Optimal Time for Acquisition of a company in the diffusion model

Transcription

1 Optimal Time for Acquisition of a company in the diffusion model Nora Muler nmuler@utdt.edu Universidad Torcuato Di Tella Coauthor: Pablo Azcue Universidad Torcuato Di Tella. Buenos Aires, Argentina. Samos. June,

2 Problem of Acquisition The model: The uncontrolled surplus of a company follows a Brownian motion with drift: 1 X t1 x 1 m 1 t 1 W t The manager of this company has the possibility at any stopping time of acquiring the portfolio of another company paying a certain price q. 2

3 We assume that after branches satisfy acquisition the uncontrolled surpluses of the two dx t 1 m 1 dt 1 dw t1, dx t 2 m 2 dt 2 dw t 2 where W t 1, W t2 is a two-dimensional Brownian motion with covariance matrix with 1, 1. 0, indicates that there is a dependence between the two branches. 3

4 We assume that after branches satisfy acquisition the uncontrolled surpluses of the two dx t 1 m 1 dt 1 dw t1, dx t 2 m 2 dt 2 dw t 2 where W t 1, W t2 is a two-dimensional Brownian motion with covariance matrix with 1, 1. 0, indicates that there is a dependence between the two branches. Givenatimeofacquisition, the total surplus can be written as * X t x m 1 t 1 W t 1 if t (before the acquisition) * X t X q m 1 m 2 t W t if t (after de acquisition). It should be X q. 4

5 Dividend Payment: The manager pays dividends, let L t be the cumulative dividend payment up to time t and the acquisition time of the second company: X t L, X t L t where L t should be predictable, L t X t and the ruin time is defined as L, min t 0:X t L, 0 for a given L, 5

6 Dividend Payment: The manager pays dividends, let L t be the cumulative dividend payment up to time t and the acquisition time of the second company: X t L, X t L t where L t should be predictable, L t X t and the ruin time is defined as L, min t 0:X t L, 0 for a given L, So the expected discounted dividend payment up to the ruin time for a dividend strategy L and acquisition time is calculated as: V L, x E x L, e cs dl s. 0 6

7 Dividend Payment: The manager pays dividends, let L t be the cumulative dividend payment up to time t and the acquisition time of the second company: X t L, X t L t where L t should be predictable, L t X t and the ruin time is defined as L, min t 0:X t L, 0 for a given L, So the expected discounted dividend payment up to the ruin time for a dividend strategy L and acquisition time is calculated as: V L, x E x 0 L, e cs dl s. The goal is to maximize the expected discounted dividend payment up to the ruin time: V x supv L, x L, 7

8 Optimal Stopping Formulation We would like to find (if it exists): The stopping time that indicates the optimal time to buy the other company ( ) an optimal dividend payment strategy L Calling now V the optimal expected discounted dividend payment of the sum of the two branches whose uncontrolled surplus satisfies: dx t m 1 m 2 dt dw t By the Dynamic Programming Principle, we have: V x supe x L, L, e cs dl s e c L, L f X L,,wheref x V x 0 Note that this formula means that for each (that indicates the time to buy the price q. second company, not necessarily the optimal one), the payment of dividends after is the optimal dividend payment for the sum of the surpluses of the two branches. 8

9 Given a stopping time : (1) For t (before acquisition) the ruin time occurs when the surplus of the first company reaches zero but For t (after acquisition) the ruin time occurs when the sum of the surpluses of the two branches reaches zero; we allow negative surplus in one of the two branches as long as the sum of the surpluses is positive. 9

10 Given a stopping time : (1) For t (before acquisition) the ruin time occurs when the surplus of the first company reaches zero but For t (after acquisition) the ruin time occurs when the sum of the surpluses of the two branches reaches zero; we allow negative surplus in one of the two branches as long as the sum of the surpluses is positive. (2) For t the surplus of the sum of the two branches can be thought as the surplus of one company in a diffusion setting so: The optimal dividend strategy for t is known, have an explicit formula and it is a barrier strategy.(see for instance Asmussen and M. Taksar (1997)) 10

11 Given a stopping time : (1) For t (before acquisition) the ruin time occurs when the surplus of the first company reaches zero but For t (after acquisition) the ruin time occurs when the sum of the surpluses of the two branches reaches zero; we allow negative surplus in one of the two branches as long as the sum of the surpluses is positive. (2) For t the surplus of the sum of the two branches can be thought as the surplus of one company in a diffusion setting so: The optimal dividend strategy for t is known, have an explicit formula and it is a barrier strategy.(see for instance Asmussen and M. Taksar (1997)) (3) We do not assume that is finite so it is not compulsory to buy the second company. 11

12 Given a stopping time : (1) For t (before acquisition) the ruin time occurs when the surplus of the first company reaches zero but for t (after acquisition) the ruin time occurs when the sum of the surpluses of the two branches reaches zero; we allow negative surplus in one of the two branches as long as the sum of the surpluses is positive. (2) For t the surplus of the sum of the two branches can be thought as the surplus of one company in a diffusion setting so: The optimal dividend strategy for t is known, have an explicit formula and it is a barrier strategy.(see for instance Asmussen and M. Taksar (1997)) (3) We do not assume that is finite so it is not compulsory to buy the second company. (4) We will see in the examples that the optimal strategy L t for t exists, it has a band structure but it is not always a barrier strategy.we will also see that the optimal value function is not necessarily concave. 12

13 V is a viscosity solution of the obstacle problem We prove that V is a viscosity solution of the HJB equation: max 2 V x m 1 V x cv x, 1 V x, 2 f x V x 0 L 1 V L 2 V L 3 V price where f x V x q. In particular if V has a second derivative it is the solution in the usual sense. Equivalently, We can think f as the obstacle because V satisfies: 1 V f. 2 If V x f x, then V is a viscosity solution of max L 1 V x, L 2 V x 0. 13

14 Note that if the function V 1 that corresponds to optimize dividends without acquisition for the first company is greater than f, then it is also the optimal value function with acquisition. This means that the optimal acquisition time (never buy the second company). 15

15 There is a natural boundary condition: V 0 0, but this condition is not enough for the uniqueness of the solution of the HJB equation ( L 1 is a second-order operator), so a verification theorem is important. Verification Theorem: If a value function of a dividend payment strategy and stopping time V L, is a viscosity solution of the HJB equation, then it is the optimal value function V. Characterization Theorem: V is the smallest of the non-decreasing Lipschitz viscosity solution of the HJB equation. 15

16 Optimal solution and optimal strategy. The optimal value function V is a viscosity solution of the HJB equation max 2 V x m 1 V x cv x, 1 V x, 2 f x V x 0 L 1 V L 2 V L 3 V The optimal strategy depends on the current surplus X t : 1. If L 3 V X t 0, that is V X t f X t V X t q, then 0 (buy immediately the second company and follow the optimal strategy for the two branches together). 2. If L 2 V X t 0, make a lump payment of dividends (and do not buy the second company at time t). 3. If L 1 V X t 0, do not pay any dividend (and do not buy the second company at time t). 17

17 The acquisition time problem with dividends is a particular case of a more general optimal stopping problem: Consider any problem such that the uncontrolled surplus follows a Brownian motion with drift, X t x mt W t and the optimal value function can be written as: V x sup L, E x L, e cs dl s e c L, f X L 0 L, where f is a continuous and non-negative and continuous function, L t an admissible dividend strategy, and stopping time. This kind of problems can be studied in the same way we did with the acquisition problem. The difference is that is a stopping time that means something that depends on the problem For instance the problem of disinvestment can be written using this formulation. 18

18 Example of Acquisition Parameters: c = 1 First Company: m 1 = 1, s 1 = 1, x 1 * = 0.76 Second Company : m 2 = 2, s 2 = 4, x 2 * = 1.85, q = V 2 Hx 2 * L -x 2 * = 0.15 After Acquisition : m = m 1 + m 2 = 3, s= s s 2 2 = 4.12, x * =2.56 VHxL-x x Red -> V 1 HxL - x Hfirst companyl Blue -> V Hx - ql - x HobstacleL

19 2 DibujosBrowniano.nb Optimal value function and value functions of some admissible strategies VHxL-x

20 4 DibujosBrowniano.nb Optimal value function and optimal strategy VHxL-x Yellow Region (L 1 HVLHxL = 0): do not pay dividends and do not buy the second branch. Light Blue Region (L 2 HVLHxL = 1 - V ' HxL = 0): pay immediatly x as dividends and do not buy the second branch. Pink Region (L 3 HVLHxL = f HxL - VHxL = 0): buy immediately the second branch and then follow the optimal strategy of the sum of the two branches.

21 DibujosBrowniano.nb 5 Derivative of the optimal value function and concavity V'HxL x At x = 0.76, V is twice continuoulsy differentiable. At x = 1.39, V is continuously differentiable but not twice continuoulsy differentiable. At x = 1.48, V is continuously differentiable but not twice continuoulsy differentiable.

22 A Related Problem : To study the merger of two companies maximizing discounted dividends. In Gerber and Shiu (NAAJ, 2006) it is study the question whether the merger of two companies is profitable. THANK YOU 18

23 The value function of a barrier strategy is as follows: Satisfies L 1 W 0 in 0, y with W 0 0, continuous and W y V y q. The value function of a multi-band strategy (2-bands) is as follows: In 0, y satisfies L 1 W 0 with W 0 0 In y 1, a satisfies W 1 (W is differentiable at y1 In a, y 2, W satisfies L 1 W 0 with W continuous at a and W y 2 V y q Note that the if the optimal value function is multi-band it should coincide with the optimal value of the first company up to a. So, since the optimal strategy of the first company is barrier this is the more complicated structure we can have. 1

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25 A comment on the price q: The "fair" value of the second company with reserve y canbecalculatedasthe value function of the optimal dividend payment policy which maximizes the cumulative expected discounted dividend pay-outs until the time of ruin, where the uncontrolled reserve evolves as V 2 y, and so we can think that the "fair price" of the portfolio is given by V 2 y y. So the maximum price possible of the portfolio (without the surplus) should be q max y 0 V 2 y y V 2 y y where y is the threshold value of the optimal dividend strategy for company. That is, q is the "fair price" for large surpluses. We took examples where the price q 0, q. 19

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27 Viscosity Solutions : A continuous function u is a viscosity subsolution of max L 1 u, L 2 u, L 3 u 0 at x if any continuously differentiable function g (if it exists) such that g x u x and g y u y for all y we have max L 1 g x, L 2 g x, L 3 g x 0 Example of such a g : uhx 0 L+pHx-x 0 L uhxl p œ D + HuLHx 0 L 21

28 A continuous function u is a viscosity supersolution of max L 1 u, L 2 u, L 3 u 0 at x if for any continuously differentiable function h (if it exists) such that h x u x and h y u y for all y we have max L 1 h x, L 2 h x, L 3 h x 0 Example of h : uhxl u Hx 0 L+pHx-x 0 L p œ D - HuLHx 0 L (1) If u is a viscosity sub- and supersolution, then it is a viscosity solution (2) There are test functions g and h at a point x 0 if, and only if, u is differentiable at 22

29 the point x 0. 23

30 Problem of disinvestment The uncontrolled joint reserve X 1 t, X 2 t of two branches of an insurance company follows the process X t1, X t2 x 1 m 1 t 1 W t1, x 2 m 2 t 2 W t2 where W t 1, W t2 is a two-dimensional Brownian motion with covariance matrix with 1, 1. The manager has the possibility of closing one of the two branches at any stopping time paying different fees depending on which branch is disinvested at any time paying a fee of c 1, c 2 0 depending on which branch is disinvested. The total reserve can be written as. X t X t 1 X t 2 x m 1 m 2 t W t if t. 24

31 . 2 X t X c 1 m 2 t 2 W t if t and the first branch is disinvested.. 1 X t X c 2 m 1 t 1 W t if t and the second branch is disinvested. The manager cannot disinvest the i th company if X c i. The manager pays dividends. Let L t be the cumulative dividend payment up to time t. The controlled total reserve is given by X t L, X t L t where L t should be predictable, L t X t and the ruin time is defined as L, min t 0:X t L, 0 The goal is to maximize the expected discounted dividend payment up to the ruin time. and V L, x E x 0 L, e cs dl s 25

32 V x supv L, x L, Let us call now V 1 and V 2 as the optimal expected discounted dividend payment of each of the branches. By the DPP, we have that the value function V canbewrittenas V x supe x e cs dl s e c f X Ḻ L, 0 where f x max V 1 x c 2, V 2 x c 1 26