Pembahasan Soal Ujian Profesi Aktuaris
Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
Mata Ujian |
: |
Matematika Keuangan |
Periode Ujian |
: |
November 2019 |
Nomor Soal |
: |
29 |
SOAL
Pak Tono memiliki hutang yang dapat dilunasi dengan pembayaran sebesar 100+X, 200+X, dan 300+X pada akhir tahun ke-1, 2, dan 3 berturut-turut. Hutang tersebut juga dapat dilunasi hanya dengan membayar 2.000 pada akhir tahun ke-4. Diketahui tingkat bunga efektif tahunan atas hutang tersebut adalah 7%. Tentukan nilai X. Bulatkan jawaban ke satuan terdekat.
a. 298
b. 317
c. 341
d. 363
e. 386
Diketahui |
- Tingkat bunga = 7%
- Skenario 1: pembayaran hutang dilakukan berkala pada akhur tahun 1,2, dan 3 sebesar 100+X , 200+X, dan 300+X berturut-turut.
- Skenario 2: pembayaran sekaligus pada akhir tahun ke-4 sebesar 2,000
Tentukan nilai X. |
Rumus yang digunakan |
Accumulated Value dengan bunga Majemuk,
\(A\left( t \right) = k{\left( {1 + i} \right)^t}\) |
Proses pengerjaan |
\(Skenario\;1 \to A\left( 4 \right) = \left( {100 + X} \right){\left( {1 + 7\% } \right)^3} + \)
\(\left( {200 + X} \right){\left( {1 + 7\% } \right)^2} + \left( {300 + X} \right)\left( {1 + 7\% } \right)\)
\(Skenario\;2 \to \;A\left( 4 \right) = 2,000\)
Lakukan eliminasi didapati,
\(2,000 = \left( {100 + X} \right){\left( {1 + 7\% } \right)^3} + \)
\(\left( {200 + X} \right){\left( {1 + 7\% } \right)^2} + \left( {300 + X} \right)\left( {1 + 7\% } \right)\)
\(2,000 = 672.4843 + 3.4499X\;\)
\(X = 385.9121\; \cong 386\) |
Jawaban |
e. 386 |
Thnnnxxx.
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