# 02 Objective functions for portfolio optim.

###### tags: `Portfolio optimization`

In general, for $n$ assets, we can combine them to the overall return $\mu$ and risk $\sigma$:
$$
\left\{
\begin{array}{rcl}
\mu &=& \boldsymbol{\mu}^T \mathbf{w}\\
\sigma^2 &=& \mathbf{w}^T \Sigma \mathbf{w}
\end{array}
\right.
$$
where $\mathbf{w}=[w_1, \dots, w_n]^T$, $\boldsymbol{\mu}=[\mu_1, \dots, \mu_n]^T$, and $\Sigma$ is the covariance matrix of these $n$ assets.

There are several basic criteria for optimization:

* Minimize risk with fixed return: Given a return $\mu$, find the weights to minimize the overall variance $\sigma^2$. (給定預期報酬值，最佳投資組合將產生最小風險。)
$$
\min_{\mathbf{w}} \sigma^2=\mathbf{w}^T \Sigma \mathbf{w} \\
s.t.
\left\{
\begin{array}{l}
\mathbf{w}^T\mathbf{1}=1\\
\boldsymbol{\mu}^T \mathbf{w}=r
\end{array}
\right.
$$
where $\mathbf{1}=[1, \dots, 1]^T$.
* Maxmimize return with fixed risk: Given a variance $\sigma^2$, find the weights to maximize the overall return $\mu$. (給定風險下，最佳投資組合將產生預期報酬最大值。)
$$
\max_{\mathbf{w}} \mu=\boldsymbol{\mu}^T \mathbf{w} \\
s.t.
\left\{
\begin{array}{l}
\mathbf{w}^T\mathbf{1}=1\\
\mathbf{w}^T \Sigma \mathbf{w}=\sigma_0^2
\end{array}
\right.
$$
* Minimize risk regardless of return: Find the weights to minimize the overall variance $\sigma^2$ regardless of the return. (讓最佳投資組合將產生最小風險，而完全不看報酬。)
$$
\min_{\mathbf{w}} \sigma^2=\mathbf{w}^T \Sigma \mathbf{w} \\
s.t.
\mathbf{w}^T\mathbf{1}=1.
$$

* Maximize the Sharpe ratio
$$
\max_{\mathbf{w}} \frac{\mu-\mu_0}{\sigma} \\
s.t.
\mathbf{w}^T\mathbf{1}=1\\
$$
* Maximize the difference between return and risk
$$
\max_{\mathbf{w}} \mu-\beta \sigma \\
s.t.
\mathbf{w}^T\mathbf{1}=1\\
$$