Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Suatu asuransi seumur hidup pada (x) dengan manfaat 1 dengan pengembalian dari “net single premium” tanpa bunga pada saat kematian. Diberikan:
(i) \({\mu _{x + t}} = 0,01\,\,for\,t > 0\)
(ii) \(\delta = 0,02\)
Hitunglah “net single premium”
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| Rumus | 
\(PV\,benefit = \left( {1 + E\left[ {{Z_x}} \right]} \right){v^{{T_x}}}\,,\,{T_x} > 0\,\)
\(PV\,benefit = {Z_x}^*\)
\(E\left[ {{Z_x}^*} \right] = \left( {1 + E\left[ {{Z_x}^*} \right]} \right)E\left[ {{v^{{T_x}}}} \right]\) |
| Step 1 | \(E\left[ {{v^{{T_x}}}} \right] = \int\limits_0^\infty {{v^t}{f_{{T_x}}}(t)\,dt} \)
\(E\left[ {{v^{{T_x}}}} \right] = \int\limits_0^\infty {{e^{ – \delta t}}\left( {{}_t{p_x}{\mu _{x + t}}} \right)\,dt} \)
\(E\left[ {{v^{{T_x}}}} \right] = \int\limits_0^\infty {{e^{ – \delta t}}\left( {\mu {e^{ – \mu t}}} \right)\,dt} \) \(\to \)dalam asumsi CFM \(E\left[ {{v^{{T_x}}}} \right] = \mu \int\limits_0^\infty {{e^{ – t\left( {\delta + \mu } \right)}}\,dt} \) \(E\left[ {{v^{{T_x}}}} \right] = \frac{\mu }{{ – \left( {\delta + \mu } \right)}}\left( {0 – 1} \right)\) \(E\left[ {{v^{{T_x}}}} \right] = \frac{\mu }{{\left( {\delta + \mu } \right)}}\) \(\mu = 0,01\,;\,\delta = 0,02\) \(E\left[ {{v^{{T_x}}}} \right] = \frac{{0,01}}{{\left( {0,01 + 0,02} \right)}}\) \(E\left[ {{v^{{T_x}}}} \right] = \frac{1}{3}\) |
| Step 2 | \(E\left[ {{Z_x}^*} \right] = \left( {1 + E\left[ {{Z_x}^*} \right]} \right)\frac{1}{3}\) \(3E\left[ {{Z_x}^*} \right] = \left( {1 + E\left[ {{Z_x}^*} \right]} \right)\) \(2E\left[ {{Z_x}^*} \right] = 1\) \(E\left[ {{Z_x}^*} \right] = \frac{1}{2}\) |
| Jawaban | a. 1/2 |