A RUIN MODEL WITH DEPENDENCE BETWEEN CLAIM SIZES AND CLAIM INTERVALS

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1 A RUIN MODEL WITH DEPENDENCE BETWEEN CLAIM SIZES AND CLAIM INTERVALS Hansjörg Albreher a, Onno J. Boxma b a Graz University of Tehnology, Steyrergasse 3, A-8 Graz, Austria b Eindhoven University of Tehnology and EURANDOM, P.O. Box 53, 56 MB Eindhoven, The Netherlands Abstrat We onsider a generalization of the lassial ruin model to a dependent setting, where the distribution of the time between two laim ourrenes depends on the previous laim size. Exat analytial expressions for the Laplae transform of the ruin funtion are derived. The results are illustrated by several examples. Introdution The lassial Cramer-Lundberg model to desribe the surplus proess of an insurane portfolio relies on the assumption of independene among laim sizes and between laim sizes and laim inter-ourrene times. However, in pratie this assumption is often too restritive and there is a need for more general models where the independene assumptions an be relaxed. Reently, various results have been obtained onerning the asymptoti behaviour of the probability of ruin for dependent laims. In the ase of light-tailed laim sizes, Nyrhinen 2, 3 derived Lundberg-type limiting results using large deviations tehniques and Müller and Pflug introdued dependene orderings to relate the limiting ruin probabilities. The behaviour of the Lundberg exponent as a funtion of a dependene measure has been investigated in Albreher and Kantor 2. For heavy-tailed laim size distributions, the asymptoti behaviour of the ruin probability with dependent laims was studied e.g. in Asmussen et al. 5 and Mikosh and Samorodnitsky 9,. However, all these results are of asymptoti nature and it is a hallenging problem to obtain results on the probability of ruin in a dependent setting, also for smaller values of the initial apital. Motivated by a related model in queueing theory (f. Boxma and Perry 6), in this paper a generalization of the lassial ruin model is onsidered, where the distribution of the time between two laim ourrenes depends on the previous laim size. presented at the 7th International Congress on Insurane: Mathematis & Eonomis in Lyon, June 25-27, 23 Supported by the Researh Counil of the K.U. Leuven and the Austrian Siene Foundation Projet S-838-MAT

2 For this speifi dependent model, we derive exat solutions for the probability of survival by means of Laplae-Stieltjes transforms. This seems to be the first exat formula for the ruin probability in a ontinuous-time risk model allowing for dependeny and thus should be viewed as a starting point for deriving analytial solutions in more general dependent senarios. For example, we would like to onsider (i) more general laim inter-ourrene distributions, and (ii) situations in whih the laim sizes and laim inter-ourrene times depend on a ommon Markov hain (f., 8). The paper is organized in the following way: In Setion 2 we introdue the risk model and derive the exat expressions for the probability of survival. In Setion 3, several related models that allow for a similar treatment are disussed. Setion 4 ontains some numerial illustrations and investigates the effet of ignoring the dependene struture. 2 The Model Let us onsider the following risk model for the surplus proess R(t) of an insurane portfolio: N(t) R(t) = x + t B j, where x is the initial apital, is the premium density whih is assumed to be onstant, B j is the size of the jth laim and N(t) is the number of laims up to time t. Let B i be a sequene of i.i.d. random variables with distribution funtion B( ), mean β and Laplae-Stieltjes transform (LST) b( ). We assume the laim ourrene proess to be of the following Markovian type: If a laim B i is larger than a threshold T i, then the time until the next laim is exponentially distributed with rate λ, otherwise it is exponentially distributed with rate λ 2. The quantities T i are assumed to be i.i.d. random variables with distribution funtion T ( ). In the sequel, B (T ) shall denote a generi laim size (threshold) with distribution B( ) (T ( )). 2. Exat Solutions We are interested in the probability of survival φ(x), i.e. P(R(t) t > R() = x). Let us assume that j= β < P(B > T ) λ + P(B T ) λ 2, () whih is the net profit ondition, and P(B > ) = P(T > ) =. Let φ i (x) (i =, 2) denote the probability of survival with initial apital x given that the first laim ours aording to the exponential distribution with rate λ i. Then we get 2

3 φ (x) = ( λ dt)φ (x + dt)+ x+ dt + λ dt P(T y)φ (x + dt y) + P(T > y)φ 2 (x + dt y) db(y). Taylor expansion and rearranging yields dφ x dx (x) λ φ (x) + λ Similarly we obtain dφ x 2 dx (x) λ 2φ 2 (x) + λ 2 Define, for Re s : P(T y)φ (x y)db(y)+ + λ x P(T y)φ (x y)db(y)+ χ (s) := Ee sb (B>T ) = χ 2 (s) := Ee sb (B T ) = + λ 2 x x= and denote the Laplae transform of φ i (x) by Note that χ (s) + χ 2 (s) = b(s). φ i (s) := x= P(T > y)φ 2 (x y)db(y) =. (2) P(T > y)φ 2 (x y)db(y) =. (3) e sx T (x)db(x), e sx ( T (x))db(x), e sx φ i (x) dx. From (2) and (3) it follows that for Re s we have φ (s) s λ + λ χ (s) + λ φ2 (s)χ 2 (s) = φ (+), φ 2 (s) s λ 2 + λ 2 χ 2 (s) + λ 2 φ (s)χ (s) = φ 2 (+), whih an further be simplified to 3

4 φ (+) s λ 2 + λ 2 χ 2 (s) λ χ 2 (s)φ 2 (+) φ (s) = s λ + λ χ (s) s λ 2 + λ 2 χ 2 (s) λ λ 2 χ (s)χ 2 (s) (4) and φ 2 (s) = φ 2 (+) s λ + λ χ (s) λ 2 χ (s)φ (+). (5) s λ + λ χ (s) s λ 2 + λ 2 χ 2 (s) λ λ 2 χ (s)χ 2 (s) Note that the denominators on the right-hand side of (4) and (5) oinide. Remark: If we set λ = λ 2 := λ in (4) we obtain φ (s) = φ (+) s λ + λχ 2 (s) λχ 2 (s)φ (+) ( s λ + λχ (s))( s λ + λχ 2 (s)) λ 2 χ (s)χ 2 (s) = φ (+) s λ + λ b(s), and thus we retain the lassial Pollazek-Khinthine formula for the independent setting. For omplete solution we now need to determine the quantities φ i (+). Sine lim φ i(x) = we have x lim s φ i (s) = (i =, 2). (6) s Using (6) w.l.o.g. in (4) (equation (5) would lead to the same result), we obtain = lim s ( s ) φ (+) s λ 2 + λ 2 χ 2 (s) λ χ 2 (s)φ 2 (+) s λ + λ χ (s) s λ 2 + λ 2 χ 2 (s) λ λ 2 χ (s)χ 2 (s) = λ 2φ (+)( + χ 2 ()) λ χ 2 ()φ 2 (+) = lim s ( s λ +λ χ (s)) ( s λ 2 +λ 2 χ 2 (s)) λ λ 2 χ (s)χ 2 (s) s λ 2 φ (+)( + χ 2 ()) λ χ 2 ()φ 2 (+). (7) λ (χ () ) + λ 2 (χ 2 () ) λ λ 2 (χ () + χ 2()) Now we an use the relations χ 2 () = P(B T ), χ () = P(B > T ) and thus χ () + χ 2 () = and also E(B (B T ) ) = χ 2(), E(B (B>T ) ) = χ () and β = χ () χ 2(). In this way (7) an be substantially simplified yielding ( φ (+)) P(B > T ) λ + ( φ 2 (+)) P(B T ) λ 2 = β. (8) 4

5 Remark: For the speial ase λ = λ 2 := λ we obtain from (8) φ (+) = φ 2 (+) = λβ, (9) whih is the well-known formula for the survival probability with zero initial apital in the lassial independent ase. We now need a seond equation for φ (+) and φ 2 (+). Using Rouhé s theorem, one an show the following: Lemma. The denominator of (4) has exatly one zero σ with Re σ >. Proof. Rewrite the denominator of (4) and (5) as s(h (s) + h 2 (s)), in whih h (s) := s λ λ 2, h 2 (s) := λ χ (s) + λ 2 χ 2 (s) + λ λ 2 β b(s). βs We wish to show that this denominator has exatly one zero for Re s > ; note that the behaviour of φ i (s) at s = has already been analysed and exploited in (7). Let us now apply Rouhé s theorem to the losed ontour C, onsisting of the imaginary axis from ir to +ir and a semi-irle in the right halfplane with radius r and origin O; we shall let r. h (s) and h 2 (s) are analyti inside C; notie that b(s) βs is the LST of x B(y) dy whih is the residual (forward reurrene) laim β size distribution. Hene it is analyti and (as will be used below) bounded by one in absolute value in the right halfplane. Furthermore, h (s) has exatly one zero inside C for r large enough. For the appliation of Rouhé s theorem it remains to show that h (s) > h 2 (s) on C. This is learly true on the semi-irle. On the imaginary axis, h (s) λ + λ 2, whereas, under the ondition (), h 2 (s) λ χ () + λ 2 χ 2 () + λ λ 2 β < λ + λ 2. () In fat, it is easy to see that σ is real, with < σ < λ +λ 2, sine h () + h 2 () < and h ( λ +λ 2 ) + h 2 ( λ +λ 2 ) >. Sine φ i (s) is an analyti funtion for Re s, σ must also be a zero of the numerators of (4) and (5). In both ases this yields the same relation between φ (+) and φ 2 (+), namely φ 2 (+) = σ λ 2 + λ 2 χ 2 (σ) φ (+) = λ χ 2 (σ) λ 2 χ (σ) σ λ + λ χ (σ) φ (+). () Combined with (4), (5) and (8), this ompletes the determination of φ i (s), i =, 2. 5

6 Remark. Note that whenever χ i (s) (i =, 2) are rational funtions (whih is e.g. fulfilled if the orresponding onditional distributions are phase-type), then the Laplae-Stieltjes transforms (4) and (5) an be inverted expliitly to yield exat formulae for φ i (x) (i =, 2) (see e.g. Spiegel 4). Sine the lass of phase-type distributions is dense (in the sense of weak onvergene) in the lass of all distributions on the positive half-line, one an approximate any given distribution arbitrarily losely by a phase-type distribution and use the exat solutions above (algorithms for phase-type fitting are e.g. disussed in Asmussen 3). Example. For the speial ase T Exp(µ) we obtain χ 2 (s) = x= χ (s) = b(s) b(s + µ). e sx e µx db(x) = b(s + µ), If in addition B Exp(ν), with ν = /β, then we have ν χ 2 (s) = ν + s + µ, χ (s) = ν ν + s ν ν + s + µ, and thus σ in () is the unique solution s with Re s > of ( λ µν )( s + (ν + s)(ν + µ + s) λ λ 2 ν ) s + ν + µ + s λ λ λ 2 µν 2 2 (ν + µ + s) 2 (ν + s) =. Sine the Laplae-Stieltjes transforms are rational funtions in this ase, they an be inverted expliitly for any given parameter values (see Setion 4 for a speifi numerial example). Example 2. For a deterministi threshold (i.e. T i = T a.s. for all i and some onstant T > ) and exponential laim sizes (B i Exp(ν)) we obtain χ (s) = ν ν + s e (ν+s)t and χ 2 (s) = ν ν + s ( e (ν+s)t ). (2) 2.2 Comparison to Model with Independene The availability of analytial solutions for the survival probability allows one to investigate the error produed by negleting a dependeny struture of the above kind. Indeed, assuming independene when in fat the dependeny struture of Model is present, an estimation of distribution of the inter-ourrene time W i would lead to the mixing density f Wi (x) = P(B i > T i )λ e λ x + P(B i T i )λ 2 e λ 2x, i.e. one would assume to have a renewal model (also alled Sparre Andersen risk model) with a hyper-exponential inter-arrival distribution. For suh a model, the Lundberg oeffiient R, given it exists, an easily be determined as the unique positive solution of b( R) w(r) =, where w( ) denotes the Laplae transform of f Wi (x) (see e.g. Asmussen 4). An illustrative example for the differene of the orresponding survival probabilities is given in Setion 4. 6

7 3 Related Models In the following, we list a number of related dependeny models for whih exat solutions for the survival probability an be derived in an analogous way: 3. Model 2 Let for every t > the risk proess be in one of the two states i =, 2, orresponding to the rate λ i of the exponential distribution for the time until the next laim ours. At the time of a laim ourrene the state of the system may hange depending on the orresponding laim size. If a laim B j is smaller than a threshold T j, then the state of the risk proess hanges, otherwise it does not. The quantities T j are again assumed to be i.i.d. random variables with distribution funtion T ( ). The net profit ondition in this model is 2β < ( λ + λ 2 ). (3) Then the analysis of φ i (x) (whih is the survival probability with initial apital x, given that the system starts out in state i) is analogous to the previous setion and we obtain and dφ x dx (x) λ φ (x) + λ dφ x 2 dx (x) λ 2φ 2 (x) + λ 2 P(T y)φ (x y)db(y)+ + λ x P(T y)φ 2 (x y)db(y)+ + λ 2 x P(T > y)φ 2 (x y)db(y) = (4) P(T > y)φ (x y)db(y) =, (5) from whih it follows that for Re s φ (+) s λ 2 + λ 2 χ (s) λ χ 2 (s)φ 2 (+) φ (s) = s λ + λ χ (s) s λ 2 + λ 2 χ (s) λ λ 2 χ 2 2(s) and (6) φ 2 (+) s λ + λ χ (s) λ 2 χ 2 (s)φ (+) φ 2 (s) =, (7) s λ + λ χ (s) s λ 2 + λ 2 χ (s) λ λ 2 χ 2 2(s) 7

8 where φ i (s) is again the Laplae transform of φ i (x). Note that the denominators on the right-hand side of (6) and (7) again oinide. Let us now determine φ (+) and φ 2 (+). As for Model, one equation for these two unknowns follows from lim s s φ i (s) =, yielding λ 2 ( φ (+)) + λ ( φ 2 (+)) = 2 λ λ 2 β. (8) A seond equation is obtained by notiing that there is a real number τ (, λ +λ 2 ) that makes the denominator of (6), and similarly (7), zero. Indeed, write the denominator of (6) and (7) as s(k (s) + k 2 (s)), in whih k (s) := s λ λ 2, k 2 (s) := (λ + λ 2 )χ (s) + λ λ 2 s ( χ (s)) 2 χ 2 (s) 2. Now observe that k () + k 2 () < if the net profit ondition (3) holds, whereas k ( λ +λ 2 ) + k 2 ( λ +λ 2 ) >. Sine φ i (s) is an analyti funtion for Re s, τ must also be a zero of the numerators of (6) and (7). In both ases this yields the same relation between φ (+) and φ 2 (+), namely φ 2 (+) = τ λ 2 + λ 2 χ (τ) φ (+) = λ χ 2 (τ) λ 2 χ 2 (τ) τ λ + λ χ (τ) φ (+). (9) We have not proved that τ is the only zero of the denominator of (6) for Re s > (appliation of Rouhé s theorem seems muh more involved here than in the ase of Model ). However, that is not needed: If (3) holds, then there should be unique solutions φ (x) and φ 2 (x) of the integro-differential equations (4) and (5). φ (s) and φ 2 (s) as given in (6) and (7) with φ (+) and φ 2 (+) given by (8) and (9) are the Laplae transforms of funtions φ (x) and φ 2 (x) that satisfy those integrodifferential equations, so we need not look further. See Cohen and Down 7 for more general ideas about handling queueing systems without taking reourse to Rouhé s theorem. Remark: For the speial ase λ = λ 2 := λ we again obtain from (8) the survival probability (9) with zero initial apital in the independent ase. If, alternatively, the state of the risk proess hanges at the time of a laim ourrene, given that B j is larger than a threshold T j and remains in its state otherwise, we get instead of (6) and (7): and φ (s) = φ (+) s λ + λ χ 2 (s) s λ 2 + λ 2 χ 2 (s) s λ 2 + λ 2 χ 2 (s) λ χ (s)φ 2 (+), (2) λ λ 2 χ 2 (s) φ 2 (+) s λ + λ χ 2 (s) λ 2 χ (s)φ (+) φ 2 (s) =, (2) s λ + λ χ 2 (s) s λ 2 + λ 2 χ 2 (s) λ λ 2 χ 2 (s) 8

9 and φ (+) and φ 2 (+) follow from (8) and φ 2 (+) = ζ λ 2 + λ 2 χ 2 (ζ) φ (+) = λ χ (ζ) λ 2 χ (ζ) ζ λ + λ χ 2 (ζ) φ (+), (22) where, similar to τ above, ζ is the real zero of the denominator of (2) in (, λ +λ 2 ). 3.2 Another Variant of Model Let us now look at the following variant of Model with appliations in reinsurane: As in Model, we take the laim intervals W i+ Exp(λ ) if B i > T i, and W i+ Exp(λ 2 ) if B i T i for all i, where T i are again i.i.d. threshold variables. However, now the atual laim payment is min(b i, T i ). Thus the threshold T i an be interpreted as the retention level of an XL-type reinsurane on the laim size (note that a deterministi threshold is a speial ase of this model). For the analysis of this model, we have to introdue the Laplae-Stieltjes transform ψ(s) := Ee st (T <B) = x= e sx ( B(x))dT (x). Note that χ 2 (s) + ψ(s) = Ee s min(b,t ) and thus Emin(B, T ) = χ 2() ψ (). A similar derivation along the lines of Setion 2. leads to φ (+) s λ 2 + λ 2 χ 2 (s) λ χ 2 (s)φ 2 (+) φ (s) = (23) s λ + λ ψ(s) s λ 2 + λ 2 χ 2 (s) λ λ 2 ψ(s)χ 2 (s) and φ 2 (s) = φ 2 (+) s λ + λ ψ(s) λ 2 ψ(s)φ (+), (24) s λ + λ ψ(s) s λ 2 + λ 2 χ 2 (s) λ λ 2 ψ(s)χ 2 (s) where φ i (+) (i=,2) are the solutions of the two equations λ 2 P(B > T )( φ (+)) + λ P(B T )( φ 2 (+)) = λ λ 2 Emin(B, T ) (25) and φ 2 (+) = γ λ 2 + λ 2 χ 2 (γ) λ 2 ψ(γ) φ (+) = λ χ 2 (γ) γ λ + λ ψ(γ) φ (+), where here γ is the unique positive zero of the denominator of (23). Note that the existene and uniqueness of γ an, as in Model, be easily shown by Rouhé type arguments. Remark. In Boxma and Perry 6 a queueing model with the above dependene struture between servie and subsequent interarrival times has been investigated. However, sample path duality between the orresponding workload proess and our risk proess does not hold for this partiular dependene struture, as an also be seen from the differene between (23) and (24) and the formulae (3.8) and (3.9) of 6. 9

10 4 Numerial Illustrations Example. Let T Exp(2), B Exp(), = 2, λ = 3, λ 2 =. The net profit ondition () is obviously fulfilled. Then the inversion of the Laplae transforms (4) and (5) yields φ (x) =.7 e 3.6 x.938 e.65 x, φ 2 (x) =.3 e 3.6 x.867 e.65 x, (26) where here and in the sequel all numerial values are rounded to their last digit (f. Figure a). Let us now ompare (26) to φ(x) in a model with the assumption of independene as desribed in Setion 2.2. The inter-arrival density in the independent model is then given by f Wi (x) = 2 e 3x + 3 e x. The Lundberg exponent in this renewal risk model is the positive solution of R ( 3 + 3R + 2 ) 3 + R =, i.e. R =.77. In this speifi example, there is even an analytial solution for the survival probability in the orresponding renewal model available, sine the laim size distribution is exponential. This solution an be derived utilizing a sample path duality to a related queuing proess (see e.g. Asmussen 4) and we obtain φ ind (x) =.923e.77x. This should be ompared with the stationary version of the dependent setting φ dep (x) = 2 3 φ (x) + 3 φ 2(x) =.6e 3.6x.95e.65x. Note that onerning the asymptoti behavior, the Lundberg exponent of φ dep (x) is smaller than the one of φ ind (x), i.e. ignoring the dependene struture underestimates the inherent risk, espeially for larger values of initial apital x (f. Figure b). Example 2. Let T Exp(), B Exp(), = 2, λ =, λ 2 = 2. Then the inversion of the Laplae transforms (4) and (5) yields φ (x) =.632 e.355 x +.7 e.889 x, φ 2 (x) =.798 e.355 x +.28 e.889 x. (27) If we again ompare (27) to φ(x) in a model with the assumption of independene, then the inter-arrival density is now given by f Wi (x) = e 2x + 2 e x. The Lundberg exponent in this renewal risk model is the positive solution of R ( 2( + R) + ) 2 + R =,

11 x x Figure : Survival probabilities in Example. Left: φ (x) (solid line) and φ 2 (x) (dashed line). Right: φ dep (x) (solid line) and φ ind (x) (dotted line) x x Figure 2: Survival probabilities in Example 2. Left: φ (x) (solid line) and φ 2 (x) (dashed line). Right: φ dep (x) (solid line) and φ ind (x) (dotted line) i.e. R =.39. Again, we even have an analytial solution for φ ind (x) in the orresponding renewal model available: φ ind (x) =.69e.39x. The stationary version of the dependent setting yields φ dep (x) = 2 φ (x) + 2 φ 2(x) =.75e.355x +.23e.889x. Note that in this ase, the Lundberg exponent of φ ind (x) is smaller than the one of φ dep (x), i.e. the independent setting is more dangerous. This is, heuristially, due to the fat that for this hoie of parameters a larger laim is likely to be followed by a longer inter-ourrene time (see also Figure 2b). Example 3. Let us again onsider the setting of Example 2, but now with a deterministi threshold T i = a.s. for all i (so the value of T i equals the expeted value of the threshold variable of Example 2). Aording to (2) we have χ (s) = +s e s and χ 2 (s) = ( +s e s ) and we obtain φ (+) =.337 and φ 2 (+) =.9. The resulting Laplae transforms (4) and (5) an easily be inverted numerially by a Bromwih ontour integration. Table illustrates the fat that the distribution of the threshold has a signifiant effet on the survival probabilities.

12 T = T Exp() x φ (x) φ 2 (x) φ (x) φ 2 (x) Referenes Table : Comparison of φ i (x) for Examples 2 and 3 I. Adan and V. Kulkarni. Single-server queue with Markov dependent interarrival and servie times. Queueing Systems, to appear, H. Albreher and J. Kantor. Simulation of ruin probabilities for risk proesses of Markovian type. Monte Carlo Methods & Appl., 8(2): 27, S. Asmussen. Matrix-analyti models and their analysis. Sand. J. Statist., 27(2):93 226, 2. 4 S. Asmussen. Ruin probabilities. World Sientifi, Singapore, 2. 5 S. Asmussen, H. Shmidli, and V. Shmidt. Tail probabilities for non-standard risk and queueing proesses with subexponential jumps. Adv. in Appl. Probab., 3(2): , O. Boxma and D. Perry. A queueing model with dependene between servie and interarrival times. European J. Oper. Res., 28(3):6 624, 2. 7 J. Cohen and D. Down. On the role of Rouhé s theorem in queueing analysis. Queueing Systems, 23:28 29, M. Combé and O. Boxma. BMAP modelling of a orrelated queue. In J. Walrand, K. Baghi, and G. Zobrist, editors, Network Performane Modeling and Simulation, pages Gordon and Breah Siene Publ., Newark, T. Mikosh and G. Samorodnitsky. Ruin probability with laims modeled by a stationary ergodi stable proess. Ann. Probab., 28(4):84 85, 2. T. Mikosh and G. Samorodnitsky. The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab., (3):25 64, 2. 2

13 A. Müller and G. Pflug. Asymptoti ruin probabilities for risk proesses with dependent inrements. Insurane: Mathematis and Eonomis, 28(3):38 392, 2. 2 H. Nyrhinen. Rough desriptions of ruin for a general lass of surplus proesses. Adv. Appl. Prob., 3:8 26, H. Nyrhinen. Large deviations for the time of ruin. J. Appl. Probab., 36(3): , M. Spiegel. Theory and problems of Laplae transforms. Shaum, New York,