Liquidity risk and optimal dividend/investment strategies

Transcription

1 Liquidity risk and optimal dividend/investment strategies Vathana LY VATH Laboratoire de Mathématiques et Modélisation d Evry ENSIIE and Université d Evry Joint work with E. Chevalier and M. Gaigi ICASQF, Cartagena, June 15-18, 2016

2 1 Introduction 2 The model 3 Characterization of the value functions 4 Numerical results

3 Motivations : a corporate finance problem Corporate governance problems : managerial decisions 1 Operational levels : marketing, product lines, internal organization... 2 Financial levels : capital/debt structure, type of financing... 3 Cash-flow levels/utilization : dividends and investments.

4 Motivations : a corporate finance problem Cash-flow (utilization) policy Dividend policy : payment to shareholders Share buyback (dividend or investment?) Dividend payment Singular problem : [1] Jeanblanc and Shiryaev (95), Choulli, Taksar and Zhou (03)

5 Motivations : a corporate finance problem Cash-flow (utilization) policy Investment policy : investment for future growth Organic growth : internal development of new products, technologies, factories... Merger/Acquisition : acquire a competitor for its product portfolio, geographic reach... Optimal switching problem : Brekke and Oksendal (94), Duckworth and Zervos (01), Hamadène and Jeanblanc (05), LV. and Pham (07), Pham, LV. and Zhou (09).

6 Motivations : a corporate finance problem Combining both problems : Dividend and investment Typical corporate dilemma : Total SA, ENI, BP : how to use the huge Cash-Flow generated? which investment projects? Three observed policies : 1 Trader Classified Media : return over 90% of stock value in cash. 2 Bouygues : return 25% of stock value in cash. 3 Microsoft : almost no dividend payment for years!

7 Introduction - motivations : the Jeanblanc-Shiryaev problem. A policy strategy is Z : a F-adapted càdlàg non-decreasing process We consider process X representing the dynamics of the cash reserve of a firm : dx t = µdt + σdw t dz t X 0 = x The value of the firm is defined as : [ T ˆV 0 (x) = sup E 0 Z Z 0 ˆV 0 the value function of a singular problem e ρt dz t ] characterized as the unique solution on (0, ) with growth linear condition to (V.I) 0 : [ min ρ ˆV ] 0 L 0 ˆV0, ˆV 0 1 = 0, x > 0, (1), ˆV 0 (0) = 0. (2)

8 Optimal dividend problem/singular control [1] Explicit solution : Jeanblanc and Shiryaev (95) or Radner and Shepp (96) ˆV O explicit expression dx = μ 0 dt + σdw dz X t 0 = x T0 Vˆ 0( x) = sup E e Z 0 t ρt t dzt dividend : x x ˆ0 Regime 0 0 ˆx 0 x Benchmark 0: a singular control problem

9 Introduction - motivations : assumptions In the previous problem, it is assumed : there is no investment opportunity the firm assets are infinitely liquid or illiquid Our objective : Study optimal dividend and investment control problem under constraints, i.e. relaxing the above assumptions L.V, Pham, and Villeneuve (2008) : the interaction between dividend policy and investment under uncertainty. A mixte singular and switching control problem Other related papers : Decamps and Villeneuve (05), Chevalier, L.V., Scotti (13), Guo and Tomecek (08)... In this study, we look at the dividend/investment problems but no longer assume that assets are infinitely illiquid or liquid. The firm may face some liquidity costs when buying or selling assets.

10 The model Let (Ω, F, P) be a probability space equipped with a filtration F = (F t ) t 0 and W and B be two correlated F-Brownian motions, A policy strategy is a singular/switching control α = ((τ i, q i ) i N, Z ) Z : F-adapted càdlàg non-decreasing process representing dividend policy. (τ i ) : an increasing sequence of stopping times representing investment decision times. (q i ) : q i are F τi -measurable variables representing the number of productive assets units bought/sold. We assume productive assets are risky assets whose value process S is solution of the following equation : ds t = S t (µdt + σdb t ), S 0 = s, (3)

11 The model the dynamics of the quantity of assets held by the firm Q t N is governed by : dq t = 0 for τ i t < τ i+1, Q τi = Q τ + q i, Q 0 = i q, for i N. (4) The dynamics of the cash reserve (or more precisely the firm s cash and equivalents) process of the firm is governed by : dx t = rx t dt + h(q t )(bdt + ηdw t ) dz t, for τ i t < τ i+1 X τi = X τ S τi f (q i )q i κ, X 0 = i 0, where b, r and η are positive constants h a non-negative, non-decreasing and concave function satisfying h(q) H with h(1) > 0 and H > 0. The non-negative, non-decreasing, function f represents the liquidity cost function (or impact function with the impact being temporary) with f (0) = 1 for i N. (5) Remarque. For a given fixed q, the model is closely related to the bachelier model used by Jeanblanc-Shiryaev.

12 The investment objective The bankruptcy time is defined as T := T y,α := inf{t 0, X t < 0}. We define the liquidation value as L(x, s, q) := x + (sf ( q)q κ) + We introduce the following notation S := R + (0, + ) N. The optimal firm value is defined on S := R + (0, + ) N, by [ ] T v(x, s, q) = sup E (x,s,q) e ρu dz u α A(x,s,q) 0 (6)

13 Trivial case where value function is infinite We now identify the trivial cases where the value function is infinite. Lemma If we have r > ρ or µ > ρ then v(y) = + on S. Proof : Let y := (x, s, q) S. We first assume that ρ < r. At time 0, by choosing to liquidate the firm s assets, we may get L(y) > 0 in cash. Then by waiting until a given time t > 0, we may obtain v(y) e (r ρ)t L(y). By letting t going to +, we have v(y) = +. Assume that ρ < µ. First, suppose that q 1. In this case, by doing nothing up to time t and then liquidate at time t, for any t > 0, we may obtain a lower bound of v(y) which we prove goes to when t goes to

14 Associated HJB equation From the Dynamic Programming Principle, we may obtain the following HJB equation : min{ρϕ(x, s, q) Lϕ(x, s, q); ϕ (x, s, q) 1; v(x, s, q) Hv(x, s, q)} = 0 on S, (7) x where we have set and Lϕ(x, s, q) = η2 h(q) 2 2 ϕ 2 x 2 + σ2 s 2 2 ϕ 2 s 2 + cσηsh(q) 2 ϕ s x + (rx + bh(q)) ϕ ϕ + µs x s Hϕ(x, s, q) = max ϕ (Γ(y, n)) n A(x,s,q) { with A(x, s, q) = n Z : n q and n(f (n)) x }, s Γ(y, n) = (x n(f (n))s, s, q + n). S = {(x, s, q) R + (0, + ) N}.

15 Sequence of auxiliary value functions We now introduce the following subsets of A(y) : A N (y) := {α = ((τ k, ξ k ) k N, Z ) A(y) : τ k = + a.s. for all k N + 1} We define [ ] T v N (y) = sup E (x,s,q) e ρu dz u, N N (8) 0 α A N (y) The different steps to follow We characterize recursively the value functions v N. We prove the convergence of v N to v. We compute numerically the different regions (continuation, buy and sell and dividend regions) since no explicit solution may be obtained.

16 Sequence of auxiliary value functions In the next Proposition, we recall explicit formulas for v 0 and the optimal strategy associated to this singular control problem. Proposition There exists x (q) [0, + ) such that { Vq(x) if 0 x x v 0 (x, s, q) := (q) x x (q) + V q(x (q)) if x x (q), where V q is the C 2 function, solution of the following differential equation η 2 h(q) 2 y + (rx + bh(q))y ρy = 0; 2 (9) y(0) = 0, y (x (q)) = 1 and y (x (q)) = 0. (10) Notice that x v 0 (x, s, q) is a concave and C 2 function on [0, + ) and that if h(0) = 0, it is optimal to immediately distribute dividends up to bankruptcy therefore v 0 (x, s, 0) = x.

17 Auxiliary functions : Optimal stopping We now are able to characterize our impulse control problem as an optimal stopping time problem, defined through an induction on the number of interventions N. Proposition For all (x, s, q, N) S N, we have T τ v N (x, s, q) = sup E{ e ρu dz u + e ρτ G N 1 (X x τ (τ,z ) T Z 0, Sτ s, q)1 {τ<t }}, (11) where T is the set of stopping times, Z the set of predictable and non-decreasing càdlàg processes, and G N 1 (x, s, q) := max N 1 (Γ(y, n)) and G 1 = 0, n a(x,s,q) with { a(x, s, q) := n Z : n q and nf (n) x κ }, s and Γ(y, n) := (x nf (n)s κ, s, q + n).

18 Bounds and convergence of v N We begin by stating a standard result which says that any smooth function, which is supersolution to the HJB equation, is a majorant of the value function. Proposition Let N N and φ = (φ q) q N be a family of non-negative C 2 functions on R + (0, + ) such that q N (we may use both notations φ(x, s, q) := φ q(x, s)), φ q(0, s) 0 for all s (0, ) and [ min ρφ(y) L N φ(y), φ(y) G N 1 (y), φ ] x (y) 1 for all y (0, + ) (0, + ) N, where we have set then we have v N φ. Corollary L N ϕ = η2 h(q) 2 2 [ σ 2 s 2 +1 {N>0} 2 ϕ ϕ + (rx + bh(q)) x 2 x Upper bound For all N N and (x, s, q) S, we have 2 2 ϕ s 2 + cσηsh(q) 2 ϕ ϕ + µs s x s 0 (12) ]. L(x, s, q) v N (x, s, q) x + sq + K where ρk = bh.

19 Bounds and convergence of v N Proposition(Convergence) For all y S, we have lim v N(y) = v(y). N + Proof : We obviously have v N v N+1 v for all N N. For y S and ε > 0, we may now consider a strategy α = ((τ k, ξ k ) k N, Z ) A(y) such that v(y) J α (y) + ε. Notice that, as (τ i ) i N is such that lim i + τ i = +, there exists N N such that which ends the proof. T τn J α (y) E[ e ρs dz s] + ε 0 v N (y) + ε,

20 Viscosity characterization of v N We turn to the characterization of the function v N as the unique function which satisfies the boundary condition v N (y) = G N 1 (y) on {0} (0, + ) N. (13) and is a viscosity solution of the following HJB equation : min{ρv N (y) Lv N (y); v N x (y) 1; v N (y) G N 1(y)} = 0 on (0, + ) 2 N, (14) It relies on the following Dynamic Programming Principle. DPP Let θ T, y := (x, s, q) S and set ν = T θ, we have (ν τ) v N (y) = sup E[ (τ,z ) T Z 0 e ρs dz s + e ρ(ν τ) v N ( X x (ν τ), S s ν τ, q ) 1l {τ<ν} ] (15)

21 Viscosity characterization of v N We are now able to establish the main results of this section. Theorem For all (N, q) N N, the value function v N (,, q) is continuous on (0, + ) 2. Moreover v N is the unique viscosity solution on (0, + ) 2 N of the HJB equation (14) satisfying the boundary condition (13) and the following growth condition for some positive constants C 1, C 2 and C 3. v N (x, s, q) C 1 + C 2 x + C 3 sq, (x, s, q) S,

22 Numerical results FIGURE: Description of different regions, in (x, s) for a fixed q 0.

23 Numerical results FIGURE: Description of different regions, in (x, s) for q 1 > q 0.