Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Matematika Aktuaria |
| Periode Ujian | : | November 2015 |
| Nomor Soal | : | 6 |
SOAL
Diberikan :
- Kematian berdistribusi seragam untuk setiap tahun usia
- \(i = 0,10\)
- \({q_x} = 0,05\)
- \({q_{x + 1}} = 0,08\)
Hitunglah \(\bar A_{x:\left. {\overline {\, 2 \,}}\! \right| }^1\) (pembulatan terdekat)
- 0,103
- 0,108
- 0,111
- 0,114
- 0,119
| Diketahui | - Kematian berdistribusi seragam untuk setiap tahun usia
- \(i = 0,10\)
- \({q_x} = 0,05\)
- \({q_{x + 1}} = 0,08\)
|
| Rumus yang digunakan | \(A_{x:\left. {\overline {\, n \,}}\! \right| }^1 = \sum\limits_{k = 0}^{n – 1} {{v^{k + 1}}{}_k{p_x} \cdot {q_{x + k}}} \) dengan \({v^k} = \frac{1}{{{{\left( {1 + i} \right)}^k}}}\) dan \({}_t{p_x} = \prod\limits_{k = 0}^{t – 1} {{p_{x + k}}} \)
\(\bar A_{x:\left. {\overline {\, n \,}}\! \right| }^1 = \frac{i}{\delta }A_{x:\left. {\overline {\, n \,}}\! \right| }^1\) dengan \(\delta = \ln \left( {1 + i} \right)\) |
| Proses pengerjaan | \(A_{x:\left. {\overline {\, 2 \,}}\! \right| }^1 = \sum\limits_{k = 0}^1 {{v^{k + 1}}{}_k{p_x} \cdot {q_{x + k}}} = v \cdot {}_0{p_x} \cdot {q_x} + {v^2} \cdot {}_1{p_x} \cdot {q_{x + 1}}\)
\(A_{x:\left. {\overline {\, 2 \,}}\! \right| }^1 = \frac{{\left( 1 \right)\left( {0.05} \right)}}{{1.10}} + \frac{{\left( {1 – 0.05} \right)\left( {0.08} \right)}}{{{{1.10}^2}}}\)
\(A_{x:\left. {\overline {\, 2 \,}}\! \right| }^1 = 0.108264\) |
| \(\bar A_{x:\left. {\overline {\, 2 \,}}\! \right| }^1 = \frac{i}{\delta }A_{x:\left. {\overline {\, 2 \,}}\! \right| }^1 = \frac{{0.10}}{{\ln \left( {1.10} \right)}}\left( {0.108264} \right)\)
\(\bar A_{x:\left. {\overline {\, 2 \,}}\! \right| }^1 = 0.113592\) |
| Jawaban | d. 0,114 |
Optimized by Seraphinite Accelerator
