Differentiation of some functionals of multidimensional risk processes and determination of optimal reserve allocation

Transcription

1 Differentiation of some fnctionals of mltidimensional risk processes and determination of optimal reserve allocation Stéphane Loisel Laboratoire de Sciences Actarielle et Financière Université Lyon 1, France Stéphane Loisel, September 5th, 25 1

2 Introdction +X t t with initial reserve Historical model: for nidimensional risk processes R t = + X t, and with X t = ct S t, where c > is the premim income rate, S t = N(t) i=1 W i, the W i are i.i.d. nonnegative random variables, independent from (N(t)) t, with the convention that the sm is zero if N(t) =. Probability of rin: ψ() = P ( t, R t < ). Stéphane Loisel, September 5th, 25 2

3 Introdction +X t 1 2 t Two lines of bsiness: classical, 1-dimensional srpls process (black), 2-dimensional process (1 for each line of bsiness, ble and red). = Stéphane Loisel, September 5th, 25 3

4 Introdction 3 main directions: +X t +X t U b (t) +X t 1 b 2 t T t t Figre 1: K-dimensional process Figre 2: Finite-time. Figre 3: Dividends. How to model the stochastic dependence between the K lines of bsiness? Which rin concept? Within finite or infinite time? Rin or severity of rin? Optimal initial reserve allocation? How to measre risk and profit (dividends)? Stéphane Loisel, September 5th, 25 4

5 Strctre of the exposé Differentiation of some fnctionals of mltidimensional risk processes and determination of optimal reserve allocation Risk and profit measres for nivariate risk processes Differentiation of some fnctionals of risk processes A general optimal reserve allocation strategy Stéphane Loisel, September 5th, 25 5

6 Risk and profit measres Differentiation of some fnctionals of mltidimensional risk processes and determination of optimal reserve allocation Risk and profit measres for nivariate risk processes Differentiation of some fnctionals of risk processes A general optimal reserve allocation strategy Stéphane Loisel, September 5th, 25 6

7 Risk measres +X t t the time to rin T = inf{t >, + X t < }, the severity of rin + X T, or the cople (T, + X T ), the time in the red (below ) from the first rin to the first time of recovery T T, where the maximal rin severity (inf t> + X t ), T = inf{t > T, + X t = }, the aggregate severity of rin ntil recovery J() = T T + X t dt,... the total time in the red τ() = + 1 {+Xt <}dt. Stéphane Loisel, September 5th, 25 7

8 Risk measres +X t t T the time to rin T = inf{t >, + X t < }, the severity of rin + X T, or the cople (T, + X T ), the time in the red (below ) from the first rin to the first time of recovery T T, where the maximal rin severity (inf t> + X t ), T = inf{t > T, + X t = }, the aggregate severity of rin ntil recovery J() = T T + X t dt,... the total time in the red τ() = + 1 {+Xt <}dt. Stéphane Loisel, September 5th, 25 8

9 Risk measres +X t T t +XT the time to rin T = inf{t >, + X t < }, the severity of rin + X T, or the cople (T, + X T ), the time in the red (below ) from the first rin to the first time of recovery T T, where the maximal rin severity (inf t> + X t ), T = inf{t > T, + X t = }, the aggregate severity of rin ntil recovery J() = T T + X t dt,... the total time in the red τ() = + 1 {+Xt <}dt. Stéphane Loisel, September 5th, 25 9

10 Risk measres +X t T T t T _ T the time to rin T = inf{t >, + X t < }, the severity of rin + X T, or the cople (T, + X T ), the time in the red (below ) from the first rin to the first time of recovery T T, where the maximal rin severity (inf t> + X t ), T = inf{t > T, + X t = }, the aggregate severity of rin ntil recovery J() = T T + X t dt,... the total time in the red τ() = + 1 {+Xt <}dt. Stéphane Loisel, September 5th, 25 1

11 Risk measres +X t t Max. Severity the time to rin T = inf{t >, + X t < }, the severity of rin + X T, or the cople (T, + X T ), the time in the red (below ) from the first rin to the first time of recovery T T, where the maximal rin severity (inf t> + X t ), T = inf{t > T, + X t = }, the aggregate severity of rin ntil recovery J() = T T + X t dt,... the total time in the red τ() = + 1 {+Xt <}dt. Stéphane Loisel, September 5th, 25 11

12 Risk measres +X t t the time to rin T = inf{t >, + X t < }, the severity of rin + X T, or the cople (T, + X T ), the time in the red (below ) from the first rin to the first time of recovery T T, where the maximal rin severity (inf t> + X t ), T = inf{t > T, + X t = }, the aggregate severity of rin ntil recovery J() = T T + X t dt,... the total time in the red τ() = + 1 {+Xt <}dt. Stéphane Loisel, September 5th, 25 12

13 Risk measres +X t T T t T _ T + the time to rin T = inf{t >, + X t < }, the severity of rin + X T, or the cople (T, + X T ), the time in the red (below ) from the first rin to the first time of recovery T T, where the maximal rin severity (inf t> + X t ), T = inf{t > T, + X t = }, the aggregate severity of rin ntil recovery J() = T T + X t dt,... the total time in the red τ() = + 1 {+Xt <}dt. Stéphane Loisel, September 5th, 25 13

14 Risk measres +X t t Consider risk measres based on some fixed time interval [, T] (T may be infinite). Simple penalty fnction (expected penalty to pay de to insolvency ntil time horizon T) : E (I T ()) = E T 1 {+Xt <} + X t dt. Note that the probability P (I T () = ) is the probability of non rin within finite time T. Stéphane Loisel, September 5th, 25 14

15 Risk measres +X t T t Consider risk measres based on some fixed time interval [, T] (T may be infinite). Simple penalty fnction (expected penalty to pay de to insolvency ntil time horizon T) : E (I T ()) = E T 1 {+Xt <} + X t dt. Note that the probability P (I T () = ) is the probability of non rin within finite time T. Stéphane Loisel, September 5th, 25 15

16 Risk measres +X t T t Consider risk measres based on some fixed time interval [, T] (T may be infinite). Simple penalty fnction (expected penalty to pay de to insolvency ntil time horizon T) : E (I T ()) = E T 1 {+Xt <} + X t dt. Note that the probability P (I T () = ) is the probability of non rin within finite time T. Stéphane Loisel, September 5th, 25 16

17 Risk and profit From an economical point of view, it seems more consistent to consider EI g,h () = E T ( 1{+Xt }g( + X t ) 1 {+Xt }h( + X t ) ) dt g h g corresponds to a reward fnction for positive reserves, and h is a penalty fnction in case of insolvency. These risk measres may be differentiated with respect to the initial reserve. Fbini s theorem. Stéphane Loisel, September 5th, 25 17

18 Differentiation theorems Differentiation of some fnctionals of mltidimensional risk processes and determination of optimal reserve allocation Risk and profit measres for nivariate risk processes Differentiation of some fnctionals of risk processes A general optimal reserve allocation strategy Stéphane Loisel, September 5th, 25 18

19 Differentiation theorems +X t T t I() = T 1 {+Xt <} + X t dt Stéphane Loisel, September 5th, 25 19

20 Differentiation theorems Th. (L., 24b): Let (X t ) t [,T) be a stochastic process with almost srely time-integrable sample paths. For R, denote by τ () the time spent in zero by the process + X t : τ () = T 1 {+Xt =}dt. Let f be defined by f() = E(I T ()) for R, where I T () = T 1 {+Xt <} + X t dt. Stéphane Loisel, September 5th, 25 2

21 Differentiation theorems Th. (L., 24b): Let (X t ) t [,T) be a stochastic process with almost srely time-integrable sample paths. For R, denote by τ () the time spent in zero by the process + X t : τ () = T 1 {+Xt =}dt. Let f be defined by f() = E(I T ()) for R, where I T () = T 1 {+Xt <} + X t dt. For R, if Eτ () =, then f is differentiable at, and f () = Eτ(). Stéphane Loisel, September 5th, 25 2-a

22 Differentiation theorems Th. (L., 24b): Let X t = ct S t, where S t is a jmp process satisfying hypothesis (H1): S t has a finite expected nmber of nonnegative jmps in every finite interval, and for each t, the distribtion of S t is absoltely continos. For example, S t might be a compond Poisson process with a continos jmp size distribtion. Define h by h() = E(τ()) for R. h is differentiable on R +, and for >, h () = 1 c EN (), where N () = Card({t [, T], + ct S t = }). Stéphane Loisel, September 5th, 25 21

23 Differentiation theorems Th. (L., 24b): Let X t = ct S t, where S t is a jmp process satisfying hypothesis (H1): S t has a finite expected nmber of nonnegative jmps in every finite interval, and for each t, the distribtion of S t is absoltely continos. For example, S t might be a compond Poisson process with a continos jmp size distribtion. Define h by h() = E(τ()) for R. h is differentiable on R +, and for >, h () = 1 c EN (), where N () = Card({t [, T], + ct S t = }). This remains valid with T = + if X t has a positive drift and τ() is integrable. In the compond Poisson case, for,. h () = 1 c 1 1 ψ() ψ() Stéphane Loisel, September 5th, a

24 Differentiation theorems Th. (L., 24b): Let X t = ct S t, where S t is a jmp process satisfying hypothesis (H1): S t has a finite expected nmber of nonnegative jmps in every finite interval, and for each t, the distribtion of S t is absoltely continos. For example, S t might be a compond Poisson process with a continos jmp size distribtion. Define h by h() = E(τ()) for R. h is differentiable on R +, and for >, h () = 1 c EN (), where N () = Card({t [, T], + ct S t = }). This remains valid with T = + if X t has a positive drift and τ() is integrable. In the compond Poisson case, for,. h () = 1 c 1 1 ψ() ψ() EI T (.) is ths well strictly convex, which will be very important for minimization. Stéphane Loisel, September 5th, b

25 Differentiation theorems Theorem: In the Poisson-Exponential(1/µ) case, ψ() = 1 µr ( ) R = 1 µ 1 λµ c. Hence, for T = +, µr e R, with Eτ() = 1 µr cµr 2 e R Gerber, Dos Reis (1993) and EI () = 1 µr cµr 3 e R L. (24b) Proof: Integration of the well-known formla for ψ(). The considered fnctions tend to as +. It is possible to derive EI () explicitly for Γ and phase-type-distribted claim amonts. Stéphane Loisel, September 5th, 25 22

26 Differentiation theorems Th. (L., 24b): Let g, h be two convex or concave fnctions in C 1 (R +, R + ), sch that for x, g(x) g() and h(x) h(). Let X t be a stochastic process sch that t g( + X t ) and t h( + X t ) are almost srely integrable on [, T]. Let I + g be the fnction from R into the space of nonnegative random variables, and defined by I + g () = T 1 {+Xt }g( + X t )dt for and let f(.) = EI g + (.) EI h (.). Define also ( 1 T L T () = lim ε 2ε 1 { +Xt <ε}dt If, for R, EI + g (), EI h (), EI + g (), EI h () < +, ). and if Eτ () =, then f is differentiable on R +, and for >, f () = EI + g () EI h () (g() + h())el T () Stéphane Loisel, September 5th, 25 23

27 strategy A general optimal reserve allocation Differentiation of some fnctionals of mltidimensional risk processes and determination of optimal reserve allocation Risk and profit measres for nivariate risk processes Differentiation of some fnctionals of risk processes A general optimal reserve allocation strategy Stéphane Loisel, September 5th, 25 24

28 strategy A general optimal reserve allocation What has to be minimized is A( 1,..., K ) = K EIT( k k ) k=1 nder the constraint K =, where [ ] T EIT( k k ) = E Rt k 1 {R k t <}dt with R k t = k + X k t Stéphane Loisel, September 5th, 25 25

29 strategy A general optimal reserve allocation What has to be minimized is A( 1,..., K ) = K EIT( k k ) k=1 nder the constraint K =, where [ ] T EIT( k k ) = E Rt k 1 {R k t <}dt with R k t = k + X k t This does not depend on the dependence strctre. Stéphane Loisel, September 5th, a

30 strategy A general optimal reserve allocation What has to be minimized is A( 1,..., K ) = K EIT( k k ) k=1 nder the constraint K =, where [ ] T EIT( k k ) = E Rt k 1 {R k t <}dt with R k t = k + X k t This does not depend on the dependence strctre. From previos differentiation theorems, A is strictly convex. On the compact space S = {( 1,..., K ) (R + ) K, K = }, A admits a niqe minimm. Stéphane Loisel, September 5th, b

31 strategy A general optimal reserve allocation Lagrange mltipliers:optimal allocation: there is a sbset J [1, K] sch that for j / J, j =, and for j, k J, Eτ j = Eτ k. Stéphane Loisel, September 5th, 25 26

32 strategy A general optimal reserve allocation Lagrange mltipliers:optimal allocation: there is a sbset J [1, K] sch that for j / J, j =, and for j, k J, Eτ j = Eτ k. In the Poisson-Exponential( 1 µ ) case, recall that EI = 1 µr cµr 3 e R. Consider a two-line model, with the following parameters: µ 1 = µ 2 = 1, c 1 = c 2 = 1, R 2 =.4 and = 1. Three vales of R 1 :different optimal allocation strategies. Stéphane Loisel, September 5th, a

33 strategy A general optimal reserve allocation x x Figre 4: Graph of A(x,1 x). When R 1 =.5 > R 2, line of bsiness 1 is safer than line 2. : 1 < 2. The optimal allocation is abot ( 1 = 3.5, 2 = 6.5.) Figre 5: Graph of A(x, 1 x). When R 1 =.8 < R 2, line of bsiness 1 is mch riskier than line 2. : 1 = = 1 and 2 =. Transfer of the whole reserve to line 1. Stéphane Loisel, September 5th, 25 27

34 strategy A general optimal reserve allocation Consider the sm where B = K Eτ k( 1,..., K ) k=1 Eτ k( 1,..., K ) = E ( T 1 {R k t <}1 { K j=1 Rj t >}dt ) B takes dependence into accont, and the following proposition prescribes to do the same kind of reasoning: Proposition: Let X t = ct S t, where S t satisfies hypothesis (H1). Define B by B( 1,..., K ) = n k=1 E(τ k ( 1,..., K )) for R K. B is differentiable on (R +) K, and for 1,..., K >, B = 1 EN k c k( 1,..., K ), k ( where Nk ( ( 1,..., K ) = Card {t [, T], R k t = ) ( K ) ) j=1 Rj t > }.. Stéphane Loisel, September 5th, 25 28

35 strategy A general optimal reserve allocation It is also possible to differentiate with respect to c instead of. Theorem: With the previos notation, consider the case X t = ct S t, where S t satisfies hypothesis (H1), and define f(c) = E(I(c)). If for all c, Eτ (c) =, then f is differentiable on R and for c R, f (c) = T tp(r t < )dt. It is interesting to look for the optimal allocation of the global premim c = c c n becase if c i is small enogh to make the safety loading negative, the process Rt i tends to. Qite often, optimizing with the c i will be easier than with the i for this reason. These examples illstrate how these differentiation reslts may be sed. The differentiation developed here is qite general and may be sefl to solve many problems involving mlti-risks models. Stéphane Loisel, September 5th, 25 29