Pembahasan Ujian PAI: A60 - No. 8 - November 2014

Pembahasan Soal Ujian Profesi Aktuaris

Institusi	:	Persatuan Aktuaris Indonesia (PAI)
Mata Ujian	:	Matematika Aktuaria
Periode Ujian	:	November 2014
Nomor Soal	:	8

SOAL

Tentukan nilai \(Var\left( {{Y_{95}}} \right)\), bila menggunakan tingkat bunga tahunan 5% dan nilai sebagai brikut: \({l_{95}} = 100\), \({l_{96}} = 70\), \({l_{97}} = 40\), \({l_{98}} = 20\), \({l_{99}} = 4\), \({l_{100}} = 0\), \({a_{95}} = 1,2352\), dan \({}^2{a_{95}} = 1,1403\)

1,0933
1,0399
2,0933
2,2352
2,2532

Kunci Jawaban & Pembahasan

Diketahui	Tingkat bunga tahunan 5% dan nilai sebagai brikut: \({l_{95}} = 100\), \({l_{96}} = 70\), \({l_{97}} = 40\), \({l_{98}} = 20\), \({l_{99}} = 4\), \({l_{100}} = 0\), \({a_{95}} = 1,2352\), dan \({}^2{a_{95}} = 1,1403\)
Rumus yang digunakan	\(Var\left( Y \right) = \frac{{{}^2{A_x} – {{\left( {{A_x}} \right)}^2}}}{{{d^2}}}\) dengan \(d = 1 – v = 1 – \frac{1}{{1 + i}}\) \({A_x} = 1 – d{\ddot a_x}\) dan \({\ddot a_x} = 1 + {a_x}\)
Proses pengerjaan	\(d = 1 – \frac{1}{{1.05}} = \frac{1}{{21}}\) dan \({d^*} = 1 – \frac{1}{{{{1.05}^2}}} = \frac{{41}}{{441}}\) \({\ddot a_{95}} = 1 + {a_{95}} = 1 + 1.2352 = 2.2352\) \({A_{95}} = 1 – d{\ddot a_{95}} = 1 – \frac{1}{{21}}\left( {2.2352} \right) = 0.893562\)
	\({}^2{\ddot a_{95}} = 1 + {}^2{a_{95}} = 1 + 1.1403 = 2.1403\) \({}^2{A_{95}} = 1 – {d^*}{}^2{\ddot a_{95}} = 1 – \frac{{41}}{{441}}\left( {2.1403} \right) = 0.801015\)
	\(Var\left( Y \right) = \frac{{{}^2{A_x} – {{\left( {{A_x}} \right)}^2}}}{{{d^2}}} = \frac{{0.8010 – {{0.8936}^2}}}{{{{\left( {\frac{1}{{21}}} \right)}^2}}} = 1.093257\)
Jawaban	a. 1,0933

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