Pembahasan Ujian PAI: F10 – No. 23 – November 2025

$$\begin{aligned}
\omega_{A} &= \frac{\left(E\left(r_{A}\right) – r_{f}\right)\sigma_{B}^{2} – \left(E\left(r_{B}\right) – r_{f}\right)\sigma_{AB}}{\left(E\left(r_{A}\right) – r_{f}\right)\sigma_{B}^{2} + \left(E\left(r_{B}\right) – r_{f}\right)\sigma_{A}^{2} – \left[\left(E\left(r_{A}\right) – r_{f}\right) + \left(E\left(r_{B}\right) – r_{f}\right)\right]\sigma_{AB}} \\
\omega_{A} + \omega_{B} &= 1
\end{aligned}$$

Rumus kovarians 2 aset:
$$\sigma_{AB} = \sigma_{A}\sigma_{B}\rho_{AB}$$

Ekspektasi dan varians imbal hasil portofolio berisiko optimal dari 2 aset:
$$\begin{aligned}
E\left(r_{p}\right) &= \omega_{A}E\left(r_{A}\right) + \omega_{B}E\left(r_{B}\right) \\
\sigma_{p}^{2} &= \omega_{A}^{2}\sigma_{A}^{2} + \omega_{B}^{2}\sigma_{B}^{2} + 2\omega_{A}\omega_{B}\sigma_{A}\sigma_{B}\rho_{AB}
\end{aligned}$$

Cari bobot masing-masing sekuritas:
$$\begin{aligned}
\sigma_{AB} &= \sigma_{A}\sigma_{B}\rho_{AB} = (30\%)(15\%)(0{,}1) = 0{,}0045 \\
\omega_{A} &= \frac{(20\% – 8\%)(15\%)^{2} – (12\% – 8\%)(0{,}0045)}{(20\% – 8\%)(15\%)^{2} + (12\% – 8\%)(30\%)^{2} – \left[(20\% – 8\%) + (12\% – 8\%)\right](0{,}0045)} \\
&= \frac{0{,}0027 – 0{,}00018}{0{,}0027 + 0{,}0036 – 0{,}00072} = \frac{0{,}00252}{0{,}00558} \\
&= 0{,}45161 \\
\omega_{B} &= 1 – 0{,}45161 = 0{,}54839
\end{aligned}$$

$$\begin{aligned}
E\left(r_{p}\right) &= (0{,}45161)(20\%) + (0{,}54839)(12\%) \\
&= 0{,}15613 = 15{,}61\% \\
\sigma_{p}^{2} &= (0{,}45161)^{2}(30\%)^{2} + (0{,}54839)^{2}(15\%)^{2} + 2(0{,}45161)(0{,}54839)(30\%)(15\%)(0{,}1) \\
&= 0{,}02735 \\
\sigma_{p} &= \sqrt{0{,}02735} = 0{,}16538 = 16{,}54\%
\end{aligned}$$

Jadi, imbal hasil yang diharapkan (\(E(r_{p})\)) dan standar deviasi dari imbal hasil (\(\sigma_{p}\)) pada portofolio berisiko optimal yang dibentuk oleh investor adalah:

$$E(r_{p}) = 15,61\% \text{ dan } \sigma_{p} = 16,54\%$$