Homogeneous risk models with equalized claim amounts

Insurance: Mathematics and Economics 26 (2000) 223–238

Homogeneous risk models with equalized claim amounts夽 F. De Vylder a , M. Goovaerts a,b,∗ a

b

Universiteit van Amsterdam, Amsterdam, Netherlands Katholieke Universiteit Leuven, CRIR, Huid Eygen Heerd, Minderbroederstraat 5, 3000 Leuven, Belgium Received 1 June 1998; received in revised form 1 September 1999; accepted 24 November 1999

Abstract We consider an homogeneous risk model on a fixed bounded time interval [0, t] and we denote by Nt the number of claims in that interval. The claim amounts are X1 , X2 , . . . , XNt . The homogeneous model is an extension of the classical actuarial risk model with Nt not necessarily Poisson distributed. In the model with equalized claim amounts, each amount Xk is replaced with Xk∼ = (X1 + · · · + XNt )/Nt . Let 9(t, u) be the ruin probability before t in the homogenous model, corresponding to the initial risk reserve u≥0 and let 9 ∼ (t, u) be the corresponding ruin probability evaluated in the associated model with equalized claim amounts. The essence of the classical Prabhu formula is that 9(t, 0)=9 ∼ (t, 0). By rather systematic numerical investigations in the classical risk model, we verify that 9 ∼ (t, u)≤9(t, u) for any value of u≥0 and that 9 ∼ (t, u) is an excellent approximation of 9(t, u). Then these conclusions must be valid in any homogeneous model and this is an interesting observation because 9 ∼ (t, u) can be calculated numerically, whereas no algorithms are yet available for the numerical evaluation of 9(t, u) in general homogeneous risk models. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Risk model; Homogeneous risk model; Ruin probability; Prabhu’s formula

1. Homogeneous risk model on a bounded time interval [0, t ] Nt is the number of claims in [0, t]. The claim instants process (T1 , T2 , . . . , TNt ) is any homogeneous point process on the fixed interval [0, t]. Its distribution is completely specified by the probabilities P(Nt =n)≥0 (n=0, 1, 2, . . . ) such that 6 n≥0 P(Nt =n)=1. The latter probabilities may be any numbers satisfying the indicated relations (see Appendix of De Vylder and Goovaerts (1999)). The claim amounts are X1 , X2 , . . . , XNt . It is assumed that X1 , X2 , . . . are i.i.d. random variables with distribution function F concentrated on [0, ∞) and that these amounts are independent from the claim instants. The risk reserve process is Rτ =u+cτ −Sτ (0≤τ ≤t), where u≥0 is the initial risk reserve, c>0 the premium income rate and Sτ the total claim amount in [0, τ ], i.e. Sτ = X1 + · · · + XNt , where Nτ is the number of claims in [0, τ ]. Of course, Sτ =0, if Nτ =0. 夽 Presented

at the Second International IME Congress, University of Lausanne, Switzerland, July 1998. author. Tel.: +32-16-323-746; fax: +32-16-323-740. E-mail address: [email protected] (M. Goovaerts) ∗ Corresponding

0167-6687/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 ( 9 9 ) 0 0 0 5 5 - 4

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We denote by U(t, u) the probability of nonruin before t corresponding to the initial risk reserve u≥0, and by Un (t, u) the corresponding conditional probability of nonruin for fixed Nt =n. Then X P (Nt = n)Un (t, u). (1) U (t, u) = n≥0

The homogeneous model with fixed Nt =n is defined by the probabilities P(Nt =n)=1 and P(Nt =m)=0 (m6=n). Then Un (t, u) is the (nonconditional) probability of ruin in the homogeneous model with fixed Nt =n. In the homogeneous model with fixed Nt =n, the claim instants are T1 < T2 < · · · < Tn and the random vector (T1 , . . . , Tn ) has a constant density on the subset Wtn = {(t1 , . . . , tn )|0 < t1 < t2 < · · · < tn < t} of Rn . The Lebesgue volume of Wtn is tn /n!. Hence the density of (T1 , . . . , Tn ) equals n!/tn on Wtn . Then, by the foregoing assumptions, the distribution of the random vector (T1 , . . . , Tn , X1 , . . . , Xn ) is completely specified in the homogeneous model with fixed Nt =n.

2. Nonruin probability before time t in the homogeneous model Let τ , y1 , y2 , . . . be any numbers. The numbers z0 =1, z1 , z2 , . . . are defined recursively as follows: −zk+1 =

z1 yk+1 k zk yk+1 1 z0 yk+1 k+1 + + ··· + (k + 1)! k! 1!

The first numbers zk are z0 = 1,

z1 = −y1 ,

1 z2 = − y2 2 + y1 y2 , 2

[yk−1 ,τk ]

(2)

  1 1 1 z3 = − y3 3 + y1 y3 2 − − y2 2 + y1 y2 y3 . 6 2 2

For all k=1, 2, . . . , we consider the k-tuple integral Z Z Z dτk dτk−1··· Ik (y1 , . . . , yk , τ ) = [yk ,τ ]

(k = 0, 1, 2, . . . ).

[y2 ,τ3 ]

Z dτ2

[y1 ,τ2 ]

dτ1 .

(3)

R R The integration domains are oriented, i.e. [a,b] = − [b,a] , if b
z0 τ k z1 τ k−1 zk−1 τ 1 zk τ 0 + +···+ + k! (k−1)! 1! 0!

(k= 1, 2, . . . ).

(4)

If yk =ky (k=1, 2, . . . ), then 0=z2 =z3 =· · · and Ik (y, 2y, . . . ,ky,τ ) =

yτ k−1 τk − k! (k − 1)!

(k= 1, 2, . . . ).

Proof. Z I1 (y1 ,τ ) =

[y1 ,τ ]

dτ1 =τ − y1 =

z0 τ 1 z1 τ 0 + . 1! 0!

Hence (4) is correct for k=1. Then the proof is completed by induction. Indeed, let (4) be correct for k. Then

(5)

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225

Z

Ik+1 (y1 , . . . ,yk+1 ,τ ) = =

[yk+1 ,τ ]

Z

X

zi

0≤i≤k

=

Ik (y1 , . . . ,yk ,τk+1 ) dτk+1 X Z τk+1 k−i dτk+1 k+1−i zi dτk+1 = [yk+1 ,τ ] (k − i)! [yk+1 ,τ ] (k+1−i)! 0≤i≤k

X zi yk+1 k+1−i X zi τ k+1−i X zi τ k+1−i X zi τ k+1−i − = +zk+1 = , (k+1−i)! (k+1−i)! (k+1−i)! (k+1−i)!

0≤i≤k

0≤i≤k

0≤i≤k

0≤i≤k+1

i.e. (4) is correct for k+1. Let us now assume that yk =ky. Then z2 =−2y2 +2y2 =0. By induction we assume that z2 = z3 = · · · = zk = 0. Then by (2), −zk+1 =

z0 yk+1 k+1 z1 yk+1 k y k+1 k k+1 y k+1 k k + = − =0 (k+1)! k! (k+1)! k!



We recall that X1 , X2 ,. . . are the i.i.d. claim amounts with distribution function F and that u≥0 is the initial risk reserve. We define Yk = (X1 + · · · +Xk − u)+

(k= 1, 2, . . . ),

(6)

where (x)+ =x, if x≥0 and (x)+ =0, if x<0. The random variables Xk are nonnegative. Hence Yk = X1 + · · · + Xk , if u=0. The random variables Z0 =1, Z1 , Z2 , . . . are defined by induction as follows: −Zk+1 =

Z0 Yk+1 k+1 Z1 Yk+1 k Zk Yk+1 1 + +···+ (k+1)! k! 1!

(k= 0, 1, 2, . . . ).

(7)

We notice that this is relation (2) with capital letters everywhere. The first random variables Zk are Z0 = 1,

Z1 = − Y1 = − (X1 − u)+ ,

Z2 = − 21 Y2 2 +Y1 Y2 = − 21 (X1 +X2 − u)+ 2 +(X1 − u)+ (X1 +X2 − u)+ , Z3 = − 16 Y3 3 + 21 Y1 Y3 2 − ( 21 Y2 2 +Y1 Y2 )Y3 = − 16 (X1 +X2 +X3 − u)+ 3 + 21 (X1 − u)+ (X1 +X2 +X3 − u)+ 2 h i − − 21 − (X1 +X2 − u)+ 2 +(X1 − u)+ (X1 +X2 − u)+ (X1 +X2 +X3 − u)+ . In Lemma 2, ϕ(X1 , . . . , Xn ) is any symmetrical function of X1 , X2 , . . . , Xn such that all expectations involved in the proof, are finite. Later in this paper, the lemma is only used with the particular function ϕ(X1 , . . . ,Xn ) = 1(X1 + · · · +Xn ≤ u+ct), where 1(.) is the indicator function of a proposition. Then all expectations are integrals with respect to dF(x1 )· · · dF(xn ) of a bounded function on a bounded integration domain of Rn , and they are finite indeed. Lemma 2. Let u=0 and let ϕ(X1 , . . . , Xn ) be a symmetrical function of X1 , . . . , Xn . Then   Yn ϕ(X1 , . . . ,Xn ) E[Z1 ϕ(X1 , . . . ,Xn )] = − E n

(8)

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and E[Zk ϕ(X1 , . . . ,Xn )] = 0

(k= 2, 3, . . . ,n)

(9)

if all involved expectations are finite. Proof. The proof results from the adequate use of the following remark. Let us consider an expectation such as E[f(X1 , . . . , Xk )ϕ(X1 , . . . , Xn )], where k≤n. By the i.i.d. assumption on X1 , X2 , . . . , we may replace X1 , . . . , Xn with any permutation Xi1 , . . . , Xin . But ϕ(X1 , . . . , Xn ) equals ϕ(Xi1 , . . . , Xin ) by the symmetry assumption on ϕ. Hence E[f (X1 , . . . ,Xk )ϕ(X1 , . . . ,Xn )] =E[f (Xi1 , . . . ,Xik )ϕ(X1 , . . . ,Xn )]

(10)

for any k different subscripts i1 , . . . , ik in the set {1, 2, . . . , n}. We can also consider various subsets {i1 , . . . , ik )⊆{1, . . . , n} and consider (10) for each subset. Summing up and dividing by the number of subsets we see that  X  f (Xi1 , . . . ,Xik ) ϕ(X1 , . . . ,Xn ) , (11) E[f (X1 , . . . ,Xk )ϕ(X1 , . . . ,Xn )] =E m where m is the number of subsets {i1 , . . . , ik }. We now prove (8). By (11)  E[Z1 ϕ(X1 , . . . ,Xn )] = − E[X1 ϕ(X1 , . . . ,Xn )] = − E

Yn n





ϕ(X1 , . . . ,Xn ) .

We now prove (9) for k=2. Considering the subsets {1, 2} and {2, 1} of {1, . . . , n}, E[Z2 ϕ(X1 , . . . ,Xn )] =E[( − 21 Y2 2 + Y1 Y2 )ϕ(X1 , . . . ,Xn )] =E[( − 21 Y2 2 + 21 Y2 Y1 ϕ(X1 , . . . ,Xn )] = 0. We now consider (9) for k=3. Then the expectation is the sum of the two terms       1 1 2 1 Y2 Y3 − Y1 Y2 Y3 ϕ(X1 , . . . ,Xn ) E − Y3 3 + Y1 Y3 2 ϕ(X1 , . . . ,Xn ) and E 6 2 2 In the first term Y1 can be replaced with 13 Y3 . Hence, it equals zero. In the last term, Y1 can be replaced with 21 Y2 . Hence, it also equals zero. A general proof by induction is direct. It is based on (7). In the last member of that relation, the first two terms must be treated jointly; all other terms can be treated separately.  In Theorem 1, F∗n is of course the distribution function of the total claim amount X1 +· · ·+Xn in the homogeneous model with fixed Nt =n. The function In is the n-tuple integral defined by (3). Of course U0 (t, u)=1 because ruin cannot occur if there are no claims. In the general homogeneous model on [0, t], we denote by Ft the distribution function of the total claim amount St =X1 +X2 + · · · +XNt in [0, t]. X (12) Ft (x) = P (Nt =n)F ∗n (x) (x ≥ 0). n≥0

Theorem 1. a. In the homogeneous model with fixed Nt =n, Un (t,u) =n!(ct)−n E[In (Y1 , . . . ,Yn ,ct)1(X1 + · · · +Xn ≤ u+ct)].

(13)

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b. In the homogeneous model with fixed Nt =n and u=0,   Z  Z h Yn yi F ∗n (y) dy. 1− 1(Yn ≤ ct) = dF ∗n (y) = (ct)−1 Un (t, 0) =E 1 − ct ct [0,ct] [0,ct] c. (Prabhu’s formula) In the general homogeneous model on [0, t] with u=0, Z Z h yi −1 dFt (y) = (ct) Ft (y) dy. 1− U (t, 0) = ct [0,ct] [0,ct]

227

(14)

(15)

Proof. a. Taking the distribution of the random vector (T1 , . . . , Tn , X1 , . . . , Xn ) into account (see Section 1), the nonruin probability Un (t, u) equals Z Z −n · · · dt1 · · · dtn dF (x1 ) · · · dF (xn ), Un (t,u) =n!t D

where the integration domain D is the subset of R2n of points (t1 , . . . , tn , x1 , . . . , xn ) satisfying the relations x1 ≥ 0, . . . ,xn ≥ 0,

0 ≤ t1 ≤ · · · ≤ tn ≤ t

x1 ≤ u+ct1 , x1 +x2 ≤ u+ct2 , . . . ,x1 +x2 + · · · +xn ≤ u+ctn . Instead of t1 , . . . , tn , let us take the new integration variables ␶1 =ct1 , . . . , ␶n =ctn . Then Z Z Un (t,u) =n!(ct)−n · · · dτ1 · · · dτn dF (x1 ) · · · dF (xn ), 1

where the integration domain 1 is the subset of R2n of points (␶1 , . . . , ␶n , x1 , . . . , xn ) satisfying the relations x1 ≥ 0, . . . ,xn ≥ 0,

0 ≤ τ1 ≤ · · · ≤ τn ≤ ct,

x1 ≤ u+τ1 , x1 +x2 ≤ u+τ2 , . . . ,x1 +x2 + · · · +xn ≤ u+τn . These relations are equivalent to the relations x1 ≥ 0, . . . ,xn ≥ 0,

x1 + · · · +xn ≤ u+ct,

(16)

(x1 + · · · +xn−1 +xn − u)+ ≤ τn ≤ ct, (x1 + · · · +xn−1 − u)+ ≤ τn−1 ≤ τn , ··· ··· ··· ··· ··· ··· ··· (x1 +x2 − u)+ ≤ τ2 ≤ τ3 , (x1 − u)+ ≤ τ1 ≤ τ2 . With the notation yk = (x1 + · · · + xk − u)+ , these relations are (16) and yn ≤ τn ≤ ct,

yn−1 ≤ τn−1 ≤ τn , . . . ,y2 ≤ τ2 ≤ τ3 ,

Then, by Fubini, Un (t,u) =n!(ct)−n

Z

Z ···

10

Z

Z ···

100

y1 ≤ τ1 ≤ τ2 .

 dτ1 · · · dτn dF (x1 ) · · · dF (xn ),

(17)

(18)

where the integration domain 10 is the subset of Rn of points (x1 , . . . , xn ) satisfying (16) and where for fixed x1 , . . . , xn , the integration domain 100 is the subset of Rn of points (␶1 , . . . , ␶n ) satisfying (17). The n-tuple integral in square brackets of (18) is In (y1 , . . . , yn ,ct) defined by the relation (3). Incorporating the integration domain 10 in the integrand by the use of the indicator function 1(.), it is clear that (18) is the same as (13).

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b. Let us now assume that u=0. By Lemmas 1 and 2, In (Y1 , . . . , Yn , ct) can be replaced with   (Yn /n)(ct)n−1 Yn (ct)n − = (ct)n (n!)−1 1 − n! (n − 1)! ct in (13). This proves the first relation (14). The second relation (14) is obvious. Then the last relation (14) results from an integration by parts. c. (15) results from (1), (12) and (14).  General expectations such as E[ϕ(X1 , . . . , Xn )] are almost impossible to evaluate numerically, fastly and precisely enough, if n is large. In particular, (13) is not useful for the practical evaluation of Un (t, u). On the contrary, expectations such as E[ϕ(X1 + · · · + Xn )] can be calculated. Indeed, they are reduced to single integrals Z E[ϕ(X1 + · · · +Xn )] = ϕ(y) dF ∗n (y). I

In practice I is a bounded interval and F the discretized, i.e. F is hold back as a long vector in the computer program. Then F∗n can be calculated iteratively and only two successive long vectors F∗k and F∗(k +1) (k=1,2, . . . , n) must be stored simultaneously in the process at each stage. If the discretized F has ν components, then the direct calculation of Z Z E[ϕ(X1 , . . . ,Xn )] = · · · ϕ(x1 , . . . ,xn ) dF (x1 ) · · · dF (xn ) D

may need the evaluation of ν n terms, to be summed up. In the case of Un (t, u) the evaluation of ϕ(x1 , . . . , xn ) goes via relations such as (4). All this must be done for values n=0, 1, 2, . . . , n0 and then U(t, u) should result from (1). In some cases ν and n0 may be pretty large, say ν=1000 and n0 =100. Clearly this rudimentary procedure is hopeless, even with the fastest computers available today.

3. Homogeneous model with equalized claim amounts We now start from an homogeneous model with time interval [0, t], called the initial model, and we replace each claim amount Xk with the average amount Xk ∼ = (X1 + · · · +XNt )/Nt . This model with equalized claim amounts is also called the associated model. The claims instants Tk (k=1, 2, . . . , Nt ) are the same in both models: Tk ∼ =Tk . The superscript ∼ is used systematically for the components of the associated model with equalized claim amounts. We notice that the risk reserve Rτ ∼ (0 ≤ τ ≤ t) cannot be determined from observations during time interval [0, τ ]. The complete trajectory Rs (0≤s≤t) is necessary in order to fix Rτ ∼ . The distribution of the trajectories Rτ ∼ (0 ≤ τ ≤ t) results from the distribution of the trajectories Rτ (0≤␶≤t). Hence, the associated model is clearly defined. In Figs. 1–4, we represent sample functions of the processes Rτ (full lines) and Rτ ∼ (stippled lines). All four following cases a, b, c and d can occur. a. No ruin in the initial model, no ruin in the associated model (Fig. 1), b. Ruin in the initial model, ruin in the associated model (Fig. 2), c. No ruin in the initial model, ruin in the associated model (Fig. 3), d. Ruin in the initial model, no ruin in the associated model (Fig. 4). Due to compensations between occurrences of these cases, the following question is relevant: is U∼ (t, u) close to U(t, u)? The most surprising answer is that in case of an initial risk reserve u=0, the compensation is perfect: U∼ (t, 0)=U(t, 0) (Theorem 2b). This throws a new light on Prabhu’s formula. The numerical investigations of Section 5 show that U∼ (t, u)≥U(t, u) and that U∼ (t, u) is rather close to U(t, u). In fact, the comparison of U∼ (t, u) with U(t, u) is possible in the classical actuarial model only, i.e. when Nt is Poisson distributed, because U(t, u) can be evaluated

F. De Vylder, M. Goovaerts / Insurance: Mathematics and Economics 26 (2000) 223–238

Fig. 1. No ruin in initial model. No ruin in associated model.

Fig. 2. Ruin in initial model. Ruin in associated model.

Fig. 3. No ruin in initial model. Ruin in associated model.

Fig. 4. Ruin in initial model. No ruin in associated model.

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Table 1 Uniform F, η=0.05, t=1 u

9

9∼

0 1 2 3 4

0.527 0.0925 0.00827 0.00049 0.00002

0.527 0.0920 0.00832 0.00050 0.00002

Table 2 Uniform F, η=0.05, t=5 u

9

9∼

0 1 2 3 4 5 6

0.777 0.346 0.113 0.0297 0.00646 0.00119 0.00019

0.777 0.320 0.105 0.0281 0.00622 0.00117 0.00019

Table 3 Uniform F, η=0.05, t=10 u

9

9∼

0 1 2 3 4 5 6 7 8

0.835 0.473 0.221 0.0900 0.0321 0.0101 0.00284 0.00072 0.00016

0.835 0.433 0.199 0.0814 0.0294 0.00940 0.00269 0.00070 0.00016

Table 4 Uniform F, η=0.05, t=20 u

9

9∼

0 1 2 3 4 5 6 7 8 9 10

0.875 0.583 0.347 0.190 0.0964 0.0452 0.0196 0.00791 0.00298 0.00105 0.00035

0.875 0.536 0.309 0.168 0.0850 0.0400 0.0176 0.00718 0.00274 0.00098 0.00033

F. De Vylder, M. Goovaerts / Insurance: Mathematics and Economics 26 (2000) 223–238 Table 5 Uniform F, η=0.25, t=1 u

9

9∼

0 1 2 3 4

0.505 0.0822 0.00734 0.00038 0.00002

0.505 0.0810 0.00685 0.00040 0.00002

Table 6 Uniform F, η=0.25, t=5 u

9

9∼

0 1 2 3 4 5 6

0.710 0.265 0.0742 0.0172 0.00335 0.00056 0.00008

0.710 0.239 0.0670 0.0158 0.00316 0.00054 0.00008

Table 7 Uniform F, η=0.25, t=10 u

9

9∼

0 1 2 3 4 5 6 7 8

0.753 0.343 0.130 0.0441 0.0134 0.00368 0.00091 0.00021 0.00004

0.753 0.300 0.111 0.0379 0.0117 0.00328 0.00083 0.00019 0.00004

Table 8 Uniform F, η=0.25, t=20 u

9

9∼

0 1 2 3 4 5 6 7 8 9 10

0.780 0.400 0.183 0.0791 0.0323 0.0124 0.00450 0.00153 0.00049 0.00015 0.00004

0.779 0.343 0.148 0.0632 0.0259 0.0101 0.00371 0.00129 0.00042 0.00013 0.00004

231

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Table 9 Exponential, η=0.05, t=1 u

9

9∼

0 2 4 6 8 10

0.470 0.122 0.0293 0.00671 0.00146 0.00031

0.470 0.120 0.0290 0.00667 0.00147 0.00031

Table 10 Exponential, η=0.05, t=5 u

9

9∼

0 2 4 6 8 10 12 14 16

0.735 0.370 0.168 0.0702 0.0274 0.0100 0.00349 0.00117 0.00038

0.735 0.325 0.146 0.0618 0.0245 0.00915 0.00325 0.00111 0.00036

Table 11 Exponential, η=0.05, t=10 u

9

9∼

0 4 8 12 16 20

0.803 0.283 0.0761 0.0165 0.00302 0.00047

0.804 0.233 0.0631 0.0141 0.00265 0.00043

Table 12 Exponential, η=0.05, t=20 u

9

9∼

0 4 8 12 16 20 24 28

0.852 0.408 0.164 0.0565 0.0168 0.00436 0.00103 0.00022

0.853 0.328 0.129 0.0447 0.0136 0.00368 0.00089 0.00020

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233

Table 13 Exponential, η=0.25, t=1 u

9

9∼

0 2 4 6 8 10

0.445 0.113 0.0268 0.00606 0.00132 0.00028

0.445 0.111 0.0265 0.00603 0.00132 0.00028

Table 14 Exponential, η=0.25, t=5 u

9

9∼

0 2 4 6 8 10 12 14 16

0.673 0.310 0.131 0.0518 0.0194 0.00688 0.00231 0.00075 0.00024

0.674 0.260 0.109 0.0435 0.0165 0.00597 0.00206 0.00069 0.00022

numerically in that case only (as far as we know). By arguments of De Vylder and Goovaerts (1999, Section 6) the validity of results in the classical model can often be extended to any homogeneous model. By the discussion at the end of foregoing Section 2 and by next formula (19) combined with the (∼)-version of (1), the nonruin probability U∼ (t, u) can be calculated in any homogeneous model. 4. Nonruin probability before t in case of equalized claim amounts Theorem 2. a. In the homogeneous risk model with fixed Nt =n and equalized claim amounts,          2Sn nSn 1Sn −u , − u ,... , − u , ct I (Sn ≤ u+ct) Un ∼ (t,u) = n!(ct)−n E In n n n + + +      Z   nx 2x 1x −u , − u ,... , − u , ct dF ∗n (x), In = n!(ct)−n n n n + [0,u+ct] + +

(19)

where Sn = X1 + · · · + Xn . b. (Interpretation of Prabhu’s formula). In the general homogenous model on[0, t] and its associated model with equalized claim amounts, Z ∼ −1 Ft (x) dx. (20) U (t, 0) =U (t, 0) = (ct) [0,ct]

Proof. a. The second relation (19) is obvious because F∗n is the distribution function of Sn . For the first relation (19) it is enough to repeat the proof of Theorem 1a. Now x1 +· · ·+xk must everywhere be replaced with k(x1 +· · ·+xn )/n (k=1, . . . , n). Hence, yk must be replaced with (k(x1 + · · · + xn )/n − u)+ and Yk with (kSn /n−u)+ .

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Table 15 Exponential, η=0.25, t=10 u

9

9∼

0 4 8 12 16 20

0.729 0.204 0.0463 0.00886 0.00148 0.00022

0.729 0.153 0.0351 0.00698 0.00120 0.00018

Table 16 Exponential, η=0.25, t=20 u

9

9∼

0 4 8 12 16 20 24 28

0.764 0.272 0.0859 0.0241 0.00603 0.00137 0.00028 0.00005

0.765 0.187 0.0568 0.0163 0.00427 0.00101 0.00022 0.00004

u

9

9∼

0 10 20 30 40

0.531 0.0124 0.00275 0.00119 0.00066

0.531 0.0124 0.00277 0.00118 0.00065

Table 17 Pareto, η=0.05, t=1

b. It is sufficient to consider the model with fixed Nt =n and equalized claim amounts and to prove that Z h xi dF ∗n (x) 1− Un ∼ (t, 0) = ct [0,ct]

(21)

because then (20) results from (15), (1) and the (∼)-version of (1). Now (kx/(n−u))+ =kx/n because u=0. Then, by (5),   nx (ct)n (x/n)(ct)n−1 1x 2x , , . . . , ,ct = − In n n n n! (n − 1)! 

and then (21) results from (19). 9 ∼ (t,

u)=1−U∼ (t,

u) the ruin probabilities before time t corresponding to We denote by 9(t, u)=1−U(t, u) and the initial risk reserve u≥0. Sometimes different values of the premium income rate c are considered simultaneously. Then we write more explicitly 9(t, u, c) and 9 ∼ (t, u, c). 4.1. Conjecture In any homogeneous risk model with time interval [0, t] and its associated model with equalized claim amounts, Ψ ∼ (t, u)≤Ψ (t, u).

F. De Vylder, M. Goovaerts / Insurance: Mathematics and Economics 26 (2000) 223–238 Table 18 Pareto, η=0.05, t=5 u

9

9∼

0 10 20 30 40 50 60 70 80

0.745 0.0798 0.0175 0.00690 0.00364 0.00224 0.00152 0.00110 0.00083

0.745 0.0742 0.0166 0.00662 0.00352 0.00219 0.00149 0.00108 0.00082

u

9

9∼

0 20 40 60 80 100 120

0.802 0.0409 0.00802 0.00323 0.00173 0.00108 0.00074

0.803 0.0365 0.00741 0.00305 0.00166 0.00104 0.00072

u

9

9∼

0 20 40 60 80 100 120 140 160

0.845 0.0908 0.0185 0.00707 0.00369 0.00226 0.00153 0.00110 0.00084

0.847 0.0758 0.0161 0.00634 0.00338 0.00211 0.00144 0.00106 0.00080

u

9

9∼

0 10 20 30 40

0.496 0.0117 0.00270 0.00117 0.00064

0.496 0.0117 0.00270 0.00116 0.00065

Table 19 Pareto, η=0.05, t=10

Table 20 Pareto, η=0.05, t=20

Table 21 Pareto, η=0.25, t=1

235

236

F. De Vylder, M. Goovaerts / Insurance: Mathematics and Economics 26 (2000) 223–238

Table 22 Pareto, η=0.25, t=5 u

9

9∼

0 10 20 30 40 50 60 70 80

0.674 0.0631 0.0151 0.00625 0.00338 0.00212 0.00146 0.00106 0.00081

0.675 0.0562 0.0138 0.00587 0.00322 0.00204 0.00141 0.00103 0.00079

u

9

9∼

0 20 40 60 80 100 120

0.720 0.0307 0.00694 0.00294 0.00162 0.00102 0.00071

0.721 0.0256 0.00612 0.00269 0.00151 0.00097 0.00068

u

9

9∼

0 20 40 60 80 100 120 140 160

0.751 0.0566 0.0138 0.00585 0.00322 0.00204 0.00140 0.00103 0.00079

0.753 0.0415 0.0109 0.00490 0.00279 0.00181 0.00127 0.00094 0.00073

Table 23 Pareto, η=0.25, t=10

Table 24 Pareto, η=0.25, t=20

Table 25 Uniform, η=0.25, t=10 u

9 sqrt

9∼

0 1 2 3 4 5 6 7

0.743 0.366 0.126 0.0637 0.0292 0.0108 0.00296 0.00055

0.743 0.308 0.128 0.0561 0.0227 0.00735 0.00170 0.00026

F. De Vylder, M. Goovaerts / Insurance: Mathematics and Economics 26 (2000) 223–238

237

Table 26 Exponential, η=0.25, t=10 u

9 sqrt

9∼

0 2 4 6 8 10 12 14 16 18 20

0.719 0.369 0.167 0.0953 0.0522 0.0273 0.0136 0.00639 0.00285 0.00121 0.00048

0.719 0.325 0.166 0.0864 0.0444 0.0221 0.0105 0.00478 0.00206 0.00085 0.00033

u

9 sqrt

9∼

0 10 20 30 40 50 60 70 80 90 100 110

0.711 0.109 0.0332 0.0135 0.00710 0.00436 0.00295 0.00214 0.00162 0.00127 0.00102 0.00084

0.711 0.103 0.0292 0.0121 0.00652 0.00407 0.00279 0.00204 0.00155 0.00122 0.00099 0.00082

Table 27 Pareto, η=0.25, t=10

This Conjecture is based on the numerical study of Section 6. By (1) and its (∼)-version, it is equivalent to the proposition: 9n ∼ (t, u) ≤ 9n (t, u) for all n=1, 2, . . . . Of course, 91 ∼ (t, u) = 91 (t, u). The proof of 92 ∼ (t, u) ≤ 92 (t, u) is rather direct by (13) and (19) and by the arguments of the proof of Lemma 2. The proof of the general case is still missing. 5. Numerical illustrations 5.1. Parameters In the classical model, we display c as c=λµ(1+η), where η is the security loading, µ the first moment of the claimsize distribution and λ the parameter of the Poisson claim instants process. We take λ=1 and we consider the values 1, 5, 10, 20 of t. We consider the values η=0.05 and η=0.25 of the security loading. For the initial risk reserve u we adopt values 0, k, 2k, . . . , mk depending on the other parameters. The integers k and m are fixed in such a way that m≤10 and that the ruin probability corresponding to the last value km is less than 1/1000. 5.2. Claimsize distributions F Atomic concentrated at 1 F(x)=1 (0≤x≤1), F(x)=1 (x>1). Then µ=1. Uniform on [0, 1] F(x)=x (0≤x≤1), F(x)=1 (x>1). Then µ= 21 .

238

F. De Vylder, M. Goovaerts / Insurance: Mathematics and Economics 26 (2000) 223–238

Table 28 Atomic F, η=0.05, t=5 u

9

9∼

0 2 4 6 8 10

0.804 0.301 0.0724 0.0125 0.00161 0.00016

0.803 0.300 0.0723 0.0124 0.00160 0.00015

Exponential Pareto

F(x)=1−e−x (x≥0). Then µ=1. F(x)=(1−x−2 )1 (x≥1) (x≥0). Then µ=2.

Distribution of Nt . In the classical model, the distribution of Nt is the exponential with parameter λt. We consider one nonclassical case. It is the homogeneous model on [0, t], t=10, with P (Nt = n) = cexp[−(10 − n)1/2 ] (n = 0, 1, 2, . . . , 20),

P (Nt = n) = 0 (n > 20),

where c results from the norm relation 6 n≥0 P(Nt =n)=1, i.e. c=0.271940. Then ENt =10 for reasons of symmetry. We take λ=ENt /t=1. The nonclassical case is treated with various claimsize distributions. 5.3. Numerical calculations The claimsize distributions are truncated and finely discretized, with conservation of first moment µ, as explained in De Vylder (1996, p. 268−271). The exact ruin probabilities in the classical model, 9(t, u)=1−U(t, u), are calculated as explained in the same book p. 257−268, with claimsize distribution and security loading adaptations. In the associated model, the ruin probabilities are evaluated via (19) and the (∼)-version of (1). In the nonclassical case, we cannot evaluate 9(t, u) numerically. Instead, we then consider the approximation 9 sqrt (t, u) defined by De Vylder and Goovaerts (1999, Section 6). 5.4. Numerical anomalies By (20), 9(t, 0) = 9 ∼ (t, 0), This relation and the conjecture of Section 4 are very slightly contradicted at a few places in following tables, because the numerical values include discretization, truncation and decimalization errors and because 9 ∼ (t, u) and 9(t, u) are obtained by completely different algorithms. For instance, in Table 28 below, we should have 9(t, u)=9 ∼ (t, u) for all values of u, because the model with equalized claims amounts is the same as the initial model when F is atomic. We have verified that with finer discretizations than those used in the tables, the contradictions disappear. Unfortunately, the finer discretizations cannot be used systematically because then the computing time would be excessive (on our 1995 P.C) Each of Tables 1–27 (classical cases, ␭=1) corresponds to the table with the same number in De Vylder and Goovaerts (1999). The atomic claimsize distribution F (Table 28), is not considered in the latter paper. References De Vylder, F.E., 1996. Advanced Risk Theory. A Self-Contained Introduction. Editions de l’Université de Bruxelles, Swiss Association of Actuaries. De Vylder, F.E., Goovaerts, M., 1999. Inequality extensions of Prabhu’s formula in ruin theory. Insurance: Mathematics Economics 24 (3), 249–272.