Some limiting Properties on the Bounds of Present
Value Functions of a Life Insurance Portfolio
Yi Zhang∗, Zhengyan Lin? , Chengguo Weng?†
†‡
Department of Mathematics,
Zhejiang University, Hangzhou 310028, China
?
Department of Statistics and Actuarial
Science, University of Waterloo, Waterloo, Canada
Abstract
Under certain assumptions on the dependence structure of the residual lives of the insureds (independent, positively/negatively associated), in this paper we establish some laws
of large numbers for the convex upper bounds, derived by the technique of comonotonicity,
of the present value function of a homogenous portfolio composed of the whole-life insurance
policies.
Keyword: Convex order, comonotonicity, present value function, Black and Scholes model,
laws of large numbers.
1
Introduction
In this paper we consider a homogeneous portfolio composed of n whole-life insurance policies. All the insureds are supposed to be aged x. We denote the residual life of the insured
l as Tl , l = 1, · · · , n, and assume that Tl have a common distribution function F (x). We
∗
(Yi Zhang, and Zhengyan Lin) Department of Mathematics, Zhejiang University, Hangzhou, 310027,
P.R.China.
†
Postal address: (Chengguo Weng) Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. E-mail address: wengcg@hotmail.com (Chengguo Weng).
†
Research is supported in part by NSFC(10471126).
1
assume that the policy provides the insured l with the benefit amount of Kl units at the instant of death, i.e., at time Tl . Moreover, we assume that the interest accumulation function,
y(t), is appropriately modelled by the Black-Scholes model, as follows:
y(t) := δt + σB(t),
0 ≤ t ≤ ∞.
Here δ > 0, σ > 0 are constants and B(t) is the standard Brownian process. Consequently,
the present value of this homogeneous life portfolio is
Sn :=
n
X
Kl e−y(Tl ) ,
(1.1)
l=1
where {Tl is assumed to be independent of the Brownian process B(t) for every l ∈ {1, · · · , n}.
It is noteworthy that Ahcan et al. (2006) considered a model similar to model (1.1) and
presented the convex upper and lower bounds for Sn . However, in Ahcan et al. (2006), Tl
in Sn was taken to be l, not random variable.
As reflected in model (1.1), recently the effects of stochastic interest rates have been
emphasized in the life insurance literature because ”durations of contracts in life insurance
and the life annuity business are typically very long (often 30 or even more years) and
the financial and investment risk—unlike the mortality risk—cannot be diversified with an
increase in the number of policies” (Hoedemakers et al. (2006)). In the actuarial literature
there are many papers covering the random interest rates. To summarize, two kinds of
methods are used to study this problem.
In most papers where the first method is used, the authors restrict themselves to calculating the first two or three moments of the present value function. For examples,Bellhouse
and Panjer (1980) and Bellhouse and Panjer (1981) used autoregressive models of order one
to compute moments of insurance and annuity functions. In Beekman and Fuelling (1993),
the authors gave expressions for the mean and the standard deviation of the future life insurance payments for certain policies. The first three moments of homogeneous portfolios of
life insurance and endowment policies were calculated in Parker (1994a, 1994b), and these
results were generalized to heterogeneous portfolios in Parker (1997). Papers involving the
calculation of the first few moments in stochastic interest rates frameworks include Boyle
(1976), Waters (1978), Wilkie (1976), Dhaene (1989), Norberg (1990) and many others.
The second method is to use the comonotonicity technique to estimate the upper and
lower bounds of the prevent values. This technique was proposed by Dhaene and Goovaerts
(1996,1997), and adopted by Wang and Dhaene (1998), Goovaerts and Redant (1999),
Goovaerts and Dhaene (1999), Dhaene et al. (2002a, 2002b), Goovaerts et al. (2000),
Simon et al. (2000), Vyncke et al.(2001), Kaas et al. (2000,2001), Ahcan et al. (2006), and
others.
2
In this paper we aim to explore the limiting properties of the upper bounds of the present
values of model (1.1), derived by the technique of comonotonicity, since an average insurer
usually has a large number of homogenous policies and the upper bounds are very informative
and useful to the insurer in making conservative estimates about the risks and calculating
premiums. However, it would clearly be difficult to discuss the limiting distributions of
the upper bounds of model (1.1) under a general dependent structure for the Kl or the Tl .
Therefore, we consider the cases in which Tl , l = 1, 2, · · · , n, are respectively independent,
negatively associated, and positively associated random variables, and establish some laws
of large numbers for each dependence structure. Note that, even if {Kl , l = 1, 2, · · · , n} and
{Tl , l = 1, 2, · · · , n} are independent random series, it is still not technically easy to establish
laws of large numbers for the present value Sn directly. The concepts of negative association
and positive association are frequently used in statistical science to describe the dependence
structure of random variables, since they encompass a wide range of dependence structures.
Of course, other dependence structures may be considered. In conclusion, in this paper we
proposes to combine the technique of commonotonicity and limiting theory to further study
present values (or other values) in actuarial science.
The outline of this paper is as follows: in Section 2 the convex bounds of the present
value function are presented; and in Section 3 we discusses some limiting properties of the
upper convex bounds of the present value functions.
2
Convex bounds of the present value functions
In this section, we focus on discussing the convex bounds of random sum (1.1). We denote by
U and V two mutual independent random variables uniformly distributed on [0, 1]. Moreover,
let Λ also be a random variable, independent of both U and V . We now recall the definition
of convex order.
Definition 2.1. Let X and Y be two random variables with finite means. Then we say that
X is less than Y in convex order (written X ≤cx Y ), if E[f (X)] ≤ E[f (Y )] for all real
convex functions f for which the expectations exist.
The following lemma will play a critical role in the subsequent discussion.
Lemma 2.1. 1. Let X = (X1 , X2 , · · · , Xn ) is a random vector. Then
n
X
i=1
E[Xi |Λ] ≤cx
n
X
Xi ≤cx
i=1
n
X
i=1
3
FX−1i (U ),
(2.1)
where FX−1 (p) = inf{x ∈ R|FX (x) ≥ p} and FX (t) is the distribution function of X.
2. Let V = (V1 , · · · , Vn ) and W = (W1 , · · · , Wn ) be nonnegative random vectors, and assume
that X = (X1 , · · · , Xn ) is independent of both V and W. If, for every x1 , x2 , · · · , xn we have
n
X
xi Vi ≤cx
i=1
n
X
xi Wi ,
i=1
then the corresponding scalar products are convex ordered, i.e.,
n
X
Xi Vi ≤cx
i=1
n
X
Xi Wi .
i=1
Proof. We refer the reader to Kaas et al. (2000) for the proof of part 1, and to Ahcan et
al. (2006) for the proof of part 2.
−Yi
Dhaene et al. (2002b) considers=ed the case in which Xi = e
in part 1 of Lemma
2.1, where (Y1 , Y2 , · · · , Yn ) constitutes a multivariate normal random vector with parameters
E[Yi ] = µi , V ar(Yi ) = σi and Cov(Yi , Yj ) = σij , i, j = 1, 2, · · · , n. Using Lemma
2.1, Dhaene
i
Pn h µj +(1/2)σ2
Pn
j · Y
et al. (2002b, Section 4) derived bounds for i=1 Xi by taking Λ = j=1 e
j ,
as in the following lemma.
Lemma 2.2. (1) Let U be uniform on [0, 1] and independent of the r.v. Λ. Then
n
X
−µi +(1/2)σi2 (1−ri2 )−σi ri Φ−1 (V )
e
≤cx
n
X
i=1
−Yi
e
≤cx
i=1
n
X
e−µi +σi Φ
−1 (U )
,
(2.2)
i=1
where
Pn
Cov(Yi , Λ)
j=1
2
eµj +(1/2)σj σij
= qP
.
ri = Corr(Yi , Λ) = p
µk +µl +(1/2)(σk2 +σl2 )
V ar(Yi )V ar(Λ)
e
·
σ
σi
kl
1≤k,l≤n
(2.3)
Now consider our model (1.1). Since E[−y(Ti )|Ti ] = −δTi , we have V ar(−y(Ti )|Ti ) =
σ 2 Ti and Cov(−y(Ti ), −y(Tj )) = σ 2 min{Ti , Tj }, i, j = 1, 2, · · · , n. Using the conditional exP
pectation technique, we know that Lemma 2.2 applies to ni=1h e−y(Ti ) with the replacements
i
√
P
2
µi = −δTi , σi = σ Ti , σij = σ 2 min{Ti , Tj }, and Λ = ni=1 e−µi −(1/2)σi · (−y(Ti )) . This
leads to the following proposition.
Proposition 2.1. In model (1.1), we have
n
X
√
−δTi +(1/2)σ 2 Ti (1−Ri2 )−σ Ti Ri Φ−1 (V )
e
≤cx
i=1
n
X
i=1
4
−y(Ti )
e
≤cx
n
X
i=1
e−δTi +σ
√
Ti Φ−1 (U )
,
(2.4)
where
Pn
Ri = q
j=1
Ti
e−δTj −(1/2)σ
2T
j
min{Ti , Tj }
.
Pn
(2.5)
−δ(Tl +Tk )−(1/2)σ 2 (Tl +Tk ) min{T , T }
l
k
1≤l,k≤n e
For notation convenience, we henceforth write
Snuu
=
n
X
FK−1i (U )e−δTi +σ
√
Ti Φ−1 (U )
,
i=1
Snu =
Snl =
n
X
i=1
n
X
Ki e−δTi +σ
√
Ti Φ−1 (U )
Ki e−δTi +(1/2)σ
,
2 T (1−R2 )−σ
i
i
√
Ti Ri Φ−1 (V )
,
i=1
Snll =
n
X
E[Ki |Θ]e−δTi +(1/2)σ
2 T (1−R2 )−σ
i
i
√
Ti Ri Φ−1 (V )
,
i=1
where Θ is a random variable independent of Λ and y(Ti ), i ≥ 1, but which may depend on
Ki , i = 1, 2, · · · , n.
Using Proposition 2.1, and repeatedly applying the result of part 2 of Lemma 2.1, we can
derive the following proposition about the bounds of present value function (1.1).
Proposition 2.2. In model (1.1), we have
Snll ≤cx Snl ≤cx Sn ≤cx Snu ≤cx Snuu .
(2.6)
Remark 2.1. Note that if Λ is chosen differently, we may obtain different bounds for the
P
present value function (1.1). For example, if we take Λ = ni=1 λ(Ti )B(Ti ) with any function
λ(x) : R+ → R+ , the low bounds are Snl , and Snll as above with Ri as follows:
Pn
n X
n
X
j=1 λ(Tj ) min(Ti , Tj )
2
√
Ri =
, σΛ =
λ(Tk )λ(Tj ) min(Tk , Tj ).
(σ Ti )σΛ
j=1 k=1
3
The limiting properties of the upper convex bounds
From the expressions for the bounds we obtained in the previous section, we seems that the
upper convex bounds are more mathematically tractable than the lower bounds. Moreover,
5
for the reason given in Section 1, the insurer may be much more interested in the results of
studying the former. In light of this, we now investigate the limiting properties of the upper
convex bounds. Specifically, we establish laws of large numbers under three kinds assumption
on the dependence structure: independent, negative association, positive association.
Let
µuu = FK−11 (U )E[e−δT1 +σ
√
T1 Φ−1 (V )
|V ],
(3.1)
and
µu = E[K1 ]E[e−δT1 +σ
√
T1 Φ−1 (V )
|V ].
(3.2)
The following theorem holds when the residual lives are independent.
Theorem 3.1. Consider model (1.1).
1. If {Tl , l = 1, 2, · · · , n} are independently and identically distributed random variables then
Snuu
→ µuu ,
n
a.s.
as n → ∞.
(3.3)
2. If {Kl , l = 1, 2, · · · , n} are independently and identically distributed random variables,
then
Snu
→ µu ,
n
a.s.
as n → ∞.
√
(3.4)
−1
Proof. To prove part 1, we let Yl = FK−1l (U )e−δTl +σ Tl Φ (V ) , Bl = σ(T1 , T2 , · · · , Tl , U, V ).
Then
n nX
o
Yl − E[Yl |Bl−1 ] , Bn , n = 1, 2, · · ·
l=1
is a martingale, and
E[Yl |Bl−1 ] = E[FK−1l (U )e−δTl +σ
√
Tl Φ−1 (V )
|U, V ] = FK−11 (U )E[e−δT1 +σ
√
T1 Φ−1 (V )
|V ] = µuu .
Hence, it follows from the strong law of large numbers for a martingale ( see Hall and Heyde
(1980), pp. 36-39 ) that
n
1 X
{ (Yl − E(Yl | Bl−1 )} → 0 a.s. as n → ∞,
n l=1
(3.5)
which completes the proof for (3.3).
To prove part 2, we let
Zl = Kl e−δTl +σ
√
Tl Φ−1 (V )
,
and Wl = E[Kl ]E[e−δTl +σ
6
√
Tl Φ−1 (V )
|V ].
Clearly, Wl = W1 = µu for all l = 1, 2, · · · . Then, similarly as the proof of (3.3) goes, we
have
n
1X
(Zl − Wl ) → 0,
n l=1
a.s n → ∞.
(3.6)
That is,
n
1X
Z l → µu ,
n l=1
a.s n → ∞.
(3.7)
As a result, (3.4) follows immediately.
Next we consider the respective cases in which {Tl , l = 1, 2, · · · , n} are mutually dependent in the following two senses.
Definition 3.1. (1) A random vector X = (X1 , X2 , · · · , Xm ) is said to be negatively associated if
Cov f (Xi , i ∈ L), g(Xj , j ∈ M ) ≤ 0
for all disjoint subsets L and M of {1, 2, · · · , m} and all increasing function f : Rd(L) → R
and g : Rd(M ) → R, where d(·) means the number of elements a set contains.
(2) A random vector X = (X1 , X2 , · · · , Xm ) is said to be positively associated if
Cov f (X), g(X) ≥ 0
for all increasing functions f, g : Rm → R.
For our further discussion, we introduce the following definition and present three lemmas.
Definition 3.2. A sequence of functions {gi (·, ·), i ≥ 1} is called to satisfy uniform Lipshitz’s
condition of order 1, if there exists a constant B > 0 such that
|gi (x + u, y + v) − gi (x, y)| ≤ B(|u| + |v|)
for any x, y, u, v ∈ R.
Lemma 3.1. (Newman(1980)) If X and Y are positively associated random variables, then
for any s, t ∈ R,
|EeisX+itY − EeisX EeitY | ≤ st|Cov(X, Y )|,
where i is the imaginary unit.
Lemma 3.2. (Lin (2003)) Suppose {Ti , i ≥ 1} is a sequence of positively associated random
variables exposed to a common density function p(·) with the characteristic function ϕ(·).
Let p1j (·, ·) and ϕ1j (·, ·) be the joint density function and the joint characteristic function of
(T1 , Tj ), j ≥ 2 respectively. Then with the following three conditions:
7
(C1) p(t) is bounded and satisfies Lipschitz’s condition of order 1,
(C2) p1j (·, ·), j ≥ 1 satisfy uniform Lipschitz’ condition of order 1,
(C3) ϕ(·) and ϕ(·, ·), j ≥ 2, are absolutely integrable,
it follows that for any D > 0
1
sup |p1j (x, y) − p(x)p(y)| ≤ 2
4π
x,y
Z
D
−D
√
6 2B(1 + A)
|ϕ1j (s, t) − ϕ(s)ϕ(t)|dsdt +
, j ≥ 2,
D
−D
Z
D
where B is a constant in condition (C2) and A = supx p(x).
Lemma 3.3. 1. (Liu(1999)) Let {Xi : i ≥ 1} be a sequence of negatively associated random
variables with a common distribution. For 0 < p < 2, the sufficient and necessary condition
such that
Pn
i=1 Xi − nE[X]
→ 0. a.s. n → ∞,
n1/p
is E|X1 |p < ∞.
2. (Birkel(1989)) Let {Xi : i ≥ 1} be a sequence of positively associated random variables
with finite variances. If
∞
X
j
−2
Cov(Xj ,
j=1
then
P
n
i=1
Xi −
j
X
Xi ) < ∞.
(3.8)
i=1
E[X
]
i /n → 0, a.s. as n → ∞.
i=1
Pn
√
−1
Theorem 3.2. Assume E e−δT1 +σ T1 Φ (U ) < ∞ and E|K1 | < ∞ .
1. If {Tl , l = 1, 2, · · · , n} are negatively associated, then (3.3) holds.
2. if, additionally, {Kl , l = 1, 2, · · · , n} are identically and independently distributed random
variables, then (3.4) holds.
Proof. 1. To prove part 1, it suffices to show that, for any u1 , u2 ∈ [0, 1],
n
h
i
√
1X
−1
f (Ti ) → FK−11 (u2 )E e−δT1 +σ T1 Φ (u1 ) ,
n i=1
where f (x) = FK−11 (u2 )e−δx+σ
√
xΦ−1 (u1 )
a.s. as n → ∞,
(3.9)
. We can decompose f (x) as f (x) = f1 (x) + f2 (x) with
(
−1
2 )
√
σΦ−1 (u1 ) 2 σ 2 [Φ−1 (u1 )]2
σΦ
(u
)
1
4δ
f1 (x) = FK−1i (u2 )e−δ( x− 2δ ) +
I x≤
2δ
(
2 )
−1
σ 2 [Φ−1 (u1 )]2
σΦ
(u
)
1
4δ
+FK−1i (u2 )e
I x>
,
(3.10)
2δ
8
and
(
σΦ−1 (u1 )
f2 (x) = FK−1i (u2 )e−δ(
I x>
2δ
)
(
2
−1
σ 2 [Φ−1 (u1 )]2
σΦ (u1 )
−1
4δ
.
−FKi (u2 )e
I x>
2δ
√
σΦ−1 (u1 ) 2 σ 2 [Φ−1 (u1 )]2
x−
) +
2δ
4δ
2 )
(3.11)
Note that both sequences {f1 (Ti ), i = 1, 2, · · · , n} and {f2 (Ti ), i = 1, 2, · · · , n} are both
negatively associated, since f1 (x) and f2 (x) are respectively increasing and decreasing in x.
Hence, by part 1 of Lemma 3.3 we can show that
(
(
−1
2 ))
n
√
σΦ−1 (u1 ) 2 σ 2 [Φ−1 (u1 )]2
1X
σΦ
(u
)
1
4δ
I T1 ≤
f1 (Ti ) → FK−1i (u2 )E e−δ( T 1 − 2δ ) +
n i=1
2δ
(
)
−1
2
σ 2 [Φ−1 (u1 )]2
σΦ (u1 )
−1
4δ
EI T1 >
, a.s n → ∞ (3.12)
+FKi (u2 )e
2δ
and that
(
(
−1
2 ))
n
√
σΦ−1 (u1 ) 2 σ 2 [Φ−1 (u1 )]2
1X
σΦ
(u
)
1
4δ
I T1 >
f2 (Ti ) → FK−1i (u2 )E e−δ( T 1 − 2δ ) +
n i=1
2δ
(
−1
2 )
σ 2 [Φ−1 (u1 )]2
σΦ
(u
)
1
4δ
−FK−1i (u2 )e
EI T1 >
, a.s n → ∞. (3.13)
2δ
Then, the combination of (3.12) and (3.13) immediately yields (3.9).
√
√
−1
−1
To prove part 2, we let Zi = Ki e−δTi +σ Ti Φ (U ) and Vi = E[Xi ]e−δTi +σ Ti Φ (U ) . Note
that Kl , l = 1, 2, · · · , n are independently and identically distributed random variables. Then
similarly to the proof of part 1 of Theorem 3.1, we find that
n
1X
(Zi − Vi ) → 0,
n i=1
a.s n → ∞,
(3.14)
from which it immediately follows that
n
1X
Vi → µl ,
n i=1
a.s n → ∞.
The proof is thus completed.
(3.15)
Theorem 3.3. 1. Let {Tj , j = 1, 2, · · · , n} be positively associated random variables satisfying condition (C1), (C2) and (C3) in Lemma 3.2. Furthermore, assume that Tj ≤ M, j =
1, 2, · · · for j = 1, 2, · · · , where M is a constant, and that
∞
X
j=1
j
−2
j
X
|Cov(Tj , Tk )|1/5 < ∞.
k=1
9
(3.16)
Then (3.3) holds.
(2) If, additionally, {Kj , j = 1, 2, · · · , n} are independently and identically distributed random variables, then (3.4) holds.
Proof. To prove part 1 it is sufficient to verify (3.9). We decompose the function f (x) in
(3.9) as f (x) = f1 (x) + f2 (x) as in the proof of Theorem 3.2. Then, if we can prove (3.12)
and (Th3.2-03), combining them immediately yields (3.9).
By the monotonicity of f1 (x) and f2 (x), we know that the sequences {f1 (Ti ), i = 1, · · · , n}
and {f2 (Ti ), i = 1, · · · , n} are positively associated. Consequently, by part 2 of Lemma 3.3,
a sufficient condition for (3.12) and (Th3.2-03) to hold is that
∞
X
j=1
j
−2
Cov(fi (Tj ),
j
X
fi (Tk )) < ∞,
i = 1, 2.
(3.17)
k=1
Now combining Lemma (3.1) and Lemma (3.2) yields
Z MZ M
fi (t1 )fi (t2 )(pjk (t1 , t2 ) − p(t1 )p(t2 ))dt1 dt2
Cov(fi (Tj ), fi (Tk )) =
0
0
Z MZ M
fi (t1 )fi (t2 )dt1 dt2
≤
0
0
√
Z DZ D
6 2B(1 + A)
1
· 2
|ϕjk (s, t) − ϕ(s)ϕ(t)|dsdt +
4π −D −D
D
√
σ 2 [Φ−1 (u1 )]2
D4
6 2B(1 + A)
2δ
{ 2 |Cov(Tj , Tk )| +
≤ M 2 [FX−11 (u2 )]2 e
}
4π
D
≤ cM 2 |Cov(Tj , Tk )|1/5 .
(3.18)
where c ≡ c(u1 , u2 ) denotes a positive constant and the last inequality results from setting
D = Cov(Ti , Tk )−1/5 . Consequently, (3.17) holds due to the assumption (3.16).
The proof of part 2 is similar to the proof of part 2 of Theorem 3.2 (2). We omit the
details.
References
[1] Ahcan, A., Darkiewicz, G., Goovaerts, M., Hoedemakers, T. (2006). Computation of
convex bounds for present value functions with random payments. Journal of Computational and Applied Mathematics 186, 23-42.
[2] Beekman, J.A. and Fuelling, C.P. (1993). One approach to dual randomness in life
insurance. Scandinavian Actuarial Journal 76(2), 173-182.
10
[3] Bellhouse, D. R. and Panjer, H. H. (1981). Stochastic Modelling of Interest Rates with
Applications to Life Contingencies-Part II. Journal of Risk and Insurance 48(4), 628-637.
[4] Boyle, P.P. (1976). Rates of Return As Random Variables. Journal of Risk and Insurance
43(4), 693-711.
[5] Dhaene, J. (1989). Stochastic interest rates and Autoregressive Integrated Moving Average Processes. ASTIN Bulletin 19(2), 131-138.
[6] Dhaene, J., Denuit, M., Goovaerts, M.J., Kass, R., Vyncke, D. (2002a). The concept
of comonotonicity in actuarial science and finance: theory. Insurance: Mathematics &
Economics 31, 3-33.
[7] Dhaene, J., Denuit, M., Goovaerts, M.J., Kass, R., Vyncke, D. (2002b). The concept of
comonotonicity in actuarial science and finance: application. Insurance: Mathematics
& Economics 31, 133-161.
[8] Dhaene, J., Goovaerts, M.J. (1996). Dependency of risks and stop-loss order. ASTIN
Bulletin 26, 201-212.
[9] Dhaene, J., Goovaerts, M.J. (1997). On the dependency of risks in the individual life
model. IInsurance: Mathematics & Economics 19, 243-253.
[10] Goovaerts, M.J., Dhaene, J. (1999). Supermodular ordering and stochastic annuities.
Insurance: Mathematics & Economics 24, 281-290.
[11] Goovaerts, M.J., Dhaene, J., De Schepper, A. (2000). Stochastic upper bounds for
present value functions. Journal of Risk and Insurance Theory 67 (1), 1-14.
[12] Goovaerts, M.J., Redant, R. (1999). On the distribution of IBNR reserves. Insurance:
Mathematics & Economics 25, 1-9.
[13] Hall, P., Heyde, C.C. (1980). Martingale Limit Theory and its Appllcation. NewYork.
[14] Hoedemakers, T., Darkiewicz, G., Dhaene, J., and Goovaerts, M. (2005). On the distribution of life annuities with stochastic interest rates. To appear in Insurance: mathematics & Economics.
[15] Kaas, R., Dhaene, J., Goovaerts, M. J. (2000). Upper and lower bounds for sums of
random variables. IInsurance: Mathematics & Economics 27, 151-168.
11
[16] Kaas, R., Dhaene, J., Vyncke, D., Goovaerts, M.J., Denuit, M. (2001). A simple geometric proof that comonotonic risks have the convex-largest sum. Research Report
Department of Applied Economics, K.U. Leuven, Belgium.
[17] Lin, Z. (2003). Asymptotic normality of kernel estimates of a density function under
association dependence. Acta Mathematica Scientia 23, Series B, 345-350.
[18] Liu, J., Gan, S., Chen, P. (1999). The Hajeck-Renyi inequality for the NA random
variables and its application1. Statistics & Probability Letters 43, 99-105.
[19] Newman, C. M. (1980). Normal fluctuations and the FKG inequalities. Comm Math
Phys 74, 119-128.
[20] Norberg, R. (1990). Payment measures, interest and discounting. An axiomatic approach
with applications to insurance. Scandinavian Actuarial Journal 73, 14-33.
[21] Panjer, H. H. and Bellhouse, D. R. (1980). Stochastic Modelling of Interest Rates with
Applications to Life Contingencies. Journal of Risk and Insurance 47, 91-110.
[22] Parker, G. (1997). Stochastic analysis of the interaction between investment and insurance risks. North American Actuarial Journal, 1(2), 55-71.
[23] Simon, S., Goovaerts, M.J., Dhaene, J. (2000).Aneasy computable upper bound for
the price of an arithmetic Asian option. Insurance: Mathematics & Economics 26, 2-3,
175-184.
[24] Vyncke, D., Goovaerts, M.J., Dhaene, J. (2001). Convex upper and lower bounds for
present value functions. Applied Stochastic Models in Business and Industry 17, 149164.
[25] Wang, S., Dhaene, J. (1998). Comonotonicity, correlation order and stop-loss premiums.
Insurance: Mathematics & Economics 22, 235-243.
[26] Waters, H.R. (1978). The Moments and Distributions of Actuarial Functions. Journal
of the Institute of Actuaries 105, 61-75.
[27] Wilkie, A.D. (1976). The Rate of Interest as a Stochastic Process: Theory and Applications. Proceedings of the 20th International Congress of Actuaries, Tokyo 1, 325-337.
12