Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Dengan tingkat bunga dan fitur anuitas yang serupa dengan soal no 7, tentukan nilai present value dari prepetuitas jika dibatasi hingga 2000. Bulatkan jawaban terdekat.
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| Diketahui | Perpetuitas pembayaran sebesar 1000 terhitung satu tahun dari sekarang. Pembayaran berikutnya akan lebih besar 5% hingga 2000. \(i = 0.09\) |
| Rumus yang digunakan | Geometri \({U_n} = a{r^{n – 1}}\) jumlah deret geometri \({S_{\left. {\overline {\, n \,}}\! \right| }} = \frac{{a\left( {1 – {r^n}} \right)}}{{1 – r}}\) perpetuity \({a_{\left. {\overline {\, n \,}}\! \right| }} = \frac{1}{i}\) |
| Proses pengerjaan | Geometri \({U_n} = a{r^{n – 1}}\) \(\Leftrightarrow 2000 = 1000{\left( {1.05} \right)^{n – 1}}\) \(\Leftrightarrow 2 = {\left( {1.05} \right)^{n – 1}}\) \(\Leftrightarrow \ln 2 = \left( {n – 1} \right)\ln \left( {1.05} \right)\) \(\Leftrightarrow n – 1 = \frac{{\ln 2}}{{\ln \left( {1.05} \right)}}\) \(\Leftrightarrow n = 1 + \frac{{\ln 2}}{{\ln \left( {1.05} \right)}}\) \(\Leftrightarrow n = 15.2067\) maka \(n = 15\) \(PV = Pv + \left( {P(1 + i)} \right){v^2} + \left( {P{{(1 + i)}^2}} \right){v^3} + … + \left( {1000{{(1 + i)}^{14}}} \right){v^{15}} + 2000{v^{16}} + 2000{v^{17}} + …\) \(PV = Pv + \left( {P(1 + i)} \right){v^2} + \left( {P{{(1 + i)}^2}} \right){v^3} + … + \left( {1000{{(1 + i)}^{14}}} \right){v^{15}} + 2000{v^{15}}(v + {v^2} + …)\) \(= 1000 \cdot {(1.09)^{ – 1}} + \left( {1000 \cdot (1.05)} \right){(1.09)^{ – 2}} + … + \left( {1000 \cdot {{(1.05)}^{14}}} \right){(1.09)^{ – 15}} + 2000{(1.09)^{ – 15}}(\frac{1}{{0.09}}){\rm{ }}\) \(= \frac{{\left( {1000 \cdot {{(1.09)}^{ – 1}}} \right)\left( {1 – {{\left( {\left( {1.05} \right){{\left( {1.09} \right)}^{ – 1}}} \right)}^{15}}} \right)}}{{1 – \left( {\left( {1.05} \right){{\left( {1.09} \right)}^{ – 1}}} \right)}} + 2000{(1.09)^{ – 15}}(\frac{1}{{0.09}})\) \(= 10731.38 + 6100.845\) \(= 16832.22\) |
| Jawaban | a. 16832 |