ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH
BÜHLMANN CREDIBILITY PREMIUM ADJUSTMENTS
Stéphane LOISEL
Université de Lyon, Université Claude Bernard Lyon 11
Julien TRUFIN
École d'Actuariat, Université Laval2
Abstract:
In this paper, we consider a discrete-time ruin model where experience rating is
taken into account. The main objective is to determine the behavior of the ultimate ruin
probabilities for large initial capital in the case of light-tailed claim amounts. The
logarithmic asymptotic behavior of the ultimate ruin probability is derived. Typical paths
leading to ruin are studied. An upper bound is derived on the ultimate ruin probability in
some particular cases. The influence of the number of data points taken into account is
analyzed, and numerical illustrations support the theoretical findings. Finally, we
investigate the heavy-tailed case. The impact of the number of data points used for the
premium calculation appears to be rather different from the one in the light-tailed case.
Keywords: Discrete-time ruin model, Bühlmann's model, light-tailed claims, large
deviation, ultimate ruin probability, Lundberg coefficient, path to ruin, heavy-tailed claims.
1.
INTRODUCTION
In risk theory, one problem of interest is to compute the probability for an insurance
company to be ruined. The main results of this field can be found in the books of Asmussen
(2000), Dickson (2005), Grandell (1990), Kaas et al. (2008) and Rolski et al. (1999).
Until recently, the increments of the surplus process of the insurance company were
often assumed to be independent. This independence assumption is however not justified
for a certain number of reasons, including the implementation of experience rating
mechanisms by the insurance company. In this case, the financial result of a given calendar
year then depends on the result(s) of one or several previous years.
1
Université de Lyon, Université Claude Bernard Lyon 1, Institut de Science Financière et d'Assurances, 50
Avenue Tony Garnier, F-69007 Lyon, France.
2
École d'Actuariat, Université Laval, Québec, Canada.
BULLETIN FRANÇAIS D’ACTUARIAT, Vol. 13, n° 25, janvier – juin 2013, pp. 73- 102
74
S. LOISEL – J. TRUFIN
In this paper, experience rating is taken into account. Allowing the premium amount
to depend on past claims experience through an appropriate credibility mechanism is in line
with insurance practice in most lines of business.
In the ruin theory literature, only a few authors have studied ruin problem with
credibility dynamics. In Asmussen (1999), the ruin model allows adapted premiums rules.
More precisely, a continuous-time risk process is considered with compound Poisson
claims, where the premium rate is exclusively calculated on the basis of past claims
statistics. In practice, one would use a mixture between a quantity based on past claim
experience and a 'a priori' premium level. The easiest way to compute credibility adjusted
premiums is to use Bühlmann's credibility model. This is what Tsai and Parker (2004) have
done in a discrete-time risk model. They have studied, by means of Monte-Carlo
simulations, the impact of the number of past claim experience years taken into account for
the calculation of the premiums on the ultimate ruin probabilities for light-tailed claims.
In this paper, we deal with a framework that is similar to the one of Tsai and Parker
(2004). One of the main observation made by Tsai an Parker (2004) is that for an insurance
company with a large initial capital and light-tailed claims, the use of a bigger (finite)
number of past claim experience years to calculate the premiums decreases the ultimate
ruin probability. Among others we derive theoretical results supporting this observation
until a certain number of past claim experience. These results should be of interest for risk
managers and people performing an ORSA (Own Risk and Solvency Assessment) of an
insurance company, as discussed in Gerber and Loisel (2012).
Some of the results established in Asmussen (1999) within the continuous-time
compound Poisson process are also extended to the discrete-time risk model discussed in
this paper. In the particular case where the claim amounts follow a compound Poisson, we
find the same results than in Asmussen (1999) for the logarithmic asymptotic behavior of
the ultimate ruin probability on the one hand, and for the typical paths leading to ruin on the
other hand.
We also show that the story is different for heavy-tailed claim amounts.
Some authors as Nyrhinen (1994) and Mikosch and Samorodnitsky (2000) have
investigated asymptotic ruin results for moving average processes. Some asymptotic results
derived for our particular process are similar to some obtained by these authors for this
related class of processes.
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
2.
75
THE MODEL
Let us consider an insurance company with an heterogeneous group of policyholders
mixing p individuals with r different risk levels. We assume that there are no observable
risk characteristics enabling to distinguish the risk level of these individuals (this group of
individuals can be seen as the result of a risk classification previously performed with the
observable risk factors).
The risk faced by the insurance company depends on the risk levels of each
individual. Therefore, the number of potential portfolios for the insurer is equal to n = r p .
Obviously, two portfolios with the same numbers of people within each risk level are
equivalent since they present the same risk.
Let us denote by j (i ) the risk level of the i th individual ( i = 1, , p ) for the j th
portfolio ( j = 1, , n ) and by qk ( k = 1, , r ) the probability for a policyholder to belong
to the risk level k . The probability for the insurance company to cover the portfolio j is
then given by p j =
∏ i =1q j (i) .
p
In such situation of practical relevance, when it is not possible to rely on additional
risk factors, the insurance company uses credibility premiums in order to determine the
pure premium corresponding to its portfolio or equivalently to each of its policyholders.
The model proposed in Bühlmann (1967) and (1969) is largely used in practice. In this
paper we focus on this well-known credibility model.
( j)
Let us denote by Yk the annual claim amounts of the company and by Yk the
independent and identically distributed annual aggregated claim amounts for portfolio j ,
with common distribution function F j . The a priori mean of the Yk 's is of course given by
μ = [Yk ] =
n
[Y ( j ) ] p j ,
(1)
j =1
where Y
( j)
is a random variable with distribution function F j , while the a priori variance
breaks up in two terms:
σ 2 = [Yk ] = a + ν ,
(2)
with
n
a=
([Y ( j) ] − μ )2 p j
j =1
and
(3)
76
S. LOISEL – J. TRUFIN
ν=
n
[Y ( j) ] p j .
(4)
j =1
Parameter a measures the part of the variance coming from the heterogeneity of the
portfolios, whereas parameter ν measures the part of hazard in the a priori variance.
As mentioned in Tsai and Parker (2004), some casualty insurers only consider a
finite number of most recent periods of claim experiences to renew the premium. Let us
denote by m ∈  + ∪ {∞} the so-called "horizon of credibility", which corresponds to the
number of periods taken into account for the calculation of the premiums.
So, depending on the choice of m , with the convention that
 i =1 = 0 , the premium
0
of the company in time k − 1 is given by
,
= (1 −
,
) +
∑
(
,
( ,
, )
(1 + ),
)
(5)
where
zk , m =
min(m, k − 1) a
(6)
ν + min(m, k − 1) a
is Bühlmann's credibility factor and η > 0 is the premium security loading. Our premium
rule depends then on the parameters μ , a and ν . For simplicity, we have assumed that no
historical data is available. This simplifying assumption does not influence the asymptotic
results derived in the sequel.
Now, if the portfolio held by the insurer is portfolio j , then the dynamics of the
insurer's surplus obeys to the equation
U k( ,jm) = u +
k
k
i =1
i =1
Ci,m − Yi( j ) ,
(7)
( j)
where the Ci,m 's are described by equation (5) with Yi = Yi , and where u is the initial
capital of the company. The corresponding ultimate ruin probability may be written as
(8)
ψ m( j ) (u ) = Pr U k( ,jm) < 0 for some k | U 0,( jm) = u  .


Immediately, it appears the existence of a subset of "bad" portfolios, denoting bm ,
( j)
such that for all j ∈ bm , ψ m
(u ) = 1 for all u . Indeed, let us define
ηm( j ) =
where
(1 + η )((1 − zm ) μ + zm [Y ( j ) ])
[Y ( j ) ]
− 1,
(9)
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
zm =
ma
ν + ma
77
(10)
.
( j)
Obviously, for m < ∞ , ηm
corresponds to the average security loading of the
( j)
premium for k > m while for m = ∞ , η m
= η , which corresponds this time to the
security loading for k → ∞ . The subset bm is then defined as
( j)
bm = { j = 1, , n : ηm
≤ 0}.
(11)
Also, in the following, the complementary subset of ("not bad") portfolios is denoted
by bm , and we shall say that j ∈ g if and only if [Y ( j ) ] ≤ μ . Clearly, we have g ⊂ bm .
The purpose of this paper is to investigate the behavior of the ultimate ruin
( j)
probabilities ψ m
(u ) for large u in the case of light-tailed annual claim amounts. First, the
( j)
logarithmic asymptotic behavior of ψ m
(u ) is derived. Then, typical paths leading to ruin
( j)
are studied. Also, an upper bound on ψ m
(u ) is derived in some particular cases.
Afterwards, we are able to determine, for each portfolio j , the influence of "strategy" m
( j)
(u ) for large u . Next, whatever the portfolio j , optimal strategies, in a sense to be
on ψ m
defined in the sequel, are deduced. As an illustration, numerical simulations are also
( j)
performed. Finally, we also study the influence of m in first order on ψ m
(u ) in the
heavy-tailed case. As we shall see, the conclusion is totally different than in the light-tailed
case.
3.
LIMIT RESULTS FOR LIGHT-TAILED CLAIMS
In this Section, for simplicity, let us assume that for all j = 1, , n , there exists
r( j) > 0
such
lim r →r ( j )[e
3.1
rY ( j )
that
[erY
( j)
]
exists
for
all
0 < r < r( j)
and
such
that
] = ∞ . Clearly, this assumption implies necessarily light-tailed claims.
The case m < ∞
( j)
(u )
3.1.1 Asymptotic behavior of ψ m
Theorem 3.1 Suppose that there exists a unique positive solution to the equation
− r (1− zm ) μ (1+η )  rY ( j ) (1−(1+η ) zm ) 
 e
e
 = 1,


(12)
( j)
which we denote ρ m
. Furthermore, assume that for all k , the sum of the first k annual
78
S. LOISEL – J. TRUFIN
losses Z k =
 i =1(Yi( j ) − Ci,m )
k
rZ
possesses a finite moment-generating function [e k ] for
( j)
0 < r < r0 , with ρ m
< r0 . Hence, we have
1
( j)
( j)
lnψ m
(u ) = − ρ m
.
u →∞ u
(13)
lim
Proof. As a consequence of the Gärtner-Ellis Theorem from large deviations (which
is due to Glynn and Whitt (1994), see also Nyrhinen (1998) and Müller and Pflug (2001)),
one only has to show the two following properties:
1
rZ
(A1) κ m (r ) := lim k →∞ ln [e k ] exists for 0 < r < r0 ,
k
( j)
( j)
(A2) ρ m
is the unique positive value such that κ m ( ρ m
)=0.
We have
ln [
] = − (1 + )
(1 −
= − (1 + )
where zk ,m =
(1 −
zk , m
min(m, k − 1)
,
,
) + ln
)+
ln
∑
(
( )(
( )
( ̅
) ̅,
(
,
̅
,
∑
⋯
( ,
̅
(
)
, ),
( )
)(
)
))
. Consequently
 rY ( j ) (1−(1+η ) zm ) 
1
rZ
ln [e k ] = −r (1 − zm ) μ (1 + η ) + ln  e
.
k →∞ k


lim
Hence, (A1) holds true. Concerning (A2), notice that κ m (r ) = ln hm (r ) , where
hm (r ) = e
− r (1− zm ) μ (1+η )
 rY ( j ) (1−(1+η ) zm ) 
 e
.


Notice that this result is known by Example 3 of Nyrhinen (1994) in case of a
moving average process.
Now, since [erY
( j)
] exists for all 0 < r < r ( j ) , result (13) requires in fact only
( j)
( j)
existence and uniqueness of ρ m
, with the condition ρm
< r( j) .
( j)
3.1.2 The Lundberg coefficient ρ m
Let us define
ηm( j ) =
(1 − zm ) μ (1 + η )
[Y ( j ) ](1 − (1 + η ) zm )
The equation (12) is equivalent to
− 1.
(14)
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
e
( j) )
− r (1− (1+η ) zm ) [Y ( j ) ](1+ηm
 r (1−(1+η ) zm )Y ( j ) 
 e
 = 1,


79
(15)
and consequently
ρ m( j ) =
ρ m( j )
,
1 − (1 + η ) zm
(16)
( j)
where ρ m
is the classical Lundberg coefficient defined as the unique positive
solution of the equation
e
( j) )
− r [Y ( j ) ](1+ηm
[e rY
( j)
(17)
] = 1.
( j)
Now, from classical results, it is then obvious that the inequality (∞ >)ηm
> 0 is a
( j)
necessary condition for the existence of ρ m
. Let us prove that the inequality
( j)
(∞ >)ηm
> 0 is equivalent to m <
ν
and j ∈ bm .
aη
Proposition 3.1
( j)
(∞ >)ηm
>0⇔m<
ν
andj ∈ bm .
aη
( j)
Proof. () : If ∞ > ηm
> 0 , then, obviously zm <
(18)
1
ν
, or equivalently m <
,
1+η
aη
and (1 − zm ) μ (1 + η ) > [Y ( j ) ](1 − (1 + η ) zm ) . Thus
ηm( j ) =
>
((1 − zm ) μ + zm [Y ( j ) ])(1 + η )
[Y ( j ) ]
−1
[Y ( j ) ](1 − (1 + η ) zm ) + zm [Y ( j ) ](1 + η )
[Y ( j ) ]
−1
= (1 − (1 + η ) zm ) + zm (1 + η ) − 1 = 0.
(⇐) :
Since
j ∈ bm ,
we
have
((1 − zm ) μ + zm [Y ( j ) ])(1 + η ) > [Y ( j ) ] .
(1 − zm ) μ (1 + η ) > [Y ( j ) ](1 − (1 + η ) zm ) . Now, since m <
ν
aη
, zm <
Thus,
1
( j)
. So, ηm
<∞
1+η
and [Y ( j ) ](1 − (1 + η ) zm ) > 0 . Consequently
( j)
∞ > ηm
=
(1 − zm ) μ (1 + η )
[Y ( j ) ](1 − (1 + η ) zm )
−1 >
[Y ( j ) ](1 − (1 + η ) zm )
[Y ( j ) ](1 − (1 + η ) zm )
− 1 = 0.
( j)
This result highlights the obvious fact that the existence of ρ m
implies that j
( j)
must belong to bm , since ψ m
(u ) = 1 for j ∈ bm .
80
S. LOISEL – J. TRUFIN
Let us explain the reason why the horizon of credibility 
ν
(c)
 , denoted by m
aη
from now, constitutes a critical value for the existence of the Lundberg coefficient and
( j)
(u ) .
consequently for the logarithmic asymptotic behavior of ψ m
( j)
From m = m(c) , the global impact of each Yk on the surplus process for k > m ,
( j)
i.e. Yk (1 − (1 + η ) zm ) , becomes negative, which is, of course, a drastic change in the
nature of the insured risk. Also, bm = ∅ for all m ≥ m(c) .
More fundamentally, in first order, the only effects selected of the credibility
dynamics compared to the classical case m = 0 are a modification of the security loading
( j)
and a decreasing of the claim amounts. Indeed, equations (15) and (16) teach us that ρ m
is identical to that of the classical ruin model with a premium security loading equals to
ηm( j ) and annual claim amounts given by Yk( j ) (1 − (1 + η ) zm ) . Therefore, in this associated
classical ruin model, from the critical value m(c) , no ruin can be observed, which explains
the reason why a Lundberg coefficient does not exist in this case.
So, since [erY
( j)
] exists for all 0 < r < r ( j ) with lim r →r ( j )[erY
( j)
] = ∞ , it
comes that the two following conditions are of course necessary but also sufficient
( j)
:
conditions to guarantee existence and uniqueness of ρ m
(C1) m < m(c) ,
(C2) j ∈ bm .
( j)
Let us recall that result (13) requires not only existence and uniqueness of ρ m
( j)
but also the additional condition ρm
< r ( j ) . Firstly, we have the following property for the
( j)
Lundberg coefficient ρ m
:
Property 3.2
ρ m( j )
1
ρ m( j )
2
<
Assume that ρ m( j ) and ρ m( j ) exist, with 0 ≤ m1 < m2 . Then,
1
2
.
Proof. Since zm < zm , we have ηm( j ) < ηm( j ) and consequently, by classical
1
results,
ρ m( j )
1
<
ρ m( j )
2
2
. By (16), we deduce
1
ρ m( j )
1
<
ρ m( j )
2
2
.
Secondly, let us define
( j)
( j)
m( j ) = min{m < m(c ) : ρ m
exist and ρ m
≥ r ( j ) }.
If
ρ m( j )
does not exist for all m < m
(c )
( j)
, then we put m
(19)
= 0 . Also, if for all
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
81
m < m(c) such that ρ m( j ) exists, we always have ρm( j ) < r ( j ) , then we put m( j ) = m(c) .
Hence, if we replace condition (C1) by the stronger condition m < m( j ) , we have now
( j)
such that
necessary and sufficient conditions to guarantee existence and uniqueness of ρ m
ρm( j ) < r ( j ) , i.e. result (13).
3.1.3 Large deviations path to ruin
Let us assume that m < m( j ) and j ∈ bm , i.e. that result (13) is valid. The goal here
is to understand how ruin occurs for large u . More particularly, we want to determine the
typical shape of a path leading to ruin when the initial capital of the company is large. To
this end, let us use large deviations results of Glynn and Whitt (1994). First of all, for large
u , we know that ruin occurs roughly at time u / κ m′ ( ρm( j ) ) . Next, the cumulant
generating function γ
γ
( j ) )  (r )
′ ( ρm
u / κ m
( j ) )  (r )
′ ( ρm
u / κ m
of Z
( j ) )
′ ( ρm
u / κ m
is asymptotically changed from
to
( j)
( j)
γ
( j ) )  ( r + ρ m ) − γ u / κ ′ ( ρ ( j ) )  ( ρ m )
′ ( ρm
u / κ m
m m
= −r (1 + η ) μ
( j ) )
′ ( ρm
u / κ m

(1 − zi,m )
i =1


( j) ( j) 
 ( r + ρm )Y 1−( zi +1,m + zi + 2,m ++ zmin(i + m, u /κ ′ ( ρ ( j ) )  ),m )(1+η )  
m m


 e


( j ) )
′ ( ρm
u / κ m


+
ln
.

 ( j) ( j) 
i =1
 ρm Y 1−( zi +1,m + zi + 2,m ++ zmin(i + m, u /κ ′ ( ρ ( j ) )  ),m )(1+η )  
m m


 e





( j)
Consequently, given that ruin occurs, the claim size distribution Yk
is
( j)
exponentially tilted by the time-dependent factor ρ m
(k ) , with
ρ m( j ) (k ) =
 ρ ( j ) ,
 m

 u / κ ′ ( ρ ( j ) )  − k

m m
 1 − 
m
 
( j)
′ ( ρm
fork = 1, , u / κ m
) − m


.
( j)
( j)
( j)
′ ( ρm
′ ( ρm
 z (1 + η )  ρ m
, fork = u / κ m
)  − m + 1, , u / κ m
)
 m



( j)
′ ( ρm
) ),
Hence, for large u , the drift of the path to ruin at time k ( k = 1, , u / κ m
denoted as δ m ( k ) , is given by
82
S. LOISEL – J. TRUFIN

ρ ( j ) ( k )Y ( j ) 
 Y ( j ) e m

 < 0.
δ m (k ) = (1 − zm ) μ (1 + η ) − (1 − (1 + η ) zm ) 
( j ) ( k )Y ( j ) 
 ρm
 e



However,
notice
that
( j)
( j)
′ ( ρm
′ ( ρm
( u / κ m
)  − m + 1, u / κ m
))
u →∞,
since
the
time
(20)
period
where δ m ( k ) decreases with k is negligible
( j)
′ ( ρm
compared with the time period (1, u / κ m
) − m) where δ m ( k ) remains constant.
This analysis highlights that ruin is caused by a succession of small claims, followed
by a succession of moderately large claims.
( j)
3.1.4 Upper bound on ψ m
(u )
In this subsection only, we take into account a possible claim experience of h years
( j)
( j)
( j)
say, i.e. ( Y−h+1, Y−h+ 2 ,, Y0 ), because, this time, as we see in the following, it will
impact the result. Hence, equation (5) becomes
k −1



Yi 

i =max( k − m,1− h) 
Ck ,m =  (1 − zk ,m ) μ + zk ,m
 (1 + η ),
min(m, h + k − 1) 








(21)
with
zk , m =
min(m, h + k − 1) a
.
ν + min( m, h + k − 1) a
(22)
Let b = min(m, h) , and let us denote
ψ m( j ) (u , y0 , , y1−b )
( j)
( j)
= Pr U k( ,jm) < 0forsomek | U 0,
= y0 , , Y1(−jb) = y1−b  .
m = u , Y0


(23)
( j)
We are now in position to derive an upper bound on ψ m
(u, y0 , , y1−b ) based on
the same approach than in Gerber (1982).
Theorem 3.3 Assume that m < m( j ) and j ∈ bm . Then
ψ m( j ) (u , y0 , , y1−b )
where
≤
e
( j )uˆ
− ρm
 − ρ ( j )Uˆ

 e m T | T < ∞ 


,
(24)
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
=
( )
,
( )
(
( )
+ (1 + )[
)(
,
with zk ,m =
̅
( ̅
+⋯+ ̅
,
+ ⋯+ ̅
,
zk , m
min(m, h + k − 1)
(
k ,m
,
)+
),
( )
( ̅
,
+⋯+ ̅
,
)], uˆ = Uˆ 0 , and T = inf{k
) + ⋯+
| U k( ,jm)
< 0} ,
.
Proof. First, we note that
Uˆ = U ( j ) + (1 + η )[Y ( j ) ( z
k
,
83
k
k +1,m
+  + zk + m,m ) + Yk(−j1) ( zk +1,m +  + zk + m −1,m )
( j)
+  + Ymax(
k − m +1,1− h ) ( zk +1,m +  + zm + max( k − m +1,1− h ),m )]
= U k( −j )1,m + Ck ,m − Yk( j )
+ (1 + η )[Yk( j ) ( zk +1,m +  + zk + m,m ) + Yk(−j1) ( zk +1,m +  + zk + m−1,m )
( j)
+  + Ymax(
k − m +1,1− h ) ( zk +1,m +  + zm + max( k − m +1,1− h ),m )]
=
( )
,
+ (1 −
,
) (1 + ) +
,
(1 + )
∑
(
,
( ,
( )
)
)
−
( )
+ (1 + η )[Yk( j ) ( zk +1,m +  + zk + m,m ) + Yk(−j1) ( zk +1,m +  + zk + m−1,m )
( j)
+  + Ymax(
k − m +1,1− h ) ( zk +1,m +  + zm + max( k − m +1,1− h ),m )]
= U k( −j )1,m + (1 + η )[Yk(−j1) ( zk ,m +  + zk + m −1,m ) + Yk(−j 2) ( zk ,m +  + zk + m − 2,m )
( j)
+  + Ymax(
k − m,1− h ) ( zk ,m +  + zm + max( k − m,1− h ), m )]
+ (1 − zk ,m ) μ (1 + η ) − Yk( j ) (1 − (1 + η )( zk +1,m +  + zk + m,m ))
= Uˆ k −1 + (1 − zk ,m )μ (1 + η ) − Yk( j ) (1 − (1 + η )( zk +1,m +  + zk + m,m )).
( j)
Hence, if we denote by Φ k the sigma-algebra generated by {Yi , i = 1, 2,, k} , we
have
( j )Y ( j ) (1− (1+η )( z

 − ρ ( j )Uˆ

− ρ ( j ) (1− zk , m ) μ (1+η )  ρ m
− ρ ( j )Uˆ
k +1, m ++ zk + m, m )) .
  e m k | Φ k −1  = e m k −1 e m
 e





Now, let us define
 rY ( j ) (1−(1+η )( zk +1, m ++ zk + m,m )) 
 e
.


Obviously, for k ≥ m − h + 1 , hk ,m (r ) = hm (r ) . Since zk −1,m ≤ zk ,m and
zk −1,m ≥ zk ,m , we clearly have hk −1,m (r ) ≤ hk ,m (r ) ≤ hm (r ) . Consequently, by (12),
hk , m (r ) = e
− r (1− zk , m ) μ (1+η )
( j)
( j)
hk ,m ( ρm
) ≤ hm ( ρm
) = 1 . Thus, we get
84
S. LOISEL – J. TRUFIN
 − ρ ( j )Uˆ

− ρ ( j )Uˆ
− ρ ( j )Uˆ
( j)
  e m k | Φ k −1  = e m k −1 hk ( ρm
) ≤ e m k −1 .


So, the process {e
( j )Uˆ
− ρm
k
} is a super-martingale.
Let w be a positive integer. By the Optional Stopping Theorem (considering the
stopping time T ∧ w = min{T , w} ),we get
 − ρ ( j )Uˆ

 − ρ ( j )Uˆ

 − ρ ( j )Uˆ

 − ρ ( j )Uˆ

− ρ ( j )Uˆ
e m 0 ≥   e m T ∧ w  =   e m T IT ≤ w  +   e m w I T > w  ≥   e m T IT ≤ w  .








Hence letting w → ∞ , we have
 − ρ ( j )Uˆ

 − ρ ( j )Uˆ
 ( j)
− ρ ( j )Uˆ
e m 0 ≥  e m T | T < ∞  Pr[T < ∞] =  e m T | T < ∞ ψ m
(u, y0 , , y1−b ).




Remark
Inequality (24) becomes an equality when h ≥ m . Indeed, in this case, for k ≥ 1 , we
have
( j )Uˆ
( j)
( j)
hk ,m ( ρm
) = hm ( ρm
) = 1 . Thus, the process {e − ρ m
k}
is a martingale.
− ρ ( j )Uˆ
Furthermore, the random variables e m w IT > w pointwise converge to zero for w → ∞
and are bounded by 1 since Uˆ > 0 for w < T .
w
3.1.5 The case m ≥ m( j ) and j ∈ bm
Now, let us consider the case m ≥ m( j ) and j ∈ bm . For the next result, let us
( j)
( j)
( j)
( j)
( j)
denote ψ m
(u ) by ψ m( j,η) (u ) , ρ m
by ρ m( j,η) , ρ m
by ρ m( j,η) , ηm
by ηm( j,η) and ηm
by
η m( j,η) to reveal explicitly the dependence in η . Obviously, notice that ηm( j,η) increases with
η . Hence, we know by classical results that ρ m( j,η) also increases with η and consequently
ρ m( j,η) by (16). Furthermore, let us define

(j)
η ( ,1) = min η ≤ η : ηm,
η >0 and m<

ν 
.
aη 
(25)
ν 
.
aη 
(26)
and

(j)
η (  ,2) = max η ≤ η : ηm,
η >0 and m<

( j)
( j)
Proposition 3.2 Assume that ρ ( ,1) < r ( j ) ≤ ρ (  ,2) . Then, we have
m,η
m,η
1
( j)
( j)
lim sup lnψ m,η (u ) ≤ − r .
u
u →∞
(27)
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
85
Proof. Obviously, there exists η (  ,1) < η ≤ η (  ,2) such that ρ m( j,η) = r ( j ) . For all
η (  ,1) ≤ η < η , we have
1
1
( j)
( j)
( j)
lim sup lnψ m,η (u ) ≤ lim lnψ m,η (u ) = − ρ m,η .
u
u →∞
u →∞ u
Letting η ↑ η , we obtain (27).
As a particular, let us assume that Y ( j )  Exp (λ j ) . In this case, [erY
for r < λ j and is equal to
λj
λj − r
( j)
] exists all
. Clearly, r ( j ) = λ j . Now, since
1
ln Pr[U1,( jm) < 0] = λ j
u →∞ u
lim
(28)
and that ψ m( j ) (u ) ≥ Pr[U1,( jm) < 0] , result (27) becomes
lim inf
u →∞
1
1
( j)
( j)
lnψ m
(u ) = lim sup lnψ m
(u ) = −λ j .
u
u
u →∞
(29)
3.1.6 Conclusion
( j)
In conclusion, on the one hand, if j ∈ bm then ψ m
(u ) = 1 for all u . On the other
( j)
(u ) is
hand, if j ∈ bm , then for m < m( j ) , the logarithmic asymptotic behavior of ψ m
given by equation (13) while for (∞) > m ≥ m( j ) , the logarithmic asymptotic behavior of
ψ m( j ) (u ) is this time characterized by equation (27).
3.2
The case m = ∞
Clearly, with zm = 1 , i.e. m = ∞ , the results of the previous section have no
meaning. This is the reason why this important case has to be treated separately.
3.2.1 Asymptotic behavior of ψ ∞( j ) (u )
Let us state an analogue of Theorem 3.1.
Theorem 3.4 Suppose that there exists a unique positive solution to the equation
1
 ( j)

dx ln  erY (1+(1+η ) ln x )  = 0,
(30)
0



rZ
which we denote ρ∞( j ) . If we assume that for all k , [e k ] exists for 0 < r < r0 (with
ρ∞( j ) < r0 ), where Z k = 
k
(Y ( j )
i =1 i
− Ci ,∞ ) , we have
86
S. LOISEL – J. TRUFIN
1
lnψ ∞( j ) (u ) = − ρ ∞( j ) .
u →∞ u
(31)
lim
1
rZ
Proof. Again, it suffices to show that κ ∞ (r ) := lim k →∞ ln [e k ] exists for
k
0 < r < r0 , and that ρ∞( j ) is the unique positive value such that κ ∞ ( ρ∞( j ) ) = 0 . We have
ln [
] = − (1 + )
(1 −
,
( )
∑
) + ln
(
) ,
( )
∑
.
Now we have
( )
( )
∑
− (1 + )
,
( )
=
−1
,
1 − (1 + )
.
Hence we get
ln [
] = − (1 + )
(1 −
,
( )
∑
) + ln
(
Thus, we have
1
1
rZ
lim ln [e k ] = − r (1 + η ) μ lim
k
k →∞
k →∞ k
k
(1 − zi,∞ )
i =1
k −1 z


 k
l +1,∞ 
 r Yi( j ) 1−(1+η )


l


 i =1
1
l =i

 .
+ lim ln   e

k →∞ k








It is obvious that
1
k →∞ k
lim
k
(1 − zi,∞ ) = 0.
i =1
Furthermore, note that
k −1 z
l +1,∞

l =i
l
=
k −1
 al +ν .
a
l =i
Since we have
l +1
l
we find that
a
a
dt ≤
≤
al +ν
al +ν
l
a
l −1 al +ν dt,
)∑
,
.
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
 ak + ν 
ln 
≤
 ai + ν 
k −1

zl +1,∞
l
l =i
 a ( k − 1) + ν
≤ ln 
 a (i − 1) + ν
87

.

Consequently, letting k → ∞ , we get
k −1 z
 ai 
l +1,∞
−
= ln 
.
l
 ak + ν 
l =i

So, we obtain
lim
1
→
ln [
1
] = lim
→
( )
∑
ln
(
)
.
Hence we have
k
1
1  ak +ν   a
rZ
lim ln [e k ] = lim 
 
k
k →∞
k →∞ k  a  i =1  ak + ν

k
 a
= lim

k →∞ i =1  ak + ν

=

 ln [e

 rY ( j ) 1+ (1+η ) ln  ai   

 ak +ν   





 ln  e







 ai  
rY ( j )  1+ (1+η ) ln 

 ak +ν   ]

 rY ( j ) (1+ (1+η ) ln x ) 
dx ln   e
.
0



1
It appears that result (31) requires only existence and uniqueness of ρ∞( j ) (since the
condition ρ∞( j ) < r ( j ) is always fulfilled).
Equation (30) and so ρ∞( j ) does not depend on parameters μ , a and ν . It means
( j)
(u ) is insensitive to the initial
that for u → ∞ , the logarithmic asymptotic behavior of ψ m
premium and to the speed with which premiums Ck ,∞ become
As a particular case, let us consider Y ( j ) 
Y1( j ) +  + Yk(−j1)
 i =1U i ,
N
k −1
(1 + η ) .
where N is Poisson
distributed with parameter β and the U i 's are i.i.d. random variables. We assume that N
and the U i 's are independent. We have
[
( )
]=
=∑
=
∑
∑
=
[
] Pr[
[
]
=
|
= ]=∑
(
∑
=
Pr[
)
Pr[
= ]=∑
= ]
Pr[
= ]
(32)
88
S. LOISEL – J. TRUFIN
since   e rN  = e β (e



1
0dx ln  e
r −1)
. So, the equation (30) becomes
rY ( j ) (1+ (1+η ) ln x ) 

1
 rU (1+(1+η ) ln x )  − β = 0.
 = β 0dx  e


(33)
We recognize the equation (2.2) in Asmussen (1999). In fact, this is not surprizing
since ρ∞( j ) is insensitive to parameters μ , a and ν .
3.2.2 The Lundberg coefficient ρ∞( j )
( j)
Since m = ∞ , we know that ηm
= η > 0 for all j , i.e. bm = ∅ .
Proposition 3.3 Assume that κ ∞′′ ( r ) > 0 . Then, ρ∞( j ) exists and is unique.
Proof. Obviously, we have κ ∞ (0) = 0 . Denoting Y ( j ) (1 + (1 + η ) ln x ) by Z x , we
have
 Z xe x 


κ ∞′ (r ) = dx 
=
rZ
0
 e x 


rZ

≥
1
1
0dx
 Z xe

rZ x
 Z xe

dx 
1
0
rZ x
IZ
 +   Z erZ x I

Z x ≤0 
 x

rZ x 

 e


x >0 
 +  Z I

 x Z x ≤0 
.
rZ
 e x 


IZ
x >0 
Hence we get

1
0

κ ∞′ (0) = [Y ( j ) ] 1 + (1 + η ) dx ln x  = −[Y ( j ) ]η < 0


since η > 0 , and κ ∞′ ( r ) becomes positive for large r . Now, since κ ∞′′ ( r ) > 0 by
assumption, the proof is over.
3.2.3 Large deviations path to ruin
As previously, we aim at knowing how ruin occurs for the case m = ∞ . Of course,
ruin occurs roughly at time u / κ ∞′ ( ρ∞( j ) ) . Now, the cumulant generating function
γ
( j ) )  (r )
′ ( ρ∞
u / κ ∞
of Z
( j ) )
′ ( ρ∞
u / κ ∞
is asymptotically changed from γ
( j ) )  (r )
′ ( ρ∞
u / κ ∞
to
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
( j)
( j)
γ
( j ) )  ( r + ρ ∞ ) − γ u / κ ′ ( ρ ( j ) )  ( ρ ∞ )
′ ( ρ∞
u /κ ∞
∞ ∞
(
+ ∑
/
(
( )
)
ln
( )
)
= −r (1 + η ) μ

i =1
( )
( ) ( )
( j ) )
′ ( ρ∞
u / κ ∞
(
(
/
)∑
)∑
/
(
(
( )
( )
)
89
(1 − zi,∞ )
,
)
,
.
(34)
( j)
Consequently, given that ruin occurs, the claim size distribution Yk is this time
exponentially tilted by the factor ρ ∞( j ) (k ) , where
ρ∞( j ) (k )
( j ) ) −1

′ ( ρ∞
u / κ ∞
zl +1,∞

=  1 − (1 + η )
l

l =k


 ( j)
 ρ∞ .


(35)
  ( j)
 ρ .
 u / κ ′ ( ρ ( j ) )   ∞
∞ ∞


(36)

Since u → ∞ , we get


ρ∞( j ) (k ) = 1 + (1 + η ) ln 


The factor
ρ ∞( j ) (k )
k
is negative as long as k <
1
−
(
)
j
+
1
u / κ ∞′ ( ρ∞ )  e η
.
We learn that this time, ruin is the consequence of atypically small claim amounts in
a first time, followed by atypically large claim amounts in a second time. This result is
similar to those obtained in Asmussen (1999) for his continuous time compound Poisson
model. A typical path to ruin is shown in Figure 1.
Figure 1: Typical path to ruin for the case m = ∞
90
S. LOISEL – J. TRUFIN
( j)
Impact of the insurer's strategy m on ψ m
(u ) for large u
3.3
( j)
( j)
3.3.1 Comparison of ψ m
(u ) and ψ m+1 (u ) for large u (with m < ∞ )
Let us assume that the insurer's strategy is to take an horizon of credibility m < ∞ .
The objective is to determine for large u if it is beneficial to take one year more to renew
( j)
( j)
the premium. In other words, the objective is to compare ψ m
(u ) and ψ m+1 (u ) for large
u.
For j ∈ bm , we have the following Proposition:
Proposition 3.4
ψ m( j+)1 (u ) ≤ ψ m( j ) (u ) = 1for all u.
Proof. It suffices to show that bm +1 ⊆ bm . Let j ∈ bm +1 . By definition,
(37)
ηm( j+)1
≤0.
( j)
Let h( x) = (1 − x) μ + x[Y ( j ) ] . We have h′( x) = [Y ( j ) ] − μ > 0 since ηm+1 ≤ 0 means
[Y ( j ) ] ≥ (1 + η )((1 − zm +1 ) μ + zm +1[Y ( j ) ]) and η > 0 . Thus, since zm+1 > zm , we have
h( zm )
h( zm+1 )
( j)
ηm( j ) = (1 + η )
− 1 < (1 + η )
− 1 = ηm
+1 ≤ 0 , or equivalently, j ∈ bm .
( j)
( j)
[Y ]
[Y ]
This means that if j ∈ bm , it is beneficial to pass from an horizon of credibility m to
an horizon m + 1 , whatever the initial capital u .
( j)
Now, for j ∈ bm , it could seem a priori that the comparison of ψ m
(u ) and
ψ m( j+)1 (u ) for large u differs according to the involved portfolio j . Indeed, two opposite a
priori behaviours seem to take shape. If
j ∈ g , then h′( x) ≤ 0 and consequently
ηm( j ) ≥ ηm( j+)1 , which seems to be in favor of ψ m( j ) (u ) ≤ ψ m( j+)1 (u ) whatever u and in
particular for large u . But, on the other hand, if j ∈ bm \ g , then h′( x) > 0 and
( j)
( j)
< ηm
consequently ηm
+1 , which seems this time to be in favor of the inequality
ψ m( j+)1 (u ) ≤ ψ m( j ) (u ) whatever u .
For m < m( j ) , we have the next Proposition:
Proposition 3.5 There exists a positive constant u0 such that for all u ≥ u0 , we
ψ m( j+)1 (u) ≤ ψ m( j ) (u).
have
Proof. We know that j ∈ bm and m < m
( j)
(38)
. Since bm +1 ⊆ bm , j ∈ bm +1 . Hence,
we have to consider two cases.
Firstly, let us assume that m + 1 < m( j ) . By Property 3.2, our result is proved.
Secondly, let us assume that m + 1 ≥ m( j ) . Hence, we know that
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
91
1
( j)
( j)
lnψ m
(u ) = − ρ m
u
u →∞
lim
and
1
( j)
( j)
lim sup lnψ m +1 (u ) ≤ − r ,
u
u →∞
( j)
and since ρm
< r ( j ) , this completes the proof.
( j)
( j)
≥ ηm
Thus, even if j ∈ g and hence ηm
+1 , (38) holds for large u . In fact, this is
not surprizing. With light-tailed claims, the event ruin occurs after many atypically large
claims. At most m is large, at most the premium reacts favorably to this large claims, i.e.
ηm( j+)1 becomes larger than ηm( j ) in such circumstances. These facts are confirmed by our
analysis of the path leading to ruin. Indeed, it teaches us that ruin is caused by a succession
( j)
of many atypically large claims more dangerous as m is large since ρ m
increases
with m .
For m ≥ m( j ) , we cannot conclude anything like that, since the only result we have
is Equation (27).
In conclusion, assuming that the strategy of the company is m , if j ∈ bm , then it is
always beneficial for large u to increase by 1 the horizon of credibility. Furthermore, if
( j)
( j)
j ∈ bm , for m < m , the same conclusion comes while for m ≥ m , we cannot certify
that this is also the case.
3.3.2 Sub-optimal strategies m for large u
Let us investigate sub-optimal strategies m for large u .
Definition 3.5 A strategy m1 is said to be sub-optimal compared to a strategy m2 if
for all portfolio j , we have ψ m( j ) (u ) ≤ ψ m( j ) (u ) for u → ∞ .
2
1
Definition 3.6 A strategy m1 is said to be sub-optimal if there exists a strategy m2
such that m1 is sub-optimal compared to m2 .
We have the next Proposition:
Proposition 3.6 m < min { j:m( j ) >0}m ( j ) like m = ∞ are sub-optimal.
Proof. In view of Propositions 3.4 and 3.5, it is clear that m < min { j:m( j ) >0}m ( j ) is
sub-optimal. For m = ∞ , it is also obvious since ρ∞( j ) < r ( j ) .
92
S. LOISEL – J. TRUFIN
As we have seen, for a given portfolio j , increasing m decreases the number of
"bad portfolios" and provides better premium adaptation to scenarios leading to ruin, at
least as long as m < m( j ) . An extension of this reasoning to all m would allow to think that
all m < ∞ is sub-optimal compared to m = ∞ . However, Proposition 3.6 shows that this is
not the case. In fact, as u → ∞ , all m < ∞ becomes negligible compared to ruin time T .
Hence, the premiums will be able to react "quickly" (relatively to T ) and "significantly" (at
least for m "large" but finite) to a succession of large claim amounts, whatever the time
where these claims occur and whatever the claim amounts observed before. But for m = ∞ ,
obviously, this is not the case anymore. Indeed, as time goes by, the premium reaction will
become less and less effective to finish by not be able anymore to thwart several important
successive claim amounts. This phenomenon is even truer if the claim amounts observed
initially are small. This is well highlighted by the analysis of the path leading to ruin.
3.3.3 Numerical illustration
Let us consider a case where n = 3 . Let us assume that Y ( j ) ( j = 1, 2, 3 ) are
Exponentially distributed, with mean 1 / λ j
2
and variance 1 / λ j . We then have
μ = (1/ λ1 ) p1 + (1/ λ2 ) p2 + (1/ λ3 ) p3 , a = (1/ λ1 − μ ) 2 p1 + (1/ λ2 − μ )2 p2 + (1 / λ3 − μ ) 2 p3
2
2
2
and ν = (1/ λ1 ) p1 + (1/ λ2 ) p2 + (1/ λ3 ) p3 .
Let us assume that 1 / λ1 = 3 / 4 , 1/ λ2 = 1 , 1/ λ3 = 5 / 4 , p1 = p2 = 1 / 3 and
η = 0.1 . We have j = 1, 2 ∈ g and j = 3 ∈ bm as long as m < 30 . The critical horizon of
credibility
m(c) = 250 . For our numerical purpose, let us compare strategies
m = 0, 2,10, 250,1000 and ∞ . Of course, for all u , ψ m(3) (u ) = 1 for m = 0, 2 and 10 .
Carrying out 100000 simulations on an horizon of 10000 time periods, we obtain the results
below, also shown on Figures 2 to 3.
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
93
ψ m(1) (u )
0
u
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
m=0
43.758% 29.996% 20.566% 14.086% 9.844%
6.754%
4.604%
3.218%
2.268%
1.566%
1.120%
m=2
44.438% 30.316% 20.572% 13.850% 9.474%
6.352%
4.232%
2.866%
1.952%
1.326%
0.890%
m = 10
45.892% 31.502% 21.348% 14.278% 9.568%
6.308%
4.072%
2.696%
1.772%
1.124%
0.724%
m = 250 46.936% 32.404% 22.034% 14.822% 9.920%
6.582%
4.252%
2.810%
1.828%
1.156%
0.758%
m = 1000 46.942% 32.408% 22.036% 14.827% 9.922%
6.583%
4.252%
2.810%
1.828%
1.156%
0.758%
6.584%
4.252%
2.810%
1.828%
1.156%
0.758%
m=∞
46.948% 32.412% 22.042% 14.832% 9.924%
u
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
m=0
0.764%
0.540%
0.386%
0.260%
0.200%
0.148%
0.086%
0.060%
0.046%
0.036%
m=2
0.608%
0.408%
0.276%
0.214%
0.150%
0.088%
0.054%
0.042%
0.018%
0.010%
m = 10
0.450%
0.308%
0.220%
0.148%
0.094%
0.046%
0.020%
0.010%
0.006%
0.000%
m = 250
0.478%
0.318%
0.236%
0.160%
0.094%
0.050%
0.024%
0.008%
0.005%
0.000%
m = 1000 0.478%
0.318%
0.236%
0.160%
0.094%
0.050%
0.024%
0.008%
0.004%
0.000%
0.318%
0.236%
0.160%
0.094%
0.050%
0.024%
0.008%
0.004%
0.000%
m=∞
0.478%
ψ m(2) (u )
u
m=0
0
1
2
3
4
5
6
7
8
9
10
82.416% 69.122% 57.854% 48.526% 40.738% 34.052% 28.682% 24.060% 20.122% 16.894% 14.206%
m = 2 82.208% 67.124% 55.578% 45.370% 37.024% 30.182% 24.640% 19.976% 16.290% 13.186% 10.594%
m = 10 82.932% 68.260% 52.710% 40.310% 30.182% 22.138% 16.162% 11.560%
8.202%
5.890% 4.152%
m = 250 91.820% 79.506% 64.450% 49.480% 36.514% 25.882% 18.026% 12.112%
7.926%
5.256% 3.402%
m = 1000
92.023% 79.808% 64.905% 49.754% 36.901% 26.228% 18.273% 12.300%
m = ∞ 92.232% 80.108% 65.160% 50.184% 37.088% 26.424% 18.486% 12.528%
8.109% 5.422% 3.529%
8.278%
5.542% 3.636%
12
13
14
15
16
17
18
19
20
m = 0 11.792%
9.918%
8.234%
6.904%
5.742%
4.812%
4.070%
3.374%
2.834%
2.372%
m=2
8.688%
7.016%
5.686%
4.618%
3.746%
3.018%
2.462%
1.970%
1.598%
1.328%
m = 10
2.866%
2.016%
1.452%
1.004%
0.738%
0.538%
0.374%
0.280%
0.188%
0.134%
m = 250 2.190%
1.360%
0.860%
0.554%
0.380%
0.224%
0.140%
0.088%
0.060%
0.038%
m = 1000
2.170%
1.304%
0.823%
0.528%
0.367%
0.203%
0.128%
0.084%
0.058%
0.037%
m=∞
2.286%
1.395%
0.865%
0.530%
0.365%
0.202%
0.127%
0.084%
0.058%
0.037%
u
11
94
S. LOISEL – J. TRUFIN
ψ m(3) (u )
0
u
2
4
6
8
10
12
14
16
18
20
m = 0 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000%
m = 2 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000%
m = 10 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000%
m = 250 99.992% 99.648% 96.894% 88.370% 73.964% 58.720% 42.814% 29.850% 20.238% 13.378% 8.816%
m = 1000 99.994%
99.790%
97.490%
89.582%
75.532%
57.010%
41.046%
28.186%
18.682%
12.006%
7.636%
7.636%
m=∞
99.996%
99.810%
96.894%
88.370%
75.124%
57.010%
41.046%
28.186%
18.682%
12.006%
u
22
24
26
28
30
32
34
36
38
40
m = 0 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000%
m = 2 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000%
m = 10 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000%
m = 250 5.714% 3.660% 2.440% 1.566% 1.034% 0.704% 0.486% 0.330% 0.240% 0.142%
m = 1000
4.708% 2.862% 1.784% 1.048% 0.644% 0.412% 0.250% 0.146% 0.098% 0.038%
m = ∞ 4.708% 2.862% 1.784% 1.048% 0.644% 0.412% 0.250% 0.146% 0.098% 0.038%
On the one hand, asymptotically, whatever the portfolio j , we see that the larger m
( j)
(u ) is, even for m ≥ m( j ) (≤ m(c ) ) . This confirms our theoretical
is, the smaller ψ m
findings and intuition, namely that (also for m ≥ min { j:m( j ) >0}m ( j ) ) m is sub-optimal
compared to m + 1 .
On the other hand, for small initial capital u , a bigger value of m may this time
increase in certain cases the ultimate ruin probability. Obviously, this observation is not
unexpected.
Remark
Clearly, the numerical illustrations can not highlight the fact that m = ∞ is suboptimal.
4.
LIMIT RESULTS FOR HEAVY-TAILED CLAIMS
We have seen that in the case where the Lundberg coefficient exists, adjusting
premium with credibility may increase this one. As typical sample paths for which ruin
occurs exhibit a sequence of moderately large claims, there is quite often enough time to
compensate early losses with credibility-adjusted future premiums. On the opposite, in the
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
95
heavy-tailed case, it is often said that ruin is likely to be caused by one large claim (see for
example Theorems 1.1 and 1.2 in Asmussen and Klüppelberg (1996) in the classical risk
model). It is logical to think that credibility plays a less important role for heavy-tailed
claim amount distributions. In this Section, we focus on the regular variation case.
Definition 4.1 The cumulative distribution function F with support (0, ∞) belongs
to the regular variation class if for some α > 0
lim
F ( xy )
x →∞
F ( x)
= y −α , for y > 0.
(39)
We note F ∈ −α . The convergence is uniform on each subset y ∈ [ y0 , ∞)
( 0 < y0 < ∞ ).
In the light-tailed case, we have studied the influence of m on the Lundberg
coefficient
1
( j)
lnψ m
(u ).
u
u →∞
ρ m( j ) = − lim
(40)
In our heavy-tailed case, the equivalent quantity to be studied is parameter
1
( j)
(41)
lnψ m
(u ).
γ m( j ) = − lim
u →∞ ln u
As we consider the case F j ∈  ( j ) , and since credibility adjustments involve
−α
implicitly that the two first moments of claim amounts exist, we restrict to the case
α ( j ) > 2 . Now, for m < m(c) , we shall prove that, if j ∈ bm , then m does not modify the
( j)
, which is in accordance with our intuition. Mikosch and Samorodnitsky
value of γ m
(2000) have proved a similar result in case of moving average processes.
( j)
Theorem 4.2 Suppose that the Yk 's are regularly varying with index α ( j ) > 2 . If
m < m(c) and j ∈ bm , i.e.
η>
[Y ( j ) ]
(1 − zm ) μ + zm [Y ( j ) ]
− 1,
1
( j)
lnψ m
(u ) = α ( j ) − 1.
u →∞ ln u
γ m( j ) = − lim
we get
(42)
(43)
Proof. In the compound Poisson risk model with sub-exponential claims, Embrechts
and Veraverbeke (1982) have shown that the (continuous-time) classical ruin probability
ψ (u ) with initial surplus u ≥ 0 (without credibility premium adjustment) behaves as
u → ∞ as
96
S. LOISEL – J. TRUFIN
λ
c − λμW
ψ (u ) 
+∞
u
(1 − FW ( x))dx,
where c is the (continuous-time) premium income rate, λ is the intensity of the Poisson
process, FW is the cumulative distribution function of the individual claim amounts and
μW is the expected individual claim amount. In the α ( j ) -regularly varying case with
α ( j ) > 2 (this means that
1 − FW ( x)  x −α
( j)
as x → ∞,
this corresponds to
ψ cont (u ) 
( j)
λ
1
u −α +1.
(
j
)
c − λμW α − 1
This result may be adapted to the discrete-time model, with constant 1 -period
premium income and independent, identically distributed (aggregate) claim amounts on
each period, with α ( j ) -regularly varying distribution: if the safety loading is positive, for
α ( j ) > 1 , there exists C > 0 such that the (discrete-time) probability of ruin ψ disc (u )
( j)
satisfies ψ disc (u )  Cu −α +1 as u → ∞ . To show this, one might for example use
arguments developed in Rullière and Loisel (2004) and Lefèvre and Loisel (2008).
To show that there exist C1 > 0 and a function ψ 1 such that
ψ m( j ) (u ) ≥ ψ1 (u ) and ψ1 (u )  C1u −α
( j ) +1
as u → ∞,
let us consider a modified model, in which future earned premiums due to each claim are
( j)
anticipated and earned at the claim instant: if claim amount Yk occurs at time k , the sum
( j)
of future credibility premiums associated to Yk corresponds to
Yk( j ) (1 + η )
k +m
 zi,m .
i = k +1
( j)
In our modified model, each claim amount Yk is replaced with
k +m


( j)
Y k = Yk( j ) 1 − (1 + η )
zi,m  ,
i = k +1



and premium income
,
= (1 −
,
) +
∑
,
(
, )
( ,
is replaced with
 k ,m = (1 − z ) μ (1 + η ).
C
k ,m
)
( )
(1 + )
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
97
For each sample path, if ruin occurs in the modified model, then it occurs as well in
the initial model as credibility adjusted premiums are the same in both models but are
received later in the modified model. Since from k = m + 1 ,
 i = k +1zi,m = zm , we have
k +m
from Embrechts and Veraverbeke (1982) that the ruin probability ψ1 (u ) in the modified
model satisfies ψ 1 (u )  C1u −α
( j ) +1
for some C1 > 0 as u → ∞ as long as the safety
loading is positive in the modified model, which is guaranteed by condition (42).
To show that there exist C2 > 0 and a function ψ 2 such that
ψ m( j ) (u ) ≤ ψ 2 (u ) and ψ 2 (u )  C2u −α
let us rewrite
ψ m( j ) (u )
( j ) +1
as u → ∞,
as
) (
(
)
ψ m( j ) (u ) = Pr  ∃1 ≤ i ≤ m,Ui(,mj ) < 0 or ∃i > m,Ui(,mj ) < 0  .

This means that
ψ m( j ) (u ) ≤ Pr  ∃1 ≤ i ≤ m,U i(, mj ) < 0  + Pr  ∃i > m,U i(,mj ) < 0  .

The first term is O(u
−α ( j )

) and so o(u
−α ( j ) +1
Pr  ∃i >



) . The second term
< 0

j)
m,U i(,m
satisfies
j)
Pr  ∃i > m,U i(,m
< 0


i
i −m


≤ Pr ∃i > m, Yk( j ) > u / 2 + u / 2 + zm (1 + η ) Yl( j ) + μ (1 + η )(1 − zm )i 


k =1
l =1
i
i −m


= Pr ∃i > m,
Yl( j ) + (1 − zm (1 + η )) Yk( j ) > u / 2 + u / 2 + (1 + η ) μ (1 − zm )i 


l =i − m +1
k =1
i −m
 i
 

= Pr[∃i > m,
Yl( j )  +  (1 − zm (1 + η )) Yk( j ) 

 

k =1
 l =i −m+1
 







δ  
δ 


>  u / 2 + i  +  u / 2 +  (1 + η ) μ (1 − zm ) −  i ],
2
2 

 

where
δ = (1 + η ) μ (1 − zm ) − [Y ( j ) ](1 − zm (1 + η )) > 0
from Condition (42).
98
S. LOISEL – J. TRUFIN
Hence, since for any sequences of nonnegative r.v.'s ( X i )i ≥1 and ( Zi )i ≥1 , and for
any sequences of nonnegative numbers (ai )i≥1 and (bi )i ≥1 , we have
Pr[∃i ≥ 1, X i + Z i > ai + bi ] ≤ Pr[∃i ≥ 1, X i > ai ] + Pr[∃j ≥ 1, Z j > b j ],
we get that
j)
Pr  ∃i > m,U i(,m
< 0


i

δ 
≤ Pr ∃i > m,
Yl( j ) > u / 2 + i 
2 

l =i − m +1
i −m

δ 

+ Pr ∃i > m,(1 − zm (1 + η )) Yk( j ) > u / 2 +  (1 + η ) μ (1 − zm ) −  i  .
2  

k =1



The second term is equivalent to C3u −α
( j ) +1
as u → ∞ for some constant C3 > 0 ,
because it corresponds to a ruin probability with claim size distribution in RV (α ( j ) ) and
with a positive safety loading (from (42)). The first term
i

δ 
Pr ∃i > m,
Yl( j ) > u / 2 + i 
2 

l =i − m +1

satisfies
i

i
δ 
δ 

Pr ∃i > m,
Yl( j ) > u / 2 + i  ≤ Pr ∃i > m, m max Yl( j ) > u / 2 + i  ,
2 
2 
l =i − m +1


l =i − m+1

which leads to
i

δ 
δ


Pr ∃i > m,
Yl( j ) > u / 2 + i  ≤ Pr ∃i > m, mYi( j ) > u / 2 + (i − m) 
2
2




l =i − m +1

δ 

= Pr ∃i > 0, mYi( j ) > u / 2 + i  .
2 

This may be rewritten as
i

δ 
u
δ 

Pr ∃i > m,
+
Yl( j ) > u / 2 + i  ≤ Pr ∃i > 0,Yi( j ) >
i .
2 
2m 2m 


l =i − m +1

Obviously, we have
u
δ 

Pr  ∃i > 0,Yi( j ) >
+
i ≤
2m 2m 

∞

δ 
 Pr Yi( j ) > 2m + 2m i  .
i =1
u
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
99
Now, by Lemma 5.2 in Foss et al. (2009), we obtain
∞

δ 
 Pr Yi( j ) > 2m + 2m i  
u
i =1
u 

Pr Y ( j ) >
asu → ∞,
δ
2m 

2m
which completes the proof.
Let us conclude in saying that, on the one hand, if
j ∈ bm
then
ψ m( j+)1 (u) ≤ ψ m( j ) (u) = 1 by Proposition 3.4 (clearly also valid for the heavy-tailed case), but
on the other hand, this time, if j ∈ bm , then an increasing of the horizon of credibility has
( j)
(u ) in first order, at least for m < m(c) .
asymptotically no impact on ψ m
ACKNOWLEDGMENTS
This work is partially funded by the Research Chair Actuariat Durable sponsored
by Milliman, the research chair Management de la modélisation sponsored by BNP Paribas
Assurance and the ANR (reference of the French ANR project : ANR-08-BLAN-0314-01).
Julien Trufin also thanks the financial support of Risk dynamic.
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NTU
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY
PREMIUM ADJUSTMENTS
FIGURES
(1)
Figure 2: Numerical results for ψ m
(u )
(2)
Figure 3: Numerical results for ψ m
(u )
101
102
S. LOISEL – J. TRUFIN
(3)
Figure 4: Numerical results for ψ m (u )