The following is a review of the statistical principles ...cfaresources.narod.ru/schweser2materials.pdf · STATISTICAL CONCEPTS AND MARKET RETURNS DEFUSCO, MCLEAVEY, PINTO & RUNKLE:

The following is a review of the statistical principles set forth in: STATISTICAL CONCEPTS AND MARKET RETURNS DEFUSCO, MCLEAVEY, PINTO & RUNKLE: CHAPTER 3

SCHWESER STUDY PROGRAM DEFUSCO, MCLEAVEY, PINTO & RUNKLE, CH. 3

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Executive Summary This reading uses descriptive statistics to summarize and describe important aspects of large sets of data. Two key areas candidates should focus on are measures of central tendency and measures of dispersion. Measures of central tendency, including the arithmetic mean, geometric mean, weighted mean, median, and mode, specify where a set of data is centered and attempts to find a single number that can represent the entire set. Measures of dispersion, including the range, mean absolute deviation, variance, and standard deviation, measure the data�s variability around its center. In the context of investments, measures of central tendency tell us where returns are centered and gives an idea of the investment�s average reward. Dispersion addresses the variability of returns and gives us an idea of the investment�s risk. For success on the Level 1 exam, candidates should also recognize a normal distribution of returns, and be able to assess departures from the normal distribution in the form of skewness, which represents a lack of symmetry, or kurtosis in which a distribution is more or less peaked.

(A) Introduction The term statistics has two different meanings. Most commonly, statistics refers to

numerical data such as a company�s earnings per share or average returns over the past five years. Statistics can also refer to the process of collecting, organizing, presenting, analyzing, and interpreting numerical data for the purpose of making decisions. Statistical methods include descriptive statistics and inferential statistics. Descriptive statistics is the study of how data can be summarized effectively to describe important aspects of large data sets. Using descriptive statistics to consolidate a mass of numerical data into useful information is the focus of this reading. Inferential statistics - making forecasts, estimates, or judgments about a larger group from a smaller group will be discussed in subsequent readings.

(B) Populations and Samples

LOS 1.B.a: Differentiate between a population and a sample.

LOS 1.B.b: Explain the concept of a parameter. 1. Populations

A population is defined as all members of a specified group. Any descriptive measure of a population characteristic is called a parameter. Populations can have many parameters, but investment analysts are usually only concerned with a few, such as the mean return, or the standard deviation of returns.


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2. Samples

A sample is defined as a portion, or subset of the population of interest. Even if it is possible to observe all members of a population, it is often too expensive or time consuming to do so. Once the population has been defined, we can take a sample of the population with the view of describing the population as a whole. Just as a parameter describes a population characteristic, a statistic is a descriptive measure of a sample characteristic.

(C) Measurement Scales

LOS 1.B.c: Explain the differences among the types of measurement scales.

Measurement scales are defined as different levels of measurement and can be classified into one of four major categories: 1. Nominal Scale. Nominal scales represent the weakest level of measurement.

Observations are classified or counted with no particular order. An example would be assigning the number �1� to a large cap value fund, the number �2� to a large cap growth fund, and so on for each style.

2. Ordinal Scale. The ordinal scale is a higher level of measurement. All observations

are placed into separate categories and the categories are placed in order with respect to some characteristic. An example would be ranking 100 large cap growth mutual funds by performance, and assigning the number 1 to the ten best performing funds and the number 10 to the ten worst performing funds. Note that we can conclude that a �1� is better than a �2,� but the scale does not tell us anything about the difference in performance.

3. Interval Scale. Interval scales provide ranking and assurance that differences

between scale values are equal. Measuring degrees of temperature is a prime example. We know that 49°C is hotter than 32°C. We can also state that difference in temperature between 49°C and 32°C is the same as the difference between 67°C and 50°C.

4. Ratio Scale. Ratio scales represent the strongest level of measurement. In addition

to providing ranking and equal differences between scale values, ratio scales have a true zero point as the origin. Money is a good example. If you have zero dollars, you have no money. If you have $4.00, you have twice as much purchasing power than the person who has $2.00.


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(D) Holding Period Return

LOS 1.B.e: Define, calculate and interpret a holding period return. Editor’s Note: When you’re reading the Reilly and Brown text, you may notice a slight discrepancy in terminology as it relates to the holding period return. In the Reilly text, the holding period yield is defined as (P1 – P0 + CF)/P0. Reilly then goes on to compute the holding period return as 1 + HPY or (P1 + CF)/P0. For the exam, focus on the DeFusco, et al., definition of HPR below.

Throughout the rest of this reading, we will apply descriptive statistics to investment returns. When analyzing rates of return, our starting point is the total return, or holding period return (HPR). HPR measures the total return for holding an investment over a certain period of time, and can be calculated using the following formula:

1

1

−

− +−=

t

tttt P

DPPR

Where: Pt = price per share at the end of time period t

Pt-1 = price per share at the end of time period t-1, the time period immediately preceding time period t

Dt = cash distributions received during time period t

Example: A stock is currently worth $60. If you purchased the stock exactly one year ago for $50 and received a $2 dividend over the course of the year, what is your holding period return? Rt = ($60 - $50 + $2)/$50 = 0.24 or 24%

The return for time period t is the capital gain (or loss) plus distributions divided by the beginning-of-period price. Note that for common stocks, the distribution is the dividend, and for bonds, the distribution is the coupon payment. The holding period return for any asset can be calculated for any time period (day, week, month, or year) simply by changing the interpretation of the time interval.

(E) Construction of a Frequency Distribution

LOS 1.B.d: Define and interpret a frequency distribution. 1. Frequency Distributions

A frequency distribution is a tabular display of data summarized into a relatively small number of groups or intervals. Frequency distributions help in the analysis of large amounts of statistical data, and they work with all types of measurement scales.


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LOS 1.B.f: Define and explain the use of intervals to summarize data. Editor’s Note: In last year’s Level 1 curriculum, an “interval” was called a “class.” In the 2002 Level 1 Sample Exam published by AIMR, there is a question that still uses the terminology “class.” Please beware of this potential discrepancy for the exam.

Step 1: The first step in building a frequency distribution is to define the intervals to which you will be assigning the data. An interval is the set of return values within which an observation falls. Each observation falls into only one interval, and the total number of intervals covers the entire population. It is important to consider the number of intervals to be used. If too few intervals are used, too much data may be summarized and we may lose important characteristics; if too many intervals are used, we may not summarize enough. Each interval has a lower limit and an upper limit. Intervals must be all-inclusive and non-overlapping. They must also be mutually exclusive so that each observation can only be placed in one interval.

Step 2: After the intervals have been defined, you must tally the observations and assign each observation to its respective interval.

Step 3: Once the data set has been tallied, you should count the number of observations that were placed in each interval. The actual number of observations in a given interval is called the absolute frequency, or simply the frequency.

Example: Use the following table to build a frequency distribution. Annual Returns for Intelco Inc. Common Stock

10.4% 22.5% 11.1% (12.4%) 9.8% 17.0% 2.8% 8.4% 34.6% (28.6%) 0.6% 5.0%

(17.6%) 5.6% 8.9% 40.4% (1.0%) (4.2%) (5.2%) 21.0%

Step 1: Define the Intervals. For Intelco stock, the range of returns is from �28.6% to 40.4%, or 69.0%. If we used return intervals of 1%, we would have 69 intervals � in this case, too many. For this example, we will use eight non-overlapping intervals of 10% width. Intervals begin with -30% ≤ R < -20% and go up to 40% ≤ R ≤ 50%.

Steps 2 and 3: Tally the observations and count the observations within each interval. Once the observations are counted, you will have constructed a frequency distribution that summarizes the pattern of annual returns on Intelco common stock.


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Interval Tallies Frequency

-30% up to �20% / 1 -20% up to �10% // 2 -10% up to 0% /// 3 0% up to 10% ///// // 7 10% up to 20% /// 3 20% up to 30% // 2 30% up to 40% / 1 40% up to 50% / 1

Total 20 2. Relative Frequency Distributions

LOS 1.B.g: Calculate relative frequencies given a frequency distribution.

Another useful way to present data is the relative frequency. Relative frequency is calculated by dividing the frequency of each return interval by the total number of observations. Simply, relative frequency is the percentage of total observations falling within each interval. Continuing with our example:

Interval Frequency Relative Frequency

-30% up to �20% 1 1/20 = 0.05, or 5% -20% up to �10% 2 2/20 = 0.10, or 10% -10% up to 0% 3 3/20 = 0.15, or 15% 0% up to 10% 7 7/20 = 0.35, or 35% 10% up to 20% 3 3/20 = 0.15, or 15% 20% up to 30% 2 2/20 = 0.10, or 10% 30% up to 40% 1 1/20 = 0.05, or 5% 40% up to 50% 1 1/20 = 0.05, or 5%

Total 20 100%

(F) Graphic Presentation

LOS 1.B.h: Describe the properties of data presented as a histogram or a frequency polygon.

1. Histogram. A histogram is the graphical equivalent of a frequency distribution. It is a bar chart of continuous data that has been grouped into a frequency distribution. The advantage of a histogram is that we can quickly see where most of the observations lie.

To construct a histogram, the intervals are scaled on the horizontal axis and the absolute frequencies are scaled on the vertical axis. Continuing with our example, the histogram would be shown as follows:


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2. Frequency Polygon. A second graphical tool used for displaying data is the

frequency polygon.

To construct a frequency polygon, we plot the midpoint of each interval on the horizontal axis and the absolute frequency for that interval on the vertical axis. Each point is then connected with a straight line. Continuing with our example, the frequency polygon would be shown as follows:

3. Cumulative Frequency Distribution. A third graphical tool used for displaying data is the cumulative frequency distribution. The cumulative frequency distribution plots either the cumulative absolute or cumulative relative frequency against the upper interval limit. It is used to see how many or what percentage of the observations lie below a certain value.

(G) Measures of Central Tendency

Measures of central tendency are used to pinpoint the center or average of a data set which can then be used to represent the typical or representative data value.

F r e q u e n c y

Interval

-30 up to -20

-20 up to �10

-10 up to 0

0 up to 10

10 up to 20

20 up to 30

30 up to 40

40 up to 50

8 6 4 2

F r e q u e n c y

Midpoint

-25 -15 -5 5 15 25 35 45

8 6 4 2


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LOS 1.B.i: Define, calculate and interpret measures of central tendency, including the population mean, sample mean, arithmetic mean, geometric mean, weighted mean, median, and mode.

1. The Population Mean (µ). A population is the entire group of objects that are being

studied. To find the population�s mean, sum up all the observed values in the population (ΣX) and divide this sum by the number of observations (N) in the population. Note that the population mean is unique; a given population only has one mean.

N

XN

ii∑

== 1µ

2. The Sample Mean ( X ). The sample mean is the sum of all the values in a sample of

a population divided by the number of values in the sample. The sample mean is used to make inferences about the population mean.

n

XX ∑=

Note: (n) is the sample size while (N) is the population size.

Example: A stock you and your research partner are analyzing has 12 years of annualized return data. The returns are as follows: 12%, 25%, 34%, 15%, 19%, 44%, 54%, 33%, 22%, 28%, 17%, 24%. Your research partner is exceedingly lazy and has decided to collect data based on only five years of returns. Given this data, calculate the population mean and calculate the sample mean (your partner�s data set is shown as the bold numbers in the original data series above).

%25.2712

241728223354441915342512Mean Population =+++++++++++==µ

%8.295

1754193425 Mean Sample =++++==X

3. Arithmetic Mean. The population mean and sample mean are both examples of the

arithmetic mean. The arithmetic mean is the sum of the observation values divided by the number of observations. It is the most widely used measure of central tendency and has the following properties:

• All interval and ratio data sets have an arithmetic mean. • All data values are considered and included in the arithmetic mean computation. • A data set has only one arithmetic mean. This says that the mean is unique. • The arithmetic mean is the only measure of central tendency where the sum of the

deviations of each value from the mean is always zero. That is:

The sum of the deviations from the mean = Σ(Xi − X ) = 0


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Example: A data set contains the following numbers: 5, 9, 4, and 10. The mean of these numbers is:

X = (5 + 9 + 4 + 10) / 4 = 7 The sum of the deviations from the mean is: Σ(Xi � X ) = (5 � 7) + (9 � 7) + (4 � 7) + (10 � 7) = -2 + 2 � 3 + 3 = 0 The arithmetic mean has the following disadvantages:

• The mean can be affected by extremes, that is, unusually large or small values. • The mean cannot be determined for an open-ended data set (i.e., n is unknown).

4. Geometric Mean. Geometric means are often used when calculating investment

returns over multiple periods, or to find a compound growth rate. The standard formula for finding the geometric mean, G, of a set of observations is:

n

nXXXG ×××= ...21 Note that this equation only has a solution if the product under the radical sign is non-negative. We can add 1.0 to values under the radical, and then subtract 1.0 from the result in order to compute the geometric mean. Example: For the last three years the returns for Acme Corporation common stock have been -9.34%, 23.45%, and 8.92%. Find the geometric mean.

3 )10892.0()12345.0()10934.0(1 +×+×+−=+ GR

06825.121903.10892.12345.19066.01 33 ==××=+ GR RG = 1.06825 � 1 = 6.825%

How do you work this on your calculator? Use the yx key. On the TI, the computation is: [1.21903] ! [yx] ! [.33333] ! [=], where .33333 represents the 1/3rd power. On the HP, the computation is: [1.21903] ! [ENTER] ! [.33333] ! [yx].


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5. Comparison of Geometric and Arithmetic Means

LOS 1.B.j: Distinguish between arithmetic and geometric means.

In the previous example, we calculated the geometric mean of Acme Corporation to be 6.825%. To calculate the arithmetic mean, we take (-9.34% + 23.45% + 8.92%)/3 = 7.67%.

Note that the value for the arithmetic mean is higher. The geometric mean will always be less than or equal to the arithmetic mean. In general, the difference between the two means increases with the variability between period-by-period observations. The only time the two means will be equal is when there is no variability in the observations (e.g. all observations are 10%).

Example: You buy stock A for $100. One year later, the stock is trading at $200. At the end of the second year, the stock price falls back to the original value of $100. This stock pays no dividends. Calculate the arithmetic and geometric mean annual returns:

Return in Year 1 = 200/100 � 1 = 100% Return in Year 2 = 100/200 � 1 = -50% Arithmetic Mean = (100% - 50%)/2 = 25% Geometric Mean = %0150.00.21)50.01()0.11( =−×=−−×+

Note two points:

• Because returns are variable, the arithmetic mean is greater than the geometric mean.

• The geometric mean value of 0% accurately reflects that the purchase price and sale price of the stock are the same. Hence, you earned nothing on the investment. The geometric mean shows us the correct compound growth rate of the security.

LOS 1.B.i: Define, calculate and interpret measures of central tendency, including the population mean, sample mean, arithmetic mean, geometric mean, weighted mean, median, and mode.

LOS 1.B.k: Define, calculate, and interpret (1) a portfolio return as a weighted mean, (2) a weighted average or mean, (3) a range and mean absolute deviation, and (4) a sample and a population variance and standard deviation.

6. Weighted Mean. The weighted mean is a special case of the mean that allows

different weights on different observations. The weighted mean of a set of numbers is computed with the following equation:

WX = (w1X1+w2X2+ �+wnXn)/(w1+w2+�+wn) or


48

∑=

=n

iiiW XwX

1

Where X1, X2, �, Xn are the observed values and w1, w2, � wn the corresponding weights associated with each of the observations.

Example: A portfolio consists of 50% common stocks, 40% bonds, and 10% cash. If the return on common stocks is 12%, the return on bonds is 7%, and the return on cash is 3%, what is the return to the portfolio?

cashcashbondsbondsstockstockW RwRwRwX ++=

WX = [(0.50 *0.12) + (0.40 * 0.07) + (0.10 * 0.03)] = 0.091, or 9.1% What you need to take away from this is that a portfolio�s return is the weighted average of the returns to the individual securities in the portfolio. The weight applied to each security is its dollar weight relative to the portfolio.

7. Median. The median is the mid-point of the data when the data is arranged from the

largest to the smallest values. Half the observations are above the median and half are below the median. To determine the median, arrange the data from the highest to the lowest (or lowest to highest) and find the middle observation.

The median is important because the arithmetic mean can be affected by extremely large or small values (outliers). When this occurs, the median is a better measure of central tendency than the mean because it is not affected by extreme values.

Example: The five-year annualized total returns for five investment managers are 30%, 15%, 25%, 21% and 23%. Find the median return for the managers.

First, rearrange the returns from highest to lowest:

30% 25% 23% 21% 15% The return observation half way down from the top or half way up from the bottom is 23% so the median return is 23%. Example: Now, let�s add a sixth manager with a return of 28%. What is the median return?

Rearranging the returns gives: 30% 28% 25% 23% 21% 15%

Here, the number of observations is even. Hence, there is no single middle observation. To find the median, take the arithmetic mean of the two middle observations: (25 + 23)/2. Thus, the median of the data set is 24.0%.


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8. Mode. The mode of a data set is the value of the observation that appears most frequently. A set of data can have more than one mode, or even no mode. When a distribution has one value that appears most frequently, it is said to be unimodal. When a set of data has two or three values that occur most frequently, it is said to be bimodal, or trimodal. When all values are different, the data set has no mode.

Example: In the following set of numbers: 30%, 28%, 25%, 23%, 28%, 15%, 5%;

28% is the mode because it is the value appearing most frequently.

9. Quantiles. Quantile is a general term for a value at or below which a stated fraction of data lies. Examples of quantiles include:

• Quartiles � the distribution is divided into quarters. • Quintile � the distribution is divided into fifths. • Decile � the distribution is divided into tenths. • Percentile � the distribution is divided into hundredths.

The formula for the position of a percentile with n data points sorted in ascending order is:

100

)1( ynLy +=

Example: Find the 3rd quartile (75% of the observations lie below) for the following distribution of returns:

8%, 10%, 12%, 13%, 15%, 17%, 17%, 18%, 19%, 23%, 24%

yL = (11+1)(75/100) = 9 The 3rd quartile is the 9th data point from the left, or 19%. 75% of all observations lie below 19%. Note that if yL is not a whole number, linear interpolation is used to find the quantile. Example: Find the 3rd quartile (75% of the observations lie below) for the following distribution of returns:

8%, 10%, 12%, 13%, 15%, 17%, 17%, 18%, 19%, 23%, 24%, 26%

yL = (12+1)(75/100) = 9.75 The 3rd quartile is the 9th data point from the left, plus 0.75*(distance between the 10th and 9th data values). 19 + 0.75(23-19) = 22. 75% of all observations lie below 22%.


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(H) Measures of Dispersion

LOS 1.B.k: Define, calculate, and interpret (1) a portfolio return as a weighted mean, (2) a weighted average or mean, (3) a range and mean absolute deviation, and (4) a sample and a population variance and standard deviation.

Dispersion is defined as the �variability around the central tendency.� Investments finance is all about reward versus variability. The central tendency is a measure of the reward of an investment and dispersion is a measure of investment risk. 1. Range. The range is the distance between the largest and the smallest value in the data

set. Range = Maximum Value � Minimum Value

Example: The five-year annualized total returns for five investment managers are 30%, 12%, 25%, 20% and 23%. What is the range of the data?

Range = 30 � 12 = 18% 2. Mean Absolute Deviation (MAD). MAD is the average of the absolute values of the

deviations of individual observations from the arithmetic mean.

n

MAD 1∑=

−=

n

ii XX

Remember that the sum of all the deviations from the mean is equal to zero. To get around this zeroing out problem, the mean deviation uses the absolute values of each deviation.

Example continued: What is the mean deviation of investment returns and how is it interpreted?

X = [30 + 12 + 25 + 20 + 23] / 5 = 22%

MAD = [|30 − 22| + |12 − 22| + |25 � 22| + |20 � 22| + |23 � 22|] / 5

MAD = [8 + 10 + 3 + 2 + 1] / 5 = 4.8%

Interpretation: On average, an individual return will deviate 4.8% from the mean return of 22%.

3. Population Variance: The variance is defined as the mean of the squared deviations

from the mean. The population variance is computed using all members of a population and is found by the formula:


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N

)( 2

12µ

σ−

=∑=

N

iiX

or,

2

11

2

2

NN

−=∑∑==

N

ii

N

ii XX

σ

Example: Assume the five-year annualized total returns for the five investment managers used in the earlier example represent all of the managers at a small investment firm. What is the population variance of returns?

=µ [30 + 12 + 25 + 20 + 23] / 5 = 22% =2σ [(30-22)2 + (12-22)2 + (25-22)2 + (20�22)2 + (23-22)2 ]/5 = 35.60%2

Interpretation: The average variation from the mean return is 35.60%2. 4. Population Standard Deviation. The major problem with using the variance is the

difficulty interpreting it. Why? The variance, unlike the mean, is in terms of units squared. How does one interpret squared percents or squared dollars? The solution to this problem is to use the standard deviation. Population standard deviation is the square root of the population variance and is found by:

N

)(1

2∑=

−=

N

iX µ

σ or,

2

11

2

NN

−=∑∑==

N

ii

N

ii XX

σ

Example continued: Find the population standard deviation. 60.35]/5 22)-(23 22)-(20 22)-(25 22)-(12 22)-[(30 22222 =++++ = 5.97%

Interpretation: Note that the standard deviation and population mean are both given in percentage units. The average deviation from the mean is 5.97%.

5. Sample Variance. The sample variance applies when we are dealing with a subset,

or sample of the total population. The sample variance is calculated using the following formulas:

1 -n

)(s

2

12XX

n

ii −

=∑= or,

1

2

12

2

−

−=

∑∑ =

nn

XX

s

n

ii

i


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The formula for the sample variance is nearly the same as that for the population variance except for the use of the sample mean, X , and the denominator. In the case of the population variance, we divide by the size of the population, N. For the sample variance, however, we divided by the sample size minus 1, or n-1. In the math of statistics, using only N in the denominator when using a sample to represent its population will result in underestimating the population variance, especially for small sample sizes. This systematic understatement causes the sample variance to be a biased estimator of the population variance. By using (n � 1) instead of �N� in the denominator, we compensate for this underestimation. Thus, by using n-1, the sample variance (s2) will be an unbiased estimator of the population variance (σ2). Example: Assume the five-year annualized total returns for the five investment managers used in the earlier example represent only a sample of the managers at a large investment firm. What is the sample variance of returns?

=X [30 + 12 + 25 + 20 + 23] / 5 = 22% 2s = [(30-22)2 + (12-22)2 + (25-22)2 + (20�22)2 + (23-22)2 ]/(5-1) = 44.5%2

Interpretation: 44.5%2 is an unbiased estimator of the population variance.

6. Sample Standard Deviation. Just as we computed a population standard deviation,

we can compute a sample standard deviation by taking the positive square root of the sample variance. The sample standard deviation, s, is defined as:

1 -n

)(s

2

1XX

n

ii −

=∑= or,

1

2

1

1

2

−

−=

∑∑ =

=

nn

XX

s

n

iin

ii

Example continued: Find the sample standard deviation.

s = 50.441) - ]/(5 22)-(23 22)-(20 22)-(25 22)-(12 22)-[(30 22222 =++++ = 6.67% Interpretation: 6.67% is an unbiased estimator of the population standard deviation.

7. Chebyshev’s Inequality

LOS 1.B.l: Calculate the proportion of items falling within a specified number of standard deviations from the mean, using Chebyshev�s inequality.

Chebyshev’s inequality states that for any set of observations (sample or population, regardless of the shape of the distribution), the proportion of the observations within k


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standard deviations of the mean is at least 1 � 1/k2 for all k > 1. If we know the standard deviation, we can use Chebyshev�s inequality to measure the minimum amount of dispersion, regardless of the shape of the distribution.

Example: What approximate percent of a distribution will lie within +/- two standard deviations of the mean?

From Chebyshev�s Inequality: 1-(1/k2) = 1 - (1/22) = 0.75 or 75%. Thus, Chebyshev�s Inequality states that for any distribution, approximately:

• 36% of observations lie within +/- 1.25 standard deviations of the mean • 56% of observations lie within +/- 1.50 standard deviations of the mean • 75% of observations lie within +/- 2 standard deviations of the mean • 89% of observations lie within +/- 3 standard deviations of the mean • 94% of observations lie within +/- 4 standard deviations of the mean

8. Relative Dispersion

LOS 1.B.m: Define, calculate, and interpret the coefficient of variation.

A direct comparison of two or more measures of dispersion may be difficult. For example, the difference between the dispersion for monthly returns on T-bills and dispersion for a portfolio of small stocks is not meaningful because the means of the distributions are far apart. In order to make a meaningful comparison, we need a relative measure. Relative dispersion is the amount of variability present in comparison to a reference point or benchmark. A common measure of relative dispersion is the coefficient of variation (CV) which is defined as:

XsCV =

The coefficient of variation expresses how much dispersion exists relative to the mean of a distribution and allows for direct comparison of dispersion across different data sets.

Example: The mean monthly return on T-bills is 0.25% with a standard deviation of 0.36%. For the S&P 500, the mean is 1.09% with a standard deviation of 7.30%. Calculate the coefficient of variation for T-bills and the S&P 500 and interpret your results.

T-bills: CV = 0.36/0.25 = 1.44 S&P 500: CV = 7.30/1.09 = 6.70

Interpretation: There is less dispersion relative to the mean in the distribution of monthly T-bill returns when compared to the distribution of monthly returns for the S&P 500 (1.44 < 6.70).


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9. Sharpe Measure of Risk-Adjusted Performance

LOS 1.B.n: Define, calculate, and interpret the Sharpe measure of risk-adjusted performance.

The Sharpe measure, or Sharpe ratio, seeks to measure excess return per unit of risk. It is defined as:

p

fp rrσ−

where: pr = portfolio return; fr = risk free return; σ = standard deviation Note that the numerator of the Sharpe measure recognizes the existence of a risk-free return. The term )( fp rr − measures the extra reward that investors receive for added risk taken and is called the excess return on portfolio p. Portfolios with large Sharpe ratios are preferred to portfolios with smaller ratios because it is assumed that rational investors prefer return and dislike risk. The Sharpe ratio is also called the reward-to-variability ratio. Example: The mean monthly return on T-bills (the risk-free rate) is 0.25%. The mean monthly return on the S&P 500 is 1.30% with a standard deviation of 7.30%. Calculate the Sharpe measure for the S&P 500 and interpret the results. Sharpe measure = (1.30 � 0.25)/7.30 = 0.144 Interpretation: The S&P 500 earned 0.144% of excess return per unit of risk, where risk is measured by standard deviation.

(I) Symmetry and Skewness in Return Distributions The degree of symmetry in a return distribution tells analysts if deviations from the mean are likely to be positive or negative.

1. Symmetrical Distributions. If a distribution is symmetrical about its mean, each

side of the distribution is a mirror image of the other. Loss and gain intervals will exhibit the same frequency. For example, for a distribution whose mean return is zero, losses from �6 percent to �4 percent occur about as frequently as gains of 4 percent to 6 percent.

The normal distribution is an example of a symmetrical distribution and has the following characteristics: • The mean and median are equal. • It is completely described by its mean and variance.


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• Roughly 66% of observations lie between +/- 1 standard deviation from the mean; 95% of observations lie between +/- 2 standard deviation from the mean; and 99% of observations lie between +/- 3 standard deviation from the mean.

LOS 1.B.p: Define and interpret skewness and explain why a distribution might be positively or negatively skewed.

2. Skewness. Skewness refers to a distribution that is not symmetrical.

• A positively skewed distribution is characterized by many outliers in its upper or right tail. Recall that an outlier is defined as an extraordinarily large outcome in absolute value. Positively skewed distributions have long right tails.

• A negatively skewed distribution is the opposite of a positively skewed distribution. A negatively skewed distribution has a disproportionately large amount of outliers on its left side. In other words, a negatively skewed distribution is said to have a long tail on its left side.

3. Location of Measures of Central Tendency

LOS 1.B.o: Describe the relative locations of the mean, median, and mode for a nonsymmetrical distribution.

• For a symmetrical distribution, the mean, median, and mode are equal. • For a positively skewed distribution, the mode is less than the median, which is

less than the mean. Recall that the mean is affected by outliers. In a positively skewed distribution, there are large, positive outliers which will tend to �pull� the mean upward. An example of a positively skewed distribution is that of housing prices. Suppose that you live in a neighborhood with 100 homes. Ninety-nine of those homes sell for $100,000 and there is one house that sells for $1,000,000. The median and the mode will be $100,000, but the mean will be $109,000. Hence, you can see that the mean has been �pulled� upward by the existence of one home in the neighborhood.

• For a negatively skewed distribution, the mean is less than the median, which is less than the mode. In this case, there are large, negative outliers which tend to �pull� the mean downward.

Negative skewness Symmetrical Positive skewness

Mean Median Mode

M M M o e e d d a e i n a n

M M M e e o a d d n i e a n

Mean = Median = Mode Mean > Median > Mode Mean < Median < Mode


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(J) Kurtosis in Return Distributions

LOS 1.B.q: Define and interpret kurtosis and explain why a distribution might have positive excess kurtosis.

Kurtosis deals with whether or not a distribution is more or less �peaked� than a normal distribution.

1. A distribution that is more peaked than normal is leptokurtic. A leptokurtic return distribution will have more returns clustered around the mean and more returns with large deviations from the mean (fatter tails). Relative to a normal distribution, a leptokurtic distribution will have a greater percentage of small deviations from the mean and a greater percentage of extremely large deviations from the mean. Most investors perceive a greater chance of large deviations from the mean as increasing risk.

2. A distribution that is less peaked, or flatter than normal is said to be platykurtic.

3. Excess Kurtosis. For all normal distributions, kurtosis is equal to three. Statisticians, however, sometimes report excess kurtosis, which is defined as kurtosis minus 3. A normal distribution has excess kurtosis equal to zero, a leptokurtic distribution has excess kurtosis greater than zero, and platykurtic distribution will have excess kurtosis less than zero.

Kurtosis is critical in a risk management setting. Most research of the distribution of securities returns has shown that returns are not normal. Actual securities returns tend to exhibit both skewness and kurtosis (sounds like fungus!). The reason that skewness and kurtosis are critical for risk management is that if securities returns are modeled after a normal distribution, then predictions from those models will take into consideration the potential for extremely large, negative outcomes. In fact, most risk managers put very little emphasis on the mean and standard deviation of a distribution and focus more on the distribution of returns in the tails of the distribution � that is where the risk is.

Leptokurtic

Normal Distribution


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(K) Using Geometric and Arithmetic Means

Earlier in the summary, we noted that variability in returns causes the arithmetic mean to be greater than the geometric mean. We also noted that the geometric mean is used anytime we need to calculate compound returns. When using the geometric and arithmetic means:

• The arithmetic mean should be used to estimate the average return over a one-period time horizon because the arithmetic mean is the average of one-period returns.

• The geometric mean is the preferred method for calculating historical returns because it calculates the rate of return that would have to be earned each year to match the actual, cumulative investment performance. The geometric mean captures how performance is linked over time.

When graphing past performance, the use of semi-logarithmic scales is based on percentage changes and provides a more realistic picture than arithmetic scales.

• Arithmetic scales are based on absolute numerical values. Equal numerical changes in an index will show up as equal sized vertical movements. For example, the movement of an index from 10 to 20 would show a vertical movement equal to the vertical movement of an index from 1,000 to 1,010.

LOS 1.B.r: Explain why a semi-logarithmic scale is often used for return performance graphs.

• Semi-logarithmic scales use an arithmetic scale on the horizontal axis, but use a logarithmic scale on the vertical axis. Hence, values on the vertical axis are spaced according to their logarithms. On a semi-logarithmic scale, equal movements on the vertical axis reflect equal percentage changes. In the example we used earlier, the change in the index value from 10 to 20 would reflect a change of 100%; the vertical difference would be 100 times as much as the change from 1,000 to 1,010, which is a change of only 1%. Hence, semi-logarithmic scales are used for performance graphs because the percentage change in an investment conveys much more information about performance than the absolute dollar change in price.


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Conceptual Overview: 1. A population includes all members of a specified group; a sample is a portion, or subset, of the

population used to draw inferences about the population. 2. Any measurable characteristic of a population is called a parameter. 3. The scale on which data are measured determines the type of analysis that can be performed on the

data. There are four measures of scale: • Nominal Scale � data is put into a category with no particular order • Ordinal Scale � data is categorized and ordered with respect to some characteristic • Interval Scale � quantifies differences in size of data • Ratio Scale � scale between data observations is meaningful and have a true zero point as the

origin 4. The holding period return (HPR) measures the total return for holding an investment for a

certain period of time and is calculated by:

1

1

−

− +−=

t

tttt P

DPPR

5. A frequency distribution is a grouping of raw data into classes, or intervals. 6. An interval is the set of return values that an observation falls between. 7. Relative frequency is the percentage of total observations falling within each interval. 8. Histograms and frequency polygons are graphical tools used for portraying frequency

distributions. 9. The population mean is the sum of all values in a population divided by the number of

observations. N

XN

ii∑

== 1µ

10. The sample mean is the sum of all values in a sample divided by the number of observations in

the sample. n

XX ∑=

11. The arithmetic mean is the sum of the values divided by the number of observations; ∑X/n 12. The geometric mean is used to find a compound return or growth rate; n

nXXXG ×××= ...21 13. The weighted mean allows different weights on different values and is used to calculate the return

on a portfolio; ∑=

=n

iiiW XwX

1

14. The median is the mid-point of a data set when the data is arranged from largest to smallest. • Given an even number of observations, the median is the arithmetic average of the middle 2

observations. 15. The mode of a data set is the value that appears most frequently. A data set may have no mode, or more

than one mode.

16. The position of a percentile in a dataset is computed as: 100

)1( ynLy +=

17. The range is the difference between the largest value and the smallest value in a data set. 18. The mean absolute deviation (MAD) is the average of absolute deviations from the arithmetic

mean. n

MAD 1∑=

−=

n

ii XX


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19. The variance is defined as the mean of the squared deviations from the arithmetic mean. The

population variance is computed as: N

)( 2

12

µσ

−=∑=

N

iiX

whereas the sample variance uses the sample

mean and n-1 in its computation: 1-n

)(s

2

12

XXn

ii −

=∑=

20. Standard deviation is the positive square root of the variance and is frequently used as a quanititative measure of risk.

21. Chebyshev’s inequality states that the proportion of the observations within k standard deviations of the mean is at least 1 � 1/k2 for all k > 1.

22. The coefficient of variation expresses how much dispersion exists relative to the mean of a distribution; XsCV /=

23. The Sharpe measure measures excess return per unit of risk; pfp rr σ/)( − 24. Skewness describes the degree to which a distribution is non-symmetric about its mean. 25. Kurtosis measures the peakedness of a distribution and provides information about the probability

of extreme outcomes. 26. Semi-logarithmic scales provide a more realistic picture when graphing past performance because

they are based on percentages. PROBLEM SET: DEFUSCO, MCLEAVEY, PINTO & RUNKLE, CHAPTER 3 1. In a frequency distribution, intervals should always:

A. Be truncated. B. Be non-overlapping. C. Have a width of 10. D. Be open ended.

2. The relative frequency of an interval is obtained by:

A. Dividing the frequency of that interval by the sum of all frequencies. B. Multiplying the frequency of the interval by 100. C. Dividing the sum of the two interval limits by 2. D. Subtracting the lower limit of the interval by the upper limit.

3. In a frequency distribution histogram, the frequency of an interval is given by the:

A. Height of the corresponding bar. B. Width of the corresponding bar. C. Height multiplied by the width of the corresponding bar. D. Horizontal logarithmic scale.


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Use the following data for questions 4-6.

Return Frequency -10% up to 0% 3 0% up to 10% 7 10% up to 20% 3 20% up to 30% 2 30% up to 40% 1

4. The number of intervals in this frequency table is:

A. 1. B. 5. C. 16. D. 50.

5. The sample size is:

A. 1. B. 5. C. 16. D. 50.

6. The relative frequency of the second class is:

A. 10.0%. B. 43.8%. C. 0.0%. D. 16.0%.

Use the following data to answer questions 7-15.

XYZ Corp. Annual Stock Prices 1995 1996 1997 1998 1999 2000 22% 5% (7%) 11% 2% 11%

7. The arithmetic mean return for XYZ stock is:

A. 7.3%. B. 8.0%. C. 11.0%. D. (7.0%).

8. The median return for XYZ stock is:

A. 7.3%. B. 8.0%. C. 11.0%. D. (7.0%).


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9. The mode return for XYZ stock is: A. 7.3%. B. 8.0%. C. 11.0%. D. (7.0%).

10. The range for XYZ stock returns is:

A. (7.0%). B. 22.0%. C. 11.0%. D. 29.0%.

11. The mean absolute deviation for XYZ stock returns is:

A. 7.3%. B. 29.0%. C. 5.2%. D. 0.0%.

12. Assuming that the distribution of XYZ stock returns is a population, what is the population

variance? A. 80.2. B. 6.8. C. 7.7. D. 5.0.

13. Assuming that the distribution of XYZ stock returns is a population, what is the population

standard deviation? A. 46.22%. B. 6.84%. C. 8.96%. D. 5.02%.

14. Assuming that the distribution of XYZ stock returns is a sample, what is the sample

variance? A. 7.4. B. 5.0. C. 72.4. D. 96.3.

15. Assuming that the distribution of XYZ stock returns is a sample, what is the sample

standard deviation? A. 7.4%. B. 9.8%. C. 72.4%. D. 96.3%.


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16. For a skewed distribution, what percentage of the observations will lie between +/- 2.5 standard deviations of the mean based on Chebyshev�s Theorem? A. 84%. B. 75%. C. 16%. D. 56%.

17. Which of the following statements is FALSE?

A. The range of a set of data is found by taking the largest data value minus the smallest data value.

B. For a skewed distribution and based on Chebyshev�s Theorem, 75% of the observations lie within +/- 2 standard deviations from the mean.

C. For a normal distribution, 95% of the observations lie within +/- 3 standard deviations from the mean.

D. The coefficient of variation is useful when comparing dispersion of data measured in different units or having large differences in their means.

18. The coefficient of variation of a distribution with a mean of 10 and a variance of 4 is:

A. 20%. B. 25%. C. 40%. D. 50%.

19. Portfolio A earned a return of 10.23% and had a standard deviation of returns of 6.22%. If the return on T-bills is 0.52% and the return to T-bonds is 4.56%, what is Sharpe measure of the portfolio? A. 0.91. B. 1.56. C. 0.56. D. 7.71.

20. A stock has annual returns of 5.60%, 22.67%, and �5.23%. What is the compound annual

growth rate for the stock? A. 6.00%. B. 7.08%. C. 17.44%. D. 8.72%.

21. An investor has a $12,000 portfolio consisting of $7,000 in stock A with an expected return

of 20% and $5,000 in stock B with an expected return of 10%. What is the investor�s expected return on the portfolio? A. 15.0%. B. 7.9%. C. 12.2%. D. 15.8%.


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22. An investor has a portfolio with 10% cash, 30% bonds, and 60% stock. If last year�s return on cash was 2.0%, the return on bonds was 9.5%, and the return on stock was �32.5%, what was the return on the investor�s portfolio? A. 33.33%. B. �7.00%. C. �16.45%. D. �33.33%.

Use the following data in answering Questions 23 and 24.

FJW Corp Annual Stock Prices 1997 1998 1999 2000 15% 19% -8% 14%

23. What is the arithmetic mean return for FJW�s common stock over the four years?

A. 8.62%. B. 10.00%. C. 14.00%. D. 15.25%.

24. What is the geometric mean return for FJW�s common stock over the four years?

A. 10.00%. B. 9.45%. C. 14.21%. D. It cannot be determined because of the negative return in 1999.

25. A distribution of returns that has a greater percentage of small deviations from the mean and a greater percentage of extremely large deviations from the mean: A. is a symmetric distribution. B. is positively skewed. C. has positive excess kurtosis. D. has negative excess kurtosis.

26. A distribution of returns that has a mean greater than its median:

A. is a symmetric distribution. B. is positively skewed. C. has positive excess kurtosis. D. has negative excess skewness.

27. Which of the following is NOT a reason that semi-logarithmic scales are often used for

historical return performance graphs? A. Arithmetic scales are based on numerical values. B. Veritical axis values of 10 to 100 and 100 to 1000 will be equally spaced because the

difference in their logarithms are equal. C. Semi-logarithmic scales are based on percentages. D. The movement of an index from 10 to 20 will not show up as the same movement of

the index from 350 to 700.


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ANSWERS 1. B. Intervals within a frequency distribution should always be non-overlapping and closed-ended so that

each data value can be placed into only one interval. Intervals have no set width � intervals should be set at a width so that data is adequately summarized without losing valuable characteristics.

2. A. The relative frequency is the percentage of total observations falling within each interval. It is found

by taking the frequency of the interval and dividing that number by the sum of all frequencies. 3. A. In a histogram, intervals are placed on horizontal axis and frequencies are placed on the vertical axis.

The frequency of the particular interval is given by the value on the vertical axis, or the height of the corresponding bar.

4. B. An interval is the set of return values that an observation falls within. Simply count the return

intervals on the table � there are 5 of them. 5. C. The sample size can be found by taking the sum of all of the frequencies in the distribution.

(3+7+3+2+1) = 16. 6. B. The relative frequency is found by dividing the frequency of the interval by the total number of

frequencies. 7/16 = 43.8%. 7. A. [22% + 5% + (7%) + 11% + 2% +11%]/6 = 7.3% 8. B. To find the median, rank the returns in order and take the middle value. (7%), 2%, 5%, 11%, 11%,

22%. In this case, because there are an even number of observations, the median is the average of the two middle values, or (5%+11%)/2 = 8.0%.

9. C. The mode is the value that appears most often, or 11%. 10. D. The range is calculated by taking the highest value � the lowest value. 22% - (7%) = 29.0%. 11. A. The mean absolute deviation is found by taking the mean of the absolute value of the deviations from

the mean. ( ) %33.76/3.7113.723.7113.773.753.722 =−+−+−+−−+−+− 12. A. The population variance is found by taking the mean of all squared deviations from the mean.

[ ] 2.806/)3.711()3.72()3.711()3.77()3.75()3.722( 222222 =−+−+−+−−+−+− 13. C. The population standard deviation is found by taking the square root of the population variance.

[ ] %96.86/)3.711()3.72()3.711()3.77()3.75()3.722( 222222 =−+−+−+−−+−+− 14. D. The sample variance uses n-1 in the denominator.

[ ] 3.96)16/()3.711()3.72()3.711()3.77()3.75()3.722( 222222 =−−+−+−+−−+−+− 15. B. The sample standard deviation is the square root of the sample variance.

[ ] %8.9)16/()3.711()3.72()3.711()3.77()3.75()3.722( 222222 =−−+−+−+−−+−+− 16. A. Using Chebyshev�s Theorem: 1-[1/(2.5)2] = 0.84, or 84%. 17. C. For a normal distribution, 95% of the observations lie within plus or minus 2 standard deviations of the

mean while 99% of the observations lie within plus or minus 3 standard deviations of the mean. All


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other statements are true. Note that 75% of observations for a skewed distribution lie within plus or minus two standard deviations of the mean using Chebyshev�s inequality.

18. A. Coefficient of variation = standard deviation/mean. The square root of the variance = standard

deviation. Square root of 4 = 2. 2/10 = 20%. 19. B. Sharpe measure = (10.23 � 0.52)/6.22 = 1.56. 20. B. Compound annual growth rate means find the geometric mean. %08.719477.02267.1056.13 =−×× 21. D. Find the weighted mean where the weights equal the proportion of $12,000. [(7000/12000)*0.20] +

[(5000/12000)*.10] = 15.8%

22. C. Find the weighted mean. (0.10 * 0.02) + (0.30*0.095) + (0.60*-0.325) = -16.45% 23. B. (15% + 19% + (8%) + 14%)/4 = 10% 24. B. %45.9114.192.019.115.14 =−××× Also note this question could have been answered very quickly

by knowing that the geometric mean must be less than the arithmetic mean. 25. C. A distribution that has a greater percentage of small deviations from the mean and a greater percentage

of extremely large deviations from the mean will be leptokurtic and will exhibit excess kurtosis (positive). The distribution will be taller with fatter tails than a normal distribution.

26. B. A distribution with a mean greater than its median is positively skewed. Note that kurtosis deals with

the height of the distribution and not the skewness. 27. D. The movement of an index from 10 to 20 WILL have the same vertical movement as the movement of

an index from 350 to 700 with a semi-logarithmic scale because both movements have the same percentage change of 100%. Under an arithmetic scale, vertical movements are based on actual numerical values. The 350 point move would be much greater than the 10 point moved based on numerical values.

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The following is a review of the statistical principles ...cfaresources.narod.ru/schweser2materials.pdf · STATISTICAL CONCEPTS AND MARKET RETURNS DEFUSCO, MCLEAVEY, PINTO & RUNKLE: