following function ( v2 (x) if D2 ≤ x < y2∗ V2 (x) = v1 (x + δ2,1 − g) if y2∗ ≤ x. V2 is a C 1 function on [D2 , +∞). We prove that V2 is the solution of the variational inequality satisfied by v2 . We obviously have V20 (x) ≥ 1 and V2 (x) ≥ v1 (x + δ2,1 − g). Moreover, we have ( v2 (x + δ1,2 − g) ≤ v1 (x) if D2 ≤ x < y2∗ + δ2,1 + g V2 (x + δ1,2 − g) = v1 (x − 2g) ≤ v1 (x) if y2∗ + δ2,1 + g ≤ x. It remains to prove that A2 V2 (x) ≤ 0 on [y2∗ , +∞). For x ≥ y2∗ , we set x = x + δ2,1 − g and we have σ 2 x2 00 v (x) + (bx − r2 D2 )v10 (x) − ρv1 (x) 2 1 σ2 = A1 v1 (x) + (x2 − x2 )v100 (x) − (∆2,1 + bg)v10 (x) 2 σ2 2 ≤ (x − x2 )v100 (x). 2
A2 V2 (x) =
21
x
As D1 < D2 , we have x2 > x2 . On the other hand, we have seen that v1 is concave so we can assert that A2 V2 (x) ≤ 0 on [y2∗ , +∞). This proves that v2 = V2 and especially ˚2 = (y ∗ , +∞). that S 2 a) If x∗2 + δ2,1 − g < x∗1 , then using Proposition 4.1, we have y2∗ > x∗2 and x∗2 6∈ S2 , so it follows from Corollary 4.1 that ρv2 (x∗2 ) = bx∗2 − r2 D2 . Moreover, we have ˚2 = (x∗ , +∞) and from Lemma 4.4, we have y ∗ = x∗ + δ1,2 + g. D 2 2 1 b) If x∗2 + δ2,1 − g = x∗1 , then using Proposition 4.1, we have y2∗ ≤ x∗2 . In this case, x∗2 ∈ S2 and it follows from Corollary 4.1 that ρv2 (x∗2 ) = bx∗2 − r2 D2 + ∆2,1 − bg. We define a∗2 as the solution of ρv2 (a∗2 ) = ba∗2 − r2 D2 and distinguish two cases: i) If a∗2 6∈ D2 , it follows from Lemma 4.3 that
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D2 = [x∗2 , +∞) and C2 = [D2 , x∗2 ). ii) Finally, we assume that a∗2 ∈ D2 . We set z2∗ = inf{x ≥ a∗2 : v20 > 1} and have D2 = [a∗2 , z2∗ ] ∪ [x∗2 , +∞) and C2 = [D2 , a∗2 ) ∪ (z2∗ , x∗2 ). 2 We now turn to the case where (b − r1 )D1 < (b − r2 )D2 . Theorem 5.2 We assume that (b − r1 )D1 < (b − r2 )D2 , 1) we have D2 = [x∗2 , +∞) where ρv2 (x∗2 ) = bx∗2 − r2 D2 ˚2 = ∅ or there exist s∗ , S ∗ ∈ (D2 + g, x∗ ) such that S ˚2 = (s∗ , S ∗ ). S 2 2 2 2 2 ˚1 = ∅ then we have 2) If S C1 = [D1 , x∗1 ), and D1 = [x∗1 , +∞) where ρv1 (x∗1 ) = bx∗1 − r1 D1 . ˚1 = ˚1 = (y ∗ , +∞) 3) If S 6 ∅ there exists y1∗ such that S 1 a) If x∗1 + δ1,2 − g < x∗2 , then y1∗ > x∗1 , y1∗ = x∗2 + δ2,1 + g and C1 = [D1 , x∗1 ), and D1 = [x∗1 , +∞) where ρv1 (x∗1 ) = bx∗1 − r1 D1 . b) If x∗2 + δ2,1 − g = x∗1 , then y1∗ ≤ x∗1 , ρv1 (x∗1 ) = bx∗1 − r1 D1 + ∆1,2 − bg. We define a∗1 as the solution of ρv1 (a∗1 ) = ba∗1 − r1 D1 and have two cases. i) If a∗1 6∈ D1 , we have D1 = [x∗1 , +∞) 22
and
C1 = [D1 , y1∗ ).
ii) If a∗1 ∈ D1 , there exists z1∗ ∈ (a∗1 , y1∗ ) such that D1 = [a∗1 , z1∗ ] ∪ [x∗1 , +∞)
C1 = [D1 , a∗1 ) ∪ (z1∗ , y1∗ ).
and
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Remark 5.2 Theorem 5.2 states that when the firm’s value is sufficiently high (above y1∗ threshold), it’s optimal to switch to a higher-debt regime which operates at a higher drift (b−ri )Di , see figure 2, case 2. However, when the firm is too small, it may be optimal not to postpone dividend payment and to operate under as a medium size company (cash-reserve lower than the threshold a∗1 , as in figure 2, case 3), i.e. to distribute dividend, whenever the cash-reserve exceed the threshold a∗1 . However, one should not switch to a lower drift regime unless it lowers the firm’s bankruptcy risk. It may happen, when the value firm dangerously approaches bankruptcy threshold, i.e. when its cash reserve stands between s∗2 and S2∗ .
Figure 2: Switching regions: case (b − r1 )D1 < (b − r2 )D2 . Regime 2
D2
x
* 2
Div2
C2
Regime 2
x
D2
S2
Div1
C2 S 2*
s2*
x2*
Div2
Div1
Regime 1
C1
Regime 1 * 1
x
D1
D1
Div1
C ase 1
C2
Regime 2
D2
S2
Div1
s
* 2
S1 * 1
x
* 1
y
Div1
C ase 2
C2 S
* 2
x2*
Div2
x
Div1
Div1
C1
Regime 1
D1
* 1
a
S1
C1 z1*
y1* x1*
Div1
C ase 3
Proof: Throughout the proof, for x ∈ R, we set x ¯ = x + δ1,2 − g and x = x + δ2,1 − g. Notice that we have x < x < x ¯.
23
x
1.) From Theorem 4.2 we have D2 = [x∗2 , +∞) where ρv2 (x∗2 ) = bx∗2 − r2 D2 ˚2 ⊂ (D2 + g, (a∗1 + δ2,1 − g) ∧ x∗2 ), S
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where a∗1 is the unique solution of the equation ρv1 (x) = (bx − r1 D1 )v10 (x). ˚2 6= ∅. We set s∗ = inf S ˚2 and S ∗ = sup S ˚2 . Now we prove that S ˚2 = (s∗ , S ∗ ). Assume that S 2 2 2 2 On [D2 , +∞), we define the following function: if x < s∗2 v2 (x) V2 (x) = v1 (x + δ2,1 − g) if s∗2 ≤ x ≤ S2∗ v (x) if x > S2∗ . 2 V2 is a continuous function on [D2 , +∞) and it is easy to see that V20 ≥ 1. For all x ∈ [D2 , +∞), we have V2 (x) ≥ v1 (x+δ2,1 −g) and for x+δ1,2 −g ∈ [s∗2 , S2∗ ], V2 (x+δ1,2 −g) = v1 (x − 2g) < v1 (x). We now prove that A2 V2 ≤ 0. Let x ∈ [s∗2 , S2∗ ], we have x2 − x2 σ 2 x2 00 x2 0 A2 V2 (x) = A1 v1 (x) + v (x) + 2 (∆1,2 + bg)v1 (x) x2 2 1 x − x2 x2 − x2 = − H1 (x). x2 We recall that H1 (x) = bx − r1 D1 −
x2 (∆1,2 + bg) v10 (x) − ρv1 (x). (x + δ1,2 + g)2 − x2
˚2 ⊂ {x ∈ (Dm , +∞) : H1 (x) ≥ 0}. We have seen in the proof of Theorem 4.2 that S Especially, we have H1 (S2∗ ) ≥ 0, with S2∗ ≤ a∗1 . Now, we prove that H1 is decreasing on (D1 , a∗1 ). As H1 is continuous, this will lead to A2 V2 ≤ 0 and allows us to assert that ˚2 = (s∗ , S ∗ ). v2 = V2 and especially that S 2 2 We may rewrite H1 : H1 (x) = U1 (x) − G1 (x), where G1 (x) = ρv1 (x) − (bx − r1 D1 )V10 (x) x2 U1 (x) = − (∆1,2 + bg)v10 (x). (x + δ1,2 + g)2 − x2 From the proof of Theorem 4.2, we have G1 is strictly non-decreasing on (D1 , a∗1 ). Furthermore, a straight study of the function U1 and recalling that on (D1 , a∗1 ], v100 (x) ≤ 0, we may obtain that U1 is non-increasing. As such, H1 is strictly non-increasing. ˚1 = ∅, then x∗ 6∈ S1 . Using the arguments from 1.) of Theorem 4.2, we may obtain 2.) If S 1 C1 = [D1 , x∗1 ), and D1 = [x∗1 , +∞) where ρv1 (x∗1 ) = bx∗1 − r1 D1 .
24
˚1 6= ∅. We set y ∗ = inf S˚1 and prove that S˚1 = (y ∗ , +∞). 3.) We now assume that S 1 1 On [D1 , +∞), we define the following function: ( v1 (x) V1 (x) = v2 (x + δ1,2 − g)
if if
x < y1∗ y1∗ ≤ x.
V1 is a C 1 function on [D1 , +∞) and it is easy to see that V10 ≥ 1. For all x ∈ [D1 , +∞), we have V1 (x) ≥ v2 (x + δ1,2 − g) and for x ≥ y1∗ , V1 (x + δ2,1 − g) = v2 (x − 2g) < v2 (x). We now prove that A1 V1 ≤ 0. Let x ∈ [y1∗ , +∞), we have (¯ x2 − x2 )σ 2 00 0 v2 (¯ x) − ∆− x) 1,2 v2 (¯ 2 (¯ x2 − x2 )σ 2 00 0 ≤ − v2 (¯ x) − ∆− x). 1,2 v2 (¯ 2
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A1 V1 (x) = A2 v2 (¯ x) −
If x ¯ ∈ D2 , we obviously have A1 V1 (x) ≤ 0. Assume that x ¯ ∈ C2 , then we have A1 V1 (x) =
x ¯ 2 − x2 H2 (¯ x). x ¯2
As H2 is decreasing, we have H2 (¯ x) ≤ H2 (y1∗ ) ≤ 0 so A1 V1 (x) ≤ 0. Finally, we assume that ˚2 . In this case, we have x ¯∈S (x − g)2 σ 2 (x − g)2 00 (x − g)2 − 0 A V (x) ≤ − v (x − g) − ∆ v (x − g) 1 1 1 x ¯ 2 − x2 2 x ¯2 − x2 1,2 1 (x − g)2 − = −ρv1 (x − g) + b(x − g) − r1 D1 − 2 ∆ v 0 (x − g) x ¯ − x2 1,2 1 (x − g)2 − v 0 (¯ x) = −ρv2 (¯ x) + b¯ x − r2 D2 − ∆1,2 − 2 ∆ x ¯ − x2 1,2 2 2 x ¯ − (x − g)2 − = H2 (¯ x) + ∆1,2 − ∆1,2 v20 (¯ x) x ¯ 2 − x2 2 x ¯ − (x − g)2 − ≤ ∆1,2 − ∆1,2 v20 (¯ x). x ¯2 − x2 However, we have (¯ x2 − (x − g)2 )∆− x2 − x2 )∆1,2 = −bg(¯ x2 − x2 ) + g(2x − g))(∆1,2 − bg) 1,2 − (¯ 2 = g 2(r1 D1 − r2 D2 )x − g∆− − b(δ − g) 1,2 1,2 ≤ 0. Therefore, A1 V1 ≤ 0 on (D1 , +∞). This allows us to assert that v1 = V1 and especially ˚1 = (y ∗ , +∞). that S 1 a) If x∗1 + δ1,2 − g < x∗2 , then using Proposition 4.1, we have y1∗ > x∗1 and x∗1 6∈ S1 . So ˚1 = it follows from Corollary 4.1 that ρv1 (x∗1 ) = bx∗1 − r1 D1 . Moreover, we have D ∗ ∗ ∗ (x1 , +∞) and from Lemma 4.4, we have y1 + δ1,2 − g = x2 . 25
b) If x∗1 + δ1,2 − g = x∗2 , then using Proposition 4.1, we have y1∗ ≤ x∗1 . In this case, x∗1 ∈ S1 and it follows from Corollary 4.1 that ρv1 (x∗1 ) = bx∗1 − r1 D1 + ∆1,2 − bg. We define a∗1 as the solution of ρv1 (a∗1 ) = ba∗1 − r1 D1 and distinguish two cases: i) If a∗1 6∈ D1 , it follows from Lemma 4.3 that D1 = [x∗1 , +∞) and C1 = [D1 , x∗1 ). ii) Finally, we assume that a∗1 ∈ D1 . We set z1∗ = inf{x ≥ a∗1 : v10 > 1} and have D1 = [a∗1 , z1∗ ] ∪ [x∗1 , +∞) and C1 = [D1 , a∗1 ) ∪ (z1∗ , x∗1 ).
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2 Remark 5.3 The arguments used to obtain the above results in Theorems 5.1 and 5.2 in the two-regime problem may also apply to higher regime problems although the required analysis involved would be much lengthier and depends on many more parameters. It is particularly the case when we reconsider our initial multi-switching problem with a slight but realistic change to our initial model: we only allow the firm to change, i.e. increase or repay, its debt level to the one immediately above or below. This latter case may be subject to further studies in the future, but we may already obtain: • The elementary decomposition of the switching regions becomes Si = Si,i−1 ∪ Si,i+1 . The system of variational inequalities becomes : 0 min −Ai vi (x) , vi (x) − 1 , vi (x) − max vj (x + (j − i)D − g) ≥ 0, x > Di (5.1) j=i−1,i+1
• With the exception of part i) of Lemma 4.1, all the other results still hold. For results obtained by using part i) of Lemma 4.1, it suffices to slightly modify the existing proofs. • The complete solution to our modified problem may be obtained by applying iteratively the results from Theorem 5.1 and 5.2. Some numerical illustrations: Below are some numerical analysis on value functions as defined in equation (2.7) versus the real value for shareholders as defined in equation (2.8) for different values of P , see Figure 3. Finally, Figure 4 shows the contribution of the management team in creating value for shareholders for different values of P .
26
Figure 3: Optimal values for managers (vi ) Vs shareholders’ value (ui ) for increasing penalty P .
R eal equity values u i (x) for different ↓ P
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V alue functions vi (x ) for different ↓ P
Figure 4: Excess shareholders’ values Vs immediate consumption.
W i ( x) = ui ( x) − ( x − Di ) for different ↓ P
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[20] Ly Vath V., H. Pham and S. Villeneuve (2008) : “A mixed singular/switching control problem for a dividend policy with reversible technology investment", Annals of Applied Probability, 2008, 18, pp. 1164-1200. [21] Pham H. (2007) : “On the smooth-fit property for one-dimensional optimal switching problem", Séminaire de Probabilités, vol XL, 187-201. [22] Pham H. (2009) : “Continuous-time Stochastic Control and Optimization with Financial Applications", Stochastic Modelling and Applied Probability, Vol. 61, 232p. [23] Pham H., V. Ly Vath, and X.Y. Zhou (2009) : “Optimal switching over multiple regimes", SIAM Journal on Control and Optimization, 48, 2217-2253. [24] Yang R. and Liu K. (2004) : “Optimal singular stochastic problem on harvesting system", Applied Mathematics E-Notes, 133-141. [25] Zervos M., T. Johnson, and F. Alazemi (2011) :“Buy-low and sell-high investment strategies", to appear in Mathematical Finance.
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