**Section 1: What is Covariance? **

Covariance is a statistical tool that examines the relationship between two variables and determines the extent to which they move or change in the same direction. In the context of finance, covariance is used to determine the extent to which the returns of two investment assets vary with each other. This brief article aims to provide a clear understanding of this econometrical tool, and dissect its uses when it comes to making investment decisions and maintaining a diversified portfolio.

**Section 2: Formula Interpretation**

Before commencing the analysis of the uses of covariance in finance and stock market navigation, it is essential to establish a clear understanding of the formula. Below is an image depicting the structure of the covariance formula:

The covariance formula utilizes variables from two distinct data sets, as identified by the letters *xi*** **and ** yi**.

**represents the mean for the values in the**

*xÌ„*

*x***data set while**

**represents the mean for the values in the**

*yÌ„***data set. For correct application of the covariance formula, the total population size of data values in each set has to be identical, and is denoted by a single character**

*y***. The sigma (Î£) symbol represents summation.**

*N*When calculating a sample covariance, you always minus 1 from the population in order to correct bias that occurs during the calculation. Usually a calculation of a sample mean, sample variance or sample standard deviation tends to be smaller than the actual population mean, variance or standard deviation. Using ** N-1** eliminates this bias by minimizing the denominator and ensuring a larger value for the sample is generated, and is as close to the population value as possible.

The formula for covariance can also be interpreted in the context of stock returns:

For two different stocks, the covariance of their returns can be calculated using the same formula in order to determine the extent to which they perform similarly.

Covariance can either be a positive or negative value. The magnitude of the number obtained isn't all that relevant but it's nature tells more of a story. A positive output suggests that variables in both data sets tend to move in the same direction, and a negative output indicates a movement in opposite directions. In the following section, we will go through an example of how to calculate the covariance from a given set of data.

**Section 3: Worked example**

Above is a random set of variables belonging to data sets ** x **and

**. To begin calculating covariance, the mean for both data sets has to be obtained (**

*y***and**

*xÌ„***). The mean is calculated by adding up all the data values in each set and dividing by the population size, which is 7 in this case. The calculated value for**

*yÌ„***approximates to 18.143 and 18.429 for**

*xÌ„***.**

*yÌ„*Now that the mean has been calculated, we have to compare the mean of each data set to their respective variables. For the first set of values, 14 and 15, you would have to do (14-18.143) for ** x **and (15-18.429) for

**. You would get -4.143 and -3.429 and these results would be multiplied, giving a value of approximately 14.204. This will be repeated for each pair of**

*y***and**

*x***data values - 18 and 16, 17 and 19, etc. The values obtained will then be added together, as indicated by the use of sigma (Î£).**

*y*In the end, the value obtained for 'Î£(** xiâˆ’xÌ„** )(

**)' amounts to about 13.571. To finally calculate the covariance, divide this number by the population size. The result is approximately 1.939, a positive figure that suggests that variables belonging to data set**

*yiâˆ’yÌ„***and**

*x***tend to change in a similar direction.**

*y***Section 4: Applications in Investment and Portfolio Management**

Covariance is a statistical metric that is very much applicable to the world of finance, and is a tool which can facilitate the construction of a diversified investment portfolio. Modern Portfolio Theory plays a huge role in understanding the importance of covariance when it comes to making investment decisions. Modern Portfolio Theory provides suitable framework for an investor to achieve maximum investment returns at their preferred level of risk (2). Under Modern Portfolio Theory, an ideal investment portfolio effectively mitigates idiosyncratic risk - risk that affects individual securities or a group of assets (3) - through diversification of the assets within the portfolio.

An exemplar investment portfolio would consist of securities and assets that share a negative covariance. Knowing the covariance between a portfolio's assets helps identify which ones carry similar risk. If all assets had a positive covariance, i.e. investment returns generally move in the same direction, a poor performance of one asset will put the whole portfolio at risk.

**Section 5: Limitations of Covariance**

One glaring limitation of covariance is that it only highlights the direction of the relationship between two variables. Another statistical measure, correlation, utilises covariance but divides it by the product of the standard deviation of both sets of data, producing a value that reflects the strength of the relationship between both variables. Covariance on its own will be useful in determining the nature of the relationship between two assets, but it has to be built upon before being able to accurately determine how closely two assets are related

Another drawback with using covariance is that it factors in outliers in the calculations. Therefore, unexpected fluctuations in the value of different shares and stocks will have a massive impact on the outputted value of covariance. Although this can be somewhat hedged through the utilisation of derivates such as forward agreements and option contracts, individuals may be led astray with the figure at their disposal and come to an unreliable conclusion about the nature of two assets.

**Section 6: Conclusion**

All in all, covariance remains a very useful tool in the world of risk management. Of course, covariance on its own is not sufficient in identifying the most profitable shares or stocks, but combined with other useful econometric tools, it can facilitate the construction of a diverse and profitable investment portfolio. Getting to grips with the fundamentals of such tools helps give individuals a clearer understanding of the nature of the financial securities under their possession, and ensures they are confident and efficient when navigating financial markets.

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[Accessed 13 August 2023].

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2. Hayes, A., 2023. *Covariance: Formula, Definition, Types, and Examples. *[Online]
Available at: __https://www.investopedia.com/terms/c/covariance.asp#toc-understanding-covariance__
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