Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Matematika Aktuaria |
| Periode Ujian |
: |
November 2015 |
| Nomor Soal |
: |
20 |
SOAL
Suatu “5-year temporary life annuity-immediate” pada \(\left( x \right)\) membayar 10 per tahun, diberikan
- \(A_{x:\left. {\overline {\, 5 \,}}\! \right| }^1 = 0,04\)
- \({}^2A_{x:\left. {\overline {\, 5 \,}}\! \right| }^1 = 0,03\)
- \({}_5{p_x} = 0,94\)
- \(i = 0,05\)
Hitunglah variansi dari “present value 5-year annuity immediate” (pembulatan terdekat)
- 53,8
- 73,8
- 120,8
- 162,8
- 200,8
| Diketahui |
Suatu “5-year temporary life annuity-immediate” pada \(\left( x \right)\) membayar 10 per tahun, diberikan
- \(A_{x:\left. {\overline {\, 5 \,}}\! \right| }^1 = 0,04\)
- \({}^2A_{x:\left. {\overline {\, 5 \,}}\! \right| }^1 = 0,03\)
- \({}_5{p_x} = 0,94\)
- \(i = 0,05\)
|
| Rumus yang digunakan |
\({A_{x:\left. {\overline {\, n \,}}\! \right| }} = A_{x:\left. {\overline {\, n \,}}\! \right| }^1 + {A_{x:\mathop {\left. {\overline {\, n \,}}\! \right| }\limits^1 }} = A_{x:\left. {\overline {\, n \,}}\! \right| }^1 + {v^n}{}_n{p_x}\)
\(Var\left[ Z \right] = \frac{{{}^2{A_{x:\left. {\overline {\, n \,}}\! \right| }} – {{\left( {{A_{x:\left. {\overline {\, n \,}}\! \right| }}} \right)}^2}}}{{{d^2}}}\) dengan \(d = \frac{i}{{1 + i}}\) |
| Proses pengerjaan |
\({A_{x:\left. {\overline {\, 5 \,}}\! \right| }} = 10\left( {A_{x:\left. {\overline {\, 5 \,}}\! \right| }^1 + {v^6}{}_5{p_x}} \right)\)
\({A_{x:\left. {\overline {\, 5 \,}}\! \right| }} = 10\left( {0.04 + \frac{{0.94}}{{{{1.05}^6}}}} \right) = 7.41442\) |
|
\({}^2{A_{x:\left. {\overline {\, 5 \,}}\! \right| }} = {10^2}\left( {{}^2A_{x:\left. {\overline {\, 5 \,}}\! \right| }^1 + {v^{2 \cdot 6}}{}_5{p_x}} \right)\)
\({}^2{A_{x:\left. {\overline {\, 5 \,}}\! \right| }} = 100\left( {0.03 + \frac{{0.94}}{{{{1.05}^{12}}}}} \right) = 55.3427\) |
|
\(Var\left[ Z \right] = \frac{{{}^2{A_{x:\left. {\overline {\, 5 \,}}\! \right| }} – {{\left( {{A_{x:\left. {\overline {\, 5 \,}}\! \right| }}} \right)}^2}}}{{{d^2}}} = \frac{{55.3427 – {{7.41442}^2}}}{{{{\left( {\frac{{0.05}}{{1.05}}} \right)}^2}}} = 162.7625\) |
| Jawaban |
d. 162,8 |
