A Note On The Erlang(λ, n) Risk Process

Transcription

1 A Note On The Erlangλ, n) Risk Process Michael Bamberger and Mario V. Wüthrich Version from February 4, 2009 Abstract We consider the Erlangλ, n) risk process with i.i.d. exponentially distributed claims severities. We prove that the ruin probability is a strictly decreasing function in n if we keep the expected interarrival times between two successive claims constant. In the limit case we obtain Lundberg s fundamental equation in the discrete time risk model ladder heights of random walks). Key words: Risk Theory, Sparre Andersen Model, Ruin Probability, Erlang Distribution, Lundberg Equation, Adjustment Coefficient. 1 Model and Results We consider a Sparre Andersen surplus process N t Ut) = u + ct X i, t 0, i=1 with non-negative initial capital u 0, premium rate c > 0, claims counting process N t ) t 0 and claims severities X i, i 1. We assume that the claims severities X i are i.i.d. exponentially distributed with parameter η > 0 and that these claims severities are independent from the claims counting process N t ) t 0. For the claims interarrival times W i, that define the claims counting process N t ) t 0, we assume that they are i.i.d. Erlangλ, n)-distributed with λ > 0 and n N. Note that in this case W i has the ETH Zurich, Department of Mathematics, 8092 Zurich, Switzerland 1

2 same distribution as the sum of n independent exponentially distributed random variables with intensity λ > 0. Henceforth, the special case n = 1 gives the homogeneous Poisson point process with intensity λ and in that case Ut) is the classical Cramér-Lundberg surplus process with exponential claims severities. The Sparre Andersen surplus process with Erlangλ, n) interarrival times W i has been widely studied in the literature, see Dickson [1], Dickson-Hipp [3, 4], Li-Garrido [8], Gerber-Shiu [5], Li-Dickson [7] and Li [6]. The main focus in these papers is the study of the ruin probability Ψu) = P [ ] inf Ut) < 0 t 0 U0) = u as well as the maximum surplus before ruin and the severity of ruin if ruin occurs. In order to have a positive survival probability we need to make sure that the so-called net profit condition NPC) is fulfilled, that is in our case 0 < E [cw i X i ] = cn λ 1 η. 1.1) From Dickson [1] see also Gerber-Shiu [5] and Li-Garrido [8], Section 8) we obtain Ψu) = 1 R ) e Ru, η where R < 0 is the unique solution r C of Lundberg s fundamental equation for exponential severities 1 c λ r ) n η + r) η = 0, which is on the negative real line all the other n solutions r C are in the right half of the complex plane, see Gerber-Shiu [5], Section 4). Note that 0 < R < η formula 4.6) in Gerber-Shiu [5] and Remark 1 in Li-Garrido [8], p. 395). Now we understand the ruin probability Ψu) as function of the parameters λ, n, η, c and u. It is decreasing in η, c and u. Our goal is to study the role of the Erlangλ, n) parameters. Note that the expected interarrival time is given by E[W i ] = n/λ, hence our goal is to keep this expected value constant and vary the parameters n and λ. That is, we keep w = E[W i ] > 0 fixed and we choose λ = λn) = n/w. Then we can study the ruin probability as a function of the Erlang parameter n, i.e. Ψu) = Ψu; n) = 1 R ) n e Rnu, 1.2) η 2

3 where R n < 0 is the unique solution r C of Lundberg s fundamental equation 1 c w n r ) n η + r) η = ) In a numerical analysis Li-Garrido [8] Figure 1) have found that this ruin probability n 1 n 2 n 3 n 4 n 5 n 6 n 8 n 10 n 25 n 50 n 0.2 Figure 1: The ruin probability Ψu; n) as a function of u for different n N. Ψu; n) is a decreasing function of n. We prove these numerical findings in the following proposition: Proposition 1.1 In the Sparre Andersen model with i.i.d. Erlangλ = n/w, n) interarrival times W i, i.i.d. exponential severities X i and satisfying the NPC 1.1) we find for all integers n 1 Ψu; n + 1) < Ψu; n) Ψu; 1) = 1 c η w e η 1 cw) u. Note that Ψ is a probability and hence bounded from below. This and the monotonicity in n implies the existence of the limit in the next corollary. 3

4 Corollary 1.2 Under the assumptions of Proposition 1.1 we have lim Ψu; n) = n e R u+cw), where R < 0 solves e cwr η + r) η = 0. Note that if we assume that V 1,..., V n are i.i.d. exponentially distributed with parameter λ > 0, then W i = n j=1 V j has an Erlangλ, n) distribution. If we choose λ = λn) = n/w as above. Then this implies that E [W i ] = w and Var W i ) = w2 n. Therefore, for n we reduce the variability of the interarrival times which reduces the ruin probability. In the limit the interarrival times converge weakly to the constant w > 0 which corresponds to a discrete time ruin model see Chapter 6 in Dickson [2] and Chapter 5 in Rolski et al. [10]). Note that Lundberg s fundamental equation 1.3) converges to the one from the discrete time model compare to 2.1) below and Section 6.5 in Dickson [2]). If we assume that the claims severities X i have a general positive distribution with density g and Laplace transform ĝ. Then the adjustment coefficient R n if it exists) is given by the solution of 1 c λ r ) n ĝr) = 1, see, for example 1.8) in Gerber-Shiu [5]. The same argument as in the proofs of Proposition 1.1 and 1.2 implies that the adustment coefficient R n if it exists) is a strictly increasing function in n and in the limit for n for λn) = n/w) we obtain Lundberg s fundamental equation in discrete time see Dickson [2], Section 6.5). Moreover, Lundberg s inequality says that the Lundberg bound is a decreasing function in the Lundberg coefficient R n see for example Mikosch [9], Theorem 4.2.3). 2 Proofs In this section we prove Proposition 1.1 and Corollary

5 Proof of Proposition 1.1. The last equality in Proposition 1.1 is clear and just corresponds to the classical Cramér-Lundberg ruin probability for exponentially distributed claims severities X i. We define the function q n r) = 1 c w n r ) n η + r) η, for r [ η, n/cw)]. From 1.2)-1.3) we know that R n is the unique negative real solution of q n r) = 0. In view of 1.2) we see that it is sufficient to prove that R n < η is a strictly increasing function in n. We calculate the first derivative of q n ) at the origin r = 0. It is given by q n0) = c w η + 1 < 0, due to the NPC 1.1). Because R n is the unique solution in [ η, 0) to q n r) = 0 and because the first derivative of q n is strictly negative in the origin, we know that q n r) > 0 for all r R n, 0). We claim that q n+1 R n ) > 0 which then implies that R n+1 < R n and proves the claim of Proposition 1.1. Hence we prove for n 1 q n+1 R n ) = 1 + c w ) n+1 n + 1 R n η R n ) η >? 0. Consider the function hx) = x log1 + 1/x) for x > 0. The derivatives are given by h x) = log1 + 1/x) 1 + x) 1 with lim h x) = 0, x h 1 x) = x1 + x) = < 0 for x > x) 2 x1 + x) 2 Hence, hx) is a strictly concave increasing function for x > 0. This implies that { )} n + 1 q n+1 R n ) = exp cwr n h η R n ) η cwr { n )} n > exp cwr n h η R n ) η = q n R n ) = 0. cwr n Henceforth, R n+1 > R n, which completes the proof. 5

6 Proof of Corollary 1.2. From the proof of Proposition 1.1 we know that R n is increasing and bounded which implies that the limit exists. So there remains to calculate the limit. Note that for fixed η < R < 0 we have q R) = lim n q n R) = e cwr η R) η. Note that q R) = 0 has a unique solution R 0, η) which implies that This completes the proof. 1 R η = e cwr. 2.1) References [1] Dickson, D.C.M. 1998). On a class of renewal risk process. North American Act. J. 7, [2] Dickson, D.C.M. 2005). Insurance Risk and Ruin. Cambridge University Press. [3] Dickson, D.C.M., Hipp, C. 1998). Ruin probabilities for Erlang2) risk process. Insurance: Math. Econom. 22, [4] Dickson, D.C.M., Hipp, C. 2001). On the time to ruin for Erlang2) risk process. Insurance: Math. Econom. 29, [5] Gerber, H.U., Shiu, E.S.W. 2005). The time value of ruin in the Sparre Andersen model. North American Act. J. 9, [6] Li, S. 2008). A note on the maximum severity of ruin in an Erlangn) risk process. Bulletin Swiss Association of Acturies, 2008, [7] Li, S., Dickson, D.C.M. 2006). The maximum surplus before ruin in an Erlangn) risk process and related problems. Insurance: Math. Econom. 38, [8] Li, S., Garrido, J. 2004). On ruin for Erlangn) risk process. Insurance: Math. Econom. 34, [9] Mikosch, T. 2006). Non-Life Insurance Mathematics. 2nd Edition. Springer. [10] Rolski, T., Schmidli, H., Schmidt, V., Teugels, J. 1998). Stochastic Processes for Insurance and Finance. Wiley. 6