Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Matematika Aktuaria |
| Periode Ujian |
: |
Mei 2017 |
| Nomor Soal |
: |
12 |
SOAL
Jika \(L = \bar L({\bar A_x})\) menyatakan nilai sekarang dari “loss random variable” pada suatu “fullycontinuous whole life model” dengan “continuous premium rate” berdasarkan prinsip equivalent.Jika \({L^*}\) menyatakan nilai sekarang dari “loss random variable” pada model yang serupa dengan“continuous annual premium rate 0,05” tentukan nilai dari \(Var({L^*})\) jika diketahui nilai dari:
\(Var(L) = 0,25\) \({\bar A_x} = 0,4\) \(\delta = 0,06\)
- 0,1025
- 0,1525
- 0,2025
- 0,2525
- 0,3025
| Step 1 |
Prinsip ekuivalen,
\(P = \frac{{{{\bar A}_x}}}{{{{\ddot a}_x}}}\)
\(P = \frac{{{{\bar A}_x}}}{{\left( {\frac{{1 – {{\bar A}_x}}}{\delta }} \right)}}\)
\(P = \frac{{0,4}}{{\left( {\frac{{1 – 0,4}}{{0,06}}} \right)}}\)
\(P = 0,04\) |
| Step 2 |
\(L = {\bar A_x} – P{\ddot a_x}\)
\(L = {\bar A_x} – P\left( {\frac{{1 – {{\bar A}_x}}}{\delta }} \right)\)
\(L = \left( {1 + \frac{P}{\delta }} \right){\bar A_x} – \frac{P}{\delta }\) |
| Step 3 |
\(Var[L] = Var\left[ {\left( {1 + \frac{P}{\delta }} \right){{\bar A}_x} – \frac{P}{\delta }} \right]\)
\(Var[L] = {\left( {1 + \frac{P}{\delta }} \right)^2}\left( {{}^2{{\bar A}_x} – {{\bar A}_x}^2} \right)\)
\(0,25 = {\left( {1 + \frac{{0,04}}{{0,06}}} \right)^2}\left( {{}^2{{\bar A}_x} – {{\bar A}_x}^2} \right)\)
\(\left( {{}^2{{\bar A}_x} – {{\bar A}_x}^2} \right) = 0,09\) |
| Maka |
\(Var[{L^*}] = {\left( {1 + \frac{{0,05}}{{0,06}}} \right)^2}\left( {{}^2{{\bar A}_x} – {{\bar A}_x}^2} \right)\)
\(Var[{L^*}] = {\left( {1 + \frac{{0,05}}{{0,06}}} \right)^2}\left( {0,09} \right)\)
\(Var[{L^*}] = 0,3025\) |
| Jawaban |
e. 0,3025 |
