Statistics for Returns (Arithmetic mean)

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Statistics for Returns:

Two such measures used with returns data are described below:

ARITHMETIC MEAN:

The arithmetic mean is the average of a sum of numbers, which reflects the central tendency of the position of the numbers. It is often used as a parameter in statistical distributions or as a result to summarize the observations of an experiment or a survey. In finance, the arithmetic mean is appropriate to support future estimates.

The best known statistic to most people is the arithmetic mean. Therefore, when someone refers to the mean return they usually are referring to the arithmetic mean unless otherwise specified. The arithmetic mean, customarily designated by the symbol X-bar, of a set of values is calculated as:

or the sum of each of the values being considered divided by the total number of values n.

For Example:

Let us take an example of a batsman who scored the following runs in his last 10 innings during last one year: 45, 65, 7, 10, 43, 35, 25, 17, 78, 91. Calculate the batsman’s average in his last 10 innings.

Arithmetic Mean is calculated using the formula given below:

Arithmetic Mean = ∑ xi / n
Arithmetic Mean = (45 + 65 + 7 + 10 + 43 + 35 + 25 + 17 + 78 + 91) / 10
Arithmetic Mean = 41.60

Therefore, the batsman’s average remained 41.60 runs per innings in his last 10 innings.\

Advantages of Arithmetic Means:

  • Fast and easy to calculate

As the most basic measure in statistics, arithmetic average is very easy to calculate. For a small data set, you can calculate the arithmetic mean quickly in your head or on a piece of paper. In computer programs like Excel, the arithmetic average is always one of the most basic and best known functions (in Excel the function is AVERAGE). Here you can see the basics of arithmetic average calculation.

  • Easy to work with and use in further analysis

Because its calculation is straightforward and its meaning known to everybody, arithmetic average is also more comfortable to use as input to further analyses and calculations. When you work in a team of more people, the others will much more likely be familiar with arithmetic average than geometric average or mode.

Disadvantage Of Arithmetic Means:

  •  Sensitive to extreme values
Arithmetic average is extremely sensitive to extreme values. Imagine a data set of 4, 5, 6, 7, and 8,578. The sum of the five numbers is 8,600 and the mean is 1,720 – which doesn’t tell us anything useful about the level of the individual numbers.

Therefore, arithmetic average is not the best measure to use with data sets containing a few extreme values or with more dispersed (volatile) data sets in general. Median can be a better alternative in such cases.
  • Not suitable for time series type of data
Arithmetic average is perfect for measuring central tendency when you’re working with data sets of independent values taken at one point of time. There was an example of this in one of the previous articles, when we were calculating average return of 10 stocks in one year.

However, in finance you often work with percentage returns over a series of multiple time periods. For calculating average percentage return over multiple periods of time, arithmetic average is useless, as it fails to take the different basis in every year into consideration (100% equals a different price or portfolio value at the beginning of each year). The more volatile the returns are, the more significant this weakness of arithmetic average is. Here you can see the example and reason why arithmetic average fails when measuring average percentage returns over time.

  •  Works only when all values are equally important

Arithmetic average treats all the individual observations equally. In finance and investing, you often need to work with unequal weights. For example, you have a portfolio of stocks and it is highly unlikely that all stocks will have the same weight and therefore the same impact on the total performance of the portfolio.

Calculating the average performance of the total portfolio or a basket of stocks is a typical case when arithmetic average is not suitable and it is better to use weighted average instead. You can find more details and an example here: Why you need weighted average for calculating total portfolio return.

Conclusion:

Arithmetic average as a measure of central tendency is simple and easy to use. But in order to take advantage of it and prevent it from doing any harm to your analysis and decision making, you should be familiar with the situations when it fails and when other tools are more useful.

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