Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
Novemmber 2015 |
| Nomor Soal |
: |
15 |
SOAL
Di bawah ini adalah tabel untuk double decrement \(l_{40}^{\left( \tau \right)} = 2.200\) dengan
| \(x\) |
\(q_x^{\left( d \right)}\) |
\(q_x^{\left( w \right)}\) |
\(q_x^{‘\left( d \right)}\) |
\(q_x^{‘\left( w \right)}\) |
| 40 |
0,24 |
0,12 |
0,26 |
\(y\) |
| 41 |
— |
— |
0,20 |
\(2y\) |
Hitunglah \(l_{42}^{\left( \tau \right)}\) (dibulatkan ke satuan terdekat)
- 803
- 822
- 840
- 860
- 880
| Diketahui |
\(l_{40}^{\left( \tau \right)} = 2.200\) dengan
| \(x\) |
\(q_x^{\left( d \right)}\) |
\(q_x^{\left( w \right)}\) |
\(q_x^{‘\left( d \right)}\) |
\(q_x^{‘\left( w \right)}\) |
| 40 |
0,24 |
0,12 |
0,26 |
\(y\) |
| 41 |
— |
— |
0,20 |
\(2y\) |
|
| Rumus yang digunakan |
\(q_x^{\left( d \right)} = q_x^{‘\left( d \right)}\left[ {1 – \frac{1}{2} \cdot q_x^{‘\left( w \right)}} \right]\) dan \(q_x^{\left( w \right)} = q_x^{‘\left( w \right)}\left[ {1 – \frac{1}{2} \cdot q_x^{‘\left( d \right)}} \right]\)
\({}_tp_x^{\left( \tau \right)} = {}_tp_x^{‘\left( d \right)} \cdot {}_tp_x^{‘\left( w \right)}\) dengan \({}_tp_x^{‘\left( j \right)} = 1 – {}_tq_x^{‘\left( j \right)}\)
\(l_{x + 1}^{\left( \tau \right)} = l_x^{\left( \tau \right)} \cdot p_x^{\left( \tau \right)}\) |
| Proses pengerjaan |
\(q_{40}^{\left( w \right)} = q_{40}^{‘\left( w \right)}\left[ {1 – \frac{1}{2} \cdot q_{40}^{‘\left( d \right)}} \right]\)
\(0,12 = y\left[ {1 – \frac{{0,26}}{2} \cdot } \right]\)
\(y = \frac{{0,12}}{{0,87}} = 0,137931\)
sehingga \(q_{40}^{‘\left( w \right)} = 0,137931\) dan \(q_{41}^{‘\left( w \right)} = 2y = 2\left( {\frac{4}{{29}}} \right) = 0,275862\) |
| \(l_{42}^{\left( \tau \right)} = l_{41}^{\left( \tau \right)} \cdot p_{41}^{\left( \tau \right)} = l_{40}^{\left( \tau \right)} \cdot p_{40}^{\left( \tau \right)} \cdot p_{41}^{\left( \tau \right)}\)
\(= l_{40}^{\left( \tau \right)} \cdot p_{40}^{‘\left( d \right)} \cdot p_{40}^{‘\left( w \right)} \cdot p_{41}^{\left( d \right)} \cdot p_{41}^{\left( w \right)}\)
\(= 2.200 \cdot \left( {1 – 0,26} \right) \cdot \left( {1 – \frac{4}{{29}}} \right) \cdot \left( {1 – 0,20} \right) \cdot \left( {1 – \frac{8}{{29}}} \right)\)
\(= 813,0321046\) |
| Jawaban |
b. 822 |
