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Aggregate survival probability of a portfolio with dependent subportfolios
Insurance: Mathematics and Economics 32 (2003) 431–443
Aggregate survival probability of a portfolio with dependent subportfolios夽 Rohana S. Ambagaspitiya∗ Department of Mathematics and Statistics, University of Calgary, Calgary, Alta., Canada T2N 1N4 Received July 2002; received in revised form March 2003 ; accepted 19 March 2003
Abstract In this paper we consider a portfolio in which claim inter-arrival times have multivariate gamma distribution. We consider a multivariate gamma distribution obtained by dimension reduction technique. We obtain explicit solution of the aggregate survival probability of the portfolio. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Dependent subportfolios; Multivariate gamma distributions; Survival probability
1. Introduction Let us assume the portfolio contains p dependent subportfolios. Then the total claim process {S(t), t ≥ 0} can be written as S(t) =
p N i (t)
Xij
(1)
i=1 j=1
Here {Ni (t), t ≥ 0}, i = 1, 2, . . . , p are dependent claim count processes, and Xij are random variables representing jth, j = 1, 2, . . . claim amount in the ith, i = 1, 2, . . . , p portfolio. We assume that the probability function of the claim sizes random variable of the ith portfolio is fXi (x), for i = 1, 2, . . . , p; i.e. the probability function of Xij , j = 1, 2, . . . , Ni (t) is fXi (x). Let us write N(t) = [N1 (t), N2 (t), . . . , Np (t)]T ,
(2)
where {N(t), t ≥ 0} is a p-variate counting process. We assume this counting process is obtained by the dimension reduction technique, i.e. we have M(t) = [M1 (t), M2 (t), . . . , Mp (t)]T 夽
This research was supported by a grant from Natural Sciences and Engineering Council of Canada. Tel.: +1-403-220-7679; fax: +1-403-282-5150. E-mail address: [email protected] (R.S. Ambagaspitiya). ∗
0167-6687/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-6687(03)00131-8
(3)
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{Mi (t), t ≥ 0}, i = 1, 2, . . . , m are independent counting processes, and N(t) = AM(t)
(4)
where the matrix A = (aij ) is a p × m binary matrix. By using the same argument as in Ambagaspitiya (1998, 1999) we can show that the total claim process S(t) in (1) can be written as m S(t) = Si (t), (5) i=1
with Si (t) =
M i (t)
Xij
j=1
where the random variables Xij have the distributions ∗(a1i )
fXi (x) = (fX
1
∗(a2i )
∗ fX
2
∗(api )
∗ · · · ∗ fX
p
)(x),
∗(a )
i = 1, 2, . . . , m. ∗(0)
Here fXi ji is the aji -fold convolution of fXi with fXi ∼ 1. The resulting {Si (t), t ≥ 0}, i = 1, 2, . . . , m are independent total claim processes. The counting processes Mi (t) represent the number of claims in (0, t] in the ith total claim processes for i = 1, 2, . . . , m. Therefore, we have decomposed p dependent subportfolios into m (m > p) independent total claim processes. From here onward we pay our attention to these independent processes. The Laplace transform fX∗ i (s) of the probability function fXi (x) takes the following form: fX∗ i (s)
=
p
[fX∗ (s)]aji
j=1
(6)
j
where fX∗ (s) is the Laplace transform of fXj (x). j
In this paper we assume that the inter-arrival time distribution associated with the counting process Mi (t) is gamma with parameters (ni , βi ) for i = 1, 2, . . . , m. Then the inter-arrival time distribution associated with the counting process N(t) is multivariate gamma. This type of multivariate gamma distributions are well known to the statistical literature [6, Chapters 47 and 48] attributes to many authors for extending the approach used by Tiecher (1954) to generate multivariate Poisson distribution. j If we denote Ti as the elapsed time between j − 1th and jth claim in the ith subportfolio the probability density function of it βini ki (t) = t ni −1 e−βi t , t > 0, i = 1, 2, . . . , m. (ni − 1)! By using the properties of gamma distribution we could write j
P[Ti > t] =
n i −1 k=0
(βi t)k −βi t . e k!
In other words we are assuming claims occur as an ordinary renewal process. Since we are interested in the time until first claim in the ith portfolio we write Ti instead of Ti1 for convenience. Following the same notation as in classical risk model we could write the expressions for ruin probability ψ(u) and the survival probability δ(u) as ψ(u) = P[u + ct − S(t) < 0; t < ∞],
δ(u) = 1 − ψ(u),
where u is the initial surplus and c is the premium rate.
R.S. Ambagaspitiya / Insurance: Mathematics and Economics 32 (2003) 431–443
433
In this model we need to consider the time until the first claim in aggregate, T(1) , which is given by T(1) = min(T1 , T2 , . . . , Tm ). Let us denote the associated claim size as X(1) . The appropriate positive security loading would be to have the premium rate c such that cE[T(1) ] > E[X(1) ]. Our purpose in this paper is to find an explicit expression for the survival probability δ(u). This is an extension of Dickson (1998), Dickson and Hipp (1998). In Section 2 we obtain this formula. In Section 3 we present two simple yet illustrative numerical examples. Finally, in the appendix we present the peripheral results that we used in the main paper. We also present a Maple program to do the manipulations presented in the paper with rational Laplace transform claim sizes.
2. Main result By considering the time and the amount of the first claim, we can write ∞ u+ct δ(u) = fT(1) ,X(1) (t, x)δ(u + ct − x) dx dt. 0
(7)
0
This is infact a generalization of the Dickson’s (1998) result. We can substitute the expression given in the appendix for the joint distribution of T(1) , X(1) to obtain the following δ(u) =
∞
t=0
m u+ct x=0
i=1
βini fX (x) e−βt (ni − 1)! i
m∗ −(n i −1)
bk (i)t ni −1+k δ(u + ct − x) dx dt.
(8)
k=0
Let us substitute s = u + ct in the integral and rearrange it slightly to obtain ni −1+k m∗ −(n m ∞ s ni i −1) βi s−u ds . bk (i) e−βu/c δ(u) = fXi (x)δ(s − x) dx e−βs/c c (n − 1)! c i u x=0 i=1
(9)
k=0
For notational convenience, define s hi (s) = fXi (x)δ(s − x) dx 0
and thus (9) becomes e
−βu/c
δ(u) =
m i=1
βini (ni − 1)!
m∗ −(n i −1) k=0
bk (i) cni +k
∞ u
e
−βs/c
(s − u)
ni −1+k
hi (s) ds .
(10)
We need to differentiate this equation with respect to u to eliminate the integral. From Lemma A.1 in Appendix A we see that if we differentiate the integral ∞ e−β(s/c) hi (s)(s − u)ni −1+k ds, k = 0, 1, 2, . . . , m∗ − (ni − 1) u
with respect to u, ni + k times it becomes (−1)ni +k (ni − 1 + k)! e−β(u/c) hi (u).
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Therefore, by differentiating (10) with respect to u, m∗ + 1 time we get D
m∗ +1
[e
−β(u/c)
δ(u)] =
m i=1
βini (ni − 1)!
m∗ −(n i −1)
bk (i)
k=0
(ni − 1 + k)! m∗ −(ni −1)−k −β(u/c) D [e hi (u)], (−c)ni +k
(11)
where D is the differential operator d/du. By using the standard properties of Laplace transforms, we have ∗
s
m∗ +1
L{e
−β(u/c)
m
βini
δ(u)} −
m
∗ −l
sm
Dl [e−β(u/c) δ(u)]|u=0
l=0
=
i=1
−
m∗ −(n i −1)
(ni − 1)!
m i=1
bk (i)
k=0 m∗ −(n i −1)
βini
(ni − 1)!
k=0
(ni − 1 + k)! m∗ −(ni −1)−k [s L{e−β(u/c) hi (u)}] (−c)ni +k
(ni − 1 + k)! bk (i) (−c)ni +k
m∗ −(n i +k)
∗ −n −k−l i
sm
Dl [e−β(u/c) hi (u)]|u=0 .
(12)
l=0
Let us define the Laplace transform of δ(u), fXi (x) and hi (x) as δ∗ (s), fX∗ i (s) and h∗i (s), respectively. An additional property of Laplace transforms gives us β β β β −β(u/c) ∗ −β(u/c) ∗ ∗ ∗ δ(u)} = δ s + L{e , L{e hi (u)} = hi s + =δ s+ fXi s + . c c c c Substituting these results in (12) and rearranging we obtain β Numer(s) ∗ δ s+ = , c Denom(s)
(13)
where Numer(s) and Denom(s) are functions of s in the following form Numer(s) =
m i=1
−
βini (ni − 1)!
m∗
∗ −l
sm
m∗ −(n i −1) k=0
(ni − 1 + k)! bk (i) (−c)(ni + k)
m∗ −(n i +k)
∗ −n −k−l i
sm
Dl [e−β(u/c) hi (u)]|u=0
l=0
Dl [e−β(u/c) δ(u)]|u=0 ,
l=0
Denom(s) =
m i=1
m∗ −(n i −1) βini β (ni − 1 + k)! m∗ −(ni −1)−k ∗ ∗ s+ bk (i) s − sm +1 . f c (ni − 1)! Xi (−c)ni +k k=0
The function Numer(s) is a l
D [e
−β(u/c)
hi (u)]|u=0
m∗
degree polynomial of s and its coefficients depend on constants
and
Dl [e−β(u/c) δ(u)]|u=0 ,
l = 0, 1, 2, . . .
yet to be determined. Therefore, we could write Numer(s) as ∗
Numer(s) =
m
a¯ i si
i=0
where a¯ i , i = 0, 1, 2, . . . are coefficient to be detemined with a¯ 0 = 0. By substituting s by s − (β/c) in (13) we could write the Laplace transform of δ(u) as m∗ a¯ i (s − (β/c))i δ∗ (s) = i=0 . (14) Denom(s − (β/c))
R.S. Ambagaspitiya / Insurance: Mathematics and Economics 32 (2003) 431–443
Since the numerator of (14) is simply a polynomial of s of order m∗ we may write (14) as m∗ i i=0 ai s ∗ . δ (s) = Denom(s − (β/c))
435
(15)
Let us simplify the function Denom(s − (β/c)). ∗ ∗ m∗ −(n m i −1) βini β m −(ni −1)+k β (ni − 1 + k)! β m +1 ∗ s− Denom s − = bk (i) − s− f (s) c (ni − 1)! Xi c c (−c)ni +k i=1 k=0 ∗ m∗ −(n m n i −1) β m +1 βi i (ni + k − 1)! ∗ = s− bk (i) − 1 . fXi (s) c (ni − 1)! (β − sc)ni +k i=1
k=0
From (A.7) in Appendix A we have ∗ ∗ β β m +1 β m +1 Denom s − = s− MY (s) − s − . c c c At first glance it appears β/c as, m∗ + 1 order, zero of Denom(s − (β/c)); however, the moment generating function ∗ ∗ of Y , MY (s), contains (β − sc)−m −1 . Therefore, multiplying MY (s) by the factor (s − (β/c))m +1 removes its m∗ + 1 order pole at s = β/c. Using these results the Laplace transform of the survival probability δ(u) becomes m∗ i i=0 ai s ∗ δ (s) = . (16) ∗ (s − (β/c))m +1 (MY (s) − 1) As in Dickson (1997) if we define L as the maximal loss random variable we can show that E[e−sL ] = sδ∗ (s). By considering lims→0 in this equation we see that ∗ i s m i=0 ai s lim = 1. (17) ∗ s→0 (s − (β/c))m +1 (MY (s) − 1) ∗ i Obviously, lims→0 , s m i=0 ai s = 0 and lim s→0 MY (s) − 1 = 0. Therefore, we need to use l’Hopital’s rule to evaluate the limit in (17). It can be easily established that m∗
d i lim s ai s = a0 . s→0 ds i=0
Since
∗ ∗ ∗ d β m β m +1 d β m +1 (MY (s) − 1) = (m∗ + 1) s − (MY (s) − 1) + s − s− MY (s) ds c c c ds
and lim
d
s→0 ds
MY (s) = E[Y ] = cE[T(1) ] − E[X(1) ]
from the l’Hopital’s rule we have ∗ β m +1 a0 = − [cE[T(1) ] − E[X(1) ]]. c
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R.S. Ambagaspitiya / Insurance: Mathematics and Economics 32 (2003) 431–443
Let us look at the denominator of δ∗ (s) again. From the part 5 of Theorem A.1 in Appendix A we know that denominator has at least m∗ positive zeros. These zeros would yield terms such as exp(Ru) with R > 0 in δ(u) and hence in ψ(u). But from the Cramer’s upper bound for ruin probabilities we know that ψ(u) < exp(−Ru) with R > 0. The only way to resolve this contradiction is that the polynomial ∗
m
ai s i
i=0
having same zeros as the positive zeros of the denominator of δ∗ (s). Since we have determined a0 already, we should have m∗ m∗ s ai si = a0 1− , si i=0
i=1
where si = 0, i = 1, 2, . . . , m∗ are the roots of ∗ β m +1 s− (MY (s) − 1) = 0 c inside the circle |s − (β/c)| = β/c. Therefore we have the main result in the following form: ∗ [cE[T(1) ] − E[X(1) ]] m i=1 (1 − (s/si )) ∗ . δ (s) = ∗ +1 m (1 − (sc/β)) (MY (s) − 1)
(18)
This is the extension of the formula for a single portfolio given in Dickson (1998) and Kalashnikov (1998) for the multiple portfolio case.
3. Numerical examples Example 3.1. Consider the case where
1 0 1 A= , ki (t) = 4t e−2t , 0 1 1
i = 1, 2, 3,
fXi (x) = 4x e−2x ,
i = 1, 2,
i.e. we have two correlated total claim processes; in each process claims occur as Erlang process with parameters (α = 4, β = 2). The moment generating function of the joint inter-arrival time distribution is in the form: 2 2 2 2 2 2 MN1 ,N2 (s1 , s2 ) = , s1 + s2 < 2. 2 − s1 2 − s2 2 − s 1 − s2 The claim size distribution is gamma with parameters (α = 2, β = 2) in each process. Using these values in the accompanying Maple program we get cE[T(1) ] − E[X(1) ] =
13 27 c
− 43 .
Therefore, the condition for positive security loading would be c > 36/13. If we use c = 4, for example, the Maple program gives the following survival probability: δ(u) = 1 + [0.020086 cos (0.76462u) + 0.044244 sin (0.76462u)] exp(−2.0824u) +0.026492 exp(−2.8153u) − 0.60916 exp(−0.50516u).
R.S. Ambagaspitiya / Insurance: Mathematics and Economics 32 (2003) 431–443
Example 3.2. 1 A = 0 0
Consider the case where 0 0 1 2i t i−1 −2t 1 0 1 , ki (t) = e , (i − 1)! 0 1 1
i = 1, 2, 3, 4,
fXi (x) =
2(2x)3−i −2x e , (3 − i)!
437
i = 1, 2, 3.
In this case we have three correlated total claim processes; in the ith process claims occur as Erlang process with parameters (α = i + 4, β = 2) for i = 1, 2, 3. The moment generating function of the joint inter-arrival time distribution is as follows: 1 2 3 4 2 2 2 2 MN1 ,N2 ,N3 (s1 , s2 , s3 ) = , s1 + s2 + s3 < 2. 2 − s1 2 − s2 2 − s3 2 − s 1 − s2 − s3 By executing the accompanying Maple program with appropriate parameters we get that the appropriate security loading is present as c > 3704/945. If we take c = 5 we get the following survival probability δ(u) = 1 + [0.027762 sin (0.81114u) + 0.037630 cos (0.81114u)] exp(−2.5814u) −0.78347 exp(−0.27498u).
Acknowledgements We are grateful to the anonymous referees and the Managing editor Rob Kaas for their valuable suggestions to an earlier version of this paper.
Appendix A Lemma A.1. Let ∞ Gn (u) = (s − u)n g(s) ds, u
n = 0, 1, . . .
(A.1)
Then n + 1th derivative of Gn (u) with respect to u is given by dn+1 Gn (u) = (−1)n+1 n!g(u). dun+1 Proof. Let us differentiate (A.1) once with respect to u: ∞ ∞ d n−1 n Gn (u) = − n(s − u) g(s) ds − (u − u) g(u) = − n(s − u)n−1 g(s) ds = −nGn−1 (u). du u u Differentiating again with respect to u we have d2 d Gn (u) = −n Gn−1 (u) = (−1)2 n(n − 1)Gn−2 (u). 2 du du Therefore, with mathematical induction we could prove dn Gn (u) = (−1)n n!G0 (u). dun Note that the derivative of G0 (u) is −g(u), hence the proof.
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R.S. Ambagaspitiya / Insurance: Mathematics and Economics 32 (2003) 431–443
Theorem A.1. Let (T1 , X1 ), (T2 , X2 ), . . . , (Tm , Xm ) be m pair of independent random variables with following properties: 1. Ti ’s and Xi ’s are independent for i = 1, 2, . . . , m. 2. The probability density function of the Ti is given by ki (t) =
βini t ni −1 e−βi t, (ni − 1)!
t ≥ 0, i = 1, 2, . . . , m,
where the parameter ni is an integer and βi > 0. 3. The probability functions of Xi takes the form fXi (x). The random variable T(1) is the minimum value of [T1 , T2 , . . . , Tm ] and the corresponding X value is X(1) . Then we have 1. The joint probability density function of T(1) and X(1) takes the following form: fT(1) ,X(1) =
m i=1
βini fX (x) e−βt (ni − 1)! i
m∗ −(n i −1)
bk (i)t ni −1+k .
(A.2)
k=0
2. The marginal distributions of T(1) and X(1) takes the form: m
fT(1) (t) =
i=1
fX(1) (t) =
βini e−βt (ni − 1)!
m i=1
m∗ −(n i −1)
βini fX (x) (ni − 1)! i
bk (i)t ni −1+k ,
(A.3)
k=0 m∗ −(n i −1)
bk (i)
k=0
(ni − 1 + k)! . βni +k
(A.4)
3. The means of T(1) and X(1) takes the following form: m
E[T(1) ] =
i=1
E[X(1) ] =
m i=1
βini (ni − 1)!
m∗ −(n i −1)
bk (i)
k=0
βini E[Xi ] (ni − 1)!
(ni + k)! , βni +k+1
m∗ −(n i −1) k=0
bk (i)
(ni − 1 + k)! . βni +k
(A.5)
(A.6)
4. The moment generating function of the random variable Y = cT(1) − X(1) takes the following form: MY (s) =
m i=1
βini fX∗ i (s)
m∗ −(n i −1) k=0
bk (i)
(ni + k − 1)! 1 . (ni − 1)! (β − sc)ni +k
5. The following function of s ∗ β m +1 s− (MY (s) − 1) = 0 c
(A.7)
(A.8)
has m∗ roots (counting multiplicities) for s with s = 0 inside the circle where the center is β/c, 0 and the radius is β/c in the complex plane provided E[Y ] > 0. The new symbols defined in these equations are as follows: ∞ fX∗ i (s) = exp(−sx)fXi (x) dx, i = 1, 2, . . . , m, 0
(A.9)
R.S. Ambagaspitiya / Insurance: Mathematics and Economics 32 (2003) 431–443
m∗ =
m
ni − m,
439
(A.10)
i=1
β=
m
βi ,
(A.11)
i=1 k
b0 (i) = 1,
bk (i) =
m βj j j=1 j=i
for k = 1, 2, . . . , m∗ − (ni − 1)
kj !
where 0 ≤ kj ≤ nj , j = 1, 2, . . . , m, j = i and of kj s; i.e. the summation contains ∗ m − (ni − 1) k
(A.12)
kj = k and the sum in (A.12) is over all possible combinations
terms. Proof. Note that ki (t)P[T1 > t, T2 , . . . , Ti−1 > t, Ti+1 > t, . . . , Tm > t] dt is the probability that the ith random variable is the smallest in the interval [t, t+dt) for i = 1, 2, . . . , m. Let us simplify this term. Since Ti , i = 1, 2, . . . , m are independent we have ki (t)P[T1 > t, T1 , . . . , Ti−1 > t, Ti+1 > t, . . . , Tm > t] = ki (t)
m
P[Tj > t] =
j=1 j=i
=e
−βt
j −1 m n βini (βj t)k −βj t t ni −1 e−βi t e (ni − 1)! k!
(A.13)
j=1 k=0 j=i
j −1 m n βini (βj t)k ni −1 t , (ni − 1)! k!
(A.14)
j=1 k=0 j=i
where β = m i=1 βi . Closer inspection of (A.14) reveals that ki (t)
m j=1 j=i
j P[T1
βini > t] = e−βt (ni − 1)!
m∗ −(n i −1)
bk (i)t ni −1+k ,
(A.15)
k=0
where m∗ , bk (i), k = 0, 1, . . . , m∗ as defined in (A.10)–(A.12). Since the joint pdf of T(1) and X(1) , fT(1) ,X(1) (t, x) is fT(1) ,X(1) (t, x) =
m
ki (t)P[T1 > t, T2 > t, . . . , Ti−1 > t, Ti+1 > t, . . . , Tm > t]fXi (x)
(A.16)
i=1
and hence the proof of (A.2). The marginal pdf of T(1) and X(1) can be obtained by integrating (A.2) with respect to x and t respectively. Hence the proof of (A.3) and (A.4). Once we have the marginal pdfs we can verify the results in (A.5) and (A.6) for E[T(1) ] and E[X(1) ] easily.
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R.S. Ambagaspitiya / Insurance: Mathematics and Economics 32 (2003) 431–443
The proof of (A.7) takes little effort. The joint moment generating function of T(1) , X(1) can be derived as MT(1) ,X(1) (s1 , s2 ) = E[e =
m i=1
=
s1 T(1) +s2 X(1)
βini (ni − 1)!
m i=1
βini
(ni − 1)!
]=
t=0 x=0
∞ x=0
∞ ∞
e
s2 x
es1 t+s2 x fT(1) ,X(1) (t, x) dt dx
fXi (x)
m∗ −(n i −1) k=0
MXi (s2 )
m∗ −(n i −1) k=0
bk (i)
bk (i)
∞ t=0
es1 t e−βt t ni −1+k dt
(ni + k − 1)! . (β − s1 )ni +k
Hence the moment generating function, MY (s), of Y = cT1 − X1 can be derived as follows: MY (s) = E[e
(cT1 −X1 )s
] = MT(1) ,X(1) (sc, −s) =
βini
m∗ −(n i −1)
m i=1
=
m i=1
(ni − 1)!
fX∗ i (s)
k=0
bk (i)
βini MXi (−s) (ni − 1)!
m∗ −(n i −1) k=0
bk (i)
(ni + k − 1)! (β − sc)ni +k
(ni + k − 1)! . (β − sc)ni +k
The last step follows from the fact that the equivalency of moment generating functions and Laplace transforms. To prove (A.8) consider the circle C s − β = β c c in the complex plane. We can denote the points on the circle C as
β β β + cos θ, sin θ . c c c
In the derivation of the joint moment generating function of T(1) , X(1) we made the implicit assumption that the integral
∞
t=0
es1 t e−βt t ni −1+k dt
converges. However it converges to (ni + k − 1)! (β − s1 )ni +k provided β − s1 > 0. Therefore, the expression for the moment generating function MY (s), of Y given in (A.7) is valid for s < β/c. But close inspection of MY (s) reveals that in fact we could write the following for the value of it on the circle C. ∗
m −(n m i −1) βini −1 (ni + k − 1)! MY (s) = bk (i) , fX∗ i (s) n +k (ni − 1)! (−β) i ( cos θ + I sin θ)ni +k i=1
k=0
R.S. Ambagaspitiya / Insurance: Mathematics and Economics 32 (2003) 431–443
where I =
√
441
−1 ∗
m −(n m i −1) βini −1 (ni + k − 1)! |MY (s)| ≤ |fX∗ i (s)| bk (i) n +k , i (ni − 1)! β |( cos θ + I sin θ)|ni +k i=1
|MY (s)| ≤
m i=1
k=0
βini −1
(ni − 1)!
|fX∗ i (s)|
m∗ −(n i −1) k=0
bk (i)
(ni + k − 1)! . βni +k
If fX∗ i (s) exist on the circle C then |fX∗ i (s)| ≤ 1, for i = 1, 2, . . . , m. ∞ ∞ m −(n m i −1) βini −1 (ni + k − 1)! |MY (s)| ≤ bk (i) = fT(1) ,X(1) (t, x) dt dx = 1 (ni − 1)! βni +k t=0 x=0 ∗
i=1
k=0
Therefore on the circle C for * > 0 ∗ ∗ β m +1 β m +1 MY (s) < s − s− (1 + *). c c ∗
∗
From the Rouche’s theorem we can conclude that (s−(β/c))m +1 (MY (s)−1−*) and −(1+*)(s−(β/c))m +1 have ∗ the same number of zeros, i.e. m∗ + 1 zeros, inside the circle C. Since s = 0 is a zero of (s − (β/c))m +1 (MY (s) − 1) ∗ +1 m (MY (s) − 1 − *) has a zero at s ≈ (*/E[Y ]). Since with multiplicity of 1, the perturbed function (s − (β/c)) E[Y ] > 0 this is inside the circle C. If * is sufficiently small this will be away from the rest of the roots; therefore we can conclude the proof by * → 0. Appendix B. Maple program The following is a listing of the Maple program that was used in Example 3.2. This program is available as a text file in “http://www.math.ucalgary/∼ sarath/papers/mapleprogram1.txt”. with(inttrans); m := 4; p := 3; # beta and ni are parameters of gamma processes beta := array(1 . . m, [2, 2, 2, 2]); ni := array(1 . . m, [1, 2, 3, 4]); A := array([[1, 0, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1]]); # Laplace transforms of claim sizes distributions #fxd are the Laplace transforms of claim sizes distributions fxd := array([(2/(2 + s))∧ 3, (2/(2 + s))∧ 2, 2/(2 + s)]); #Default value of premium rate is 2. C := 2; #For different cases one may change m, p, beta, ni, A, fxd and C #appropriately #Anything below this line should not be changed # # Use equation(6) to compute the Laplace transforms # #
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fx := array(1 . . m); for i from 1 to m do fx[i] := product(‘fxd[j]∧ A[j, i]’, ‘j’ = 1 . . p) od : #mst is the value defined in (A.9) mst := sum(‘ni[i]−i’, ‘i’ = 1 . . m); # to extract coefficients in (A.10), (A.11) use explicit multiplication and # simplification bki := array(0 . . mst, 1 . . m); # Bet is beta in the paper Bet := sum(‘beta[i]’, ‘i’ = 1 . . m); CDFPRD := product(‘sum(‘(beta[j]∗t)∧ k/k!’, ‘k’ = 0 . . ni[j]−1)’, ‘j’ = 1 . . m); for i from 1 to m do CDI := sum(‘(beta[i]∗t)∧ k/k!’, ‘k’ = 0 . . ni[i]−1); TEMP := collect(simplify(CDFPRD/CDI), t); for k from 0 to mst((ni[i]−1) do bki[k, i] := coeff(TEMP, t, k); od : od : # MY is the moment generating function of Y MY := sum(‘beta[i]∧ ni[i]/(ni[i]−1)!∗fx[i]∗(sum(‘bki[k, i]∗(ni[i] + k−1)!/ (Bet−s∗c)∧ (ni[i] + k)’, ‘k’ = 0 . . mst−ni[i] + 1))’, ‘i’ = 1 . . m); # ETX is the expected value of Y ETX := eval(limit(diff(MY, s), s = 0)); # If the security loading is not sufficient use the smallest possible #value + 1 as the security loading. ETX1 := subs(c = C, ETX); if ETX1<0then C := round(solve(ETX = 0, c)) + 1; print(“Given premium rate is too low”); print(“C = ”, C, “is selected”); fi; # Separate the numerator and denominator of the denominator in (17) # We are using the fact that $s = 0$ is a root of the denominator of (17). DNM := numer(simplify((1−s∗C/Bet)∧ (mst + 1)∗(subs(c = C, MY)−1)/s)); DDM := denom(simplify((1−s∗C/Bet)∧ (mst + 1)∗(subs(c = C, MY)−1))); #Make the accuracy of floating point calculations to 100 digits Digits := 100; # Compute the roots inside the contour and in the first quadrant # if a root is complex its conjugate is also a root. AIS := evalf(allvalues(RootOf(DNM, s, 0 . . 2∗Bet/C∗(1 + I)))); PRD := 1; for x in AIS do if Im(x)< > 0 then PRD := PRD∗(1 + (s∧ 2−2∗Re(x)∗s)/abs(x)∧ 2); else PRD := PRD∗(1−s/x); fi
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od : DNM1 := evalf(s∗DNM/PRD); # DS is Laplace transform of the survival probability. DS := simplify(subs(c = C, ETX)∗DDM/DNM1); DEL := collect(simplify(evalf(invlaplace(DS, s, u))), [exp, sin, cos]); #For final results round everything to 5 digits. Digits := 5; # Clean up the expression by deleting the terms with # small coefficients as a result of floating point computations DEL := fnormal(evalf(DEL)); print(DEL); References Ambagaspitiya, R.S., 1998. On the distribution of a sum of correlated aggregate claims. Insurance: Mathematics and Economics 23, 15–19. Ambagaspitiya, R.S., 1999. On the distributions of two classes of correlated aggregate claims. Insurance: Mathematics and Economics 24, 301–338. Dickson, D.C.M., 1998. On a class of renewal risk process. North American Actuarial Journal 2 (3), 60–68. Dickson, D.C.M., Hipp, C., 1998. Ruin Probabilities for Erlang(2) risk process. Insurance: Mathematics and Economics 22, 251–262. Kalashnikov, V., 1998. The discussion of the paper “On a class of renewal risk process by David C.M. Dickson”. North American Actuarial Journal 2 (3), 70–71. Kotz, S., Balakrishnan, N., Johnson, N.L., 2000. Continuous Multivariate distributions, vol. 1, Models and Applications, Wiley, New York. Tiecher, H., 1954. On the multivariate Poisson distribution. Skandinavisk Aktuarietidskrift 37, 1–9.
Aggregate survival probability of a portfolio with dependent subportfolios夽 Rohana S. Ambagaspitiya∗ Department of Mathematics and Statistics, University of Calgary, Calgary, Alta., Canada T2N 1N4 Received July 2002; received in revised form March 2003 ; accepted 19 March 2003
Abstract In this paper we consider a portfolio in which claim inter-arrival times have multivariate gamma distribution. We consider a multivariate gamma distribution obtained by dimension reduction technique. We obtain explicit solution of the aggregate survival probability of the portfolio. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Dependent subportfolios; Multivariate gamma distributions; Survival probability
1. Introduction Let us assume the portfolio contains p dependent subportfolios. Then the total claim process {S(t), t ≥ 0} can be written as S(t) =
p N i (t)
Xij
(1)
i=1 j=1
Here {Ni (t), t ≥ 0}, i = 1, 2, . . . , p are dependent claim count processes, and Xij are random variables representing jth, j = 1, 2, . . . claim amount in the ith, i = 1, 2, . . . , p portfolio. We assume that the probability function of the claim sizes random variable of the ith portfolio is fXi (x), for i = 1, 2, . . . , p; i.e. the probability function of Xij , j = 1, 2, . . . , Ni (t) is fXi (x). Let us write N(t) = [N1 (t), N2 (t), . . . , Np (t)]T ,
(2)
where {N(t), t ≥ 0} is a p-variate counting process. We assume this counting process is obtained by the dimension reduction technique, i.e. we have M(t) = [M1 (t), M2 (t), . . . , Mp (t)]T 夽
This research was supported by a grant from Natural Sciences and Engineering Council of Canada. Tel.: +1-403-220-7679; fax: +1-403-282-5150. E-mail address: [email protected] (R.S. Ambagaspitiya). ∗
0167-6687/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-6687(03)00131-8
(3)
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{Mi (t), t ≥ 0}, i = 1, 2, . . . , m are independent counting processes, and N(t) = AM(t)
(4)
where the matrix A = (aij ) is a p × m binary matrix. By using the same argument as in Ambagaspitiya (1998, 1999) we can show that the total claim process S(t) in (1) can be written as m S(t) = Si (t), (5) i=1
with Si (t) =
M i (t)
Xij
j=1
where the random variables Xij have the distributions ∗(a1i )
fXi (x) = (fX
1
∗(a2i )
∗ fX
2
∗(api )
∗ · · · ∗ fX
p
)(x),
∗(a )
i = 1, 2, . . . , m. ∗(0)
Here fXi ji is the aji -fold convolution of fXi with fXi ∼ 1. The resulting {Si (t), t ≥ 0}, i = 1, 2, . . . , m are independent total claim processes. The counting processes Mi (t) represent the number of claims in (0, t] in the ith total claim processes for i = 1, 2, . . . , m. Therefore, we have decomposed p dependent subportfolios into m (m > p) independent total claim processes. From here onward we pay our attention to these independent processes. The Laplace transform fX∗ i (s) of the probability function fXi (x) takes the following form: fX∗ i (s)
=
p
[fX∗ (s)]aji
j=1
(6)
j
where fX∗ (s) is the Laplace transform of fXj (x). j
In this paper we assume that the inter-arrival time distribution associated with the counting process Mi (t) is gamma with parameters (ni , βi ) for i = 1, 2, . . . , m. Then the inter-arrival time distribution associated with the counting process N(t) is multivariate gamma. This type of multivariate gamma distributions are well known to the statistical literature [6, Chapters 47 and 48] attributes to many authors for extending the approach used by Tiecher (1954) to generate multivariate Poisson distribution. j If we denote Ti as the elapsed time between j − 1th and jth claim in the ith subportfolio the probability density function of it βini ki (t) = t ni −1 e−βi t , t > 0, i = 1, 2, . . . , m. (ni − 1)! By using the properties of gamma distribution we could write j
P[Ti > t] =
n i −1 k=0
(βi t)k −βi t . e k!
In other words we are assuming claims occur as an ordinary renewal process. Since we are interested in the time until first claim in the ith portfolio we write Ti instead of Ti1 for convenience. Following the same notation as in classical risk model we could write the expressions for ruin probability ψ(u) and the survival probability δ(u) as ψ(u) = P[u + ct − S(t) < 0; t < ∞],
δ(u) = 1 − ψ(u),
where u is the initial surplus and c is the premium rate.
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433
In this model we need to consider the time until the first claim in aggregate, T(1) , which is given by T(1) = min(T1 , T2 , . . . , Tm ). Let us denote the associated claim size as X(1) . The appropriate positive security loading would be to have the premium rate c such that cE[T(1) ] > E[X(1) ]. Our purpose in this paper is to find an explicit expression for the survival probability δ(u). This is an extension of Dickson (1998), Dickson and Hipp (1998). In Section 2 we obtain this formula. In Section 3 we present two simple yet illustrative numerical examples. Finally, in the appendix we present the peripheral results that we used in the main paper. We also present a Maple program to do the manipulations presented in the paper with rational Laplace transform claim sizes.
2. Main result By considering the time and the amount of the first claim, we can write ∞ u+ct δ(u) = fT(1) ,X(1) (t, x)δ(u + ct − x) dx dt. 0
(7)
0
This is infact a generalization of the Dickson’s (1998) result. We can substitute the expression given in the appendix for the joint distribution of T(1) , X(1) to obtain the following δ(u) =
∞
t=0
m u+ct x=0
i=1
βini fX (x) e−βt (ni − 1)! i
m∗ −(n i −1)
bk (i)t ni −1+k δ(u + ct − x) dx dt.
(8)
k=0
Let us substitute s = u + ct in the integral and rearrange it slightly to obtain ni −1+k m∗ −(n m ∞ s ni i −1) βi s−u ds . bk (i) e−βu/c δ(u) = fXi (x)δ(s − x) dx e−βs/c c (n − 1)! c i u x=0 i=1
(9)
k=0
For notational convenience, define s hi (s) = fXi (x)δ(s − x) dx 0
and thus (9) becomes e
−βu/c
δ(u) =
m i=1
βini (ni − 1)!
m∗ −(n i −1) k=0
bk (i) cni +k
∞ u
e
−βs/c
(s − u)
ni −1+k
hi (s) ds .
(10)
We need to differentiate this equation with respect to u to eliminate the integral. From Lemma A.1 in Appendix A we see that if we differentiate the integral ∞ e−β(s/c) hi (s)(s − u)ni −1+k ds, k = 0, 1, 2, . . . , m∗ − (ni − 1) u
with respect to u, ni + k times it becomes (−1)ni +k (ni − 1 + k)! e−β(u/c) hi (u).
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Therefore, by differentiating (10) with respect to u, m∗ + 1 time we get D
m∗ +1
[e
−β(u/c)
δ(u)] =
m i=1
βini (ni − 1)!
m∗ −(n i −1)
bk (i)
k=0
(ni − 1 + k)! m∗ −(ni −1)−k −β(u/c) D [e hi (u)], (−c)ni +k
(11)
where D is the differential operator d/du. By using the standard properties of Laplace transforms, we have ∗
s
m∗ +1
L{e
−β(u/c)
m
βini
δ(u)} −
m
∗ −l
sm
Dl [e−β(u/c) δ(u)]|u=0
l=0
=
i=1
−
m∗ −(n i −1)
(ni − 1)!
m i=1
bk (i)
k=0 m∗ −(n i −1)
βini
(ni − 1)!
k=0
(ni − 1 + k)! m∗ −(ni −1)−k [s L{e−β(u/c) hi (u)}] (−c)ni +k
(ni − 1 + k)! bk (i) (−c)ni +k
m∗ −(n i +k)
∗ −n −k−l i
sm
Dl [e−β(u/c) hi (u)]|u=0 .
(12)
l=0
Let us define the Laplace transform of δ(u), fXi (x) and hi (x) as δ∗ (s), fX∗ i (s) and h∗i (s), respectively. An additional property of Laplace transforms gives us β β β β −β(u/c) ∗ −β(u/c) ∗ ∗ ∗ δ(u)} = δ s + L{e , L{e hi (u)} = hi s + =δ s+ fXi s + . c c c c Substituting these results in (12) and rearranging we obtain β Numer(s) ∗ δ s+ = , c Denom(s)
(13)
where Numer(s) and Denom(s) are functions of s in the following form Numer(s) =
m i=1
−
βini (ni − 1)!
m∗
∗ −l
sm
m∗ −(n i −1) k=0
(ni − 1 + k)! bk (i) (−c)(ni + k)
m∗ −(n i +k)
∗ −n −k−l i
sm
Dl [e−β(u/c) hi (u)]|u=0
l=0
Dl [e−β(u/c) δ(u)]|u=0 ,
l=0
Denom(s) =
m i=1
m∗ −(n i −1) βini β (ni − 1 + k)! m∗ −(ni −1)−k ∗ ∗ s+ bk (i) s − sm +1 . f c (ni − 1)! Xi (−c)ni +k k=0
The function Numer(s) is a l
D [e
−β(u/c)
hi (u)]|u=0
m∗
degree polynomial of s and its coefficients depend on constants
and
Dl [e−β(u/c) δ(u)]|u=0 ,
l = 0, 1, 2, . . .
yet to be determined. Therefore, we could write Numer(s) as ∗
Numer(s) =
m
a¯ i si
i=0
where a¯ i , i = 0, 1, 2, . . . are coefficient to be detemined with a¯ 0 = 0. By substituting s by s − (β/c) in (13) we could write the Laplace transform of δ(u) as m∗ a¯ i (s − (β/c))i δ∗ (s) = i=0 . (14) Denom(s − (β/c))
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Since the numerator of (14) is simply a polynomial of s of order m∗ we may write (14) as m∗ i i=0 ai s ∗ . δ (s) = Denom(s − (β/c))
435
(15)
Let us simplify the function Denom(s − (β/c)). ∗ ∗ m∗ −(n m i −1) βini β m −(ni −1)+k β (ni − 1 + k)! β m +1 ∗ s− Denom s − = bk (i) − s− f (s) c (ni − 1)! Xi c c (−c)ni +k i=1 k=0 ∗ m∗ −(n m n i −1) β m +1 βi i (ni + k − 1)! ∗ = s− bk (i) − 1 . fXi (s) c (ni − 1)! (β − sc)ni +k i=1
k=0
From (A.7) in Appendix A we have ∗ ∗ β β m +1 β m +1 Denom s − = s− MY (s) − s − . c c c At first glance it appears β/c as, m∗ + 1 order, zero of Denom(s − (β/c)); however, the moment generating function ∗ ∗ of Y , MY (s), contains (β − sc)−m −1 . Therefore, multiplying MY (s) by the factor (s − (β/c))m +1 removes its m∗ + 1 order pole at s = β/c. Using these results the Laplace transform of the survival probability δ(u) becomes m∗ i i=0 ai s ∗ δ (s) = . (16) ∗ (s − (β/c))m +1 (MY (s) − 1) As in Dickson (1997) if we define L as the maximal loss random variable we can show that E[e−sL ] = sδ∗ (s). By considering lims→0 in this equation we see that ∗ i s m i=0 ai s lim = 1. (17) ∗ s→0 (s − (β/c))m +1 (MY (s) − 1) ∗ i Obviously, lims→0 , s m i=0 ai s = 0 and lim s→0 MY (s) − 1 = 0. Therefore, we need to use l’Hopital’s rule to evaluate the limit in (17). It can be easily established that m∗
d i lim s ai s = a0 . s→0 ds i=0
Since
∗ ∗ ∗ d β m β m +1 d β m +1 (MY (s) − 1) = (m∗ + 1) s − (MY (s) − 1) + s − s− MY (s) ds c c c ds
and lim
d
s→0 ds
MY (s) = E[Y ] = cE[T(1) ] − E[X(1) ]
from the l’Hopital’s rule we have ∗ β m +1 a0 = − [cE[T(1) ] − E[X(1) ]]. c
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Let us look at the denominator of δ∗ (s) again. From the part 5 of Theorem A.1 in Appendix A we know that denominator has at least m∗ positive zeros. These zeros would yield terms such as exp(Ru) with R > 0 in δ(u) and hence in ψ(u). But from the Cramer’s upper bound for ruin probabilities we know that ψ(u) < exp(−Ru) with R > 0. The only way to resolve this contradiction is that the polynomial ∗
m
ai s i
i=0
having same zeros as the positive zeros of the denominator of δ∗ (s). Since we have determined a0 already, we should have m∗ m∗ s ai si = a0 1− , si i=0
i=1
where si = 0, i = 1, 2, . . . , m∗ are the roots of ∗ β m +1 s− (MY (s) − 1) = 0 c inside the circle |s − (β/c)| = β/c. Therefore we have the main result in the following form: ∗ [cE[T(1) ] − E[X(1) ]] m i=1 (1 − (s/si )) ∗ . δ (s) = ∗ +1 m (1 − (sc/β)) (MY (s) − 1)
(18)
This is the extension of the formula for a single portfolio given in Dickson (1998) and Kalashnikov (1998) for the multiple portfolio case.
3. Numerical examples Example 3.1. Consider the case where
1 0 1 A= , ki (t) = 4t e−2t , 0 1 1
i = 1, 2, 3,
fXi (x) = 4x e−2x ,
i = 1, 2,
i.e. we have two correlated total claim processes; in each process claims occur as Erlang process with parameters (α = 4, β = 2). The moment generating function of the joint inter-arrival time distribution is in the form: 2 2 2 2 2 2 MN1 ,N2 (s1 , s2 ) = , s1 + s2 < 2. 2 − s1 2 − s2 2 − s 1 − s2 The claim size distribution is gamma with parameters (α = 2, β = 2) in each process. Using these values in the accompanying Maple program we get cE[T(1) ] − E[X(1) ] =
13 27 c
− 43 .
Therefore, the condition for positive security loading would be c > 36/13. If we use c = 4, for example, the Maple program gives the following survival probability: δ(u) = 1 + [0.020086 cos (0.76462u) + 0.044244 sin (0.76462u)] exp(−2.0824u) +0.026492 exp(−2.8153u) − 0.60916 exp(−0.50516u).
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Example 3.2. 1 A = 0 0
Consider the case where 0 0 1 2i t i−1 −2t 1 0 1 , ki (t) = e , (i − 1)! 0 1 1
i = 1, 2, 3, 4,
fXi (x) =
2(2x)3−i −2x e , (3 − i)!
437
i = 1, 2, 3.
In this case we have three correlated total claim processes; in the ith process claims occur as Erlang process with parameters (α = i + 4, β = 2) for i = 1, 2, 3. The moment generating function of the joint inter-arrival time distribution is as follows: 1 2 3 4 2 2 2 2 MN1 ,N2 ,N3 (s1 , s2 , s3 ) = , s1 + s2 + s3 < 2. 2 − s1 2 − s2 2 − s3 2 − s 1 − s2 − s3 By executing the accompanying Maple program with appropriate parameters we get that the appropriate security loading is present as c > 3704/945. If we take c = 5 we get the following survival probability δ(u) = 1 + [0.027762 sin (0.81114u) + 0.037630 cos (0.81114u)] exp(−2.5814u) −0.78347 exp(−0.27498u).
Acknowledgements We are grateful to the anonymous referees and the Managing editor Rob Kaas for their valuable suggestions to an earlier version of this paper.
Appendix A Lemma A.1. Let ∞ Gn (u) = (s − u)n g(s) ds, u
n = 0, 1, . . .
(A.1)
Then n + 1th derivative of Gn (u) with respect to u is given by dn+1 Gn (u) = (−1)n+1 n!g(u). dun+1 Proof. Let us differentiate (A.1) once with respect to u: ∞ ∞ d n−1 n Gn (u) = − n(s − u) g(s) ds − (u − u) g(u) = − n(s − u)n−1 g(s) ds = −nGn−1 (u). du u u Differentiating again with respect to u we have d2 d Gn (u) = −n Gn−1 (u) = (−1)2 n(n − 1)Gn−2 (u). 2 du du Therefore, with mathematical induction we could prove dn Gn (u) = (−1)n n!G0 (u). dun Note that the derivative of G0 (u) is −g(u), hence the proof.
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Theorem A.1. Let (T1 , X1 ), (T2 , X2 ), . . . , (Tm , Xm ) be m pair of independent random variables with following properties: 1. Ti ’s and Xi ’s are independent for i = 1, 2, . . . , m. 2. The probability density function of the Ti is given by ki (t) =
βini t ni −1 e−βi t, (ni − 1)!
t ≥ 0, i = 1, 2, . . . , m,
where the parameter ni is an integer and βi > 0. 3. The probability functions of Xi takes the form fXi (x). The random variable T(1) is the minimum value of [T1 , T2 , . . . , Tm ] and the corresponding X value is X(1) . Then we have 1. The joint probability density function of T(1) and X(1) takes the following form: fT(1) ,X(1) =
m i=1
βini fX (x) e−βt (ni − 1)! i
m∗ −(n i −1)
bk (i)t ni −1+k .
(A.2)
k=0
2. The marginal distributions of T(1) and X(1) takes the form: m
fT(1) (t) =
i=1
fX(1) (t) =
βini e−βt (ni − 1)!
m i=1
m∗ −(n i −1)
βini fX (x) (ni − 1)! i
bk (i)t ni −1+k ,
(A.3)
k=0 m∗ −(n i −1)
bk (i)
k=0
(ni − 1 + k)! . βni +k
(A.4)
3. The means of T(1) and X(1) takes the following form: m
E[T(1) ] =
i=1
E[X(1) ] =
m i=1
βini (ni − 1)!
m∗ −(n i −1)
bk (i)
k=0
βini E[Xi ] (ni − 1)!
(ni + k)! , βni +k+1
m∗ −(n i −1) k=0
bk (i)
(ni − 1 + k)! . βni +k
(A.5)
(A.6)
4. The moment generating function of the random variable Y = cT(1) − X(1) takes the following form: MY (s) =
m i=1
βini fX∗ i (s)
m∗ −(n i −1) k=0
bk (i)
(ni + k − 1)! 1 . (ni − 1)! (β − sc)ni +k
5. The following function of s ∗ β m +1 s− (MY (s) − 1) = 0 c
(A.7)
(A.8)
has m∗ roots (counting multiplicities) for s with s = 0 inside the circle where the center is β/c, 0 and the radius is β/c in the complex plane provided E[Y ] > 0. The new symbols defined in these equations are as follows: ∞ fX∗ i (s) = exp(−sx)fXi (x) dx, i = 1, 2, . . . , m, 0
(A.9)
R.S. Ambagaspitiya / Insurance: Mathematics and Economics 32 (2003) 431–443
m∗ =
m
ni − m,
439
(A.10)
i=1
β=
m
βi ,
(A.11)
i=1 k
b0 (i) = 1,
bk (i) =
m βj j j=1 j=i
for k = 1, 2, . . . , m∗ − (ni − 1)
kj !
where 0 ≤ kj ≤ nj , j = 1, 2, . . . , m, j = i and of kj s; i.e. the summation contains ∗ m − (ni − 1) k
(A.12)
kj = k and the sum in (A.12) is over all possible combinations
terms. Proof. Note that ki (t)P[T1 > t, T2 , . . . , Ti−1 > t, Ti+1 > t, . . . , Tm > t] dt is the probability that the ith random variable is the smallest in the interval [t, t+dt) for i = 1, 2, . . . , m. Let us simplify this term. Since Ti , i = 1, 2, . . . , m are independent we have ki (t)P[T1 > t, T1 , . . . , Ti−1 > t, Ti+1 > t, . . . , Tm > t] = ki (t)
m
P[Tj > t] =
j=1 j=i
=e
−βt
j −1 m n βini (βj t)k −βj t t ni −1 e−βi t e (ni − 1)! k!
(A.13)
j=1 k=0 j=i
j −1 m n βini (βj t)k ni −1 t , (ni − 1)! k!
(A.14)
j=1 k=0 j=i
where β = m i=1 βi . Closer inspection of (A.14) reveals that ki (t)
m j=1 j=i
j P[T1
βini > t] = e−βt (ni − 1)!
m∗ −(n i −1)
bk (i)t ni −1+k ,
(A.15)
k=0
where m∗ , bk (i), k = 0, 1, . . . , m∗ as defined in (A.10)–(A.12). Since the joint pdf of T(1) and X(1) , fT(1) ,X(1) (t, x) is fT(1) ,X(1) (t, x) =
m
ki (t)P[T1 > t, T2 > t, . . . , Ti−1 > t, Ti+1 > t, . . . , Tm > t]fXi (x)
(A.16)
i=1
and hence the proof of (A.2). The marginal pdf of T(1) and X(1) can be obtained by integrating (A.2) with respect to x and t respectively. Hence the proof of (A.3) and (A.4). Once we have the marginal pdfs we can verify the results in (A.5) and (A.6) for E[T(1) ] and E[X(1) ] easily.
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The proof of (A.7) takes little effort. The joint moment generating function of T(1) , X(1) can be derived as MT(1) ,X(1) (s1 , s2 ) = E[e =
m i=1
=
s1 T(1) +s2 X(1)
βini (ni − 1)!
m i=1
βini
(ni − 1)!
]=
t=0 x=0
∞ x=0
∞ ∞
e
s2 x
es1 t+s2 x fT(1) ,X(1) (t, x) dt dx
fXi (x)
m∗ −(n i −1) k=0
MXi (s2 )
m∗ −(n i −1) k=0
bk (i)
bk (i)
∞ t=0
es1 t e−βt t ni −1+k dt
(ni + k − 1)! . (β − s1 )ni +k
Hence the moment generating function, MY (s), of Y = cT1 − X1 can be derived as follows: MY (s) = E[e
(cT1 −X1 )s
] = MT(1) ,X(1) (sc, −s) =
βini
m∗ −(n i −1)
m i=1
=
m i=1
(ni − 1)!
fX∗ i (s)
k=0
bk (i)
βini MXi (−s) (ni − 1)!
m∗ −(n i −1) k=0
bk (i)
(ni + k − 1)! (β − sc)ni +k
(ni + k − 1)! . (β − sc)ni +k
The last step follows from the fact that the equivalency of moment generating functions and Laplace transforms. To prove (A.8) consider the circle C s − β = β c c in the complex plane. We can denote the points on the circle C as
β β β + cos θ, sin θ . c c c
In the derivation of the joint moment generating function of T(1) , X(1) we made the implicit assumption that the integral
∞
t=0
es1 t e−βt t ni −1+k dt
converges. However it converges to (ni + k − 1)! (β − s1 )ni +k provided β − s1 > 0. Therefore, the expression for the moment generating function MY (s), of Y given in (A.7) is valid for s < β/c. But close inspection of MY (s) reveals that in fact we could write the following for the value of it on the circle C. ∗
m −(n m i −1) βini −1 (ni + k − 1)! MY (s) = bk (i) , fX∗ i (s) n +k (ni − 1)! (−β) i ( cos θ + I sin θ)ni +k i=1
k=0
R.S. Ambagaspitiya / Insurance: Mathematics and Economics 32 (2003) 431–443
where I =
√
441
−1 ∗
m −(n m i −1) βini −1 (ni + k − 1)! |MY (s)| ≤ |fX∗ i (s)| bk (i) n +k , i (ni − 1)! β |( cos θ + I sin θ)|ni +k i=1
|MY (s)| ≤
m i=1
k=0
βini −1
(ni − 1)!
|fX∗ i (s)|
m∗ −(n i −1) k=0
bk (i)
(ni + k − 1)! . βni +k
If fX∗ i (s) exist on the circle C then |fX∗ i (s)| ≤ 1, for i = 1, 2, . . . , m. ∞ ∞ m −(n m i −1) βini −1 (ni + k − 1)! |MY (s)| ≤ bk (i) = fT(1) ,X(1) (t, x) dt dx = 1 (ni − 1)! βni +k t=0 x=0 ∗
i=1
k=0
Therefore on the circle C for * > 0 ∗ ∗ β m +1 β m +1 MY (s) < s − s− (1 + *). c c ∗
∗
From the Rouche’s theorem we can conclude that (s−(β/c))m +1 (MY (s)−1−*) and −(1+*)(s−(β/c))m +1 have ∗ the same number of zeros, i.e. m∗ + 1 zeros, inside the circle C. Since s = 0 is a zero of (s − (β/c))m +1 (MY (s) − 1) ∗ +1 m (MY (s) − 1 − *) has a zero at s ≈ (*/E[Y ]). Since with multiplicity of 1, the perturbed function (s − (β/c)) E[Y ] > 0 this is inside the circle C. If * is sufficiently small this will be away from the rest of the roots; therefore we can conclude the proof by * → 0. Appendix B. Maple program The following is a listing of the Maple program that was used in Example 3.2. This program is available as a text file in “http://www.math.ucalgary/∼ sarath/papers/mapleprogram1.txt”. with(inttrans); m := 4; p := 3; # beta and ni are parameters of gamma processes beta := array(1 . . m, [2, 2, 2, 2]); ni := array(1 . . m, [1, 2, 3, 4]); A := array([[1, 0, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1]]); # Laplace transforms of claim sizes distributions #fxd are the Laplace transforms of claim sizes distributions fxd := array([(2/(2 + s))∧ 3, (2/(2 + s))∧ 2, 2/(2 + s)]); #Default value of premium rate is 2. C := 2; #For different cases one may change m, p, beta, ni, A, fxd and C #appropriately #Anything below this line should not be changed # # Use equation(6) to compute the Laplace transforms # #
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fx := array(1 . . m); for i from 1 to m do fx[i] := product(‘fxd[j]∧ A[j, i]’, ‘j’ = 1 . . p) od : #mst is the value defined in (A.9) mst := sum(‘ni[i]−i’, ‘i’ = 1 . . m); # to extract coefficients in (A.10), (A.11) use explicit multiplication and # simplification bki := array(0 . . mst, 1 . . m); # Bet is beta in the paper Bet := sum(‘beta[i]’, ‘i’ = 1 . . m); CDFPRD := product(‘sum(‘(beta[j]∗t)∧ k/k!’, ‘k’ = 0 . . ni[j]−1)’, ‘j’ = 1 . . m); for i from 1 to m do CDI := sum(‘(beta[i]∗t)∧ k/k!’, ‘k’ = 0 . . ni[i]−1); TEMP := collect(simplify(CDFPRD/CDI), t); for k from 0 to mst((ni[i]−1) do bki[k, i] := coeff(TEMP, t, k); od : od : # MY is the moment generating function of Y MY := sum(‘beta[i]∧ ni[i]/(ni[i]−1)!∗fx[i]∗(sum(‘bki[k, i]∗(ni[i] + k−1)!/ (Bet−s∗c)∧ (ni[i] + k)’, ‘k’ = 0 . . mst−ni[i] + 1))’, ‘i’ = 1 . . m); # ETX is the expected value of Y ETX := eval(limit(diff(MY, s), s = 0)); # If the security loading is not sufficient use the smallest possible #value + 1 as the security loading. ETX1 := subs(c = C, ETX); if ETX1<0then C := round(solve(ETX = 0, c)) + 1; print(“Given premium rate is too low”); print(“C = ”, C, “is selected”); fi; # Separate the numerator and denominator of the denominator in (17) # We are using the fact that $s = 0$ is a root of the denominator of (17). DNM := numer(simplify((1−s∗C/Bet)∧ (mst + 1)∗(subs(c = C, MY)−1)/s)); DDM := denom(simplify((1−s∗C/Bet)∧ (mst + 1)∗(subs(c = C, MY)−1))); #Make the accuracy of floating point calculations to 100 digits Digits := 100; # Compute the roots inside the contour and in the first quadrant # if a root is complex its conjugate is also a root. AIS := evalf(allvalues(RootOf(DNM, s, 0 . . 2∗Bet/C∗(1 + I)))); PRD := 1; for x in AIS do if Im(x)< > 0 then PRD := PRD∗(1 + (s∧ 2−2∗Re(x)∗s)/abs(x)∧ 2); else PRD := PRD∗(1−s/x); fi
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od : DNM1 := evalf(s∗DNM/PRD); # DS is Laplace transform of the survival probability. DS := simplify(subs(c = C, ETX)∗DDM/DNM1); DEL := collect(simplify(evalf(invlaplace(DS, s, u))), [exp, sin, cos]); #For final results round everything to 5 digits. Digits := 5; # Clean up the expression by deleting the terms with # small coefficients as a result of floating point computations DEL := fnormal(evalf(DEL)); print(DEL); References Ambagaspitiya, R.S., 1998. On the distribution of a sum of correlated aggregate claims. Insurance: Mathematics and Economics 23, 15–19. Ambagaspitiya, R.S., 1999. On the distributions of two classes of correlated aggregate claims. Insurance: Mathematics and Economics 24, 301–338. Dickson, D.C.M., 1998. On a class of renewal risk process. North American Actuarial Journal 2 (3), 60–68. Dickson, D.C.M., Hipp, C., 1998. Ruin Probabilities for Erlang(2) risk process. Insurance: Mathematics and Economics 22, 251–262. Kalashnikov, V., 1998. The discussion of the paper “On a class of renewal risk process by David C.M. Dickson”. North American Actuarial Journal 2 (3), 70–71. Kotz, S., Balakrishnan, N., Johnson, N.L., 2000. Continuous Multivariate distributions, vol. 1, Models and Applications, Wiley, New York. Tiecher, H., 1954. On the multivariate Poisson distribution. Skandinavisk Aktuarietidskrift 37, 1–9.