Statistical Concepts and Market Return

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1 Statistical Concepts and Market Return 2014 Level I Quantitative Methods IFT Notes for the CFA exam

2 Contents 1. Introduction Some Fundamental Concepts Summarizing Data Using Frequency Distributions The Presentation of Data Graphic Measures of Central Tendency Other Measures of Location: Quantiles Measures of Dispersion Symmetry and Skewness in Return Distributions Kurtosis in Return Distributions Using Geometric and Arithmetic Means This document should be read in conjunction with the corresponding reading in the 2014 Level I CFA Program curriculum. Some of the graphs, charts, tables, examples, and figures are copyright 2013, CFA Institute. Reproduced and republished with permission from CFA Institute. All rights reserved. Required disclaimer: CFA Institute does not endorse, promote, or warrant the accuracy or quality of the products or services offered by Irfanullah Financial Training. CFA Institute, CFA, and Chartered Financial Analyst are trademarks owned by CFA Institute. Copyright Irfanullah Financial Training. All rights reserved. Page 1

3 1. Introduction Statistical methods provide a powerful set of tools for analyzing data and drawing conclusions from them. These are particularly useful when we are analyzing asset returns, earning growth rates, commodity prices, or any other financial data. Descriptive statistics is the branch of statistics that deals with describing and analyzing data. In this reading, we will study statistical methods that allow us to summarize return distributions. Specifically, we will explore four properties of return distributions: Where the returns are centered (central tendency) How far returns are dispersed from their center (dispersion) Whether the distribution of returns is symmetrically shaped or lopsided (skewness) Whether extreme outcomes are likely (kurtosis) 2. Some Fundamental Concepts 2.1 The Nature of Statistics The term statistics can have two broad meanings, one referring to data and the other to method. Statistical methods include: Descriptive statistics: Study of how data can be summarized effectively to describe the important aspects of large data sets. Statistical inference: Making forecasts, estimates, or judgments about a larger group from the smaller group actually observed. 2.2 Populations and Samples A population is defined as all members of a specified group. Any descriptive measure of a population characteristic is called a parameter. A sample is a subset of a population. Any descriptive measure of a sample characteristic is called a sample statistic (statistic, for short). Copyright Irfanullah Financial Training. All rights reserved. Page 2

4 2.3 Measurement Scales To choose the appropriate statistical method for summarizing and analyzing data, we need to distinguish among different measurement scales. All data measurements are taken on one of the following scales: Nominal scales: These scales categorize data but do not rank them. Hence, they are often considered the weakest level of measurement. An example could be if we assigned integers to mutual funds that follow different investment strategies. Number 1 might refer to a small-cap value fund, number 2 might refer to a large-cap value fund, and so on for each possible style. Ordinal scales: These scales sort data into categories that are ordered with respect to some characteristic. An example is Standard & Poor s star ratings for mutual funds. One star represents the group of mutual funds with the worst performance. Similarly, groups with two, three, four and five stars represent groups with increasingly better performance. Interval scales: These scales not only rank data, but also ensure that the differences between scale values are equal. The Celsius and Fahrenheit scales are examples of such scales. The difference in temperature between 10 o C and 11 o C is the same amount as the difference between 40 o C and 41 o C. The zero point of an interval scale does not reflect complete absence of what is being measured. Hence, it is not a true zero point or natural zero. Ratio scales: These scales have all the characteristics of interval scales as well as a true zero point as the origin. This is the strongest level of measurement. The rate of return on an investment is measured on a ratio scale. A return of 0% means the absence of any return. Worked Example 1: Identifying Scales of Measurement Note: this example has been reproduced from the curriculum. State the scale of measurement for each of the following: 1. Credit ratings for bond issues. 2. Cash dividends per share. Copyright Irfanullah Financial Training. All rights reserved. Page 3

5 3. Hedge fund classification types. 4. Bond maturity in years. Solution to 1: Credit ratings are measured on an ordinal scale. A rating places a bond issue in a category, and the categories are ordered with respect to the expected probability of default. But the difference in the expected probability of default between AA and A+, for example, is not necessarily equal to that between BB and B+. In other words, letter credit ratings are not measured on an interval scale. Solution to 2: Cash dividends per share are measured on a ratio scale. For this variable, 0 represents the complete absence of dividends; it is a true zero point. Solution to 3: Hedge fund classification types are measured on a nominal scale. Each type groups together hedge funds with similar investment strategies. In contrast to credit ratings for bonds, however, hedge fund classification schemes do not involve a ranking. Thus such classification schemes are not measured on an ordinal scale. Solution to 4: Bond maturity is measured on a ratio scale. 3. Summarizing Data Using Frequency Distributions A frequency distribution is a tabular display of data summarized into a relatively small number of intervals. In order to construct a frequency distribution, we can follow the following procedure: Sort the data in ascending order. Copyright Irfanullah Financial Training. All rights reserved. Page 4

6 Calculate the range of the data, defined as Range = Maximum value Minimum value. Decide on the number of intervals in the frequency distribution, k. Determine interval width as Range/k. Determine the intervals by successively adding the interval width to the minimum value to determine the ending points of intervals. Stop after reaching an interval that includes the maximum value. Count the number of observations falling in each interval. Construct a table of the intervals listed from smallest to largest that shows the number of observations falling in each interval. The following example illustrates the process. Worked Example 2: Construction of a Frequency Table Say you are evaluating 100 stocks with prices ranging from 46 to 65. Stock Price (Absolute) Frequency Cumulative Frequency Relative Frequency Cumulative Relative Frequency In order to summarize this data, we have divided the stock prices into 4 intervals of stock price each having a width of 5. The actual number of observations in a given interval is called the absolute frequency, or simply the frequency. For example, there are 25 stocks falling in the interval of price range from The relative frequency is the absolute frequency of each interval divided by the total number of observations. The cumulative relative frequency cumulates the relative frequencies as we move from the first to the last interval. It tells us the fraction of observations that are less than the upper limit of each interval. So there are 60 observations less than the stock price of 55. The frequency distribution gives us a sense of where most of the observations lie and also whether the distribution is evenly distributed, lopsided, or peaked. Copyright Irfanullah Financial Training. All rights reserved. Page 5

7 Frequency Statistical Concepts and Market Return Irfanullah.co 4. The Presentation of Data Graphic A graphical display allows us to visualize important characteristics of a data set. In this section we discuss the histogram, frequency polygon, and the cumulative frequency distribution. 4.1 The Histogram A histogram is a bar chart of data that have been grouped into a frequency distribution. The advantage of the visual display is that we can quickly see where most of the observations lie. Consider the histogram shown below Histogram 15 Frequency Stock Price The height of each bar in the histogram represents the absolute frequency for each interval. 4.2 The Frequency Polygon and the Cumulative Frequency Distribution The frequency polygon is constructed when we plot the midpoint of each interval on the x-axis and the absolute frequency for that interval on the y-axis. We then connect the neighboring points with a straight line. The figure below is an example of a frequency polygon. Copyright Irfanullah Financial Training. All rights reserved. Page 6

8 40 Frequency Polygon Frequency Another graphical tool is the cumulative frequency distribution. Such a graph can plot either the cumulative absolute or cumulative relative frequency against the upper interval limit. The cumulative frequency distribution allows us to see how many or what percent of the observations lie below a certain value. The figure below is an example of a cumulative frequency distribution. 120 Cumulative Frequency Cumulative Frequency Copyright Irfanullah Financial Training. All rights reserved. Page 7

9 5. Measures of Central Tendency A measure of central tendency specifies where the data are centered. Measures of location include not only measures of central tendency but other measures that illustrate the location or distribution of data. As a basis for understanding the measures of central tendency, let us consider the stock returns of a company over the last 10 years: 2%, 5%, 4%, 7%, 8%, 8%, 12%, 10%, 8%, and 5%. We will use this data set to explain various measures of central tendency. 5.1 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 Population Mean The population mean is the arithmetic mean computed for a population. A given population has only one mean. For a finite population, the population mean is: µ = N i=1 X i N where N is the number of observations in the entire population and X i is the ith observation. For the dataset described above, µ = 10 µ = 6.9% The Sample Mean The sample mean is calculated like the population mean, except we use the sample values. X = n i=1 X i n where n is the number of observations in the sample. If the sample data is: 8, 12, 10, 8 and 5, the sample mean can be calculated as: Copyright Irfanullah Financial Training. All rights reserved. Page 8

10 X = X = Properties of the Arithmetic Mean As analysts, we often use the mean return as a measure of the typical outcome for an asset. Some of the advantages of the arithmetic mean are: Uses all information about the size and magnitude of the observations Easy to work with and compute mathematically One of the drawbacks of the arithmetic mean is its sensitivity to extreme values. Because all observations are used to compute the mean, the arithmetic mean can be pulled sharply upward or downward by extremely large or small observations, respectively. Unusually large or small observations are called outliers. 5.2 The Median The median is the value of the middle item of a set of items that has been sorted into ascending or descending order. In an odd numbered sample of n items, the median occupies the (n + 1)/2 position. In an even numbered sample, we define the median as the mean of the values of items occupying the n/2 and (n + 2)/2 positions (the two middle items). Sorting the sample data given above we have: 5%, 8%, 8% 10%, 12%. Here n = 5. The position or location of the median number is given by (n + 1)/2 = 3 and the median value is 8%. A distribution has only one median. An advantage of the median is that, unlike the mean, extreme values do not affect it. The median, however, does not use all the information about the size and magnitude of the observations. It focuses only on the relative position of the ranked observations. Another disadvantage is that it is more tedious to compute as compared to the mean. Copyright Irfanullah Financial Training. All rights reserved. Page 9

11 5.3 The Mode The mode is the most frequently occurring value in a distribution. For the following data set: 5%, 8%, 8% 10%, 12%, the mode is 8%. A distribution can have more than one mode, or even no mode. When a distribution has one most frequently occurring value, the distribution is said to be unimodal. If a distribution has two modes, it is bimodal. The mode is the only measure of central tendency that can be used with nominal data. Stock return data and other data from continuous distributions may not have a modal outcome. When such data are grouped into intervals, however, we often find an interval (possibly more than one) with the highest frequency. This is called the modal interval (or intervals). 5.4 Other Concepts of Mean The arithmetic mean is a fundamental concept for describing the central tendency of data. Other concepts of mean are also important and are discussed below. The Weighted Mean In the arithmetic mean, all observations are equally weighted by the factor 1/n. In working with portfolios, we need the more general concept of weighted mean to allow different weights on different observations. The formula for the weighted mean is: n X w = w i X i i=1 where the sum of the weights equals 1; that is n i=1 w i = 1 Consider an investor with a portfolio of three stocks. 40 is invested in A, 60 in B and 100 in C. If returns were 5% on A, 7% on B and 9% on C, we can compute the portfolio return by using the weighted mean: (40/200) x 5% + (60/200) x 5% + (100/200) x 9% = 7% The Geometric Mean The general formula for calculating the geometric mean is: G =(X1 X2 X3 Xn) 1/n with X i 0 for i =1, 2, n. Copyright Irfanullah Financial Training. All rights reserved. Page 10

12 As an example, the geometric mean of 7, 8 and 9 will be: (7 x 8 x 9) 1/3 = The most common application of the geometric mean is to calculate the average return of an investment. The formula is: R G =[(1+R1)(1+R2).(1+Rn)] 1/ n 1 We will illustrate the use of this formula through a simple scenario: For a given stock the return over the last four periods is: 10%, 8%, -5% and 2%. The geometric mean is calculated as: [( )( )(1 0.05)( )] 1/4 1 = = 3.58% Given the returns shown above, $1.00 invested at the start of period 1 grew to: $1.00 x 1.10 x 1.08 x 0.95 x 1.02 = If the investment had grown at 3.58% every period, $1.00 invested at the start of period 1 would have increased to: $1.00 x x x x = As expected, both scenarios give the same answer. 3.58% is simply the average growth rate per period. The geometric mean is always less than or equal to the arithmetic mean. The only time that the two means will be equal is when there is no variability in the observations i.e. when all the observations are the same. Note: In the reading on Discounted Cash Flow Applications we used the geometric mean to calculate the time-weighted rate of return. The Harmonic Mean The harmonic mean is a special type of weighted mean in which an observation s weight is inversely proportional to its magnitude. The formula for a harmonic mean is: n X H = n / ( 1 X i ) i=1 with X i > 0 for i = 1,2, n, and n is the number of observations. Copyright Irfanullah Financial Training. All rights reserved. Page 11

13 This concept can be applied when we invest the same amount every month in a particular stock and want to calculate the average purchase price. Suppose an investor purchases $1,000 of a security each month for three months. The share prices are $10, $15 and $20 at the three purchase dates. The average purchase price is simply the harmonic mean of 10, 15 and 20. The harmonic mean is: 3 / (1/10 + 1/15 + 1/20) = The harmonic mean is generally less than the geometric mean, which is in turn less than the arithmetic mean. To illustrate this fact take three numbers: 10, 15, and 20. It has just been shown that the harmonic mean is The geometric mean is (10 x 15 x 20) 1/3 = The arithmetic mean is simply 15. If all the observations in a dataset are the same then the three means are the same. 6. Other Measures of Location: Quantiles A quantile is a value at or below which a stated fraction of the data lies. 6.1 Quartiles, Quintiles, Deciles and Percentiles Quartiles divide the distribution into quarters, quintiles into fifths, deciles into tenths, and percentiles into hundredths. Given a set of observations, the y th percentile is the value at or below which y percent of observations lie. Often we need to approximate the value of a percentile. To do so we arrange the data in ascending order and locate the position of the percentile within the set of observations. We then determine (or estimate) the value associated with that position. The formula to calculate the percentile in such a way is: L y = (n+1) y /100 Where y is the percentage point at which we are dividing the distribution, n is the number of observations and L y is the location (L) of the percentile (P y ) in an array sorted in ascending order. Some important points to remember are: Copyright Irfanullah Financial Training. All rights reserved. Page 12

14 When the location, L y, is a whole number, the location corresponds to an actual observation. When L y is not a whole number or integer, L y lies between the two closest integer numbers (one above and one below) and we use linear interpolation between those two places to determine P y. Worked Example 3: Calculating Percentiles Given below is the return data on 20 mutual funds arranged in ascending order. Number Return in % Number Return in % At a given percentile, y = 10%, with n = 20 and the data sorted in ascending order, the location of the observation is given by: L 10 = (20 + 1) (10/100) = 2.1 With a small data set, such as this one, the location calculated using the above formula is approximate. As the data set becomes larger, the formula gives a more precise location. 6.2 Quantiles in Investment Practice Quantiles can be used to rank the performance of portfolios and even investment managers. In investment research, analysts often refer to the set of companies with returns falling below the 10 th percentile cutoff point as the bottom return decile. It is also common to place funds in quartiles based on performance in a given period. A top quartile fund means that relative to comparable funds, the performance of this fund is in the top 25%. Copyright Irfanullah Financial Training. All rights reserved. Page 13

15 7. Measures of Dispersion Dispersion is the variability around the central tendency. Absolute dispersion is the amount of variability present without comparison to any reference point or benchmark. Range, mean absolute deviation, variance, and standard deviation are all examples of absolute dispersion. 7.1 The Range The range is the difference between the maximum and minimum values in a data set. Consider the same data set we used before: 2%, 5%, 4%, 7%, 8%, 8%, 12%, 10%, 8%, and 5%. Here the maximum return is 12% and the minimum return is 4%. The range is 12% 4% = 8%. The range is easy to calculate but uses only two pieces of information from the distribution. It cannot tell us how the data are distributed i.e. the shape of the distribution. 7.2 The Mean Absolute Deviation The dispersion around the mean is a fundamental piece of information used in statistics. However, if we take an arithmetic average of the deviations around the mean, we encounter a problem: such an arithmetic average always sums to 0. One solution to this is to examine the absolute deviations around the mean as in the mean absolute deviation. n MAD = [ X i X ] /n i=1 where X is the sample mean and n is the number of observations in the sample. Consider the following data set: 8, 12, 10, 8 and 5. X = ( )/5 = ) MAD = = 0 5 Copyright Irfanullah Financial Training. All rights reserved. Page 14

16 7.3 Population Variance and Population Standard Deviation Variance is defined as the average of the squared deviations around the mean. Standard deviation is the positive square root of the variance. Population variance is the arithmetic average of the squared deviations around the mean. N σ 2 = (X i μ) 2 / N i=0 where µ is the population mean and N is the size of the population. For the data set: 2%, 5%, 4%, 7%, 8%, 8%, 12%, 10%, 8%, and 5%, the variance is given by: σ 2 = [(2 6.9)2 +(5 6.9) 2 +(4 6.9) 2 +(7 6.9) 2 +(8 6.9) 2 +(8 6.9) 2 +(12 6.9) 2 +(10 6.9) 2 +(8 6.9) 2 + (5 6.9) 2 ] 10 σ 2 = 7.89 Because variance is measured in squared units, we need a way to return to the original units. We can solve this problem by using standard deviation, the square root of the variance. The population standard deviation is defined as the positive square root of the population variance. For the data given above, σ = 7.89 = 2.81%. Both the population variance and standard deviation are examples of parameters of a distribution. We often do not know the mean of a population of interest. We then estimate the population mean with a mean from a sample drawn from the population. Next, we calculate the sample variance and standard deviation. 7.4 Sample Variance and Sample Standard Deviation When we deal with samples, the summary measures are called statistics. The statistic that measures dispersion in a sample is called the sample variance. n s 2 = (X i X ) 2 / (n 1) i=0 Copyright Irfanullah Financial Training. All rights reserved. Page 15

17 where X is the sample mean and n is the number of observations in the sample. The steps to calculate a sample variance are: Calculate the same mean, X Calculate each observation s squared deviation from the sample mean, (X i X ) 2 Sum the squared deviations from the mean: n i=0(x i X ) 2 Divide the sum of squared deviations from the mean by n-1: n (X i X ) 2 / (n 1) i=0 By using n - 1 (rather than n) as the divisor, we improve the statistical properties of the sample variance. Consider the following data set: 8, 12, 10, 8 and 5. The sample variance is calculated as follows: s 2 = [(8 8.6)2 + (12 8.6) 2 + (10 8.6) 2 + (8 8.6) 2 + (5 8.6) 2 ] 5 1 s 2 = 6.80% The sample standard deviation is the positive square root of the sample variance. For the sample data given above, s = 6.80 = 2.61%. The population and sample standard deviation can easily be computed using a financial calculator. Assume the following data set: 10%, -5%, 10%, 25%, the calculator key strokes are show below: Keystrokes Explanation Display [2nd] [DATA] Enter data entry mode [2nd] [CLR WRK] Clear data registers X01 10 [ENTER] X01 = 10 [ ] [ ] 5+/- [ENTER] X02 = -5 [ ] [ ] 10 [ENTER] X03 = 10 [ ] [ ] 25 [ENTER] X04 = 25 Copyright Irfanullah Financial Training. All rights reserved. Page 16

18 Keystrokes Explanation Display [2nd] [STAT] [ENTER] Puts calculator into stats mode. [2nd] [SET] Press repeatedly till you see 1-V [ ] Number of data points N = 4 [ ] Mean X = 10 [ ] Sample standard deviation Sx = [ ] Population standard deviation σx = Notice that the calculator gives both the sample and the population standard deviation. On the exam we will have to determine whether we are dealing with population or sample data. 7.5 Semivariance, Semideviation, and Related Concepts Note: Semivariance and semideviation are not emphasized in the learning outcomes and have a very low probability of being tested on the Level I exam. Nevertheless, a brief explanation is given below. Variance and standard deviation of returns take account of returns above and below the mean, but investors are concerned only with downside risk, for example returns below the mean. As a result, analysts have developed semivariance, semideviation and related dispersion measures that focus on downside risk. Semivariance is defined as the average squared deviation below the mean. Semideviation is the positive square root of semivariance. When return distributions are symmetric, semivariance and variance are effectively equivalent. For asymmetric distributions, variance and semivariance rank prospects risk differently. 7.6 Chebyshev s Inequality According to Chebyshev s inequality, the proportion of the observations within k standard deviations of the arithmetic mean is at least 1-1/k 2 for all k > 1. To find out what percent of the observations must be within 2 standard deviations of the mean we simply plug into the formula and get: 1 1/2 2 = 1 ¼ = 0.75 = 75%. Hence at least 75% of the data will be between 2 Copyright Irfanullah Financial Training. All rights reserved. Page 17

19 standard deviations of the mean. To understand this concept, consider a distribution with a mean value of 10 and a standard deviation of 3. For this distribution at least 75% of the data will be between 4 and 16. Note that 4 is two standard deviations (2 x 3) less than the mean (10) and 16 is two standard deviations greater than the mean. Plugging in a value of k = 3 in Chebyshev s inequality shows us that at least 89% of the population data will lie within three standard deviations of the mean. Chebyshev s inequality holds for samples and populations, and for discrete and continuous data regardless of the shape of the distribution. Worked Example 4: Chebyshev s Inequality Note: this example has been reproduced from the curriculum. The arithmetic mean monthly return and standard deviation of monthly returns on the S&P 500 were 0.97 percent and 5.65 percent, respectively, during the period, totaling 924 monthly observations. Using this information, address the following: 1. Calculate the endpoints of the interval that must contain at least 75 percent of monthly returns according to Chebyshev s inequality. 2. What are the minimum and maximum number of observations that must lie in the interval computed in Part 1, according to Chebyshev s inequality? Solution to 1: According to Chebyshev s inequality, at least 75 percent of the observations must lie within two standard deviations of the mean, X ± 2s. For the monthly S&P 500 return series, we have 0.97% ± 2(5.65%) = 0.97% ± 11.30%. Thus the lower endpoint of the interval that must contain at least 75 percent of the observations is 0.97% 11.30% = 10.33%, and the upper endpoint is 0.97% % = 12.27%. Solution to 2: Copyright Irfanullah Financial Training. All rights reserved. Page 18

20 For a sample size of 924, at least 0.75(924) = 693 observations must lie in the interval from 10.33% to 12.27% that we computed in Part Coefficient of Variation Sometimes we may find it difficult to interpret what the standard deviation means in terms of the relative degree of variability of different data sets. This can be because the data sets are significantly different or because the data sets have different units of measurement. The coefficient of variation can be useful in such situations. It is the ratio of the standard deviation of a set of observations to their mean value. CV = s/x When the observations are returns, the coefficient of variation measures the amount of risk (standard deviation) per unit of mean return. Hence, it allows us to directly compare dispersion across different data sets. Consider a simple example. Investment A has a mean return of 7% and a standard deviation of 5%. Investment B has a mean return of 12% and a standard deviation of 7%. The coefficients of variation can be calculated as follows: CV A = 5% 7% = 0.71 CV B = 7% 12% = 0.58 This metric shows that Investment A is more risky than Investment B. 7.8 The Sharpe Ratio If we use an inverse of the CV, we get a measure for the return per unit of risk of an investment. A more precise return-risk measure is the Sharpe ratio. The Sharpe ratio is the ratio of excess return to standard deviation of return for a portfolio, p. Excess return refers to the return above the risk free rate. The formula for calculating the Sharpe ratio is: S p = R p R F s p where Copyright Irfanullah Financial Training. All rights reserved. Page 19

21 R p = Mean return to the portfolio R F = Mean return to a risk-free asset s p = Standard deviation of return on the portfolio Worked Example 5: Calculating Sharpe Ratio The table below provides data for two portfolios. Given that the mean annual risk free rate is 10.5%, which portfolio has the higher Sharpe ratio? Portfolio Arithmetic mean Variance of (%) return (%) Portfolio A 16.4% 4.9% Portfolio B 12.6% 3.5% Solution: Portfolio A: Portfolio B: = = Portfolio A offers a higher excess return per unit of risk relative to Portfolio B. 8. Symmetry and Skewness in Return Distributions While mean and variance are useful, we need to go beyond measures of central tendency and dispersion to reveal other important characteristics of a distribution. One important characteristic of interest to analysts is the degree of symmetry in return distributions. If a return distribution is symmetrical about its mean, then each side of the distribution is a mirror image of the other. A distribution that is not symmetrical is called skewed. A return distribution with positive skew has frequent small losses and a few extreme gains. A return distribution with negative skew has frequent small gains and a few extreme losses. The figures below show these distributions. Copyright Irfanullah Financial Training. All rights reserved. Page 20

22 Distribution Skewed to the Right (Positively Skewed) Distribution Skewed to the Left (Negatively Skewed) For the positively skewed unimodal distribution, the mode is less than the median, which is less than the mean. For the negatively skewed unimodal distribution, the mean is less than the median, which is less than the mode. All else equal, if investment returns have negative skew, that is considered more risky than symmetric and positively skewed distributions. A negative skew implies a fat left tail and hence a relatively high probability of extreme losses. A skewness of greater than 0.5 or less than -0.5 is considered significant. The curriculum presents formulas for calculating skewness. However, it is extremely unlikely that we ll be tested on these formulas at Level I. Consequently the formulas are not being reproduced in these notes. 9. Kurtosis in Return Distributions A return distribution might differ from a normal distribution by having more returns clustered closely around the mean (being more peaked) and more returns with large deviations from the mean (having fatter tails). Kurtosis is the statistical measure that tells us when a distribution is more or less peaked than a normal distribution. A distribution that is more peaked than normal is called leptokurtic. A distribution that is less peaked than normal is called platykurtic. A distribution identical to the normal distribution is called mesokurtic. Examples of a mesokurtic distribution (normal) and leptokurtic distribution (fat tails) are shown below. Copyright Irfanullah Financial Training. All rights reserved. Page 21

23 For all normal distributions, kurtosis is equal to 3. Excess kurtosis is kurtosis minus 3. Hence, a mesokurtic distribution has excess kurtosis equal to 0. A leptokurtic distribution has excess kurtosis greater than 0, and a platykurtic distribution has excess kurtosis less than 0. For a sample of 100 or larger taken from a normal distribution, a sample excess kurtosis of 1.0 or larger would be considered unusually large. A leptokurtic distribution is considered more risky than a normal distribution because it has fatter tails and hence a higher probability of extreme losses. 10. Using Geometric and Arithmetic Means For reporting historical returns (time series data), the geometric mean is attractive because it is the rate of growth of return we would have had to earn each year to match the actual, cumulative investment performance. Consequently, to estimate the average returns over more than one period, we should use the geometric mean because it captures how the total returns are linked over time. On the other hand, if we want to estimate the average return of multiple investments over a one-period horizon (cross-sectional data), we should use the arithmetic mean. Copyright Irfanullah Financial Training. All rights reserved. Page 22

24 Summary Note: This summary has been adapted from the CFA Program curriculum. A population is defined as all members of a specified group. A sample is a subset of a population. A parameter is any descriptive measure of a population. A sample statistic (statistic, for short) is a quantity computed from or used to describe a sample. Data measurements are taken using one of the four major scales: nominal, ordinal, interval, or ratio. Nominal scales categorize data but do not rank them. Ordinal scales sort data into categories that are ordered with respect to some characteristic. Interval scales provide not only ranking but also assurance that the differences between scale values are equal. Ratio scales have all the characteristics of interval scales as well as a true zero point as the origin. The scale on which data are measured determines the type of analysis that can be performed on the data. A frequency distribution is a tabular display of data summarized into a relatively small number of intervals. Frequency distributions permit us to evaluate how data are distributed. The relative frequency of observations in an interval is the number of observations in the interval divided by the total number of observations. The cumulative relative frequency cumulates (adds up) the relative frequencies as we move from the first interval to the last, thus giving the fraction of the observations that are less than the upper limit of each interval. A histogram is a bar chart of data that have been grouped into a frequency distribution. A frequency polygon is a graph of frequency distributions obtained by drawing straight lines joining successive points representing the class frequencies. Sample statistics such as measures of central tendency, measures of dispersion, skewness, and kurtosis help with investment analysis, particularly in making probabilistic statements about returns. Measures of central tendency specify where data are centered and include the (arithmetic) mean, median, and mode (most frequently occurring value). The mean is the sum of the observations divided by the number of observations. The median is the value of the middle item (or the mean of the values of the two middle items) when the items in a set are sorted Copyright Irfanullah Financial Training. All rights reserved. Page 23

25 into ascending or descending order. The mean is the most frequently used measure of central tendency. The median is not influenced by extreme values and is most useful in the case of skewed distributions. The mode is the only measure of central tendency that can be used with nominal data. A portfolio s return is a weighted mean return computed from the returns on the individual assets, where the weight applied to each asset s return is the fraction of the portfolio invested in that asset. The geometric mean, G, of a set of observations X1, X2,... Xn is G =(X1X2X3 Xn) 1/n The geometric mean is especially important in reporting compound growth rates for time series data. When calculating the average return over a given period the following formula is used: R G =[(1+R1) (1+R2) (1+Rn)] 1/ n 1 The harmonic mean is a special type of weighted mean in which an observation s weight is inversely proportional to its magnitude. The formula is: n X H = n / ( 1 X i ) i=1 For any data set where the values are not the same, arithmetic mean > geometric mean > harmonic mean. Quantiles such as the median, quartiles, quintiles, deciles, and percentiles are location parameters that divide a distribution into halves, quarters, fifths, tenths, and hundredths, respectively. Dispersion measures such as the variance, standard deviation, and mean absolute deviation (MAD) describe the variability of outcomes around the arithmetic mean. Range is defined as the maximum value minus the minimum value. Range has only a limited scope because it uses information from only two observations. MAD is average of the absolute deviation from the mean. This can be expressed as: n MAD = [ X i X ] /n i=1 The variance is the average of the squared deviations around the mean, and the standard deviation is the positive square root of variance. In computing sample variance (s 2 ) and Copyright Irfanullah Financial Training. All rights reserved. Page 24

26 sample standard deviation, the average squared deviation is computed using a divisor equal to the sample size minus 1. According to Chebyshev s inequality, the proportion of the observations within k standard deviations of the arithmetic mean is at least 1 1/k 2 for all k > 1. Chebyshev s inequality permits us to make probabilistic statements about the proportion of observations within various intervals around the mean for any distribution with finite variance. As a result of Chebyshev s inequality, a two-standard-deviation interval around the mean must contain at least 75 percent of the observations, and a three-standard-deviation interval around the mean must contain at least 89 percent of the observations, no matter how the data are distributed. The coefficient of variation, CV, is the ratio of the standard deviation of a set of observations to their mean value. A scale-free measure of relative dispersion, by expressing the magnitude of variation among observations relative to their average size, the CV permits direct comparisons of dispersion across different data sets. The Sharpe ratio for a portfolio, p, based on historical returns, is defined as: (return on portfolio risk free rate) / standard deviation of portfolio. It gives the excess return per unit of risk. Skew describes the degree to which a distribution is not symmetric about its mean. A return distribution with positive skewness has frequent small losses and a few extreme gains. A return distribution with negative skewness has frequent small gains and a few extreme losses. Zero skewness indicates a symmetric distribution of returns. Kurtosis measures the peakness of a distribution and provides information about the probability of extreme outcomes. A distribution that is more peaked than the normal distribution is called leptokurtic; a distribution that is less peaked than the normal distribution is called platykurtic; and a distribution identical to the normal distribution in this respect is called mesokurtic. Excess kurtosis is kurtosis minus 3, the value of kurtosis for all normal distributions. Next Steps Make sure you are comfortable using the financial calculator for statistical calculations. Copyright Irfanullah Financial Training. All rights reserved. Page 25

27 Work through the examples presented in the curriculum. Solve the practice problems in the curriculum. Solve the IFT Practice Questions associated with this reading. Review the learning outcomes presented in the curriculum. Make sure that you can perform that actions implied by learning outcome. Copyright Irfanullah Financial Training. All rights reserved. Page 26

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