Means and their average use cases.

The data sets that we work with on almost all of todays practical landscapes are extremely dynamic, and susceptible to either internal or external factors or both. Hence, investing; more precisely, analyzing the various investment opportunities becomes an overwhelmingly difficult task. This includes comparing and drawing conclusions regarding the security’s market prices, returns, P/E ratios, sales growth, etc. over a defined time horizon. As a solution to this quest for quantifying all of this expansive data into a single numerical, which gives us an approximation idea regarding the performance estimate of that metric, means are the best, no-nonsense quantifiers that come to mind.

But what do these means mean, exactly?

Means are the averages, a type of a statistical measures, which give us a synopsis of the whole set of underlying observations. They summarize the entire data set in a single value, which furthers our inferences with respect to the given data.

Some of the means which have the widest use cases in the world of finance are Arithmetic Mean, Geometric Mean and Harmonic Mean. For as long as the question of how to deal with outliers goes, two specialized categorizations of Arithmetic mean, namely, Wincorized Mean and Trimmed Mean, as well as Harmonic Mean would come to the rescue. Outliers are nothing but the observation points which are situated extremely far from the cluster of most of the observations. For e.g the S&P 500 index returns from years 2018-2022 are -6.4%, 22.88%, 16.26%, 26.89%, -19.44%, respectively. Here, a seemingly obvious outlier appears to be the 2022 return that is -19.44%.

Let’s dive into the world of means!

Arithmetic Mean is the simplest average to calculate in the world of statistics. It is computed by dividing the sum of all the observations by the number of observations.

Arithmetic Mean= (∑ni=1 xi)/n

For an ‘n’ number of observations, an arithmetic mean is most likely to be its middle most value. Why most likely?

Consider this example: You’ve a portfolio of 5 stocks, and you’ve 1 share of each stock. The returns on these fictitious 5 stocks are as follows: 5%, 7%, 9%, 11% and 13% for a given time period. If you had to find average return of your overall portfolio, its simple- all you’ve to do is find out its arithmetic mean, which is 9%. This means that your portfolio, as a whole, generated you a return on 9%.

Now consider the same example, but with a slight alteration: You still have a portfolio of 5 stocks but now you’ve 12, 25, 55, 86, 10 number shares of each stock, respectively. The returns on these fictitious 5 stocks remain constant at 5%, 7%, 9%, 11% and 13%, respectively. A simple arithmetic average would still be 9%. However, since you’ve varying quantities of each stock, the total returns generated by each stock become varying too. We call these quantities held per share as the ‘weight’ of that stock in our portfolio. Weight of the first stock, for example, would be 12/(Sum total of all the weights= 188) = 0.0638= 6.38%.

Hence to dabble with the questions of arithmetically averaging a return when weights given to various stocks (assets) are unequal, we use the weighted mean:

Weighted Mean = ∑ni=1 (xi*wi)/∑ni=1wi

Thus, upon integrating the values into the formula, we find that the weighted average is 9.6%, which is a slightly better and more realistic approximation than arithmetic mean.

Geometric Mean is the average measure that is the most widely used in computation of interest rates, CAGRs (Compounding annual growth rates), IRRs (Internal Rates of Returns). This is because geometric mean takes into consideration the effects of compounding. In real life scenarios, almost all the rates we come across especially in the finance world, are compounded on annual, semiannual, quarterly, bimonthly, monthly, weekly, or even on continuous basis. Interest rates are an excellent example of such measures. Alongside the inclusion of the effects of compounding, geometric mean is the best option while dealing with data points having serial correlation or autocorrelation. Serial correlation is the statistical relationship that exists between data points over various time intervals. In finance, for example, this relationship is exploited to determine how well the past prices or interest rates incidences determine the future ones.

Geometric Mean = {[(1+ X1) x (1+ X2) …. x (1 + Xn)] ^ (1/n)} - 1

Geometric means are used in cases to get an average with respect to fluctuating interest rates or to calculate average returns received over unequal time periods. Furthermore, Geometric means are also employed in the computations of a specific kind of IRR called as Money Weighted Rate of Return and also Time Weighted Rate of Returns which is essentially geometric mean of all the Holding Period Returns achieved during the period. Geometric mean is an excellent indicator of the volatility (risk) of a stock. How? Higher the geometric mean (higher the concentration in the area of central tendency), lower the volatility or risk (lower the dispersion, i.e., lower the standard deviation); and vice versa.

Harmonic Means are essentially the tools to deal with presence of outliers which are frequently occurring in our observation sets. Taking a look at the formula for Harmonic Mean would make sense of this statement better.

Harmonic Mean = n / (1/X1 + 1/X2 …. + 1/Xn)

As proven, we divide the number of observations by the sum of all the reciprocals of each of our price figures (observations). Naturally, as a result of this, larger values are given smaller weights and smaller values are given significant weights; thus, the linearity of the approximation is maintained and a more accurate figure is estimated, especially in cases of outliers.

Another interesting way to deal with outliers is making use of these extremely niche versions of arithmetic means: Wincorized Mean and Trimmed Mean.

In Wincorized Means, we completely replace all the outlier values with the value closest to them, falling in the area of the data set’s central tendency. We add these values and then further add them with the sum of remaining observation points. We divide this bigger sum by total number of observations present (including outliers). The impact of outliers is restricted due to replacing the outliers with values found in the central tendency.

source: https://images.app.goo.gl/ZJP1S35SqWaXjrSx6

As observed, the outlier values in the original data have been replaced by value closest to them falling in the central tendency region (it can be observed by the increase in the frequencies of extreme right and left ends of the central tendency regions in the second graph)

In Trimmed Mean, the outlier values are completely eliminated from the calculation. Rest of the observations are added and their sum is divided by the number of observations considered (that is, total number of observations – number of outliers omitted). The impact of outliers is wholly ignored in this calculation. Hence, trimmed mean gives a very territorial and restricted view on the data set because the presence of outliers is not taken into consideration at all.

Source: www.wallstreetmojo.com

One of Trimmed Mean’s use cases is the calculation of LIBOR- London Inter-Bank Offered Rate, which used to greatly serve as the most popular benchmark rate on basis of which, many global banks borrowed from each other until June 30th, 2023. The way this rate used to calculated was, every day 18 banks would submit their ideas for what the rates must be for the next day, to ICE (Intercontinental Exchange). ICE, in turn, would omit the highest 4 and lowest 4 rates, and compute the trimmed average of the middlemost 10 rates.

Means can be used in variety of combinations depending upon the eccentricities of the situation you are handling. In instances of cost averaging and SIPs, Harmonic Mean proves to be the most idealistic measure; As does the geometric mean in instances where average rates of return offered historically by a particular security needs to be calculated. Finance professionals are very precise with their use of means, as these measures are further employed in their statistical models to draw conclusions on various metrics such as volatility, interrelation with performance of some other asset(s),regression models to predict future values of the securities, etc. However, it is important to note that means are merely an approximation measure, and purpose of using correct means is not to arrive at the precise values, but to get as close to the precise values as possible.