Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Matematika Keuangan |
| Periode Ujian |
: |
Mei 2018 |
| Nomor Soal |
: |
5 |
SOAL
Tentukan kondisi yang harus dipenuhi oleh \(n,p,q,{\rm{ dan }}r\) sehingga persamaan berikut benar.
\({a_{\left. {\overline {\, n \,}}\! \right| }} = ({a_{\left. {\overline {\, p \,}}\! \right| }} + {S_{\left. {\overline {\, q \,}}\! \right| }}){v^r}\)
- \(p + q + r = n\)
- \(p + q = n{\rm{ dan }}n = r\)
- \(p + q = n{\rm{ dan }}p = r\)
- \(p + q = n{\rm{ dan }}q = r\)
- Tidak ada kondisi yang benar
| Diketahui |
\({a_{\left. {\overline {\, n \,}}\! \right| }} = ({a_{\left. {\overline {\, p \,}}\! \right| }} + {S_{\left. {\overline {\, q \,}}\! \right| }}){v^r}\) |
| Rumus yang digunakan |
\({a_{\left. {\overline {\, n \,}}\! \right| }} = \frac{{1 – {v^n}}}{i}\)
\({S_{\left. {\overline {\, n \,}}\! \right| }} = \frac{{{{(1 + i)}^n} – 1}}{i} = \frac{{{v^{ – n}} – 1}}{i}\)
\({v^t} = {(1 + i)^{ – t}}\) |
| Proses Pengerjaan |
\({a_{\left. {\overline {\, n \,}}\! \right| }} = ({a_{\left. {\overline {\, p \,}}\! \right| }} + {S_{\left. {\overline {\, q \,}}\! \right| }}){v^r}\)
\(\Leftrightarrow \frac{{1 – {v^n}}}{i} = (\frac{{1 – {v^p}}}{i} + \frac{{{v^{ – q}} – 1}}{i}){v^r}\)
\(\Leftrightarrow \frac{{1 – {v^n}}}{i} = (\frac{{{v^{ – q}} – {v^p}}}{i}){v^r}\)
\(\Leftrightarrow 1 – {v^n} = ({v^{ – q}} – {v^p}){v^r}\)
\(\Leftrightarrow 1 – {v^n} = {v^{ – q + r}} – {v^{p + r}}\)
\(\Leftrightarrow 1 – {v^n} = {v^{ – q + r}} – {v^{p + r}}{\rm{ (*)}}\)
\({\rm{dari (*) diperoleh}}\)
\({v^{ – q + r}} = 1\)
\(\Leftrightarrow – q + r = 0\)
\(\Leftrightarrow r = q\)
\({\rm{selanjutnya }}\)
\({v^n}{\rm{ = }}{v^{p + q}}\)
\(\Leftrightarrow n = p + q\) |
| Jawaban |
d. \(p + q = n{\rm{ dan }}q = r\) |
