Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Matematika Aktuaria |
| Periode Ujian | : | November 2018 |
| Nomor Soal | : | 19 |
SOAL
Diketahui:
| \(x\) | 70 | 71 | 72 | 73 | 74 | 75 | 76 |
| \({l_x}\) | 100 | 90 | 70 | 50 | 40 | 20 | 0 |
Tingkat suku bunga efektif adalah 8% per tahun.
Dengan mengasumsikan UDD, tentukanlah \({\bar A_{70:\left. {\overline {\, 3 \,}}\! \right| }}!\)
- 0,836
- 0,612
- 0,448
- 0,250
- 0,061
| Rumus | \({\bar A_{70:\left. {\overline {\, 3 \,}}\! \right| }} = {\bar A_{\mathop {70}\limits^| :\left. {\overline {\, 3 \,}}\! \right| }} + {}_3{E_{70}}\) |
| Step 1 | \(_3{E_{70}}\) = \({v^3}\,{}_3{p_{70}}\) \(_3{E_{70}}\) = \({(1,08)^{ – 3}}\left( {\frac{{50}}{{100}}} \right)\) \(_3{E_{70}}\) = \(0,3969161205\) |
| Step 2 | \({\bar A_{\mathop {70}\limits^| :\left. {\overline {\, 3 \,}}\! \right| }} = \frac{i}{\delta }{A_{\mathop {70}\limits^| :\left. {\overline {\, 3 \,}}\! \right| }}\) \({\bar A_{\mathop {70}\limits^| :\left. {\overline {\, 3 \,}}\! \right| }} = \frac{{0,08}}{{\ln (1,08)}}\left( {\sum\limits_{k(x) = 0}^2 {{v^{k(x) + 1}}P(K(x) = k)} } \right)\) \({\bar A_{\mathop {70}\limits^| :\left. {\overline {\, 3 \,}}\! \right| }} = \frac{{0,08}}{{\ln (1,08)}}\left( {v{q_{70}} + {v^2}{p_{70}}{q_{71}} + {v^3}{}_2{p_{70}}{q_{72}}} \right)\) \({\bar A_{\mathop {70}\limits^| :\left. {\overline {\, 3 \,}}\! \right| }} = \frac{{0,08}}{{\ln (1,08)}}\left( {v\left( {\frac{{100 – 90}}{{100}}} \right) + {v^2}\left( {\frac{{90}}{{100}}} \right)\left( {\frac{{90 – 70}}{{90}}} \right) + {v^3}\left( {\frac{{70}}{{100}}} \right)\left( {\frac{{70 – 50}}{{70}}} \right)} \right)\) \({\bar A_{\mathop {70}\limits^| :\left. {\overline {\, 3 \,}}\! \right| }} = 0,4395229572\) |
| Maka | \({\bar A_{70:\left. {\overline {\, 3 \,}}\! \right| }} = {\bar A_{\mathop {70}\limits^| :\left. {\overline {\, 3 \,}}\! \right| }} + {}_3{E_{70}}\) \({\bar A_{70:\left. {\overline {\, 3 \,}}\! \right| }} = 0,4395229572 + 0,3969161205\) \({\bar A_{70:\left. {\overline {\, 3 \,}}\! \right| }} = 0,836439077\) \({\bar A_{70:\left. {\overline {\, 3 \,}}\! \right| }} = 0,836\,\) |
| Jawaban | a. 0,836 |