Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Matematika Aktuaria |
| Periode Ujian |
: |
Juni 2016 |
| Nomor Soal |
: |
10 |
SOAL
Diberikan sebagai berikut:
- \(Z\left( n \right)\) adalah “present value random variabel for n-year term insurance on a life age \(x\)” berdasarkan yield curve sekarang
- \(E\left[ {Z\left( 1 \right)} \right] = 0,014354\) dan \(E\left[ {Z\left( 2 \right)} \right] = 0,032308\)
- “current one-year spot rate” adalah 4,50%
- \({q_{x + 1}} = 0,02\)
Hitunglah “current two year spot rate” (pembulatan terdekat)
-
- 4,55%
- 4,75%
- 4,95%
- 5,15%
- 5,35%
| Diketahui |
Diberikan sebagai berikut:
- \(Z\left( n \right)\) adalah “present value random variabel for n-year term insurance on a life age \(x\)” berdasarkan yield curve sekarang
- \(E\left[ {Z\left( 1 \right)} \right] = 0,014354\) dan \(E\left[ {Z\left( 2 \right)} \right] = 0,032308\)
- “current one-year spot rate” adalah 4,50%
- \({q_{x + 1}} = 0,02\)
|
| Rumus yang digunakan |
\(E\left[ Z \right] = \sum\limits_{k = 0}^{n – 1} {{v^{k + 1}} \cdot {}_k{p_x} \cdot {q_{x + k}}} \) |
| Proses pengerjaan |
\(E\left[ {Z\left( 1 \right)} \right] = \sum\limits_{k = 0}^0 {{v^{k + 1}} \cdot {}_k{p_x} \cdot {q_{x + k}}} = {v_{4.5\% }} \cdot {q_x}\)
\(0.014354 = \frac{{{q_x}}}{{1.045}}\)
\({q_x} = 0.015\)
\({p_x} = 0.985\)
\(E\left[ {Z\left( 2 \right)} \right] = \sum\limits_{k = 0}^1 {{v^{k + 1}} \cdot {}_k{p_x} \cdot {q_{x + k}}} = {v_{4.5\% }} \cdot {q_x} + {v^2} \cdot {p_x} \cdot {q_{x + 1}}\)
\(0.032308 = 0.014354 + \frac{{\left( {0.985} \right)\left( {0.02} \right)}}{{{{\left( {1 + i} \right)}^2}}}\)
\(i = \sqrt {\frac{{\left( {0.985} \right)\left( {0.02} \right)}}{{0.032308 – 0.014354}}} – 1\)
\(i = 0.0475 = 4.75\% \) |
| Jawaban |
B. 4,75% |
