Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | A60 – Matematika Aktuaria |
| Periode Ujian | : | November 2017 |
| Nomor Soal | : | 11 |
SOAL
Manakah dari pernyataan berikut yang benar dari \(\frac{d}{{dt}}{}_t\bar V({\bar A_x})\)
- \(\frac{{{{\bar A}_{x + t}} + {{\bar a}_{x + t}}{\mu _{x + t}}}}{{{{\bar a}_x}}}\)
- \(\frac{{{{\bar A}_{x + t}} – {{\bar a}_{x + t}}{\mu _{x + t}}}}{{{{\bar a}_x}}}\)
- \(\frac{{1 – \delta {{\bar a}_{x + t}} – {{\bar a}_{x + t}}{\mu _{x + t}}}}{{{{\bar a}_{x + t}}}}\)
- \(\frac{{1 – \delta {{\bar a}_{x + t}} + {{\bar a}_{x + t}}{\mu _{x + t}}}}{{{{\bar a}_{x + t}}}}\)
- \(\frac{{1 – {\mu _{x + t}}}}{{{{\bar a}_{x + t}}}}\)
PEMBAHASAN
| Rumus | \({}_t\bar V({\bar A_x})\, = \,1 – \frac{{{{\bar a}_{x + t}}}}{{{{\bar a}_x}}}\)
\(\frac{d}{{dt}}{\bar a_{x + t}} = {\bar a_{x + t}}\left( {{\mu _{x + t}} + \delta } \right) – 1\)
\(1 – \delta {\bar a_{x + t}} = {\bar A_{x + t}}\) |
| Kalkulasi | \(\frac{d}{{dt}}{}_t\bar V({\bar A_x}) = \frac{d}{{dt}}\left( {1 – \frac{{{{\bar a}_{x + t}}}}{{{{\bar a}_x}}}} \right)\)
\(\frac{d}{{dt}}{}_t\bar V({\bar A_x}) = – \frac{1}{{{{\bar a}_x}}}\left( {\frac{d}{{dt}}{{\bar a}_{x + t}}} \right)\)
\(\frac{d}{{dt}}{}_t\bar V({\bar A_x}) = – \frac{1}{{{{\bar a}_x}}}\left( {{{\bar a}_{x + t}}\left( {{\mu _{x + t}} + \delta } \right) – 1} \right)\)
\(\frac{d}{{dt}}{}_t\bar V({\bar A_x}) = \frac{{1 – \delta {{\bar a}_{x + t}} – {{\bar a}_{x + t}}{\mu _{x + t}}}}{{{{\bar a}_x}}}\)
\(\frac{d}{{dt}}{}_t\bar V({\bar A_x}) = \frac{{{{\bar A}_{x + t}} – {{\bar a}_{x + t}}{\mu _{x + t}}}}{{{{\bar a}_x}}}\) |
| Jawaban | b. \(\frac{{{{\bar A}_{x + t}} – {{\bar a}_{x + t}}{\mu _{x + t}}}}{{{{\bar a}_x}}}\) |
