Quantitative Analysis, Risk Management, Modelling, Algo Trading, and Big Data Analysis

Final Frontier of Efficient Frontier


Have you ever considered why in the risk-return space the efficient frontier takes its shape which resembles some of the Bézier curves? And also, why everybody around suggests you that if you approach the efficient frontier from the right-hand side, your selection of a new investment could be the best choice?

Well, as for the latter question the answer seems to be easy. First, you plot all your stocks or the combination of stocks into a number of various portfolios (i.e. you derive the expected return and volatility of a given stock or portfolio, respectively; check out my article on 2-Asset Portfolio Construction for some fundamentals in that field) in the expected risk-return space. Your result can be similar to that one that follows:

where red dots correspond to specific positions of securities which you actually consider for a future investment. The data in the plot are annualized, what simply means we are looking for a potential return and risk (volatility) in a 1-year time horizon only. Secondly, from the plot you can read out that UTX, based on historical data, offers the highest potential return but at quite high risk. If you are not a risk-taker, the plot suggest immediately to choose MMM or JNJ but at lower expected return. So far so good.

The blue line represent so called efficient frontier in the risk-return space. The investor wants to choose the portfolio on the efficient frontier becasue – for each volatility – there is portfolio on the efficient frontier that has a higher rate of return than any other portfolio with the same volatility. In addition, by holding a portfolio on the efficient frontier the investor benefits from the diversification (see also Riskless Diversification). That brings us to an attempt of answering the very first question we posed at the beginning: about the shape, therefore about:

THE MATHEMATICS OF EFFICIENT FRONTIER


As we will show below, the efficient frontier is a solution to the problem of efficient selection between two securities. It is sufficient to conduct the basic calculation based on two assets considered together. For the higher number of potential securities or portfolios of securities (as presented in the plot above), the finding of the efficient frontier line is the subject to more advanced optimization.

Let’s start from the simple case of two securities, A and B, respectively. In our consideration we will omit the scenario where short sales are allowed. The expected portfolio return would be:
$$
\bar{R_P} = x_A\bar{R_A} + x_B\bar{R_B}
$$
where $x_A+x_B=1$ represents the sum of shares (weights), therefore $x_B=1-x_A$ simply means that the our whole initial capital has been fully invested into both securities, and we firstly decided to put a part of $x_A$ of our money into security A (say, $x_A=0.75$ or 75%). By rewriting the above we get:
$$
\bar{R_P} = x_A\bar{R_A} + (1-x_A)\bar{R_B} .
$$
The volatility (standard deviation) of that portfolio is therefore equal:
$$
\sigma_P = [x_A^2\sigma_A^2 + (1-x_A)^2\sigma_B^2 + 2\sigma_A\sigma_B\rho_{AB}x_A(1-x_A)]^{1/2}
$$
where by $\rho_{AB}$ we denoted a linear correlation between the expected rates of return for both securities, also possible to denote as $corr(\bar{R_A},\bar{R_B})$. Please also note the the third term can be jotted down as:
$$
2\sigma_A\sigma_B\rho_{AB}x_A(1-x_A) = 2x_A(1-x_A)cov(\bar{R_A},\bar{R_B})
$$
where the relation between covariance and correlation holds as follows:
$$
cov(\bar{R_A},\bar{R_B}) = \sigma_A\sigma_B corr(\bar{R_A},\bar{R_B}) .
$$
The correlation can be a free parameter if we are given from the historical data of securities A and B the following parameters: $\sigma_A, \sigma_B$ and $\bar{R_A}, \bar{R_B}$. This is so important to keep it in mind!

Below, we will consider four cases leading us to the final mathematical curvature of the efficient frontier.

CASE 1: Perfect Positive Correlation ($\rho_{AB}=+1$)

Considering two assets only, A and B, respectively, the perfect positive correlation of $\rho_{AB}=+1$ leads us to the portfolio volatility:
$$
\sigma_P = [x_A^2\sigma_A^2 + (1-x_A)^2\sigma_B^2 + 2\sigma_A\sigma_B\times 1 \times x_A(1-x_A)]^{1/2}
$$
where we recognize $a^2+b^2+2ab=(a+b)^2$ what allows us to rewrite it as follows:
$$
\sigma_P = [x_A\sigma_A + (1-x_A)\sigma_B] \ \ \mbox{for}\ \ \rho_{AB}=+1 .
$$
Solving for $x_A$ we get:
$$
x_A = \frac{\sigma_P-\sigma_B}{\sigma_A-\sigma_B} \ \ \mbox{i.e.}\ \ x_A \propto \sigma_P .
$$
Substituting that into the formula for the expected portfolio return of 2-assets, we arrive at:
$$
\bar{R_P} = x_A\bar{R_A} + (1-x_A)\bar{R_B} \\
= \frac{\sigma_P-\sigma_B}{\sigma_A-\sigma_B}\bar{R_A} + \left(1-\frac{\sigma_P-\sigma_B}{\sigma_A-\sigma_B}\right)\bar{R_B} \\
… \\
= \sigma_P\left(\frac{\bar{R_A}-\bar{R_B}}{\sigma_A-\sigma_B}\right) – \sigma_B\left(\frac{\bar{R_A}-\bar{R_B}}{\sigma_A-\sigma_B}\right)+\bar{R_B} .
$$
Therefore, the solution for that case can be summarized by the linear relation holding between the expected portfolio return and the portfolio volatility if the portfolio is constructed based on two assets perfectly positively correlated:
$$
\bar{R_P}(\sigma_P) \propto \sigma_P .
$$

CASE 2: Perfect Negative Correlation ($\rho_{AB}=-1$)

In this case, we can denote the portfolio volatility as:
$$
\sigma_P = [x_A^2\sigma_A^2 + (1-x_A)^2\sigma_B^2 + 2\sigma_A\sigma_B\times (-1) \times x_A(1-x_A)]^{1/2}
$$
where similarly as in Case 1, we recognize $a^2+b^2-2ab=(a-b)^2$ what lead us to double possible solutions, namely:
$$
\sigma_P = x_A\sigma_A – (1-x_A)\sigma_B \ \ \ \ \ \ \ \mbox{or} \\
\sigma_P = -x_A\sigma_A + (1-x_A)\sigma_B \ .
$$
Since $\sigma_P$ must always be greater than zero, thus one of the solutions holds anytime.

Please also note that $\sigma_P$(Case 2) is always less than $\sigma_P$(Case 1) for all $x_A\in\langle 0,1\rangle$. That allows us to formulate a very important note: The risk on the portfolio with $\rho_{AB}=-1$ is smaller than for the portfolio with $\rho_{AB}=+1$. Interesting, isn’t it? Hold on and check this out!

It is possible to obtain a portfolio with zero risk by setting $\sigma_P$(Case 2) to zero and solving for $x_A$ (i.e. what fraction of our money we need to invest in asset A to minimize the risk) as follows:
$$
x_A\sigma_A – (1-x_A)\sigma_B = 0 \\
x_A^2\sigma_A – \sigma_B – x_A\sigma_B =0 \\
x_A(\sigma_A+\sigma_B) = \sigma_B
$$
what provide us with a solution of:
$$
x_A = \frac{\sigma_B}{\sigma_A+\sigma_B} .
$$
It simply means that for such derived value of $x_A$ the minimal risk of the portfolio occurs, i.e. $\sigma_P=0$.

CASE 3: No Correlation between returns on the assets ($\rho_{AB}=0$)

In this case the cross-term vanishes leaving us with the simplified form of the expected portfolio volatility:
$$
\sigma_P = [x_A^2\sigma_A^2 + (1-x_A)^2\sigma_B^2 ]^{1/2} .
$$
The graphical presentation is as follows:

The point marked by a dot is peculiar. This portfolio has a minimum risk. It is possible to derive for what sort of value of $x_A$ this is achievable. In order to do it, for a general form of $\sigma_P$,
$$
\sigma_P = [x_A^2\sigma_A^2 + (1-x_A)^2\sigma_B^2 + 2x_A(1-x_A)\sigma_A\sigma_B\rho_{AB}]^{1/2} ,
$$
we find $x_A$ by calculating the first derivative,
$$
\frac{d{\sigma_P}}{d{x_A}} = \left(\frac{1}{2}\right) \frac{2x_A\sigma_A^2-2\sigma_B^2+2x_A\sigma_B^2+2\sigma_A\sigma_B\rho_{AB}-4x_A\sigma_A\sigma_B\rho_{AB}}
{[x_A^2\sigma_A^2+(1-x_A)^2\sigma_B^2+2x_A(1-x_A)\sigma_A\sigma_B\rho_{AB}]^{1/2}} \\
= \left(\frac{1}{2}\right) \frac{2x_A\sigma_A^2-2\sigma_B^2+2x_A\sigma_B^2+2\sigma_A\sigma_B\rho_{AB}-4x_A\sigma_A\sigma_B\rho_{AB}}{\sigma_P} ,
$$
and by setting it to zero:
$$
\frac{d{\sigma_P}}{d{x_A}} = 0
$$
we get:
$$
x_A = \frac{\sigma_B^2-\sigma_A\sigma_B\rho_{AB}}{\sigma_A^2+\sigma_B^2-2\sigma_A\sigma_B\rho_{AB}}
$$
what in case of $\rho_{AB}=0$ reduces to:
$$
x_A = \frac{\sigma_B^2}{\sigma_A^2+\sigma_B^2} .
$$
This solution is also know as the minimum variance portfolio when two assets are combined in a portfolio.

CASE 4: All other correlations between two assets (e.g. $\rho_{AB}=0.4$)

Given $\sigma_A$ and $\sigma_B$ at $\rho_{AB}$ not equal to zero, $-1$ or $1$, one can find that the formula for the expected variance of 2-asset portfolio will take, in general, the following form:
$$
\sigma_P = \sqrt{c_1x_A^2 + c_2}
$$
where $c_1$ and $c_2$ denote some coefficients and it can be presented in the graphical form as follows:

There is particular point of interest here and the shape of the curve is simply concaved. Since the investor is always to choose the asset which offers a bigger rate of return (say, $\bar{R}_A\gt\bar{R}_B$) at lower risk ($\sigma_B\lt\sigma_A$), therefore the best combination of portfolios we get every time following these rules are called as efficient frontier, in general represented by the Case 4’s line.

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