Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
November 2018 |
| Nomor Soal |
: |
17 |
SOAL
Diberikan informasi
\({l_x} = 10.000,{\rm{ }}{{\rm{L}}_{x + 1}} = 8.000,{\rm{ }}{q_{x + 1}} = 0,25\)
\({l_{x + 1}} = 8.100,{\rm{ }}{{\rm{L}}_{x + 2}} = 6.000,{\rm{ }}{m_{x + 2}} = 0,3645\)
Tentukanlah \({}_2{p_{x + 0,5}}\) dengan menggunakan metode eksponensial (constant force)
- 0,44
- 0,46
- 0,54
- 0,56
- 0,64
| Diketahui |
\({l_x} = 10.000,{\rm{ }}{{\rm{L}}_{x + 1}} = 8.000,{\rm{ }}{q_{x + 1}} = 0,25\)
\({l_{x + 1}} = 8.100,{\rm{ }}{{\rm{L}}_{x + 2}} = 6.000,{\rm{ }}{m_{x + 2}} = 0,3645\) |
| Rumus yang digunakan |
\({q_x} = 1 – {p_x}\)
\(= 1 – \frac{{{l_{x + 1}}}}{{{l_x}}}\)
\({m_x} = \frac{{{d_x}}}{{{L_x}}}\)
\(= \frac{{{l_x} – {l_{x + 1}}}}{{{L_x}}}\)
\({}_t{p_{x + s}} = \frac{{{l_{x + s + t}}}}{{{l_{x + s}}}}\)
\(= \frac{{{l_{x + t}}{{\left( {{p_{x + t}}} \right)}^s}}}{{{l_x}{{\left( {{p_x}} \right)}^s}}}\) |
| Proses Pengerjaan |
Pertama,
\({q_{x + 1}} = 1 – {p_{x + 1}}\)
\(= 1 – \frac{{{l_{x + 2}}}}{{{l_{x + 1}}}}\)
\(0,25 = \frac{{{l_{x + 1}} – {l_{x + 2}}}}{{{l_{x + 1}}}}\)
\(0,25 \cdot {l_{x + 1}} = {l_{x + 1}} – {l_{x + 2}}\)
\({l_{x + 2}} = {l_{x + 1}} – 0,25 \cdot {l_{x + 1}}\)
\(= 0,75 \cdot {l_{x + 1}}\)
\(= 0,75 \cdot 8.100 = 6.075\) |
|
Kedua,
\({m_x} = \frac{{{d_x}}}{{{L_x}}} = \frac{{{l_x} – {l_{x + 1}}}}{{{L_x}}}{\rm{ }}\) sehingga
\({m_{x + 2}} = \frac{{{l_{x + 2}} – {l_{x + 3}}}}{{{L_{x + 2}}}}\)
\(0,3645 = \frac{{6.075 – {l_{x + 3}}}}{{6.000}}\)
\({l_{x + 3}} = 6.075 – 0,3645 \cdot 6.000\)
\(= 3.888\) |
|
ketiga,
\({}_2{p_{x + 0,5}} = \frac{{{l_{x + 0,5 + 2}}}}{{{l_{x + 0,5}}}}\)
\(= \frac{{{l_{x + 2}}{{\left( {{p_{x + 2}}} \right)}^{\frac{1}{2}}}}}{{{l_x}{{\left( {{p_x}} \right)}^{\frac{1}{2}}}}}\)
\(= \frac{{{l_{x + 2}}{{\left( {\frac{{{l_{x + 3}}}}{{{l_{x + 2}}}}} \right)}^{\frac{1}{2}}}}}{{{l_x}{{\left( {\frac{{{l_{x + 1}}}}{{{l_x}}}} \right)}^{\frac{1}{2}}}}}\)
\(= \frac{{6.075{{\left( {\frac{{3.888}}{{6.075}}} \right)}^{\frac{1}{2}}}}}{{10.000{{\left( {\frac{{8.100}}{{10.000}}} \right)}^{\frac{1}{2}}}}} = \frac{{4860}}{{9000}} = 0,54\) |
| Jawaban |
c. 0,54 |
