Vathana Ly Vath - Mathematics | U-M LSA

IntroductionThe model

Characterization of the value functionsNumerical Results

Optimal dividend and capital policy with debt covenants

Vathana Ly VathLaboratoire de Mathématiques et Modélisation d’Evry

ENSIIE

Financial and Actuarial Mathematics Seminar

University of MichiganAnn Arbor, November 1, 2017

Joint works with :Etienne Chevalier, University of EvryAlexandre Roch, UQAM, Montréal

Vathana Ly Vath Optimal dividend and capital policy with debt covenants



Motivations : a corporate finance problem

Corporate governance problems : managerial decisions

Operational levels : marketing, product lines, internal organization...

Cash-flow levels/utilization : dividends and investments.

Financial levels : capital structure, debt and equity




Motivations : Cash-flow (utilization) policy

→ Investment policy : capital expenditure/investment for future growth.

- Operationnal growth : increase/decrease rates of production...

- Organic growth : internal development of new products, technologies, factories...

- Merger/Acquisition : acquire another company

→ Real Options : Dixit & Pendyck (94) : Optimal stopping/investing

→ Optimal switching problems : Brekke and Oksendal (94), Duckworth and Zervos(01), Hamadène and Jeanblanc (05), Pham, LV and Zhou (09), Hamadène and Zang(10), and Hu and Tang (10)

→ Dividend policy : payment to shareholders

→ Regular or/and singular control problems : Jeanblanc and Shiryaev (95), Radner

and Shepp (96), Choulli, Taksar and Zhou (03),




Seminal model

Z is the set of dividend strategies i.e.

Z := Z : Z is a F-adapted, càdlàg, non-decreasing process

The cash reserve of the firm X has the following dynamics :dXt = µdt + σdWt − dZtX0− = x ≥ 0,

The value of the firm is defined as : V (x) = supZ∈Z E[∫ T

0 e−ρt dZt

].

6

-

Value Function

Cash reserve

-

(((((((

((((((((

x∗ Dividend Region

x − x∗

V (x) = x − x∗ + V (x∗)

Continuation Region

LV − ρV = 0




Motivations : Financial levels

→ Capital structure :

Liability management : How to increase the capital that may be used for investmentprojects? How to fullfill regulatory constraints?

- Dividend distribution→ decrease in equity

- Capital issuance for investment and/or regulatory constraints

- Debt control for investment or commercial expansion (banks deposits)

→ Mixed control problems

Sethi & Taksar (02), Lokka & Zervos (08) : optimal dividend and capital issuance→ two singular controls in different models

Peura & Keppo (06) : optimal dividend and capital issuance for banks→ mixed singular and impulse controls with delay

Leland (94) (debt with infinite maturity), Leland & Toft (96) (Roll-over debt)Décamps & Villeneuve (14),...




Optimal dividend and capital policy

A is the set of dividend and capital strategies i.e.

A := (Z ,K ) : Z and K are F-adapted, càdlàg, non-decreasing process

The cash reserve of the firm X has the following dynamics :

dXt = µdt + σdWt − dZt + dKt

The value of the firm is V (x) = supZ∈A E[∫ T

0 e−ρt[(1− κ)dZt − (1 + κ)dKt

]].

6

-

Value Function

Cash reserve

-

((((((((

(((((((


x − x∗

V (x) = (1− κ)(x − x∗) + V (x∗)

Continuation Region

LV − ρV = 0




Optimal dividend and capital policy

A is the set of dividend and capital strategies i.e.

A := (Z ,K ) : Z and K are F-adapted, càdlàg, non-decreasing process

The cash reserve of the firm X has the following dynamics :

dXt = µdt + σdWt − dZt + dKt

The value of the firm is V (x) = supZ∈A E[∫ T

0 e−ρt[(1− κ)dZt − (1 + κ)dKt

]].

6

-

Value Function

Cash reserve

-

((((((((

(((((((


x − x∗

V (x) = (1− κ)(x − x∗) + V (x∗)

Continuation Region

LV − ρV = 0*

Capital issuanceV ′(0) = 1 + κ




Motivations : Modelling bankruptcy

I Optimize firm management, price defaultable bonds, hedge credit derivatives...

→ Structural models (usual models in corporate finance/ ruin theory)

Bankruptcy : First time an underlying process hits or crosses a lower bound.

Cash or equity process : Black, Cox (‘76), Leland, Toft (‘96),...

Occupation time : Yildirim (‘06), Makarov & als (‘15),...

Bankruptcy time may be predictable

→ Reduced models (usual models in credit derivatives valuation problems)

Bankruptcy : First jump time of a counting process.

Lando (‘94), Jarrow & Turnbull (‘95), ...

Bankruptcy time is not controlled

→ Information based models : Duffie, Lando (‘01), Cetin & al. (‘04), ...




Motivations : Modelling bankruptcy

I Distinction between liquidation and default (Chapter 7 and 11)

→ A structural model : ( Broadie et al. (‘07))

Default :→ occurs when an underlying process hits a bound d > 0.→ At that time, the firm enters in grace period (Chapter 11) of lenght δ.

Liquidation :→ happens when the process hits a lower boundary ` (0 ≤ ` < d) or has notreached an upper boundary ` (0 < d < `) before the end of the grace period.

→ A mixed model (Optimization of dividend distribution and capital issuance.)

A reduced model for default :→ occurs at the first jump of a counting process when an underlying process isbelow a bound d > 0.

A structural model for liquidation :→ happens when the process hits 0 or has not reached an upper boundary dbefore the end of the grace period




1 The model

2 Characterization of the value functions

3 Numerical Results




Cash flows and controls

I Let (Ω,F,P) a probability space and W a F-Brownian motion

I Cash flows process :dXt = µ(Xt )dt + σ(Xt )dBt + (1− κt )dKt − (1 + κ′)dZt ,X0 = x ∈ R+,

where 0 < κt , κ′, µ and σ lipschitz with linear growth and (K ,Z ) ∈ A.

I A is the set of admissible strategies α = (K ,Z ) such that

Capital issuance process : K is a F-adapted, càdlàg, non-decreasing process

Dividend process : Z is a F-adapted, càdlàg, non-decreasing process such that

∀t ≥ 0; Zt −Zt− ≤ (Xt− −D)+ and∫ +∞

01lXt≤DdZt dt = 0, where D > 0.

→ Dividends are not allowed to be distributed when the firm is in financial distress.




Default and liquidation

I Default indicator

dIt = 1Xt<D(1− It )dNt − It d1Xt≥D

where N is a Poisson process with intensity parameter λ.

Capital issuance cost : We set κ(t) = κIt , where 0 < κ0, κ1.

I Duration of default

τt = t − sups ≤ t : Is = 0, with the convention sup ∅ = 0.

I Liquidation time

T = inft ≥ 0 : Xt < 0 or τt > δ,

where 0 < δ is the fixed duration of the grace period.




Domain

6

Default duration

Cash reserve

D

δLiquidation

Clearance of default

Financial distress

Liqu

idat

ion

-

S0 := [0,+∞)

S1 := [0, δ]× [0,D]




The value function

I Firm value before default : For x ∈ S0,

v0(x) = sup(Z ,K )∈A

E0,x,0

∫ T x0

0e−ρs (dZs − dKs),

where we have setT x

0 = inft ≥ 0 : X xu < 0 or τ0

u > δ

I Firm value after default : For (t , x) ∈ S1,

v1(ζ, x) := sup(Z ,K )∈A

Eζ,x,1∫ T (ζ,x)

1

0e−ρs d(Zs − Ks),

where we have setT (ζ,x)

1 = infu ≥ t : X xu < 0 or τζu > δ.




1 The model

2 Characterization of the value functions

3 Numerical Results




Dynamic Programming Principle

I Let x ∈ S0 and ξ the first default time. For all stopping time ν ≥ 0,

v0(x) = sup(Z ,K )∈A E[ ∫ ν∧ξ∧T

0 e−ρs d(Zs − Ks) +e−ρνv0(Xν)1ν<ξ∧T

+e−ρξv1(0,Xξ)1ξ≤ν∧T

]

I Let (ζ, x) ∈ S1 and set θD,+ = infs ≥ 0 : Xs ≥ D. For all stopping time ν ≥ 0,

v1(ζ, x) = sup(Z ,K )∈A E[−∫ ν∧θD,+∧T

0 e−ρt dKt +e−ρνv1(τν ,Xν)1ν<T∧θD,+

+e−ρθD,+

v0(XθD,+ )1θD,+≤T∧ν

]




Analytical properties

Limits on the boundary of S1

For all 0 ≤ x ≤ D,

v1(δ, x) = g(x) := max(

v0(D)−D − x1− κ1

; 0),

For all 0 ≤ ζ ≤ δ,

limh↓0

v1(ζ,D − h) = v0(D) = v1(ζ,D) and limh↓0

v1(ζ, h) exists.

Monotonicity

For any ζ ∈ [0, δ], the functions v0 and v1(ζ, x) are non-decreasing

For any x ∈ [0,D], the function v1(·, x) is non-increasing on [0, δ]

Continuity

The value function v1 is continuous on S1, and v0 is continuous on S0.




HJB equation (1)

I Infinitesimal generator : We set

Lv = −ρv + µ(x)∂v∂x

+12σ2(x)

∂2v∂x2

I HJB for v0 : On S0, we have

0 = min(−Lv0, v ′0−

11 + κ′

,1

1− κ0− v ′0

)on [D,∞), (1)

0 = min(−Lv0 − λ

[v1(0, ·)− v0

],

11− κ0

− v ′0

)on (0,D), (2)

I Boundary conditions satisfied by v0 :

limx↑∞

v0(x + 1)− v0(x) =1

1 + κ′(3)

min(

v0(0),1

1− κ− v ′0(0)

)= 0, (4)




HJB equation (2)

I HJB for v1 : On S1, we have

0 = min(−Lv1 −

∂v1

∂t,

11− κ

−∂v1

∂x

)on S1, (5)

I Boundary conditions satisfied by v1 :

min(

v1,1

1− κ−∂v1

∂x

)= 0 on 0 × [0, δ], (6)

limx↑D

v1(t , x) = v0(D) for t ∈ (0, δ], (7)

limt↑δ

v1(t , x) = g(x) := max(v0(D)− (D − x)/(1− κ), 0) for x ∈ [0,D].(8)

Theorem

The value functions (v0, v1) are the unique viscosity solutions of the system ofvariational inequalities :




Optimal dividend strategy

I We introduce the dividend region

D := z ∈ [D,+∞) :∂v0

∂x(z−) =

11 + κ

Dividend region

There exists a threshold x∗0 ≥ D such that D = [x∗0 ,+∞). If v0(D) ≥ µ(1+κ)ρ

thenx∗0 = D else x∗0 is the unique solution in (D,+∞) of the following equation :

ρv0(x∗0 ) =µ

1 + κ.

Moreover, on (D,+∞), v0 is twice continuously differentiable.




Optimal capital injection strategies

I We introduce the following capital injection and continuation regions

Ki := z ∈ S i :∂vi

∂x(z+) =

11− κi

C i :=(Ki ∪ D

)c

Optimal capital injection when v0(D) ≥ h(D) := 11−k1

(D + µ

ρ

)In this case, we have K0 = [0,D) and K1 = S1. Hence,

v0(x) = v0(D)−D − x1− κ0

on [0,D) and v1(t , x) = g(x) on S1




Optimal capital injection strategies when v0(D) < h(D)

Upper bound

For all (t , x) ∈ S1, we have v1(t , x) ≤ h(x) := µρ(1−κ1)

+ x1−κ1

.

Optimal capital injection under default when v0(D) < h(D)

For all t ∈ [0, δ], we set

N 1t := x ≥ 0 :

∂v1

∂x(t , x+) <

11− κ1

.

x∗1 (t) := supN 1t is the smallest solution of the equation v1(t , x) = g(x)

We have x∗1 (t) := infN 1t = 0.

Boundaries monotonicities

The function t → x∗1 (t) is non increasing on [0, δ].

There exists a threshold t∗ ∈ [0, δ] such that

v1(t , 0) > 0 for 0 ≤ t < t∗ and v1(t , 0) = 0 for t∗ ≤ t < δ.




Optimal strategies when v0(D) < h(D) (2)

6

Default duration

Cash reserve

D

δ

-x∗0 D

Dividends

Capital-

-

-

-

-

-

Cap

ital

t∗

-

-

-




An example of optimal strategy

0.5 1 1.5 2

x

0.5

0.375

0.25

0.125

0

t

FIGURE – D = 2, δ = 0.5, κ = 0.0909, κ′ = 0.1, λ = 5, µ = 0.05, ρ = 0.1 and σ = 1. Blackareas represent regions where it is optimal to inject capital in S1.




Sensibility analysis of the value function v0(x)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

FIGURE – D taking values in 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1, 1.5 (from left to right in thegraph). δ = 0.3, κ = 0.25, κ′ = 0.1, λ = 1, µ = 0.07, ρ = 0.1 and σ = 0.3. Blue squaresrepresent the value function at x = D. Dashed line is at level µ

ρ(1+κ′) .




Sensibility analysis of the derivative of the value function v0(x) in termsof x .

0 0.1 0.2 0.3 0.4 0.5 0.6

x

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

FIGURE – D takes values in 0.1, 0.2, 0.3 from left to right.δ = 0.3, κ = 0.25, κ′ = 0.1, λ = 1, µ = 0.07, ρ = 0.1 and σ = 0.3. Dashed lines are at levels

11−κ = 1.3333 and 1

1+κ′ = 0.0909. Blue squares represent the value of the derivative at x = D.




Sensibility analysis of the derivative of the value function v0(x) in termsof x .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x

0

0.2

0.4

0.6

0.8

1

1.2

1.4

FIGURE – D takes values in 0.4, 0.5, 0.6, 0.8, 1, 1.5 from left to rightδ = 0.3, κ = 0.25, κ′ = 0.1, λ = 1, µ = 0.07, ρ = 0.1 and σ = 0.3. Dashed lines are at levels

11−κ = 1.3333 and 1

1+κ′ = 0.0909. Blue squares represent the value of the derivative at x = D.


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Vathana Ly Vath - Mathematics | U-M LSA