Pembahasan Ujian PAI: A60 – No. 6 – Juni 2016

1. Premi bruto adalah 130
2. \({q_{80 + k}} = 0,01\left( {k + 1} \right)\),\(k = 0,1,2, \ldots ,5\)
3. \(v = 0,95\)
4. \(1000{A_{86}} = 683\)
5. \({}_3L\) adalah “prospective loss random variable” saat ,\(t = 3\) berdasarkan premi bruto

• \(E\left[ {{}_nL} \right] = {A_x} – P \cdot {\ddot a_{x:\left. {\overline {\, n \,}}\! \right| }}\)
• \({A_x} = A_{x:\left. {\overline {\, n \,}}\! \right| }^1 + {}_n{E_x} \cdot {A_{x + n}}\)
• \({\ddot a_{x:\left. {\overline {\, n \,}}\! \right| }} = \sum\nolimits_{k = 0}^{n – 1} {{v^k} \cdot {}_k{p_x}} \)

• Expected present value dari future benefit adalah
  \(1000{A_{83}} = 1000\left( {A_{83:\left. {\overline {\, 3 \,}}\! \right| }^1 + {}_3{E_{83}} \cdot {A_{86}}} \right)\)
  \(1000{A_{83}} = 1000\left[ {v \cdot {q_{83}} + {v^2} \cdot {p_{83}} \cdot {q_{84}} + {v^3} \cdot {}_2{p_{83}} \cdot {q_{85}} + {v^3} \cdot {}_3{p_{83}} \cdot {A_{86}}} \right]\)
  \(1000{A_{83}} = 1000[0.95\left( {0.04} \right) + {0.95^2}\left( {0.96} \right)\left( {0.05} \right) + \) \({0.95^3}\left( {0.96} \right)\left( {0.95} \right)\left( {0.06} \right) + {0.95^3}\left( {0.96} \right)\left( {0.95} \right)\left( {0.94} \right)\left( {683} \right)]\)
  \(1000{A_{83}} = 630.2477\)
• Pada saat \(t = 3\) hanya ada kekurangan 2 premi yang harus di bayar, sehingga expected present value-nya adalah
  \(P \cdot {{\ddot a}_{83:\left. {\overline {\, 3 \,}}\! \right| }} = 130 \cdot \left( {1 + v \cdot {p_{83}}} \right)\)
  \(P \cdot {{\ddot a}_{83:\left. {\overline {\, 3 \,}}\! \right| }} = 130\left[ {1 + \left( {0.95} \right)\left( {0.96} \right)} \right] = 248.56\)

\(E\left[ {{}_3L} \right] = 1000{A_{83}} – P \cdot {\ddot a_{83:\left. {\overline {\, 3 \,}}\! \right| }} = 630.2477 – 248.56 = 381.6877\)