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Pengantar Matematika Asuransi

Pengantar Matematika Asuransi. Winita Sulandari. What is insurance mathematics?. Insurance mathematics is the area of applied mathematics that studies different risks to individuals, property and businesses, and ways to manage these risks. What is insurance and w hy do people need insurance ?.

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Pengantar Matematika Asuransi

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  1. PengantarMatematikaAsuransi WinitaSulandari

  2. What is insurancemathematics? • Insurance mathematics is the area of applied mathematics that studies different risks to individuals, property and businesses, and ways to manage these risks.

  3. What is insurance and why do people need insurance?

  4. Asuransi • Asuransi adalah sebuah sistem untuk merendahkan kehilangan finansial dengan menyalurkan resiko kehilangan dari seseorang atau badan ke lainnya.

  5. Istilahdalamasuransi • Polis • Insurer • Insured • Premi • Klaim • Risk : averse, neutral and loving/seeking  Perjanjian antara kedua badan  Penanggung: badan yang menerima resiko  Tertanggung: badan yang menyalurkan resiko • Biaya yang yang dibayar oleh tertanggung kpd penanggung • utk resiko yang ditanggung tindakanuntukmemperolehhakataskerugian yang diderita olehseseorangsesuaidenganjanji yang terteradi polis asuransi

  6. Jenis asuransi • Asuransi jiwa • Asuransi kerugian • Yang membedakan adalah pada obyek pertanggungannya

  7. Asuransi jiwa • Asuransi jiwa dalah perjanjian tentang pemberian santunan dalam jumlah tertentu yang berhubungan dengan hidup matinya seseorang • Tujuan asuransi jiwa adalah perlindungan pendapatan (sementara) keluarga (ahli waris) nasabah

  8. Asuransi kerugian • Asuransi kerugian adalah asuransi yang memberikan ganti rugi kepada tertanggung yang menderita kerugian barang atau benda miliknya.

  9. Bagaimana dengan asuransi beasiswa? • Apa bedanya dengan asuransi jiwa?

  10. Asuransi beasiswa • Asuransi beasiswa adalah asuransi yang mengcover resiko biaya sekolah anak, apabila terjadi resiko pada si tertanggung.

  11. Ilustrasi asuransi kerugian

  12. Effect of insurance • Example:

  13. Effect of insurance • Prob of loss is 0,5 • Expected level of wealth: 90.000 (0,5 x 100.000 + 0,5 x 80.000) • Misal : Asuransi 10.000 denganharga 5.000 premium = insurer’s expected claim cost = 0,5 x 10.000

  14. Effect of insurance • Example: • * with insurance

  15. Effect of insurance • Perhatikan bagan berikut • Asuransi wealth (LOSS occurs) wealth (No LOSS occurs)

  16. Risk Aversion, neutral and loving Example: Expected payoff = $50

  17. Risk Aversion, neutral and loving Risk attitudes: • Risk averse: Accept a payoff of less than $50 with no uncertainty Taking gamble and possibly receiving nothing

  18. Risk Averse • Lebih memilih sesuatu yang lebih pasti, dan meminimalkan resiko • Contoh: investor memilih return lbh tinggi utk resiko yang sama atau investor lebih memilih resiko yg lbh rendah utk return yang sama

  19. Risk Aversion, neutral and loving • Risk neutral: indifferent between the bet and a certain $50 payment

  20. Risk Aversion, neutral and loving • Risk loving (or risk seeking/loss aversion) Guaranteed payment must be more than $50 to induce him to take the guaranteed option, rather than taking gamble and possibly winning $100

  21. Loss aversion Kondisidimana investor tidakinginmengalamikerugiansehingga investor tersebutbertindaksecaratidakrasional. Merekalebihmemperhatikankondisiuntungataurugi dan tidakmemperhatikanbesarkecilnyanilaiuntungataupunrugi

  22. Contoh tindakan Loss Averse • Mengambilinvestasi yang berisikotinggidenganharapanuntukmengembalikankerugian yang sudahterjadi. • Investor lebih memilih pilihan dg expected return adalah -25% daripada -20% Return -50% dg p=0,5 atau tdk rugi apapun dg p=0,5 Return -20%

  23. Utility theory uncertainty Decision making problem Expected value principle Inadequacy of expected value principle Open and read Bowers, et al (1997) on p. 3-4

  24. Utility theory • Utility is often assumed to be a function of profit or final portofolio wealth, with positive first derivative. • ` • 1. risk seeking • 2. risk neutral • 3.risk averse

  25. Utility function • w : wealth • u(w):utility function • Linear transformation: u*(w) = au(w) + b, a> 0

  26. Utility function • Suppose you face a loss of 20.000 with prob 0,5 and no loss with prob 0,5. • What is the max amount G to pay for complete insurance protection? u(20.000-G)=0,5u(20.000)+0,5u(0) =0,5(0) + 0,5(-1) =-0,5

  27. Utility function • Utility function can be used to compare economic prospects (X and Y). • If decision maker has wealth w, he will select X if E[u(w+X)]>E[u(w+Y)] and indifferent if E[u(w+X)]=E[u(w+Y)]

  28. Insurance and utility • Pure or net premium: price for full insurance coverage, i.e expected loss E(X)= • Loaded premium: H = (1+a) +c a>0, c>0

  29. Insurance and utility expected utility of not buying insurance, current wealth is w U(w-G) = E[u(w-X)] Expected utility of paying G for complete financial protection

  30. Insurance and utility If owner have u(w) = bw + d with b > 0, owner prefers or indiferrents to the insurance if u(w-G) = b(w-G)+d ≥E[u(w-X)]=E[b(w-X)+d] b(w-G)+d ≥b(w- )+d G ≤ 

  31. Jensen’s inequalities For a random variable X, function u(w) if u”(w)<0 then E[u(X)] ≤ u(E[X]) if u”(w)>0 then E[u(X)] ≥ u(E[X]) Proof: see Bowers, et al (1997) on p. 9

  32. Jensen’s inequality Applying Jensen’s inequality to U(w-G) = E[u(w-X)] We have u(w-G) = E[u(w-X)] ≤ u(w-) because u’(w)>0,u(w) is increasing function w-G ≤ w- G ≥ 

  33. Utility function for the insurer H: acceptable premium; X: random loss u(w)=E[u(w+H-X)] Jensen’s inequality u(w)=E[u(w+H-X)] ≤ u(w+H-) We can conclude that H ≥ 

  34. Exponential utility function Check on page 10-11 u(w) = - exp(-w) for all w and a fixed  > 0 G doesn’t depend on w

  35. Contoh 1 • Open and read example 1.3.1 p.11 Diketahui : u(w)= -(exp(-5w)), dan X,Y adlh hsl yg mungkin di capai ? X  N(5,2) Y  N(6,2.5)

  36. Contoh 1 Solution E[u(X)]=-1 > E[u(Y)] = - (exp(1,25)) Dengan demikian, Distribusi X lebih disukai daripada distribusi Y

  37. Family of fractional power utility functions • u(w) = w risk averse cek u’(w) dan u’’(w) 

  38. Contoh 2 • Misal • w= 10 dan kehilangan random X UNIF (0,10) Max amount will pay for complete insurance againts the random loss? diperoleh G = 5.5556 , G > E [X]

  39. Family of Quadratic Utility Function u(w) = w - w2 , w < (2)-1 and >0 cek u’(w) dan u’’(w)

  40. Contoh 3 • Diketahui: u(w) = w - w2 , w< 50  retain wealth w with prop p and  financial loss with prop (1-p) Sehingga u(w-G) = pu(w) + (1-p)u(w-c)

  41. Contoh 4 Diketahui: Prop properti tdk ada kerugian : 0,75 • Pdf kerugian f(x) = 0,25(0,01exp(-0,01x)), x>0 u(w) = -exp(-0,005w) Hitung: E[X] dan G Cek Ex 1.3.4

  42. Contoh 5 • Diskusikan contoh 1.3.5

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