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In this article, we'll explore the process of determining the statistically optimal allocation of your investments. We'll delve into various investment constellations and the importance of diversification in order to maximize your returns while minimizing potential losses. By the end of this article, you'll have a better understanding of how to strategically allocate your money to achieve your financial goals. We explore the concept of return correlation in the portfolio example. We've attempted to minimize the math involved, making it easier for beginners to grasp the concept.
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Expected returns
Investment returns (R1, R2, ...) are random variables, which means they have an expected value, variance, distribution, and so on. In the figure, you can see the distribution for the return on a security. This security has a positive expected return, but a loss cannot be ruled out. The probability of a loss is represented by the light blue color. We can easily calculate this probability as the area under the density curve.
Figure 1 - Expected return (Inspired by [1])
The concept of expected value might raise some questions. To better understand its meaning, let's consider adding a second security to our portfolio. This second security has a higher expected value compared to the first one. In general, a higher expected value indicates a higher potential return on the investment.
Figure 2 - Two securities: Expected returns (Inspired by [1])
In the context of securities, the expected value represents the average return an investor can anticipate over a long period, taking into account the probability of different outcomes. It is essential to remember that the expected value is not a guarantee of the actual return, but rather a useful tool for comparing and evaluating different investment opportunities.
When comparing the two securities, the one with the higher expected value may seem more attractive due to its potential for higher returns. However, it is crucial for investors to also consider other factors, such as the associated risks and the overall diversification of their portfolio.
Risk of Securities
Variance plays a crucial role in evaluating investment opportunities, as it represents the dispersion in the return on securities. To illustrate this concept, let's assume we have two securities to compare.
Figure 3 - Risk of two securities (Inspired by [1])
We can see that both securities have the same expected value. However, the second security has a significantly higher variance.
In this context, a higher variance indicates that the risk is greater for the second security. The return on this security can be very high or very low (marked in green), with substantial deviations from the mean return. This variation in returns is what makes the security riskier. The variance of securities is often referred to as the risk of the security.
The second security's increased risk is also evident when examining the loss probability, which is significantly higher for the second security compared to the first. The following figure illustrates this.
Figure 4 - Loss Probability (Inspired by [1])
This means that when investing in the second security, there's a higher chance of facing a loss. Investors should carefully weigh the potential for higher returns against the increased risk associated with the second security.
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Mu Sigma Diagram
The Mu Sigma diagram, also known as the risk-return plot, serves as an effective tool for illustrating optimal portfolio allocation, providing investors with a clear visual representation of the trade-off between risk and return. In general, the standard deviation is used directly to measure portfolio risk, rather than the variance, as it offers a more intuitive understanding of the risk involved.
As an investor, your primary goal is to achieve the highest possible return while minimizing the risk associated with your investments. The Mu Sigma diagram helps you visualize this balance by plotting the expected return (Mu) on the vertical axis and the standard deviation (Sigma) on the horizontal axis. In this diagram, each investment option is represented as a point, with its position determined by its expected return and associated risk.
When analyzing the Mu Sigma diagram, the top left quadrant is always the most desirable, as it represents investments with high returns and low risk. Conversely, the bottom right quadrant is the least preferable, as it contains investments with low returns and high risk.
Figure 5 - Mu Sigma Diagram (Inspired by [1])
By examining the diagram, investors can identify the optimal portfolio allocation that aligns with their risk tolerance and return objectives.
Now, let's examine what happens when we invest in two securities. We assume the following conditions for the example.
Figure 6 - Example with two securities (Inspired by [1])
First, let's look at the extreme cases.
Figure 7 - Extreme cases (Inspired by [1])
Additionally, we can include both securities in the portfolio. The Mu-Sigma diagram displays the so-called possibility line. This line illustrates the different allocations of the two securities in a portfolio that can be achieved, each with its unique risk and return characteristics. By using a smart allocation strategy, investors can minimize the overall risk of their portfolio while still achieving their desired return objectives.
Figure 8 - Smart Portfolio Allocation (Inspired by [1])
The point marked in yellow on the diagram represents the minimum risk portfolio, which is the allocation that results in the lowest possible risk for the given set of securities.
Understanding the correlations between securities is crucial for managing your assets intelligently. When two securities are positively correlated, their prices tend to move in the same direction, while a negative correlation indicates that their prices move in opposite directions. By combining securities with different correlations, investors can reduce the overall risk of their portfolio through diversification.
It's important to note that, in reality, the correlation between securities is typically not constant, as illustrated in our example. The correlation can also change over time due to various factors, such as changes in market conditions, industry trends, or company-specific events. This dynamic nature of correlations makes it essential for investors to regularly monitor and adjust their portfolio allocations to maintain an optimal balance between risk and return.
Now that you have understood the basics of portfolio allocation and the importance of considering correlations between securities, you can move on to more advanced techniques. By continually refining your investment strategy and staying informed about market developments, you can make more informed decisions and work towards achieving your financial goals.
Conclusion
Efficient financial allocation across multiple investments involves understanding the expected returns, the associated risks, and the correlations between different securities. By leveraging tools such as the Mu Sigma diagram, investors can visually comprehend the trade-off between risk and return, and identify the optimal portfolio allocation that aligns with their risk tolerance and return objectives. It's crucial to remember that market conditions, industry trends, and company-specific events can affect your investments.
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[1] YouTube Video: Korri erklärt: Rendite-Korrelation im Portofoliobeispiel (German - Accessed on 09.01.2023)
Disclaimer:*All texts, notes and information provided by tinztwins do not constitute investment advice or a recommendation to buy or sell securities. They are for personal information only and reflect the opinion of the authors. No recommendation for a specific investment strategy is given. The authors are NOT LIABLE for any damages caused by the software. The use of the software is at your own risk.*
