Dividend Problems in Insurance: From de Finetti to Today

Transcription

1 Dividend Problems in Insurance: From de Finetti to Today Nicole Bäuerle based on joint works with A. Jaśkiewicz Strasbourg, September 2014

2 Outline Generic Dividend Problem Motivation Basic Models and Results Generalizations Risk-sensitive Dividend Problems

3

4 Generic Dividend Problem Free risk reserve: (R t ). Accumulated dividend process: (D t ). Net risk reserve: Technical time of ruin: Optimization problem: sup D X t = R t D t, X 0 = x > 0. τ := inf{t > 0 : X t < 0}. [ τ IE x e βt dd t ], β > 0. 0

5 Motivation De Finetti 1957: Gordon 1962: If a company is not ruined in the Cramér-Lundberg model, it gets infinitely rich. Value of a firm = Sum of expected, discounted dividends. Survey papers: Avanzi 2009, Albrecher and Thonhauser 2009.

6 Simple Discrete Model (Setup) De Finetti Free risk reserve: R t = x + Z Z t, t, x N with Z 1, Z 2,... iid and P(Z 1 = +1) = q 1, P(Z 1 = 1) = q 1 = 1 q 1. Let d t = Dividend paid at time t. (w.l.o.g. d t N 0 ) Optimization problem: sup d [ τ 1 ] IE x β k d k =: J(x), β (0, 1). k=0

7 Simple Discrete Model (Solution) Optimal dividend policy: Barrier-type, i.e. { 0, X d t = f (X t ) = t b X t b, X t > b 2 where b = 1 + log r log s log ( ) 1 s r 1 and r > 1, s < 1 are the roos of q 1 βx 2 x + q 1 β = 0. Value function: J(x) = r x s x (r 1)r b (s 1)s b, x N 0.

8 General Discrete Model (Setup) Miyasawa 1962, Tekeuchi 1962, Morill Free risk reserve: R t = x + Z Z t, t, x N with Z 1, Z 2,... iid and P(Z 1 = k) = q k, k Z. Further P(Z1 < 0) > 0, IE Z + 1 <. Optimization problem: sup d [ τ 1 ] IE x β k d k =: J(x). k=0 back

9 General Discrete Model (Solution Method) Bellman equation: J(x) = [ max d + β d {0,1,...,x} Convergence condition: k=d x lim n βn sup IE π x J(X n ) = 0. π J(x d + k)q k ]. Then: J is largest d-subharmonic solution of the Bellman equation and maximizer of J yields optimal dividend policy. Schmidli 2008, B. and Rieder 2011.

10 General Discrete Model (Solution) Optimal dividend policy: Band-policy, i.e. there exists n N 0 and 0 c 0 b 1 c 1 b 2... b n c n s.t. b k c k 1 2 and d t = f (X t ) = 0, if X t c 0 X t c k, if c k < X t < b k+1 0, if b k+1 X t c k+1 X t c n, if X t > c n.

11 Properties of Band Policy Finite number of bands. If P(Z 1 z 0 ) = 1 for some z 0 N, then the length of the bands is b k c k 1 z 0. If P(Z 1 1) = 1, then the optimal policy reduces to a barrier policy.

12 Cramér-Lundberg Model (Setup) Gerber 1969, Azcue and Muler Free risk reserve: R t = x + ct N t k=1 Y k, t, x > 0 with c > 0 premium rate, (Nt ) Poisson process with intensity λ > 0, Y 1, Y 2,... 0 iid claim sizes, IE(Y 1 ) <. Optimization problem: sup D [ τ ] IE x e βt dd t =: J(x). 0

13 Cramér-Lundberg Model (Solution) Hamilton-Jacobi Bellman equation: { 0 = max 1 V (x), cv (x) (λ+β)v (x)+λ x 0 } V (x y)df Y (y). Optimal dividend strategy: Band-strategy, i.e. R + = A + B + C where x A : pay out premium income, x B : lump sum payment to closest point in A, smaller that x, x C : no payment.

14 Typical Path under Band Strategy X t no payment lump sum payment no payment t

15 Properties of Band Strategy If density of Y 1 is completely monotone, then optimal strategy collapses to Barrier ( Loeffen 2008), e.g. F Y = Exp. Same, if density of Y 1 is log-convex (Kyprianou 2009). Number of connecting components in A can be infinite.

16 Diffusion Model (Setup) Shreve et al. 1984, Jeanblanc-Picqué and Shiryaev 1995, Asmussen and Taksar Free risk reserve: R t = x + µt + σw t, t, x > 0. Optimization problem: sup D [ τ IE x e βt dd t ]. Hamilton-Jacobi Bellman equation: 0 0 = max {1 V (x), µv (x) + σ2 } 2 V (x) βv (x). Optimal dividend strategy: Barrier.

17 Further Generalizations Non-Markovian models (Sparre-Anderson model). Albrecher and Hartinger Spectrally negative Lévy processes. Avram et al. 2007, Loeffen Consider constraints. Borch 1967, Waldmann 1988, Thonhauser and Albrecher Different ruin definition. Kulenko and Schmidli 2008, Albrecher et al Return on investment. Paulsen and Gjessing 1997, Højgaard and Taksar 2001, Albrecher and Thonhauser Tax and Transaction cost. Jeanblanc-Picqué and Shiryaev 1995, Paulsen 2007.

18 Risk-sensitive Dividend Problems Risk-sensitive Dividend Problem

19 Risk-sensitive Dividend Problems How to incorporate risk aversion in decision problems? Consider risk constraints (Markowitz 1952). Use risk measures (B/Mundt 2009, Ruszczyński 2010, B/Ott 2011, Shen et al. 2014). Use utility functions, certainty equivalents (since 1930ies; Howard/Matheson 1972): where U 1( IE [ U(Y ) ]) IE Y 1 2 l U(IE Y )Var[Y ] l U (y) = U (y) U (y) is the Arrow-Pratt function of absolute risk aversion.

20 Risk-sensitive Dividend Problems Risk-Sensitive Dividend Pay-Out Consider the general discrete setup with optimization criterion: sup d sup d IE x [ 1 γ exp ( ) ] β k d k, γ < 0 τ 1 γ k=0 [ ] ( τ 1 ) γ IE x β k d k, γ (0, 1). k=0 Gerber and Shiu 2004, Grandits et al. 2007

21 Risk-sensitive Dividend Problems Policies Histories H 0 := Z, H n := (Z N 0 ) n Z, ie. h n = (x 0, d 0, x 1,..., d n 1, x n ) H n. A decision rule at time n is a measurable mapping f n : H n N 0 such that f n (h n ) {0, 1,..., x n }. A policy is a sequence of decision rules π = (f 0, f 1,...).

22 Risk-sensitive Dividend Problems Exponential Case For a history-dependent policy π = (f 0, f 1,...) and θ [γ, 0) we define ( J π (x, θ) := IE x [exp θ τ 1 β k f k (h k )) ]. k=0 J(x, θ) = inf π J π(x, θ).

23 Risk-sensitive Dividend Problems Bellman Equation - Exponential Case Theorem For every (x, θ) Z [γ, 0), the function J is a solution to the following discounted optimality equation J(x, θ) = min a {0,...,x} [ e θa ( k=a x )] a x 1 J(x a + k, θβ)q k + q k k=

24 Risk-sensitive Dividend Problems Properties of Minimizer Let f be the largest minimizer of J. Theorem Let ξ(θ) := sup{x N 0 : f (x, θ) = 0}. Then ξ := sup θ [γ,0) ξ(θ) < and f (x, θ) = x ξ(θ), for all x > ξ(θ). Consider the following policy π := (g 0, g 1,...), where g n(h n ) := f (x n, γβ n ).

25 Risk-sensitive Dividend Problems Structure of Optimal Policy Definition A function g is called a band-function, if there exists n N 0 and 0 c 0 b 1 c 1 b 2... b n c n s.t. d k c k 1 2 and 0, if x c 0 x c g(x) = k, if c k < x < b k+1 0, if b k x c k x c n, if x > c n. π = (g n ) n N0 is a band-policy, if g n is a band-function for all n. Theorem The policy π is optimal and a band-policy.

26 Risk-sensitive Dividend Problems Numerical Tool: Policy Improvement Start with pay-out rule f and fixed s s.t. Compute J f. f (x, θ) x s, for x > s, θ Determine minimizer h of a x 1 a e θa J f (x a + k, θβ)q k + k=a x If h f then J h J f. If h = f then J f = J. k= q k.

27 Risk-sensitive Dividend Problems Bellman Equation - power utility case Let J(x, y) := sup π [ ( τ 1 IE x k=0 ] ) γ β k f k (h k ) + y, (x, y) Z R +. Theorem For every (x, y) Z R +, the function J is a solution to the following discounted optimality equation [ ( J(x, y) = β γ max J x a + k, a + y ) ] q k a {0,...,x} β k=a x

28 Risk-sensitive Dividend Problems Properties of Maximizer Let f be the largest maximizer of J. Theorem Let ξ(y) := sup{x N 0 : f (x, y) = 0}. Then ξ := sup y>0 ξ(y) <. The policy π constructed from f by π := (g 0, g 1,...), g n(h n ) := f (x n, y n ), y n := y 0 + n 1 k=0 a k β n is optimal.

29 Risk-sensitive Dividend Problems Thank you very much for your attention!