Judul: Determination of Premium on Reversionary Annuity with Two Dependent Future Lifetimes using Markov Model
Abstrak: Pada umumnya, premi untuk asuransi multilife ditentukan menggunakan asumsi saling bebas. Namun, asumsi ini tidak realistis, contohnya untuk pasangan suami istri. Tujuan dari karya ilmiah ini adalah menentukan premi tunggal bersih pada reversionary annuity dengan memperhitungkan kebergantungan masa hidup di antara pasangan suami istri menggunakan batas Frechet-Hoeffding dan model Markov, membandingkan premi tunggal bersih yang memperhitungkan kebergantungan dengan premi yang menggunakan asumsi saling bebas, serta memeriksa pengaruh nilai parameter kebergantungan terhadap premi tunggal bersih. Penghitungan didasarkan pada Tabel Mortalitas Indonesia (TMI) IV mengikuti sebaran Makeham. Hasil penghitungan menunjukkan bahwa dengan model Markov nilai premi tunggal bersih lebih rendah daripada dengan model standar yang menggunakan asumsi saling bebas. Nilai premi tunggal bersih akan menurun seiring meningkatnya nilai parameter kebergantungan masa hidup lakilaki terhadap istrinya yang masih hidup. Nilai premi tunggal bersih akan meningkat seiring meningkatnya nilai parameter kebergantungan masa hidup perempuan terhadap suaminya yang masih hidup. Nilai premi tunggal bersih akan menurun seiring meningkatnya nilai parameter kebergantungan masa hidup perempuan terhadap suaminya yang sudah meninggal., In general, premiums for multilife insurance are determined by assuming the independence of the future lifetimes of insured persons. However, this assumption is unrealistic, such as for husbands and wives. The aims of this paper are to determine the net single premium on reversionary annuity by taking into account the dependence of the future lifetimes of husbands and wives using the FrechetHoeffding bounds and Markov model, to compare the net single premium that takes into account the dependence with the premium using independent assumption, as well asto examine the effect of the value of dependence parameters on the net single premium. The calculation is based on Indonesian Mortality Table IV following the Makeham distribution. The calculation results show that with Markov model the net single premium value is lower than the standard model that assumes independence. The parameter net single premium value will decrease as the value of man’s lifetime dependency parameter towards his wife who is still alive increases. The parameter net single premium value will increase as the value of woman's lifetime dependency towards her husband who is still alive increases. The parameter net single premium value will decrease as the value of woman's lifetime dependency parameter towards her deceased husband increases.
Keyword: Frechet-Hoeffding bounds, lifetime dependency, Makeham distribution, Markov model, reversionary annuity

Judul: Determining Net Single Premium of Reversionary  Annuity for Married Couple with Copula Model
Abstrak: Salah satu contoh produk anuitas multi-life ‘mencakup lebih dari satu orang’ adalah reversionary annuity, yaitu produk anuitas hidup untuk dua orang tertanggung yang pembayaran anuitas akan dimulai setelah salah satu tertanggung yang telah ditentukan di kontrak meninggal terlebih dahulu sampai pihak tertanggung lainnya meninggal juga. Perhitungan premi anuitas tersebut biasanya dilakukan dengan mengasumsikan kebebasan di antara pihak tertanggung. Namun, hal tersebut tidak relevan dengan keadaan sesungguhnya karena pasangan suami istri seharusnya saling terkait kelangsungan hidupnya. Pada karya ilmiah inidipertimbangkan keterkaitan tersebut dengan Frank’s copula pada saat memodelkan sebaran bersamanya yang diasumsikan sebaran marginalnya menyebar Makeham dengan parameter yang diestimasi dari Tabel Mortalitas Indonesia IV 2019. Penerapan copula terjadi pada saat umur suami dan istri menikah. Secara umum, hasil perhitungan premi tunggal bersih dengan sebaran bersama Frank’s copula menghasilkan nilai lebih rendah dibandingkan dengan perhitungan asumsi kebebasan untuk semua kasus penerima manfaat., One of the multi-life annuity products is a reversionary annuity, which is a life annuity product for two insured people. The annuity payment for this product will begin after one of the insured who has been specified in the contract dies first until the other insured dies as well. The calculation of the annuity premium is usually done by assuming independence between the insured parties. However, this is not relevant to the actual situation, because the husband and wife are interrelated with their lives. This undergraduate thesis considered this relationship when modelling the joint distribution using Frank’s copula which the marginal distribution assumed to follow a Makeham distribution with parameters estimated by using the 2019 Indonesian Mortality Table IV. The joint survival distribution modelled by copula was constructed exactly when a husband and wife are married. In general, the results of the calculation of the net single premium with the Frank’s copula produced a lower value than the calculation of the assumption of independence for all beneficiary cases.
Keyword: Frank’s copula, Makeham mortality law, married couple, net single  premium, reversionary annuity

Judul: Characterizations of low alloy steel sae 4422  plated with chromic acid (Cro3)
Abstrak: Characterizations of low alloy steel SAE 4422 plated with chromic acid (CrOrO3) was use Optical Microscopy (OM), Scanning Electron Microscopy (SEM), X-Ray Diffraction (XRD), RD),Energy Dispersive Analysis of X-Ray (EDAX), Hardness Test and Roughness Test. Microstructures analysis exhibits cracks and grain boundaries microcracks as a characteristics of chrome metal found at plating metal. Perlite and Ferrite as grains of steel found at base metal with grain boundaries around it. Phase analysis of plating metal appropriate to chrome metal with lattice constant (a) is 2.559 Å and phase analysis of base metal appropriate to iron ferrite with lattice constant (a) is 3.318 Å. Both plating and base metal have cubic system. Chemical analysis of plating metal at surface section exhibits that cracks area have highest value of carbon element compare with grains and grains boundaries area. While chemical analysis of base metal at cross section exhibits that perlite have highest value of carbon element compare with ferrite and grain boundaries. The hardness value of plating metal at surface section and cross section are 753 and 637 VHN. While, the hardness value of base metal and interface area at cross section are 205 and 373 VHN. The roughness value at measurement points of chrome surface for as received sample are 0.445 μm. Observation used Optical Microscopy and Scanning Electron Microscopy exhibits surface irregularities combine to form surface texture of plating metal at surface section.
Keyword: