isbn prosiding seminar tnasional lt mahasiswa i r 53

ISBN : 978-979-17979-0-0

r' ''

ffiitor:UuslimAnsorilsnrail $akariaDhoriv* Urwatul lfilutsqaAgus Maman AbadiH- Andy Rudhito

$mu So'adahKaryatiltasih Flatiwi

dl

ir(w

Prosiding

Seminar Nasionaltlt I rMahasiswa 53 Matematika

REVITALISA$I DAN $O$IALI$A$I DIRIUNTUK BERPTRAN AKTIF DALAMPENINGKATAN KUALITA$ PTNELITIAN &PENDIDIKAN MATEMATIKA DI INDONESIA

l.i.l '!'i,lrA':q'!irif"

&'*t

::"r, c

:: repository.unisba.ac.id ::

ISBN =

978-979-L7979-O-O

PROSIDINGSEMINAR NASIONALMAHASISWA 53 MATEMATIKA INDONESIA 2OO8

EDITOR:Muslim AnsoriIsmail DjakariaDhoriva Urwatul WutsqaAgus Maman AbadiM. Andy RudhitoUmu Sa'adahKaryatiHasih Pratiwi

PENERBIT:JURUSAN MATEMATIKA FMIPA UGM

ALAMAT:Sekip Utara Unit III YogyakartaTelp. O27 4-552243, Fax. O27 4-555131


KATA PENGANTAR

Puji slukur kami panjatkan kehadirat Allah SWT atas terselenggarakannya Seminar

Nasional Mahasiswa 53 Matematika Indonesia pada tanggal 31 Mei 2008 di Jurusan

Matematika FMIPA UGM Yogyakarta. Kegiatan ini merupakan kegiatan ilmiah rutin yang

diadakan oleh Forum Komunikasi Mahasiswa 53 Matematika lndonesia sejak tahun 2005.

Seminar Nasional ini diselenggarakan dengan tujuan (1) memberikan wawasan dan

berbagi informasi diantara mahasiswa matematika,/statistika dan pendidikan matematika, atau

peminat matematika tentang hasil penelitian yang telah atau sedang dilakukan; (2)

mendorong terciptanya keqasama keilmuan antara mahasiswa 53 matematika./satistika dan

pendidikan matematika, atau antara mahasiswa 53 matematiktstatistika,/pendidikan

matematika dengan para pakar matematika/statistika/pendidikan matematika lainnya di

Indonesia; dan (3) membentuk korps keilmuan untuk pengembangan profesi ke depan dan

menjalin silaturahmi antara sesama mahasiswa dan alumni 33

matematika/statistika/pendidikan maternatika se-lndonesia.

Peserta yang berpartisipasi dalam kegiatan ini be{umlah 90 orang yang berasal dari

berbagai institusi/instansi, yairu ITB, t ry, UPI, UT, LINMER Malang, INMUL, LINNES,

tlM, IPB, UNIMED, USD, ITS, LINSOED, UNTAD, IAIN Antasari Banjarsari, UNISBA,

Universitas Widya Dharma Klaten, UNESA, SMA Sang Timur Yogyakarta, UBAYA,

Universitas Veteran Bangun Nusantara Sukohafo, IINIKA Parahyangan, LINTAD.

Universitas Mahasaraswati, UI, UII, UNPAD, STAIN Purwokerto, IINHALU, UNSRAT,

Universitas PGRI Palembang, UNCEN, UNPAD, STKIP YASIKA Majalengka, UNEJ,

STKIP PGRI Pontianak, UIN Syarif Hidayatullah, IINPATI, STT Garut, UNILA, Universitas

Negeri Gorontalo, SMKN 2 Ygyakarla, BATAN, UBINUS, Universitas Muhammadiyah

Bengkulu, UNAND, SKIP Siliwangi Bandung, Sekolah Tinggi Sandi Negara, dan UGM. Di

antam peserta ini, yang membawakan makalah berjumlah 50 orang. Pembicara utama pada

seminar ini adalah Prof. Dr. Muchlas Samani (Direktur Ketenagaan Ditjen Dikti Depdiknas)

dan Dr. Supama, M.Si. (Peneliti Bidang Analisis Jurusan Matematika FMIPA Universitas

Gadiah Mada).

111


Terselenggaranya Seminar Nasional ini tidak lepas dari dukungan berbagai pihak.

untuk itu terira kasih kami sampaikan kepada peserta Seminar Nasional atas partisipasinya

yang aktif serta kepada Fakultas MIPA UGM dan Jurusan Matematika FMIPA UGM atas

fasilitas dan dukungan dananya. Ucapan terima kasih juga kami sampaikan kepada Forum

Komunikasi Mahasiswa 53 Matematika Indonesia atas dukungan dan publikasinya' Terakhir'

terima kasih juga kami sampaikan kepada rekan-rekan Panitia dan staf Juusan Matematika

FMIPA UGM atas dukungan dan kerjasamanya sehingga Seminar Nasional ini dapat

terselenggara dengan lancar.

Yogyakarta, Mei 2008

Ketua Panitia

iv


SAMBUTAN:PENGELOLA S2lS3 MATEMATIKA FMIPA UGM

Yang terhormat Dekan Fakultas MIPA,Yang terhormat Direktur Ketenagaan Ditjen Dikti Depdiknas,Yang terhormat Staf Pengajar 33 Matematika FMIPA UGM,Yang terhormat Hadirin Peserta Seminar Nasional,Assalamu' alaikum Warahmatullaahi Wabarakaatuh,

Selamat pagi dan salam sejahtera bagi kita semua.Pertama-tama marilah kita memanjatkan puji dan slukur kehadirat Tuhan Yang Maha Kuasakarena atas perkenan-Nya kita dapat berkumpul pada acara seminar yang penting ini, yaituSeminar Nasional Mahasiswa 53 Matematika se-Indonesia. Kami menyambut baik acarayang diprakarsai dan diselenggarakan oleh mahasiswa 53 Matematika karena dapatmendorong terciptanya ke{asama keilmuan antara mahasiswa 53 Matematika./Satistika danPendidikan Matematika, atau antam mahasiswa 33 Matematika/Statistika/PendidikanMatematika dengan para pakar Matematika,/Statistika,/Pendidikan Matematika lainnya diIndonesia.

Pada kesempatan ini, kami mengucapkan selamat mengikuti rangkaian Seminar NasionalMahasiswa 53 Matematika se-Indonesia yang sangai penting ini_ Semoga upaya yang kitalaksanakan ini dapat menghasilkan sumbangan yang berharga dan menjadi bekal bagi kitasemua dalam mencapai masyarakat lndonesia yang andal, cerdas, mempunyai komitmenmoral dan semangat pengabdian.

Lebih jauh lagi, kami mengharapkan Seminar Nasional ini dapat memberikan wawasan danberbagi informasi diantara mahasiswa Matematika,/Statistika dan Pendidikan Matematika,atau Peminat Matematika tentang hasil penelitian yang telah atau sedang dilakukan.

Demikian sambutan ini kami sampaikan, atas perhatian seluruh peserta Seminar Nasional,kami ucapkan terimakasih dan semoga Tuhan Yang Maha Kuasa memberkahi seluruh upayabersama yang terus kila la}<ukan ini.

Was s a I amu' a laikum Wara h ma tu I I a a h i ll/a b a raka atuh.Yogyakarta, 31 Mei 2008

Pengelola S2/S3 Matematika FMIPA UGM


DAFTAR ISI

Makalah Bidang Aljabar

The Sufficient Conditions for R[x,o ] to be a GCD-rinE (Santi lrawati) I

Lapangan yang Memenuhi Sifat Unit I -Stable Range(Indriati Nurul Hidayah) . ... ...... . . ... ... ... ..... I 0

Aljabar Max-Plus Ifiewal (M. Andy Rudhito, Sri Wahyuni, Ari Suparwamto,

F. Susilo, SJ) ...................... 14

Matriks Atas Aljabar Max-Plus Interval (M. Andy Rudhito, Si lYahyuni,

Ari partuamto, F. Susilo, SJ) . . . . . . . . . ..... ... ... ... .. . 23

Syarat Perlu dan Cukup Ideal Prime Terkait (Sfficient and Nessecary Condition ofThe Associated Prime) (Suprapto, Sri Wahyuni, Indah Emilia llijayanti, Irawdti) ... ... 33

Makalah Bidang Analisis

Operator Integral Fraksional dan Ketaksamaan Olsen di Ruang Morrey Tak Homogenyang Diperumum (Idha Sihwaningntm, Hendra Gunawan,

dan Wono Setya Budhi) ..........40Barisan Fibonacci Ditinjau dari Teori Bilangan (Sangadji) ......... 48

Fungsi Riernann ZeIa (Sangadji) ....................... 53

Konvergensi Metode Newton dan Metode Turun Tercuram dalam Menyelesaikan

Sistem Persamaan Non-Linear (Zusla Krismiyati Budiasih) ......... 60

Representation Of SM-Operators on Classical Sequences 4r, 1 < p < * (Muslim Ansoi,Soeparna Datmawijaya, Supama) ...................... 68

Representasi Operator-SM.od pada Ruang Barisan 0 o, 1 < p < at (Muslim Ansoi,Soeparna Darmawijaya, Supama) . . .. ... . . . . . . . . . . .. . .. 78

Makalah Bidang Pendidikan Matematika

Kemampuan Pemahaman Konsep dan Penalaran Matematika Melalui PenggunaanKalkulator Grafik pada Mahasiswa Jurusan Matematika Universitas Negeri Malang(Indriati Ntrrul Hidayah, Santi lra*-ati,Ipung Yuv'ono) . ..... . . . . . . . . . . . . . . . . . . . . . . 92

Kontrol Proses Pembelajaran pada Mata Pelajaran yang Di-Ebtanasfuan-Kan TP20002001-200612007 pada SMPN 6 Medan (Menggunakan Pendekatan AnalisisMultivariat) (Hasratuddin) ......103

Pemahaman Matematis dan Penggunaan Komputer dalam Pembelajaran Matematika(Abd. Qohar) ... ... ................ 1 14

vl


i28Penilaian Pembelajaran Matematika dengan Pendekatan Ink-uiri(Muhamad Sa birin) ..................Peran Metakognisi dalam Menyelesaikan Masalah Matemattka (TheresiaKriswianti Nugrahaningsih) ......................137Peningkatan Motivasi Belajar Dan Pemahaman Keruangan Siswa MelaluiPembelajaran Geometri Berbantuan Program Komput er (MG. Erni Harmiati danAgustin Rahayu) ... ...... ...... 148

Meningkatkan Daya Matematika Siswa Melalui Paikem Berbantuan TeknologiM]oltimedia (ktnandi'S ...........159

Meningkatkan Kualitas Pembelajaran Matematika Di Sekolah Melalui PenelitianTindakan Kelas (Zaenuri Mastur dan Nurkaromah Dwidayati) ..........................114

Merancang Tugas untuk Mendorong Berpikir Kreatif Siswa dalam Belajar Matematika(Tatag Yuli Eko Siswono\ ... ... 1 88

Proses Berfikir Mahasiswa dalam Mengkonstruksi Konsep Grup(Herry Agus Strsanto) ..........201

Adversity Quotient: Kajian Kemungkinan Penginterasiannya dalam PembalajaranMalemattka (Sudar"man) .........209

Visualisasi Geometris dalam Pemahaman Masalah Matematika (1 Wayan Ponter) ... ...219

Explicit Values ln Statistics E<iucation: A case of Informatics Engineering Students(Bambang Suharjo) ... ... ... ......230

Matematika Melalui Perkuliahan Berbasis Masalah ( Djamilah Bondan llidiajanti) ... ...240

Karakteristik Proses Kognitif dalam Pemecahan Masalah Matematika (WilmintjieMataheru) .. . .. ... .. . ... . . . .....251

Penalaran Visuospatial dalam Mengkonstruk Bentuk Bangun Geometri(Ronaldo Kho) . .......... ...263

Proses Berpikir Anak Tunanetra dalam Menyelesaikan Permasalahan Matematika(Susanto) . . . . . . . . . . . . . . . . . . . . .268

Analisis Kontrol Proses Pembelajaran melalui Data Ujian Nasional di SMA PGRICiranjang Cianjnr (Rudy Kurniawan) ... . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . ..28 1

Metakognisi dalam Pemecahan Masalah Matematika Kontekstual(Mustamin Anggo) .... -........ .....................296ImplementasiZessonStudypad.aMatakuliahAnalisisF.eall(Manuharawati).............310

Implementasi Pendekatan PMRI dalam Pembelajaran Materi Pecahan di SekolahDasar (Rini Setianingsih) ............................317

vii


Makalah Bidang Statistika

Estimating Conditional Intensity of Point Processes Models (Nurtiti Sanusi,

Sutawanir D.,I Wayan M,I(ahyu 7) ............328

Generator Frame Wavelet Ketat dari Kelas Box Spline M:ultivariate (Mahmud Yunus,

Jartny Lindiarni, Hendra Gunawan) ............334

Generalized Regression dalam Pendugaan Area Kecll (Anang Kurnia, Khairil A.

Notodiputro, Asep Saefuddin, dan I l(ayan Mangku) ... ... ... . . . .. ... . . . . . . ... ... ... .. .346

Chain Ladder Method as a Gold Standard to Estimate Loss Reserves (Aceng K.

Mutaqin, Dumaria R. Tampubolon, Sutawanir Darwis) . ....... ......................356

Optimalisasi Respon Ganda pada Metode Respon Permukaan (Response Surface)dengan PendekatanFtngsi Desirability (Hari Sakti Wibowo, I Made Sumertajaya,Hari Wijayanto) ..........365

Sifat Penaksir Simulated MLE pada Respon Multinomial (Jaka Nugraha) .. . . . . ... .. . ..376

Estimasi Model Probit pada Respons Biner Multivariat Menggunakan MLE dan. . .. ....336

Analisis Komponen Utama Temporar (Ismail Djakaria) .. . ...........399

Pendekatan Dynamic Linear Model atau State Space Model pada Pendugaan Area

Kectl (Small Area Estimation\ (Kusman Sadik dan Khaiil Anwar Notodiputro) ... . . . .41 I

Pengembangan Algoritma EM untuk Data Tidak Lenglap (Incomplete Data) pada

Model Log-Linear (Kusman Sadik) . . . . . . .. . . . ........422

CRUISE sebagai Metode Berstruktur Pohon (Tree-Structurefl pada DataNon-Biner {Kasman Sadikl . . ..............+llModel Proportional Hazards Cox dengan Missing Covariates (Nurkaromah

Dwidayati, Sri Haryatmi, Subanar) ..................447

Aplikasi Mixture Distribution dalam Pemodelan Resiko Aktuaria (Adhi4ta RonnieFlfan/l;2\ 1{o

Analisis Komponen Utama Probabilistik pada Data Mrr sing (Ismail Djakaia) .. . . . . ..464

Malakah Bidang Terapan

Bilangan Ramsey untuk Graf Bintang 56 dan Graf Bipartit Lengkap K2.1". N:2,3,4(Isnaini Rosyida) . ..............476

l1-supermagic labelings of c copies of some graphs (Tita Khalis Maryati, A. N. M.Salman, Edy Tri Baskoro, Irawati) .................485

v1l1


Pendekatan Teori Operator Semigrup Solusi Masalah Nilai Awal Model DissipatifDua Kanal (Sumardi, Soeparna Darmatvijaya, Lina Aryati) . . . . . . . . . . ..493

On size multipartite Ramsey numbers for stars versus cycles (Syafrizal Sy, ET. Baskoro,S. Utnnggadewa, H. Assiyatun) ............. .... -......503


-!

II

I

II

I

Chain Ladder Method as a GoldStandard to Estimate Loss Reserves

Aceng K. Mutaqint), Dumaria R. Tampubolon2), Sutawanir Darwis2)

AbstractThis paper review the chain ladder method to estimate loss reserves in generalinsurance and standard error ofchain ladder reserve estimates that formulated by Mack(1993). The example to illustrate the method is also given. The authors provide aMATLAB program to implement the method.

Key words: run-of triangle dat.t, loss reseryes, chain ladder method, MATLABprogrqm

I . Introduction

The correct estimation of the amount of money an insurance company should set aside to

meet claims that arise in the future on written policies represents an important task for the

rnsurance company. This estimation is often named as the loss reserve (or claim resereve).

An insurance company needs to hold loss reserve because the timing of premiums receipt and

claims payment do not coincide. There is the delay between the claim event and the claim

settlement date. It could take a long time from the loss event to the reporting ofthe loss to the

company, or it could take a long time from the day of reporting until the company knows tne

ultimate cost of the claim.

hr the end of every reporting period (usually year and/or quader) the insurance company

should present in the accounting how much of the money that is allocated for this incurred

but not settled loss. It is necessary for business planning, budgeting and product pricing

(Olofsson, 2006).

The problem of how to calculate the loss reserve is usually solved with statistical methods.

Since the amount and timing of future claims are unknown, this creates an uncertainty over

the amount of reserves that is necessary. The degee of uncertainty depends on the class ofbusiness written, where we have shofi-tail and long-tail classes (Olofsson, 2006). A short-tail

'' A PhD student in Depatment ofMatelnatics, Institut Teknologi Bandung') Lecturer in Departrnent of Matematics, Institut Teknologi Bandung


-

Cfiain Lailer flLetfiof as a Qo[d Stan[art to ...

class is a business where the delay between the occurrence of a claim and the settlement is

short, often less than a year, such as motor insurance that, for example, covers fire and theft

(Olofsson, 2006). A long-tail business is a business where the delay between the occurrence

of a claim and it being settle is long, more than couple of years, such as motor TPL (hird

pafiy liability) that covers personal injury (Olofsson, 2006), and marine insurance (Herlig,

1985).

It is conmon to estimate loss resewe for long-tail business based on run-off triangle data (for

example, see De Jong, 2006; England, dan Verrall, 2002: Y erlall, 2002; Mack, 1993;

Pinheiro, E Silva, dan Centeno, 2003; De AJba, 2006; Panning, 2004: dan Olofsson, 2006).

Run-offtriangle data provides a summary ofunderlying data set with individual claim figures

(Antonio, Beirlant, Hoedemakers, dan Verlaak,2006). There are different statistical methods

to estimate loss reserve based on run-off triangle data. The chain ladder method is probably

the most popular one for estimating loss reserve. The main reason for this is its simplicity and

the fact that it is distribution-free. It is frequently used as a gold standard (benchmark)

because of its generalized use an ease to apply. This method is deterministic and we only get

a point estimate.

Is it well-known that chain ladder reserve estimates are very sensitive to variations in the data

observed (Mack, 1993). Therefore it would be very helpful to know the standard error ofthe

chain ladder reserve estimates as a measure of the uncertainty contained in the data. Mack

(1993) developed a very simple formula and distribution-free for the standard error of chain

ladder reserve estimates. This paper review the chain ladder method and standard error of

chain ladder reserve estimates that formulated by Mack (1993).

The remainder of the paper is structured as follows. Section 2 presents the chain ladder

method to estimate loss reserve. The standard error of the chain ladder reserve estimates that

formulated by Mack (1993) is discussed in Section 3. Section 4 provides an example to

illustrate the method. The implementation of the method in MATLAB program are given in

Section 5.

Ei[ang Statistifor 357


flcengK gvLutaqin, [F,k

2. Chain Ladder Method

Let Cr denote the cumulative total claims amount of accident year i, 1 < I < r, paid up to

development year j,1 < j < n- The cumulative total claims amount Ci; have been observed for

i + j < n + i (run-off triangle data), the other amounts, especially the ultimate claims amount

Cin, i > 1, have to be predicted. The aim is to estimate the the ultimate claims amount and the

outstandins claims resewe

Ri: Cn- Cun r ii for accident year I :2, ..., n

Run-off triangle data are presented in Table 1 .

(1)

The basic ass.rrption of the chain ladder method is that there are development factors f , ..',

f,-\ > 0 with E(Cij*tlCu, ..., Ct) = Cefj,1 < i < n, | < j <n - l.

The chain ladder method consists of estimating thef by

The total reserve is

B-\'r:'\- / .'\; (2)

(3)

n-JSr-

^ ,/i" k.j+l

ft=ft-;forl<j<n-1,Yr-

and the ultimate claims amount C1, by

Table 1 Run-off Triangle Data

AccidentPeiod

Deyelopment Pefiod2 n- |

I Ct r- 12 Cri Cr,'r Cn

2 \'21 Czz C2j Cz.,-t

I Cil ci

n-1 C-'., Cn-t.2

n c,.t

358 S eminar 9{asiona[ fu atematiforTKlvls 3*fi 200 8


Cfrain Lad[er Metfrof as a Gof[Stan[art to ...

e * = C,,**,i,*r-, " "' i, ,

or equivalently the reserve R; by

(4)

D _/- /-r\t - \- i, - Li,n+t_i

= C,,,*,-,(i^*r-, '"'' i,-', -t) (5)

Because the chain ladder method does not take into account any dependencies between

accident years, Mack (1993) added assumption that the variables C,; of different accident

years, i.e. {C1, ..., C*}, { C^, ..., Ck"}, i + k, are independent. This must be regarded as a

further implicit assumption of the chain ladder method.

Mack (1993) proved that

\. E(Ch I D) = C,.*r-, f ̂ .r-, ..... f ,-r, where D = {C,jli + j < n + I } is the set of all

data observed so far.12. The estimators fr,forl<i<r- 1, are unbiased and uncorrelated off.

3. The estimato., d,,, = C,,,*r-,j,*r-, ""'f,-t, for 2 ! i < n, are unbiased ofE(C,, I D) = Ci,^*tif ,*ri' "' f ,-r.

4. The reserve estimators R,=C, -Ci,u*t i, for 2 < i ( n, are unbiased of the true

reserves Ri Cr, - C,,, t '.

3. Standard Enor of Chain Ladder Reserve Estimates

The mean squared error of the estimator e,n of C; is defined to be

,: , -,,;mse(C,^)= E(C,, - C,^)' lD)whereD= {C1ili + j <n+ 1} isthe set of all data observed so far.

The mean squared error of the estimator -4, of R1 is

mse(&.,1 = E(Rr - R, )'z I D)

=E(e ,,-c,,,)' lD)

= mse\L in )

= Var(C;, I D) + (E(C,, I D) - e)'

Eifang Statistifot 359


flcengK flLutaqin, fEfr

which shows that the mean squared error is the sum ofthe stochastic error (process variance)

and of the estimation error.

Mack (1993) introduced an assumption that

Var(C,.,*, I C i1, "', C ij) = C,,ol, t < i < n, 1 < j < n - l,

with unknown parameters ol , t =j 3 n- 1. This is the variance assumption which is

implicitly underlying the chain ladder method.

The parameters ol , I <i < " - 1, are estimated by

1 n-t lC. ^)td.? = ' Ic,.l "'''.' - 7. | .t.i<"-2. (6)-t --i-1 4-'tl f rll'"-r-- -'

tt-J-t t_.1 \ -t )

ff ?,-1 =1' and if the claims development is believed to be finished after n - 1 years, then

6*t=0,otherwise

ti, , = min(rif,, / 6l-r,rnin(6i-r,e3))

Mack (1993) proved that

i. The mean squared error mse(i<,) can be estimated by

(7)

(8)i^h=e?" i +t=n+ t-t J j

l1. 7 |7--- |L,i Sr I' L'4 |&=1 J

*h"re eo -- C,,,*r-r?,*r-, ..... f ,-r,i +i> n + l, are the estimated values of the

.Atuflrre La.and Li.,+r_; = Ur.n_r_i.

The mean souared enor of the total reserve estimate R can b. estimated by

#lro-i= i{,qn,r,''"1.

f " I nr*e,"1 Ie," I F

Ll=i+l Ji=n+1-i

(e)

where se(fr,) is the standard error of R,, i.e. the squared root of mse(R,).

360 S eminar ltrasianaf MatematifoL-fKfrls 3 *lI 200 8


Cfrain La[der %.etfiof as a GoUstundart to ...

4. Example

In this paper, we use the data which were used by Mack (1993) in the first example to apply

the chain ladder reserve estimates and its standard errors. The data are presented in Table 2.

Table 2 Run-off Triangle Data (Cumulative Figures)

i CT c,l c,. Cit C C,6, C CA Cn c,"I 357848 1124'188 1735330 22t8270 2'745596 3319994 3466336 1606286 3833515 390t 463

2 352118 1236139 2110033 3353322 3799067 4120063 464186',7 4914039 5319085

3 290507 1292306 2218525 32351'79 3985995 4132918 46289t0 4909315

4 I r0608 l4 1885 8 219504'7 3'7 5744',7 4029929 4381982 4588268

5 443160 I116350 2128333 2891821 34026'12 387331 lo 396132 t3312l7 2180715 2985752 36917],2,7 M0832 t288463 2419861 34831308 359480 l4)l t28 2864498

3',76686 t363294l0 344014

This yields the parameter estimates i , and 6l , Q : 1 ,2, ...,9):

Table 4 shows the ultimate claims, the estimated reserves, the standard errors ofthe estimated

reserves, and the standard errors ofthe estimated reserves in % of R, fori=2.3....' 10. The

last row in Table 4 contains the estimated total reserve and its standard error.

Table 4 Summarv of The Chain Ladder Reserve Estimates

UltimateClaims

EstimatedReserves

StandardErrors

Standard Errorsin%ofi1

5433'119 94634 75535 0.80

5378826 469511 t2t699 0.265297906 709638 133549 0.19

4858200 984889 26t406 0.275llt17l 1419459 4 t l0l0 0.29

5660'77 t 2177641 55 8317 0.26

t:8 6'/84't99 3920301 875328 0.22

5642266 42789'12 97 | 258 0.23

t= l0 496982s 4625811 ll63l55 0.29

Overall 18680856 2447095 0.13

Table 3 The Parameter Estimales

=3 =83.49061 1 .'7 4733 | .45141 1. t7385 1.10382 1.0862'7 1.05387 t.0'/656 1.0t 17 2

160-2 80 37'736.86 4t965.2 15182.9 llTll.l 8185.77 416.617 I t4'1.3',7 446.6t'7

cBi[ang StatistifuL JO I


ncengK %utaqin, [ftf,

function oueCLM(C)o/o This program is designed to implement the chain ladder method.o/o The input program is C : a matrix contains the run-off trianglc data

% in cumulative formo/o The output program is out : a matrix contains

% column 1: Ultimate claims% column 2: Reserve estimatesyo (including total reserve estimate in the last row)% columl3: Stardard errors ofres€rve estimates

% (including for total reserve estimate in the last row)% column 4: Standard errors in o/o ofreserve estimates

% (columa 3/column 2)n:size(C,1); o/o number of accident years/development years

m:n-1;for j:1:m

f0,l)=sum(C(: j+1))/(sum(C(:j))-C(n+l-j j)); % development factorsend

len+l;for i:2:n

k:k-l;forj:k:n

c 1(ij):c 1(ij-1)*f[- I,l );endout(i,l):Cl(i,n)l o%UltimateClaims

out(i,2):C1(i,n)-C1(i,k-1); o/o Reserve Estimatesendout(n+1,2):sum(out(2:n,2)); o/o Total Reserve Estimatem2:12;forj:1:nO

sigma2(,1):sum(C( l:n-jj).*(C(1:n-jj+1)./C(1:n-jj)-f[,1)).^2)/(n-j-1);endiff(n-1,1).:1

sigma2(m2+1,1):0;else

a:(sigma2(n-2, 1 ))"2 I sigma?(n-3,1);b:min(sigma2(n-3, I ),sigma2(n-2, 1 ));sigma2(m2+ 1

" 1 ):min(a,b);

endfor i:2:n

kl=n+l-i;k2:n- I ;s1:0;for lekl:k2

b=sum(C1(1:n-k,k));s1:s1+sigma2(k,1)/f(k, 1)^2+(1/C I (i,k)+1,/b);

endout(i,3):sqrt(C I (i,n) 2*sl); V. Standard Errors ofReserve Estimates

ends2:0;lbr i:2:n

362 S emin ar $[asionaf gvtat em atifut -qrKfuls 39il1 2 00 8


Cfrain Lailer lMetfrod as a Q ofd Stanfart to ...

kl=n+1-i;k2:n-l;al=sum(C 1(i+ 1:n,n));a2=0;for lckl:k2

b=sum(C I ( l:n-k,k));a2= a2+ 2* sigma2(k,l ) / f (k, | ) 2 b :

ends2:s2+out(i,3)^2+C1 (i,n)*a1 +a2;

endout(n+ 1,3 ):sqrt(s2); 7o Standard Errors ofTotal Reserve Estimates

for i:2:n+ Iou(i,4)=out(i,3)/ou(i,2); % Standard Errors in o/o of Reserve Estimates

end

Figure IMATLAB Program to Implement The Chain Ladder Method

5. Implementing the Chain Ladder Method in MATLAB

The chain ladder method can be implemented using MATLAB program (see Figure 1). This

program can be used to calculate the ultimate claims, the estimated reserves, the standard

errors of the estimated reserves, and the standard errors of the estimated reserves in y; of the

estimated reserves. The values in Table 4 are generated using the MATLAB program in

Figure 1.

References

Antonio, K., Beirlant, J., Hoedemakers, T., and Verlaak, R. (2006). Lognormal MixedModels for Reported Claims Reserv es. North American Actuarial Journal Vol. 10,No. 1:

30-48.

De Alba, E. (2006). Claim Reserving When There are Negative Values in Runoff Triangle:Bayesian Analysis Using the Three-Parameter Log-Normal Distribution. North American

Actuarial Joutnal Vol. 10, No. 3: 45-59.

De Jong, P. (2006). Forecasting RunoffTriangle. North American Actuarial Joumal Vol. 10,

No. 2: 28-38.

England, P. D., and Verrall, R. J. (2002). Stochastic Claims Reserving in General Insurance

http//www.actuaries.org.uk/fi les/pdfl sessional/smO20 l.pdi

Hedig, J. (1985). A Statistical Approach to IBNR-Reserves in Marine Reinsurance. ISZNBULLETIN. Vol. 15. No. 2:171-183.

Gi[ang Statistifot JOJ


flceng K gLutaqin, [f;$,

Mack, T. (1993). Distribution-free Calculation of the Standard Error of Chain Ladder

Reserves Estimates. ASTIN BULLEZN, Vol. 23, No. 2: 213-225.

Olofsson, M. (2006). Stochastic Loss Reserving Testing the New Guidelines from the

Australian Prudential Regulation Authority (APRA) on Swedish Portfolio Data Using a

Bootstrap Simulation and Distribution-Free Method by Thomas Mack.

http://www.math.su.se/mathstat/reports/serieb/2006/rep 13/report.pdf'

Panning, W. H. (2004). Measuring Loss Reserving Uncertainty. http://www.actuaries.org/

ASTIN/Colloquia/Zurich,?anning.pdf.

Pinheiro, P. J. R., E Silva, J. M. A., and Centeno, M. D. L. (2003). Bootstrap Methodology inClaim Reserving. The Journal of Risk and Insurance Yol. 70, No. 4:701-714.

Verrall, R. J. (2002). Obtaining Predictive Distributions for Reserves Which IncorporateE xp e rt Opi n i o n. http ://wv'w.casact.org/pubs/forum/04fforum/O4ff283.pdf.

364 Seminar !(asionaf tulatematifut-FWs3*ll 2008


'.

I

t.

t


isbn prosiding seminar tnasional lt mahasiswa i r 53

Documents