VOLUME 14 issue 2 australian actuarial journal 2008 - Institute of

114 Pages·2008·4.34 MB·English

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Feature Articles
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The Moments of the Present Value of Total Dividends in the Compound
Binomial Model Under a Constant Dividend Barrier and Stochastic Interest 175
Rates – S Li
Risk Equalisation by Mixed Schemes – W Neuhaus 193
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On the Use of MCMC Simulation for Stochastic Reserving – J Li 227
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Mission Statement
The Australian Actuarial Journal is committed to promoting the work of actuaries, researchers and
other experts in financial risk assessment, with a particular focus on Australian and Asia-Pacific issues.
The focus of the Journal is in publishing contributions on:
Editor
• Original research in actuarial science, finance and investment, and related areas.
Andrew P Leung MSc PhD FIA FIAA • Applications of theory that are of interest to practitioners.
• Surveys or reviews of current research that may appear in the literature.
• Significant presentations at the Institute’s public forums.
Associate Editors All published papers, other than those that report the Institute’s proceedings, are subject to a rigorous
process of peer review. Contributions that appear as Feature Articles are assessed by two independent
reviewers. Contributions that appear as Notes may be reviewed by members of the Editorial
Anthony Asher BBusS FIA FIAA Geoff Morley BSc BCom FIAA Committee.
Rade T Musulin BA Sachi Purcal Bec MA
The Journal is published in hardcopy edition on a quarterly basis, or at such frequency appropriate
Rob Thomson PhD FIA to the submission of quality contributions. Papers published in the Journal are also accessible in
electronic format from the Institute’s website at www.actuaries.asn.au.
Contributions from prospective authors are invited in the areas of the Journal’s focus. Full authoring
guidelines may be downloaded from the Institute’s website.
Editorial Policy
The Australian Actuarial Journal publishes papers of interest to actuaries and other experts in financial
risk, with a particular focus on Australian, New Zealand and Asian topics. All papers which appear in
the Feature Article section of the Journal, with the exception of the annual presidential addresses, are
assessed by two independent reviewers. The Notes section of the Journal contains less weighty items,
which are reviewed by members of the editorial team.
Guidelines for Intending Authors
ISSN 1442-3065
The editor welcomes contributions to the Journal, either as Feature Articles or as Notes. Papers may
© Institute of Actuaries of Australia 2008
be practical or theoretical in nature. Full guidelines for authors are available from the Institute of
Actuaries of Australia or can be downloaded from the Institute’s website www.actuaries.asn.au
The Australian Actuarial Journal is published by
The Institute of Actuaries of Australia
Level 7 Challis House Subscriptions
4 Martin Place Sydney NSW 2000 Australia
The Australian Actuarial Journal is published bi-annually and is issued to members of the Institute of
Telephone: [phone] Actuaries of Australia. Subscription rates for non-members are AUS$100 per year, including postage
and handling. For subscription inquiries, contact the:
www.actuaries.asn.au
Membership Coordinator
For subscription information and guidelines for authors please see inside of back cover. Institute of Actuaries of Australia
Level 7 Challis House
The opinions expressed in this journal are not necessarily those of the Editor, the 4 Martin Place Sydney NSW 2000 Australia
Telephone: [phone]
Associate Editors, or the Institute of Actuaries of Australia. Email: [email protected]
The Editor, the Associate Editors and the Institute of Actuaries of Australia do not
accept any responsibility for those opinions or for the accuracy of any statements
appearing in the journal.
Mission Statement
The Australian Actuarial Journal is committed to promoting the work of actuaries, researchers and
other experts in financial risk assessment, with a particular focus on Australian and Asia-Pacific issues.
The focus of the Journal is in publishing contributions on:
Editor
• Original research in actuarial science, finance and investment, and related areas.
Andrew P Leung MSc PhD FIA FIAA • Applications of theory that are of interest to practitioners.
• Surveys or reviews of current research that may appear in the literature.
• Significant presentations at the Institute’s public forums.
Associate Editors All published papers, other than those that report the Institute’s proceedings, are subject to a rigorous
process of peer review. Contributions that appear as Feature Articles are assessed by two independent
reviewers. Contributions that appear as Notes may be reviewed by members of the Editorial
Anthony Asher BBusS FIA FIAA Geoff Morley BSc BCom FIAA Committee.
Rade T Musulin BA Sachi Purcal Bec MA
The Journal is published in hardcopy edition on a quarterly basis, or at such frequency appropriate
Rob Thomson PhD FIA to the submission of quality contributions. Papers published in the Journal are also accessible in
electronic format from the Institute’s website at www.actuaries.asn.au.
Contributions from prospective authors are invited in the areas of the Journal’s focus. Full authoring
guidelines may be downloaded from the Institute’s website.
Editorial Policy
The Australian Actuarial Journal publishes papers of interest to actuaries and other experts in financial
risk, with a particular focus on Australian, New Zealand and Asian topics. All papers which appear in
the Feature Article section of the Journal, with the exception of the annual presidential addresses, are
assessed by two independent reviewers. The Notes section of the Journal contains less weighty items,
which are reviewed by members of the editorial team.
Guidelines for Intending Authors
ISSN 1442-3065
The editor welcomes contributions to the Journal, either as Feature Articles or as Notes. Papers may
© Institute of Actuaries of Australia 2008
be practical or theoretical in nature. Full guidelines for authors are available from the Institute of
Actuaries of Australia or can be downloaded from the Institute’s website www.actuaries.asn.au
The Australian Actuarial Journal is published by
The Institute of Actuaries of Australia
Level 7 Challis House Subscriptions
4 Martin Place Sydney NSW 2000 Australia
The Australian Actuarial Journal is published bi-annually and is issued to members of the Institute of
Telephone: [phone] Actuaries of Australia. Subscription rates for non-members are AUS$100 per year, including postage
and handling. For subscription inquiries, contact the:
www.actuaries.asn.au
Membership Coordinator
For subscription information and guidelines for authors please see inside of back cover. Institute of Actuaries of Australia
Level 7 Challis House
The opinions expressed in this journal are not necessarily those of the Editor, the 4 Martin Place Sydney NSW 2000 Australia
Telephone: [phone]
Associate Editors, or the Institute of Actuaries of Australia. Email: [email protected]
The Editor, the Associate Editors and the Institute of Actuaries of Australia do not
accept any responsibility for those opinions or for the accuracy of any statements
appearing in the journal.
2008 VOLUME 14
issue 2
Feature Articles
The Moments of the Present Value of Total Dividends in the Compound
Binomial Model Under a Constant Dividend Barrier and Stochastic Interest 175
Rates – S Li
Risk Equalisation by Mixed Schemes– W Neuhaus 193
On the Use of MCMC Simulation for Stochastic Reserving – J Li 227
Feature Articles
2008 VOLUME 14
issue 2
australian actuarial journal 2008 Volume 14 Issue 2 pp. 175–192
The Moments of the Present Value of Total
Dividends in the Compound Binomial Model
Under a Constant Dividend Barrier and
Stochastic Interest Rates
S Li*
Abstract
We study the moments of the present value of total dividend payments in the
compound binomial risk model in the presence of a constant dividend barrier.
In the evaluation of the present value of dividends, the interest rates are
assumed to follow a Markov chain with finite state space. A system of difference
equations with certain boundary conditions for the moments of the present
value of total dividend payments prior to ruin is derived and solved. Explicit
results can be obtained when the claim sizes are rationally distributed or the
claim-size distributions have finite support. Numerical results are given for two
special claim-size distributions: the geometric distribution and the degenerate
distribution at 2.
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Keywords: Compound binomial risk model; Discounted dividend ustr
A
payments; Stochastic interest rate; Markov chain; Time of of
ruin es
ari
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ct
A
*Contact: Shuanming Li , Center for Actuarial Studies, Department of Economics e of
The University of Melbourne, Victoria 3010. Email: [email protected] ut
nstit
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australian actuarial journal
1 Introduction
Consider the classical compound binomial risk model
N(n)
U(n)=u+n− ∑ X , n=1,2,…,
i
i=1
where u∈N is the initial surplus. TheX ’s are i.i.d. positive integer-
i
valued random variables with common probability function (p.f.)
f(x)=P(X =x), for x=1,2,…, denoting claim sizes. Denote by
µ =E[Xk] the k-th moment of X and by fˆ(s)=∑∞ sif(i),s∈C, its
k i=1
probability generating function (p.g.f.). Note that we adopt the standard
assumption in the compound binomial model that the premium is 1 per
unit time.
The counting process {N(n);n∈N} denotes the number of claims up
to time n and is defined as N(n)=max{k:H +H ++H ≤n}, where
1 2 k
the claim inter-arrival times H ’s are assumed to be i.i.d. positive
i
integer-valued random variables with common probability function .
h(x)=q(1−q)x−1, for x=1,2,… and 0<q<1, that is to say, the probability
of having a claim in each period of time is q and the probability of
having no claims is 1−qFurther assume that {H;i∈N+} and {X ;i∈N+}
i i
are independent, and E(H )=(1+η)E(X )=(1+η)µ, or 1=(1+η)qµ
1 1 1 1
, where η is the positive loading factor. The compound binomial risk
model, as a discrete analogue of the classical compound Poisson risk
model, was first proposed by Gerber (1988) and further studied in
a number of papers, see Pavlova and Willmot (2004) and references
therein for details. Extensions to the compound binomial model can be
found in Yuen and Guo (2001), Cossette, Landriault and Marceau (2003)
and Li (2005a, b).
8 In this paper, a barrier strategy is considered by assuming that there
0
0
a 2 is a horizontal barrier of level b≥u and b∈N+: at the end of each
ustrali period (immediately after possible claim settlement), when the surplus
of A is greater than level b, dividends are paid such that the surplus stays at
es level b until it becomes less than b by a claim. It implies that a dividend
ari
ctu of 1 is payable at time n only if the surplus was at b at time n−1 and
A
e of there are no claims at time n, for n=1,2,…. Let Ub(n) be the modified
ut surplus process with initial surplus U (0)=u under the above barrier
nstit strategy. Then b
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The Moments of the Present Value of Total Dividends
N(n)
U (n)=u+n− ∑ X −D(n), n=1,2,…, (1)
b i
i=1
where D(n) is the sum of the total dividend payments in the first n
periods, with the definition
n
D(n)=∑D,
i
i=1
with
D =max{U (i)−b,0}
i b
being the amount of dividend paid out at the end of period i. Define
T =inf{n∈N:U (n)<0}
u,b b
to be the time of ruin. See Figure 1 below for a typical sample path of
the surplus process {U (n):n∈N}.
b
Figure 1. A sample path for the surplus process U (n)
b
8
The barrier strategy was initially proposed by De Finetti (1957) for a 00
2
binomial model. More general barrier strategies for a compound Poisson alia
risk process have been studied in a number of papers and books. The ustr
A
main focus is on optimal dividend payouts and problems associated with of
es
the time of ruin, under various barrier strategies and other economic ari
u
conditions. Act
of
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nstit
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australian actuarial journal
Claramunt, Mármol and Alegre (2003) calculate the expected
present value of dividends in a discrete time risk model with a barrier
dividend strategy. Dickson and Waters (2004) show how to use the
compound binomial risk model to approximate the classical compound
Poisson risk model in calculating the moments of discounted dividend
payments. Tan and Yang (2006) study the ruin probability and the
distribution of the surplus before ruin and the deficit at ruin in the
compound binomial risk model with randomized decisions on paying
dividends in which the dividend of 1 unit is paid with certain probability
when the surplus process is greater than a constant barrier. Landriault
(2008) studies the expected discounted penalty function for the
compound binomial risk model under the Markov chain interest rates.
In the actuarial literature, it is well known that the optimal dividend
barrier level which maximizes the expectation of the discounted
dividends until the time of ruin is independent of the initial surplus in
the classical compound Poisson risk model and its discrete counterpart,
the compound binomial risk model, if the force of interest or the
discount factor per period is assumed to be a constant.
In this paper, we assume that the interest rates {R ,n∈N} with
n
R being the interest rate in the interval (n,n+1] follow a time
n
homogenous Markov chain with finite state space {r,r,…,r }. The one
1 2 m
period transition probability matrix is given by P=(p )m , where
i,j i,j=1
p =P(R =r |R =r), for n∈N. The one period discount factors are
i,j n+1 j n i
denoted by v,v ,…, respectively, where v =1/(1+r).
1 2 i i
Under the interest rate model described above, the present value of
total dividends until the time of ruin T given that the initial surplus is
u,b
u is denoted by
8
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20 Tu,b  k 1 
a D =∑D ∏ , u=0,1,2,…,b.
ustrali u,b k=1 ki=11+Ri
A
of
aries Define
u
ct
of A Vi,n(u;b)=E[Dun,b|R0=ri], n∈N+,i=1,2,…,m,u=0,1,2,…,b,
e
ut
nstit
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The Moments of the Present Value of Total Dividends
to be the n-th moment of the present value of dividend payments up to
the time of ruin, given that the initial interest rate R =r.
0 i
The moments and distributions of present values of stochastic or
deterministic payment streams under stochastic interest rates have
been studied in many actuarial papers. Pollard (1971) was the first to
consider the problem of finding the distribution of a life annuity when
both mortality and the discount rates are random. Several authors have
studied the same problem since, including Boyle (1976), Waters (1978),
Panjer and Bellhouse (1980), Dufresne (1989, 1990, 1992, 2007), Parker
(1994a, b), Norberg (1995a, b), and references therein. The majority
of the contributions focused on calculating the moments and the
distributions of life annuities and annuities certain. Yuen, Wang and Li
(2007) study the expected discounted dividend payments until the time
of ruin in the compound Poisson risk model with constant interest rates
to the surplus process and a constant dividend barrier.
The aim of this paper is to calculate V (u;b), the moments of the
i,n
present value of a sequence of dividend payments until the time of ruin
under stochastic interest rates to the dividends, for some special claim-
size distributions so to determine whether the optimal dividend barrier
level is still independent of the initial surplus.
2 A system of difference equations with boundary
conditions and its solution
In this section, we will first show that V (u;b),i=1,2,…,m, satisfy
i,n
a system of difference equations with certain boundary conditions.
Then we will solve the system of difference equations with boundary
conditions to express the n-th moment of Du,b in matrix form. 08
0
2
a
Conditioning on the occurrence (or not) of a claim in the first period, ali
the amount of the claim, and the interest rate at time 1, we have for Austr
u=0,1,2,…,b−1 and i=1,2,…,m that es of
ari
u
ct
A
of
e
ut
nstit
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