Pembahasan Ujian PAI: F10 – No. 28 – November 2025

• \(E(r_{A}) = 10\%\) ; \(\sigma_{A} = 5\%\)
• \(E(r_{B}) = 15\%\) ; \(\sigma_{B} = 10\%\)
• \(\rho_{AB} = -1\)

$$\omega = \frac{\sigma_{B}}{\sigma_{A} + \sigma_{B}}$$

Rumus bobot optimal untuk portofolio dengan 2 aset:

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

Hitung bobot setiap aset:

$$\begin{aligned}
\omega_{A} &= \frac{\sigma_{B}}{\sigma_{A} + \sigma_{B}} \\
\omega_{A} &= \frac{10\%}{5\% + 10\%} \\
\omega_{A} &= \frac{10\%}{15\%} = 0{,}67
\end{aligned}$$

Maka \(\omega_{B} = 1 – 0,67 = 0,33\)

$$\begin{aligned}
\sigma_{AB} &= \sigma_{A}\sigma_{B}\rho_{AB} \\
\sigma_{AB} &= (5\%)(10\%)(-1) \\
\sigma_{AB} &= -0{,}005
\end{aligned}$$

$$\begin{aligned}
\omega_{A} &= \frac{\left(E\left(r_{A}\right) – r_{f}\right)\sigma_{B}^{2} – \left(E\left(r_{B}\right) – r_{f}\right)\sigma_{AB}}{\left(E\left(r_{A}\right) – r_{f}\right)\sigma_{B}^{2} + \left(E\left(r_{B}\right) – r_{f}\right)\sigma_{A}^{2} – \left(\left(E\left(r_{A}\right) – r_{f}\right) + \left(E\left(r_{B}\right) – r_{f}\right)\right)\sigma_{AB}} \\
0{,}67 &= \frac{\left(10\% – r_{f}\right)(10\%)^{2} – \left(15\% – r_{f}\right)(-0{,}005)}{\left(10\% – r_{f}\right)(10\%)^{2} + \left(15\% – r_{f}\right)(5\%)^{2} – \left(\left(10\% – r_{f}\right) + \left(15\% – r_{f}\right)\right)(-0{,}005)} \\
r_{f} &= 0{,}1025 = 10{,}25\%
\end{aligned}$$

Jadi, tingkat suku bunga bebas risiko yakni sebesar 10,25%.