Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | November 2017 |
| Nomor Soal | : | 5 |
SOAL
Diketahui tabel mortalita dengan periode seleksi 2 tahun sebagai berikut:
| \(x\) | \({q_{\left[ x \right]}}\) | \({q_{\left[ x \right] + 1}}\) | \({q_{x + 2}}\) | \(x + 2\) |
| 50 | 0,0060 | 0,0053 | 0,0070 | 52 |
| 51 | 0,0070 | 0,0063 | 0,0080 | 53 |
| 52 | 0,0080 | 0,0073 | 0,0090 | 54 |
| 53 | 0,0090 | 0,0083 | 0,0100 | 55 |
Jika force of mortality adalah konstan, hitunglah \(1.000{}_{2,5}{q_{\left[ {50} \right] + 0,4}}\) (dibulatkan 2 desimal).
- 11,17
- 12,96
- 14,35
- 15,13
- 16,42
| Diketahui | | \(x\) | \({q_{\left[ x \right]}}\) | \({q_{\left[ x \right] + 1}}\) | \({q_{x + 2}}\) | \(x + 2\) | | 50 | 0,0060 | 0,0053 | 0,0070 | 52 | | 51 | 0,0070 | 0,0063 | 0,0080 | 53 | | 52 | 0,0080 | 0,0073 | 0,0090 | 54 | | 53 | 0,0090 | 0,0083 | 0,0100 | 55 |
|
| Rumus yang digunakan | \({}_t{p_x} = {}_{t + x}{p_0}\)
\({}_t{p_x} = {p_x}{p_{x + 1}}{p_{x + 2}} \cdots {p_{x + t – 1}}\)
Untuk force of mortality konstan dan usia bukan bilangan bulat
\({}_s{p_x} = {\left( {{p_x}} \right)^s}\)
\({}_s{p_{x + t}} = \frac{{{}_{s + t}{p_x}}}{{{}_t{p_x}}} = \frac{{{}_{s + t}{p_x}}}{{{{\left( {{p_x}} \right)}^t}}}\) |
| Proses pengerjaan | \({}_{2,5}{q_{\left[ {50} \right] + 0,4}} = 1 – {}_{2,5}{p_{\left[ {50} \right] + 0,4}}\)
\(= 1 – \frac{{{}_{2,9}{p_{\left[ {50} \right]}}}}{{{{\left( {{p_{\left[ {50} \right]}}} \right)}^{0,4}}}}\)
\(= 1 – \frac{{{p_{\left[ {50} \right]}} \cdot {p_{\left[ {50} \right] + 1}} \cdot {{\left( {{p_{52}}} \right)}^{0,9}}}}{{{{\left( {1 – {q_{\left[ {50} \right]}}} \right)}^{0,4}}}}\)
\(= 1 – \frac{{\left( {1 – {q_{\left[ {50} \right]}}} \right) \cdot \left( {1 – {q_{\left[ {50} \right] + 1}}} \right) \cdot {{\left( {1 – {q_{52}}} \right)}^{0,9}}}}{{{{\left( {1 – {q_{\left[ {50} \right]}}} \right)}^{0,4}}}}\)
\(= 1 – \frac{{\left( {1 – 0,006} \right) \cdot \left( {1 – 0,0053} \right) \cdot {{\left( {1 – 0,007} \right)}^{0,9}}}}{{{{\left( {1 – 0,006} \right)}^{0,4}}}}\)
\(= 0,0151314\)
\(1000{}_{2,5}{q_{\left[ {50} \right] + 0,4}} = 15,1314\) |
| Jawaban | d. 15,13 |
