APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY
Appl. Stochastic Models Bus. Ind. 2008; 24:109–128
Published online 6 November 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/asmb.694
Upper bound for ruin probabilities under optimal investment and
proportional reinsurance
Zhibin Liang1, 2, ∗, † and Junyi Guo2
1 Institute
of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University,
Jiangsu 210097, China
2 School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
SUMMARY
In this paper, we consider the optimal investment and reinsurance from an insurer’s point of view to
maximize the adjustment coefficient. We obtain the explicit expressions for the optimal results in the
diffusion approximation (D-A) case as well as in the jump-diffusion (J-D) case. Furthermore, we derive
a sharper bound on the ruin probability, from which we conclude that the case with investment is always
better than the case without investment. Some numerical examples are presented to show that the ruin
probability in the D-A case sometimes underestimates the ruin probability in the J-D case. Copyright q
2007 John Wiley & Sons, Ltd.
Received 3 March 2007; Revised 18 July 2007; Accepted 30 July 2007
KEY WORDS:
ruin probability; Lundberg’s inequality; investment; proportional reinsurance; adjustment
coefficient
1. INTRODUCTION
Recently, the optimization problem of an insurer has been studied extensively in the literature,
see, for example, Browne [1], Hipp and Plum [2, 3], Liang [4], Schmidli [5, 6], Liu and Yang [7]
and Gerber and Shiu [8]. In these works, stochastic control theory and related tools have been
widely used. They assume that the aggregate claims process is a compound Poisson process or a
Brownian motion with drift, where the variables, such as reinsurance, new businesses, investment
∗ Correspondence
to: Zhibin Liang, Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing
Normal University, Jiangsu 210097, China.
†
E-mail: [email protected]
Contract/grant sponsor: National Natural Science Foundation of China; contract/grant number: 10701082
Contract/grant sponsor: Hunan Provincial Natural Science Foundation of China; contract/grant number: 06JJ2019
Contract/grant sponsor: 973 project of MOSTC
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Z. LIANG AND J. GUO
and dividend, are adjusted dynamically. Under some assumptions, they are able to obtain closedform solutions for the optimal strategy and the value function in the sense of maximizing (or
minimizing) a certain objective function under different constraints, for example, minimizing the
ruin probability (or maximizing the survival probability), see Schmidli [5, 6], maximizing the
expected discounted penalty of ruin, see Browne [1], and maximizing the expected discounted
dividend, see Gerber and Shiu [8].
Browne [1], Liang [4], Schmidli [5], Promislow and Young [9] consider one of (or both) the
two controls to minimize the probability of ruin: (1) investing in a risky asset and (2) purchasing
proportional reinsurance. They derive the explicit expression for the optimal values from the
Brownian motion (the so-called diffusion approximation (D-A, for short)) model. Yang and Zhang
[10] consider the optimal investment problem in the jump-diffusion (J-D, for short) model to
maximize the expected utility. They also derive the closed-form expression for the optimal strategy
and the optimal value function.
Since the explicit expression of the ruin probability (u) is very difficult to obtain in the J-D
case, the estimation of ruin probabilities has been a central topic in risk theory. Liu and Yang [7]
and Yang and Zhang [10] discuss it by a numerical method. Some other literature focuses on the
asymptotic behavior of the ruin probability and obtains the following results (see, for example,
Hipp and Schmidli [11], Gaier et al. [12]):
lim (u)eRu = (1)
(u)Ce−Ru
(2)
u→∞
or
where and C are constants, and R is the adjustment coefficient. The adjustment coefficient R is
therefore a very important measure of risk. Some literature focuses on finding an optimal strategy
to maximize the adjustment coefficient, see, for example, Waters [13], Centeno [14], Centeno
and Simões [15], Hald and Schmidli [16] and Liang and Guo [17]. In this paper, we consider
optimal investment and proportional reinsurance in the D-A case as well as in the J-D case. The
closed-form expressions of the optimal results and an exponential bound for the ruin probability
are given. Since the adjustment coefficient in this bound is the maximal, we obtain a smallest
exponential bound on the ruin probability. Furthermore, two amazing conclusions are presented:
one is that the case with investment is always better than the case without investment, and the
other is that the ruin probability in the D-A case sometimes underestimates the ruin probability in
the J-D case.
The paper is organized as follows. In Section 2, the model and assumptions are given. The
closed-form expressions of the optimal values and an exponential bound for the ruin probability
are obtained in Section 3. In Section 4, we prove that the case with investment is always better
than the case without investment, and we analyze the optimal values by numerical examples in
Section 5. Finally, in Section 6, we conclude this paper by giving a concise summary.
2. THE MODEL
Under the classical risk model, the surplus process {Ut }t 0 is given by
Ut = u +ct − St
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where u0 is the initial surplus, c is the premium rate and St represents the aggregate claims up to
N (t)
time t. We assume that St = i=1 Yi is a compound Poisson process, i.e. N (t) is a homogeneous
Poisson process with intensity ; Yi , i1 is a sequence of positive i.i.d. random variables with
common distribution F(y), mean value = E(Yi ) and moment generating function MY (r ) = Eer Y1 .
The claim number process N (t) is also independent of the claim amount Yi , i1. An introduction
to classical ruin theory can be found, for instance, in Grandell [18].
In this paper, we consider the risk process that is perturbed by a Brownian motion; we also
suppose that the insurer has the possibility to choose proportional reinsurance with level q ∈ (0, 1].
The premium rate for the reinsurance is (1+)(1−q), where is the safety loading of the
reinsurer. Let = c/−1 be the safety loading of the insurer. Without the loss of generality, we
assume >. Then the surplus process of the insurer becomes
q
Rt = u +[q(1+)−(−)]t +Wt −q St
(4)
where 0 is a constant, Wt is a standard Brownian motion independent of the claim number
process N (t) and of Yi , i1. The diffusion term Wt represents the additional uncertainty associated
with the insurance market or the economic environment. The uncertainty is not necessarily related
to the claims; therefore, in this paper, we consider only the case where Wt is not affected by
reinsurance at all.
In order that the net profit condition is fulfilled, i.e.
[q(1+)−(−)]−q>0
we need
q>q̄ = 1−
Otherwise, the probability of ruin will be one for any initial surplus u0.
In Liang and Guo [17], under the criterion of maximizing the adjustment coefficient, the optimal
problem of model (4) has been discussed, and some useful conclusions have been derived. Here,
we assume that there is one risky asset available for the insurer in the financial market, whose
price at time t is denoted by P(t), and modelled as a geometric Brownian motion
dP(t) = a P(t) dt +P(t) dBt
(5)
where a and are positive constants and represent the expected instantaneous rate of return of the
risky asset and the volatility of the risky asset, respectively. Bt , t0, is another standard Brownian
motion independent of the claim number process N (t) and of Yi , i1.
Let A be the total amount of money invested in the higher risky asset. Here, the investment
strategy A and the retention level q are constants in time. Let X t denote the wealth of the company
at time t. If it follows the reinsurance strategy q as well as the investment strategy A, then this
process can be expressed as
A,q
Xt
= u + A(at +Bt )+[q(1+)−(−)]t +Wt −q St
(6)
Denote by the correlation coefficient of Bt with Wt , i.e. E[Bt Wt ] = t. In the following context,
we consider only the case ||<1 (by the same method, we can discuss the case || = 1).
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Now define
A,q
A,q = inf{t0 : X t
0}
to be the ruin time, and
A,q
A,q (u) = P(
A,q <∞|X 0
= u)
to be the probability of ultimate ruin.
3. THE OPTIMAL STRATEGY AND RUIN PROBABILITY
3.1. Optimal results in the D-A case
In this subsection, we assume that the aggregate claims process follows a Brownian motion with
drift, i.e.
Ŝt = a0 t −0 Wt0
(7)
Here, Wt0 is another standard Brownian motion independent of Bt and Wt . Ŝt can be seen as the
D-A of compound Poisson process St with a0 = , 20 = 2 = EY12 (see Grandell [18]).
Replacing St of (6) with Ŝt and simplifying yield a new surplus process
A,q
X̂ t
= u + A(at +Bt )+[(−)+q]t +Wt +q0 Wt0
(8)
where q ∈ (q̄, 1].
Let
A,q
A,q
D = inf{t0 : X̂ t
0}
be the ruin time, and
A,q
A,q
A,q
D (u) = P(
D <∞| X̂ 0
= u)
be the probability of ultimate ruin.
By the ‘martingale approach’, we obtain the following theorem.
Theorem 3.1
A,q
The process {e−R D (A,q) X̂ t , t0} is a martingale. Furthermore,
D (u) = e−R D (A,q)u
A,q
(9)
where
R D (A, q) =
2[Aa +(−+q)]
A2 2 +2 +q 2 20 +2A
(10)
Proof
From (8), it is not difficult to obtain
A,q
E(e−r ( X̂ t
Copyright q
−u)
1 2 2 2
(A +2 +q 2 20 +2A)}t
) = e{−r [Aa+(−+q)]+ 2 r
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UPPER BOUND FOR RUIN PROBABILITIES
Let
g(r ) = −r [Aa +(−+q)]+ 12 r 2 (A2 2 +2 +q 2 20 +2A)
Then equation g(r ) = 0 has a positive root:
2[Aa +(−+q)]
R D (A, q) =
A2 2 +2 +q 2 20 +2A
A,q
A,q
Then we have E(e−R D (A,q)( X̂ t −u) ) = 1; thus the process {e−R D (A,q) X̂ t , t0} is a martingale.
Here, R D (A, q) is an adjustment coefficient, or the so-called Lundberg exponent.
By the standard martingale approach as in Grandell [18, pp. 10–12], we can directly derive
D (u) = e−R D (A,q)u
A,q
Now we are to find the optimal strategy (A∗ , q ∗ ) to minimize the ruin probability, i.e.
D (u) :=
A,q
inf
(A,q)∈(0,∞)×(q̄,1]
D (u)
From (9), finding (A∗ , q ∗ ) to minimize the ruin probability is equivalent to finding (A∗ , q ∗ ) to
maximize the adjustment coefficient. Hence, we try to find R D such that
RD =
sup
(A,q)∈(0,∞)×(q̄,1]
R D (A, q)
Note that the function g(r ) is non-negative at r = R D , i.e. R D is the solution to
inf
{−r [Aa +(−+q)]+ 12 r 2 (A2 2 +2 +q 2 20 +2A)} = 0
(A,q)∈(0,∞)×(q̄,1]
(11)
Now differentiating g(r ) with respect to A gives
a 1 · −
2 r
and the minimum of g(r ) over q is attained at
A∗ =
qmax =
1
·
20 r
Hence, we obtain
q ∗ = qmax ∧1
When q ∗ = qmax <1, substituting A∗ and q ∗ into (11), we obtain
(−)−a/± [(−)−a/]2 +2 (1−2 )(a 2 /2 +2 2 2 /20 )
R +,− =
2 (1−2 )
Since D (u) is to be minimized, only the positive root R + is valid. Thus, the minimum ruin
probability is
A,q
D (u) = e−R
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and the optimal reinsurance strategy is
q∗ =
where
R+ =
1
·
20 R +
(−)−a/+ [(−)−a/]2 +2 (1−2 )(a 2 /2 +2 2 2 /20 )
2 (1−2 )
When q ∗ = 1, inserting A∗ and q ∗ = 1 into (11) yields
−a/+ [−a/]2 +[2 (1−2 )+20 ](a 2 /2 )
R1 =
2 (1−2 )+20
(12)
(13)
and the minimum ruin probability is
D (u) = e−R1 u
To summarize, we have the following theorem.
Theorem 3.2
The optimal strategies to minimize the ruin probability of model (8) are
A∗ =
and
a 1
·
−
2 R D
1
∧1
q =
·
20 R +
(14)
∗
(15)
and the minimum ruin probability is
D (u) = e−R D u
where
RD =
R+
if q̄<q ∗ <1
R1
if q ∗ = 1
(16)
(17)
3.2. Optimal results in the J-D case
In this subsection, we discuss the optimal investment and reinsurance strategy of an insurer with
a J-D risk process (6). Since it is not easy to obtain the explicit expression for the ruin probability
in the J-D case, we consider here the optimal strategy to maximize the adjustment coefficient.
Let R J (A, q) be the adjustment coefficient in this case. Then R J (A, q) satisfies the following
equation
[Aa +(1+)q −(−)]r − 12 (2 A2 +2 +2A)r 2 −[MY (qr )−1] = 0
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Our goal is to maximize R J (A, q), i.e. to find the value of R J such that
RJ =
sup
(A,q)∈(0,∞)×(q̄,1]
R J (A, q)
We can solve the problem using the same method as in Section 3.1, although it is more difficult
to deal with.
Note that the function on the left-hand side of (18) is non-positive at r = R J , i.e. R J is the
solution to
{[Aa +(1+)q −(−)]r
sup
(A,q)∈(0,∞)×(q̄,1]
1
− (2 A2 +2 +2A)r 2 −[MY (qr )−1]} = 0
2
(19)
In order to find the adjustment coefficient, the claim size distribution should have an exponentially
decreasing tail F̄(y)(= 1− F(y)). Here, exponentially decreasing tail means that the tail F̄(y)
satisfies F̄(y) = o(e−sy ) for some s>0. It is, for instance, fulfilled if there is r∞ >0 such that
limr ↑r∞ MY (r ) = ∞. This is exactly the condition needed for the Cramér–Lundberg approximation
in the classical risk model (see, for instance, the book by Asmussen [19] or Gerber [20]).
Then the maximum over A in (18) is attained at
A∗ =
a 1 · −
2 r
Substituting A∗ into (19) yields another equation:
a2 1
a
+(1+)q −(−) r + 2 − 2 (1−2 )r 2 −[MY (qr )−1] = 0
−
sup
2
2
q∈(q̄,1]
Let
(20)
a2 1
a
f˜1 (q) = −
+(1+)q −(−) r + 2 − 2 (1−2 )r 2 −[MY (qr )−1]
2
2
Differentiating f˜1 (q) with respect to q yields
(1+) = E[Y eqr Y ] := MY (qr )
Let m = qr , then the above equation becomes
(1+) = MY (m)
(21)
which is the same as (5) in Hald and Schmidli [16].
Suppose (21) has a unique solution , then we have the following lemma, which plays a key
role in our results.
Lemma 3.1
Let be the unique positive root of (21), then we have
MY ()−1<(1+)
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Proof
Let
f (r ) = (1+)r −(MY (r )−1)
we have
f (r ) = (1+)− MY (r )
f () = 0
f (r ) = −MY (r ) = −E[Y 2 er Y ]0
This means that f (r ) is a concave function and its maximum is attained at r = . Hence, we can
obtain that f ()>0, i.e.
MY ()−1<(1+)
Now let q1 denote the argument where f˜1 (q) attains its maximum, then = q1r . Substituting
r = /q1 into (20) yields another equation for q1 :
a
a2
1
−
−(−) q1 + (1+)−[MY ()−1]+ 2 q12 − 2 (1−2 )2 = 0
2
2
Therefore,
q1+,− ()=
[a/+(−)]± [a/+(−)]2 2 +22 (1−2 )2 [(1+)−[MY ()−1]+a 2 /22 ]
2[(1+)−[MY ()−1]+a 2 /22 ]
From Lemma 3.1, we know that
(1+)>[MY ()−1]
Omitting the negative root, we have
q1 ()=
[a/+(−)]+ [a/+(−)]2 2 +22 (1−2 )2 [(1+)−[MY ()−1]+a 2 /22 ]
2[(1+)−[MY ()−1]+a 2 /22 ]
(22)
If we know the distribution of Y , then the closed-form expression of can be obtained and the
optimal policy q ∗ can thus be given as
q ∗ = q1 ()∧1
Substituting q ∗ into Equation (20) and simplifying, we have
a
a2 1
(−)−
r +(1+)+ 2 − 2 (1−2 )r 2 = [MY ()−1],
2
2
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q ∗ <1
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UPPER BOUND FOR RUIN PROBABILITIES
and
(1+)−
a2 1
a
r + 2 − 2 (1−2 )r 2 = [MY (r )−1],
2
2
q∗ = 1
(24)
In the following proposition, we will verify that the above equations about r have a unique
positive solution.
Proposition 3.1
Assume that F̄(y) is exponentially decreasing, then there is a unique positive solution of Equations
(23) and (24).
Proof
If q ∗ <1, we have
a2 1
a
r +(1+)+ 2 − 2 (1−2 )r 2 = [MY ()−1]
(−)−
2
2
Let
a
a2 1
f (r ) = (−)−
r + 2 − 2 (1−2 )r 2
2
2
Since f (r ) is a concave function, equation f (r ) = 0 has two roots: one is positive and the other is
negative. Thus, by Lemma 3.1, it is easily seen that Equation (23) has a unique positive solution
R J , which is given by
RJ =
(−)−a/+ [(−)−a/]2 −22 (1−2 )[[MY ()−1]−(1+)−a 2 /22 ]
2 (1−2 )
If q ∗ = 1, we have
a
a2 1
(1+)−
r + 2 − 2 (1−2 )r 2 = [MY (r )−1]
2
2
Let
a
a2 1
f (r ) = (1+)−
r + 2 − 2 (1−2 )r 2
2
2
and
g(r ) = [MY (r )−1]
Then f (r ) is a concave function, and equation f (r ) = 0 has two roots: one is positive and the
other is negative. Since g(r ) is an increasing convex function and g(0) = 0, g(r ) and f (r ) have a
unique point of intersection at some r >0. Thus Equation (24) has a unique positive solution and
the proof is complete.
By the above results, the following theorem is directly obtained.
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Theorem 3.3
Assume that F̄(y) is exponentially decreasing. Let be the positive root of (21), q1 () be given
as in (22). Then the optimal strategies to maximize the adjustment coefficient are
A∗ =
a 1
·
−
2
RJ
(25)
and
q ∗ = q1 ()∧1
(26)
The maximal adjustment coefficient in the case q ∗ <1 is
RJ =
(−)−a/+ [(−)−a/]2 −22 (1−2 )[[MY ()−1]−(1+)−a 2 /22 ]
2 (1−2 )
(27)
and the maximal adjustment coefficient in the case q ∗ = 1 is the unique positive solution of (24).
Remark 3.1
From (25) and (14), we can observe that the optimal investment strategies in the two cases look
quite similar, but it is greatly dependent on the risk process chosen because different risk processes
yield different maximal adjustment coefficients.
By Theorem 3.3 and the ‘martingale approach’ as in Grandell [18, pp. 10–12], it is not difficult
to obtain the following theorem.
Theorem 3.4
Let (A∗ , q ∗ ) be the optimal strategy, and R J be the maximal adjustment coefficient of model (6),
A∗ ,q ∗
then the process {e−R J X t
} is a martingale and
A
∗ ,q ∗
(u)e−R J u e−R J (A,q)u
(28)
where (A, q) is an arbitrary admissible strategy.
In the following theorem, we will show that the Lundberg’s inequality of Theorem 3.4 also
holds for D (u).
Theorem 3.5
A,q
Let R J be the maximal adjustment coefficient of X t in model (6), and D (u) be the minimum
A,q
ruin probability of X̂ t in model (8). Then we have
D (u)e−R J u
A,q
where the process X̂ t
(29)
A,q
is the approximation of the process X t
.
To verify the theorem, we first prove the following lemma that plays a key role in Theorem 3.5.
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UPPER BOUND FOR RUIN PROBABILITIES
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Lemma 3.2
Let R J (A, q) be the adjustment coefficient in the J -D case, R D (A, q) be the adjustment coefficient
in the D-A case and (A, q) be an arbitrary strategy. Then we have
R J (A, q)R D (A, q)
(30)
Proof
From Theorem 3.1 and Equation (18), we can observe that R D (A, q) satisfies the following equation
[Aa +(−)+q]r − 12 (A2 2 +2 +2A)r 2 = 12 q 2 20r 2
(31)
and R J (A, q) satisfies another equation
[Aa +(1+)q−(−)]r − 12 (A2 2 +2 +2A)r 2 −[MY (qr )−1] = 0
or equivalently
[Aa +(−)+q]r − 12 (A2 2 +2 +2A)r 2 = [MY (qr )−1]−qr
(32)
Comparing Equations (31) and (32), we can observe that the left-hand side of (31) is the same as
the left-hand side of (32). Let
g(r ) = [Aa +(−)+q]r − 12 (A2 2 +2 +2A)r 2
Then equation g(r ) = 0 has two roots: one is zero and the other is
r1 =
2[Aa +q+(−)]
A2 2 +2 +2A
>0
To prove that R J (A, q)R D (A, q), it is sufficient to consider functions h 1 and h 2 defined as
h 1 (r ) = 12 q 2 20r 2
and
h 2 (r ) = [MY (qr )−1]−qr
Then as shown in Figure 1, if h 1 (r )h 2 (r ) for all r 0, then R J (A, q)R D (A, q).
Note that
h 1 (r ) = 12 q 2 20r 2
+∞
1 2 2 2
r q y f (y) dy
=
2
0
and
h 2 (r ) = [MY (qr )−1]−qr
+∞
(eqr y −1−qr y) f (y) dy
=
0
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Z. LIANG AND J. GUO
0.6
0.5
h1(r)
0.4
0.3
h2(r)
0.2
g(r)
0.1
0
RD(A,q)
RJ(A,q)
−0.1
−0.2
0
0.5
1
r
1.5
2
Figure 1. Functions h 1 and h 2 giving R D (A, q)R J (A, q).
Hence, we have
h 2 (r )−h 1 (r ) = +∞ qr y
e
0
1
−1−qr y − r 2 q 2 y 2
2
f (y) dy
It is not difficult to prove that
eqr y −1−qr y − 12 r 2 q 2 y 2 0
for all r 0. Therefore we have h 1 (r )h 2 (r ), and hence R J (A, q)R D (A, q).
Remark 3.2
The method used here as well as in Section 4 is inspired by Dickson [21].
Now we can prove Theorem 3.5.
The proof of Theorem 3.5
From Lemma 3.2, we can observe that the inequality R J (A, q)R D (A, q) holds for arbitrary
admissible strategy (A, q). Hence, we have
R J = R J (A∗ , q ∗ )R D (A∗ , q ∗ )R D
(33)
where (A∗ , q ∗ ) is the optimal strategy in the J-D case, and R D is the maximal adjustment coefficient
in the D-A case.
From (33) and Theorem 3.2, we can directly derive
D (u) = e−R D u e−R J u
and the proof is complete.
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UPPER BOUND FOR RUIN PROBABILITIES
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Remark 3.3
From Theorem 3.4, we obtain an exact analogue of the estimate for the ruin probability with
∗ ∗
investment and reinsurance, i.e. an exponential inequality: A ,q (u)e−R J u . Since R J is the
maximal adjustment coefficient, we obtain a smallest exponential bound on the ruin probability.
From Theorem 3.5, we can observe that the Lundberg’s inequality also holds for D (u). Although
∗ ∗
this means that D (u) cannot overestimate A ,q (u) too much, D (u) may very well—and this
∗
∗
is more serious—underestimate A ,q (u).
4. COMPARISON BETWEEN THE CASES: WITH AND WITHOUT INVESTMENT
In this section, we show that the case with investment is always better than the case without
investment. For notational convenience, we denote by R the maximal adjustment coefficient in the
case with investment and by R0 the maximal adjustment coefficient in the case without investment.
Let w = a/ denote the return of the unit risk, which is a very important factor in risky investment.
In the following, we discuss only the case q ∗ <1 (using the same method, we can derive the same
conclusions for the case q ∗ = 1).
Now we first discuss the D-A case. From Section 3.1, we know R satisfies the following
equation:
2 2 2
2 (1−2 )r 2 +2[w−(−)]r − w 2 +
=0
(34)
20
and R0 satisfies the equation
2r 2 −2(−)r −
2 2 2
=0
20
(35)
In fact, (35) is the special case of (34) with w = 0, = 0.
To prove that RR0 , it is sufficient to consider functions g1 and g2 defined as
2 2 2
2
2 2
2 g1 (r ) = (1− )r +2[w−(−)]r − w +
20
and
g2 (r ) = 2r 2 −2(−)r −
2 2 2
20
Then as shown in Figure 2, if g1 (r )g2 (r ) for all r 0, then RR0 .
To see that g1 (r )g2 (r ) is indeed true, note that
g1 (r )− g2 (r ) = −2 2r 2 +2wr −w 2
= −(r −w)2 0
and hence RR0 .
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DOI: 10.1002/asmb
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Z. LIANG AND J. GUO
1.5
1
g2(r)
0.5
g1(r)
0
R0
R
−0.5
−1
−1.5
0
0.2
0.4
0.6
0.8
1
r
Figure 2. Functions g1 and g2 giving RR0 .
Now we consider the J-D case. From Section 3.2, we know R satisfies the following equation
− 12 2 (1−2 )r 2 −[w−(−)]r + 12 w 2 +(1+)−[MY ()−1] = 0
(36)
and R0 satisfies the equation
− 12 2r 2 +(−)r +(1+)−[MY ()−1] = 0
(37)
Equation (37) is also the special case of (36) with w = 0, = 0.
To prove that RR0 , it is sufficient to consider functions f 1 and f 2 defined as
f 1 (r ) = − 12 2 (1−2 )r 2 −[w−(−)]r + 12 w 2 +(1+)−[MY ()−1]
and
f 2 (r ) = − 12 2r 2 +(−)r +(1+)−[MY ()−1]
Then as shown in Figure 3, if f 1 (r ) f 2 (r ) for all r 0, then RR0 .
To see that f 1 (r ) f 2 (r ) is indeed true, note that
f 1 (r )− f 2 (r ) = 12 2 2r 2 −wr + 12 w 2
= 12 (r −w)2 0
and hence RR0 .
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UPPER BOUND FOR RUIN PROBABILITIES
0.8
0.6
0.4
0.2
0
R0
R
−0.2
f1(r)
−0.4
f2(r)
−0.6
−0.8
−1
0
0.2
0.4
0.6
0.8
1
r
Figure 3. Functions f 1 and f 2 giving RR0 .
Hence, we derive the following theorem.
Theorem 4.1
Whenever w0, RR0 holds.
5. NUMERICAL EXAMPLES
In this section, we assume that the claim sizes {Yi } are exponentially distributed with the parameter
1/. For notational convenience, we denote by 1 (u) the ruin probability in the J-D case when
q = 1. We first give the following results:
Lemma 5.1
Assume that the claim sizes are exponentially distributed and = 1, q ≡ 1 and a = 0, then the ruin
probability is
1 (u) = C1 e−r1 u +C2 e−r2 u
(38)
where
C1 =
Copyright q
(r1 −1)r2
,
r1 −r2
2007 John Wiley & Sons, Ltd.
C2 =
(r2 −1)r1
r2 −r1
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DOI: 10.1002/asmb
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Z. LIANG AND J. GUO
and
r1 =
r2 =
(1+)2 + 12 (1−2 )− [(1+)2 + 12 (1−2 )]2 −2(1−2 )2
(1−2 )
(1+)2 + 12 (1−2 )+ [(1+)2 + 12 (1−2 )]2 −2(1−2 )2
(1−2 )
Proof
See Dufresne and Gerber [22, pp. 56–57].
Lemma 5.2
Assume that the claim sizes are exponentially distributed, then the positive solution of (21) is
given as
1
1
=
1−
(39)
1+
Lemma 5.3
Assume that the claim sizes are exponentially distributed, then the optimal strategies to maximize
the adjustment coefficient are
and
A∗ =
a 1
·
−
2
RJ
1
q∗ =
1−
1
1+
(40)
R J ∧1
(41)
where
RJ =
√
(−)−a/+ [(−)−a/]2 +22 (1−2 )[(2+−2 1+)+a 2 /22 ]
2 (1−2 )
(42)
is the maximal adjustment coefficient when q ∗ <1. When q ∗ = 1, the maximal adjustment coefficient
R J is the unique positive root, which satisfies R J <1/, of the following equation:
a2 1
r
a
r + 2 − 2 (1−2 )r 2 =
(43)
(1+)−
2
1−r
2
Proof
Substituting (39) into Theorem 3.3, the results of Lemma 5.3 can be easily obtained.
Example 5.1
Let a0 = = 3, = 0.3, = 0.4, 20 = 22 = 6. The results are shown in Figures 4–6.
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DOI: 10.1002/asmb
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UPPER BOUND FOR RUIN PROBABILITIES
1.8
w=0.7
w=0.3
without investment
1.6
1.4
1.2
R
1
0.8
0.6
0.4
0.2
0
0
1
2
3
β
4
5
Figure 4. The effects of on R in the J-D case.
1.4
w=0.3
w=0.7
without investment
1.2
1
q*
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
β
2
2.5
3
Figure 5. The effect of on q ∗ in the J-D case.
From Figure 4 ( = −0.6), we can observe that the adjustment coefficient with investment
is larger than the one without investment, which is just the natural consequence as mentioned
in Theorem 4.1. Furthermore, a larger return of the unit risk w will yield a larger adjustment
coefficient.
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DOI: 10.1002/asmb
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Z. LIANG AND J. GUO
6
6
a=0.1
a=0.3
a=0.7
5
5
4
A*
4
A*
a=0.1
a=0.3
a=0.7
3
3
2
2
1
1
case a: ρ=0.6
0
0
0.5
σ
case b: ρ=−0.6
1
0
0
0.5
σ
1
Figure 6. The effect of on A∗ in the J-D case.
From Figure 5 ( = −0.6), we can observe that a higher return of unit risk w yields a lower
retention level. Furthermore, the retention level with investment is always less than the one without
investment.
From Figure 6, we can observe that a larger volatility will yield less investment. Whereas, a
larger expected instantaneous rate of return a will not necessarily yield more investment, especially
in the case <0 (see Figure 6(b)).
Example 5.2
Let a0 = = 3, = 0.3, = 0.4, 20 = 6(20 = 0.16), = 1, q ≡ 1, = 0.6, a = 0. The results are
shown in Figure 7.
In Figure 7, we compare the following three results: the ruin probability 1 (u) (dashed line), the
upper bound e−R J u (thin line), and the ruin probability D (u) (heavy line). From Figure 7(a), we can
observe that when uu 0 , D (u)<1 (u)<e−R J u , which means that D (u) indeed underestimates
1 (u) and that is really dangerous in actuarial practice. Hence, it is a better way of estimating
the 1 (u) by e−R J u than by D (u) when VarS(1) is large. However, from Figure 7(b), we can
observe that the results in the three cases are almost the same. That is to say, when VarS(1) is
very small, D (u) and e−R J u are both good estimations for 1 (u).
Remark 5.1
From Figure 7, we can conclude that, to guarantee the safety of the insurance company, the
exponential bound e−R J u is a better estimation for the ruin probability in the J-D case than the one
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DOI: 10.1002/asmb
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UPPER BOUND FOR RUIN PROBABILITIES
1
1
case a: σ 2=6
0.8
0.8
0.7
0.7
0.6
0.5
0.4
(u0,ψ(u0))
0.3
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
0
case b: σ 2=0.16
0.9
the ruin probability ψ(u)
the ruin probability ψ(u)
0.9
5
10
15
the initial surplus u
0
0
1
2
3
the initial surplus u
4
Figure 7. The effect of u on the ruin probability (u).
resulting from the Brownian motion model. This stresses once more the importance of a proper
asset-liability management of an insurance company.
6. CONCLUSIONS
We briefly summarize the main results of this paper. The optimization problem of investment and
proportional reinsurance from an insurer’s point of view is considered. We obtain the explicit
expressions for the optimal values in the D-A case as well as in the J-D case and derive an
∗ ∗
∗ ∗
exponential inequality for the ruin probability A ,q (u), i.e. A ,q (u)e−R J u . Since R J is the
maximal adjustment coefficient, we obtain a smallest bound on the ruin probability. Furthermore,
we prove that the Lundberg’s inequality also holds for D (u), i.e. D (u)e−R J u . Although this
∗ ∗
means that D (u) cannot overestimate A ,q (u) too much, D (u) may very well—and this is
∗ ∗
more serious—underestimate A ,q (u). Besides, we obtain an amazing conclusion that the case
with investment is always better than the case without investment.
Some numerical examples are presented to show that D (u) sometimes indeed underestimate
∗ ∗
A ,q (u), which is really dangerous in the actuarial practice. Therefore, we can conclude that, to
guarantee the safety of the insurance company, the exponential bound e−R J u is a better estimation
for the ruin probability in the J-D case than the one resulting from the Brownian motion model.
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2007 John Wiley & Sons, Ltd.
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DOI: 10.1002/asmb
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Z. LIANG AND J. GUO
ACKNOWLEDGEMENTS
The authors would like to thank the anonymous referee for the careful reading and helpful comments on
an earlier version of this paper, which led to a considerable improvement of the presentation of the work.
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DOI: 10.1002/asmb