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Sains Malaysiana
50(7)(2021): 2109-2121
http://doi.org/10.17576/jsm-2021-5007-24
Risk Analysis of the Copula Dependent Aggregate Discounted
Claims with Weibull Inter-Arrival Time
(Analisis
Risiko Agregat Tuntutan Terdiskaun yang Bersandar Secara Kopula dengan antara Waktu Ketibaan
Bertaburan Weibull)
SITI NORAFIDAH MOHD RAMLI1, SHARIFAH FARAH SYED
YUSOFF ALHABSHI1* & NUR ATIKAH MOHAMED ROZALI2
1Department of
Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan
Malaysia, 43600 UKM Bangi, Selangor Darul
Ehsan, Malaysia 2Actuarial Services
Department, Malaysian Reinsurance Berhad, 11th Floor, Bangunan Malaysian Re., No.
17, Lorong Dungun, Damansara Height, 50490 Kuala Lumpur, Federal Territory, Malaysia
Received: 17 July 2020/Accepted:
16 November 2020
ABSTRACT
We model the recursive moments of aggregate discounted
claims, assuming the inter-claim arrival time follows a Weibull distribution to
accommodate overdispersed and underdispersed data set. We use a copula to
represent the dependence structure between the inter-claim arrival time and its
subsequent claim amount. We then use the Laplace inversion via the Gaver-Stehfest algorithm to solve numerically the
first and second moments, which takes the form of a Volterra integral equation
(VIE). We compute the average and variance of the aggregate discounted claims
under the Farlie-Gumbel-Morgenstern (FGM) copula and conduct a sensitivity
analysis under various Weibull inter-claim parameters and claim-size
parameters. The comparison between the equidispersed, overdispersed and
underdispersed counting processes shows that when claims arrive at times that
vary more than is expected, insured lives can expect to pay higher premium, and vice versa for the case of claims arriving at times that vary less than
expected. Upon comparing the Weibull risk process with an equivalent Poisson
process, we also found that copulas with a wider range of dependency parameter
such as the Frank and Heavy Right Tail (HRT), have a greater impact on the
value of moments as opposed to modeling under FGM copula with weak dependence
structure.
Keywords: FGM copula; Gaver-Stehfest algorithm; Laplace
transform; Volterra integral equation; Weibull count model
ABSTRAK
Kajian ini memodelkan momen rekursif tuntutan agregat
terdiskaun, dengan andaian bahawa waktu ketibaan antara tuntutan mengikut
taburan Weibull bagi memenuhi keperluan set data yang terlebih atau terkurang
serak. Kajian ini menggunakan kopula untuk mewakili struktur kebersandaran
antara waktu ketibaan antara tuntutan dan jumlah tuntutan berikutnya. Kajian
ini mengggunakan songsangan Laplace melalui algoritma Gaver-Stehfest untuk
menyelesaikan secara berangka momen pertama dan kedua dalam bentuk persamaan kamiran
Volterra (VIE). Kajian ini menghitung purata dan varians tuntutan agregat
terdiskaun di bawah kopula Farlie-Gumbel-Morgenstern (FGM) dan analisis
kepekaan dijalankan dengan mengubah parameter Weibull, waktu ketibaan antara
tuntutan dan parameter saiz tuntutan. Perbandingan antara proses pengiraan sama
serakan, terlebih serakan atau terkurang serakan menunjukkan bahawa apabila
tuntutan tiba pada waktu yang bervariasi lebih dari yang dijangkakan, pihak
yang diinsurans perlu membayar premium yang lebih tinggi dan sebaliknya bagi
kes tuntutan yang tiba pada waktu yang bervariasi kurang dari yang dijangkakan.
Selain perbandingan proses risiko Weibull dengan proses Poisson yang setara,
kajian ini juga mendapati bawah kopula dengan julat parameter kebersandaran
yang lebih luas seperti Frank dan Heavy Right Tail (HRT), memberi kesan yang
lebih besar kepada nilai momen berbanding di bawah kopula FGM dengan struktur
kebersandaran yang lemah.
Kata kunci: Algoritma Gaver-Stehfest; kopula FGM; model kira
Weibull; persamaan kamiran Volterra; transformasi Laplace
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