Efficient Portfolio

This is a project for dealing with an efficient portfolio by using python.

1. Definition of Efficient Portfolio

An efficient portfolio is a portfolio (a combination of various financial assets) that offers the highest expected return for a given level of risk, or the lowest level of risk for a given expected return.

We can express the above sentence as the next mathematical formula

$$ \min_{\mu_{p}=c}\sigma_{p}^{2}\text{ or }\max_{\sigma_{p}^{2}=c}\mu_{p} $$

2. Virtual example of Efficient Portfolio

To understand the above condition formulas vividly, I'm going to set a portfolio comprised of two virtual stocks: $A$ and $B$

Let $x_{A}$ and $x_{B}$ as portions of stock A and B each. That is, $x_{A} + x_{B} = 1$. So portfolio $R = x_{A}A + x_{B}B$. A mean and variance of $R$ (a.k.a return and volatility) is as follow:

1. $\mu_{R}=x_{A}\mu_{A}+x_{B}\mu_{B}$
2. $Var(R)=x_{A}^{2}\sigma_{A}^{2}+x_{B}^{2}\sigma_{B}^{2}+2x_{A}x_{B}\sigma_{AB}$

We need to assume several information about stock $A$ and $B$. Means and Variances are as follow:

1. $\mu_{A}=9,\mu_{B}=3$ ($)
2. $\sigma_{A}=2,\sigma_{B}=1,\sigma_{AB}=1$ ($)

If an investor wants to design an efficient portfolio given a return of 6 dollars, below is the condition formula for getting what he wants.

$$ \min_{9x_{A}+3x_{B}=6}4x_{A}^{2}+x_{B}^{2}+2x_{A}x_{B}\times1 $$

This is a simple optimization problem.

$$ \begin{align*} x_{B}=2-3x_{A}\Longrightarrow & 4x_{A}^{2}+(2-3x_{A})^{2}+2x_{A}(2-3x_{A})\\ = & 7x_{A}^{2}-8x_{A}+4 \end{align*} $$

In summary, what an inverstor has to find is the root of the next formula

$$ \arg\min7x_{A}^{2}-8x_{A}+4, \quad \frac{1}{3}\leq x_{A}\leq\frac{2}{3} $$

Because $X_{B} = 2 - 3x_{A}$ and $0\leq x_{B} \leq 1$, so the range of $x_{A}$ is like that. Fortunately, the root of the above formula is $\frac{4}{7}$, whici is in the range of $x_{A}$. In conclusion, an efficieint portfolio is comprised of $\frac{4}{7}A$ and $\frac{3}{7}B$

3. Generalized Efficient Portfolio

We don't have to confine ourselves to a portfolio comprised of two shares. We can deal with arbitrary numbers of stocks through matrix algebra.

Let's assume that stocks $S_i, i\in I=(1,2,...,k)$ has its own mean and standard deviation ($\mu_i, \sigma_i$). So a mean vector and variance-covariance matrix is as follows.

$$ \mu=\left[\begin{array}{c} \mu_{1}\\ \mu_{2}\\ \vdots\\ \mu_{k} \end{array}\right],\sum=\left[\begin{array}{cccc} \sigma_{1}^{2} & \sigma_{12} & \cdots & \sigma_{1k}\\ \sigma_{21} & \sigma_{2}^{2} & \cdots & \sigma_{2k}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{k1} & \sigma_{k2} & \cdots & \sigma_{k}^{2} \end{array}\right] $$

And if we set a weight vector as

$$ \boldsymbol{x}=\left[\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{k} \end{array}\right],\sum_{i\in I}x_{i}=1 $$

The mean and the variance of portfolio $R$ is in such a way

$$ \begin{align*} \mu_{R} & =\boldsymbol{x^{t}}\mu=\sum_{i\in I}x_{i}\mu_{i}\\ \sigma_{R}^{2} & =\boldsymbol{x^{t}}\sum\boldsymbol{x}=\left[\begin{array}{cccc} x_{1} & x_{2} & \cdots & x_{k}\end{array}\right] \sum\left[\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{k} \end{array}\right]\\ & =\sum_{i\in I}x_{i}^{2}\sigma_{i}^{2}+2\sum_{i\neq j}x_{i}x_{j}\sigma_{ij} \end{align*} $$

So, to find out the most efficient portfolio, if we fix $\mu_{R} = \mu_0$, the below condition has to be solved.

4. Realistic example of Efficient Portfolio

In this project, I will tackle korean five companys:

1. LG Electronics
2. Naver
3. KEPCO (Korea Electric Power Corporation)
4. Samsung Electronics
5. Yuhan

I generated 50000 portfolio simulations and plotted them. Also I drew an efficient frontier as below