Dividend problems in the dual risk model

Transcription

1 Dividend problems in the dual risk model Lourdes B. Afonso, Rui M.R. Cardoso 1 & Alfredo D. Egídio dos Reis 2 CMA and FCT, New University of Lisbon & CEMAPRE and ISEG, Technical University of Lisbon ASTIN 2011 Madrid 1 This work was partially supported by CMA/FCT/UNL, under Financiamento Base 2009 ISFL from FCT/MCTES/PT 2 The authors gratefully acknowledge nancial support from FCT-Fundação para a Ciência e a Tecnologia (Project reference PTDC/EGE-ECO/108481/2008) LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 1 / 26

2 Dual Risk Model Key reference: Avanzi, Gerber & Shiu (2007). U(t) = u ct + S(t), t 0 Accumulated reserve up to time t : U(t) u : initial reserve, surplus; Individual random gains: X i _ P(x), E[X i ] = p 1 ; Accumulated gains up to t : S(t) = N (t) i=0 X i, X 0 0. fx i g, i.i.d. fx i g independent of N(t); fs(t), t 0g :compound Poisson process, parameters λ and p.d.f. p(x); c : constant rate of expenses, µ = E [S(1)] c = λp 1 c > 0, λp 1 > c. b u: (Upper) dividend barrier (absorbing); Ruin barrier (absorbing). LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 2 / 26

3 Dual Risk Model U(t) = u ct + S(t), t 0 LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 3 / 26

4 Classical vs Dual LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 4 / 26

5 Problem: Dividend calculation We have 2 absorbing barriers: Dividend barrier b and Ruin level 0. A dividend can be paid if the process is not ruined. b is xed. Once reached a dividend is paid and process re-starts, from b. If the process is ruined, the investment is insolvent. Let δ (> 0) be the force of interest. Problem: Calculate the discounted future dividends, e.g., Expected. Dividends can be multiple. There are results on expected, moments of discounted dividends: Avanzi et al. (2007), Cheung & Drekic (2008), Ng (2009, 2010) We go further, with a new approach: connection the classical and the dual LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 5 / 26

6 Solutions for the dividend calculation Probability of ruin, with no dividend barrier: known. Setting a dividend barrier, we evaluate: Expected discounted future dividend payments: known, but new approach; higher moments also available; The probability of getting a dividend payment; Distribution for the amount of a dividend payment; Distribution for the number of dividends (in nite time). LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 6 / 26

7 De nitions Let, process free of the barrier, and continue if it is ruined: Time to ruin, from initial surplus x: τ x = inf ft > 0 : U(t) = 0jU(0) = xg ; (τ x = if U(t) 0 8t 0) Probability of ultimate ruin ψ(x) = Pr fτ x < ju(0) = xg = e Ru where R > 0: λ R 0 e Rx p(x)dx 1 = cr. Time to reach an upper level b x 0, from x: T x = inf ft > 0 : U(t) > bju(0) = xg Probability of ruin before reaching b: ξ(u, b) = Pr (T x > τ x ). Probability of reaching b before ruin: χ(u, b) = Pr (T x < τ x ) Single dividend amount: D u = fu(t u ) b ^ T u < τ u g. D.f.: G (u, b; x) = Pr(T u < τ u and U(T u ) b + x)ju, b) M: number of dividends to be distributed. LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 7 / 26

8 Total discounted dividends Total discounted dividends: D(u, b, δ) " # V (u; b) = E[D(u, b, δ)] = E e δ( i j=1 T (j)) D(i) i=1, 0 u b. LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 8 / 26

9 Expected discounted dividends V (u; b) = E[D(u, b, δ)] From Avanzi et al.(2007), solutions from solving, 0 = cv 0 (u; b) + (λ + δ)v (u; b) λ λ Z b u Z b 0 u V (u + y, b)p(y)dy (1) V (u b + y, b)p(y)dy λv (b, b) [1 P(b u)] Also, Laplace transforms from there; Instead, we propose solving V (u; b) = E[D(u, b, δ)] = E " i=1 e δ( i j=1 T (j)) D(i) # n o T (i), D (i), sequence of i.i.d. random pairs. i=2 d= T (i), D (i) (Tb, D b ) d= Independent of T (1), D (1). Let T (1), D (1) (Tu, D u ). LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 9 / 26

10 Expected discounted dividends Independence gives " # V (u; b) = E e δ( i j=1 T (j)) D(i) = i=1 = E e δt (1) D (1) + E e δt (1) T (i), D (i) + E E e δt (1) e δt (2) i=1 E i E he δ( i j=1 T (j)) D(i) E e δt (2) D (2) e δt (3) D (3) +... Now, (T b, D b ) = d, i = 2, 3,...; (T u, D u ) = T d (1), D (1), V (u; b) = E e δt u D u + E e δt u E e δt b D b E e δt b + E e δt b +... = E e δt u D u + E e δt u E e δt b D b 1 E (e δt. b ) LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 10 / 26

11 2nd moment of discounted dividends Similarly, V 2 (u; b) = E[D(u, b, δ) 2 ] = Z i = e δ( i j=1 T (j)) D(i) E[Z 2 i=1 i ] + 2 i=1 j=i+1 E[Z i Z j ] i=1 i=2 j=i+1 E[Zi 2 ] = E e 2δT u Du 2 + E e 2δT u E e 2δT b Db 2 1 E (e 2δT b ) E e 2δT b D b E e δt b D b E[Z i Z j ] = E e 2δT u [1 E (e δt b )] [1 E (e 2δT b )] LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 11 / 26

12 with boundary condition (φ n (b, b, δ) = 0) we nd φ n (0, b, δ). LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 12 / 26 Discounted Dividend Expectations We need the expectations E e δt u D u, E e δt b D b, E e δt u and E e δt u... Adapt from the classical severity, Dickson & Waters (2004), φ n (u, b, δ) = E[e δt u D n u ], n = 0, 1, 2,..., u = b u d du φ n (u, b, δ) = λ+δ c φ n (u R λ u, b, δ) c φ 0 n (u y, b, δ)p(y)dy R λ c (y u u ) n p(y)dy. Laplace transform of φ n (u, b, δ) w.r.t. u ( 0, extended ) φ n (s, b, δ) = cφ n (0, b, δ) λp n 1 p n(s) s. (2) cs (λ + δ) + λ p(s)

13 Probability of a single dividend payment De nition χ(u, b) = Pr [T u < τ u ] and ξ(u, b) = Pr [T u > τ u ], u b, Solutions through Integro-di erential equations or Laplace transforms: Standard approach, (t 0 : u ct 0 = 0), 0 < u < b. Z t0 ξ(u, b) = e λt Z b (u ct) λe λt p(x)ξ(u 0 ct + x, b) dxdt, di erentiating and rearranging (Exact solutions in some cases) λξ(u, b) + c d Z b u du ξ(u, b) = λ p(x)ξ(u + x, b) dx (3) 0 d du log ξ(u, b) = λ Z b u p(x)ξ(x, b u) dx 1 c Boundary condition: ξ(0, b) = 1. LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 13 / 26 0

14 Probability of a single dividend payment Laplace transform: Change of variable: z = b u and E(z, b) = ξ(b z, b) = ξ(u, b) λe(z, b) c E(z, b) z λ Z z 0 p(z y)e(y, b) dy = 0. In E(z, b) extend the range of z from 0 z b to 0 z, resulting function by ɛ(z) Compute its Laplace transform, ɛ(s), [ p(s) is transform of p(x)] ɛ(s) = cs cɛ(0) λ + λ p(s) (4) ɛ(0) = ξ(b, b), [ɛ(b) = E(b, b) = ξ(0, b) = 1]. LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 14 / 26

15 Single dividend amount distribution De nition G (u, b; x) = Pr(T u < τ u and U(T u ) b + x)ju, b) Ruin probability ψ(x) = e Rx can be written as ψ(u) = ξ(u, b) + Z = ξ(u, b) + ψ(b) 0 g(u, b; x)ψ(b + x)dx Z 0 g(u, b; x)ψ(x)dx ξ(u, b) = e Ru e Rb ḡ(u, b; R) (5) Integro-di erential equation for G (u, b; x), using usual procedure, G (u, b; x) = Z t0 Z b (u ct) λe λt p(y)g (u 0 0 Z b (u ct)+x + b (u ct) p(y)dy dt. ct + y, b; x) dy LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 15 / 26

16 Single dividend amount distribution Rearranging and di erentiating. λg (u, b; x) + c u G (u, b; x) = λ Z b u p(y Boundary condition G (0, b; x) = 0. Similarly, λg(u, b; x) + c g(u, b; x) = u = λ Z b 0 u u)g (y, b; x) dy +λ [P(b u + x) P(b u)]. p(y)g(u + y, b; x) dy + λp(b u + x). Let G(z, b; x) = G (b z, b; x). Then G(b, b; x) = G (0, b; x) = 0 LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 16 / 26

17 Single dividend amount distribution Use of Laplace transforms λg(z, b; x) c Z z G(z, b; x) z λ p(z 0 λ(p(z + x) P(z)) = 0 y)g(y, b; x) dy Let ρ(z, x) be the function resulting from extending the range of z. Taking Laplace transforms, where ˆp(s, x) = λ ρ(s, x) c [s ρ(s, x) ρ(0, x)] λ ρ(s, x) p(s) p(s) +λ ˆp(s, x) = 0, s cρ(0, x) + λ [ p(s)/s ˆp(s, x)] ρ(s, x) = cs λ + λ p(s) Z 0 Z e sz P(z + x) dz = e sx e sy P(y) dy. LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 17 / 26 x (6)

18 Using the Classical Model Consider the process continuing even if ruin occurs. The process can cross the upper dividend level before or after ruin. De nition (proper) Distribution of the amount by which the process rst upcrosses b, H(u, b; x) = Pr [U(T u ) b + x] H(u, b; x) = H(u, b; xjt u < τ u )χ(u, b) + H(u, b; xjt u > τ u )ξ(u, b) = G (u, b; x) + ξ(u, b)h(0, b; x). Equation above simply means that H(u, b; x) equals the probability of a dividend claim less or equal than x plus the probability of a similar amount but that cannot be a dividend LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 18 / 26

19 Using the Classical Model Compute now H(u, b; x), through expressions of the severity of ruin from the classical model G (b u; x) = G (u, b; x) + ξ(u, b)g (b; x) G (u, b; x) = G (b u; x) ξ(u, b)g (b; x) For u = b, G (b, b; x) = G (0; x) ξ(b, b)g (b; x) g(b, b; x) = g (0; x) ξ(b, b)g (b; x) Calculate Laplace transforms at R and (5) 1 g ξ(b, b) = (0; R) e Rb 1 g (b; R)e Rb with g (0; x) = p1 1 [1 R P(x)] and g (0; R) = 1 Rp e Rx p(x)dx = c/λp 1. LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 19 / 26

20 Number of dividend payments De nition Let M : number of dividends to be claimed, or the number of times the process upcrosses the upper level b. M follows a zero-modi ed geometric distribution Pr[M = 0] = ξ(u, b) Pr[M = k] = χ(u, b)χ(b, b) k 1 ξ(b, b), k = 1, 2,... Hence, the total amount of dividend payments (not discounted) follows a zero modi ed compound distribution. LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 20 / 26

21 Illustration Example p(x) = αe αx, x > 0, (α > 0) E e δt u Du n = n! λ α n c e r 2u e r 1u (r 1 + α)e r 2b (r 2 + α)e r 1b e r 2u e r 1u V (u, b, δ) = λ α (δ cr 2 )e r 2b (δ cr 1 )e r 1b, where r 1 < 0 e r 2 > 0 are solutions of the equation (7) s 2 + α s λ + δ c αδ c = 0. (7) LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 21 / 26

22 Illustration (cont d) λe Ru χ(u, b) = λ λ αce Rb ; ξ(u, b) = λe Ru αce Rb λ αce Rb. LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 22 / 26

23 Illustration (cont d) λe Ru χ(u, b) = λ λ αce Rb ; ξ(u, b) = λe Ru αce Rb λ αce Rb. G (u, b; x) = (1 e αx ) G (u, b; x) χ(u, b) = 1 e αx. λ λe Ru λ αce Rb. LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 22 / 26

24 Illustration (cont d) λe Ru χ(u, b) = λ λ αce Rb ; ξ(u, b) = λe Ru αce Rb λ αce Rb. G (u, b; x) = (1 e αx ) G (u, b; x) χ(u, b) = 1 e αx. λ λe Ru λ αce Rb. E e δt u Du n jt u < τ u = E (Du n jt u < τ u ) E e δt u jt u < τ u LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 22 / 26

25 Illustration (cont d) Let λ = α = 1, c = 0.75 and δ = 0.02 u b Table: Values for V (u, b, 0.02) for u =1, 3, 5, 10, 15, 20; b =2, 3, 6, 10, 30, 40 LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 23 / 26

26 Illustration (cont d) Let λ = α = 1, c = 0.75 and δ = 0.02 u b Table: Values for E e δt u D u for u =1, 3, 5, 10, 15, 20; b =2, 3, 6, 10, 30, 40 LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 24 / 26

27 Illustration (cont d) Let λ = α = 1, c = 0.75 and δ = 0.02 u b Table: Values for χ(u, b) for u =1, 3, 5, 10, 15, 20; b =2, 3, 6, 10, 30, 40 LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 25 / 26

28 References Avanzi (2009). Strategies for dividend distribution: A review, North American Actuarial Journal 13(2), Avanzi, Gerber & Shiu (2007). Optimal dividends in the dual model, Insurance: Mathematics and Economics, 41(1), Cheung & Drekic (2008). Dividend moments in the dual risk model: exact and approximate approaches, ASTIN Bulletin 38 (2), Dickson & Waters (2004). Some optimal dividends problems, ASTIN Bulletin 34 (1), Gerber (1979). An Introduction to Mathematical Risk Theory, S.S. Huebner Foundation for Insurance Education, University of Pennsylvania, Philadelphia, Pa , USA. Ng (2009), On a dual model with a dividend threshold, Insurance: Mathematics and Economics 44, Ng (2010). On the upcrossing and downcrossing probabilities of a dual risk model with phase-type gains, ASTIN Bulletin 40 (1) LRA (FCT, UNL & ISEG, UTL) Dividends Dual Risk Model ASTIN 2011 Madrid 26 / 26