# 2024 Level 1 CFA® Exam:

Return Measures

The arithmetic mean can be most simply defined as the sum of all observations divided by the number of observations.

If we're dealing with a sample, the arithmetic mean can be computed using the following formula:

\(\bar{X}=\sum_{i=1}^n\frac{X_i}{n}\)

- \(\bar{X}\) - sample mean
- n - number of observations in a sample
- \(X_i\) - i observation value

Note: When you multiply the arithmetic mean by the number of observations, the result will be the sum of the observations. Also, remember that the sum of deviations of individual elements of a set from the mean equals zero.

The arithmetic mean is a popular and frequently used measure, as it is easy to calculate and can be interpreted intuitively. We should remember, however, that it doesn't always properly reflect the characteristics of a dataset. One of the reasons is that the arithmetic mean takes into account all elements of a dataset including outliers.

Outliers are extreme observations, that is observations extremely different from the majority of observations for a variable. So, outliers take either extremely high or extremely low values.

In the case of a large difference between the highest or the lowest value and the central value, the arithmetic mean can also be very high or very low, which may distort the characteristics of the examined data.

On the other hand, the fact that the mean includes all elements of a dataset is an advantage when compared to such measures of central tendency as the mode or the median.

When we detect outliers, the first thing we should do is to check for possible errors in our data. We might draw outliers from a different population or erroneously record an outlier. If there are no errors, we generally have two options – either we leave the data as it is, namely we don’t remove the outliers, or we remove the outliers.

If we decide to remove outliers, two solutions come in handy:

- trimmed mean, and
- winsorized mean.

The following example shows how the trimmed mean and winsorized mean are calculated.

Our sample dataset includes the following 60 numbers:

-111, -33, -12, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 327, 576, 5012

You notice the outliers in the dataset and decide to remove them. How will the dataset for 10% trimmed mean and 90% winsorized mean look like?

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The geometric mean is often used to compute the average rate of return over a series of periods or to calculate the growth rate. When calculating a rate of return, the formula for the geometric mean looks as follows:

\(R_G=\sqrt[T]{\prod_{t=1}^T(1+R_t)}-1\)

- \(R_G\) - geometric mean return
- T - number of periods
- \(R_t\) - return in period t

The geometric mean is often used to compute the average rate of return over a series of periods or to calculate the growth rate.

The last type of mean we're going to discuss here is the harmonic mean. The harmonic mean has fewer applications than the arithmetic mean and the geometric mean. It can be used, however, to determine the average purchase price paid for stocks if we bought them in several periods for the same amount or to calculate the average time necessary for the production of a given product. The harmonic mean applies reciprocals of the values of observations. It can be represented with this expression:

\(\bar{X}_H=\frac{n}{\sum_{i=1}^n\frac{1}{X_i}}\)

- \(\bar{X}_H\) - harmonic mean
- \(X_i\) - value of i observation, \(X_i>0\)
- n - number of observations

**1) **The harmonic mean can be used e.g. to determine the average purchase price paid for stocks bought in several months for the same monthly budget OR to calculate the average time necessary for a given product to be produced.

**2) **Harmonic mean (H) is always less or equal to geometric mean (G), which is always less or equal to arithmetic mean (A).

\(H\le{G}\le{A}\)

The harmonic mean equals the number of observations divided by the sum of the inverse values of observations.

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