Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | Mei 2018 |
| Nomor Soal | : | 2 |
SOAL
Untuk sebuah tabel double decrement, diberikan
| Usia x | \(l_x^{\left( \tau \right)}\) | \(d_x^{\left( 1 \right)}\) | \(d_x^{\left( 2 \right)}\) |
| 20 | 1000 | 70 | 100 |
| 21 | | 66 | |
| 22 | 650 | | |
setiap decrement menyebar seragam untuk tiap usia. Hitunglah \(q’\)\({}_{21}^{\left( 2 \right)}\)
- 0,1434
- 0,156
- 0,176
- 0,18
- 0,2
| Diketahui | | Usia x | \(l_x^{\left( \tau \right)}\) | \(d_x^{\left( 1 \right)}\) | \(d_x^{\left( 2 \right)}\) | | 20 | 1000 | 70 | 100 | | 21 | | 66 | | | 22 | 650 | | |
dan setiap decrement menyebar seragam (UDD) |
| Rumus yang digunakan | Untuk kasus UDD maka
\(_tp_x^{‘(j)} = 1{ – _t}q_x^{‘(j)}\), \(d_x^{(\tau )} = l_x^{(\tau )} – l_{x + 1}^{(\tau )}\), \(l_x^{(\tau )} = \sum\limits_{j = 1}^n {l_x^{(j)}} \)latex , \(d_x^{(\tau )} = \sum\limits_{j = 1}^m {d_x^{(j)}} \)
\(d_x^{(j)} = l_x^{(\tau )}q_x^{(j)},d_x^{(\tau )} = l_x^{(\tau )}q_x^{(\tau )},\)
\(_{{\rm{ }}s}P{‘_x}^{(j)} = {{(_s}{P_x}^{(\tau )})^{q_x^{(j)}/q_x^{(\tau )}}}\) |
| Proses pengerjaan | Dengan mengaplikasi rumus di atas diperoleh
\(d_{20}^{(\tau )} = \sum\limits_{j = 1}^2 {d_x^{(j)} = 170} \)
\(d_{20}^{(\tau )} = l_{20}^{(\tau )} – l_{21}^{(\tau )}\) maka
\(l_{21}^{(\tau )} = l_{20}^{(\tau )} – d_{20}^{(\tau )} = 1000 – 170 = 830\) selanjutnya
\(d_{21}^{(\tau )} = l_{21}^{(\tau )} – l_{22}^{(\tau )} = 830 – 650 = 180\)
\(d_{21}^{(\tau )} = d_{21}^{(1)} + d_{21}^{(2)}\) maka
\(d_{21}^{(2)} = 180 – 66 = 114\)
\(q_{21}^{(2)} = \frac{{d_{21}^{(2)}}}{{l_{21}^{(\tau )}}} = \frac{{114}}{{830}}\)
\(q_{21}^{(\tau )} = \frac{{d_{21}^{(\tau )}}}{{l_{21}^{(\tau )}}} = \frac{{180}}{{830}}\) Selanjutnya dipunyai \(q’\)\({}_{21}^{\left( 2 \right)}\)
artinya
\(q’\)\({}_{21}^{\left( 2 \right)}\)\(= 1 – \) \({p’}\)\({}_{21}^{\left( 2 \right)}\)
\(= 1 – {({P_{21}}^{(\tau )})^{q_{21}^{(2)}/q_{21}^{(\tau )}}}\)
\(= 1 – {(\frac{{650}}{{830}})^{\frac{{(\frac{{114}}{{830}})}}{{(\frac{{180}}{{830}})}}}}\)
\(= 0.1434\) |
| Jawaban | a. 0,1434 |
