Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Matematika Aktuaria |
| Periode Ujian | : | Mei 2018 |
| Nomor Soal | : | 19 |
SOAL
Diketahui untuk sebuah select and ultimate mortality model dengan periode seleksi 1 tahun, bahwa \({P_{\left[ x \right]}} = \left( {1 + k} \right){p_x}\) untuk suatu konstanta k. Jika \({\ddot a_{x:\left. {\overline {\, n \,}}\! \right| }} = 21,854\) dan \({\ddot a_{\left[ x \right]:\left. {\overline {\, n \,}}\! \right| }} = 22,167\)
tentukanlah k!
- 0,015
- 0,020
- 0,025
- 0,030
- 0,035
| Diketahui | - Seleksi 1 tahun
- \({\ddot a_{x:\left. {\overline {\, n \,}}\! \right| }} = 21,854\)
- \({\ddot a_{\left[ x \right]:\left. {\overline {\, n \,}}\! \right| }} = 22,167\)
- \({P_{\left[ x \right]}} = \left( {1 + k} \right){p_x}\)
|
| Step 1 | \({\ddot a_{x:\left. {\overline {\, n \,}}\! \right| }} = \sum {{v^n}{ \cdot _n}{p_x}} = 1 + v \cdot {p_x} + {v^2} \cdot {p_x} \cdot {p_{x + 1}} + …\) |
| Step 2 | \({\ddot a_{\left[ x \right]:\left. {\overline {\, n \,}}\! \right| }} = \sum {{v^n}{ \cdot _n}{p_{\left[ x \right]}}} \) \(= 1 + v \cdot {p_{\left[ x \right]}} + {v^2} \cdot {p_{\left[ x \right]}} \cdot {p_{x + 1}} + …\)
\(= 1 + v \cdot \left( {1 + k} \right){p_x} + {v^2} \cdot \left( {1 + k} \right){p_x} \cdot {p_{x + 1}} + …\)
\(= \left( {1 + k} \right)\left[ {\underbrace {1 + v \cdot {p_x} + {v^2} \cdot {p_x} \cdot {p_{x + 1}} + …}_{{{\ddot a}_{x:\left. {\overline {\, n \,}}\! \right| }}}} \right] – k\)
\(22,167 = \left( {1 + k} \right) \cdot 21,854 – k\)
\(22,167 = 21,854 + 21,854k – k\)
\(22,167 – 21,854 = 20,854k\)
\(\frac{{0,313}}{{20,854}} = k\)
\(0,015 = k\) |
| Jawaban | a. 0,015 |
