Pembahasan Ujian PAI: A10 – No. 13 – Juli 2024

$$\bar{a}_{\overline{n}|} = \frac{1 – v^{n}}{\delta}$$

FV anuitas kontinu:

$$\bar{s}_{\overline{n}|} = \frac{(1 + i)^{n} – 1}{\delta}$$

$$ \bar{a}_{\overline{n}|} = 4$$
$$ \frac{1 – v^{n}}{\delta} = 4$$

Untuk FV:

$$ \bar{s}_{\overline{n}|} = 12$$

$$ \frac{(1 + i)^{n} – 1}{\delta} = 12$$

Bandingkan kedua persamaan:

$$\frac{\dfrac{1 – v^{n}}{\delta}}{\dfrac{(1 + i)^{n} – 1}{\delta}} = \frac{4}{12}$$
$$\frac{1 – v^{n}}{(1 + i)^{n} – 1} = \frac{1}{3}$$
$$\frac{1 – v^{n}}{v^{-n} – 1} = \frac{1}{3}$$
$$\frac{v^{n}\left(1 – v^{n}\right)}{1 – v^{n}} = \frac{1}{3}$$
$$v^{n} = \frac{1}{3}$$

Diperoleh:

$$\frac{1 – v^{n}}{\delta} = 4$$
$$\frac{1 – \frac{1}{3}}{\delta} = 4$$
$$\delta = \frac{1 – \frac{1}{3}}{4} = \frac{\frac{2}{3}}{4} = \frac{1}{6}$$

Jadi nilai \(\delta\) yakni sebesar 1/6.