Pembahasan Ujian PAI: A50 – No. 16 – Juni 2016

1. 1.000 orang masuk dalam pengamatan tepat pada umur 80
2. 40 orang meninggal dunia pada umur 80,30
3. 100 orang baru masuk dalam pengamatan pada umur 80,60
4. 20 orang meninggal dunia pada umur 80,80

\(\hat q = 1 – \exp \left( { – \frac{{{d_j}}}{{{e_j}}}} \right)\)

Actuarial exposure

\(\hat q = \frac{{{d_j}}}{{{e_j}}}\)

\({y_i}\) = tanggal awal pengamatan – tanggal lahir
\({z_i}\) = tanggal akhir pengamatan – tanggal lahir
\(\theta \) = tanggal meninggal – tanggal lahir
\({\phi _i}\) = tanggal withdraw – tanggal lahir

\({r_i} = \left\{ {\begin{array}{*{20}{c}} {0\_{\rm{jika\_}}{y_i} \le x}\\ {{y_i} – x\_{\rm{jika\_}}x < {y_i} < x + 1} \end{array}} \right.\)

\({s_i} = \left\{ {\begin{array}{*{20}{c}} {{z_i} – x\_{\rm{jika\_}}x < {z_i} < x + 1}\\ {1\_{\rm{jika\_}}{z_i} \ge x + 1} \end{array}} \right.\)

\({\iota _i} = \left\{ {\begin{array}{*{20}{c}} {0\_{\rm{jika\_}}{\theta _i} = 0}\\ {{\theta _i} – x\_{\rm{jika\_}}x < {\theta _i} \le x + 1}\\ {0\_{\rm{jika\_}}{\theta _i} > x + 1} \end{array}} \right.\)

\({\kappa _i} = \left\{ {\begin{array}{*{20}{c}} {0\_{\rm{jika\_}}{\phi _i} = 0}\\ {{\phi _i} – x\_{\rm{jika\_}}x < {\phi _i} \le x + 1}\\ {0\_{\rm{jika\_}}{\phi _i} > x + 1} \end{array}} \right.\)

\({\varepsilon _{eksak}} = \left\{ {\begin{array}{*{20}{c}} {{s_i} – {r_i}{\rm{\_jika\_seseorang\_tidak\_meninggal\_dan\_withdraw}}}\\ {{\kappa _i} – {r_i}{\rm{\_jika\_seseorang\_withdraw}}}\\ {{\iota _i} – {r_i}{\rm{\_jika\_seseorang\_meninggal}}} \end{array}} \right.\)

\({\varepsilon _{aktuaria}} = \left\{ {\begin{array}{*{20}{c}} {{s_i} – {r_i}{\rm{\_jika\_seseorang\_tidak\_meninggal\_dan\_withdraw}}}\\ {{\kappa _i} – {r_i}{\rm{\_jika\_seseorang\_withdraw}}}\\ {1 – {r_i}{\rm{\_jika\_seseorang\_meninggal}}} \end{array}} \right.\)

\(\hat q = 1 – \exp \left( { – \frac{{{d_j}}}{{{e_j}}}} \right)\)
\(= 1 – \exp \left( { – \frac{{60}}{{1000 – 0,7\left( {40} \right) + 0,4\left( {100} \right) + 0,2\left( {20} \right)}}} \right)\)
\(= 0,0577869\)

\(\hat q = \frac{{{d_j}}}{{{e_j}}}\)
\(= \frac{{60}}{{1000 + 0,4\left( {100} \right)}}\)
\(= 0,0576923\)