Simulation analysis of ruin capital in Sparre Andersen’s model of risk

Accepted Manuscript Simulation analysis of ruin capital in Sparre Andersen’s model of risk Vsevolod K. Malinovskii, Ksenia O. Kosova PII: DOI: Reference:

S0167-6687(14)00118-8 http://dx.doi.org/10.1016/j.insmatheco.2014.09.004 INSUMA 1983

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Insurance: Mathematics and Economics

Received date: December 2013 Revised date: August 2014 Accepted date: 4 September 2014 Please cite this article as: Malinovskii, V.K., Kosova, K.O., Simulation analysis of ruin capital in Sparre Andersen’s model of risk. Insurance: Mathematics and Economics (2014), http://dx.doi.org/10.1016/j.insmatheco.2014.09.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

SIMULATION ANALYSIS OF RUIN CAPITAL IN SPARRE ANDERSEN’S MODEL OF RISK Vsevolod K. Malinovskii and Ksenia O. Kosova

Abstract. Ruin capital is a function of premium rate set to render the probability of ruin within finite time equal to a given value. The analytical studies of this function in the classical Lundberg model of risk with exponential claim sizes done in Malinovskii (2014) have shown that the ruin capital’s shape is surprisingly simple. This work presents the results of related simulation studies. They are focused on the question whether this shape remains similar in Sparre Andersen’s model of risk.

1. Introduction and definitions Sparre Andersen’s model of risk is a well known generalization of the classical Lundberg model. Its mathematical innovation is a move to renewal processes1. In seeking for economical rationale, its originator emphasized (see Sparre Andersen (1957), p. 219) “contagion, which may be . . . characterized by the property that a claim is more likely (or, if that should be wanted, less likely) to occur shortly after another claim, and that the probability of occurrence of claims depends on the time elapsed since the last claim and only on this quantity.” This paper deals with Sparre Andersen’s model of risk generated by i.i.d. interclaim times Ti , i = 1, 2, . . . , and i.i.d. claim sizes Yi , i = 1, 2, . . . , assumed mutually independent. Developing the classical model which paramount special case is yielded by assuming both interclaim times and claim sizes exponential with parameters λ > 0 and µ > 0 respectively, Sparre Andersen’s model focuses on the non-exponential case. The risk reserve process is a continuous-time random process Rs = u + cs − Vs ,

(1.1) P Ns where u > 0 is the initial risk reserve, c > 0 is the premium rate, © Vs =Pni=1 Yi , orª0, if T1 > s, is the aggregate claims payout process and Ns = max n > 0 : i=1 Ti 6 s , or 0, if T1 > s, is the claims arrival process. The probability of ruin within time t is defined as ψ t (u, c) = P{inf 0 0 small and fixed, the solution with respect to u of the equation ψ t (u, c) = α

s > 0,

(1.2)

Key words and phrases. Ruin capital, Finite time ruin probability, Sparre Andersen’s model of risk, Simulation study. This work was supported by RFBR (grant No. 14-06-00017-a). 1 Other established generalizations are based on L´ evy, Ammeter, and Cox processes. 1

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VSEVOLOD K. MALINOVSKII AND KSENIA O. KOSOVA

is called α-level initial capital, or α-level ruin capital, or simply ruin capital. We denote it by uα,t (c). By uα (c) we denote the solution (w.r.t. u) of the equation2 ψ +∞ (u, c) = α. The ruin capital is paramount in modeling solvency. It lays grounds for investigation of controlled multi-period models of insurance business (see, e.g., Malinovskii (2012)). In the framework of the classical model with exponential claim sizes, the analytical investigations done in Malinovskii (2012)–Malinovskii (2014) have established the following properties of the ruin capital3. As t is sufficiently large, it is • almost linear for 0 < c < λ/µ, • convex for c > λ/µ, • asymptotically independent on t and close to uα (c) for c > c∗α,t , where c∗α,t is a certain value larger than λ/µ. The extension of these analytical results to Sparre Andersen’s model is still an open and quite a difficult problem. In this paper, we conduct simulation studies in order to see to what extent the ruin capital in Sparre Andersen’s model preserves these properties4. This method of action is renowned and is often used in ruin theory and its applications (see, e.g., Asmussen and Albrecher (2010) for an overview of related simulation literature). This paper is organized as follows. In Section 2, we present more insight into this matter. In particular, we discuss the paramount case of c close to the critical point EY1 /ET1 . The latter separates the premium rates for which the insurance process is profitable from the premium rates for which it is unprofitable. We put forth some ideas to check the simulation results. We formulate a set of conjectures about uα,t (c) which we seek to explore. In Section 3, we introduce several families of probability distributions appropriate for modeling the interclaim time and/or claim size distributions. Regarded as mixtures of exponential distributions, they are natural extensions of these ones. In Section 4, we present an algorithm of numerical simulation of the ruin capital. We do not discuss the optimality of this algorithm, including the advantages and deficiencies of pseudo-random number generators5 used for simulation, which would require much more space. In our algorithm, we rely on the standard (see Knuth (1981)) linear congruence random number generator. 2In the classical case with exponential claim sizes, we have ψ +∞ (u, c) = (λ/(cµ)) exp{−u(cµ−λ)/c}

for λ/cµ < 1 and 1 elsewhere, and the explicit formula for uα (c) is straightforward. It is noteworthy that the endeavor to determine such time-dependent premium rate that the probability of ultimate ruin remains bounded from above by a predetermined constant is the paper Constantinescu et al. (2012). 3The similar results for all t > 0 rather than for t sufficiently large hold true in the diffusion model of risk (see Malinovskii (2014a)). Being transparent, their proofs seem instructive. 4Besides a rather trivial idea “let’s compute it and see if we obtain anything reasonable”, our guiding motives are the following two observations. First (see Pentikainen (1975), p. 45), that “as a compromise between simulation and analytical methods it may be advisable to perform some simplified calculations analytically first. By finding suitable special assumptions the analytical treatment of the problem can perhaps be carried out. How far these analytical results are valid also as a solution of the original problem, which is based on more general and more realistic assumptions, probably cannot be estimated. However, the decision variables obtained analytically can serve initial variables for simulation and the problem of optimization by means of simulation can, perhaps, be made easier”. Second (see Pentikainen (1988), p. 30), that “the merit of the simulation approach is its great flexibility to allow for quite general conditions and assumptions. . . Its drawback is that the results are subject to sample inaccuracy, which may require large and time consuming sample sizes.” 5Every random number generator has its advances and deficiencies, see Hellekalek (1998). Quoting from Section 10 of this paper which discusses criteria for good random number generators, we agree that “random number generators are like antibiotics. Every type of generator has its unwanted side-effects. There is no safe generators. Good random number generators are characterized by theoretical support, convincing empirical evidence, and positive practical aspects. They will produce correct results in many, though in not all, situations.”

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In Section 5, we report the simulation studies of the ruin capital in semi-classical models. So we call the models with the interclaim time and/or claim size distributions only slightly different from exponential. To be particular, we take mixtures of a finite number of exponential distributions and the Erlang distributions which are produced by a sum of a finite number of independent exponential random variables. Despite the closeness of semi-classical models to ones with exponential interclaim time and claim size distributions, to extend the analytical results from the latter to the former is always a challenge6. In Section 6, we address the ruin capital in the case of non-Poissonian claims’ arrivals. We deal with exponential claim sizes, being therefore within the framework of Cram´er’s theory which treats light-tailed claim sizes, and with interclaim times markedly different from exponential. They belong to Pareto and Kummer7 families discussed in Section 3. These families are two-parametric, heavy-tailed mixtures of exponentials, and are known to be suitable to model the non-Poissonian claims’ arrivals. In Section 7, we go beyond the framework of Cram´er’s theory. We are interested in the case of Poisson claims’ arrival and claim sizes markedly different from exponential. Being heavy-tailed, they are more relevant to reinsurance models. As such, we focus again on the Pareto and Kummer families discussed in Section 3. To conclude, we say that the simulation analysis in no case can give the full understanding of the structure of the ruin capital in Sparre Andersen’s model. But all results presented in Sections 5–7 support definitively the conjecture that the ruin capital uα,t (c) regarded as a function of c behaves similarly in both classical, allowing an analytical solution, and Sparre Andersen’s models. Further analytical investigation is quite demanding, though not developed yet, and our simulation analysis stimulates this study. 2. Rationale, motivation and main issues to be investigated The main conjectures about the ruin capital uα,t (c) regarded as a function of c in Sparre Andersen’s model stem from the information available in the classical model. We present it and formulate a set of conjectures to be examined. We discuss why the standard theoretical results about the probabilities of ruin are deficient in our context. 2.1. Ruin capital in classical model with exponential claim sizes. The intuition suggests that the ruin capital uα,t (c) monotone decreases to zero, as c increases to infinity. In addition, it is smooth for all c > 0, including the neighborhood of the critical point c = EY1 /ET1 . This point is known to separate the premium rates for which the insurance process is unprofitable from those rates for which it is profitable. In the classical model, uα,t (c) has a surprisingly simple structure fully consistent with the intuition. It looks different on the left and on the right of the critical point8 c = λ/µ. It can be seen from the following two theorems. Proposition 2.1 (Theorem 3.2 in Malinovskii (2012)). In the classical Lundberg model with exponential claim sizes, we have √ √ ³ µ(λ/µ − c)√t ´i  2tλ h µ(λ/µ − c) t   √ √ + U , c 6 λ/µ, α,t  µ 2λ 2λ uα,t (c) = √ √ ³  − c) t ´   2tλ Uα,t µ(λ/µ √ , c > λ/µ,  µ 2λ

6See, e.g., Wang and Liu (2002) where Theorem 2.1 of Malinovskii (1998) was extended to Y with 1 p.d.f. pλ1 e−λ1 y +qλ2 e−λ2 y , 0 < λ1 < λ2 , p+q = 1, 0 6 p, q 6 1, i.e. to the mixture of two exponentials. 7 This family was introduced in Malinovskii (1998). 8Recall that in the classical model EY = 1/µ and ET = 1/λ. 1 1

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VSEVOLOD K. MALINOVSKII AND KSENIA O. KOSOVA

140 120 100 80 60 40 20 0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure 1. Ruin capital uα,t (c) numerically evaluated using Eq. (2.2) (solid) and simulated (dots) in the case of exponential claim sizes with µ = 3/5 and exponential interclaim times with λ = 4/5. Vertical line: EY1 /ET1 = λ/µ = 4/3. Horizontal line: uα,t (λ/µ) = 59.9033. For 0 < c < 4/3, shown are the upper and lower bounds and for c > 4/3, shown are the upper bounds of Propositions 2.1, 2.2. Here α = 0.05, t = 200.

where the function Uα,t (v) is continuous and monotone increasing, as v increases from −∞ to 0, and monotone decreasing, as v increases from 0 to +∞, and such that9 lim Uα,t (v) = 0,

v→−∞

lim Uα,t (v) = κα

v→+∞

and Uα,t (0) = κα/2 (1 + o(1)), as t → ∞.

For brevity, we set u∗α,t = uα,t (λ/µ). In the classical Lundberg model with exponential claim sizes, we have (see Theorem 3.1 of Malinovskii (2012)) √ 2tλ ∗ uα,t = κα/2 (1 + o(1)), t → ∞. µ Proposition 2.2 (Theorem 3.1 in Malinovskii (2012) and Theorem 3.3 in Malinovskii (2014)). In the classical Lundberg model with exponential claim sizes, as t → ∞, we have  ∗ ∗ c∗α,t ln(αc∗α,t µ/λ) λ  λ − µcα,t + λ ln(αcα,t µ/λ) ∗  (c − c ) + < c 6 c∗α,t , ,  α,t  ∗ ∗ 2 (λ − µcα,t ) λ − µcα,t µ uα,t (c) 6  ln(αcµ/λ)   − , c > c∗α,t , µ − λ/c where c∗α,t = c∗α,t (λ, µ) is a solution (w.r.t. c) of the non-linear equation λ + cµ ³ µc ´ ln α = µu∗α,t − 1. λ − cµ λ

Propositions 2.1 and 2.2 are illustrated in Fig. 1. They strongly enhance the intuition. The proof of Proposition 2.1 relies on the convexity of uα,t (c), as c > λ/µ and t is 9We write κ = Φ−1 (1 − γ) for the (1 − γ)-quantile of the standard normal c.d.f. Φ γ (0,1) (x). Since (0,1)

0 < α < 1/2, we have 0 < κα < κα/2 < 1.

SIMULATION ANALYSIS IN SPARRE ANDERSEN’S MODEL OF RISK

5

sufficiently large. This convexity which is not at all intuitive, was proved in Malinovskii (2014) by checking that d2 uα,t (c) > 0 (2.1) dc2 for t sufficiently large. It allows us to claim that a line tangent to the function uα (c) at the point c∗α,t and starting from the point with abscissa λ/µ and ordinate u∗α,t is an upper bound for uα,t (c), as λ/µ < c < c∗α,t . In the endeavor to write down the derivative in (2.1), the key result was the explicit formula for the probability ψ t (u, c). It allows us to apply the well known theorem on differentiation of the implicit function. To be particular, the explicit formula (see, e.g., Section 2 in Malinovskii (2008)) is ψ t (u, c) = F (uµ, tλ, λ/cµ),

(2.2)

where

Z ∞ X υn τ λ vn+1 (x | ρ) dx n! 0 n=0 √ −(1+1/ρ)x and vn+1 (x | ρ) = ρ(n+1)/2 n+1 In+1 (2x/ ρ). Here In+1 is the modified Bessel x e function of the first kind of order n + 1. The formula (2.2) valid in the classical Lundberg model with exponential claim sizes is quite unique. In more general classical Lundberg model and in Sparre Andersen’s model no such explicit result is known. F (υ, τ, ρ) = e−u

2.2. Ruin capital in a neighborhood of the critical point. We will demonstrate that the analysis of uα,t (c) in the neighborhood of the critical point c = λ/µ is quite uneasy even in the classical case with exponential claim sizes. We draw a particular attention to the fact that the most well known approximations are rather useless for this. To illustrate the nature of the difficulties, for x > 0, ρ > 0 we set (see Section 3 in Malinovskii (2008)) ( ρ−(n+1) vn+1 (x | ρ), 0 < ρ < 1, fn+1 (x, ρ) = vn+1 (x | ρ), ρ>1 R∞ and note that 0 fn+1 (x, ρ) dx = 1 for all ρ > 0. Moreover, ∗(n+1)

fn+1 (x, ρ) = f1

where10 f1 (x, ρ) =

(x, ρ),

( ρ−1 v1 (x | ρ), ρ < 1, v1 (x | ρ),

ρ > 1.

The functions f1 (x, ρ) are p.d.f. of the positive random variables ξ1+ (ρ), as ρ < 1, as ρ > 1, and ξ1◦ , as ρ < 1, i.e. Z x + P{ξ1 (ρ) 6 x} = f1 (y, ρ) dy, ρ < 1, Z0 x P{ξ1− (ρ) 6 x} = f1 (y, ρ) dy, ρ > 1, 0 Z x P{ξ1◦ 6 x} = f1 (y, 1) dy.

ξ1− (ρ),

0

10These functions are known to be p.d.f. of the first hitting moment in one model of a random walk, see Feller (1971), Chapter II, Section 7, Eq. (7.13), (7.14) and Feller (1971), Chapter XIV, Section 6, Eq. (6.6), (6.9).

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VSEVOLOD K. MALINOVSKII AND KSENIA O. KOSOVA

Express F (υ, τ, ρ) through these random variables. It is easy to verify that11  n+1 ∞ o X  υ n ρn+1 n X +   P ξk (ρ) 6 τ e−υ   n!  n=0 k=1   nP o   Nυρ +1 +   = ρe−υ(1−ρ) P ξk (ρ) 6 τ , ρ < 1,  k=1  n+1 ∞ F (υ, τ, ρ) = (2.3) υ +1 o n NX o X υn n X ◦  −υ ◦  e P ξ 6 τ = P ξ 6 τ , ρ = 1,  k k   n!  n=0 k=1 k=1    ∞ υ +1  o n NX o n n n+1 X X  υ  −υ − −  P ξ (ρ) 6 τ = P ξ (ρ) 6 τ , ρ > 1, e  k k  n! n=0 k=1

k=1

where all random summands are i.i.d. The random variables Nυρ and Nυ are Poisson with parameters υρ and υ and independent on these summands. Since12 I1 (x) = ex (2πx)−1/2 (1 + o(1)), as x → ∞, the random variable ξ1◦ has no power moments of order greater than 1/2. This is in contrast to the random variables ξ1+ (ρ) and ξ1− (ρ), for which Eξ1+ (ρ) = ρ/(1 − ρ) and Eξ1− (ρ) = ρ/(ρ − 1), E(ξ1+ (ρ))2 = 2ρ2 /(1 − ρ)3 and E(ξ1− (ρ))2 = 2ρ3 /(ρ − 1)3 , E(ξ1+ (ρ))3 = 6ρ3 (1 + ρ)/(1 − ρ)5 and E(ξ1− (ρ))3 = 6ρ4 (1 + ρ)/(ρ − 1)5 , and so on. This observation is a useful hint. It sheds light on why the famous Cram´er’s approximations hold for c > λ/µ (i.e., for ρ < 1) and c < λ/µ (i.e., for ρ > 1), but not for c = λ/µ (i.e., for ρ = 1). For completeness, we formulate them as Proposition 2.3. Proposition 2.3. In the classical Lundberg model with exponential claim sizes, introduce C = λ/cµ > 0, κ = (cµ/λ − 1)λ/c > 0, m⊕ = 1/(c(cµ/λ − 1)) > 0,

mª = −µ/(λ(cµ/λ − 1)) > 0,

2 D⊕ = 2µ/(λ2 (cµ/λ − 1)3 ) > 0,

2 Dª = −µ(2 + (cµ/λ − 1)2 )/(λ2 (cµ/λ − 1)3 ) > 0.

For 0 < c < λ/µ, we have ¯ sup ¯ψ t (u, c) − Φm t>0

and for c > λ/µ, we have ¯ sup ¯ψ t (u, c) − Ce−κu Φ(m t>0

¯ ¯ = o(1),

u,D 2 u) (t)

ª

ª

u,D 2 u) (t) ⊕ ⊕

u → ∞,

¯ ¯ = o(1),

u → ∞.

It is noteworthy that Proposition 2.3 is straightforward from Eqs. (2.2), (2.3) and from the central limit theorem for the sums with random limit of summation. Various extensions of Proposition 2.3 are known in Sparre Andersen’s risk theory. We do not present them here. We mention only the paper Malinovskii (2000), where approximations for ψ t (u, c), as t → ∞, were considered in the case when the safety loading tends to zero, as t → ∞. In other words, this is the case of c > λ/µ dependent on t and converging with a certain rate to the critical point λ/µ, as t → ∞. 2.3. Checks of simulation results. Once simulations are carried out, a careful check has to be done in order to obtain solid results. The first check is based on the observation13 that (EY1 )2 DT1 + (ET1 )2 DY1 EY1 t+ κα t1/2 + o(t1/2 ), t → ∞. (2.4) uα,t (0) = ET1 (EY1 )3

11We draw attention to the fact that this formula is c.d.f. of a compound Poisson distribution. 12This is well known in the theory of Bessel functions. 13It may be extended to any non-arithmetic and non-degenerated i.i.d. interclaim times and non-

degenerated i.i.d. claim sizes, not obligatory mutually independent.

SIMULATION ANALYSIS IN SPARRE ANDERSEN’S MODEL OF RISK

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Indeed, the paths of the risk reserve process©(1.1) with c = 0 ªare (a.s.) step functions with unit jumps down. Therefore, the set inf 0 u}, and we have (2.5) ψ (u, 0) = P{Vt > u}. ¡ t ¢ √ 14 It is approximated by 1−Φ(0,1) (u−EVt )/ DVt , where EVt = (EY1 /ET1 )t+EY1 (DT1 − (ET1 )2 )/(2(ET1 )2 )+o(1) and DVt = (((EY1 )2 DT1 +(ET1 )2 DY1 )/(EY1 )3 )t+o(t), as t → ∞. For simulations in Sections 5–7, the check based on Eq. (2.4) is reported in Table 2. The second check is based on the observation that uα (c) is the upper bound15 for the ruin capital uα,t (c), as c > EY1 /ET1 . In the framework of Cram´er’s theory, i.e. when the positive solution16 κ = κ(c) of the Cram´er–Lundberg equation Eeκ(Y1 −cT1 ) = 1

(2.6)

exists for c > EY1 /ET1 , we have ψ t (u, c) 6 ψ +∞ (u, c) 6 e for all u > 0. It follows that uα (c) 6 u ˆα (c) = − ln α/κ(c) for all c > EY1 /ET1 . It is noteworthy that in the case of heavy-tailed claim size distribution satisfying certain regularity conditions, we have18 Z ∞ 1 P{Y1 > y} dy, u → ∞, ψ t (u, c) 6 ψ +∞ (u, c) ≈ (c/(λEY1 ) − 1)EY1 u 17

−κ(c)u

i.e. ψ +∞ (u, c) is essentially determined by the tail P{Y1 > y} of the claim size distribution for values of y large.

2.4. Conjectures examined by simulation. The major aim of this paper is to examine the following conjectures. • The shape of uα,t (c) in Sparre Andersen’s model of risk remains similar to its shape in the classical model with exponential claim sizes. In particular, convexity of uα,t (c) for c > EY1 /ET1 holds true. • The function uα,t (c) varies little with certain small changes in the model.

• The shape of uα,t (c) for t small or medium remains the same as it is for t large. We briefly comment on these three conjectures. The first among them is the key issue of this article. The second one is formulated rather vaguely. To be more concrete, by changes in the model we mean taking different but coordinated in a certain way claim size and interclaim time distributions. In particular, interesting is to address the impact of their form, as the mean values and variances are fixed. We do it in Sections 6 and 7. To this end, different heavy-tailed distributions are selected with the same mean values and, when feasible, the same variances. About the third conjecture, it is unclear whether it is true or false even in the case of the classical model with exponential claim sizes. Indeed, Proposition 2.2 has been proven rigorously only for large t. 3. Basic probability distributions Sparre Andersen’s model emphasizes the interclaim times Ti , i = 1, 2, . . . , with c.d.f. n Z t o PT (t) = 1 − exp − g(z)dz , t > 0, 0

14See, e.g., Gut (1988). 15Recall that u (c) is the solution of the equation ψ α +∞ (u, c) = α. 16It is called Lundberg’s exponent or adjustment coefficient. 17See, e.g., Embrechts et al. (1997), Section 1.2. It is well known that ψ −κ(c)u , as +∞ (u, c) ≈ Ce

u → ∞, where 0 < C < 1 is the Cram´ er’s constant. 18See, e.g., Theorem 1.3.6. in Embrechts et al. (1997). We assume here that the claims arrival process is Poisson with parameter λ.

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VSEVOLOD K. MALINOVSKII AND KSENIA O. KOSOVA

where g(z), z > 0, is given. In Sparre Andersen (1957), the function g(z), z > 0, is called contagion rate. In the case of no-contagion, when g(z) ≡ λ > 0, one has PT (t) = 1 − exp{−tλ}, t > 0, and the interclaim times are exponential19. Contagion occurs if g(z) differs from a constant. Recall the following well-known result. Proposition 3.1. Let the c.d.f. PT (t) of the positive random variable T1 be absolutely continuous with respect to Lebesgue measure. There exists a non-negative integrable g(z), z > 0, such that Z 1 Z ∞ g(z)dz < ∞, g(z)dz = ∞ 0

0

and

PT (t) = 1 − exp

n

−

Z

t 0

o g(z)dz ,

t > 0.

(3.1)

Plainly, by taking g(z) ≡ λ > 0 in Eq. (3.1), we have PT (t) exponential. Using Theorem 3.1, it is easy to express the probability density function pT (t) through the contagion rate g(z) and vice versa, i.e. g(z) = and

p (z) R zT , 1 − 0 pT (x) dx

pT (t) = g(t) exp

n

−

Z

0

t

z > 0,

o g(z)dz ,

t > 0.

(3.2)

(3.3)

A powerful tool to get a wide range of distributions is mixing. Proposition 3.2. If the probability density function of a positive random variable T1 is expressed as Z ∞ pT (t) = exp{−tx−1 }x−1 fξ (x) dx, t > 0, (3.4) 0

i.e. as a mixture of exponential probability density functions with mixing function fξ (x), then pT (t) is represented in the form (3.3), where R∞ exp{−tx−1 } x−1 fξ (x) dx g(t) = 0R ∞ . (3.5) exp{−tx−1 } fξ (x) dx 0

If p.d.f. pT (t) is given, then we have the contagion rate g(z) by applying Eq. (3.2). Vice versa, if g(z) is given, then we have p.d.f. pT (t) by applying Eq. (3.3). This is especially useful in the case of exponential mixtures (3.4) since we can use Eq. (3.5) instead of (3.2). Our attention is attracted to several distributions of such a type. Following Malinovskii (1998), besides rather standard light-tailed • mixture of a finite number of exponential distributions, • Erlang distribution, we address heavy-tailed • Pareto distribution, or a gamma mixture of exponential distributions20, • Kummer distribution, or a Fisher–Snedecor mixture of exponential distributions.

19This characteristic feature of the exponential law is called “lack of memory” or “lack of ageing”: no matter which is the present age, the residual lifetime is unaffected by the past and has the same distribution as the lifetime itself. 20This fact is well known in statistics (see, e.g., Harris (1968)).

SIMULATION ANALYSIS IN SPARRE ANDERSEN’S MODEL OF RISK

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We recall two definitions. First, for a > 1 and b > 0, we set21 −b

fξ (z | a, b) = (Γ(a)ba )−1 z −(a+1) e−z ,

ga,b (z) = ab(zb + 1)−1 ,

z > 0.

We have22 (a) (b)

pT (t) = pT (t) =

ab , a+1 Z(tb + 1)

t > 0,

∞

exp{−tz −1 }z −1 fξ (z | a, b) dz, t > 0, ¾ ½ Z t ga,b (z) dz , t > 0. pT (t) = ga,b (t) exp − 0

(c)

0

This is called Pareto distribution. Second, for d > 2, l > 0 and for23 ¡ ¢ Γ d+l d+l d l d fξ (z | d, l) = ¡ d ¢ 2 ¡ l ¢ d 2 l 2 z 2 −1 (l + dz)− 2 , z > 0, Γ 2 Γ 2 ¡ ¢ d U 1 + 2l , 2 − d2 , dl z ¡ ¢ , z > 0, gd,l (z) = 2 U 2l , 1 − d2 , dl z R∞ where24 U (a, b, z) = Γ(a)−1 0 e−zt ta−1 (1 + t)b−a−1 dt is a confluent hypergeometric, or Kummer function, we have ¢ µ ¡ ¶ l d Γ d+l d d 2 ¡ ¢ U 1 + , 2 − , t , t > 0, (a) pT (t) = 2 Γ d 2 2 l Z ∞ 2 −1 −1 exp{−tz }z fξ (z | d, l) dz, t > 0, (b) pT (t) = 0 n Z t o (c) pT (t) = gd,l (t) exp − gd,l (z) dz , t > 0. 0

This is called (see Malinovskii (1998)) Kummer distribution. The difference between heavy-tailed and light-tailed interclaim times may be illustrated in terms of contagion rate g(z). Note that for k = 1, 2, . . . Z ∞ n Z t o k k t exp − g(z) dz g(t) dt. (3.6) E(T1 ) = 0

Rt

0

If 0 g(z)dz = O(t ) for any s > 0, then the exponential factor in Eq. (3.6) decreases to zero and dominates the rest, whatever R tk may be selected: the exponential function grows faster than any power function. If 0 g(z)dz = O(ln t), then the exponential factor reduces to a power. Therefore, it is possible to choose k so large that the integral (3.6) becomes divergent. It is easily verified by direct calculation25 that both contagion rates ga,b (z), gd,l (z) are of order O(z −1 ), as z → +∞. Consequently, both integrals Z t Z t ga,b (z) dz and gd,l (z) dz s

0

0

21This f (z | a, b) is p.d.f. of inverted Gaussian distribution. ξ 22We consider positive Pareto random variables. The transform z = t + z , z = 1/b > 0 converts 0 0

the expression in the right hand of (a) into the standard form of Pareto probability density function (see, e.g., Cramer (1957), 19.3.2), i.e. to pT (z) = za ( zz0 )a+1 , as z > z0 , and to 0, as z 6 z0 . 0 23Note that the mixing function f (z | d, l) is p.d.f. of Fisher–Snedecor distribution. ξ 24See, e.g., Eq. (13.2.5) in Abramowitz and Stegun (1972). 25Use Eq. (13.1.8) in Abramowitz and Stegun (1972) and observe that U (a, b, z) = z −a (1+O(z −1 )), as z → +∞.

10

VSEVOLOD K. MALINOVSKII AND KSENIA O. KOSOVA

Table 1. Theoretical and simulated tail probabilities P {ξ > x} for Pareto ξ

with parameters a = 3, b = 2/5 and Kummer ξ with parameters d = 4, l = 10 (in both cases Eξ = 5/4 = 1.25 and Dξ = 75/16 = 4.6875; each simulation uses 100 000 pseudo-random numbers).

x

Kummer theoretical

Kummer simulated

Pareto theoretical

Pareto simulated

5 10 20

0.044 0.00997 0.001436

0.044 0.0098 0.00118

0.037 0.008 0.00137

0.0376 0.0086 0.00147

are of order O(ln t), as t → ∞. So, both Pareto and Kummer distributions are heavytailed. It suggests that both corresponding renewal claims arrival processes behave quite unlike the Poisson process. Comparing these two heavy-tailed distributions, it is useful to note that the choice a = 3, b = 2/5 for Pareto and d = 4, l = 10 for Kummer distributions yields the same expectations and variances. The former equals to 5/4, the latter equals to 75/16. Plainly, the Pareto distribution with such parameters has only moments up to the third order inclusive, while the Kummer distribution has moments up to the fifth order inclusive. So, the tails of the former distribution are heavier than the tails of the latter one. 4. Algorithm of ruin capital’s numerical simulation The starting point for the entire simulation process is the standard (see Knuth (1981)) linear congruence random number generator26. It is based on the equation xn+1 = (kxn + a)

mod m,

where k = 23456789 is the multiplier, a = 22185 is the increment, and m = 232 is the modulus. The initial seed x0 was selected using a build-in Maple procedure. Each successive term is transformed into the next. The pseudo-random terms are in the range from 1 to m − 1. To get floating point numbers between 0 and 1, a floating point division by m was done. It is known that the quality, i.e. the matching of the numbers thus produced to a sample from the uniform distribution, depends heavily on k and m. Using this uniform pseudo-random number generator, the Pareto27 pseudo-random numbers were obtained using the direct method of inverse transforms (see, e.g., Devroye (1986)). Bearing in mind that the Kummer c.d.f. is a mixture of exponentials with the Fisher–Snedecor mixing distribution, each Kummer pseudo-random number was obtained by means of a two-step procedure called sometimes method of decomposition. First, Fisher–Snedecor pseudo-random number z was obtained using a build-in Maple procedure. Second, exponential with parameter 1/z pseudo-random number was obtained using the method of inverse transforms. Comprehensive testing of the quality of thus obtained Pareto and Kummer pseudorandom numbers lies outside the scope of this paper. However, bearing in mind the common warning (see, e.g., Hellekalek (1998)) that every type of generator has its unwanted side-effects and there is no safe generators, and wishing to be on the safe side, we made some simplistic tests of fit. 26 Though presumably some built-in pseudo-random number generators implemented in most standard symbolic computation packages such as Maple may be in some cases superior to that pseudo-random number generator, we use it to avoid “black boxes” in the description of the algorithm. 27 Here (see Section 3) c.d.f. PT (t) = 1 − (1 + bt)−a , t > 0, is meant.

SIMULATION ANALYSIS IN SPARRE ANDERSEN’S MODEL OF RISK

11

1.0

0.8

0.6

0.4

0.2

0.0 0

2

4

6

8

10

Figure 2. Solid line: Pareto c.d.f. PT P (t) with parameters a = 3, b = 2/5. N −1 Dotted line: the values of PN (t) = N i=1 1{ξi
0.8

0.6

0.4

0.2

0.0 0

2

4

6

8

10

Figure 3. Solid line: Kummer c.d.f. PP T (t) with parameters d = 4, l = 10. N −1 Dotted line: the values of PN (t) = N i=1 1{ξi
PN In Fig. 2 and Fig. 3 we draw28 PN (t) = N −1 i=1 1{ξi x} for Pareto ξ with parameters a = 3, b = 2/5 and Kummer ξ with parameters d = 4, l = 10 (in both cases the mean is 5/4 and variance is 75/16) with the empirical values. To calculate each of them, 100 000 pseudo-random numbers were taken. The fit demonstrated in Table 1 is sensibly well. 28Evidently, P (t) is set similar to the empirical distribution function well known in statistics. N

12

VSEVOLOD K. MALINOVSKII AND KSENIA O. KOSOVA

80

60

40

20

0 0

50

100

150

Figure 4. Five simulated trajectories (X-axis is time; claim sizes are exponential with µ = 3/5, interclaim times are exponential with λ = 4/5) from a bundle used for calculation of ψ t (uj , ci ) with ci = 3/2, uj = 39. One of these trajectories falls below zero. The point ci = 3/2, uj = 39 is such that 0.045 = α − ε 6 ψ t (uj , ci ) 6 α + ε = 0.055. Here t = 200, α = 0.05, ε = 0.005. 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

10

20

30

40

50

60

70

Figure 5. The function ψ t (u, ci ), as ci = 3/2 (X-axis is u; claim sizes are exponential with µ = 3/5, interclaim times are exponential with λ = 4/5), calculated numerically using Eq. (2.2) (solid) and simulated (dots) for uj from the lattice U. Horizontal lines: α + ε = 0.055, α − ε = 0.045. Here t = 200, α = 0.05, ε = 0.005.

To evaluate the ruin capital uα,t (c) using numerical simulation, we address the interval [0, cmax ] on the abscissa axis, where cmax > EY1 /ET1 is sufficiently large. We introduce the lattice C = {ci , i = 0, 1, . . . , nC } with the span ∆c > 0, i.e. put c0 = 0 and ci = ci−1 + ∆c, i = 1, 2, . . . , [cmax /∆c] + 1. Second, we address the interval [0, umax ] on the ordinate axis, choosing umax close to uα,t (0) (see (2.4)). We introduce the lattice U = {uj , j = 0, 1, . . . , nU }

SIMULATION ANALYSIS IN SPARRE ANDERSEN’S MODEL OF RISK

13

8

6

4

2

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Figure 6. Three p.d.f. pT (t) for mixtures of two exponentials. Stretched along X-axis the most corresponds to Case 1. Adjacent to Y -axis the most corresponds to Case 3. Vertical line: ET1 = 5/4.

with the span ∆u > 0, i.e. put u0 = 0 and ui = ui−1 + ∆u, i = 1, 2, . . . , [umax /∆u] + 1. Starting with c0 , we iterate through the nodes of C. Dealing with the node ci , we simulate the values ψ t (uj , ci ), j = 0, 1, . . . , nU on the basis of the definition of the ruin probability. Namely, for each j we simulate the bundle (see Fig. 4) consisting of N trajectories of the risk reserve process (1.1) with premium rate ci and initial risk reserve uj . Then we pick up ψ t (uji , ci ) closest to α and declare uji a solution of Eq. (1.2) in the node ci . To control the accuracy of simulation, we are always seeking to have this value lying inside the strip [α − ε, α + ε]. Here ε > 0 is a sufficiently small preselected value, the same for the entire simulation process. Plainly, the values ∆c, ∆u and the number N used to achieve such accuracy when ψ t (uj , ci ), j = 0, 1, . . . , nU , are simulated, are all dependent on ε. They must be so that at least one simulated point falls (see Fig. 5) within the strip [α − ε, α + ε]. The end-product of the simulation algorithm in the case of classical risk model is shown in Fig. 1. Here ε = 0.005, cmax = 40, umax = 400, ∆c = 0.1, ∆u = 1, N = 1000. This algorithm, which starts from iterating through ci ∈ C, i = 0, 1, . . . , nC , can be modified in order to avoid unnecessary calculations. It refers to ψ t (uj , ci ) for such uj in the lattice U, for which it is a priori seen that respective points do not fall into the target strip [α + ε, α − ε]. Sensible as well is to shift the bundle of trajectories corresponding to ci by altering u in order to make the ratio of the number of trajectories fallen below zero to the total number of trajectories equal to α. This modification of the algorithm may yield a considerable gain in computation time. Being not focused in this paper on the optimality of the computational procedure, we leave it aside. 5. Simulated ruin capital in semi-classical models In the classical model with exponential claim sizes, we have illustrated in Fig. 1 both analytical results of Propositions 2.1, 2.2 and the results of simulation analysis described in Section 4. Now we address semi-classical models where the simulation analysis requires a little more efforts, while the analytical results are absent.

14

VSEVOLOD K. MALINOVSKII AND KSENIA O. KOSOVA

5.1. Mixture of two exponentials. We start with the case when Y1 is exponential with parameter µ > 0 and T1 is a mixture of two exponentials. For 0 < λ1 < λ2 < ∞ and for p, q such that p + q = 1, 0 6 p, q 6 1, we set pλ1 e−λ1 z + qλ2 e−λ2 z , z > 0. pe−λ1 z + qe−λ2 z For this contagion rate, it is easily seen that o n Z t g(z)dz = 1 − pe−λ1 t − qe−λ2 t , 1 − exp − g(z) =

(5.1)

0

so that

PT (t) = 1 − pe−λ1 t − qe−λ2 t , t > 0, (5.2) and 0, as t 6 0. Plainly, it is a mixture of two exponentials. Note that if λ1 = λ2 , then there is no contagion. Note also that ET1 = p/λ1 + q/λ2 , DT1 = (qλ21 + pλ22 + pq(λ1 − λ2 )2 )/(λ21 λ22 ), ET13 = 6p/λ31 + 6q/λ32 . The Cram´er–Lundberg equation (2.6) can be written in the form c2 κ 2 − (µc2 − (λ1 + λ2 )c)κ − (pλ2 + qλ1 )µc + λ1 λ2 = 0.

For c > EY1 /ET1 , the adjustment coefficient29 κ is µ λ1 + λ2 1p 2 2 κ= − + µ c + 2µc(λ1 − λ2 )(q − p) + (λ1 − λ2 )2 . 2 2c 2c Taking Y1 exponential with µ = 3/5 and T1 with c.d.f. PT (t) set in (5.2), we present the typical simulation results for the ruin capital uα,t (c). We pick up three sets of parameters in (5.2) which yield the same mean ET1 = 5/4, as follows: Case 1: λ1 = 0.16, λ2 = 1.6, and p = 0.1111 which yields DT1 = 7.8125, Case 2: λ1 = 0.14, λ2 = 10.0, and p = 0.1633 which yields DT1 = 15.1161, Case 3: λ1 = 0.10, λ2 = 56.8, and p = 0.123457 which yields DT1 = 23.1294. The most dangerous of all three cases in the sense of ruin is Case 3. Indeed, when p.d.f. pT (t) is the most closely adjacent to Y -axis (see Fig. 6), short interclaim intervals are most likely to appear. So, cumulative claims largest of all three cases are most likely to appear within a finite time interval. The simulated ruin capital uα,t (c) in Case 3 is shown in Fig. 8 by the upper line (squares). The middle line (crosses) is the ruin capital in Case 2. The lower line (dots) is the ruin capital in Case 1. The solid lines show the respective upper bounds u ˆα,t (c), as c > EY1 /ET1 = 4/3. Simulations reported in Fig. 7 and Fig. 8 yield evidence30 supporting the conjecture that the shape of the ruin capital uα,t (c) remains similar (cf. Fig. 1) to its shape in the classical model with exponential claim sizes. 5.2. Erlang. We proceed with the case when Y1 is exponential with parameter µ > 0 and T1 is Erlang with parameters λ > 0 and k integer. Recall that the Erlang p.d.f. is defined by the equation λk tk−1 , t > 0, (5.3) Γ(k) and is 0, as t 6 0. It is a particular case of the Gamma p.d.f. The mean of T1 with this Erlang distribution is ET1 = k/λ and the variance is DT1 = k/λ2 . pT (t) = e−tλ

29Recall that κ is a positive solution of Cram´ er–Lundberg equation.

Bearing in mind that EY1 /ET1 = λ1 λ2 /(µ(qλ1 + pλ2 )), the inequality κ > 0 is yielded by direct algebra. 30Because of limited space, we do not show here a number of similar results yielding more evidence.

SIMULATION ANALYSIS IN SPARRE ANDERSEN’S MODEL OF RISK

300

15

+ ++ + + + + + + + + + + + + ++ +++++ ++++++++++++++++++++++++++++++++++++++

250 200 150 100 50 0 0

1

2

3

4

5

6

Figure 7. Simulated ruin capital uα,t (c) (dots and crosses) and numerically evaluated upper bound u ˆα,t (c) (solid) in the case of exponential claim sizes with µ = 3/5 and mixture of two exponentials interclaim times (see Eq. (5.2)) with (dots) λ1 = 0.232575, λ2 = 4.43839, p = 0.251496 which yields ET1 = 5/4, DT1 = 7.8125, ET 3 = 120; with (crosses) λ1 = 0.00637369, λ2 = 0.81306, p = 0.00012898 which yields ET1 = 5/4, DT1 = 7.8125, ET 3 = 3000. Vertical line: EY1 /ET1 = 4/3. Horizontal lines: simulated uα,t (EY1 /ET1 ), see Table 2. à 350 + æ

300 250 200 150 100 50

à + æà + æ à + æà + æà +à æ+à æ+à æ+ à æ+à à æ+ à + à æ + à æ + à à à æ + à à æ + + àààà æ àà ++ + æ ++++ à à à à à à æ æ + +++++ à à à à à à à à à à à à à à à à à à à à à à à à à à à à æ ææ +++++++++++++ ææææ +++++++++++++++++ ææææ æææææææ ææææææææææææææææææææææææ

0 0

1

2

3

4

5

6

Figure 8. Simulated ruin capital uα,t (c) (squares, crosses, and dots) and numerically evaluated upper bound u ˆα,t (c) (solid) for exponential claim sizes with µ = 3/5 and mixture of two exponentials interclaim times (see Eq. (5.2)) with (squares) λ1 = 0.1, λ2 = 56.8, p = 0.123457 which yields ET1 = 5/4, DT1 = 23.1294; with (crosses) λ1 = 0.14, λ2 = 10, p = 0.1633 which yields ET1 = 5/4, DT1 = 15.1161; with (dots) λ1 = 0.16, λ2 = 1.6, p = 1/9 which yields ET1 = 5/4, DT1 = 7.8125. Vertical line: EY1 /ET1 = 4/3. Horizontal lines: simulated uα,t (EY1 /ET1 ), see Table 2.

In Fig. 9, the lines in dots and crosses show the simulated ruin capital uα,t (c) for exponential claim sizes with parameter µ = 3/5 and Erlang interclaim times. The upper line (dots) corresponds to the case λ = 8/5, k = 2 which yields ET1 = 5/4, DT1 = 25/32. The lower line (crosses) corresponds to the case λ = 320, k = 400 which yields ET1 = 5/4,

16

VSEVOLOD K. MALINOVSKII AND KSENIA O. KOSOVA

300 +

+ +

250

+ +

200

+ + +

150

+ + +

100

+ +

50

+

0 0.0

0.5

1.0

+

+++ 1.5

++++++++++++ 2.0

2.5

3.0

Figure 9. Simulated ruin capital uα,t (c) (dots and crosses) and numerically evaluated upper bound u ˆα,t (c) (solid) for exponential claim sizes with µ = 3/5 and Erlang interclaim times with (dots) λ = 8/5, k = 2 which yields ET1 = 5/4, DT1 = 25/32; with (crosses) λ = 320, k = 400 which yields ET1 = 5/4, DT1 = 1/256. Vertical line: EY1 /ET1 = 4/3. Horizontal lines: simulated uα,t (EY1 /ET1 ), see Table 2.

DT1 = 1/256. The solid lines show u ˆα,t (c) which are the upper bounds for uα,t (c), as c > EY1 /ET1 = 4/3. Note that in both cases EY1 = 5/3, ET1 = 5/4, while the variances are different. 6. Simulated ruin capital for non-Poisson claims’ arrival In this section, staying within the framework of Cram´er’s theory, we address Pareto and Kummer interclaim times discussed in Section 3, which are markedly different from exponential. Being two-parametric, they are heavy-tailed and mixtures of exponential distributions. The claim sizes are exponential. Table 2. Simulated uα,t (EY1 /ET1 ) and results of check at c = 0.

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

7 (dots) 7 (crosses) 8 (squares) 8 (crosses) 8 (dots) 9 (dots) 9 (crosses) 10 (dots) 10 (crosses) 11 (dots) 11 (crosses) 12 (dots) 12 (crosses) 13 (dots) 13 (crosses)

Theoretical uα,t (0)

Simulated uα,t (0)

Simulated uα,t (EY1 /ET1 )

351.6065 351.6065 404.5155 378.8388 351.6065 309.1366 301.3865 336.0198 318.6815 378.4339 336.0198 424.9087 336.0198 367.1689 191.6377

341 319 355 349 345 309 298 331 318 379 333 421 332 333 190

101 63 151 129 97 48 39 76 61 78 78 80 80 102 36

SIMULATION ANALYSIS IN SPARRE ANDERSEN’S MODEL OF RISK

300

17

+ + + +

250

+ +

200

+ + +

150

+ + +

100

+ +

50

+

+

++

0 0

++ + +++ ++ +++++++

1

2

+++++++

3

Figure 10. Simulated ruin capital uα,t (c) (dots and crosses) and numerically evaluated upper bound u ˆα,t (c) (solid) for exponential claim sizes with µ = 3/5 and Pareto interclaim times with (dots) a = 3, b = 2/5 which yields ET1 = 1.25, DT1 = 4.6875; with (crosses) a = 10, b = 4/45 which yields ET1 = 1.25, DT1 = 1.9531. Vertical line: EY1 /ET1 = 4/3. Horizontal lines: simulated uα,t (EY1 /ET1 ), see Table 2. + +

300

+ +

250

+

+ +

200

+ + +

150

+ +

100

+ +

50

+

+

++

+++

0 0

1

2

+ ++ ++++ +++

++++++++ +

3

Figure 11. Simulated ruin capital uα,t (c) (dots and crosses) and numerically evaluated upper bound u ˆα,t (c) (solid) for exponential claim sizes with µ = 3/5 and Kummer interclaim times with (dots) d = 4, l = 30 which yields ET1 = 15/14, DT1 = 2.56083; with (crosses) d = 4, l = 10 which yields ET1 = 1.25, DT1 = 4.6875. Vertical lines: EY1 /ET1 = 1.5556 (dots case) and EY1 /ET1 = 1.3333 (crosses case). Horizontal lines: simulated uα,t (EY1 /ET1 ), see Table 2.

6.1. Pareto. In Fig. 10, shown are the simulated ruin capital uα,t (c) for exponential claim sizes with parameter µ = 3/5 and Pareto interclaim times. The upper line (dots) corresponds to the case a = 3, b = 2/5 which yields ET1 = 1.25, DT1 = 4.6875. The lower line (crosses) corresponds to the case a = 10, b = 4/45 which yields ET1 = 1.25, DT1 = 1.9531.

18

VSEVOLOD K. MALINOVSKII AND KSENIA O. KOSOVA

Table 3. Adjustment coefficients κ = κ(c) used to calculate uˆα (c) = − ln α/κ(c) for Pareto T1 (Fig. 10) and Kummer T1 (Fig. 11).

Pareto (dots) Pareto (crosses) Kummer (dots) Kummer (crosses)

c = 1.6

1.8

2.0

2.4

2.8

3.2

3.6

4.0

0.0627 0.0909 0.0104 0.0568

0.1048 0.1430 0.0524 0.0938

0.1416 0.1856 0.0935 0.1261

0.2017 0.2507 0.1510 0.1794

0.2482 0.2981 0.1980 0.2215

0.2852 0.3342 0.2356 0.2555

0.3152 0.3626 0.2664 0.2836

0.3400 0.3855 0.2922 0.3071

The solid lines show u ˆα,t (c) which are the upper bounds for uα,t (c), as c > EY1 /ET1 = 4/3. Note that in both cases EY1 = 5/3, ET1 = 5/4, while the variances are different. Note also (see Section 2.3 and Table 2) that uα,t (0) calculated by using Eq. (2.4) are 336.0198 and 318.6815. The simulated values are 331 and 318 respectively, which is a rather good fit. 6.2. Kummer. In Fig. 11, shown are the simulated ruin capital uα,t (c) for exponential claim sizes with parameter µ = 3/5 and Kummer interclaim times. The upper line (dots) corresponds to the case d = 4, l = 30 which yields ET1 = 15/14, DT1 = 2.5608. The lower line (crosses) corresponds to the case d = 4, l = 10 which yields ET1 = 1.25, DT1 = 4.6875. Note that, unlike in Figs. 7–10, the critical points in Fig. 11 are different: c = EY1 /ET1 = 14/9 = 1.5556 (dots case) and c = EY1 /ET1 = 4/3 = 1.3333 (crosses case). The solid lines show u ˆα,t (c) which are the upper bounds for uα,t (c) on the right of the critical points. Note also (see Section 2.3 and Table 2) that the values for uα,t (0) calculated by using Eq. (2.4) are 378.4339 and 336.0198. The simulated values are 379 and 333 respectively, which is a rather good fit. 7. Simulated ruin capital for non-exponential claims In this section, we address Pareto and Kummer claim sizes which are heavy-tailed and markedly different from exponential. The interclaim times are exponential, and the claims arrival process is Poisson. 7.1. Pareto. In Fig. 12, shown are the simulated ruin capitals uα,t (c) for exponential interclaim times with parameter λ = 4/5 and Pareto claim sizes with (dots) a = 10, b = 0.05 which yields EY1 = 20/9 = 2.2222, DY1 = 6.1728, and with (crosses) a = 3, b = 0.3 which yields EY1 = 5/3 = 1.6667, DY1 = 8.3333. Vertical lines are EY1 /ET1 = 16/9 = 1.7778 (dots case) and EY1 /ET1 = 4/3 = 1.3333 (crosses case). The simulated values of uα,t (0) are 421 (dots case) and 332 (crosses case), which is sensibly close to the theoretical values which are 424.9087 and 336.0198 respectively. The simulated values of uα,t (EY1 /ET1 ) in both cases are 80. 7.2. Kummer. In Fig. 13, shown are the simulated ruin capitals uα,t (c) for exponential interclaim times with parameter λ = 4/5 and Kummer claim sizes with (dots) d = 5, l = 5 which yields EY1 = 5/3 = 1.6667, DY1 = 20.5556, and with (crosses) d = 200, l = 200 which yields EY1 = 100/99 = 1.0101, DY1 = 1.0617. Vertical lines are EY1 /ET1 = 4/3 = 1.3333 (dots case) and EY1 /ET1 = 80/99 = 0.8081 (crosses case). The simulated values of uα,t (0) are 333 (dots case) and 190 (crosses case), which is sensibly close to the theoretical values which are 367.1689 and 191.6377 respectively. The simulated values of uα,t (EY1 /ET1 ) are 102 and 36.

References

19

+ +

300

+ + +

250

+ +

200

+ +

150

+

+ +

100

+

+ +

50 0 0

1

++

+++

++++ 2

++ ++ ++ +++++++ +++ 3

Figure 12. Simulated ruin capital uα,t (c) (dots and crosses) for exponential interclaim times with λ = 4/5 and Pareto claim sizes with (dots) a = 10, b = 0.05 which yields EY1 = 2.2222, DY1 = 6.1728; with (crosses) a = 3, b = 0.3 which yields EY1 = 1.6667, DY1 = 8.3333. Vertical lines: EY1 /ET1 = 1.7778 (dots case) and EY1 /ET1 = 1.3333 (crosses case). Horizontal lines: simulated uα,t (EY1 /ET1 ), see Table 2. 350 300 250 200

+ +

150

+ + +

100

+ + +

50 0 0

+

++ 1

++ ++ +++++++ ++++++++++++++++++ 2

3

Figure 13. Simulated ruin capital uα,t (c) (dots and crosses) for exponential interclaim times with λ = 4/5 and Kummer claim sizes with (dots) d = 5, l = 5 which yields EY1 = 1.6667, DY1 = 20.5556; with (crosses) d = 200, l = 200 which yields EY1 = 1.0101, DY1 = 1.0617. Vertical lines: EY1 /ET1 = 1.3333 (dots case) and EY1 /ET1 = 0.8081 (crosses case). Horizontal lines: simulated uα,t (EY1 /ET1 ), see Table 2.

References Abramowitz, M., Stegun, I.A. (1972) Handbook of Mathematical Functions, 10-th ed., Dover, New York. Andersen, E. Sparre (1957) On the collective theory of risk in case of contagion between the claims. In book: Transactions of the 15-th International Congress of Actuaries, New York 1957, vol. 2, 219–229.

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References

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