Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Matematika Aktuaria |
| Periode Ujian | : | November 2017 |
| Nomor Soal | : | 22 |
SOAL
Untuk suatu anuitas yang dibayarkan semi tahunan, diberikan sebagai berikut:
- Kematian berdistribusi “uniform” untuk setiap usia
- \({q_{69}} = 0,03\)
- \(i = 0,06\)
- \(1.000{\bar A_{70}} = 530\)
Hitunglah nilai dari \(\ddot a_{69}^{(2)}\)
- 8,35
- 8,47
- 8,59
- 8,72
- 8,85
| Rumus | \(\ddot a_x^{(m)} = {\ddot a_x} – \frac{{(m – 1)}}{{2m}}\)
\({A_x} = 1 – d\,{\ddot a_x}\)
\({A_x} = v\,{q_x} + v\,{p_x}\,{A_{x + 1}}\)
\({\bar A_x} = \frac{i}{\delta }{A_x}\) |
| Step 1 | \({\bar A_{70}} = \frac{i}{\delta }{A_{70}}\)
\({\bar A_{70}} = \frac{{0,06}}{{\ln (1 + 0,06)}}{A_{70}}\) |
| \(1.000{\bar A_{70}} = 530\)
\({\bar A_{70}} = 0,53\) |
| \(0,53 = \frac{{0,06}}{{\ln (1 + 0,06)}}{A_{70}}\)
\({A_{70}} = \frac{{\ln (1 + 0,06)}}{{0,06}}\left( {0,53} \right)\)
\({A_{70}} \cong {\rm{0}}{\rm{,51471}}\) |
| Step 2 | \({A_{69}} = v\,{q_{69}} + v\,{p_{69}}\,{A_{70}}\)
\({A_{69}} = \,\frac{{0,03}}{{(1 + 0,06)}} + \frac{{(1 – 0,03)}}{{(1 + 0,06)}}\,\left( {0,51471} \right)\)
\({A_{69}} \cong {\rm{0}}{\rm{,49931}}\) |
| Step 3 | \({A_{69}} = 1 – d\,{\ddot a_{69}}\)
\(0,49931 = 1 – \frac{{0,06}}{{1,06}}{\ddot a_{69}}\)
\({\ddot a_{69}} = \left( {1 – 0,49931} \right)\frac{{1,06}}{{0,06}}\)
\({\ddot a_{69}} \cong {\rm{8}}{\rm{,84552}}\) |
| Step 4 | \(\ddot a_{69}^{(2)} = {\ddot a_{69}} – \frac{{(2 – 1)}}{{2(2)}}\)
\(\ddot a_{69}^{(2)} = (8,84552) – \frac{1}{4}\)
\(\ddot a_{69}^{(2)} = {\rm{8}}{\rm{,59552}}\)
\(\ddot a_{69}^{(2)} \cong {\rm{8}}{\rm{,59}}\) |
| Jawaban | C. 8,59 |
