Non-Life Insurance Mathematics. Christel Geiss and Stefan Geiss Department of Mathematics and Statistics University of Jyväskylä

Transcription

1 Non-Life Insurance Mathematics Christel Geiss and Stefan Geiss Department of Mathematics and Statistics University of Jyväskylä November 5, 218

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3 Contents 1 Introduction The ruin of an insurance company Solvency II Directive Idea of the mathematical model Some facts about probability Claim number process models The homogeneous Poisson process A generalization of the Poisson process: the renewal process The inhomogeneous Poisson process The total claim amount process S(t) The renewal model and the Cramér-Lundberg-model Properties of S(t) Premium calculation principles Classical premium calculation principles Net principle Expected value principle The variance principle The standard deviation principle The constant income principle Modern premium calculation principles The Exponential Principle The Esscher Principle The Quantile Principle Reinsurance treaties Random walk type reinsurance

4 4 CONTENTS Extreme value type reinsurance Probability of ruin: small claim sizes The risk process Bounds for the ruin probability Fundamental integral equation Proof of Cramér s ruin bound Probability of ruin: large claim sizes Tails of claim size distributions Subexponential distributions Definition and basic properties Examples Another characterization of subexponential distributions An asymptotics for the ruin probability Conditions for F X,I S Summary Distributions QQ-Plot The distribution of S(t) Compound Poisson random variables Applications in insurance The Panjer recursion Approximation of F S(t) Monte Carlo approximations of F S(t)

5 Chapter 1 Introduction Insurance Mathematics might be divided into life insurance, health insurance, non-life insurance. Life insurance includes for instance life insurance contracts and pensions, where long terms are covered. Non-life insurance comprises insurances against fire, water damage, earthquake, industrial catastrophes or car insurance, for example. Non-life insurances cover in general a year or other fixed time periods. Health insurance is special because it is differently organized in each country. The course material is based on the textbook Non-Life Insurance Mathematics by Thomas Mikosch [7]. 1.1 The ruin of an insurance company Solvency II Directive In the following we concentrate ourselves on non-life insurance. There is a the Solvency II Directive of the European Union. Published: 29 Taken into effect: 1/1/216 5

6 6 CHAPTER 1. INTRODUCTION Contents: Defines requirements for insurance companies. One of these requirements is the amount of capital an insurer should hold, or in other words, the Minimum Capital Requirement (MCR): The probability that the assets of an insurer are greater or equal to its liabilities (in other words, to avoid the ruin) has to be larger or equal than 99,5 %. In the lecture we will treat exactly this problem. Here we slightly simplify the problem by only looking at one particular insurance contract instead of looking at the overall company (this is a common approach in research as well). What are the key parameters for a non-life insurance contract for a certain class of claims? 1. How often does this event occur? Significant weather catastrophes in Europe: 2 per year Accidents in public transportation in Berlin in 216: Amount of loss or the typical claim size: Hurricane Niklas, 215, Europe: 75 Mio Euro Storm Ela, 214, Europe, 65 Mio Euro Damage on a parking area: 5-15 Euro Opposite to the Solvency II Directive an insurance company needs to have reasonable low premiums and fees to attract customers. So there has to be a balance between the Minimum Capital Requirement and the premiums Idea of the mathematical model We will consider the following situation: (1) Insurance contracts (or policies) are sold. The resulting premium (yearly or monthly payments of the customers for the contract) form the income of the insurance company. (2) At times T i, T 1 T 2... claims happen. The times T i are called the claim arrival times.

7 1.2. SOME FACTS ABOUT PROBABILITY 7 (3) The i-th claim arriving at time T i causes the claim size X i. Mathematical problem: Find a stochastic model for the T i s and X i s to compute or estimate how much an insurance company should demand for its contracts and how much initial capital of the insurance company is required to keep the probability of ruin below a certain level. 1.2 Some facts about probability We shortly recall some definitions and facts from probability theory which we need in this course. For more information see [12], or [2] and [3], for example. (1) A probability space is a triple (Ω, F, P), where Ω is a non-empty set, F is a σ-algebra consisting of subsets of Ω, and P is a probability measure on (Ω, F). (2) A function f : Ω R is called a random variable if and only if for all intervals (a, b), < a < b < we have that f 1 ((a, b)) : {ω Ω : a < f(ω) < b} F. (3) By B(R) we denote the Borel σ-algebra. It is the smallest σ-algebra on R which contains all open intervals. The σ-algebra B(R n ) is the Borel σ- algebra, which is the smallest σ-algebra containing all the open rectangles (a 1, b 1 )... (a n, b n ). (4) The random variables f 1,..., f n are independent if and only if P(f 1 B 1,..., f n B n ) P(f 1 B 1 ) P(f n B n ) for all B k B(R), k 1,..., n. If the f i s have discrete values, i.e. f i : Ω {x 1, x 2, x 3,...}, then the random variables f 1,..., f n are independent if and only if P(f 1 k 1,..., f n k n ) P(f 1 k 1 ) P(f n k n ) for all k i {x 1, x 2, x 3...}.

8 8 CHAPTER 1. INTRODUCTION (5) If f 1,..., f n are independent random variables such that f i has the density function h i (x), i.e. P(f i (a, b)) b h a i(x)dx, then P((f 1,..., f n ) B) 1I B (x 1,..., x n )h 1 (x) h n (x n )dx 1 dx n R n for all B B(R n ).. (6) The function 1I B (x) is the indicator function for the set B, which is defined as { 1 if x B 1I B (x) if x B. (7) A random variable f : Ω {, 1, 2,...} is Poisson distributed with parameter λ > if and only if λ λk P(f k) e k!. This is often written as f P ois(λ). (8) A random variable g : Ω [, ) is exponentially distributed with parameter λ > if and only if for all a < b P(g (a, b)) λ b a 1I [, ) (x)e λx dx. The picture below shows the density λ1i [, ) (x)e λx for λ 3. density for lambda3 y x

9 1.2. SOME FACTS ABOUT PROBABILITY 9 (9) How to characterize distributions? We briefly recall how to describe the distribution of a random variable. Let (Ω, F, P) be a probability space. (a) The distribution of a random variable f : Ω R can be uniquely described by its distribution function F : R [, 1], F (x) : P({ω Ω : f(ω) x}), x R. (b) Especially, it holds for g : R R, such that g 1 (B) B(R) for all B B(R), that Eg(f) g(x)df (x) R in the sense that, if either side of this expression exists, so does the other, and then they are equal, see [8], pp (c) The distribution of f can also be determined by its characteristic function (see [12]) ϕ f (u) : Ee iuf, u R, or by its moment-generating function m f (h) : Ee hf, h ( h, h ) provided that Ee h f < for some h >. We also recall that for independent random variables f and g it holds that ϕ f+g (u) ϕ f (u)ϕ g (u).

10 1 CHAPTER 1. INTRODUCTION

11 Chapter 2 The Models for the claim number process N(t) In the following we will introduce three processes which are used as claim number processes: the Poisson process, the renewal process and the inhomogeneous Poisson process. 2.1 The homogeneous Poisson process with parameter λ > Definition (homogeneous Poisson process). A stochastic process N (N(t)) t [, ) is a Poisson process if the following conditions are fulfilled: (P1) N() a.s. (almost surely), i.e. P({ω Ω : N(, ω) }) 1. (P2) The process N has independent increments, i.e. for all n 1 and t < t 1 <... < t n < the random variables N(t n ) N(t n 1 ), N(t n 1 ) N(t n 2 ),..., N(t 1 ) N(t ) are independent. (P3) For any s and t > the random variable N(t+s) N(s) is Poisson distributed, i.e. λt (λt)m P(N(t + s) N(s) m) e, m, 1, 2,... m! (P4) The paths of N, i.e. the functions (N(t, ω)) t [, ) for fixed ω, are almost surely right continuous and have left its. One says N has càdlàg (continue à droite, ite à gauche) paths. 11

12 12 CHAPTER 2. CLAIM NUMBER PROCESS MODELS Remark One can show that the definition implies the following: there is a set Ω F with P(Ω ) 1 such that for all ω Ω one has that (1) N(, ω) 1, (2) the paths t N(t, ω) are càdlàg, take values in {, 1, 2,...}, are nondecreasing, (3) all jumps have the size 1. In the following we prove that the Poisson process does exist. Lemma Assume W 1, W 2,... are independent and exponentially distributed with parameter λ >. Then, for any t > we have n 1 P(W W n t) 1 e λt k (λt) k. k! Consequently, the sum of independent exponentially distributed random variables is a Gamma distributed random variable. The proof is subject to an Exercise. Definition Let W 1, W 2,... be independent and exponentially distributed with parameter λ >. Define T n : W W n, ˆN(t, ω) : #{i 1 : T i (ω) t}, t. Lemma For each n, 1, 2,... and for all t > it holds P({ω Ω : ˆN(t, ω) n}) e λt (λt)n, n! i.e. ˆN(t) is Poisson distributed with parameter λt. Proof. From the definition of ˆN it can be concluded that {ω Ω : ˆN(t, ω) n} {ω Ω : Tn (ω) t < T n+1 (ω)} {ω Ω : T n (ω) t} \ {ω Ω : T n+1 (ω) t}.

13 2.1. THE HOMOGENEOUS POISSON PROCESS 13 Because of T n T n+1 we have the inclusion {T n+1 t} {T n t}. This implies P( ˆN(t) n) P(T n t) P(T n+1 t) n 1 1 e λt (λt) k 1 + e λt k! k λt (λt)n e. n! n (λt) k Theorem (1) ˆN(t)t [, ) is a Poisson process with parameter λ >. (2) Any Poisson process N(t) with parameter λ > can be written as N(t) #{i 1, T i t}, t, where T n W W n, n 1, and W 1, W 2,... are independent and exponentially distributed with λ >. Proof. (1) We check the properties of the Definition (P1) From (8) of Section 1.2 we get that P(W 1 > ) P(W 1 (, )) λ k 1I [, ) (y)e λy dy 1. This implies that ˆN(, ω) if only < T 1 (ω) W 1 (ω) but W 1 > holds almost surely. Hence ˆN(, ω) a.s. (P2) We only show that ˆN(s) and ˆN(t) ˆN(s) are independent, i.e. P( ˆN(s) l, ˆN(t) ˆN(s) m)p( ˆN(s) l)p( ˆN(t) ˆN(s) m) (1) for l, m. The independence of arbitrary many increments can be shown similarly. It holds for l and m 1 that P( ˆN(s) l, ˆN(t) ˆN(s) m) P( ˆN(s) l, ˆN(t) m + l) By defining functions f 1, f 2, f 3 and f 4 as f 1 : T l P(T l s < T l+1, T l+m t < T l+m+1 ). k!

14 14 CHAPTER 2. CLAIM NUMBER PROCESS MODELS f 2 : W l+1 f 3 : W l W l+m f 4 : W l+m+1, and h 1,..., h 4 as the corresponding densities, it follows that P(T l s < T l+1, T l+m t < T l+m+1 ) P(f 1 s < f 1 + f 2, f 1 + f 2 + f 3 t < f 1 + f 2 + f 3 + f 4 ) P( f 1 < s, s f 1 < f 2 <, f 3 < t f 1 f 2, : I 1 s t x 1 x 2 s x 1 h 4 (x 4 )dx 4 h 3 (x 3 )dx 3 t x 1 x 2 x } 3 {{} I 4 (x 1,x 2,x 3 ) }{{} I 3 (x 1,x 2 ) t (f 1 + f 2 + f 3 ) < f 4 < ) } {{ } I 2 (x 1 ) h 2 (x 2 )dx 2 h 1 (x 1 )dx 1 By direct computation and rewriting the density function of f 4 W l+m+1, I 4 (x 1, x 2, x 3 ) t x 1 x 2 x 3 λe λx 4 1I [, ) (x 4 )dx 4 e λ(t x 1 x 2 x 3 ). Here we used t x 1 x 2 x 3. This is true because the integration w.r.t. x 3 implies x 3 t x 1 x 2. The density of f 3 W l W l+m is Therefore, I 3 (x 1, x 2 ) x m 2 3 h 3 (x 3 ) λ m 1 (m 2)! 1I [, )(x 3 )e λx 3. t x 1 x 2 λ m 1 x m 2 3 (m 2)! e λx 3 e λ(t x 1 x 2 x 3 ) dx 3 1I [,t x1 )(x 2 )e λ(t x 1 x 2 ) λ m 1 (t x 1 x 2 ) m 1. (m 1)!

15 2.1. THE HOMOGENEOUS POISSON PROCESS 15 The density of f 2 W l+1 is This implies I 2 (x 1 ) h 2 (x 2 ) 1I [, ) (x 2 )λe λx 2. 1I [,t x1 )(x 2 )e λ(t x1 x2) λ m 1 (t x 1 x 2)m 1 (m 1)! s x 1 λ m e λ(t x 1) (t s)m. m! Finally, from Lemma we conclude If we sum I 1 s λ m e λ(t x 1) λ m λ l λt (t s)m e ( m! (λs) l l! (t s)m m! s l l! λ l x l 1 1 ) ( ) (λ(t s)) e λs m e λ(t s) m! P( ˆN(s) l)p( ˆN(t s) m). λe λx 2 dx 2 (l 1)! 1I [, )(x 1 )e λx 1 dx 1 P( ˆN(s) l, ˆN(t) ˆN(s) m) P( ˆN(s) l)p( ˆN(t s) m) over l N we get P( ˆN(t) ˆN(s) m) P( ˆN(t s) m) (2) and hence (1) for l and m 1. The case m can deduced from that above by exploiting and P( ˆN(s) l, ˆN(t) ˆN(s) ) P( ˆN(s) l) P( ˆN(t s) ) 1 P( ˆN(s) l, ˆN(t) ˆN(s) m) m1 P( ˆN(t s) m). m1

16 16 CHAPTER 2. CLAIM NUMBER PROCESS MODELS (P3) follows from Lemma and (2) and (P4) follows from the construction. (2) This part is subject to an exercise. Poisson, lambda5 X T Poisson, lambda1 X T 2.2 A generalization of the Poisson process: the renewal process To model windstorm claims, for example, it is not appropriate to use the Poisson process because windstorm claims happen rarely, sometimes with years

17 2.2. A GENERALIZATION OF THE POISSON PROCESS: THE RENEWAL PROCESS17 in between. The Pareto distribution, for example, which has the distribution function ( ) α κ F (x) 1 for x κ + x with parameters α, κ > will fit better when we use this as distribution for the waitig times, i.e. ( ) α κ P(W i x) for x. κ + x For a Pareto distributed random variable it is more likely to have large values than for an exponential distributed random variable. Definition (Renewal process). Assume that W 1, W 2,... are i.i.d. (independent and identically distributed) random variables such that W n (ω) > for all n 1 and ω Ω. Then { T : T n : W W n, n 1 is a renewal sequence and N(t) : #{i 1 : T i t}, t, is the corresponding renewal process. In order to study the it behavior of N we need the Strong Law of Large Numbers (SLLN): Theorem (SLLN). If the random variables X 1, X 2,... are i.i.d. with E X 1 <, then X 1 + X X n n EX 1 a.s. n Theorem (SLLN for renewal processes). Assume N(t) is a renewal process. If EW 1 <, then N(t) t t 1 EW 1 a.s.

18 18 CHAPTER 2. CLAIM NUMBER PROCESS MODELS Proof. Because of {ω Ω : N(t, ω) n} {ω Ω : T n (ω) t < T n+1 (ω)} for n N we have for N(t)(ω) > that T N(t,ω) (ω) N(t, ω) t N(t, ω) < T N(t,ω)+1(ω) N(t, ω) T N(t,ω)+1(ω) N(t, ω) + 1 N(t, ω) + 1. (3) N(t, ω) Note that Ω {ω Ω : T 1 (ω) < } {ω Ω : sup N(t) > }. t Theorem implies that T n n EW 1 (4) holds on a set Ω with P(Ω ) 1. Hence n T n on Ω and by definition of N also t N(t) on Ω. From (4) we get t N(t,ω)> Finally (3) implies that t N(t,ω)> T N(t,ω) N(t, ω) EW 1 for ω Ω. t N(t, ω) EW 1 for ω Ω. In the following we will investigate the behavior of EN(t) as t. Theorem (Elementary renewal theorem). Assume that (N(t)) t is a renewal process and that < EW 1 <. Then EN(t) t t 1 EW 1. (5) Remark If the W i s are exponentially distributed with parameter λ >, W i Exp(λ), i 1, then N(t) is a Poisson process and EN(t) λt.

19 2.2. A GENERALIZATION OF THE POISSON PROCESS: THE RENEWAL PROCESS19 Since EW i 1, it follows that for all t > that λ EN(t) t 1 EW 1. (6) If the W i s are not exponentially distributed, then the equation (6) holds only for the it t. In order to prove Theorem we formulate the following Lemma of Fatou type: Lemma Let Z (Z t ) t [, ) be a stochastic process such that Z t : Ω [, ) for all t and inf s t Z s : Ω [, ) is measurable for all t. Then E inf t Z t inf t EZ t. Proof. By monotone convergence, since t inf s t Z s is non-decreasing, we have E t inf s t Z s t E inf s t Z s. Obviously, E inf s t Z s EZ u for all u t which allows us to write This implies the assertion. E inf s t Z s inf u t EZ u. Proof of Theorem Let λ 1 EW 1. From Theorem we conclude λ t N(t) t N(s) inf t s t s Since Z t : N(t) for t > and Z t : fulfills the requirements of Lemma we have N(s) λ E inf inf t s t s EN(t). t t So, we only have to verify that sup t E N(t) λ. Let t >. Recall that t N(t) #{i 1 : T i t}. We introduce the filtration (F n ) n given by a.s. F n : σ(w 1,..., W n ), n 1, F : {, Ω}.

20 2 CHAPTER 2. CLAIM NUMBER PROCESS MODELS Then the random variable τ t : N(t) + 1 is a stopping time w.r.t. (F n ) n i.e. it holds {τ t n} F n, n. Let us verify this. n yields to {τ t } {N(t) 1} F, n 1 yields to {τ t 1} {N(t) } {t < W 1 } F 1, and for n 2 we have {τ t n} {T n 1 t < T n } {W W n 1 t < W W n } F n. By definition of N(t) we have that T N(t) t. Hence we get ET N(t)+1 E(T N(t) + W N(t)+1 ) t + EW 1 <. (7) On the other hand it holds τ t ET N(t)+1 E W i i1 τt K E W i K i1 by monotone convergence. Since Eτ t K < and the W i s are i.i.d with EW 1 < we may apply Wald s identity This implies τ t K E i1 W i E(τ t K)EW 1. > ET N(t)+1 K E(τ t K)EW 1 Eτ t EW 1. This relation is used in the following computation to substitute Eτ t EN(t) + 1: sup t EN(t) t sup t sup t sup t EN(t) + 1 t Eτ t t ET N(t)+1 t EW 1

21 2.3. THE INHOMOGENEOUS POISSON PROCESS 21 sup t where (7) was used for the last estimate. t + EW 1 t EW 1 1 EW 1, 2.3 The inhomogeneous Poisson process and the mixed Poisson process Definition Let µ : [, ) [, ) be a function such that (1) µ() (2) µ is non-decreasing, i.e. s t µ(s) µ(t) (3) µ is càdlàg. Then the function µ is called a mean-value function. µ(t)t y t µ(t) continuous y t

22 22 CHAPTER 2. CLAIM NUMBER PROCESS MODELS µ(t) càdlàg y t Definition (Inhomogeneous Poisson process). A stochastic process N N(t) t [, ) is an inhomogeneous Poisson process if and only if it has the following properties: (P1) N() a.s. (P2) N has independent increments, i.e. if t < t 1 <... < t n, (n 1), it holds that N(t n ) N(t n 1 ), N(t n 1 ) N(t n 2 ),..., N(t 1 ) N(t ) are independent. (P inh. 3) There exists a mean-value function µ such that for s < t (µ(t) µ(s)) (µ(t) µ(s))m P(N(t) N(s) m) e, m! where m, 1, 2,..., and t >. (P4) The paths of N are càdlàg a.s. Theorem (Time change for the Poisson process). If µ denotes the mean-value function of an inhomogeneous Poisson process N and Ñ is a homogeneous Poisson process with λ 1, then (1) (N(t)) t [, ) d (Ñ(µ(t))) t [, ) (2) If µ is continuous, increasing and t µ(t), then N(µ 1 (t)) t [, ) d (Ñ(t)) t [, ).

23 2.3. THE INHOMOGENEOUS POISSON PROCESS 23 Here µ 1 (t) denotes the inverse function of µ and f d g means that the two random variables f and g have the same distribution (but one can not conclude that f(ω) g(ω) for ω Ω). Definition (Mixed Poisson process). Let ˆN be a homogeneous Poisson process with intensity λ 1 and µ be a mean-value function. Let θ : Ω R be a random variable such that θ > a.s., and θ is independent of ˆN. Then N(t) : ˆN(θµ(t)), t is a mixed Poisson process with mixing variable θ. Proposition It holds ( var( ˆN(θµ(t))) E ˆN(θµ(t)) 1 + var(θ) ) Eθ µ(t). Proof. We recall that E ˆN(t) var( ˆN(t)) t and therefore E ˆN(t) 2 t + t 2. We conclude var( ˆN(θµ(t))) E ˆN(θµ(t)) 2 [ E ˆN(θµ(t)) ] 2 E ( θµ(t) + θ 2 µ(t) 2) (Eθµ(t)) 2 µ(t) (Eθ + varθµ(t)). The property var(n(t)) > EN(t) is called over-dispersion. inhomogeneous Poisson process, then If N is an var(n(t)) EN(t).

24 24 CHAPTER 2. CLAIM NUMBER PROCESS MODELS

25 Chapter 3 The total claim amount process S(t) 3.1 The renewal model and the Cramér- Lundberg-model Definition The renewal model (or Sparre-Anderson-model) considers the following setting: (1) Claims happen at the claim arrival times T 1 T 2... of a renewal process N(t) #{i 1 : T i t}, t. (2) At time T i the claim size X i happens and it holds that the sequence (X i ) i1 is i.i.d., X i. (3) The processes (T i ) i1 and (X i ) i1 are independent. The renewal model is called Cramér-Lundberg-model if the claims happen at the claim arrival times < T 1 < T 2 <... of a Poisson process 25

26 26 CHAPTER 3. THE TOTAL CLAIM AMOUNT PROCESS S(T ) 3.2 Properties of the total claim amount process S(t) Definition The total claim amount process is defined as N(t) S(t) : X i, t. i1 The insurance company needs information about S(t) in order to determine a premium which covers the losses represented by S(t). In general, the distribution of S(t), i.e. P({ω Ω : S(t, ω) x}), x, can only be approximated by numerical methods or simulations while ES(t) and var(s(t)) are easy to compute exactly. One can establish principles which use only ES(t) and var(s(t)) to calculate the premium. This will be done in chapter 4. Proposition One has that ES(t) EX 1 EN(t), var(s(t)) var(x 1 )EN(t) + var(n(t)(ex 1 ) 2. Consequently, one obtains the following relations: (1) Cramér-Lundberg-model: It holds (i) ES(t) λtex 1, (ii) var(s(t)) λtex 2 1. (2) Renewal model: Let EW 1 1 λ (, ) and EX 1 <. (i) Then t ES(t) t λex 1. (ii) If var(w 1 ) < and var(x 1 ) <, then var(s(t)) t t λ ( var(x 1 ) + var(w 1 )λ 2 (EX 1 ) 2).

27 3.2. PROPERTIES OF S(T) 27 Proof. (a) Since by a direct computation, 1 1I Ω (ω) 1I {N(t)k}, k N(t) ES(t) E E i1 k X i ( ( k ) X i )1I {N(t)k} i1 E(X X k ) E1I }{{} {N(t)k} }{{} k kex 1 P(N(t)k) EX 1 kp(n(t) k) k EX 1 EN(t). In the Cramér-Lundberg-model we have EN(t) λt. For the general case we use the Elementary Renewal Theorem (Thereom 2.2.4) to get the assertion. (b) We continue with ES(t) 2 N(t) E E i1 X i 2 ( ( k ) ) 2 E X i 1I {N(t)k} k ( k ) 2 X i 1I {N(t)k} k k i,j1 EX 2 1 i1 k E ( ) X i X j 1I {N(t)k} i1 kp(n(t) k) + (EX 1 ) 2 k k1 EX 2 1 EN(t) + (EX 1 ) 2 (EN(t) 2 EN(t)) var(x 1 )EN(t) + (EX 1 ) 2 EN(t) 2. k(k 1)P(N(t) k)

28 28 CHAPTER 3. THE TOTAL CLAIM AMOUNT PROCESS S(T ) It follows that var(s(t)) ES(t) 2 (ES(t)) 2 ES(t) 2 (EX 1 ) 2 (EN(t)) 2 var(x 1 )EN(t) + (EX 1 ) 2 var(n(t)). For the Cramér-Lundberg-model it holds EN(t) var(n(t)) λt, hence we have var(s(t)) λt(var(x 1 ) + (EX 1 ) 2 ) λtex1. 2 For the renewal model we get var(x 1 )EN(t) var(x 1 )λ. t t The relation is shown in [6, Theorem 2.5.2]. var(n(t)) t t var(w 1) (EW 1 ) 3. Theorem (SLLN and CLT for renewal model). (1) SLLN for (S(t)): If EW 1 1 λ < and EX 1 <, then S(t) t t λex 1 a.s. (2) CLT for (S(t)): If var(w 1 ) <, and var(x 1 ) <, then ( ) P S(t) ES(t) x Φ(x) t, var(s(t)) sup x R where Φ is the distribution function of the standard normal distribution, Φ(x) 1 2π x e y2 2 dy. Proof. (1) We follow the proof of [7, Theorem ]. We have shown that N(t) t t λ a.s.

29 3.2. PROPERTIES OF S(T) 29 and therefore it holds N(t) t a.s. Because of S(t) X 1 + X X N(t) and, by the SSLN we get S(t) t t (2) See [5, Theorem ]. X X n n n t N(t) t t EX 1 a.s., S(t) N(t) λex 1 a.s.

30 3 CHAPTER 3. THE TOTAL CLAIM AMOUNT PROCESS S(T )

31 Chapter 4 Premium calculation principles The standard problem in insurance is to determine that amount of premium such that the losses S(t) are covered. On the other hand, the price of the premiums should be low enough to be competitive and attract customers. In the following we let ρ(t) [, ) be the cumulative premium income up to time t [, ) in our stochastic model. Below we review some premium calculation principles. 4.1 Classical premium calculation principles First approximations of S(t) are given by ES(t) and var(s(t)), and the classical principles are based on these quantities. Intuitively, we have: ρ(t) < ES(t) insurance company loses on average ρ(t) > ES(t) insurance company gains on average Net principle The Net Principle ρ NET (t) ES(t) defines the premium to be a fair market premium. However, this usually leads to the ruin for the company as we will see later. 31

32 32 CHAPTER 4. PREMIUM CALCULATION PRINCIPLES Expected value principle In the Expected Value Principle the premium is calculated by p EV (t) (1 + ρ)es(t) where ρ > is called the safety loading The variance principle The Variance Principle is given by p VAR (t) ES(t) + αvar(s(t)), α >. This principle is in the renewal model asymptotically the same as p EV (t), since by Proposition we have that t p EV (t) p VAR (t) t (1 + ρ)es(t) ES(t) + αvar(s(t)) (1 + ρ) var(s(t)) 1 + α t ES(t) is a constant. This means that α plays the role of the safety loading ρ The standard deviation principle This principle is given by p SD (t) ES(t) + α var(s(t)), α > The constant income principle Here we simply ρ const (t) : ct, c >. In the case of the Cramér-Lundberg-modelthis principle coincides with the expected value principle by setting c : (1 + ρ)λex 1. In the case of the renewal model it is asymptotically the expected value principle as by Proposition we have ES(t) [λex 1 ]t. In Definition we introduce the Net Profit Condition that gives the necessary range for c (or ρ).

33 4.2. MODERN PREMIUM CALCULATION PRINCIPLES Modern premium calculation principles In the following principles the expected value E(g(S(t)) needs to be computed for certain functions g(x) in order to compute ρ(t). This means it is not enough to know ES(t) and var(s(t)), the distribution of S(t) is needed as well The Exponential Principle The Exponential Principle is defined as ρ exp (t) : 1 δ log EeδS(t), for some δ >, where δ is the risk aversion constant. The function p exp (t) is motivated by the so-called utility theory. By Jensen s inequality one checks that ρ exp (t) 1 δ log EeδS(t) ES(t) as the function x e x is convex The Esscher Principle The Esscher principle is defined as As a homework we show that ρ Ess (t) : ES(t)eδS(t) Ee δ(s(t)), δ >.. ρ Ess (t) ES(t)eδS(t) Ee δ(s(t)) ES(t) The Quantile Principle Denote by F t (x) : P({ω : S(t, ω) x}), x R, the distribution function of S(t). Given < ε < 1, we let ρ quant (t) : min{x : P({ω Ω : S(t, ω) x}) 1 ε}.

34 34 CHAPTER 4. PREMIUM CALCULATION PRINCIPLES This principle is called (1 ε) quantile principle. Note that P(S(t) > ρ quant (t)) ε. This setting is related to the theory of Value at Risk. 4.3 Reinsurance treaties Reinsurance treaties are mutual agreements between different insurance companies to reduce the risk in a particular insurance portfolio. Reinsurances can be considered as insurance for the insurance company. Reinsurances are used if there is a risk of rare but huge claims. Examples of these usually involve a catastrophe such as earthquake, nuclear power station disaster, industrial fire, war, tanker accident, etc. According to Wikipedia, the world s largest reinsurance company in 29 is Munich Re, based in Germany, with gross written premiums worth over $31.4 billion, followed by Swiss Re (Switzerland), General Re (USA) and Hannover Re (Germany). There are two different types of reinsurance: Random walk type reinsurance 1. Proportional reinsurance: The reinsurer pays an agreed proportion p of the claims, R prop (t) : ps(t). 2. Stop-loss reinsurance: The reinsurer covers the losses that exceed an agreed amount of K, where x + max{x, }. R SL (t) : (S(t) K) +, 3. Excess-of-loss reinsurance: The reinsurer covers the losses that exceed an agreed amount of D for each claim separately, where D is the deductible. N(t) R ExL : (X i D) +, i1

35 4.3. REINSURANCE TREATIES Extreme value type reinsurance Extreme value type reinsurances cover the largest claims in a portfolio. Mathematically, these contracts are investigated with extreme value theory techniques. The ordering of the claims X 1,..., X N(t) is denoted by X (1)... X (N(t)). 1. Largest claims reinsurance: The largest claims reinsurance covers the k largest claims arriving within time frame [, t], R LC (t) : k X (N(t) i+1). i1 2. ECOMOR reinsurance: (Excédent du coût moyen relatif means excess of the average cost). Define k N(t)+1 2. Then R ECOMOR (t) N(t) (X (N(t) i+1) X (N(t) k+1) ) + i1 k 1 X (N(t) i+1) (k 1)X (N(t) k+1) i1 Treaties of random walk type can be handled like before. For example, P( R SL (t) }{{} (S(t) K) + x) P(S(t) K) + P(K < S(t) x + K), so if F S(t) is known, so is F RSL (t).

36 36 CHAPTER 4. PREMIUM CALCULATION PRINCIPLES

37 Chapter 5 Probability of ruin: small claim sizes 5.1 The risk process In this chapter, if not stated differently, we use the following assumptions and notation: The renewal model is assumed. Total claim amount process: S(t) : N(t) i1 X i with t. Premium income function: ρ(t) ct where c > is the premium rate. The risk process or surplus process is given by U(t) : u + ρ(t) S(t), t, where U(t) is the insurer s capital balance at time t and u is the initial capital. 37

38 38 CHAPTER 5. PROBABILITY OF RUIN: SMALL CLAIM SIZES risk process U(t) U(t) U() t Definition (Ruin, ruin time, ruin probability). We let ruin(u) : {ω Ω : U(t, ω) < for some t > } the event that U ever falls below zero, Ruin time T : inf{t > : U(t) < } the time when the process falls below zero for the first time. The ruin probability is given by Remark ψ(u) P(ruin(u)) P(T < ). (1) T : Ω R { } is an extended random variable (i.e. T can also take the value ). (2) In the literature ψ(u) is often written as ψ(u) P(ruin U() u) to indicate the dependence on the initial capital u. (3) Ruin can only occur at the times t T n, n 1. This implies ruin(u) {ω Ω : T (ω) < } {ω Ω : inf U(t, ω) < } t>

39 5.1. THE RISK PROCESS 39 {ω Ω : inf n 1 U(T n(ω), ω) < } {ω Ω : inf n 1 (u + ct n S(T n )) < }, where the last equation yields from the fact that U(t) u + ct S(t). Since in the renewal model it was assumed that W i >, it follows that and where which imply that By setting and it follows that N(T n ) #{i 1 : T i T n } n S(T n ) N(T n) i1 X i n X i, i1 T n W W n, { ( ) } ω Ω : inf u + ct n S(T n ) < n 1 { ( ) } n ω Ω : inf u + ct n X i <. n 1 i1 Z n : X n cw n, n 1 G n : Z Z n, n 1, G :, {ω Ω : T (ω) < } { } ω Ω : inf ( G n) < u n 1 { } ω Ω : sup G n > u n 1 and for the ruin probability the equality it holds that ( ) ψ(u) P sup G n (ω) > u n 1. First we state the theorem that justifies the Net Profit Condition introduced below:

40 4 CHAPTER 5. PROBABILITY OF RUIN: SMALL CLAIM SIZES Theorem If EW 1 <, EX 1 <, and EZ 1 EX 1 cew 1, then ψ(u) 1 for all u >, i.e. ruin occurs with probability one independent from the initial capital u. Proof. (a) EZ 1 > : By the Strong Law of Large Numbers, G n n n EZ 1 almost surely. Because we assumed EZ 1 >, one gets that G n a.s., n, because G n nez 1 for large n. This means ruin probability ψ(u) 1 for all u >. (b) The case EZ 1 we show under the additional assumption that EZ1 2 <. Let { } Z Z n Ā m : sup m. n n Notice that for fixed ω Ω and n 1 we have iff Hence sup n sup n n Ā m Z 1 (ω) Z n (ω) n m Z n (ω) Z n (ω) n m. n 1 σ(z n, Z n +1,...). The sequence (Z n ) n 1 consists of independent random variables. By the 1 law of Kolmogorov (see [4, Proposition 2.1.6]) we conclude that P(Ām) {, 1}. Since ( Z Z P sup n n n ) m P(Ām)

41 5.2. BOUNDS FOR THE RUIN PROBABILITY 41 it suffices to show P(Ām) >. We have { Ā m sup n n1 kn Z Z n n { Z Z k k By Fatou s Lemma and the Central Limit Theorem, ( Z Z P(Ām) sup P k k k e x2 dx 2σ 2 > 2πσ 2 where σ 2 EZ 2 1. m } m } m. ) m Definition (Net profit condition). The renewal model satisfies the net profit condition (NPC) if and only if EZ 1 EX 1 cew 1 <. (NP C) The consequence of (NP C) is that on average more premium flows into the portfolio of the company than claim sizes flow out: We have G n ρ(t n ) + S(T n ) c(w W n ) + X X n which implies EG n nez 1 <. Theorem implies that any insurance company should choose the premium ρ(t) ct in such a way that EZ 1 <. In that case there is hope that the ruin probability is less than Lundberg inequality and Cramér s ruin bound In this section it is assumed, that the renewal model is used and the net profit condition holds, that means EX 1 cew 1 <.

42 42 CHAPTER 5. PROBABILITY OF RUIN: SMALL CLAIM SIZES Definition For a random variable f : Ω R on some probability space (Ω, F, P) the function m f (h) Ee hf, is called the moment-generating function if it is finite for h ( h, h ) for some h >. Remark (1) The map h Ee hf is called two-sided Laplace transform. (2) If E f k e hf < on ( h, h ) for all k,..., m, then m f (h) exists, is m-times differentiable, and one has Therefore d m dh m m f(h) Ef m e hf. d m dh m m f() Ef m. Definition (Small claim size condition). We say that the small claim size condition with paramer h > is satisfied if m X1 (h) Ee hx 1 and m cw1 (h) Ee hcw 1 exist for h < h. Theorem (Lundberg inequality). Assume that the small claim size condition with paramer h > is satisfied. If there is an r (, h ) with m Z1 (r) Ee r(x 1 cw 1 ) 1, then ψ(u) e ru for all u >. Definition The exponent r in Theorem is called Lundberg coefficient. The result implies, that if the small claim condition holds and the initial capital u is large, there is in principal no danger of ruin.

43 5.2. BOUNDS FOR THE RUIN PROBABILITY 43 Remark (1) If m Z1 exists in ( h, h ) for some h >, then and P(Z 1 λ) P(e εz 1 e ελ ) e ελ m Z1 (ε) P( Z 1 λ) e ελ m Z1 (ε) e ελ m Z1 ( ε) for all ε ( h, h ). This implies P( Z 1 λ) e ελ [m Z1 (ε) + m Z1 ( ε)]. (2) If the Lundberg coefficient r exists, then it is unique. First we observe that which follows from the fact that m Z1 is convex: We have e (1 θ)r Z 1 +θr 1 Z 1 (1 θ)e r Z 1 + θe r 1Z 1. Moreover, m Z1 () 1 and by Jensen s inequality, m Z1 (h) Ee Z 1h e EZ 1h such that (assuming (NPC) holds) we get m Z 1 (h) h h e EZ 1( h). If m Z1 exists in ( ε, ε) and m Z1 (h) 1 for some h {r, s} (, ε] then, by convexity, m Z1 (h) 1 h [, r s]. From (1) we have and it holds E Z 1 n P( Z 1 > λ) ce λ c for some c > P( Z 1 n > λ)dλ n n P( Z 1 > λ)λ n 1 dλ ce λ/c λ n 1 dλ ( ) n 1 λ nc n e λ/c dλ c

44 44 CHAPTER 5. PROBABILITY OF RUIN: SMALL CLAIM SIZES Because of n for hc < 1, the function h n n! E Z 1 n n m Z1 (h) Ee hz 1 E nc n+1 e λ λ n 1 dλ n!c n+1. h n n! n!cn+1 < (hz 1 ) n is infinitely often differentiable for h < 1/c. Moreover the function is constant on [, r s] so that, for < h < min{1/c, r s}, one has n n! d2 dh 2 m Z 1 (h) EZ 2 1e hz 1. This implies EZ 2 1e hz 1 and Z 1 a.s. (3) In practice, r is hard to compute from the distributions of X 1 and W 1. Therefore it is often approximated numerically or by Monte Carlo methods. Proof of Theorem We use Z n X n cw n and set G k : Z Z k. We consider ψ n (u) : P ( max 1 k n G k > u ), u >. Because of ψ n (u) ψ(u) for n it is sufficient to show For n 1 we get the inequality by ψ n (u) e ru, n 1, u >. ψ 1 (u) P(Z 1 > u) P(e rz 1 > e ru ) e ru Ee rz 1 e ru. Now we assume that the assertion holds for n. We have ψ n+1 (u) P ( max G k > u ) 1 k n+1

45 5.2. BOUNDS FOR THE RUIN PROBABILITY 45 P(Z 1 > u) + P ( max G k > u, Z 1 u ) 1 k n+1 P(Z 1 > u) + P ( max (G k Z 1 ) > u Z 1, Z 1 u ) 2 k n+1 P(Z 1 > u) + u P ( max 1 k n G k > u x ) df Z1 (x) where we have used for the last line that max 2 k n+1 (G k Z 1 ) and Z 1 are independent. We estimate the first term P(Z 1 > u) df Z1 (x) e r(x u) df Z1 (x), (u, ) (u, ) and proceed with the second term as follows: P ( max G k > u x ) df Z1 (x) 1 k n (,u] ψ n (u x)df Z1 (x) (,u] e r(u x) df Z1 (x). (,u] Consequently, ψ n+1 (u) e r(x u) df Z1 (x) + e r(u x) df Z1 (x) e ru. (u, ) (,u] We consider an example where it is possible to compute the Lundberg coefficient: Example Let X 1, X 2,... Exp(γ) and W 1, W 2,... Exp(λ). Then m Z1 (h) Ee h(x 1 cw 1 ) Ee hx 1 Ee hcw 1 γ λ γ h λ + ch for λ c < h < γ since Ee hx 1 e hx γe γx dx γ γ h.

46 46 CHAPTER 5. PROBABILITY OF RUIN: SMALL CLAIM SIZES The (NPC) condition reads as > EZ 1 EX 1 cew 1 1 γ c λ or cγ > λ. Hence m Z1 exists on ( λ, γ) and for r > we get c γ λ γ r λ + cr 1 γλ γλ + γcr λr cr 2 r γ λ c. Consequently, ψ(u) e ru e (γ λ c )u. Applying the expected value principle ρ(t) (1 + ρ)es(t) (1 + ρ)λex 1 t we get This implies γ λ c γ λ (1 + ρ) λ γ ψ(u) e ru e uγ γ ρ 1 + ρ. ρ 1+ρ, where one should notice that even ρ does not change the ruin probability considerably! The following theorem considers the special case, the Cramér-Lundbergmodel: Theorem (Cramér s ruin bound). Assume the Cramér-Lundbergmodel with the net profit condition (NPC) and with ρ(t) ct (1 +ρ)es(t). Suppose that the distribution of X 1 has a density, that m X1 (h) exists in a neighborhood ( h, h ) of the origin, and that r (, h ) is the Lundberg coefficient. Then one has with c : ρ EX 1 r u eru ψ(u) c, ( ye ry (1 F X1 (y))dy) 1. To prove this theorem introduce in the next section the fundamental integral equation for the survival probability.

47 5.3. FUNDAMENTAL INTEGRAL EQUATION Fundamental integral equation for the survival probability We introduce the survival probability ϕ(u) 1 ψ(u). Theorem (Fundamental integral equation for the survival probability). Assume that for the Cramér-Lundberg-model (NPC) holds and EX 1 <. Suppose that ρ(t) ct (1 + ρ)es(t). Then 1 ϕ(u) ϕ() + (1 + ρ)ex 1 u F X1 (y)ϕ(u y)dy (1) with F X1 (y) P(X 1 > y). Remark Let the assumptions of Theorem hold. (1) The assertion can be reformulated as follows. Let F X1,I(y) : 1 EX 1 y F X1 (z)dz, y. The function F X1,I(y) is a distribution function since F X 1,I(y) y 1 EX 1 1 EX 1 F X1 (z)dz P(X 1 > z)dz 1. Hence equation (1) can be written as ϕ(u) ϕ() ρ u ϕ(u y)df X1,I(y). (2) It holds that u ϕ(u) 1. This can be seen as follows: ϕ(u) u u u (1 ψ(u)) ( ( 1 P sup G k > u k 1 ))

48 48 CHAPTER 5. PROBABILITY OF RUIN: SMALL CLAIM SIZES ( ) P sup G k u u k 1 where G k Z Z k. Since EZ 1 < the SLLN implies G k k a.s. Therefore we have sup k 1 G k < a.s. and ( ) P sup G k u 1. u k 1 (3) It holds that ϕ() ρ 1 for ρ. Indeed, because of (1), and (2), 1+ρ and Theorem we may conclude that 1 ϕ() ρ ϕ() ρ ϕ() ρ ϕ() ρ. u u df X1,I(y) 1I [,u] (y)ϕ(u y)df X1,I(y) ( 1I[,u] (y)ϕ(u y) ) df X1,I(y) The interpretation of ϕ() is the survival probability when starting with Euro initial capital. Proof of Theorem (a) We first show that ϕ(u) λ c e λu c e λy c ϕ(y x)df X1 (x)dy. (2) To do this, we consider [u, ) [,y] ϕ(u) ( ) P sup G n u n 1 ( ) P Z 1 u, G n Z 1 u Z 1 for n 2

49 5.3. FUNDAMENTAL INTEGRAL EQUATION 49 [, ) [,u+cw] where we used for the last line that ( ) P G n Z 1 u (x cw) for n 2 df X1 (x)df W1 (w) x cw u and x x u + cw. We use that G n Z 1 Z Z n 1 and substitute y : u + cw in order to obtain ( ) ϕ(u) P G n u (x cw) for n 1 df X1 (x)λe λw dw [, ) [,u+cw] ϕ(u x + cw)df X1 (x)λe λw dw [, ) [,u+cw] y u λ ϕ(y x)df X1 (x)λe c d y c. [u, ) [,y] (b) Differentiation of (2) leads to ϕ (u) λ c ϕ(u) so that λ c ϕ(u) λ c t [,y] ϕ(y x)df X1 (x) λ c [,u] ϕ(t) ϕ() λ ϕ(u)du c λ t ϕ(u x)df X1 (x)du c [,u] λ t [ u ϕ(u x)f X1 (x) c + [,u] t [ ϕ()f X1 (u) ϕ(u)f X1 () + λ c λ t c ϕ() F X1 (u)du λ c λ c t ϕ(t x)f X1 (x)dx. t ϕ(u x)df X1 (x), [x,t] y u e λ c yu ] ϕ (u x)f X1 (x)dx du [,u] ] ϕ (u x)f X1 (x)dx du ϕ (u x)f X1 (x)dudx

50 5 CHAPTER 5. PROBABILITY OF RUIN: SMALL CLAIM SIZES This implies ϕ(t) ϕ() λ c t ϕ(t x)(1 F X1 (x))dx. Using that gives λ c 1 (1+ρ)EX 1 ES(t) λtex 1 and ct (1 + ρ)es(t) (1 + ρ)λtex 1 which yields the assertion. Now we can deduce a representation of the survival probability: Theorem Assume the Cramér-Lunberg model with the NPCcondition. Assume for the claim size distribution X 1 : Ω (, ) that EX 1 <, let (X I,k ) k 1 be i.i.d. random variables with the distribution function F X1,I, and define ϕ(u) : ρ [ ρ ] (1 + ρ) n P(X I, X I,n u) n1 Then ϕ is the unique solution to ϕ(u) ϕ() ρ u ϕ(u y)df X1,I(y) for u R. in the class G : { g : R [, ) : non-decreasing, bounded, { for x < right-continuous with g(x) ρ for x 1+ρ }. Proof. Uniqueness: Assume ϕ 1, ϕ 2 are solutions and ϕ ϕ 1 ϕ 2. Then ϕ(u) ρ ρ u u ϕ(u y)df X1,I(y) ϕ(u y) F X 1 (y) dy EX 1

51 5.3. FUNDAMENTAL INTEGRAL EQUATION 51 1 (1 + ρ)ex 1 u ϕ(y)f X1 (u y)dy and 1 u ϕ(u) ϕ(y) dy. (1 + ρ)ex 1 Gronwall s Lemma implies that ϕ(u) for u R. Verification that ϕ is a solution: Here we get ϕ() ρ [ ] 1 + (1 + ρ) n P(X I, X I,n ) 1 + ρ and ϕ() ρ ρ 1 + ρ ρ ρ 1 + ρ ρ u [ 1 + ρ 1 + ρ u u n1 ϕ(u y)df X1,I(y) ϕ(u y)df X1,I(y) (1 + ρ) n P (X I, X I,n u y) n1 ρ 1 + ρ ρ [ ρ F X1,I(u) ρ ρ 1 + ρ ρ [ ρ F X1,I(u) ρ ρ 1 + ρ ρ [ ρ ρ 1 + ρ ] df X1,I(y) ] u (1 + ρ) n P (X I, X I,n + y u) df X1,I(y) n1 ] (1 + ρ) n P (X I, X I,n+1 u) n1 (1 + ρ) 1 P(X I,1 u) + [ ρ ρ ] (1 + ρ) (n+1) P (X I, X I,n+1 u) n1 ] (1 + ρ) (n+1) P (X I, X I,n u) n1

52 52 CHAPTER 5. PROBABILITY OF RUIN: SMALL CLAIM SIZES ϕ(u). 5.4 Proof of Cramér s ruin bound Lemma (Smith s renewal equation). It holds e ru ψ(u) qe ru FX1,I(u) + u where q : 1, r is the Lundberg coefficient, and 1+ρ F (r) X 1 (x) : q EX 1 e r(u x) ψ(u x)df (r) X 1 (x) x e ry FX1 (y)dy. Definition The function F (r) X 1 F X1. is called the Esscher transform of Proof of Lemma (a) We first show that F (r) X 1 In fact, we have is a distribution function. F (r) x X 1 (x) q e ry FX1 (y)dy EX 1 q 1 EX 1 r (EerX 1 1), since Ee rx r P(e rx 1 > t)dt P(e rx 1 > e ry )re ry dy re ry dy + F X1 (y)e ry dy. P(X 1 > y)re ry dy

53 5.4. PROOF OF CRAMÉR S RUIN BOUND 53 From Ee r(x 1 cw 1 ) 1 we conclude F (r) x X 1 (x) ( q 1 EX 1 r q 1 EX 1 r qc EX 1 1 λ (b) From Theorem we conclude that ψ(u) q F X1,I(u) + u ) 1 Ee 1 rcw 1 ( ) rc + λ 1 λ c 1 (1 + ρ)ex 1 λ 1. ψ(u y)d(qf X1,I)(y). Indeed, by equation (1) and Remark we have Hence ϕ(u) ρ 1 + ρ ρ u ϕ(u y) F X1 (y) dy. EX 1 1 ψ(u) 1 ϕ(u) 1 + ρ 1 ϕ(u y) F X1 (y) dy 1 + ρ EX 1 u F X1 (y) u q q dy + q ψ(u y) F X1 (y) dy EX 1 EX 1 q F X1,I(u) + u u ψ(u y)d(qf X1,I)(y). This equation would have the structure of a renewal equation (see the renewal equation (3) below) only if q 1. Therefore we consider e ru ψ(u) e ru q F X1,I(u) + e ru qf X1,I(u) + u u e r(u y) ψ(u y)e ry d(q F X1,I)(y) e r(u y) ψ(u y)df (r) X 1 (y). Since F (r) is a distribution function, we have indeed a renewal equation. The following assertion is a generalization of the Blackwell Renewal Lemma.

54 54 CHAPTER 5. PROBABILITY OF RUIN: SMALL CLAIM SIZES Lemma (Smith s key renewal lemma). Let H : R R be a continuous distribution function such that H(x) for x and < xdh(x) <. Let R (1) k : R [, ) be Riemann integrable such that k(x) for x <, (2) x k(x). Then R(u) u k(u y)dh(y) is in the class of all functions on (, ) which are bounded on finite intervals the only solution to the renewal equation and it holds R(u) k(u) + u R(u) 1 u xdh(x) R The proof can be found in [9, pp. 22]. R(u y)dh(y) (3) Proof of Theorem We apply Lemma for For α : 1 R xdf (r) (x) we get u eru ψ(u) α α α R(u) : e ru ψ(u), k(u) : qe ru FX1,I(u), H(x) : F (r) (x). qe ry FX1,I(y)dy [ qe ry 1 1 qe ry 1 EX 1 EX 1 y y k(u)du. ] F X1 (z)dz dy F X1 (z)dzdy

55 5.4. PROOF OF CRAMÉR S RUIN BOUND 55 q α EX 1 α [ q r EX 1 z α r [1 q] α r e ry dy F X1 (z)dz e rz FX1 (z)dz q EX 1 ρ 1 + ρ. ] F X1 (z)dz Finally 1 α R xdf (r) (x) q xe rx FX1 (x)dx EX 1 1 xe rx FX1 (x)dx. (1 + ρ)ex 1 implies u eru ψ(u) ρex 1 r 1 xe rx F X1 (x)dx.

56 56 CHAPTER 5. PROBABILITY OF RUIN: SMALL CLAIM SIZES

57 Chapter 6 Probability of ruin: large claim sizes 6.1 Tails of claim size distributions So far we did not discuss how to separate small and large claim sizes, and how to choose the distributions to model the claim sizes (X i )? If one analyzes data of claim sizes that have happened in the past, for example by a histogram or a QQ-plot (see Chapter 7), it turns out that the distribution is either light-tailed or heavy-tailed, the latter case is more often the case. Definition LetX : Ω R be a random variable with X(ω) for all ω Ω with distribution function F, i.e. F (x) P({ω Ω : X 1 (ω) x}). (1) The distribution function F or the random variable X is called lighttailed if and only if there is some λ > such that one has sup x e λx P(X > x) sup e λx [1 F (x)] <. x (2) The distribution function F or the random variable X is called heavytailed if and only if for all λ > one has inf x eλx P(X > x) inf x eλx [1 F (x)] >. 57

58 58 CHAPTER 6. PROBABILITY OF RUIN: LARGE CLAIM SIZES Example (1) The exponential distribution Exp(α) is light-tailed for all α >, since the distribution function is F (x) 1 e αx, x >, and 1 F (x) e λx and by choosing < λ < α, e αx e λx e(λ α)x, sup e (λ α)x e (λ α)n, as n. x n (2) The Pareto distribution is heavy-tailed. The distribution function is κ α F (x) 1 (κ + x) α for x, α >, κ >, or F (x) 1 ba x a for x b >, a >. Proposition If X : Ω R is light-tailed, then there is an h >, such that the moment generating function is finite on ( h, h ). h Ee hx Proof. It is sufficient to find an h > such that Ee h X <. We get for < h < λ. Ee hx P(e hx > x)dx 1 + P(e hx > x)dx < (1, ) (1, ) (1, ) 1 + C 1 + C (1, ) (1, ) ( P X > 1 ) log(x) dx h [ ( 1 1 F [ e λ )] log(x) dx h )] dx ( 1 log(x) h [x λ h ] dx

59 6.2. SUBEXPONENTIAL DISTRIBUTIONS Subexponential distributions Definition and basic properties In Theorem below we need subexponential distributions: Definition A distribution function F : R [, 1] such that F () and F (x) < 1 for all x > is called subexponential if and only if for i.i.d. (X i ) i1 and P(X i u) F (u), u R, it holds that P(X X n > x) x P(max 1 k n X k > x) 1 for all n 2. We denote the class of subexponential distribution functions by S. We start with an equivalence that can be used to define S as well. Proposition (Equivalent conditions for S, part I). Assume i.i.d. (X i ) i1 with P(X i u) F (u), u R. Then F S if and only if P(X X n > x) x P(X 1 > x) Proof. It holds for S n X X n that n for all n 1. P(S n > x) P(max 1 k n X k > x) P(S n > x) 1 P(max 1 k n X k x) P(S n > x) 1 P(X 1 x) n P(S n > x) 1 (1 P(X 1 > x)) n P(S n > x) P(X 1 > x)n(1 + o(1)). Next we continue with properties of subexponential distributions that motivate the name subexponential: Proposition Assume i.i.d. random variables (X i ) i1 and a random variable X with P(X u) P(X i u) F (u), u R. Then one has the following assertions:

60 6 CHAPTER 6. PROBABILITY OF RUIN: LARGE CLAIM SIZES (1) For F S it holds F (x y) x F (x) (2) If F S, then for all ε > it holds 1 for all y >. e εx P(X > x) for x. (3) If F S, then for all ε > there exists a K > such that P(S n > x) P(X 1 > x) K(1 + ε)n n 2 and x. For the proof we need the concept of slowly varying functions: Definition (Slowly varying functions). A measurable function L : [, ) (, ) is called slowly varying if L(cξ) ξ L(ξ) 1 for all c >. Proposition (Karamata s representation). Any slowly varying function can be represented as ( ξ ) ε(t) L(ξ) c (ξ) exp dt for all ξ ξ t ξ for some ξ > where c, ε : [ξ, ) R are measurable functions with c (ξ) c > and ε(t). ξ t Corollary For any slowly varying function L it holds ξ ξδ L(ξ) for all δ >. Proof. We can enlarge ξ such that sup t ξ ε(t) < δ. With this choice we get ( ξ ) ε(t) ξ ξδ c (ξ) exp dt ξ t

61 6.2. SUBEXPONENTIAL DISTRIBUTIONS 61 ( ξ c (ξ) exp δ log ξ + ξ c (ξ) exp ξ c (ξ) exp ξ. ξ ( δ log ξ sup ) ε(t) dt t ξ ) dt t ε(t) t ξ ξ ( δ log ξ (log ξ log ξ ) sup ε(t) t ξ Proof of Proposition (1) For y x < we have P(X 1 + X 2 > x) P(t + X > x)df (t) R P(X 1 > x) P(X 1 > x) P(X 1 > x) + P(t + X > x)df (t) (,y] P(X 1 > x) P(t + X > x)df (t) (y,x] + P(X 1 > x) F (x y) 1 + F (y) + (F (x) F (y)). F (x) We choose x large enough such that F (x) F (y) > and observe that 1 as x. F (x y) F (x) ( P(X1 + X 2 > x) P(X 1 > x) ) 1 F (y) ) 1 F (x) F (y) 1 (2) Let L(ξ) : F (log ξ). It follows from (1) that for all c > one has L(cξ) ξ L(ξ) F (log c + log ξ) ξ F (log ξ) By definition, L is slowly varying. Therefore, where we use Corollary ξ ξδ F (log ξ) e δx F (x) x (3) The proof can be found in [5][Lemma 1.3.5].

62 62 CHAPTER 6. PROBABILITY OF RUIN: LARGE CLAIM SIZES Examples In order to consider fundamental examples we need the next lemma: Lemma Let X 1, X 2 be independent positive random variables such that for some α > F Xi (x) L i(x) x α where L 1, L 2 are slowly varying. Then F X1 +X 2 (x) x α (L 1 (x) + L 2 (x))(1 + o(1)). Proof. For < δ < 1 2 we have and hence {X 1 + X 2 > x} {X 1 > (1 δ)x} {X 2 > (1 δ)x} P(X 1 + X 2 > x) {X 1 > δx, X 2 > δx} F X1 ((1 δ)x) + F X2 ((1 δ)x) + F X1 (δx)f X2 (δx) [F X1 ((1 δ)x) + F X2 ((1 δ)x)][1 + F X1 (δx)] [F X1 ((1 δ)x) + F X2 ((1 δ)x)][1 + o(1)] [ L1 ((1 δ)x) ((1 δ)x) + L ] 2((1 δ)x) [1 + o(1)] α ((1 δ)x) α From this we get [ F X1 (x) L 1((1 δ)x) + F X2 (x) L 2((1 δ)x) L 1 (x) L 2 (x) ] [1 + o(1)](1 δ) α. sup x P(X 1 + X 2 > x) F X1 (x) + F X1 (x) sup x (1 δ) α. F X1 (x) L 1((1 δ)x) L 1 (x) P(X 1 + X 2 > x) + F X2 (x) L 2((1 δ)x) L 2 (x) As this is true for all < δ < 1, we may conclude 2 sup x P(X 1 + X 2 > x) F X1 (x) + F X1 (x) 1.

63 6.2. SUBEXPONENTIAL DISTRIBUTIONS 63 On the other hand, P(X 1 + X 2 > x) P({X 1 > x} {X 2 > x}) and hence Consequently, P(X 1 > x) + P(X 2 > x) P(X 1 > x)p(x 2 > x) F X1 (x) + F X2 (x) F X1 (x)f X2 (x) [ F X1 (x) + F X2 (x) ][ 1 F X1 (x) ] inf x x P(X 1 + X 2 > x) F X1 (x) + F X2 (x) 1. P(X 1 + X 2 > x) F X1 (x) + F X2 (x) 1. Definition If there exists a slowly varying function L and some α > such that for a positive random variable X it holds F X (x) L(x) x α, then F X is called regularly varying with index α or of Pareto type with exponent α. Proposition If F X is regularly varying with index α, then F X is subexponential. Proof. An iteration of Lemma implies F X X n (x) F X (x) L(x) L(x) L(x) n. Example (1) The exponential distribution with parameter λ > is not subexponential.