Risk and Return
Scalar Form
$$E(r_{i})=\sum_{j=1}^M {P_{ij} r_{ij}}$$
$$\sigma_{i}^2 = \sum_{j=1}^M {P_{ij}(r_{ij}-E(r_{i}))^2}$$
- $E$ : the expected return
- $\sigma$ : the variance of return
- $r_{ij}$ : the return of a security $i$ during a time period $j$
- $P_{ij}$ : the possibility of the return $j$ for the security $i$ $$r_{\mathcal{P}j} = \sum_{i^{*}=1}^N {\omega_{i^{*}}r_{i^{*}j}}$$ $$E(r_{\mathcal{P}j}) = \sum_{i^{*}=1}^N {\omega_{i^{*}}E(r_{i^{*}})}$$ $$\sigma_{\mathcal{P}}^2 = \sum_{i^{*}=1}^N {\omega_{i^{*}}^2\sigma_{i^{*}}^2} + \sum_{i^{*}=1}^N \sum_{j=1,j \ne i^{*}}^N {\omega_{i^{*}}\omega_{j}\sigma_{i^{*}j}}$$
- $r_{\mathcal{P}j}$ : the return of a portfolio
- $\mathcal{P}$ : a portfolio which include $N$ securities
Vector Form
$$E(r_{\mathcal{P}}) = \mathbf{r^T \omega}$$
$$\sigma_{\mathcal{P}}^2 = \mathbf{\omega^T V \omega}$$
$$\sigma_{ij} = \frac{1}{M} \sum_{t=1}^M {[r_{it}-E(r_{i})][r_{jt}-E(r_{j})]}$$
Portfolio Diversification
Uncorrelated Securities
$$\begin{align*}
\sigma_{\mathcal{P}}^{2} &= \sum_{i=1}^{N} \omega_{i}^{2}\sigma_{i}^{2} \\
&= \sum_{i=1}^{N} \left ( \frac{1}{N} \right )^{2} \sigma_{i}^{2} \\
&= \frac{1}{N} \sum_{i=1}^{N} \left ( \frac{\sigma_{i}^{2}}{N} \right ) \\
&= \frac{1}{N} \bar{\sigma}_{i}^{2}
\end{align*}$$
Correlated Securities
$$\begin{align*}
\sigma_{\mathcal{P}} &= \sum_{i=1}^{N} \left ( \frac{1}{N} \right )^{2} \sigma_{i}^{2}
+ \sum_{i=1}^{N}\sum_{j=1, j \ne i}^{N} \left ( \frac{1}{N} \right )^{2} \sigma_{ij} \\
&= \frac{1}{N} \left ( \sum_{i=1}^{N} \frac{\sigma_{i}^{2}}{N} \right )
+ \frac{N-1}{N} \left ( \sum_{i=1}^{N}\sum_{j=1, j \ne i}^{N} \frac{\sigma_{ij}}{N(N-1)} \right ) \\
&= \frac{1}{N} \bar{\sigma}_{i}^{2} + \frac{N-1}{N}\bar{\sigma}_{ij} \\
&= \frac{1}{N} \left ( \bar{\sigma}_{i}^{2} - \bar{\sigma}_{ij} \right ) + \bar{\sigma}_{ij} \\
\end{align*}$$
Calculating Efficient Frontiers
Risk-Free Security
$$
r_{FP} = \omega_{\mathcal{P}}E(r_{\mathcal{P}}) + (1-\omega_{\mathcal{P}})r_{F}
$$
Portfolio standard deviation
$$\begin{align*}
\sigma_{F\mathcal{P}} &= \sqrt{\omega_{\mathcal{P}}^{2}\sigma_{\mathcal{P}}^{2} + (1-\omega_{\mathcal{P}})^{2} \sigma_{F}^{2} + 2\omega_{\mathcal{P}}(1-\omega_{\mathcal{P}})\rho_{F\mathcal{P}}\sigma_{F}\sigma_{\mathcal{P}} } \\
&= \omega_{\mathcal{P}}\sigma_{\mathcal{P}}\\
\end{align*}$$
$$\begin{align*}
\omega_{\mathcal{P}} &= \frac{\sigma_{F\mathcal{P}}}{\sigma_{\mathcal{P}}} \\
r_{F\mathcal{P}} &= r_{F} + \frac{E(r_{\mathcal{P}}) - r_{F}}{\sigma_{\mathcal{P}}} \sigma_{F\mathcal{P}}
\end{align*}$$
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