Portfolio Optimization

The portfolio Optimization is a problem looks into identify the optimal number of shares of each stock to purchase in order to minimize risk(variance) and maximize returns, while staying under some specified spending budget.

Problem Definition

Consider a set of $n$ types of stockds to choose from, with an average monthly return per dollar spent of $r_i$ for wach stock $i$. Let $\sigma_{i,j}$ be the covariance of the returns of stocks $i$ and $j$. For a spending budget of $B$ dollars, let $x_i$ denote the number of shares of stock $i$ purchased at price $p_i$ per share.

The problem and be formaulated as

$$\begin{array}{ll} \text{minimize or maximize} & \text{objective function}\\ \text{subject to} & \text{constraint(s)} \end{array}$$

For example, if we choose $n=3$, our decision vriables must satisfied

$$\sum_{i=1}^{n=3} x_{i} \leq \text{budget},\ x_{i}\geq 0.$$

Let's follow saome assumptions below: 1. Fraction share is NOT acceptable. 2. No short-selling is allowed. 3. No transcation costs.

We denote an average monthly return per dollar spent of $r_i$ for wach stock $i$, then the return (profit) on $x_i$ dollars invested in stock $i$ is $r_{i}x_{i}$ and the total (random) return on our investment is $\sum_{i=1}^{3}r_{i}x_{i}$.

The expected return on our investment is then $\mathbb{E}[\sum_{i=1}^{3} r_{i}x_{i}] = \sum_{i=1}^{3}\overline{r_{i}}x_{i}$, where $\overline{r_{i}}$ is the expected value of the $r_{i}$. Since we want to have an expected return of at least $ $50.00$, $x_{i}'s$ must be such that:

$$\sum_{i=1}^{3}\overline{r_{i}}x_{i} \geq 50.00.$$

Now, lets try to minimize the risk. The risk can be closely approximated by minimizing the variance of the return of the investment portfolio. The variance is given by:

$$\begin{array}{ll} \text{Var}[\sum_{i=1}^{3}r_{i}x_{i}] & = \mathbb{E}\bigg[\bigg(\sum_{i=1}^{3}r_{i}x_{i} - \sum_{i=1}^{3}\overline{r_{i}}x_{i} \bigg)\bigg]\\ & = \mathbb{E}\bigg[\bigg(\sum_{i=1}^{3}(r_{i} - \overline{r_{i}})x_{i}\bigg)\bigg(\sum_{j=1}^{3}(r_{j} - \overline{r_{j}})x_{j}\bigg)\bigg]\\ & = \sum_{i=1}^{3}\sum_{j=1}^{3}x_{i}x_{j}\mathbb{E}[(r_{i} - \overline{r_{i}})(r_{j} - \overline{r_{j}})] \\ & = \sum_{i=1}^{3}\sum_{j=1}^{3}x_{i}x_{j}\sigma_{ij}, \end{array}$$

where $\sigma_{ij}$ is the covariance of the return of stock $i$ with stock $j$. Therefore, our problem can be defined, in a general form, as:

$$\begin{array}{ll} \text{min} & \sum_{i}^{I}\sum_{j}^{J} x_{i}x_{j}\sigma_{ij}, \\ \text{subject to} & \sum_{i}^{I} x_{i} \leq B, \\ & \sum_{i}^{I} x_{i} \geq R, \\ & x_{i} \geq 0, \forall i. \end{array}$$

where $B$ is the total budget, $R$ is the expected return after a time period.

We can also write our formula into a matrices and vectors,

$$

$$

where $x$ is the decision vector of size $n$ ($n$ is the number of stocks), $e$ is an $n$-vector of ones, $\overline{r}$ is the $n$-vector of expected returneds of the stocks, and $Q$ is the $n\times n$ covariance matrix (whose $i-j$th element $Q_{ij} = \sigma{ij}$)

Constructing Hamiltonian

From the problem definination, we can construct our Hamiltonian by introducing penalty terms from our constraints listed above.

$$H = \sum_{i}\sum_{j}x_{i}x_{j}\sigma_{ij} + \lambda_{1}\bigg(\sum_{i}x_{i}-B\bigg)^{2} + \lambda_{2}\bigg(R-\sum_{i}\overline{r_i}x_{i} \bigg)^{2}$$

where $\lambda_{1}$ and $\lambda_{2}$ are penalty coefficients.