The Different Types of Means in Finance

Not all mean values in finance are calculated the same way - whether it's an average cost per share or an average return over time, you might already know that there are multiple types of means. Today we will explore why "no one mean fits all" in finance. We'll examine several widely used approaches for calculating means and discuss their differences and ideal applications in the finance field.

1. Arithmetic Mean

The first type of mean I want to talk about is the simple arithmetic mean. This type of mean is the most widespread—the most used and abused one. I don't blame people for throwing this mean at every use case, but especially within finance, it's not always the most optimal one, as we'll see. This type of mean is the purest one; it's a simple average and acts as an unbiased estimator of the true mean of the underlying statistical distribution.

$$\overline{x}=\dfrac{(x_1+x_2+...+x_n)}{n}$$

Probably the biggest issue—and the reason why we just can't use the arithmetic mean everywhere in finance—is that its accuracy decreases as the volatility of the underlying observations grows.

2. Geometric Mean

The mean with probably the most use cases in finance is the geometric mean. This is because this mean captures powerful concepts: the effects of compounding and volatility. We encounter these two at every step in our finance journey, with the easiest example being the average (annualized) returns calculation.

$$\overline{x}_{geo}=\sqrt[n]{(1+x_1)\,\,...\,\,(1+x_n)} -1$$

The multiplication between the values of $1+x$ accounts for the compounding happening between the various measurements of what, in our case, could be a return. If we had to modify this for a finance use case, we’d need to replace the simple $n$ of periods with the value of $n/m$, where $m$ is the number of periods in a desired timeframe (e.g., a year).

2.1. Calculating the average annual return from non-annualized returns

I think the complexity that the periods might have introduced will be slightly more understandable with an example. Let’s say that we’re looking to calculate the average return on our investment in a certain equity. We have quarterly rates of return for this equity spanning the last two years, and we want to calculate a yearly rate of return. Here is where the mismatch happens.

$$\overline{x}_{geo}=\sqrt[8/4]{(1+x_1)\,\, ... \,\,(1+x_8)}-1$$

If we just took the $8\,(=2\,\,years\cdot4)$ quarters as $n$, we would be calculating an average quarterly rate of return. Here is where the $n/m$ relation comes into play. If we take the $n=8$ quarters and divide it by $m=4$, being the quarters in a year, we get 2, which represents the years.

3. Harmonic Mean

This type of mean, while not quite as popular as the previous two, is an essential helper in finance when it comes to averaging ratios. One great example within equities is calculating the average price at which we bought a security over time.

$$\overline{x}_{har}=\dfrac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}=\frac{n}{\sum_{i=1}^{n}{\frac{1}{x_n}}}$$

3.2. Cost averaging the price

Let's say we buy $500 worth of shares of $AAPL over the course of three months, for $196, $200, and $211 per share, respectively. To calculate with the most accuracy the average price per share which we paid, we can simply go and compute it as follows.

$$\overline{P}_{AAPL}=\dfrac{3}{\frac{1}{196}+\frac{1}{200}+\frac{1}{211}}= 202.1375$$

This gives us the most exact result of the average price per share during our purchases. We can verify this by calculating the amount of shares we bought each month.

$$\#\,\,shares=\dfrac{500}{196}+\frac{500}{200}+\frac{500}{211}=7.4206$$

And afterward, dividing the total amount of money we spent on these buys by the number of shares we actually were able to buy over time, calculated in the previous equation.

$$\overline{P}_{AAPL}=1500\div7.4206=202.1375$$

As we can see, the values match exactly, and the harmonic mean saved us a ton of laborious calculations along the way.

4. Trimmed Mean

Outliers can be tricky to handle and often pose significant challenges in data analysis. They can skew results, particularly affecting measures like the mean. When it's justifiable to exclude outliers, one effective method is to use the trimmed mean.

The trimmed mean works by removing a specified percentage of data from both the lower and upper extremes. Analysts commonly trim 5%, 10%, or even 20% from each end, which helps reduce the influence of extreme values on the overall mean. However, this comes at the cost of throwing away potentially valuable information.

It's considered good practice to compare the regular mean (with all data points included) and the trimmed mean. Doing so can offer deeper insights into the distribution and impact of outliers. Additionally, outliers themselves often carry meaningful information. Rather than being dismissed outright, they should be investigated—more often than not, they have a story to tell!

5. Winsorized Mean

Another method of handling outliers when calculating the mean is through the use of the winsorized mean. Unlike the trimmed mean, which removes extreme values entirely from the dataset, the winsorized mean replaces these outliers with the nearest values that fall within a specified percentile range. This technique maintains the original sample size while reducing the impact of extreme values on the overall mean.

To illustrate how this works, consider a set of returns, $R$, recorded over time for 20 independent portfolio managers:

$$R = [\,\,-1,\, 0,\, 1,\, 2,\, 3,\, ...\,\, 5,\, 5,\, 6,\, 11,\, 13\,\,]$$

If we decide for a 10% winsorized mean, we identify the lowest 10% and the highest 10% of values and replace them with the closest observations that lie within the remaining central range. Since the dataset contains 20 observations, 10% corresponds to two values at each end.

Thus, the two lowest values, −1 and 0, are replaced with the next smallest non-outlier, which is 1. Similarly, the two largest values, 11 and 13, are replaced with the next largest value within the acceptable range, which is 6. The resulting winsorized dataset becomes:

$$R = [\,\,1,\, 1,\, 1,\, 2,\, 3,\, \, ...\,\, 5,\, 5,\, 6,\, 6,\, 6\,\,]$$

To compute the winsorized mean, we simply take the arithmetic mean of this adjusted series. Since the total number of observations remains unchanged at 20, the winsorized mean is given by the sum of the transformed values divided by 20.

This approach offers a more robust estimate of central tendency by limiting the influence of outliers, while still preserving the integrity and size of the original dataset.

6. How Much Difference Does It Make Anyway

That’s the question many of us, including me, have asked. Well, now that we know how to calculate each one of these means, we can actually test them and see how different—and accurate to the supposed value—they turn out. We can use the last example from the harmonic mean to compare how different types of means would perform.

We can see that the most precise one in this case is the harmonic mean. In general, we can assume that Harmonic Mean < Geometric Mean < Arithmetic Mean.

7. So Which One to Use When

As a general rule of thumb, as suggested by the CFA curriculum and other sources, in a financial setting one should use the means as shown in the following table.

One important thing before concluding is that we also need to account for outliers in the data. Average returns, depending on which mean we use, can be significantly impacted by unusually high or low observations. For this case, there are other methods of dealing with outliers by employing more sophisticated techniques, such as the trimmed or winsorized mean or simply using the median instead.

8. Conclusion

Understanding the differences between these types of means can be crucial in financial analysis. Each mean has its strengths and is best suited to specific scenarios. Misapplying them can lead to inaccurate interpretations and misguided decisions. By choosing the right type of mean for the right context we can ensure our analyses are not just mathematically sound but also financially meaningful.