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LM03 Statistical Measures of Asset Returns 2025 Level I Notes © IFT. All rights reserved 1 LM03 Statistical Measures of Asset Returns 1. Introduction...........................................................................................................................................................2 2. Measures of Central Tendency and Location............................................................................................2 3. Measures of Dispersion.....................................................................................................................................6 4. Measures of Shape of a Distribution .........................................................................................................10 5. Correlation Between Two Variables .........................................................................................................13 Summary...................................................................................................................................................................16 Required disclaimer: IFT is a CFA Institute Prep Provider. Only CFA Institute Prep Providers are permitted to make use of CFA Institute copyrighted materials which are the building blocks of the exam. We are also required to create / use updated materials every year and this is validated by CFA Institute. Our products and services substantially cover the relevant curriculum and exam and this is validated by CFA Institute. In our advertising, any statement about the numbers of questions in our products and services relates to unique, original, proprietary questions. CFA Institute Prep Providers are forbidden from including CFA Institute official mock exam questions or any questions other than the end of reading questions within their products and services. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of the product and services offered by IFT. CFA Institute®, CFA® and “Chartered Financial Analyst®” are trademarks owned by CFA Institute. © Copyright CFA Institute Version 1.0
LM03 Statistical Measures of Asset Returns 2025 Level I Notes © IFT. All rights reserved 2 1. Introduction In this learning module we will learn how to summarize and analyze important aspects of financial returns. This learning module covers: • Measures of central tendency and location • Measures of dispersion • Measures of the shape of return distributions • Covariance and correlation between two variables 2. Measures of Central Tendency and Location A ‘population’ is defined as all members of a specified group. A ‘parameter’ describes the characteristics of a population. A ‘sample’ is a subset drawn from a population. A ‘sample statistic’ describes the characteristic of a sample. For example, all stocks listed on a country’s exchange refers to a population. If 30 stocks are selected from the listed stocks, then this refers to a sample. Sample statistics—such as measures of central tendency, measures of dispersion, skewness, and kurtosis—help make probabilistic statements about investment returns. ‘Measures of central tendency’ specify where data are centered. ‘Measures of location’ include not only measures of central tendency but other measures that explain the location or distribution of data. Measures of Central Tendency The Arithmetic Mean The arithmetic mean is the sum of the observations divided by the number of observations. It is the most frequently used measure of the middle or center of data. The Sample Mean The sample mean is the arithmetic mean calculated for a sample. It is expressed as: X̅ = ∑ Xi n i=1 n where: n is the number of observations in the sample. If the sample data is: 2, 4, 4, 6, 10, 10, 12, 12, and 12 the sample mean can be calculated as: X̅ = 2 + 4 + 4 + 6 + 10 + 10 + 12 + 12 + 12 9 = 8 A drawback of the arithmetic mean is that it is sensitive to extreme values (outliers). It can be pulled sharply upward or downward by extremely large or small observations,
LM03 Statistical Measures of Asset Returns 2025 Level I Notes © IFT. All rights reserved 3 respectively. The Median The median is the midpoint of a data set that has been sorted into ascending or descending order. For odd number of observations: 2,5,7,11,14 → Median = 7 For even number of observations: 3, 9, 10, 20 → Median = (9 + 10)/2 = 9.5 As compared to a mean, a median is less affected by extreme values (outliers). The Mode The mode is the most frequently occurring value in a distribution. For the following data set: 2, 4, 5, 5, 7, 8, 8, 8, 10, 12 → Mode = 8 A distribution can have more than one mode, or even no mode. When a distribution has one mode it is said to be unimodal. If a distribution has two or three modes, it is called bimodal or trimodal respectively. When working with continuous data such as stock returns, ‘modal interval’ is often used instead of a mode. The data is divided into bins and the bin with the highest frequency is considered the modal interval. The following exhibit demonstrates this concept by plotting a histogram of the daily returns on an index. The highest bar in the histogram ‘0.0 to 0.9%’ is the modal interval. Dealing with Outliers When data contains outliers, there are three options to deal with the extreme values: Option 1: Do nothing; use the data without any adjustment.
LM03 Statistical Measures of Asset Returns 2025 Level I Notes © IFT. All rights reserved 4 Option 2: Delete all the outliers. Option 3: Replace the outliers with another value. Option 1 is appropriate in cases when the extreme values are genuine. Option 2 excludes extreme observations. A trimmed mean excludes a stated percentage of the lowest and highest values and then calculates the arithmetic mean of the remaining values. For example, a 5% trimmed mean discards the lowest 2.5% and the highest 2.5% of values and computes the mean of the remaining 95% of values. Option 3 replaces extreme observations with observations closest to them. A winsorized mean assigns a stated percentage of the lowest values equal to one specified low value and a stated percentage of the highest values equal to one specified high value, and then computes a mean from the restated data. For example, a 95% winsorized mean sets the bottom 2.5% of values equal to the value at or below which 2.5% of all the values lie (the “2.5th percentile” value) and the top 2.5% of values equal to the value at or below which 97.5% of all the values lie (the “97.5th percentile” value). Measures of Location Quartiles, Quintiles, Deciles, and Percentiles A quantile is a value at or below which a stated fraction of the data lies. Some examples of quantiles include: • Quartiles: The distribution is divided into quarters. • Quintiles: The distribution is divided into fifths. • Deciles: The distribution is divided into tenths. • Percentile: The distribution is divided into hundredths. The formula for the position of a percentile in a data set with n observations sorted in ascending order is: Ly = (n + 1)y 100 where: y is the percentage point at which we are dividing the distribution. n is the number of observations. Ly is the location (L) of the percentile (Py) in an array sorted in ascending order. Some important points to remember are: • When the location, Ly, is a whole number, the location corresponds to an actual observation. • When Ly is not a whole number or integer, Ly lies between the two closest integer numbers (one above and one below) and we use linear interpolation between those two places to determine Py. • Interquartile range is the difference between the third and the first quartiles.