Pembahasan Ujian PAI: F10 – No. 22 – November 2025

$$\mu = \begin{pmatrix} 0{,}20 \\ 0{,}15 \\ 0{,}10 \end{pmatrix}$$

Matriks varians-kovarians:

$$\boldsymbol{\Sigma} = \begin{pmatrix}
0{,}36 & 0{,}084 & 0{,}105 \\
0{,}084 & 0{,}1225 & 0{,}07 \\
0{,}105 & 0{,}07 & 0{,}0625
\end{pmatrix}$$

Dari matriks tersebut dapat diketahui:

• \(\sigma_{A}^{2} = 0{,}36\) ;
• \(\sigma_{B}^{2} = 0{,}1225\) ;
• \(\sigma_{C}^{2} = 0{,}0625\);
• \(\sigma_{AB} = 0{,}084\) ;
• \(\sigma_{AC} = 0{,}105\) ;
• \(\sigma_{BC} = 0{,}07\)

$$E\left(r_{p}\right) = \mathbf{w}^{T}\boldsymbol{\mu}$$

dengan \(\boldsymbol{\omega}\) merupakan matriks bobot aset.

Varians portofolio untuk 3 aset:

$$\sigma_{p}^{2} = \omega_{A}^{2}\sigma_{A}^{2} + \omega_{B}^{2}\sigma_{B}^{2} + \omega_{C}^{2}\sigma_{C}^{2} + 2\omega_{A}\omega_{B}\sigma_{AB} + 2\omega_{A}\omega_{C}\sigma_{AC} + 2\omega_{B}\omega_{C}\sigma_{BC}$$

Misalkan matriks bobot aset pada portofolio:

$$\mathbf{w} = \begin{pmatrix} \omega_{A} \\ \omega_{B} \\ \omega_{C} \end{pmatrix}$$

Berlaku: \(\omega_{A} + \omega_{B} + \omega_{C} = 1\)

Diketahui expected return portofolio 16%, dapat dituliskan:

$$\begin{aligned}
0{,}20\,\omega_{A} + 0{,}15\,\omega_{B} + 0{,}10\,\omega_{C} &= 0{,}16 \\
0{,}20\,\omega_{A} + 0{,}15\left(1 – \omega_{A} – \omega_{C}\right) + 0{,}10\,\omega_{C} &= 0{,}16 \\
0{,}05\,\omega_{A} – 0{,}05\,\omega_{C} &= 0{,}01 \\
\omega_{A} &= 0{,}20 + \omega_{C}
\end{aligned}$$

Selanjutnya dapat dituliskan:

$$\begin{aligned}
\omega_{A} + \omega_{B} + \omega_{C} &= 1 \\
\left(0{,}20 + \omega_{C}\right) + \omega_{B} + \omega_{C} &= 1 \\
\omega_{B} &= 0{,}80 – 2\omega_{C}
\end{aligned}$$

Matriks bobot portfolio:

$$\mathbf{w} = \begin{pmatrix} 0{,}20 + \omega_{C} \\ 0{,}80 – 2\omega_{C} \\ \omega_{C} \end{pmatrix}$$

Selanjutnya substitusi besaran yang diketahui ke formula varians portofolio 3 aset:

$$\begin{aligned}
\sigma_{p}^{2} &= \omega_{A}^{2}\sigma_{A}^{2} + \omega_{B}^{2}\sigma_{B}^{2} + \omega_{C}^{2}\sigma_{C}^{2} + 2\omega_{A}\omega_{B}\sigma_{AB} + 2\omega_{A}\omega_{C}\sigma_{AC} + 2\omega_{B}\omega_{C}\sigma_{BC} \\
\sigma_{p}^{2} &= \omega_{A}^{2}(0{,}36) + \omega_{B}^{2}(0{,}1225) + \omega_{C}^{2}(0{,}0625) + \omega_{A}\omega_{B}(0{,}168) + \omega_{A}\omega_{C}(0{,}21) + \omega_{B}\omega_{C}(0{,}14)
\end{aligned}$$

Substitusi nilai dari matriks bobot aset:

$$\begin{aligned}
\sigma_{p}^{2} &= \omega_{A}^{2}(0{,}36) + \omega_{B}^{2}(0{,}1225) + \omega_{C}^{2}(0{,}0625) \\
&\quad + \omega_{A}\omega_{B}(0{,}168) + \omega_{A}\omega_{C}(0{,}21) + \omega_{B}\omega_{C}(0{,}14) \\
&= \left(0{,}20 + \omega_{C}\right)^{2}(0{,}36) + \left(0{,}80 – 2\omega_{C}\right)^{2}(0{,}1225) + \omega_{C}^{2}(0{,}0625) \\
&\quad + \left(0{,}20 + \omega_{C}\right)\left(0{,}80 – 2\omega_{C}\right)(0{,}168) \\
&\quad + \left(0{,}20 + \omega_{C}\right)\omega_{C}(0{,}21) + \left(0{,}80 – 2\omega_{C}\right)\omega_{C}(0{,}14) \\
&= 0{,}5065\,\omega_{C}^{2} – 0{,}0268\,\omega_{C} + 0{,}11968
\end{aligned}$$

Minimum Variance Portfolio:

Untuk meminimumkan varians, fungsi varians portofolio tersebut diturunkan lalu disamakan dengan nol:

$$\begin{aligned}
\frac{d\sigma_{p}^{2}}{d\omega_{C}} &= 0 \\
\frac{d}{d\omega_{C}}\left(0{,}5065\,\omega_{C}^{2} – 0{,}0268\,\omega_{C} + 0{,}11968\right) &= 0 \\
1{,}013\,\omega_{C} – 0{,}0268 &= 0 \\
\omega_{C} &= \frac{0{,}0268}{1{,}013} = 0{,}02646
\end{aligned}$$

Diperoleh:

$$\begin{aligned}
\sigma_{p}^{2} &= 0{,}5065\,\omega_{C}^{2} – 0{,}0268\,\omega_{C} + 0{,}11968 \\
&= 0{,}5065(0{,}02646)^{2} – 0{,}0268(0{,}02646) + 0{,}11968 \\
&= 0{,}11932 \\
\sigma_{p} &= \sqrt{0{,}11932} = 0{,}34543 = 34{,}54\%
\end{aligned}$$

Jadi, standar deviasi dari imbal hasil portofolio yang dibentuk oleh Pak Asep yakni sebesar 34,54%.