- Email: [email protected]
Inequalities for the ruin probability in a controlled discrete-time risk process
European Journal of Operational Research 204 (2010) 496–504
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
Inequalities for the ruin probability in a controlled discrete-time risk process M. Diasparra a,*, R. Romera b a b
Department of Pure and Applied Mathematics, Universidad Simón Bolívar, Bolivarian Republic of Venezuela Department of Statistics, Universidad Carlos III de Madrid, Spain
a r t i c l e
i n f o
Article history: Received 12 December 2008 Accepted 10 November 2009 Available online 24 November 2009 MSC: 91B30 60J05 60K10
a b s t r a c t Ruin probabilities in a controlled discrete-time risk process with a Markov chain interest are studied. To reduce the risk of ruin there is a possibility to reinsure a part or the whole reserve. Recursive and integral equations for ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived given a constant stationary policy. The relationships between these inequalities are discussed. To illustrate these results some numerical examples are included. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Risk process Ruin probability Proportional reinsurance Lundberg’s inequality
1. Introduction This paper studies an insurance model where the risk process can be controlled by proportional reinsurance. The performance criterion is to choose reinsurance control strategies to bound the ruin probability of a discrete-time process with a Markov chain interest. Controlling a risk process is a very active area of research, particularly in the last decade; see [8,9,11,12], for instance. Nevertheless obtaining explicit optimal solutions is a difficult task in a general setting even for the classical risk process. In fact, the explicit solution in the classical controlled risk process is known only in a very few cases. This is the main reason why we deal with the optimal control problem by using an alternative method than dynamic programming. Hence, an alternative method commonly used in ruin theory is to derive inequalities for ruin probabilities (see [1,4–6,12,13]). Following Cai [2] and Cai and Dickson [3], we model the interest rate process as a denumerable state Markov chain. This model can be in fact a discrete counterpart of the most frequently occurring effect observed in continuous interest rate process, e.g., mean-reverting effect. Stochastic inequalities for the ruin probabilities are derived by martingales and inductive techniques. The inequalities can be used to obtain upper bounds for the ruin probabilities. We use the proposal by Gaier et al. [4]
* Corresponding author. Tel.: +58 212 761 6143; fax: +58 212 761 5945. E-mail addresses: [email protected] (M. Diasparra), [email protected] (R. Romera). 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.11.015
and Schmidli [12], we restrict ourselves to use constant control policies (see Remark 1). Explicit condition are obtained for the optimality of employing no reinsurance. The outline of the paper is as follows. In Section 2 the risk model is formulated. Some important special cases of this model are briefly discussed. In Section 3 we derive recursive equations for finite-horizon ruin probabilities and integral equations for the ultimate ruin probability. In Section 4 we obtain upper bounds for the ultimate probability of ruin. An analysis of the new bounds and a comparison with the Lundberg’s inequality is also included. Finally, in Section 5 we illustrate our results on the ruin probability in a risk process with a heavy tail claims distribution under proportional reinsurance and a Markov interest rate process. We conclude in Section 6 with some general comments and some suggestions further research. 2. The model We consider a discrete-time insurance risk process in which the surplus X n varies according to the equation
X n ¼ X n1 ð1 þ In Þ þ Cðbn1 Þ Z n hðbn1 ; Y n Þ;
ð1Þ
for n P 1, with X 0 ¼ x P 0. Following Schmidli [12, p. 21], we introduce an absorbing (cemetery) state ,, such that if X n < 0 or X n ¼ ,, then X nþ1 ¼ ,. We denote the state space by X ¼ R [ f,g. Let Y n be the nth claim payment, which we assume to form a sequence of i.i.d. random variables with common probability distribution function
497
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
(p.d.f.) F. The random variable Z n stands for the length of the nth period, that is, the time between the occurrence of the claims Y n1 and Y n . We assume that fZ n g is a sequence of i.i.d. random variables with p.d.f. G. This case includes a controlled version of the Cramér– Lundberg model if we assume that the claims occur as a Poisson process. Of course, we can also think of the case where Z n ¼ 1 is deterministic. In addition, we suppose that fY n gnP1 and fZ n gnP1 are independent. The process can be controlled by reinsurance, that is, by choosing the retention level (or proportionality factor or risk exposure) b 2 B of a reinsurance contract for one period, where B :¼ ½bmin ; 1, and bmin 2 ð0; 1 will be introduced below. Let fIn gnP0 be the interest rate process; we suppose that In evolves as a Markov chain with a denumerable (possibly finite) state space I consisting of non-negative rational numbers and the process fIn gnP0 is independent of fZ n gnP1 1 and fY n gnP1 . The function hðb; yÞ with values in ½0; y specifies the fraction of the claim y paid by the insurer, and it also depends on the retention level b at the beginning of the period. Hence y hðb; yÞ is the part paid by the reinsurer. The retention level b ¼ 1 stands for the control action no reinsurance. In this article, we consider the case of proportional reinsurance, which means that
hðb; yÞ :¼ b y;
ð2Þ
with retention level b 2 B. The premium (income) rate c is fixed. Since the insurer pays to the reinsurer a premium rate, which depends on the retention level b, we denote by CðbÞ the premium left for the insurer if the retention level b is chosen, with
0 6 cmin < c 6 CðbÞ 6 c;
b 2 B;
P
where pij P 0 and j pij ¼ 1 for all i; j 2 I. The ruin probability when using the policy p, given the initial surplus x, and the initial interest rate I0 ¼ i is defined as p
w ðx; iÞ :¼ P
p
1 [
! fX k < 0gjX 0 ¼ x; I0 ¼ i ;
ð6Þ
k¼1
which we can also express as
wp ðx; iÞ ¼ Pp ðX k < 0 for some k P 1jX 0 ¼ x; I0 ¼ iÞ:
ð7Þ
Similarly, the ruin probabilities in the finite horizon case are given by
wpn ðx; iÞ :¼ Pp
n [
! fX k < 0gjX 0 ¼ x; I0 ¼ i :
ð8Þ
k¼1
Thus,
wp1 ðx; iÞ 6 wp2 ðx; iÞ 6 6 wpn ðx; iÞ 6 ; and
lim wpn ðx; iÞ ¼ wp ðx; iÞ:
n!1
The following lemma is used below to simplify some calculations. Lemma 1. For any given policy p, there is a function wp ðxÞ such that
wp ðx; iÞ 6 wp ðxÞ for every initial state x > 0 and initial interest rate I0 ¼ i. Proof. By (1) and (2), the risk model is given by
where c denotes the minimal value of the premium considered by the insurer and cmin corresponds to b ¼ 0 in CðbÞ. We define bmin :¼ minfb 2 ð0; 1jCðbÞ P c g. Since CðbÞ P c > cmin P 0, there exists a bmin > 0. Moreover, CðbÞ is an increasing function that we will calculate according to the expected value principle with added safety loading h from the reinsurer:
E½Y CðbÞ ¼ c ð1 þ hÞð1 bÞ ; E½Z
ð3Þ
where E½Y denotes the mean claim and E½Z denotes the average which we astime between claims. Note that, cmin ¼ c ð1 þ hÞ E½Y E½Z sume that to be greater or equal than zero. We define Markovian control policies p ¼ fan gnP1 , which at each time n depend only on the current state, that is, an ðX n Þ :¼ bn for n P 0. Abusing notation, we will identify functions a : X ! B with stationary strategies, where B ¼ ½bmin ; 1 is the decision space. However, in this study we will focus on stationary constant policies following the arguments in Remark 1. Consider an arbitrary initial state X 0 ¼ x P 0 (note that the initial value is not stochastic) and a control policy p ¼ fan gnP1 . Then, by iteration of (1) and assuming (2), and (3), it follows that for n P 1, X n satisfies
Xn ¼ x
n Y l¼1
ð1 þ Il Þ þ
n X l¼1
ðCðbl1 ÞZ l bl1 Y l Þ
n Y
! ð1 þ I m Þ :
ð4Þ
m¼lþ1
Since In P 0, we have
X n ¼ X n1 ð1 þ In Þ þ Cðbn1 ÞZ n bn1 Y n P X n1 þ Cðbn1 ÞZ n bn1 Y n :
ð9Þ
Define recursively
e n :¼ X e n1 þ Cðbn1 ÞZ n bn1 Y n ; X
ð10Þ
e 0 ¼ x. Hence, X n P X e n for all n 2 N. Clearly, if X n < 0, with X 0 ¼ X e n < 0. then X Let
) 1 [ and E1 :¼ x 2 X fX n ðxÞ < 0g n¼1 ( ) [ 1 n o fn ðxÞ < 0 ; E2 :¼ x 2 X X n¼1 (
and note that E1 E2 . Therefore,
P
1 [
p
! fX n < 0gjI0 ¼ i
n¼1
6P
p
1 [
! e f X n < 0gjI0 ¼ i ;
n¼1
e n do not depend on In , we obtain from (6) and since the X
Let ðpij Þ be the matrix of transition probabilities of fIn g, i.e.,
pij :¼ PðInþ1 ¼ jjIn ¼ iÞ;
X n ¼ X n1 ð1 þ In Þ þ Cðbn1 ÞZ n bn1 Y n :
ð5Þ
wp ðx; iÞ ¼ Pp
1 [
!
fX n < 0gjX 0 ¼ x; I0 ¼ i
n¼1 1
This hypothesis is necessary when one works with a methodology based on martingale theory, as it is the case. Moreover, from the financial point of view, it is not difficult to find particular scenarios for which the interest rate and the waiting time are independent. For example, if one invest in a predictable asset, which is typical on stochastic models of interest rates in order to preserve risk-neutral pricing, and if one considers that the number of events before the maturity may be zero or one, which is not restrictive because these maturity periods are often very short on applications, then, to consider independence between capitalization and waiting time seems quite natural.
6P
p
1 [
! e n < 0gjX 0 ¼ x fX
¼: wp ðxÞ:
n¼1
We denote by P the policy space. A control policy p is said to be optimal if for any initial values ðX 0 ; I0 Þ ¼ ðx; iÞ, we have
wp ðx; iÞ 6 wp ðx; iÞ
498
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
for all p 2 P. Schmidli [12] and Schäl [11] show the existence of an optimal control policy for some special cases of the model risk (1). However, even in these special cases it is extremely difficult to ob tain closed expressions for wp ðx; iÞ. We are thus led to consider bounds for the ruin probabilities, which we do in Sections 3–5. Remark 1. For the reinsurance problem in the case of small claims, Schmidli [12] obtains that the optimal strategy depends on the current wealth, in the following sense: If bðxÞ is the optimal reinsurance policy for x small enough, bðxÞ ¼ 1, that is, no reinsurance, and if x tends to infinity the reinsurance bðxÞ tends to some appropriate constant k. Then, we can argue to consider constant policies of reinsurance in a more general setting, based on this result of Schmidli, in the sense that, for small claims the optimal strategy is asymptotically constant. A second additional argument, is the following one. In Gaier et al. [4], they obtain upper bounds for the ruin probability for a model with investment in a risky asset (no reinsurance) by considering a constant policy, and show that this constant policy is asymptotically optimal (when the initial capital tends to infinity). Finally, a third argument for using constant policies is the following. First, we assume that PðbY > CðbÞZÞ > 0 for all b 2 B. Because, if there is some be 2 B such that Pðbe Y > Cðbe ÞZÞ ¼ 0, the ruin can be prevented by retention level be and the risk process considered in this case becomes trivial. Secondly, we assume the net profit condition Ep ½CðbÞZ bY > 0 for some p 2 P. Otherwise, ruin cannot be prevented because the surplus would be decreasing in time for all reinsurance treaties. Therefore, using the law of large numbers, we have n 1X ½CðbÞZ i bY i ! Ep ½CðbÞZ bY a: e:; n i¼1
this implies that for the stationary strategy bn ¼ b the process X n tends to infinity (in particular, inf n X n > 1). Hence, there is an initial capital X 0 ¼ x such that Pðinf n X n P 0jX 0 ¼ xÞ > 0. Because there is a strictly positive probability that from initial capital zero the set ½x; 1Þ is reached before the set ð1; 0Þ, we get also that Pðinf n X n P 0jX 0 ¼ 0Þ > 0. Finally, we have a stationary strategy for which ruin is not certain. Moreover, we will assume in this article that bE½Y < CðbÞE½Z (where Y and Z are independent) and Ep ½etY < 1 for t 2 ð1; .Þ with . 2 ð0; 1, then there exists a constant R0 R0 ðbÞ > 0 satisfying
Ep eR0 ðCðbÞZbY Þ ¼ 1:
ð11Þ
Note that our model is quite general, and contains the following particular cases: 1. The classical risk process.2 2. The classical risk process with investment in a Markov chain interest rate. 3. The model with reinsurance in the above cases. 4. The model with reinsurance and dividends.
or equivalently,
X n ¼ X n1 þ Cðbn1 ÞZ n bn1 Y n : The corresponding ruin probability is
1. If In ¼ 0 for all n P 1, then the risk model (4) reduces to the discrete-time risk model with proportional reinsurance:
Xn ¼ x
n X
ðbt1 Y t Cðbt1 ÞZ t Þ
t¼1
2
See, Diasparra and Romera [7].
ð12Þ
! fX n < 0gjX 0 ¼ x :
n¼1
Assuming, constant stationary strategies, say bn ¼ b0 , then, by (12), p
w ðxÞ ¼ P
1 [
p
(
n¼1
) ! n X ½b0 Y t Cðb0 ÞZ t > x jX 0 ¼ x : t¼1
Therefore, by the classical Lundberg inequality for ruin probabilities (see [1,5,13]), for x P 0
wp ðxÞ 6 eR0 x :
ð13Þ
2. Exponentially distributed length periods. Our process (1) a controlled version of Cramér–Lundberg model if the claims occur as a Poisson process, in which case the Z n are exponentially distributed, say Z n ’ ExpðkÞ. We suppose that In ¼ 0 for all n P 1, and that moment generating function M Y ðsÞ is finite in an open interval ð1; .Þ with . 2 ð0; 1. Thus, Y n has a compound distribution with expectation kl and moment generation function R1 e½kðMY ðsÞ1Þ . Let M Y ðb; rÞ :¼ 0 ebry dFðyÞ be the moment generating function of the part of the claim the insurer has to pay if the retention level b is chosen. We assume constant stationary strategies, say bn ¼ b0 for all n P 1. Moreover, we assume that Cðb0 Þ > b0 kl and M Y ðb0 ; rÞ < 1 for some r > 0 and b0 2 B. It P is clear that the risk process X n x ¼ nk¼1 ðCðbÞZ n bY k Þ satisfies all the hypotheses of Theorem 14 in [5, p. 10]. Then
Ep eR0 ½CðbÞbY n ¼ eR0 Cðb0 Þ e½kðMY ðb0 ;R0 Þ1Þ : Then, by (11), we have that the adjustment coefficient R0 ¼ R0 ðb0 Þ fulfils
R0 Cðb0 Þ þ kðM Y ðb0 ; R0 Þ 1Þ ¼ 0:
ð14Þ
By Lemma 4.1 Schmidli [12], R0 is unimodal and it attains its maximum value at a point b0 2 B. Then, it is easy to see that it is optimal to have no reinsurance (b0 ¼ 1) if and only if the safety loading h is too high in the sense that
1þhP
M 0Y ð1; R0 Þ
l
¼
E½YeR0 Y : E½Y
ð15Þ
with M0 denoting the derivative of M. 3. Let dn be the constant, short-term dividend rate in the nth period (the dividends are payments made by a corporation to its shareholder members). Then the discrete-time risk model with stochastic interest rate and dividends is given by
X n ¼ X n1 ð1 þ In Þ þ Cðbn1 ÞZ n hðbn1 ; Y n Þ dn X n ; with hðb; yÞ as in (2). Thus, rearranging terms,
X n ¼ X n1
To conclude this section, let us briefly review some particular cases of the risk model (1). Special cases.
1 [
wp ðxÞ :¼ Pp
1 þ In 1 þ dn
þ
Cðbn1 Þ hðbn1 ; Y n Þ Zn : ð1 þ dn Þ ð1 þ dn Þ
Yn In dn Let Y 0n :¼ ð1þd and I0n :¼ ð1þd . Since fIn g and fY n g are indepennÞ nÞ n1 Þ . Then the dent, then so are fI0n g and fY 0n g. Let C0 ðbn1 Þ :¼ Cðb ð1þdn Þ model becomes
X n ¼ X n1 1 þ I0n þ C0 ðbn1 ÞZ n hðbn1 ; Y 0n Þ; which from a statistical viewpoint is essentially the same as the model without dividends (1) and can be analyzed in a similar way. 4. As an extension of the latter case, some companies have dividend reinvestment plans (or DRIPs). These plans allow share-
499
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
holders to use dividends to systematically buy small amounts of stock. Let e d n be the short-term dividend reinvestment rate in the nth period, e d n 2 ½0; 1Þ. Then, the discrete-time risk model with stochastic interest rate and dividends reinvestment is given by
X n ¼ X n1 ð1 þ In Þ þ Cðbn1 ÞZ n hðbn1 ; Y n Þ þ e dnXn:
X n ¼ X n1
!
Cðbn1 Þ hðbn1 ; Y n Þ Zn : e ð1 d Þ ð1 e d Þ
þ
n
Let Y 00n :¼
n
In þe dn
, I00n :¼
Yn
ð1e dn Þ
n
, and C00 ðbn1 Þ :¼ Cðbn1 Þ. It follows that ð1e dn Þ
00
X n ¼ X n1 1 þ In þ C ðbn1 ÞZ n
fX k < 0gjY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ i
¼ 1;
ð19Þ
k¼1
while if 0 6 uðy; zÞ 6 h1 , then
Pp ðX 1 < 0jY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ iÞ ¼ 0:
Pp
ð20Þ
n[ þ1
!
fX k < 0gjY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ i
k¼1
ð1e dn Þ
00
n[ þ1
e n g ; fZ e n g , and feI n g Let f Y nP1 nP1 nP0 be independent copies of fY n gnP1 ; fZ n gnP1 , and fIn gnP0 , respectively. e k :¼ b0 Y e k . Thus, (20) and (4) yield that for e k Cðb0 Þ Z Let U 0 6 uðy; zÞ 6 h1 ,
Hence, rearranging terms, we obtain
1 þ In 1e d
Pp
!
hðbn1 ; Y 00n Þ;
¼P
p
nþ1 [ k¼2
which, again, is essentially the same as the model (1).
¼P
p
nþ1 [
! fX k < 0gjY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ i (
) k k k Y Y X ðh1 uðy; zÞÞ ð1 þ Il Þ Ul ð1 þ Im Þ < 0
k¼2
Let us go back to the original risk model (1). In the next section, we will derive recursive equations for the ruin probabilities and integral equations for the ultimate ruin probability associated to the model (1).
l¼1
!
l¼1
m¼lþ1
jX 0 ¼ x; I1 ¼ j ¼P
Remark 2. Given a p.d.f. G, we denote the tail of G by G, that is, GðxÞ :¼ 1 GðxÞ.
n [
p
(
) k k k Y X Y e e e Ul ðh1 uðy; zÞÞ ð1 þ I l Þ ð1 þ I m Þ < 0
k¼1
l¼1
!
jX 0 ¼ x; eI 0 ¼ j
l¼1
m¼lþ1
¼ wpn ðh1 uðy; zÞ; jÞ ¼ wpn ðxð1 þ jÞ uðy; zÞ; jÞ
3. Recursive and integral equations for ruin probabilities In this section, we first derive a recursive equation for wpn ðx; iÞ. Secondly, we give an integral equation for wp ðx; iÞ. Finally, we obtain an equation for the ruin probability with horizon n ¼ 1 given I0 ¼ i; X 0 ¼ x and a stationary policy p. These results, which are valid for any initial interest rate, are summarized in the following lemma.
where the third equality follows from the Markov property of fIn gnP0 , and the independence of fY n gnP1 ; fZ n gnP1 and fIn gnP0 . Let us now consider the event A ¼ fY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ ig, and recall that FðyÞ ¼ PðY 6 yÞ and GðzÞ ¼ PðZ 6 zÞ. From (8) and (4) we obtain p
wnþ1 ðx; iÞ ¼ P
wp1 ðx; iÞ ¼
X
pij
Fðsj ðzÞÞdGðzÞ;
ð16Þ
X
pij
j2I
pij
Z
wnþ1 ðx; iÞ ¼ Z sj ðzÞ
1
0
0
X
pij
wpn ðxð1 þ jÞ X j2I
þ pij
Z
Z
Fðsj ðzÞÞdGðzÞ:
ð17Þ
¼
0
X
wp ðx; iÞ ¼
X
pij
j2I
þ
X j2I
1
0
pij
Z sj ðzÞ
þ
wp ðxð1 þ jÞ uðy; zÞ; jÞdFðyÞdGðzÞ
¼
1
Fðsj ðzÞÞdGðzÞ:
ð18Þ
X j2I
0
Proof. Let U k :¼ uðY k ; Z k Þ ¼ b0 Y k Cðb0 ÞZ k . Given Y 1 ¼ y; Z 1 ¼ z, the control strategy p, and I1 ¼ j, by definition we have U 1 ¼ uðy; zÞ. Therefore,
X 1 ¼ xð1 þ I1 Þ U 1 ¼ h1 uðy; zÞ; where h1 ¼ xð1 þ jÞ Thus, if uðy; zÞ > h1 then
Pp ðX 1 < 0jY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ iÞ ¼ 1: This implies that for uðy; zÞ > h1
þ
Z
Z sj ðzÞ
P
n[ þ1
p
nþ1 [
!
fX k < 0gjA dFðyÞdGðzÞ !
)
fX k < 0gjA dFðyÞdGðzÞ
k¼1
0
wpn ðxð1 þ jÞ uðy; zÞ; jÞdFðyÞdGðzÞ )
1
dFðyÞdGðzÞ
Z
0
1
Z sj ðzÞ 0
wpn ðxð1 þ jÞ uðy; zÞ; jÞdFðyÞdGðzÞ
Fðsj ðzÞÞdGðzÞ :
1
fX k < 0gjA dFðyÞdGðzÞ:
k¼1
Z sj ðzÞ
sj ðzÞ
pij
!
k¼1
Pp
1
Z
1 0
0
Z
Z
nþ1 [
0
0
j2I
Moreover,
1
1
Z
p
0 Þz sj ðzÞ ¼ xð1þjÞþCðb and from (19) we have b0
sj ðzÞ
pij
P
0
0
1
1
0
(Z Z
1
Z
0
j2I
uðy; zÞ; jÞdFðyÞdGðzÞ þ
Z
1
Then, recalling that p
Z
X j2I
and for n ¼ 1; 2; . . .
wpnþ1 ðx; iÞ ¼
¼
1
0
j2I
fX k < 0gjX 0 ¼ x; I0 ¼ i
k¼1
Lemma 2. Let uðy; zÞ :¼ b0 y Cðb0 Þz, where b0 is the initial retention level. Let sj ðzÞ :¼ ðxð1 þ jÞ þ Cðb0 ÞzÞ=b0 ; X 0 ¼ x P 0, and pij as in (5). Then
Z
!
nþ1 [
p
ð21Þ
0
This gives (17). In particular,
wp1 ðx; iÞ ¼
X j2I
pij
Z
1
Fðsj ðzÞÞdGðzÞ:
0
Finally, letting n ! 1 in (21) and using dominated convergence we obtain limn!1 wpnþ1 ðx; iÞ ¼ wp ðx; iÞ, and (18) follows. h Remark 3. If we consider the risk model without reinsurance, that is, b ¼ 1, we obtain similar results to those in Cai and Dickson [3].
500
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
and hence
4. Bounds for ruin probabilities We will use the results obtained in Section 3 to find upper bounds for the ruin probabilities with infinite horizon taking into account the information contributed by the Markov chain of the interest rate process. We derive a functional for the ultimate ruin probability in terms of the new worse than used in convex (NWUC) ordering; see Remark 4, below. This idea was first introduced by Willmot and Lin [13] and has been generalized by other authors. We will present two upper bounds for the ruin probabilities. The first bound is obtained by an inductive approach, and the second by a martingale approach. These bounds are discussed in Remark 5, at the end of this section.
bE½eR0 ½xð1þjÞþCðb0 Þzb0 yð1þI1 Þ jI0 ¼ j 6 beR0 ½xð1þjÞþCðb0 Þzb0 y : Therefore, replacing (26) in (17), we get p
wnþ1 ðx; iÞ 6
X
pij b
Z
þ
1
e
X
Z pij b
X
1
eR0 ½xð1þjÞþCðb0 Þz
0
Z pij b
eR0 ½xð1þjÞþCðb0 Þz
¼ bEp ½eR0 bY 1
pij Ep ½eR0 xð1þjÞ ¼ bEp ½eR0 ½xð1þI1 Þ jI0 ¼ i;
ð22Þ
where b ¼ bðb0 Þ and is given by 1
b
R1 t
¼ inf
eR0 b0 y dFðyÞ
eR0 b0 t FðtÞ
tP0
:
!1 dFðyÞ
R0 b0 y
1
ð23Þ
#
for any # P 0. This implies that for every x P 0, i P 0, and b0 2 B, by (16) and (23) we have p
w1 ðx; iÞ ¼
X
pij
j2I
6
X
Z
1
Fðsj ðzÞÞdGðzÞ
0
p R0 bY 1
pij bE ½e
p R0 bY 1
¼ bE ½e
j2I
p R0 bY 1
¼ bE ½e
X
1
R0 b0
e
xð1þjÞþCðb0 Þz b0
!
pij
Z
¼ bE ½e
1
eR0 b0 y dFðyÞdGðzÞ
0
X
pij
Z
1
eR0 ½xð1þjÞþCðb0 Þz dGðzÞ
0
e
Remark 4. A distribution F concentrated on ð0; 1Þ is said to be new worse than used in convex (NWUC) ordering if, for all x; y P 0
Z
1
FðzÞdz P FðyÞ
Z
R0 ½xð1þjÞþCðb0 Þz
pij E e
FðzÞdz:
x
For example, let F a phase-type distribution with parameters ða; TÞ (see [1, pp. 215–222]). Then F is NWUC if and only if T 1 and ! T 1 eTy ðI 1 aÞ are both non-negative or non-positive definite simultaneously for all y P 0 (where I represent the identity matrix ! and 1 is the column vector of ones). Corollary 4. Under the hypotheses of Theorem 3, and assuming that Ep ½eR0 bY 1 < 1 for all b 2 B, and that, in addition, F is a NWUC distribution, we have
ð27Þ
Proof. Following Willmot and Lin [13, pp. 96–97], let r :¼ R0 b > 0. Therefore
dGðzÞ
0
p R0 ½xð1þjÞþCðbÞZ1
1
dGðzÞ
1
jI0 ¼ i
b1 :¼ inf
R1 t
tP0
ery dFðyÞ
ert FðtÞ
¼
Z
1
ery dFðyÞ;
0
that is, b1 ¼ Ep ½eR0 bY 1 . Finally, replacing this equality in (22), we obtain (27). h
Ep ½eR0 ½xð1þI1 ÞþCðbÞZ1 jI0 ¼ i
¼ bEp ½eR0 bY 1 Ep ½eR0 CðbÞZ1 Ep ½eR0 xð1þI1 Þ jI0 ¼ i ¼ bEp ½eR0 ½CðbÞZ1 bY 1 Ep ½eR0 xð1þI1 Þ jI0 ¼ i
4.2. Bounds by means of the martingale approach
¼ bEp ½eR0 xð1þI1 Þ jI0 ¼ iðbyð11ÞÞ: This shows that the desired result holds for n ¼ 1. To prove the result for general n P 1, the induction hypothesis is that, for some n P 1, and every x P 0 and i 2 I,
wpn ðx; iÞ 6 bEp ½eR0 xð1þI1 Þ jI0 ¼ i:
eR0 b0 y dFðyÞdGðzÞ
Hence, (24) holds for any n ¼ 1; 2; . . .. Finally, letting n ! 1 in (24) we obtain (22). h
j2I
p R0 bY 1
Z sj ðzÞ
wp ðx; iÞ 6 ðEp ½eR0 bY 1 Þ1 Ep ½eR0 xð1þI1 Þ jI0 ¼ i:
0
j2I
X
Z
dFðyÞdGðzÞ
¼ bEp ½eR0 xð1þI1 Þ jI0 ¼ i:
xþy
Z
e eR0 b0 # eR0 b0 y dFðyÞ eR0 b0 # Fð#Þ # Z 1 6 beR0 b0 # eR0 b0 y dFðyÞ 6 beR0 b0 # Ep ½eR0 bY 1 #
Fð#Þ ¼
e
As an application of Theorem 3, we next consider the special case in which the claim distribution is in the class of NWUC distributions [13, p. 25], which are defined as follows.
Proof. It suffices to show that the rightmost term in (22) is an upper bound for wpn ðx; iÞ, for all n P 1. We will prove this by induction. First note that
R1
R0 b0 y
¼ bEp ½eR0 bY 1 Ep ½eR0 CðbÞZ1 Ep ½eR0 xð1þI1 Þ jI0 ¼ i
Theorem 3. Let R0 > 0 be the constant satisfying (11). Then, for all x P 0 and i 2 I, j2I
Z
0
j2I
!
1
0
1
j2I
wp ðx; iÞ 6 b
Z
sj ðzÞ
j2I
¼
R0 ½xð1þjÞþCðb0 Þz
0
j2I
4.1. Bounds obtained by the inductive approach
X
ð26Þ
ð24Þ
Now, let 0 6 y 6 sj ðzÞ, with sj ðzÞ as in Lemma 2. Further, in (24) replace x and i by xð1 þ jÞ þ Cðb0 Þz b0 y and j, respectively, to obtain
wpn ðxð1 þ jÞ þ Cðb0 Þz b0 y; jÞ 6 bEp ½eR0 ½xð1þjÞþCðbÞzbyð1þI1 Þ jI0 ¼ j; ð25Þ
Another way for deriving upper bounds for ruin probabilities is n Q ð1 þ Il Þ1 with
the martingale approach. To this end, let V n :¼ X n
l¼1
n P 1, be the so-called discounted risk process. The ruin probabilities wpn in (8) associated to the process fV n ; n ¼ 1; 2 . . .g are
wpn ðx; iÞ ¼ Pp
n [
!
ðV k < 0ÞjX 0 ¼ x; I0 ¼ i :
k¼1
In the classical risk model, feR0 X n gnP1 is a martingale. However, for our model (4), there is no constant r > 0 such that ferX n gnP1 is a martingale. Still, there exists a constant r > 0 such that ferV n gnP1
501
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
is a supermartingale, which allows us to derive probability inequalities by the optional stopping theorem. Such a constant is defined in the following proposition. Proposition 5. Assume that for each i 2 I, there exists qi > 0 satisfying that
Thus, for any n P 1,
Ep ½Snþ1 jY 1 ; . . . Y n ; Z 1 ; . . . Z n ; I1 ; . . . In 2 3 n þ1 Q R1 ðCðb0 ÞZ nþ1 b0 Y nþ1 Þ ð1þIt Þ1 t¼1 ¼ Sn E4e jY 1 ; . . . Y n ; Z 1 ; . . . Z n ; I1 ; . . . In 5 2
h i 1 Ep eqi ½CðbÞZ1 bY 1 ð1þI1 Þ jI0 ¼ i ¼ 1:
R1 ðCðb0 ÞZ nþ1 b0 Y nþ1 Þð1þInþ1 Þ1
ð28Þ
¼ Sn E4e
ð29Þ
h
n Q
ð1þIt Þ1
t¼1
3 jI1 ; . . . In 5
Assume as well that Ep ½CðbÞZ bY > 0.3 Then
R1 :¼ inf qi P R0 i2I
and, furthermore, for all i 2 I
6 Sn E
R1 ðCðb0 ÞZ nþ1 b0 Y nþ1 Þð1þInþ1 Þ1
e
jI1 ; . . . In
i
n Q
ð1þIt Þ1
t¼1
6 Sn :
Proof. For each i 2 I and r > 0, let
This implies that fSn gnP1 is a supermartingale. Let T i ¼ minfn : V n < 0jI0 ¼ ig, where V n is given by (32). Then T i is a stopping time and n ^ T i :¼ minfn; T i g is a finite stopping time. Thus, by the optional stopping theorem for martingales, we get
h i 1 li ðrÞ :¼ Ep er½CðbÞZbYð1þI1 Þ jI0 ¼ i 1:
Ep ðSn^T i Þ 6 Ep ðS0 Þ ¼ eR1 x :
Then the first derivative of li ðrÞ at r ¼ 0 is
Hence,
p
h
R1 ½CðbÞZ 1 bY 1 ð1þI1 Þ1
E e
0 li ð0Þ
i
jI0 ¼ i 6 1:
ð30Þ
h i ¼ E ½ðCðbÞZ bYÞ E ð1 þ I1 Þ1 jI0 ¼ i
eR1 x P Ep ðSn^T i Þ P Ep ððSn^T i ÞIðT i 6nÞ Þ P Ep ððST i ÞIðT i 6nÞ Þ
p
¼ Ep ðeR1 V T i IðT i 6nÞ Þ P Ep ðIðT i 6nÞ Þ ¼ wpn ðx; iÞ;
< 0 ðby independenceÞ; and the second derivative is
2 1 00 li ðrÞ ¼ Ep ðCðbÞZ bYÞð1 þ I1 Þ1 er½CðbÞZbYð1þI1 Þ jI0 ¼ i > 0: This shows that li ðrÞ is a convex function. Let qi be the unique positive root of the equation li ðrÞ ¼ 0 on ð0; 1Þ. Further, if 0 < q 6 qi , then li ðqÞ 6 0. However,
h i h i X 1 1 E eR0 ½Cðb0 ÞZb0 Yð1þI1 Þ jI0 ¼ i ¼ pij E eR0 ½Cðb0 ÞZb0 Yð1þjÞ j2I
ðby Jensen’s inequalityÞ 6
X
ð1þjÞ1 pij E eR0 ½Cðb0 ÞZ1 b0 Y 1 :
ð33Þ
where (33) follows because V T i < 0. Thus, by letting n ! 1 in (33) we obtain (31). h Remark 5. Summarizing, we have three upper bounds for the ruin probabilities with infinite horizon. First, the Lundberg bound, which only depends on R0 , the Lundberg exponential in (11), (13). Second, the inductive bound (22) which depends on R0 and also on the interest rate process. Third, the martingale bound in (31), which depends on R1 . Note that the last two bounds are sharper than the Lundberg bound. Observe also that the number of operations to get R1 in (31) is higher than that to get R0 in (22). In the next section we present some numerical results.
j2I
Consequently, by (11), we have E½eR0 ½Cðb0 ÞZ1 b0 Y 1 ¼ 1. Hence, since P j2I pij ¼ 1,
h i 1 Ep eR0 ½CðbÞZbYð1þI1 Þ jI0 ¼ i 6 1:
5. Numerical results To illustrate the bounds given by Theorems 3 and 6 we present two numerical examples that use Matlab and Maple implementations. Without loss of generality we can work in monetary units equal to E½Y in all examples.
This implies that li ðR0 Þ 6 0. Moreover, R0 6 qi for all i, and so
R1 :¼ inf qi P R0 : i2I
Thus, (29) holds. In addition R1 6 qi for all i 2 I, which implies that li ðR1 Þ 6 0. This yields (30). h
5.1. Exponentially distributed claims
Theorem 6. Under the hypotheses of Proposition 5, for all i 2 I and x P 0;
Let consider the special case 2 in Section 2, in which Z n and Y n are exponentially distributed with parameters k and 1=l, respectively. In addition, we will consider an interest model with three possible interest rates:
wp ðx; iÞ 6 eR1 x :
I ¼ f6%; 8%; 10%g:
ð31Þ k Q
Proof. By (4), the discounted risk process V k :¼ X k ð1 þ Il Þ1 satl¼1 isfies that
V k :¼ x þ
k X
ðCðb0 ÞZ 1 b0 Y l Þ
l Y
! ð1 þ It Þ
t¼1
l¼1
Q
nþ1
3
R1 ðCðb0 ÞZ nþ1 b0 Y nþ1 Þ
See, Remark 1.
t¼1
:
ð32Þ
0
0:2
0:8
0
1
B C @ 0:15 0:7 0:15 A: 0 0:8 0:2 Thus, our interest rate model incorporates mean reversion to a level of 8%. If h is too high, in the sense that
Let Sn ¼ eR1 V n . Then
Snþ1 ¼ Sn e
1
The transition matrix (see (5)) is given by
1 þ h P ð1 lR0 Þ2 ;
ð1þIt Þ1
:
then the optimal policy is given by p ¼ fan gnP1 with an ¼ 1 for all n. If we assume that c > kl, then we have that the ruin probability for the Cramér–Lundberg model is
502
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
kl xðl1kcÞ : wp ðxÞ ¼ e c Recalling (14), the Cramér–Lundberg exponent R0 is the solution of equation
k þ cR0 ¼ kð1 lR0 Þ1 : p
p
By Lemma 1 and (13), we have that w ðx; iÞ 6 w ðxÞ 6 eR0 x . In the case that Y has NWUC distribution, then the inductive bound is given by (27). The martingale bound can be obtained from Theorem 6. Table 1 shows the numerical results when k ¼ 1; l ¼ 2; h ¼ 3; c ¼ 4, and x ¼ 1. Note that
wp ðx; iÞ 6 0:3817 < wp ðxÞ: The numerical results in Table 1 show that the upper bound in (22) can be tighter than that in (31). This suggests that the upper bounds derived by the inductive approach are tighter than the upper bounds obtained by supermartingales. In addition, the upper bounds derived by the inductive approach are tighter than the ruin probability without interest rate. Moreover, Table 1 shows that the upper bounds derived in this article are sharper than the Lundberg upper bound.
The Lundberg bound: In this example we can guarantee that the Lundberg bound (13) holds for each b 2 B. Then there exists a constant R0 such that (11) is achieved. Moreover, solving
Ep ½eR0 bY 1 Ep ½eR0 CðbÞZ1 ¼ 1; is equivalent to find the Cramér–Lundberg adjustment coefficient such that
1 þ CðbÞR0 ¼ aðbR0 I TÞ1 t: Then the Lundberg bound for the ruin probability is
wb ðxÞ 6 eR0 x ; for x P 0: Fig. 1 shows the relation between R0 and b in this inequality is inversely proportional. Table 2 presents numerical values of the bounds obtained for several admissible decision policies. The Induction bound: Here, the claim distribution is a NWUC (see [13, p. 24]) and such that Ep ½eR0 bY 1 ¼ M Y ðR0 bÞ < 1 for each b 2 B. Then Corollary 4 applies and for each i 2 I and x P 0, we have
wp ðx; iÞ 6 ðEp ½eR0 bY 1 Þ1 Ep ½eR0 xð1þI1 Þ jI0 ¼ i h i1 X 1 pik eR0 xð1þkÞ : 6 aðbR0 I T Þ t k2I
5.2. Claims with phase-type distribution We consider claim distributions of the phase-type because this class is a generalization of the exponential distribution such that they and their moments can be written in a closed form, various quantities of interest can be evaluated with relative ease, and furthermore, the set of phase-type distributions is dense in the set of all distributions with support in ½0; 1Þ (see [1]). Suppose that the claim size Y has a phase-type density with parameters ða; TÞ where
T¼
1
0
0
2
;
See Table 2 for numerical values of this bound obtained for several admissible decision policies. As it is to be expected we get induction bounds smaller than the Lundberg bounds for the same decision policies. The Martingale bound: By the condition (28) of Proposition 5 and Theorem 6, we get the martingale bound (31). Observe that
h i 1 E eqi ðCðbÞZ1 bY 1 Þð1þI1 Þ jI0 ¼ i ¼ 1 which is equivalent to the following condition for each i 2 I:
and a ¼ ð1=2; 1=2Þ:
140 120
Let
I¼
1 0 ; 0 1
1 ! 1 ¼ ; 1
! and t ¼ T 1 ¼
1
100
2
80
R0
In this case,
60
M Y ðsÞ ¼ E½esY ¼ aðsI T Þ1 t: d MY ðsÞjs¼0 ds
40
2
Thus, E½Y ¼ ¼ aðTÞ t ¼ 0:75, and Y has NWUC distribution. Let ZwExpð1Þ; E½Z ¼ 1, and MZ ðsÞ ¼ E½esZ ¼ ð1 sÞ1 . We consider an interest model with three possible interest rates: I ¼ f6%; 8%; 10%g. We would like to have an idea of the dependence of our bounds on the transition probability matrix of the interest rate process. To this end, we consider two transition probability matrices, namely,
0
0
0:9 0:1
B P 1 ¼ @ 0:8 0:2
0:9 0:1
1
0
0:3 0:7
C B 0 A and P2 ¼ @ 0 0 0
0
0
0.2
0.4
0.6
1
0.8
b Fig. 1. Relation between R0 and b.
1
0
C 0:2 0:8 A: 0:1 0:9
We fix the premium income rate c ¼ 0:975 and the safety loading h ¼ 0:1 of the reinsurer. In addition, B ¼ ð0; 1. In this case (15) is not satisfied.
Table 1 Table of upper bounds for ruin probabilities, with x ¼ 1 and i ¼ 8%.
20
Lundberg
wp ðxÞ
Inductive
Martingale
R0
R1
0.7788
0.3894
0.3817
0.4366109286
0.25
0.8287128040
Table 2 Numerical bounds of ruin probability. Pj
b
Lundberg
Induction
Martingale
R0
R1
P1 P1 P1 P1 P1
0.5 0.75 0.85 0.95 1
0.323e7 0.111e4 0.434e4 0.126e3 0.2e3
0.369e8 0.196e5 0.846e5 0.268e4 0.436e4
0.448e9 0.586e6 0.317e5 0.120e4 0.212e4
3.4491 2.2810 2.0086 1.7943 1.7034
4.3048 2.8697 2.5321 2.2655 2.1522
P2 P2 P2 P2 P2
0.5 0.75 0.85 0.95 1
0.323e7 0.111e4 0.434e4 0.126e3 0.2e3
0.213e8 0.136e5 0.614e5 0.201e4 0.333e4
0.382e9 0.527e6 0.288e5 0.110e4 0.195e4
3.4491 2.2810 2.0086 1.7943 1.7034
4.3368 2.8911 2.5509 2.2824 2.1683
503
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504 −5
−4
x 10
x 10
6
Lundberg bound Induction bound (P ) 1
Induction bound (P )
5 Ruin Probability
Ruin Probability
2
Martingale bound (P 1) Martingale bound (P )
2
2
1
4 3 2 1 0
0 0.5
0.6
0.7
0.8
0.9
1
0.75
0.8
The retention level (b)
0.85
0.9
0.95
1
The retention level (b)
Fig. 2. Bounds for the ruin probabilities. Left panel: b 2 ½0:5; 1. Right panel: b 2 ½0:75; 1.
X k2I
1
pik eqi ð1þkÞ M Y
bqi CðbÞqi MZ ¼ 1: 1þk 1þk
In our example we solve R1 ¼ inf i2I qi P R0 , and then we obtain wp ðx; i1 Þ 6 eR1 x for x P 0. Numerical results of this bound are reported in Table 2. It is obvious that this martingale bound improves the results of the induction bound. We run numerical experiments to compare, for a fixed retention level b, the ruin probability bounds that could be achieved. Fig. 2 shows the upper bounds of ruin probability from different approaches with the initial state x ¼ 5 and i ¼ 8%. Finally, we find of special interest the case of small reinsurers for which the retention level could be restricted by economic considerations. Table 2 shows the numerical values of the bounds from different values of b when b is increasing towards 1. Recall, that b ¼ 1 stands for the control action no reinsurance. Clearly, the best results are obtained in the case where the transition interest rate matrix is P 2 . The numerical results in Table 2 show that the upper bound in (31) can be tighter than that in (22). This suggests that the upper bounds derived by the martingale approach are tighter than the upper bounds obtained by induction. In addition, Table 2 also shows that the upper bounds derived in this article are sharper than the Lundberg upper bound.
a. Is it possible to obtain bounds tighter than those in Theorems 3 and 6? b. Actually, what do we need to obtain the ruin probabilities in closed form? c. Let s :¼ inffk P 1jX k < 0g be the time of ruin. Can we calculate or estimate quantities such as E½s, or Pðs 6 TÞ for given T > 0? These are just a few of the many questions that we can ask ourselves. But two immediate queries are: i. Since fIn g in (1) is supposed to be a Markov chain, can we rewrite the minimization of the ruin probability as a Markov decision problem? ([8–10], for instance). ii. Suppose that in (1) we include an investment process. What can we say about these models? Further research in some of these directions is in progress. Acknowledgements The authors are grateful to the anonymous referees for helpful comments and suggestions and in particular for detailed reports. The first author thank Onésimo Hernández-Lerma and Henryk Gzyl for several useful discussions and stimulating suggestions.
6. Concluding remarks
References
Our main results in this paper, Theorems 3 and 6, give upper bounds for the probability of ruin of a certain risk process, which (as shown in Section 2) includes as special cases several relevant models. To obtain these results, first, we present an important preliminary result, Lemma 2, which gives recursive equations for finite-horizon ruin probabilities and an integral equation for the ultimate ruin probability. We illustrate our results with an application to the ruin probability in a risk process with a heavy tail claims distribution under proportional reinsurance and a Markov interest rate process. This application suggests that the upper bounds derived by inductive approach are tighter than the ruin probability without interest rate (the function considered in Lemma 1). In addition, the upper bounds derived in this article are sharper than the Lundberg upper bound. Our paper leaves, of course, many open issues. For instance:
[1] S. Asmussen, Ruin Probabilities, World Scientific, Singapore, 2000. [2] J. Cai, Ruin probabilities with dependent rates of interest, Journal of Applied Probability 39 (2002) 312–323. [3] J. Cai, D. Dickson, Ruin probabilities with a Markov chain interest model, Insurance Mathematics and Economics 35 (2004) 513–525. [4] J. Gaier, P. Grandits, W. Schachermayer, Asymptotic ruin probabilities and optimal investment, Annals of Applied Probability 13 (3) (2003) 1054– 1076. [5] J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991. [6] P. Grandits, An analogue of the Cramér–Lundberg approximation in the optimal investment case, Applied Mathematics and Optimization 50 (2004) 1– 20. [7] M. Diasparra, R. Romera, Bounds for the ruin probability of a discrete-time risk process, Journal of Applied Probability 46 (2009) 99–112. [8] O. Hernández-Lerma, J.B. Lasserre, Discrete-Time Markov Control Processes: Basic Optimality Criteria, Springer-Verlag, New York, 1996. [9] O. Hernández-Lerma, J.B. Lasserre, Further Topics on Discrete-Time Markov Control Processes, Springer-Verlag, New York, 1999. [10] O. Hernández-Lerma, J.B. Lasserre, Markov Chains and Invariant Probabilities, Birkhäuser, Basel, 2003.
504
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
[11] M. Schäl, On dicrete-time dynamic programming in insurance: Exponential utility and minimizing the ruin probability, Scandinavian Actuarial Journal 3 (2004) 189–210. [12] H. Schmidli, Stochastic Control in Insurance, Springer-Verlag, London, 2008.
[13] G. Willmot, X. Lin, Lundberg Approximations for Compound Distributions with Insurance Applications, Lectures Notes in Statistics, vol. 156, Springer-Verlag, 2001.
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
Inequalities for the ruin probability in a controlled discrete-time risk process M. Diasparra a,*, R. Romera b a b
Department of Pure and Applied Mathematics, Universidad Simón Bolívar, Bolivarian Republic of Venezuela Department of Statistics, Universidad Carlos III de Madrid, Spain
a r t i c l e
i n f o
Article history: Received 12 December 2008 Accepted 10 November 2009 Available online 24 November 2009 MSC: 91B30 60J05 60K10
a b s t r a c t Ruin probabilities in a controlled discrete-time risk process with a Markov chain interest are studied. To reduce the risk of ruin there is a possibility to reinsure a part or the whole reserve. Recursive and integral equations for ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived given a constant stationary policy. The relationships between these inequalities are discussed. To illustrate these results some numerical examples are included. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Risk process Ruin probability Proportional reinsurance Lundberg’s inequality
1. Introduction This paper studies an insurance model where the risk process can be controlled by proportional reinsurance. The performance criterion is to choose reinsurance control strategies to bound the ruin probability of a discrete-time process with a Markov chain interest. Controlling a risk process is a very active area of research, particularly in the last decade; see [8,9,11,12], for instance. Nevertheless obtaining explicit optimal solutions is a difficult task in a general setting even for the classical risk process. In fact, the explicit solution in the classical controlled risk process is known only in a very few cases. This is the main reason why we deal with the optimal control problem by using an alternative method than dynamic programming. Hence, an alternative method commonly used in ruin theory is to derive inequalities for ruin probabilities (see [1,4–6,12,13]). Following Cai [2] and Cai and Dickson [3], we model the interest rate process as a denumerable state Markov chain. This model can be in fact a discrete counterpart of the most frequently occurring effect observed in continuous interest rate process, e.g., mean-reverting effect. Stochastic inequalities for the ruin probabilities are derived by martingales and inductive techniques. The inequalities can be used to obtain upper bounds for the ruin probabilities. We use the proposal by Gaier et al. [4]
* Corresponding author. Tel.: +58 212 761 6143; fax: +58 212 761 5945. E-mail addresses: [email protected] (M. Diasparra), [email protected] (R. Romera). 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.11.015
and Schmidli [12], we restrict ourselves to use constant control policies (see Remark 1). Explicit condition are obtained for the optimality of employing no reinsurance. The outline of the paper is as follows. In Section 2 the risk model is formulated. Some important special cases of this model are briefly discussed. In Section 3 we derive recursive equations for finite-horizon ruin probabilities and integral equations for the ultimate ruin probability. In Section 4 we obtain upper bounds for the ultimate probability of ruin. An analysis of the new bounds and a comparison with the Lundberg’s inequality is also included. Finally, in Section 5 we illustrate our results on the ruin probability in a risk process with a heavy tail claims distribution under proportional reinsurance and a Markov interest rate process. We conclude in Section 6 with some general comments and some suggestions further research. 2. The model We consider a discrete-time insurance risk process in which the surplus X n varies according to the equation
X n ¼ X n1 ð1 þ In Þ þ Cðbn1 Þ Z n hðbn1 ; Y n Þ;
ð1Þ
for n P 1, with X 0 ¼ x P 0. Following Schmidli [12, p. 21], we introduce an absorbing (cemetery) state ,, such that if X n < 0 or X n ¼ ,, then X nþ1 ¼ ,. We denote the state space by X ¼ R [ f,g. Let Y n be the nth claim payment, which we assume to form a sequence of i.i.d. random variables with common probability distribution function
497
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
(p.d.f.) F. The random variable Z n stands for the length of the nth period, that is, the time between the occurrence of the claims Y n1 and Y n . We assume that fZ n g is a sequence of i.i.d. random variables with p.d.f. G. This case includes a controlled version of the Cramér– Lundberg model if we assume that the claims occur as a Poisson process. Of course, we can also think of the case where Z n ¼ 1 is deterministic. In addition, we suppose that fY n gnP1 and fZ n gnP1 are independent. The process can be controlled by reinsurance, that is, by choosing the retention level (or proportionality factor or risk exposure) b 2 B of a reinsurance contract for one period, where B :¼ ½bmin ; 1, and bmin 2 ð0; 1 will be introduced below. Let fIn gnP0 be the interest rate process; we suppose that In evolves as a Markov chain with a denumerable (possibly finite) state space I consisting of non-negative rational numbers and the process fIn gnP0 is independent of fZ n gnP1 1 and fY n gnP1 . The function hðb; yÞ with values in ½0; y specifies the fraction of the claim y paid by the insurer, and it also depends on the retention level b at the beginning of the period. Hence y hðb; yÞ is the part paid by the reinsurer. The retention level b ¼ 1 stands for the control action no reinsurance. In this article, we consider the case of proportional reinsurance, which means that
hðb; yÞ :¼ b y;
ð2Þ
with retention level b 2 B. The premium (income) rate c is fixed. Since the insurer pays to the reinsurer a premium rate, which depends on the retention level b, we denote by CðbÞ the premium left for the insurer if the retention level b is chosen, with
0 6 cmin < c 6 CðbÞ 6 c;
b 2 B;
P
where pij P 0 and j pij ¼ 1 for all i; j 2 I. The ruin probability when using the policy p, given the initial surplus x, and the initial interest rate I0 ¼ i is defined as p
w ðx; iÞ :¼ P
p
1 [
! fX k < 0gjX 0 ¼ x; I0 ¼ i ;
ð6Þ
k¼1
which we can also express as
wp ðx; iÞ ¼ Pp ðX k < 0 for some k P 1jX 0 ¼ x; I0 ¼ iÞ:
ð7Þ
Similarly, the ruin probabilities in the finite horizon case are given by
wpn ðx; iÞ :¼ Pp
n [
! fX k < 0gjX 0 ¼ x; I0 ¼ i :
ð8Þ
k¼1
Thus,
wp1 ðx; iÞ 6 wp2 ðx; iÞ 6 6 wpn ðx; iÞ 6 ; and
lim wpn ðx; iÞ ¼ wp ðx; iÞ:
n!1
The following lemma is used below to simplify some calculations. Lemma 1. For any given policy p, there is a function wp ðxÞ such that
wp ðx; iÞ 6 wp ðxÞ for every initial state x > 0 and initial interest rate I0 ¼ i. Proof. By (1) and (2), the risk model is given by
where c denotes the minimal value of the premium considered by the insurer and cmin corresponds to b ¼ 0 in CðbÞ. We define bmin :¼ minfb 2 ð0; 1jCðbÞ P c g. Since CðbÞ P c > cmin P 0, there exists a bmin > 0. Moreover, CðbÞ is an increasing function that we will calculate according to the expected value principle with added safety loading h from the reinsurer:
E½Y CðbÞ ¼ c ð1 þ hÞð1 bÞ ; E½Z
ð3Þ
where E½Y denotes the mean claim and E½Z denotes the average which we astime between claims. Note that, cmin ¼ c ð1 þ hÞ E½Y E½Z sume that to be greater or equal than zero. We define Markovian control policies p ¼ fan gnP1 , which at each time n depend only on the current state, that is, an ðX n Þ :¼ bn for n P 0. Abusing notation, we will identify functions a : X ! B with stationary strategies, where B ¼ ½bmin ; 1 is the decision space. However, in this study we will focus on stationary constant policies following the arguments in Remark 1. Consider an arbitrary initial state X 0 ¼ x P 0 (note that the initial value is not stochastic) and a control policy p ¼ fan gnP1 . Then, by iteration of (1) and assuming (2), and (3), it follows that for n P 1, X n satisfies
Xn ¼ x
n Y l¼1
ð1 þ Il Þ þ
n X l¼1
ðCðbl1 ÞZ l bl1 Y l Þ
n Y
! ð1 þ I m Þ :
ð4Þ
m¼lþ1
Since In P 0, we have
X n ¼ X n1 ð1 þ In Þ þ Cðbn1 ÞZ n bn1 Y n P X n1 þ Cðbn1 ÞZ n bn1 Y n :
ð9Þ
Define recursively
e n :¼ X e n1 þ Cðbn1 ÞZ n bn1 Y n ; X
ð10Þ
e 0 ¼ x. Hence, X n P X e n for all n 2 N. Clearly, if X n < 0, with X 0 ¼ X e n < 0. then X Let
) 1 [ and E1 :¼ x 2 X fX n ðxÞ < 0g n¼1 ( ) [ 1 n o fn ðxÞ < 0 ; E2 :¼ x 2 X X n¼1 (
and note that E1 E2 . Therefore,
P
1 [
p
! fX n < 0gjI0 ¼ i
n¼1
6P
p
1 [
! e f X n < 0gjI0 ¼ i ;
n¼1
e n do not depend on In , we obtain from (6) and since the X
Let ðpij Þ be the matrix of transition probabilities of fIn g, i.e.,
pij :¼ PðInþ1 ¼ jjIn ¼ iÞ;
X n ¼ X n1 ð1 þ In Þ þ Cðbn1 ÞZ n bn1 Y n :
ð5Þ
wp ðx; iÞ ¼ Pp
1 [
!
fX n < 0gjX 0 ¼ x; I0 ¼ i
n¼1 1
This hypothesis is necessary when one works with a methodology based on martingale theory, as it is the case. Moreover, from the financial point of view, it is not difficult to find particular scenarios for which the interest rate and the waiting time are independent. For example, if one invest in a predictable asset, which is typical on stochastic models of interest rates in order to preserve risk-neutral pricing, and if one considers that the number of events before the maturity may be zero or one, which is not restrictive because these maturity periods are often very short on applications, then, to consider independence between capitalization and waiting time seems quite natural.
6P
p
1 [
! e n < 0gjX 0 ¼ x fX
¼: wp ðxÞ:
n¼1
We denote by P the policy space. A control policy p is said to be optimal if for any initial values ðX 0 ; I0 Þ ¼ ðx; iÞ, we have
wp ðx; iÞ 6 wp ðx; iÞ
498
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
for all p 2 P. Schmidli [12] and Schäl [11] show the existence of an optimal control policy for some special cases of the model risk (1). However, even in these special cases it is extremely difficult to ob tain closed expressions for wp ðx; iÞ. We are thus led to consider bounds for the ruin probabilities, which we do in Sections 3–5. Remark 1. For the reinsurance problem in the case of small claims, Schmidli [12] obtains that the optimal strategy depends on the current wealth, in the following sense: If bðxÞ is the optimal reinsurance policy for x small enough, bðxÞ ¼ 1, that is, no reinsurance, and if x tends to infinity the reinsurance bðxÞ tends to some appropriate constant k. Then, we can argue to consider constant policies of reinsurance in a more general setting, based on this result of Schmidli, in the sense that, for small claims the optimal strategy is asymptotically constant. A second additional argument, is the following one. In Gaier et al. [4], they obtain upper bounds for the ruin probability for a model with investment in a risky asset (no reinsurance) by considering a constant policy, and show that this constant policy is asymptotically optimal (when the initial capital tends to infinity). Finally, a third argument for using constant policies is the following. First, we assume that PðbY > CðbÞZÞ > 0 for all b 2 B. Because, if there is some be 2 B such that Pðbe Y > Cðbe ÞZÞ ¼ 0, the ruin can be prevented by retention level be and the risk process considered in this case becomes trivial. Secondly, we assume the net profit condition Ep ½CðbÞZ bY > 0 for some p 2 P. Otherwise, ruin cannot be prevented because the surplus would be decreasing in time for all reinsurance treaties. Therefore, using the law of large numbers, we have n 1X ½CðbÞZ i bY i ! Ep ½CðbÞZ bY a: e:; n i¼1
this implies that for the stationary strategy bn ¼ b the process X n tends to infinity (in particular, inf n X n > 1). Hence, there is an initial capital X 0 ¼ x such that Pðinf n X n P 0jX 0 ¼ xÞ > 0. Because there is a strictly positive probability that from initial capital zero the set ½x; 1Þ is reached before the set ð1; 0Þ, we get also that Pðinf n X n P 0jX 0 ¼ 0Þ > 0. Finally, we have a stationary strategy for which ruin is not certain. Moreover, we will assume in this article that bE½Y < CðbÞE½Z (where Y and Z are independent) and Ep ½etY < 1 for t 2 ð1; .Þ with . 2 ð0; 1, then there exists a constant R0 R0 ðbÞ > 0 satisfying
Ep eR0 ðCðbÞZbY Þ ¼ 1:
ð11Þ
Note that our model is quite general, and contains the following particular cases: 1. The classical risk process.2 2. The classical risk process with investment in a Markov chain interest rate. 3. The model with reinsurance in the above cases. 4. The model with reinsurance and dividends.
or equivalently,
X n ¼ X n1 þ Cðbn1 ÞZ n bn1 Y n : The corresponding ruin probability is
1. If In ¼ 0 for all n P 1, then the risk model (4) reduces to the discrete-time risk model with proportional reinsurance:
Xn ¼ x
n X
ðbt1 Y t Cðbt1 ÞZ t Þ
t¼1
2
See, Diasparra and Romera [7].
ð12Þ
! fX n < 0gjX 0 ¼ x :
n¼1
Assuming, constant stationary strategies, say bn ¼ b0 , then, by (12), p
w ðxÞ ¼ P
1 [
p
(
n¼1
) ! n X ½b0 Y t Cðb0 ÞZ t > x jX 0 ¼ x : t¼1
Therefore, by the classical Lundberg inequality for ruin probabilities (see [1,5,13]), for x P 0
wp ðxÞ 6 eR0 x :
ð13Þ
2. Exponentially distributed length periods. Our process (1) a controlled version of Cramér–Lundberg model if the claims occur as a Poisson process, in which case the Z n are exponentially distributed, say Z n ’ ExpðkÞ. We suppose that In ¼ 0 for all n P 1, and that moment generating function M Y ðsÞ is finite in an open interval ð1; .Þ with . 2 ð0; 1. Thus, Y n has a compound distribution with expectation kl and moment generation function R1 e½kðMY ðsÞ1Þ . Let M Y ðb; rÞ :¼ 0 ebry dFðyÞ be the moment generating function of the part of the claim the insurer has to pay if the retention level b is chosen. We assume constant stationary strategies, say bn ¼ b0 for all n P 1. Moreover, we assume that Cðb0 Þ > b0 kl and M Y ðb0 ; rÞ < 1 for some r > 0 and b0 2 B. It P is clear that the risk process X n x ¼ nk¼1 ðCðbÞZ n bY k Þ satisfies all the hypotheses of Theorem 14 in [5, p. 10]. Then
Ep eR0 ½CðbÞbY n ¼ eR0 Cðb0 Þ e½kðMY ðb0 ;R0 Þ1Þ : Then, by (11), we have that the adjustment coefficient R0 ¼ R0 ðb0 Þ fulfils
R0 Cðb0 Þ þ kðM Y ðb0 ; R0 Þ 1Þ ¼ 0:
ð14Þ
By Lemma 4.1 Schmidli [12], R0 is unimodal and it attains its maximum value at a point b0 2 B. Then, it is easy to see that it is optimal to have no reinsurance (b0 ¼ 1) if and only if the safety loading h is too high in the sense that
1þhP
M 0Y ð1; R0 Þ
l
¼
E½YeR0 Y : E½Y
ð15Þ
with M0 denoting the derivative of M. 3. Let dn be the constant, short-term dividend rate in the nth period (the dividends are payments made by a corporation to its shareholder members). Then the discrete-time risk model with stochastic interest rate and dividends is given by
X n ¼ X n1 ð1 þ In Þ þ Cðbn1 ÞZ n hðbn1 ; Y n Þ dn X n ; with hðb; yÞ as in (2). Thus, rearranging terms,
X n ¼ X n1
To conclude this section, let us briefly review some particular cases of the risk model (1). Special cases.
1 [
wp ðxÞ :¼ Pp
1 þ In 1 þ dn
þ
Cðbn1 Þ hðbn1 ; Y n Þ Zn : ð1 þ dn Þ ð1 þ dn Þ
Yn In dn Let Y 0n :¼ ð1þd and I0n :¼ ð1þd . Since fIn g and fY n g are indepennÞ nÞ n1 Þ . Then the dent, then so are fI0n g and fY 0n g. Let C0 ðbn1 Þ :¼ Cðb ð1þdn Þ model becomes
X n ¼ X n1 1 þ I0n þ C0 ðbn1 ÞZ n hðbn1 ; Y 0n Þ; which from a statistical viewpoint is essentially the same as the model without dividends (1) and can be analyzed in a similar way. 4. As an extension of the latter case, some companies have dividend reinvestment plans (or DRIPs). These plans allow share-
499
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
holders to use dividends to systematically buy small amounts of stock. Let e d n be the short-term dividend reinvestment rate in the nth period, e d n 2 ½0; 1Þ. Then, the discrete-time risk model with stochastic interest rate and dividends reinvestment is given by
X n ¼ X n1 ð1 þ In Þ þ Cðbn1 ÞZ n hðbn1 ; Y n Þ þ e dnXn:
X n ¼ X n1
!
Cðbn1 Þ hðbn1 ; Y n Þ Zn : e ð1 d Þ ð1 e d Þ
þ
n
Let Y 00n :¼
n
In þe dn
, I00n :¼
Yn
ð1e dn Þ
n
, and C00 ðbn1 Þ :¼ Cðbn1 Þ. It follows that ð1e dn Þ
00
X n ¼ X n1 1 þ In þ C ðbn1 ÞZ n
fX k < 0gjY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ i
¼ 1;
ð19Þ
k¼1
while if 0 6 uðy; zÞ 6 h1 , then
Pp ðX 1 < 0jY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ iÞ ¼ 0:
Pp
ð20Þ
n[ þ1
!
fX k < 0gjY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ i
k¼1
ð1e dn Þ
00
n[ þ1
e n g ; fZ e n g , and feI n g Let f Y nP1 nP1 nP0 be independent copies of fY n gnP1 ; fZ n gnP1 , and fIn gnP0 , respectively. e k :¼ b0 Y e k . Thus, (20) and (4) yield that for e k Cðb0 Þ Z Let U 0 6 uðy; zÞ 6 h1 ,
Hence, rearranging terms, we obtain
1 þ In 1e d
Pp
!
hðbn1 ; Y 00n Þ;
¼P
p
nþ1 [ k¼2
which, again, is essentially the same as the model (1).
¼P
p
nþ1 [
! fX k < 0gjY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ i (
) k k k Y Y X ðh1 uðy; zÞÞ ð1 þ Il Þ Ul ð1 þ Im Þ < 0
k¼2
Let us go back to the original risk model (1). In the next section, we will derive recursive equations for the ruin probabilities and integral equations for the ultimate ruin probability associated to the model (1).
l¼1
!
l¼1
m¼lþ1
jX 0 ¼ x; I1 ¼ j ¼P
Remark 2. Given a p.d.f. G, we denote the tail of G by G, that is, GðxÞ :¼ 1 GðxÞ.
n [
p
(
) k k k Y X Y e e e Ul ðh1 uðy; zÞÞ ð1 þ I l Þ ð1 þ I m Þ < 0
k¼1
l¼1
!
jX 0 ¼ x; eI 0 ¼ j
l¼1
m¼lþ1
¼ wpn ðh1 uðy; zÞ; jÞ ¼ wpn ðxð1 þ jÞ uðy; zÞ; jÞ
3. Recursive and integral equations for ruin probabilities In this section, we first derive a recursive equation for wpn ðx; iÞ. Secondly, we give an integral equation for wp ðx; iÞ. Finally, we obtain an equation for the ruin probability with horizon n ¼ 1 given I0 ¼ i; X 0 ¼ x and a stationary policy p. These results, which are valid for any initial interest rate, are summarized in the following lemma.
where the third equality follows from the Markov property of fIn gnP0 , and the independence of fY n gnP1 ; fZ n gnP1 and fIn gnP0 . Let us now consider the event A ¼ fY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ ig, and recall that FðyÞ ¼ PðY 6 yÞ and GðzÞ ¼ PðZ 6 zÞ. From (8) and (4) we obtain p
wnþ1 ðx; iÞ ¼ P
wp1 ðx; iÞ ¼
X
pij
Fðsj ðzÞÞdGðzÞ;
ð16Þ
X
pij
j2I
pij
Z
wnþ1 ðx; iÞ ¼ Z sj ðzÞ
1
0
0
X
pij
wpn ðxð1 þ jÞ X j2I
þ pij
Z
Z
Fðsj ðzÞÞdGðzÞ:
ð17Þ
¼
0
X
wp ðx; iÞ ¼
X
pij
j2I
þ
X j2I
1
0
pij
Z sj ðzÞ
þ
wp ðxð1 þ jÞ uðy; zÞ; jÞdFðyÞdGðzÞ
¼
1
Fðsj ðzÞÞdGðzÞ:
ð18Þ
X j2I
0
Proof. Let U k :¼ uðY k ; Z k Þ ¼ b0 Y k Cðb0 ÞZ k . Given Y 1 ¼ y; Z 1 ¼ z, the control strategy p, and I1 ¼ j, by definition we have U 1 ¼ uðy; zÞ. Therefore,
X 1 ¼ xð1 þ I1 Þ U 1 ¼ h1 uðy; zÞ; where h1 ¼ xð1 þ jÞ Thus, if uðy; zÞ > h1 then
Pp ðX 1 < 0jY 1 ¼ y; Z 1 ¼ z; I1 ¼ j; X 0 ¼ x; I0 ¼ iÞ ¼ 1: This implies that for uðy; zÞ > h1
þ
Z
Z sj ðzÞ
P
n[ þ1
p
nþ1 [
!
fX k < 0gjA dFðyÞdGðzÞ !
)
fX k < 0gjA dFðyÞdGðzÞ
k¼1
0
wpn ðxð1 þ jÞ uðy; zÞ; jÞdFðyÞdGðzÞ )
1
dFðyÞdGðzÞ
Z
0
1
Z sj ðzÞ 0
wpn ðxð1 þ jÞ uðy; zÞ; jÞdFðyÞdGðzÞ
Fðsj ðzÞÞdGðzÞ :
1
fX k < 0gjA dFðyÞdGðzÞ:
k¼1
Z sj ðzÞ
sj ðzÞ
pij
!
k¼1
Pp
1
Z
1 0
0
Z
Z
nþ1 [
0
0
j2I
Moreover,
1
1
Z
p
0 Þz sj ðzÞ ¼ xð1þjÞþCðb and from (19) we have b0
sj ðzÞ
pij
P
0
0
1
1
0
(Z Z
1
Z
0
j2I
uðy; zÞ; jÞdFðyÞdGðzÞ þ
Z
1
Then, recalling that p
Z
X j2I
and for n ¼ 1; 2; . . .
wpnþ1 ðx; iÞ ¼
¼
1
0
j2I
fX k < 0gjX 0 ¼ x; I0 ¼ i
k¼1
Lemma 2. Let uðy; zÞ :¼ b0 y Cðb0 Þz, where b0 is the initial retention level. Let sj ðzÞ :¼ ðxð1 þ jÞ þ Cðb0 ÞzÞ=b0 ; X 0 ¼ x P 0, and pij as in (5). Then
Z
!
nþ1 [
p
ð21Þ
0
This gives (17). In particular,
wp1 ðx; iÞ ¼
X j2I
pij
Z
1
Fðsj ðzÞÞdGðzÞ:
0
Finally, letting n ! 1 in (21) and using dominated convergence we obtain limn!1 wpnþ1 ðx; iÞ ¼ wp ðx; iÞ, and (18) follows. h Remark 3. If we consider the risk model without reinsurance, that is, b ¼ 1, we obtain similar results to those in Cai and Dickson [3].
500
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
and hence
4. Bounds for ruin probabilities We will use the results obtained in Section 3 to find upper bounds for the ruin probabilities with infinite horizon taking into account the information contributed by the Markov chain of the interest rate process. We derive a functional for the ultimate ruin probability in terms of the new worse than used in convex (NWUC) ordering; see Remark 4, below. This idea was first introduced by Willmot and Lin [13] and has been generalized by other authors. We will present two upper bounds for the ruin probabilities. The first bound is obtained by an inductive approach, and the second by a martingale approach. These bounds are discussed in Remark 5, at the end of this section.
bE½eR0 ½xð1þjÞþCðb0 Þzb0 yð1þI1 Þ jI0 ¼ j 6 beR0 ½xð1þjÞþCðb0 Þzb0 y : Therefore, replacing (26) in (17), we get p
wnþ1 ðx; iÞ 6
X
pij b
Z
þ
1
e
X
Z pij b
X
1
eR0 ½xð1þjÞþCðb0 Þz
0
Z pij b
eR0 ½xð1þjÞþCðb0 Þz
¼ bEp ½eR0 bY 1
pij Ep ½eR0 xð1þjÞ ¼ bEp ½eR0 ½xð1þI1 Þ jI0 ¼ i;
ð22Þ
where b ¼ bðb0 Þ and is given by 1
b
R1 t
¼ inf
eR0 b0 y dFðyÞ
eR0 b0 t FðtÞ
tP0
:
!1 dFðyÞ
R0 b0 y
1
ð23Þ
#
for any # P 0. This implies that for every x P 0, i P 0, and b0 2 B, by (16) and (23) we have p
w1 ðx; iÞ ¼
X
pij
j2I
6
X
Z
1
Fðsj ðzÞÞdGðzÞ
0
p R0 bY 1
pij bE ½e
p R0 bY 1
¼ bE ½e
j2I
p R0 bY 1
¼ bE ½e
X
1
R0 b0
e
xð1þjÞþCðb0 Þz b0
!
pij
Z
¼ bE ½e
1
eR0 b0 y dFðyÞdGðzÞ
0
X
pij
Z
1
eR0 ½xð1þjÞþCðb0 Þz dGðzÞ
0
e
Remark 4. A distribution F concentrated on ð0; 1Þ is said to be new worse than used in convex (NWUC) ordering if, for all x; y P 0
Z
1
FðzÞdz P FðyÞ
Z
R0 ½xð1þjÞþCðb0 Þz
pij E e
FðzÞdz:
x
For example, let F a phase-type distribution with parameters ða; TÞ (see [1, pp. 215–222]). Then F is NWUC if and only if T 1 and ! T 1 eTy ðI 1 aÞ are both non-negative or non-positive definite simultaneously for all y P 0 (where I represent the identity matrix ! and 1 is the column vector of ones). Corollary 4. Under the hypotheses of Theorem 3, and assuming that Ep ½eR0 bY 1 < 1 for all b 2 B, and that, in addition, F is a NWUC distribution, we have
ð27Þ
Proof. Following Willmot and Lin [13, pp. 96–97], let r :¼ R0 b > 0. Therefore
dGðzÞ
0
p R0 ½xð1þjÞþCðbÞZ1
1
dGðzÞ
1
jI0 ¼ i
b1 :¼ inf
R1 t
tP0
ery dFðyÞ
ert FðtÞ
¼
Z
1
ery dFðyÞ;
0
that is, b1 ¼ Ep ½eR0 bY 1 . Finally, replacing this equality in (22), we obtain (27). h
Ep ½eR0 ½xð1þI1 ÞþCðbÞZ1 jI0 ¼ i
¼ bEp ½eR0 bY 1 Ep ½eR0 CðbÞZ1 Ep ½eR0 xð1þI1 Þ jI0 ¼ i ¼ bEp ½eR0 ½CðbÞZ1 bY 1 Ep ½eR0 xð1þI1 Þ jI0 ¼ i
4.2. Bounds by means of the martingale approach
¼ bEp ½eR0 xð1þI1 Þ jI0 ¼ iðbyð11ÞÞ: This shows that the desired result holds for n ¼ 1. To prove the result for general n P 1, the induction hypothesis is that, for some n P 1, and every x P 0 and i 2 I,
wpn ðx; iÞ 6 bEp ½eR0 xð1þI1 Þ jI0 ¼ i:
eR0 b0 y dFðyÞdGðzÞ
Hence, (24) holds for any n ¼ 1; 2; . . .. Finally, letting n ! 1 in (24) we obtain (22). h
j2I
p R0 bY 1
Z sj ðzÞ
wp ðx; iÞ 6 ðEp ½eR0 bY 1 Þ1 Ep ½eR0 xð1þI1 Þ jI0 ¼ i:
0
j2I
X
Z
dFðyÞdGðzÞ
¼ bEp ½eR0 xð1þI1 Þ jI0 ¼ i:
xþy
Z
e eR0 b0 # eR0 b0 y dFðyÞ eR0 b0 # Fð#Þ # Z 1 6 beR0 b0 # eR0 b0 y dFðyÞ 6 beR0 b0 # Ep ½eR0 bY 1 #
Fð#Þ ¼
e
As an application of Theorem 3, we next consider the special case in which the claim distribution is in the class of NWUC distributions [13, p. 25], which are defined as follows.
Proof. It suffices to show that the rightmost term in (22) is an upper bound for wpn ðx; iÞ, for all n P 1. We will prove this by induction. First note that
R1
R0 b0 y
¼ bEp ½eR0 bY 1 Ep ½eR0 CðbÞZ1 Ep ½eR0 xð1þI1 Þ jI0 ¼ i
Theorem 3. Let R0 > 0 be the constant satisfying (11). Then, for all x P 0 and i 2 I, j2I
Z
0
j2I
!
1
0
1
j2I
wp ðx; iÞ 6 b
Z
sj ðzÞ
j2I
¼
R0 ½xð1þjÞþCðb0 Þz
0
j2I
4.1. Bounds obtained by the inductive approach
X
ð26Þ
ð24Þ
Now, let 0 6 y 6 sj ðzÞ, with sj ðzÞ as in Lemma 2. Further, in (24) replace x and i by xð1 þ jÞ þ Cðb0 Þz b0 y and j, respectively, to obtain
wpn ðxð1 þ jÞ þ Cðb0 Þz b0 y; jÞ 6 bEp ½eR0 ½xð1þjÞþCðbÞzbyð1þI1 Þ jI0 ¼ j; ð25Þ
Another way for deriving upper bounds for ruin probabilities is n Q ð1 þ Il Þ1 with
the martingale approach. To this end, let V n :¼ X n
l¼1
n P 1, be the so-called discounted risk process. The ruin probabilities wpn in (8) associated to the process fV n ; n ¼ 1; 2 . . .g are
wpn ðx; iÞ ¼ Pp
n [
!
ðV k < 0ÞjX 0 ¼ x; I0 ¼ i :
k¼1
In the classical risk model, feR0 X n gnP1 is a martingale. However, for our model (4), there is no constant r > 0 such that ferX n gnP1 is a martingale. Still, there exists a constant r > 0 such that ferV n gnP1
501
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
is a supermartingale, which allows us to derive probability inequalities by the optional stopping theorem. Such a constant is defined in the following proposition. Proposition 5. Assume that for each i 2 I, there exists qi > 0 satisfying that
Thus, for any n P 1,
Ep ½Snþ1 jY 1 ; . . . Y n ; Z 1 ; . . . Z n ; I1 ; . . . In 2 3 n þ1 Q R1 ðCðb0 ÞZ nþ1 b0 Y nþ1 Þ ð1þIt Þ1 t¼1 ¼ Sn E4e jY 1 ; . . . Y n ; Z 1 ; . . . Z n ; I1 ; . . . In 5 2
h i 1 Ep eqi ½CðbÞZ1 bY 1 ð1þI1 Þ jI0 ¼ i ¼ 1:
R1 ðCðb0 ÞZ nþ1 b0 Y nþ1 Þð1þInþ1 Þ1
ð28Þ
¼ Sn E4e
ð29Þ
h
n Q
ð1þIt Þ1
t¼1
3 jI1 ; . . . In 5
Assume as well that Ep ½CðbÞZ bY > 0.3 Then
R1 :¼ inf qi P R0 i2I
and, furthermore, for all i 2 I
6 Sn E
R1 ðCðb0 ÞZ nþ1 b0 Y nþ1 Þð1þInþ1 Þ1
e
jI1 ; . . . In
i
n Q
ð1þIt Þ1
t¼1
6 Sn :
Proof. For each i 2 I and r > 0, let
This implies that fSn gnP1 is a supermartingale. Let T i ¼ minfn : V n < 0jI0 ¼ ig, where V n is given by (32). Then T i is a stopping time and n ^ T i :¼ minfn; T i g is a finite stopping time. Thus, by the optional stopping theorem for martingales, we get
h i 1 li ðrÞ :¼ Ep er½CðbÞZbYð1þI1 Þ jI0 ¼ i 1:
Ep ðSn^T i Þ 6 Ep ðS0 Þ ¼ eR1 x :
Then the first derivative of li ðrÞ at r ¼ 0 is
Hence,
p
h
R1 ½CðbÞZ 1 bY 1 ð1þI1 Þ1
E e
0 li ð0Þ
i
jI0 ¼ i 6 1:
ð30Þ
h i ¼ E ½ðCðbÞZ bYÞ E ð1 þ I1 Þ1 jI0 ¼ i
eR1 x P Ep ðSn^T i Þ P Ep ððSn^T i ÞIðT i 6nÞ Þ P Ep ððST i ÞIðT i 6nÞ Þ
p
¼ Ep ðeR1 V T i IðT i 6nÞ Þ P Ep ðIðT i 6nÞ Þ ¼ wpn ðx; iÞ;
< 0 ðby independenceÞ; and the second derivative is
2 1 00 li ðrÞ ¼ Ep ðCðbÞZ bYÞð1 þ I1 Þ1 er½CðbÞZbYð1þI1 Þ jI0 ¼ i > 0: This shows that li ðrÞ is a convex function. Let qi be the unique positive root of the equation li ðrÞ ¼ 0 on ð0; 1Þ. Further, if 0 < q 6 qi , then li ðqÞ 6 0. However,
h i h i X 1 1 E eR0 ½Cðb0 ÞZb0 Yð1þI1 Þ jI0 ¼ i ¼ pij E eR0 ½Cðb0 ÞZb0 Yð1þjÞ j2I
ðby Jensen’s inequalityÞ 6
X
ð1þjÞ1 pij E eR0 ½Cðb0 ÞZ1 b0 Y 1 :
ð33Þ
where (33) follows because V T i < 0. Thus, by letting n ! 1 in (33) we obtain (31). h Remark 5. Summarizing, we have three upper bounds for the ruin probabilities with infinite horizon. First, the Lundberg bound, which only depends on R0 , the Lundberg exponential in (11), (13). Second, the inductive bound (22) which depends on R0 and also on the interest rate process. Third, the martingale bound in (31), which depends on R1 . Note that the last two bounds are sharper than the Lundberg bound. Observe also that the number of operations to get R1 in (31) is higher than that to get R0 in (22). In the next section we present some numerical results.
j2I
Consequently, by (11), we have E½eR0 ½Cðb0 ÞZ1 b0 Y 1 ¼ 1. Hence, since P j2I pij ¼ 1,
h i 1 Ep eR0 ½CðbÞZbYð1þI1 Þ jI0 ¼ i 6 1:
5. Numerical results To illustrate the bounds given by Theorems 3 and 6 we present two numerical examples that use Matlab and Maple implementations. Without loss of generality we can work in monetary units equal to E½Y in all examples.
This implies that li ðR0 Þ 6 0. Moreover, R0 6 qi for all i, and so
R1 :¼ inf qi P R0 : i2I
Thus, (29) holds. In addition R1 6 qi for all i 2 I, which implies that li ðR1 Þ 6 0. This yields (30). h
5.1. Exponentially distributed claims
Theorem 6. Under the hypotheses of Proposition 5, for all i 2 I and x P 0;
Let consider the special case 2 in Section 2, in which Z n and Y n are exponentially distributed with parameters k and 1=l, respectively. In addition, we will consider an interest model with three possible interest rates:
wp ðx; iÞ 6 eR1 x :
I ¼ f6%; 8%; 10%g:
ð31Þ k Q
Proof. By (4), the discounted risk process V k :¼ X k ð1 þ Il Þ1 satl¼1 isfies that
V k :¼ x þ
k X
ðCðb0 ÞZ 1 b0 Y l Þ
l Y
! ð1 þ It Þ
t¼1
l¼1
Q
nþ1
3
R1 ðCðb0 ÞZ nþ1 b0 Y nþ1 Þ
See, Remark 1.
t¼1
:
ð32Þ
0
0:2
0:8
0
1
B C @ 0:15 0:7 0:15 A: 0 0:8 0:2 Thus, our interest rate model incorporates mean reversion to a level of 8%. If h is too high, in the sense that
Let Sn ¼ eR1 V n . Then
Snþ1 ¼ Sn e
1
The transition matrix (see (5)) is given by
1 þ h P ð1 lR0 Þ2 ;
ð1þIt Þ1
:
then the optimal policy is given by p ¼ fan gnP1 with an ¼ 1 for all n. If we assume that c > kl, then we have that the ruin probability for the Cramér–Lundberg model is
502
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
kl xðl1kcÞ : wp ðxÞ ¼ e c Recalling (14), the Cramér–Lundberg exponent R0 is the solution of equation
k þ cR0 ¼ kð1 lR0 Þ1 : p
p
By Lemma 1 and (13), we have that w ðx; iÞ 6 w ðxÞ 6 eR0 x . In the case that Y has NWUC distribution, then the inductive bound is given by (27). The martingale bound can be obtained from Theorem 6. Table 1 shows the numerical results when k ¼ 1; l ¼ 2; h ¼ 3; c ¼ 4, and x ¼ 1. Note that
wp ðx; iÞ 6 0:3817 < wp ðxÞ: The numerical results in Table 1 show that the upper bound in (22) can be tighter than that in (31). This suggests that the upper bounds derived by the inductive approach are tighter than the upper bounds obtained by supermartingales. In addition, the upper bounds derived by the inductive approach are tighter than the ruin probability without interest rate. Moreover, Table 1 shows that the upper bounds derived in this article are sharper than the Lundberg upper bound.
The Lundberg bound: In this example we can guarantee that the Lundberg bound (13) holds for each b 2 B. Then there exists a constant R0 such that (11) is achieved. Moreover, solving
Ep ½eR0 bY 1 Ep ½eR0 CðbÞZ1 ¼ 1; is equivalent to find the Cramér–Lundberg adjustment coefficient such that
1 þ CðbÞR0 ¼ aðbR0 I TÞ1 t: Then the Lundberg bound for the ruin probability is
wb ðxÞ 6 eR0 x ; for x P 0: Fig. 1 shows the relation between R0 and b in this inequality is inversely proportional. Table 2 presents numerical values of the bounds obtained for several admissible decision policies. The Induction bound: Here, the claim distribution is a NWUC (see [13, p. 24]) and such that Ep ½eR0 bY 1 ¼ M Y ðR0 bÞ < 1 for each b 2 B. Then Corollary 4 applies and for each i 2 I and x P 0, we have
wp ðx; iÞ 6 ðEp ½eR0 bY 1 Þ1 Ep ½eR0 xð1þI1 Þ jI0 ¼ i h i1 X 1 pik eR0 xð1þkÞ : 6 aðbR0 I T Þ t k2I
5.2. Claims with phase-type distribution We consider claim distributions of the phase-type because this class is a generalization of the exponential distribution such that they and their moments can be written in a closed form, various quantities of interest can be evaluated with relative ease, and furthermore, the set of phase-type distributions is dense in the set of all distributions with support in ½0; 1Þ (see [1]). Suppose that the claim size Y has a phase-type density with parameters ða; TÞ where
T¼
1
0
0
2
;
See Table 2 for numerical values of this bound obtained for several admissible decision policies. As it is to be expected we get induction bounds smaller than the Lundberg bounds for the same decision policies. The Martingale bound: By the condition (28) of Proposition 5 and Theorem 6, we get the martingale bound (31). Observe that
h i 1 E eqi ðCðbÞZ1 bY 1 Þð1þI1 Þ jI0 ¼ i ¼ 1 which is equivalent to the following condition for each i 2 I:
and a ¼ ð1=2; 1=2Þ:
140 120
Let
I¼
1 0 ; 0 1
1 ! 1 ¼ ; 1
! and t ¼ T 1 ¼
1
100
2
80
R0
In this case,
60
M Y ðsÞ ¼ E½esY ¼ aðsI T Þ1 t: d MY ðsÞjs¼0 ds
40
2
Thus, E½Y ¼ ¼ aðTÞ t ¼ 0:75, and Y has NWUC distribution. Let ZwExpð1Þ; E½Z ¼ 1, and MZ ðsÞ ¼ E½esZ ¼ ð1 sÞ1 . We consider an interest model with three possible interest rates: I ¼ f6%; 8%; 10%g. We would like to have an idea of the dependence of our bounds on the transition probability matrix of the interest rate process. To this end, we consider two transition probability matrices, namely,
0
0
0:9 0:1
B P 1 ¼ @ 0:8 0:2
0:9 0:1
1
0
0:3 0:7
C B 0 A and P2 ¼ @ 0 0 0
0
0
0.2
0.4
0.6
1
0.8
b Fig. 1. Relation between R0 and b.
1
0
C 0:2 0:8 A: 0:1 0:9
We fix the premium income rate c ¼ 0:975 and the safety loading h ¼ 0:1 of the reinsurer. In addition, B ¼ ð0; 1. In this case (15) is not satisfied.
Table 1 Table of upper bounds for ruin probabilities, with x ¼ 1 and i ¼ 8%.
20
Lundberg
wp ðxÞ
Inductive
Martingale
R0
R1
0.7788
0.3894
0.3817
0.4366109286
0.25
0.8287128040
Table 2 Numerical bounds of ruin probability. Pj
b
Lundberg
Induction
Martingale
R0
R1
P1 P1 P1 P1 P1
0.5 0.75 0.85 0.95 1
0.323e7 0.111e4 0.434e4 0.126e3 0.2e3
0.369e8 0.196e5 0.846e5 0.268e4 0.436e4
0.448e9 0.586e6 0.317e5 0.120e4 0.212e4
3.4491 2.2810 2.0086 1.7943 1.7034
4.3048 2.8697 2.5321 2.2655 2.1522
P2 P2 P2 P2 P2
0.5 0.75 0.85 0.95 1
0.323e7 0.111e4 0.434e4 0.126e3 0.2e3
0.213e8 0.136e5 0.614e5 0.201e4 0.333e4
0.382e9 0.527e6 0.288e5 0.110e4 0.195e4
3.4491 2.2810 2.0086 1.7943 1.7034
4.3368 2.8911 2.5509 2.2824 2.1683
503
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504 −5
−4
x 10
x 10
6
Lundberg bound Induction bound (P ) 1
Induction bound (P )
5 Ruin Probability
Ruin Probability
2
Martingale bound (P 1) Martingale bound (P )
2
2
1
4 3 2 1 0
0 0.5
0.6
0.7
0.8
0.9
1
0.75
0.8
The retention level (b)
0.85
0.9
0.95
1
The retention level (b)
Fig. 2. Bounds for the ruin probabilities. Left panel: b 2 ½0:5; 1. Right panel: b 2 ½0:75; 1.
X k2I
1
pik eqi ð1þkÞ M Y
bqi CðbÞqi MZ ¼ 1: 1þk 1þk
In our example we solve R1 ¼ inf i2I qi P R0 , and then we obtain wp ðx; i1 Þ 6 eR1 x for x P 0. Numerical results of this bound are reported in Table 2. It is obvious that this martingale bound improves the results of the induction bound. We run numerical experiments to compare, for a fixed retention level b, the ruin probability bounds that could be achieved. Fig. 2 shows the upper bounds of ruin probability from different approaches with the initial state x ¼ 5 and i ¼ 8%. Finally, we find of special interest the case of small reinsurers for which the retention level could be restricted by economic considerations. Table 2 shows the numerical values of the bounds from different values of b when b is increasing towards 1. Recall, that b ¼ 1 stands for the control action no reinsurance. Clearly, the best results are obtained in the case where the transition interest rate matrix is P 2 . The numerical results in Table 2 show that the upper bound in (31) can be tighter than that in (22). This suggests that the upper bounds derived by the martingale approach are tighter than the upper bounds obtained by induction. In addition, Table 2 also shows that the upper bounds derived in this article are sharper than the Lundberg upper bound.
a. Is it possible to obtain bounds tighter than those in Theorems 3 and 6? b. Actually, what do we need to obtain the ruin probabilities in closed form? c. Let s :¼ inffk P 1jX k < 0g be the time of ruin. Can we calculate or estimate quantities such as E½s, or Pðs 6 TÞ for given T > 0? These are just a few of the many questions that we can ask ourselves. But two immediate queries are: i. Since fIn g in (1) is supposed to be a Markov chain, can we rewrite the minimization of the ruin probability as a Markov decision problem? ([8–10], for instance). ii. Suppose that in (1) we include an investment process. What can we say about these models? Further research in some of these directions is in progress. Acknowledgements The authors are grateful to the anonymous referees for helpful comments and suggestions and in particular for detailed reports. The first author thank Onésimo Hernández-Lerma and Henryk Gzyl for several useful discussions and stimulating suggestions.
6. Concluding remarks
References
Our main results in this paper, Theorems 3 and 6, give upper bounds for the probability of ruin of a certain risk process, which (as shown in Section 2) includes as special cases several relevant models. To obtain these results, first, we present an important preliminary result, Lemma 2, which gives recursive equations for finite-horizon ruin probabilities and an integral equation for the ultimate ruin probability. We illustrate our results with an application to the ruin probability in a risk process with a heavy tail claims distribution under proportional reinsurance and a Markov interest rate process. This application suggests that the upper bounds derived by inductive approach are tighter than the ruin probability without interest rate (the function considered in Lemma 1). In addition, the upper bounds derived in this article are sharper than the Lundberg upper bound. Our paper leaves, of course, many open issues. For instance:
[1] S. Asmussen, Ruin Probabilities, World Scientific, Singapore, 2000. [2] J. Cai, Ruin probabilities with dependent rates of interest, Journal of Applied Probability 39 (2002) 312–323. [3] J. Cai, D. Dickson, Ruin probabilities with a Markov chain interest model, Insurance Mathematics and Economics 35 (2004) 513–525. [4] J. Gaier, P. Grandits, W. Schachermayer, Asymptotic ruin probabilities and optimal investment, Annals of Applied Probability 13 (3) (2003) 1054– 1076. [5] J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991. [6] P. Grandits, An analogue of the Cramér–Lundberg approximation in the optimal investment case, Applied Mathematics and Optimization 50 (2004) 1– 20. [7] M. Diasparra, R. Romera, Bounds for the ruin probability of a discrete-time risk process, Journal of Applied Probability 46 (2009) 99–112. [8] O. Hernández-Lerma, J.B. Lasserre, Discrete-Time Markov Control Processes: Basic Optimality Criteria, Springer-Verlag, New York, 1996. [9] O. Hernández-Lerma, J.B. Lasserre, Further Topics on Discrete-Time Markov Control Processes, Springer-Verlag, New York, 1999. [10] O. Hernández-Lerma, J.B. Lasserre, Markov Chains and Invariant Probabilities, Birkhäuser, Basel, 2003.
504
M. Diasparra, R. Romera / European Journal of Operational Research 204 (2010) 496–504
[11] M. Schäl, On dicrete-time dynamic programming in insurance: Exponential utility and minimizing the ruin probability, Scandinavian Actuarial Journal 3 (2004) 189–210. [12] H. Schmidli, Stochastic Control in Insurance, Springer-Verlag, London, 2008.
[13] G. Willmot, X. Lin, Lundberg Approximations for Compound Distributions with Insurance Applications, Lectures Notes in Statistics, vol. 156, Springer-Verlag, 2001.