Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Matematika Keuangan |
| Periode Ujian |
: |
September 2012 |
| Nomor Soal |
: |
4 |
SOAL
Nyatakan \({d^{\left( 4 \right)}}\) sebagai fungsi dari \({i^{\left( 3 \right)}}\)
- \(4\left[ {1 – {{\left( {1 + \frac{{{i^{\left( 3 \right)}}}}{3}} \right)}^{ – \frac{3}{4}}}} \right]\)
- \(3\left[ {1 – {{\left( {1 + \frac{{{i^{\left( 4 \right)}}}}{4}} \right)}^{ – \frac{3}{4}}}} \right]\)
- \(4\left[ {1 + {{\left( {1 + \frac{{{i^{\left( 3 \right)}}}}{3}} \right)}^{\frac{3}{4}}}} \right]\)
- \(3\left[ {1 + {{\left( {1 + \frac{{{i^{\left( 4 \right)}}}}{4}} \right)}^{ – \frac{3}{4}}}} \right]\)
| Rumus yang digunakan |
\({\left( {1 – \frac{{{d^{\left( p \right)}}}}{p}} \right)^{ – p}} = {\left( {1 + \frac{{{i^{\left( n \right)}}}}{n}} \right)^n}\) untuk n dan p bilangan bulat |
| Proses pengerjaan |
\({\left( {1 – \frac{{{d^{\left( 4 \right)}}}}{4}} \right)^{ – 4}} = {\left( {1 + \frac{{{i^{\left( 3 \right)}}}}{3}} \right)^3}\)
\(\left( {1 – \frac{{{d^{\left( 4 \right)}}}}{4}} \right) = {\left( {1 + \frac{{{i^{\left( 3 \right)}}}}{3}} \right)^{ – \frac{3}{4}}}\)
\(– \frac{{{d^{\left( 4 \right)}}}}{4} = {\left( {1 + \frac{{{i^{\left( 3 \right)}}}}{3}} \right)^{ – \frac{3}{4}}} – 1\)
\(\frac{{{d^{\left( 4 \right)}}}}{4} = 1 – {\left( {1 + \frac{{{i^{\left( 3 \right)}}}}{3}} \right)^{ – \frac{3}{4}}}\)
\({d^{\left( 4 \right)}} = 4\left[ {1 – {{\left( {1 + \frac{{{i^{\left( 3 \right)}}}}{3}} \right)}^{ – \frac{3}{4}}}} \right]\) |
| Jawaban |
a. \(4\left[ {1 – {{\left( {1 + \frac{{{i^{\left( 3 \right)}}}}{3}} \right)}^{ – \frac{3}{4}}}} \right]\) |
