Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Matematika Aktuaria |
| Periode Ujian |
: |
Juni 2015 |
| Nomor Soal |
: |
29 |
SOAL
Diberikan sebagai berikut
- \(\begin{array}{*{20}{c}} {{\mu _{x + t}} = 0,01;}&{0 \le t < 5} \end{array}\)
- \(\begin{array}{*{20}{c}} {{\mu _{x + t}} = 0,02;}&{5 \le t} \end{array}\)
- \(\delta = 0,06\)
Hitunglah \({\bar a_x}\)
- 12,5
- 13,0
- 13,4
- 13,9
- 14,3
| Diketahui |
- \(\begin{array}{*{20}{c}} {{\mu _{x + t}} = 0,01;}&{0 \le t < 5} \end{array}\)
- \(\begin{array}{*{20}{c}} {{\mu _{x + t}} = 0,02;}&{5 \le t} \end{array}\)
- \(\delta = 0,06\)
|
| Rumus yang digunakan |
Untuk \(\delta \) konstan
\({{\bar a}_x} = {{\bar a}_{x:\left. {\overline {\, n \,}}\! \right| }} + {}_{\left. n \right|}{{\bar a}_x}\)
\({{\bar a}_{x:\left. {\overline {\, n \,}}\! \right| }} = {{\bar a}_x}\left( {1 – {}_n{E_x}} \right) = \frac{{1 – \exp \left[ { – n\left( {\mu + \delta } \right)} \right]}}{{\mu + \delta }}\)
\({}_{\left. n \right|}{{\bar a}_x} = {}_n{E_x} \cdot {{\bar a}_x} = \frac{{\exp \left[ { – n\left( {\mu + \delta } \right)} \right]}}{{\mu + \delta }}\) |
| Proses pengerjaan |
Bagi annuitas \({\bar a_x}\) menjadi 2 bagian annuitas yaitu 5-years temporary life dan 5 years deffered life.
- Temporray life
\({\bar a_{x:\left. {\overline {\, 5 \,}}\! \right| }} = \frac{{1 – \exp \left[ { – 5\left( {0.01 + 0.06} \right)} \right]}}{{0.01 + 0.06}} = 4.21874\)
- Deffered dengan memperhitungkan dari tahun ke-5 ke tahun ke-0 dengan \(\mu = 0.01\)
\({}_{\left. 5 \right|}{\bar a_x} = \frac{{\exp \left[ { – 5\left( {0.01 + 0.06} \right)} \right]}}{{0.02 + 0.06}} = 8.80860\) |
|
\({\bar a_x} = {\bar a_{x:\left. {\overline {\, 5 \,}}\! \right| }} + {}_{\left. 5 \right|}{\bar a_x} = 4.21874 + 8.80860 = 13.0273\) |
| Jawaban |
b. 13,0 |
