Optimal dividend and reinvestment policies when payments are subject - PowerPoint PPT Presentation

Introduction to the problem The solution A financial example Optimal dividend and reinvestment policies when payments are subject to both fixed and proportional costs Jostein Paulsen October 15, 2008

Introduction to the problem The solution A financial example Talk based on the paper Optimal dividend payments and reinvestments of di ff usion processes with both fixed and proportional costs SIAM Journal of Control and Optimization , 2008, Vol. 47, No 5, 2201-2226.

Introduction to the problem The solution A financial example Introduction to the problem 1 The solution 2 A financial example 3

Introduction to the problem The solution A financial example The model Income process without payments dX t = µ ( X t ) dt + σ ( X t ) dW t . Standing assumptions: A1. | µ ( y ) | + | σ ( y ) | ≤ K (1 + y ) for all y ≥ 0 and some K > 0. A2. µ and σ are continuously di ff erentiable and the derivatives µ ′ and σ ′ are Lipschitz continuous for all y ≥ 0. A3. σ 2 ( y ) > 0 for all y ≥ 0. A4. µ ′ ( y ) ≤ r for all y ≥ 0. Here r is a discount factor. Let Lg ( y ) = 1 2 σ 2 ( y ) g ′′ ( y ) + µ ( y ) g ′ ( y ) − rg ( y ) .

Introduction to the problem The solution A financial example Comments on Assumption A4 A4: µ ′ ( y ) ≤ r for all y ≥ 0. Here r is a discount factor. Consider the special case dX t = ( µ 0 + µ 1 X t ) dt + σ ( X t ) dW t , X 0 = x. Here µ ′ ( x ) = µ 1 and furthermore � x + µ 0 � e ( µ 1 − r ) t − µ 0 E x [ e − rt X t ] = e − rt . µ 1 µ 1 If µ 1 ≤ r this stabilizes, but if µ 1 > r it grows to infinity and therefore it is clearly better to wait. The right quantities to compare are therefore µ ′ ( x ) and r , one representing the geometric growth rate and the other the geometric discounting rate. The condition µ ′ ( x ) ≤ r just says that in no state should growth rate exceed discounting rate.

Introduction to the problem The solution A financial example The problem Total dividends paid up to time t is D t . When reserves hit zero reinvestments are made, total reinvestments up to time t is C t . Both C and D are nondecreasing and RCLL. Associated costs are d ¯ C t = c 0 1 {△ C t > 0 } + c 1 dC t , 0 ≤ c 1 ≤ 1 , d ¯ D t = d 0 1 {△ D t > 0 } + d 1 dD t , where c 0 , c 1 , d 0 and d 1 all are nonnegative constants. Therefore dY t = µ ( Y t ) dt + σ ( Y t ) dW t + (1 − c 1 ) dC t − (1 + d 1 ) dD t − c 0 1 {△ C t > 0 } − d 0 1 {△ D t > 0 } , with Y 0 − = y .

Introduction to the problem The solution A financial example The problem For given ( C, D ) let �� ν n − � E y e − rt dA t V C,D ( y ) = lim sup , n →∞ 0 − where A = D − C and ν n = inf { t : C t ∨ D t > n } . We want to find V ∗ ( y ) = sup V C,D ( y ) . ( C,D ) and also, if it exists, the optimal policy ( C ∗ , D ∗ ) .

Introduction to the problem The solution A financial example Literature Shreve, Lehoczky and Gaver (1984). Same model as here, but without fixed costs. Richard (1977), Constantinides and Richard (1978), Harrison, Sellke and Taylor (1983). With fixed costs, but only linear Brownian motion. Avram, Palmowski and Pistorius (2007). Spectrally negative Lévy process, but no fixed costs. Porteus (1977). Discrete time Papers with absorbtion at zero Paulsen (2007). Same model and expenses as in this paper Jeanblanc-Picqué and Shiryaev (1995) . Linear Brownian motion.

Introduction to the problem The solution A financial example Literature Papers written for combinations of dividend payments, investment policies and reinsurance policy, but restricted to Brownian motion are Cadenillas, Sarkar and Zapatero (2007) , Cadenillas, Choulli, Taksar and Zhang (2006) .

Introduction to the problem The solution A financial example General considerations Why is it possible to give a complete solution for such a general model? Consider again the equation Lg ( y ) = 0. Four (or five) basic solutions

Introduction to the problem The solution A financial example General considerations 6 10 4 value value 8 2 6 0 4 0 2 4 6 8 10 0 2 4 6 8 10 x x −0.7 1.0 value value −1.0 0.8 −1.3 0.6 0 2 4 6 8 10 0 2 4 6 8 10 x x

Introduction to the problem The solution A financial example The variational problem Consider the variational problem for unknown V , y ∗ , γ ∗ ∈ (0 , y ∗ ) and δ ∗ ∈ (0 , y ∗ ),

Introduction to the problem The solution A financial example The variational problem 0 < y < y ∗ , LV ( y ) = 0 , V (0) + γ ∗ + c 0 V ( γ ∗ ) = , 1 − c 1 1 V ′ ( γ ∗ ) = , 1 − c 1 V ( y ∗ − δ ∗ ) + δ ∗ − d 0 V ( y ∗ ) = , 1 + d 1 1 V ′ ( y ∗ − δ ∗ ) = , 1 + d 1 1 V ′ ( y ∗ ) = , 1 + d 1 V ( y ∗ ) + y − y ∗ y > y ∗ . V ( y ) = , 1 + d 1

Introduction to the problem The solution A financial example The variational problem a) If this has a solution this solution is unique and V ( y ) = V ∗ ( y ) , y ≥ 0 . The optimal policy is to pay δ ∗ in dividends whenever Y t − = y ∗ and to reinvest γ ∗ whenever Y t − = 0. b) If this has no solution there is no optimal policy, but V ∗ ( y ) = lim y →∞ V ¯ y ) ( y ) y,γ ( ¯ y ) ,δ ( ¯ ¯ and this limit exists and is finite for every y ≥ 0.

Introduction to the problem The solution A financial example The variational problem Proposition 1 a) Assume there is no optimal solution. Then there exists a solution g 2 of Lg = 0 so that y →∞ g ′ y →∞ g 2 ( y ) = lim lim 2 ( y ) = 0 . Furthermore, for any other independent solution g 1 , g 1 ( y ) y →∞ g ′ lim 1 ( y ) = lim = ¯ g 1 y y →∞ for some positive and finite ¯ g 1 .

Introduction to the problem The solution A financial example The variational problem b) Assume that there are two solutions g 1 and g 2 of Lg = 0 so that y →∞ g ′ ¯ lim 1 ( y ) = g 1 , y →∞ g 2 ( y ) lim = 0 , where ¯ g 1 is finite and nonzero. Assume in addition that � g 1 ( y ) � > µ (0) − y − d 0 . lim g 1 ¯ r y →∞ Then there is no optimal solution.

Introduction to the problem The solution A financial example The variational problem c) Assume there is a solution g of Lg = 0 so that g ( y ) lim = ∞ y y →∞ or equivalently y →∞ g ′ ( y ) = ∞ . lim Then there is an optimal solution.

Introduction to the problem The solution A financial example Linear Brownian Motion Let the income process without dividends follow dX t = µdt + σdW t , It is easy to verify that Lg ( y ) = 0 has the independent solutions g i ( y ) = e θ i y , i = 1 , 2 , where �� � 1 µ 2 + 2 rσ 2 − µ θ 1 = σ 2 �� � − 1 µ 2 + 2 rσ 2 + µ θ 2 = . σ 2 Clearly θ 1 > 0, hence an optimal solution exists by Proposition 1.c. This is the main result of Harrison & al. (1983).

Introduction to the problem The solution A financial example A useful comparison result Lemma Assume A2 and A3. Let f i ( y ), i = 1 , 2 solve 1 2 σ 2 ( y ) f ′′ i ( y ) + µ i ( y ) f ′ i ( y ) − rf i ( y ) = 0 , y ≥ 0 , where µ 1 ( y ) > µ 2 ( y ) for all y ≥ 0 and f ′ f i (0) = f 0 and i (0) = f 1 ≥ 0 , i = 1 , 2 . Then f ′ 1 ( y ) < f ′ 2 ( y ) for all y > 0, which in turn implies that f 1 ( y ) < f 2 ( y ) for all y > 0.

Introduction to the problem The solution A financial example A useful comparison result Proposition 2 Assume there is no optimal policy, and let V be the value function. Consider the equation (in ¯ γ ). 1 V ′ ( ¯ γ ) = , (1) 1 − c 1 ¯ γ + c 0 V ( ¯ γ ) = V (0) + . (2) 1 − c 1 Furthermore, with g 1 and g 2 as in Proposition 1, write V ( y ) = a 1 g 1 ( y ) + a 2 g 2 ( y ) . a) We have 1 y →∞ V ′ ( y ) = lim . 1 + d 1

Introduction to the problem The solution A financial example A useful comparison result b) If c 1 + d 1 > 0 then (1) has a unique solution. Furthermore 1 1 a 1 = , 1 + d 1 g 1 ¯ g ′ 1 ( ¯ γ ) 1 1 1 1 γ ) − a 2 = γ ) . g ′ g ′ 1 − c 1 1 + d 1 ¯ 2 ( ¯ g 1 2 ( ¯ g 1 = lim y →∞ g ′ Here ¯ 1 ( y ) and ¯ γ is the solution of 1 − c 1 1 c 0 = ( g 1 ( y ) − g 1 (0)) 1 + d 1 g 1 ¯ g ′ � � 1 ( y ) 2 ( y ) − 1 − c 1 1 1 + ( g 2 ( y ) − g 2 (0)) − y. g ′ g ′ 1 + d 1 g 1 ¯ 2 ( y )

Introduction to the problem The solution A financial example A useful comparison result c) If c 1 = d 1 = 0 there are two possibilities. (i) The equation (1) has a unique solution and then a 1 , a 2 and γ are is in part b above. ¯ (ii) The equation (1) has no solution, but 1 a 1 = , g 1 ¯ � g 1 ( y ) g ′ � 1 (0) lim y →∞ − y − − c 0 g 1 g 1 ¯ a 2 = . g 2 (0)

Introduction to the problem The solution A financial example A financial example Income process without dividends assumed to be a linear Brownian motion with drift µ and di ff usion σ , but money can be invested in risk free assets with return r . Investment costs are incurred with rate α ( Y t ) so that total investment costs have intensity α ( Y t ) Y t . Assume that this consists of a fixed part α 0 and a part that is proportional with the amount invested α 1 , i.e. α ( y ) y = α 0 + α 1 y. This gives dX t = ( µ 0 + ( r − α 1 ) X t ) dt + σdW t , where µ 0 = µ − α 0 . Assume that µ 0 > 0 and 0 ≤ α 1 < r . When α 0 = 0 and α 1 = r , this is Brownian motion. The generator is Lg ( y ) = 1 2 σ 2 g ′′ ( y ) + ( µ 0 + ( r − α 1 ) y ) g ′ ( y ) − rg ( y ) = 0 .