Pembahasan Ujian PAI: F10 – No. 13 – November 2025

• \(M = 0{,}5\,A + 0{,}5\,B \Longleftrightarrow \omega_{A} = \omega_{B} = 50\%\)
• \(E\left(r_{M}\right) = 11\%\)
• \(\sigma_{A} = 20\%\)
• \(\sigma_{B} = 40\%\)
• \(\rho_{AB} = 0{,}75\)
• \(r_{f} = 2\%\)

• \(E\left(r_{i}\right) = r_{f} + \beta_{i}\left(E\left(r_{Mkt}\right) – r_{f}\right)\)
• \(\beta_{i} = \frac{\mathrm{Cov}_{i,M}}{\sigma_{M}^{2}}\)
• \(\sigma_{M}^{2} = \omega_{A}^{2}\sigma_{A}^{2} + \omega_{B}^{2}\sigma_{B}^{2} + 2\omega_{A}\omega_{B}\sigma_{A}\sigma_{B}\rho_{AB}\)
• Kovarians \(\mathrm{Cov}_{A,M} = \omega_{A}\sigma_{A}^{2} + \omega_{A}\sigma_{A}\sigma_{B}\rho_{AB}\)
• Kovarians \(\mathrm{Cov}_{B,M} = \omega_{B}\sigma_{B}^{2} + \omega_{B}\sigma_{A}\sigma_{B}\rho_{AB}\)

Hitung terlebih dulu nilai dari kovarians tiap aset:

$$\begin{aligned}
\mathrm{Cov}_{A,M} &= \omega_{A}\sigma_{A}^{2} + \omega_{B}\sigma_{A}\sigma_{B}\rho_{AB} \\
&= (50\%)(20\%)^{2} + (50\%)(20\%)(40\%)(0{,}75) \\
&= 0{,}02 + 0{,}03 = 0{,}05 \\[6pt] \mathrm{Cov}_{B,M} &= \omega_{B}\sigma_{B}^{2} + \omega_{A}\sigma_{A}\sigma_{B}\rho_{AB} \\
&= (50\%)(40\%)^{2} + (50\%)(20\%)(40\%)(0{,}75) \\
&= 0{,}08 + 0{,}03 = 0{,}11
\end{aligned}$$

Selanjutnya dihitung nilai dari varians portofolio pasar:

$$\begin{aligned}
\sigma_{M}^{2} &= \omega_{A}^{2}\sigma_{A}^{2} + \omega_{B}^{2}\sigma_{B}^{2} + 2\omega_{A}\omega_{B}\sigma_{A}\sigma_{B}\rho_{AB} \\
&= (50\%)^{2}(20\%)^{2} + (50\%)^{2}(40\%)^{2} + 2(50\%)(50\%)(20\%)(40\%)(0{,}75) \\
&= 0{,}01 + 0{,}04 + 0{,}03 \\
&= 0{,}08
\end{aligned}$$

Hitung nilai beta setiap aset:

$$\begin{aligned}
\beta_{A} &= \frac{\mathrm{Cov}_{A,M}}{\sigma_{M}^{2}} = \frac{0{,}05}{0{,}08} = 0{,}625 \\[6pt] \beta_{B} &= \frac{\mathrm{Cov}_{B,M}}{\sigma_{M}^{2}} = \frac{0{,}11}{0{,}08} = 1{,}375
\end{aligned}$$

Hitung masing-masing nilai \(E\left(r_{A}\right)\) dan \(E\left(r_{B}\right)\):

$$\begin{aligned}
E\left(r_{A}\right) &= r_{f} + \beta_{A}\left(E\left(r_{M}\right) – r_{f}\right) \\
&= 2\% + 0{,}625\,(11\% – 2\%) \\
&= 2\% + 5{,}625\% = 7{,}625\% \\
E\left(r_{B}\right) &= r_{f} + \beta_{B}\left(E\left(r_{M}\right) – r_{f}\right) \\
&= 2\% + 1{,}375\,(11\% – 2\%) \\
&= 2\% + 12{,}375\% = 14{,}375\%
\end{aligned}$$

Sehingga diperoleh:

$$\begin{aligned}
E\left(r_{B}\right) – E\left(r_{A}\right) &= 14{,}375\% – 7{,}625\% \\
&= 6{,}75\%
\end{aligned}$$

Jadi, nilai \(E\left(r_{B}\right) – E\left(r_{A}\right)\) sebesar 6,75%, yakni: Lebih dari 6,5%, tapi kurang dari 7%.