On ruin probabilities for risk models with ordered claim ... · Orderedclaimarrivalprocess Orderedaggregateclaimprocess Orderedinsuranceriskmodel Twoparticularriskmodels On ruin probabilities

Ordered claim arrival processOrdered aggregate claim process

Ordered insurance risk modelTwo particular risk models

On ruin probabilities for risk modelswith ordered claim arrivals

Claude Lefèvre

Université Libre de Bruxelles

7th Conference in Actuarial Science and Finance, Samos

Joint work with Philippe Picard, Université de Lyon

C. Lefèvre, P. Picard (ULB, ISFA)On ruin probabilities for risk models with ordered claim arrivals 1/ 53



1 Ordered claim arrival processHomogeneous point processTime scale transformation

2 Ordered aggregate claim processi.i.d. claim amountsExchangeable claim amountsEqualized claim amounts

3 Ordered insurance risk modelFinite-time ruin probability

4 Two particular risk modelsWhen Ft is linearWhen Ft is exponentialOver an infinite horizon




From the papers

Lefèvre,C. and Picard, P. (2011). Insurance : Mathematics andEconomics 49, 512-519.

Lefèvre,C. and Picard, P. (2012). Working paper.




Homogeneous point processTime scale transformation

1. ORDERED CLAIM ARRIVAL PROCESS

De Vylder and Goovaerts (1999), (2000) proposed a claim arrivalprocess on a time interval [0, t], named homogeneous. We revisitthis model and propose a simple extension useful in insurance.





HOMOGENEOUS POINT PROCESS

1) A point process on an interval [0, t] is constructed as follows.

Let N(t) be the total number of claims in [0, t]. The distribution ofN(t) may be chosen arbitrarily. If N(t) = n, the successive claiminstants form a sequence T1 < . . . < Tn of continuous r.v.

The associated densities πt,n(t1, . . . , tn) are defined by

πt,n(t1, . . . , tn)dt1 . . . dtn =

P[N(t) = n, T1 ∈ (t1, t1 + dt1), . . . ,Tn ∈ (tn, tn + dtn)],

on the set Dt,n = {0 < t1 < . . . < tn ≤ t}.





Attention is then focused on the special case of a homogeneouspoint process on [0, t], in which all the densities πt,n are constant(i.e. independent of t1, . . . , tn).

Since the Lebesgue measure of Dt,n is equal to tn/n!,

πt,n = P[N(t) = n]n!

tn , n ≥ 0.

A point process on [0, t] defines a point process on anysubinterval [0, s] ⊆ [0, t]. In the homogeneous case, the restrictedprocess is again a homogeneous process. The associated densitiesπs,k have to satisfy the compatibility conditions

πs,k =∞∑

n=k

πt,n(t − s)n−k

(n − k)!, k ≥ 0.





2) Consider the process Nt(s), 0 ≤ s ≤ t that counts thenumber of claims during (0, t). From above, one sees that

[Nt(s)|N(t) = n] =d Bin(n,

st

).





Alternatively, the constant densities assumption implies that theconditional claim arrival times

[T1, . . . ,Tn|N(t) = n]

are distributed as the order statistics of a sample of n independent(0, t)-uniform random variables :

[U1:n(0, t), . . . ,Un:n(0, t)].





Examples

(a) Suppose that N(t) has a mixed Poisson law with mixing rv Xand parameter t. Then, N(s) has a mixed Poisson law with mixingrv X and parameter s.

(b) Suppose that N(t) has a negative binomial law with parameterst and p (P[N(t) = n] =

(t+n−1n

)(1− p)tpn). Then, N(s) has a

negative binomial law with parameters t andpt(s) ≡ ps/[(1− p)t + ps].

(c) Suppose that N(t) has a binomial law with parameters t and p.Then, N(s) has a binomial law with parameters t and ps/t.





TIME SCALE TRANSFORMATION

An extended model is obtained just by making a transformationof the time axis.

Let Ft(s), 0 ≤ s <≤ t, be a nondecreasing continuous functionfrom Ft(0) = 0 to Ft(t) = 1.

The counting claim arrival process Nt(s), 0 ≤ s ≤ t is definedconditionally by

[Nt(s)|N(t) = n] =d Bin(n,Ft(s)).





An equivalent representation is that the conditional claim arrivaltimes

[T1, . . . ,Tn|N(t) = n]

are distributed as the order statistics

[V1:n(0, t), . . . ,Vn:n(0, t)].

of a sample of n i.i.d. random variables on (0, t) with Ff (.) asdistribution function.





Examples

(a) Suppose that N(t) has a mixed Poisson law with mixing rv Xand parameter t. Then, Nt(s) has a mixed Poisson law with mixingrv X and parameter tFt(s).

(b) Suppose that N(t) has a negative binomial law with parameterst and p. Then, Nt(s) has a negative binomial law with parameterst and pFt(s)/[1− p + pFt(s)].

(c) Suppose that N(t) has a binomial law with parameters t and p.Then, Nt(s) has a binomial law with parameters t and pFt(s).





On [0, t∗] or R+

Point processes with such an order statistic structure on [0, t∗] orR+ have been studied and characterized (see e.g. Puri (1992)).

Here are three particular of special interest.

(a) Suppose that {N(t)} is a Poisson process with rate λ. Then,N(t) has a Poisson distribution with rate λt, and

Ft(s) = s/t.

This remains true for a mixed Poisson process with random rate Λ.





(b) Suppose that {N(t)} is a linear birth process withimmigration with birth rate b and immigration rate a. Then, N(t)has a negative binomial distribution with parameters a/b and e−bt ,and

Ft(s) = (ebs − 1)/(ebt − 1).

In this model, a claim arrival increases the likelihood of furtherclaim arrivals (this translates an effect of contagion or clustering)





(c) Suppose that {N(t)} is a linear death-counting process withdeath rate b and initial size m. Then, N(t) has a binomialdistribution with parameters m and 1− e−bt , and

Ft(s) = (1− e−bs)/(1− e−bt).

This arises, for instance, with a group life insurance covering aclosed group of m individuals.




i.i.d. claim amountsExchangeable claim amountsEqualized claim amounts

2. ORDERED AGGREGATE CLAIM PROCESS

Suppose that each j-th claim is of random severity Xjindependent of the claim arrival times. Consider a time interval(0, t). For s < t, denote by St(s) the aggregate claim amount untiltime s, i.e.

[St(s)|N(t) = n] =n∑

j=1

Xj I [Vj :n(0, t) < s]

=

[Nt(s)|N(t)=n]∑j=1

Xj .





Our aim is to illustrate how the variability of N(t) can influencethe dispersion and the dependence structures of the processes

Nt(s), and St(s).

A similar question was discussed by Balakrishnan andKozubowski (2008) for a so-called weighted Poisson process.

We first suppose that the Xj ’s are i.i.d. Then, two cases withdependence are considered.





I.I.D. CLAIM AMOUNTS

Suppose that the Xj ’s are i.i.d. (distributed as X ). As inBalakrishnan and Kozubowski (2008), we measure the relativedispersion of a r.v. Y trough its index of dispersionV (Y ) ≡ Var(Y )/E (Y ). This parameter takes the value 1 for aPoisson variable, so that a random variable Y is said to be over(under)-dispersed when V (Y ) > 1 (< 1).





Result 1

By usual conditional arguments, one gets

V [St(s)] = E (X ) + V (X ) + E (X )Ft(s){V [N(t)]− 1}.

Thus, if N(t) is over (under)-dispersed, then V [St(s)], the relativedispersion of St(s), increases (decreases) with time s. In particular,over (under)-dispersion of N(t) implies the same property for Nt(s).





Similarly, for 0 ≤ s1 ≤ s2,≤ t,

Cov [St(s1), St(s2)− St(s1)]

E [St(s1)]= E (X )[Ft(s2)−Ft(s1)]{V [N(t)]−1},

showing that two consecutive increments are positively (negatively)correlated if N(t) is over (under)-dispersed. Also,

Cov [St(s1), St(s2)]

E [St(s1)]= E (X ) + V (X ) + E (X )Ft(s2){V [N(t)]− 1}.

Thus, in case of over (under)-dispersion of N(t), the covarianceincreases (decreases) with the horizon s2 (for fixed s1).





Examples. These conclusions can be applied to the cases where

N(t) has a Poisson distribution, for which V [N(t)] = 1,

N(t) has a negative binomial distribution, for which V [N(t)] > 1,

N(t) has a binomial distribution, for which V [N(t)] < 1.





Result 2

An alternative approach to see the effect of the variability ofN(t) is by means of the convex order ≤cx .

Consider another similar aggregate claim process St(s), s < t,defined with the same claim amounts Xj but a total number ofclaims N(t) instead of N(t).

Then, one can show that

N(t) ≤cx N(t) implies

Nt(s) ≤cx Nt(s), St(s) ≤cx St(s).





Examples. It is known that N(t), a Poisson variable with parameterλt, ≤cx N(t), a negative binomial variable with parameters t andλ/(λ+ 1). Moreover, for λ < 1, N(t) ≥cx N∗(t), a binomialvariable with parameters t and λ.

Thus, among these three cases, Var [St(s)] will be the largest (resp.smallest) when N(t) has a negative binomial (resp. binomial)distribution.





EXCHANGEABLE CLAIM AMOUNTS

Suppose that the Xj ’s are exchangeable ... The results are lesssimple ...

From the expression of Cov [St(s1), St(s2)], one sees that thiscovariance increases with s2 if

1 +Cov(X ,Y )

[E (X )]2>

E [N(t)]

E [N(t)] + V [N(t)]− 1.

So, when N(t) is overdispersed, this condition is satisfied ifCov(X ,Y ) is positive or, at least, not too much negative. WhenN(t) is underdispersed, Cov(X ,Y ) has to be sufficiently positive.





EQUALIZED CLAIM AMOUNTS

De Vylder and Goovaerts (2000) proposed that, given N(t) = n,the n severities X1, . . . ,Xn are replaced by an equal amount thatcorresponds to their average Sn = Sn/n. This "equalized" modelprovides, in some sense, a smooth approximation to the originalone.

Let Set (s), s < t, be the associated aggregate claim process.

Here too, ... Result 1 ...





Result 2

The equalized model is expected to be less variable than theinitial one. In fact, both models can be compared in the convexsense, namely

Set (s) ≤cx St(s).

This follows from the well-known property that for exchangeabler.v., the sequence of successive averages Sn is decreasing in theconvex sense.




Finite-time ruin probability

3. ORDERED INSURANCE RISK MODEL

Consider a risk model where

(i) the aggregate claim amounts are represented by the previousordered process St(s), for 0 < s < t ;

(ii) the claim amounts Xj are strictly positive random variables ;they are allowed to be discrete or continuous, independent orcorrelated, but are independent of the claim arrival process.Working with the partial sums Sj , let F (y1, . . . , yj) be the jointdistribution function of (S1, . . . , Sj) ;

(iii) the cumulated premiums, including the initial reserves, arerepresented by a function h(t), non-negative and non-decreasing,continuous or not, with h(0) = u and h(t)→∞ as t →∞.





FINITE-TIME RUIN PROBABILITY

The surplus of the company at time s is given by

Ut(s) = h(s)− St(s), 0 < s < t.

The problem under concern is the evaluation of the non-ruinprobability during a time interval [0, t].

Ruin is said to occur at time T (> 0) as soon as the reservesbecome negative or null. So, the probability of non-ruin until time tis given by

φ(t) = P(T > t) = P[St(s) < h(s), for 0 < s < t].





Our central result provides a simple and compact formula forφ(t), or φ(t), in terms of a special family of Appell polynomials.Such a family is associated to some real sequence U = {ui , i ≥ 0} ;it is denoted

{An(t|U), n ≥ 0},

where An(t|U) is a polynomial of degree n in t, with A0 = 1.

Appell polynomials are well-known in mathematics.





Their key property is that

A′n = An−1, with An(un−1) = 0, n ≥ 1.

They can be expressed under an integral form. This allows one tosee directly that

P[U1:n(0, 1) ≥ u1, . . . ,Un:n(0, 1) ≥ un, and Un:n(0, 1) ≤ x ]

= n! An(x |u1, . . . , un)

For their numerical evaluation, it is convenient to use the following(Taylor) expansion :

An(t|U) =n∑

j=0

An−j(0|U) t j/j!,

where the coefficients An−j(0|U) are obtained from the recursion

An(0|U) = −n∑

j=1

An−j(0|U) ujn−1/j!, n ≥ 1,





In ruin theory, Appell polynomials were first exploited in Picardand Lefèvre (1997). An extension of these polynomials in terms ofpseudopolynomials (Picard and Lefèvre (1996), (2003)) was alsoused. Various applications are in e.g. Picard, Lefèvre and Coulibaly(2003) and Lefèvre and Loisel (2009). Appell polynomials are alsoused in e.g. Ignatov and Kaishev (2000), Ignatov and Kaishev(2004), Kaishev and Dimitrova (2006). They are underlying in DeVylder and Goovaerts (2000).

For a different approach based on Laplace transforms, see e.g.Perry, Stadje and Zacks (1999), (2003) and Zacks (2005).





Given any level y for the aggregate claim amount, define theinstant r(y) when the premium income becomes large enough toavoid ruin, i.e.

r(y) = inf{s : h(s) > y}, y ≥ 0.

In particular, if the premium rate is a constant c and the initialreserves are equal to u, then r(y) = 0 if y < u, otherwiseu + cr(y) = y , so that

r(y) = (y − u)+/c , y ≥ 0.





Central result

For t ≥ 0, φ(t) is given by

E{N(t)! AN(t)

[1|Ft(r(S1)), . . . ,Ft(r(SN(t)))

]I[SN(t) < h(t)

]}.

This result generalizes formulas given in e.g. Picard and Lefèvre(1997), Ignatov and Kaishev (2000), De Vylder and Goovaerts(2000), Lefèvre and Loisel (2009).





Proof.

(1)

φ(t) =∞∑

n=0

P[N(t) = n] P[T > t|N(t) = n]

(2)

P[(1)] =

∫Dh(t),n

P[T > t|N(t) = n, S1 = y1, . . . , Sn = yn] dF (y1, . . . , yn)

(3) Let T1, . . . ,Tn denote the n consecutive claim occurrencetimes :

P[(2)] = P[T1 ≥ r(y1), . . . ,Tn ≥ r(yn)|N(t) = n].





(4) (T1, . . . ,Tn) is distributed, conditionally on [N(t) = n], as[F−1t (U1:n(0, 1)), . . . ,F−1t (Un:n(0, 1))] :

P[(3)] = P[U1:n(0, 1) ≥ Ft(r(y1)), . . . ,Un:n(0, 1) ≥ Ft(r(yn))].

(5)

P[U1:n(0, 1) ≥ u1, . . . ,Un:n(0, 1) ≥ un, and Un:n(0, 1) ≤ u]

= n! An(u|u1, . . . , un)

Thus,P[(4)] = n! An[1|Ft(r(y1)), . . . ,Ft(r(yn))].




When Ft is linearWhen Ft is exponentialOver an infinite horizon

TWO PARTICULAR RISK MODELS

In this part, it is assumed that

1) the premium rate is a constant c ,

2) and the d.f. Ft(s), 0 ≤ s ≤ t, is

(a) either of a linear form

st,

(b) or of an exponential form

ebs − 1ebt − 1

, b real.





WHEN Ft IS OF LINEAR FORM

As for a (mixed) Poisson process.

Lemma

Let Sj = X1 + . . .+ Xj , j ≥ 1, where the Xj ’s are nonnegativeexchangeable random variables. Then,

E [An(x |S1, . . . , Sn) | Sn] = (x − Sn)xn−1

n!.





From the general formula, we know that P(T ≥ t|N(t) = n),denoted by φ(t|n), is given by

φ(t|n) = n! E{An[1|(S1 − u)+/ct, . . . , (Sn − u)+/ct

]I [Sn < u + ct]

}.

Using the lemma and some properties of Appell polynomials, weobtain two different expressions for these probabilities.

Notation : let G be the d.f. of the Xj ’s (discrete or continuous),and denote

mk(y |t) =

∫[0,u+ct−y)

(1− z

u + ct − y

)G ∗k(dz),

bl (y |t, n) =

(nl

)(y − u

ct

)l (1− y − u

ct

)n−l

.





Proposition. For t > 0 and n ≥ 1,

φ(t|n) =∑

k+l=n

∫[0,u]

bl (y |t, n) mk(y |t) G ∗l (dy).

An alternative formula is

φ(t|n) = P(Sn < u+ct)−∑

k+l=n, l≥1

∫(u,u+ct)


Remark. If u = 0, this gives the well-known ballot type formula

φ(t|n) = E (1− Sn/ct)+.





Special cases

Consider the compound Poisson model with parameter λ andmeasure Q. Define for t > 0 and any Borel set A,

µt(A) ≡ P[S(t) ∈ A] =∞∑

n=0

e−λt (λt)n

n!Q∗n(A).

Observe that µt(A) may be considered as an analytic function forall reals t, and not only for t ≥ 0. Of course, µt(A) looses itsprobabilistic interpretation when t < 0. Nevertheless, one easilychecks that even when t < 0, the well-known Panjer algorithmallows us to determine the pseudo-distribution µt .





Two compact formulas for φ(t) for the compound Poisson model :

Corollary. For t > 0,

φ(t) =

∫[0,u]

µ y−uc

(dy)

∫[0,u+ct−y)

(1− z

u + ct − y

)µt+ u−y

c(dz).


φ(t) = µt([0, u + ct))−∫(u,u+ct)

µ y−uc

(dy)

∫[0,u+ct−y)

(1− z

u + ct − y

)µt+ u−y

c(dz).





The second formula is of Seal type and is traditionally presented inthe literature.

The first formula is much less standard. By comparison, it isespecially convenient since y is restricted to [0, u] (independently oft). Note that the time index in µ(y−u)/c is nonpositive, which isallowed as explained above.Different variants of this formula, under additional assumptions,were derived in Picard and Lefèvre (1997), De Vylder (1999), DeVylder and Goovaerts (1999), Lefèvre and Loisel (2009).





Consider the equalized risk model.

Corollary. For t > 0,

φ(t|n) =

∫[0,u+ct)

{ ∑k+l=n

(nl

)[−((l/n)y − u)−

ct]l [1− (l/n)y − u

ct]k−1

}

(1− y − uct

) G ∗n(dy).


φ(t|n) = P(Sn < u + ct) −∫(u,u+ct)

∑k+l=n, l≥1

(nl

)[((l/n)y − u)+

ct]l [1− (l/n)y − u

ct]k−1

(1− y − u

ct) G ∗n(dy).





Remark. If u = 0, φ(t|n) has the same value as in the initial i.i.d.model (ballot formula).

De Vylder and Goovaerts (2000) conjectured that when u > 0, thenon-ruin probability is larger in the equalized version ... this seemsplausible and is indeed observed through many numericalexperiments ... we are trying to prove it.





WHEN Ft IS OF EXPONENTIAL FORM

As for a linear birth process with immigrationor a linear death-counting process.

Lemma

Let Πj = X1 . . .Xj , j ≥ 1, where the Xj ’s are nonnegative i.i.d.random variables (distributed as X ). For any reals α, x ,

E [(Πn)α An(x |Π1, . . . ,Πn)] = An[xE (Xα)|E (Xα+1), . . . ,E (Xα+n)],

provided E (Xα+i ), i ≥ 0, exist.





In this exponential case, the general formula becomes

φ(t|n) = n! E{An

[1|(eβSi − eβu)+/(eβ(u+ct) − eβu), 1 ≤ i ≤ n

]I [Sn < u + ct]}.

Introducing some notation/definitions :

bl (y |t, n) = . . . ,

mk(y |t) = . . . ,

we get two formulas for conditional probabilities of non-ruin thathave the same structure as in the linear case :





Proposition. For t > 0 and n ≥ 1,

φ(t|n) =∑

k+l=n

∫[0,u]



φ(t|n) = P(Sn < u+ct)−∑

k+l=n, l≥1

∫(u,u+ct)






OVER AN INFINITE HORIZON

(a) N(t) is a Poisson process

Proposition

limt→∞

φ(t) = (1− λµ

c)+

∫[0,u]

µ y−uc

(dy),


limt→∞

φ(t) = I (c > λµ)− (1− λµ

c)+

∫(u,∞)

µ y−uc

(dy).

Here too, the first formula is non-standard, with a negative indexin µ(y−u)/c . For some special cases, see e.g. Gerber (1988), Shiu(1988) and Picard and Lefèvre (1997).





(b) N(t) is a linear birth process with immigration

Proposition. Ruin is a.s.

This was expected : the premium income is linear while N(t) growsat an exponential rate as t →∞ (e.g. Resnick (1992)).





(c) N(t) is a linear death-counting process

Proposition. limt→∞ φ(t) is given by

= (−1)mm∑

l=0

m!

l !Bm−l (∞)

∫[0,u]

e−mβ(u−y)[1− eβ(u−y)]lG ∗l (dy),

where Bn(∞) satisfy the recursion :

Bn(∞) = −n−1∑k=0

bn−kn

(n − k)!Bk(∞), n ≥ 1.

with B0 = 1 and bn = E (enβXj ).






= 1−m∑

l=1

(−1)m−l m!

l !Bm−l (∞)

∫(u,∞)

eβ(m−l)(y−u)[1−eβ(y−u)]lG ∗l (dy).

EXTENSIONS AND VARIANTS ... for a next time ...





From the papers

Lefèvre,C. and Picard, P. (2011). Insurance : Mathematics andEconomics 49, 512-519.

Lefèvre,C. and Picard, P. (2012). Working paper.

A few references -partial list-

Balakrishnan, N. and Kozubowski, T.J. (2008). Statistics andProbability Letters 78, 2346-2352.

De Vylder, F.E. and Goovaerts, M.J. (1999). Insurance :Mathematics and Economics 24, 249-271.

De Vylder, F.E. and Goovaerts, M.J. (2000). Insurance :Mathematics and Economics 26, 223-238.





Ignatov, Z.G. and Kaishev, V.K. (2004). Journal of AppliedProbability 41, 570-578.

Kaishev, V.K. and Dimitrova, D.S. (2006). Insurance :Mathematics and Economics 39, 376-389.

Lefèvre, C. and Loisel, S. (2009). Methodology and Computing inApplied Probability 11, 425-441.

Picard, P. and Lefèvre, C. (1997). Scandinavian Actuarial Journal1, 58-69.

Picard, P. and Lefèvre, C. (2003). Stochastic Processes and theirApplications 104, 217-242.

Puri, P.S. (1982). Journal of Applied Probability 19, 39-51.


Documents

On ruin probabilities for risk models with ordered claim ... · Orderedclaimarrivalprocess Orderedaggregateclaimprocess Orderedinsuranceriskmodel Twoparticularriskmodels On ruin probabilities