Pembahasan Soal Ujian Profesi Aktuaris
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. \(4\left[ {1 – {{\left( {1 + \frac{{{i^{\left( 3 \right)}}}}{3}} \right)}^{ – \frac{3}{4}}}} \right]\) |