Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | November 2018 |
| Nomor Soal | : | 30 |
SOAL
Diberikan sebuah table mortalita dengan 2 tahun seleksi
| \(\left[ x \right]\) | \({l_{\left[ x \right]}}\) | \({l_{\left[ x \right] + 1}}\) | \({l_{x + 2}}\) | \(x + 2\) |
| 65 | | | 8200 | 67 |
| 66 | | | 8000 | 68 |
| 67 | | | 7700 | 69 |
Juga diberikan:
- \(3{q_{\left[ x \right] + 1}} = 4{q_{\left[ {x + 1} \right]}}\)
- \(4{q_{x + 2}} = 5{q_{\left[ {x + 1} \right] + 1}}\)
Hitunglah \({l_{\left[ {67} \right]}}\)
- 7940
- 8000
- 8060
- 8130
- 8200
| Diketahui | \({l_{67}} = 8200,{\rm{ }}{l_{68}} = 8000,{\rm{ }}{l_{69}} = 7700\)
\(3{q_{\left[ x \right] + 1}} = 4{q_{\left[ {x + 1} \right]}}\) dan \(4{q_{x + 2}} = 5{q_{\left[ {x + 1} \right] + 1}}\) |
| Rumus yang digunakan | \({q_x} = \frac{{{l_x} – {l_{x + 1}}}}{{{l_x}}},{\rm{ }}{q_{\left[ x \right] + 1}} = \frac{{{l_{\left[ x \right] + 1}} – {l_{x + 2}}}}{{{l_{\left[ x \right] + 1}}}},{\rm{ }}{q_{\left[ x \right]}} = \frac{{{l_{\left[ x \right]}} – {l_{\left[ x \right] + 1}}}}{{{l_{\left[ x \right]}}}}\) |
| Proses pengerjaan | \({q_{67}} = \frac{{{l_{67}} – {l_{68}}}}{{{l_{67}}}} = \frac{{8200 – 8000}}{{8200}} = 0,02439\)
\({q_{68}} = \frac{{{l_{68}} – {l_{69}}}}{{{l_{68}}}} = \frac{{8000 – 7700}}{{8000}} = 0,0375\)
\({q_{\left[ {66} \right] + 1}} = \frac{4}{5}{q_{67}}\)
\(= \frac{4}{5}\left( {0,02439} \right)\)
\(= 0,01951\)
\({q_{\left[ {67} \right] + 1}} = \frac{4}{5}{q_{68}}\)
\(= \frac{4}{5}\left( {0,0375} \right)\)
\(= 0,03\)
\({q_{\left[ {67} \right]}} = \frac{3}{4}{q_{\left[ {66} \right] + 1}}\)
\(= \frac{3}{4}\left( {0,01951} \right)\)
\(= 0,01463\)
\({q_{\left[ {67} \right] + 1}} = \frac{{{l_{\left[ {67} \right] + 1}} – {l_{69}}}}{{{l_{\left[ {67} \right] + 1}}}}\)
\(0,03 = \frac{{{l_{\left[ {67} \right] + 1}} – 7700}}{{{l_{\left[ {67} \right] + 1}}}}\)
\(0,03 \cdot {l_{\left[ {67} \right] + 1}} = {l_{\left[ {67} \right] + 1}} – 7700\)
\(0,97 \cdot {l_{\left[ {67} \right] + 1}} = 7700\)
\({l_{\left[ {67} \right] + 1}} = \frac{{7700}}{{0,97}}\)
\(= 7938,144\)
\({q_{\left[ {67} \right]}} = \frac{{{l_{\left[ {67} \right]}} – {l_{_{\left[ {67} \right] + 1}}}}}{{{l_{\left[ {67} \right]}}}}\)
\(0,01463 = \frac{{{l_{\left[ {67} \right]}} – 7938,144}}{{{l_{\left[ {67} \right]}}}}\)
\(0,01463 \cdot {l_{\left[ {67} \right] + 1}} = {l_{\left[ {67} \right] + 1}} – 7938,144\)
\(0,98537 \cdot {l_{\left[ {67} \right] + 1}} = 7938,144\)
\({l_{\left[ {67} \right] + 1}} = \frac{{7938,144}}{{0,98537}} = 8056,037\) |
| Jawaban | c. 8060 |
