Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
November 2015 |
| Nomor Soal |
: |
14 |
SOAL
Data di bawah ini diekstrak dari table mortalita select dan ultimate dengan periode seleksi 2 tahun:
| \(x\) |
\({l_{\left[ x \right]}}\) |
\({l_{\left[ x \right] + 1}}\) |
\({l_{x + 2}}\) |
\(x + 2\) |
| 60 |
80.625 |
79.954 |
78.839 |
62 |
| 61 |
79.137 |
78.402 |
77.252 |
63 |
| 62 |
77.575 |
76.770 |
75.578 |
64 |
Hitunglah \({}_{0,9}{q_{\left[ {60} \right] + 0,6}}\) (dibulatkan 4 desimal)
- 0,0102
- 0,0103
- 0,0104
- 0,0105
- 0,0106
| Diketahui |
| \(x\) |
\({l_{\left[ x \right]}}\) |
\({l_{\left[ x \right] + 1}}\) |
\({l_{x + 2}}\) |
\(x + 2\) |
| 60 |
80.625 |
79.954 |
78.839 |
62 |
| 61 |
79.137 |
78.402 |
77.252 |
63 |
| 62 |
77.575 |
76.770 |
75.578 |
64 |
|
| Rumus yang digunakan |
\({q_x} = \frac{{{l_x} – {l_{x + 1}}}}{{{l_x}}},{\rm{ }}{q_{\left[ x \right] + 1}} = \frac{{{l_{\left[ x \right] + 1}} – {l_{x + 2}}}}{{{l_{\left[ x \right] + 1}}}},{\rm{ }}{q_{\left[ x \right]}} = \frac{{{l_{\left[ x \right]}} – {l_{\left[ x \right] + 1}}}}{{{l_{\left[ x \right]}}}}\)
\({}_t{p_x} = {}_{t + x}{p_0}\)
\({}_t{p_x} = {p_x}{p_{x + 1}}{p_{x + 2}} \cdots {p_{x + t – 1}}\)
Untuk UUD konstan dan usia bukan bilangan bulat
\({}_s{p_x} = 1 – s \cdot {q_x}\)
\({}_s{p_{x + t}} = \frac{{{}_{s + t}{p_x}}}{{{}_t{p_x}}} = \frac{{{}_{s + t}{p_x}}}{{1 – t \cdot {q_x}}}\) |
| Proses pengerjaan |
\({}_{0,9}{q_{\left[ {60} \right] + 0,6}} = 1 – {}_{0,9}{p_{\left[ {60} \right] + 0,6}}\)
\(= 1 – \frac{{{}_{1,5}{p_{\left[ {60} \right]}}}}{{{}_{0,6}{p_{\left[ {60} \right]}}}}\)
\(= 1 – \frac{{{p_{\left[ {60} \right]}} \cdot {}_{0,5}{p_{\left[ {60} \right] + 1}}}}{{{}_{0,6}{p_{\left[ {60} \right]}}}}\)
\(= 1 – \frac{{{p_{\left[ {60} \right]}} \cdot \left( {1 – 0,5 \cdot {q_{\left[ {60} \right] + 1}}} \right)}}{{\left( {1 – 0,6 \cdot {q_{\left[ {60} \right]}}} \right)}}\)
\(= 1 – \frac{{\frac{{79.954}}{{80.625}} \cdot \left( {1 – 0,5 \cdot \frac{{79.954 – 78.839}}{{79.954}}} \right)}}{{\left( {1 – 0,6 \cdot \frac{{80.625 – 79.954}}{{80.625}}} \right)}}\)
\(= 0,010295\) |
| Jawaban |
b. 0,0103 |
