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 |
