r07 statistical concepts and market returns

7/23/2019 R07 Statistical Concepts and Market Returns

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Quantitative Methods

Statistical Concepts and Market Return

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Graphs, charts, tables, examples, and figures are copyright 2012, C

Institute. Reproduced and republished with permission from CFA In

All rights reserved.


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Contents

1. Introduction

2. Some Fundamental Concepts

3. Summarizing Data using Frequency Distributions

4. The Graphic Presentation of Data

5. Measures of Central Tendency

6. Other Measures of Location: Quantiles

7. Measures of Dispersion8. Symmetry and Skewness in Return Distributions

9. Kurtosis in Return Distributions

10. Using Geometric and Arithmetic Means

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Video Le47 minu

Video Le

34 minut


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1. Introduction

Statistics is a powerful tool for analyzing data and dra

conclusions

Focus on statistical methods which allow us to summ

return distributions

Central tendency

Dispersion

Skewness

Kurtosis

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2. Some Fundamental Concepts

1. The Nature of Statistics

2. Population and Samples

3. Measurement Scales

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2.1 The Nature of Statistics

Statistics has two broad meanings

Data

Methods to collect and analyze data

Statistical methods include

Descriptive statistics describe the properties of a large data set Inferential statistics use a sample from a population to make pro

statements about the characteristics of a population

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2.2 Populations and Samples

Population: all members of a specified group

A parameter describes a characteristics of a

population

Sample: subset drawn from a population

A sample statistic describes a characteristic of a

sample

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.


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2.3 Measurement Scales

Nominal only names make sense

Ordinal order makes sense

Intervals interval make sense

Ratio ratios make sense (absolute zero)

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Practice Question

State the scale of measurement for each of the following:

1. Credit rating for corporate bond

2. Coupon rate

3. Mutual fund classification types

4. Bond maturity

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3. Summarizing Data Using Frequency Distr

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Stock

Price

(Absolute)

Frequency

Cumulative

Frequency

Relative

Frequency

Cumulative

RelativeFrequency

46-50 25 25 0.25 0.25

51-55 35 60 0.35 0.60

56-60 29 89 0.29 0.89

61-65 11 100 0.11 1.00

Summary data for 100 stocks with prices ranging between 46 and 65

Constructing a frequency distribution:

1. Define the intervals

2. Tally the observations (assign observations to intervals)

3. Count the observations in each interval

P i Q i


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Practice Question

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The actual number of observations in a given interval is called the:

A. Absolute frequency

B. Relative frequency

C. Cumulative absolute frequency

Answer: A

The actual number of observations in a given interval is known as absolute frequenc

frequency is the absolute frequency of each interval divided by the total number of obs

Cumulative absolute frequency is the running total of all absolute frequencies.

P ti Q ti


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Practice Question

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Which of the following is mostlikely to be accurate?

A. An observation can fall in more than one interval.

B. The data is sorted in a descending order for the construction of a frequ

distribution.C. The cumulative relative frequency tells the observer the fraction of the

observations that are less than the upper limit of each interval.

Answer: C

The cumulative relative frequency tells the observer the fraction of the observations tha

than the upper limit of each interval. An observation cannot fall in more than one interva

data is sorted in an ascending order for the construction of a frequency distribution.

4 Th G hi P t ti f D t


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4. The Graphic Presentation of Data

Histogram: bar chart of

data that has been

grouped into a frequencydistribution

On the right

Histogram of S&P 500

Monthly Total Returns

1926 - 2002

Allows us to quickly see where

most of the data lies

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Frequency Polygon of S&P 500 Monthly Total Returns: Jan 1926 Dec 20


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Cumulative Absolute Frequency of S&P 500 Monthly Total Returns: Jan 1926 D

Practice Q estion


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Practice Question

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Which of the following statements is most likely to be inaccurate about his

A. A histogram is the graphical equivalent of a frequency distribution.

B. A histogram is a form of a bar chart.C. In a histogram, the height represents the relative frequency for each in

Answer: C

In a histogram, the height represents the absolute frequency for each interval.

Practice Question


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Practice Question

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Which of the following graphical representation of data requires that mid

are plotted for each interval?

A. Frequency polygon

B. HistogramC. Cumulative frequency curve

Answer: A

For a frequency polygon, the mid points for each interval are plotted on the x-axis.

Practice Question


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Practice Question

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Month Return (%) Month Return (%)

January 10% July 5%

February 15% August 6%

March 14% September 7.5%

April 11% October 9%

May 8% November 12%

June 3% December 11%

The following table gives the average monthly returns of

a portfolio over the past one year.

Interval

1 < r < 3

4 < r < 6

7 < r < 9

10 < r < 12

13 < r < 15

What is the cumulative relative frequency for the interval 10 < r < 12?


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Interval Frequency Relative Frequency Cumulative Relative Frequency

1 < r < 3 1 8.33% 8.33%

4 < r < 6 2 16.67% 25%

7 < r < 9 3 25% 50%10 < r < 12 4 33.33% 83.33%

13 < r < 15 2 16.67% 100%

Solution

Relative Frequency = Frequency/Total Observations

Cumulative relative frequency is the sum of subsequent relative frequenciThus the cumulative relative frequency is 83.33% for the interval 10 r 1

5 Measures of Central Tendency


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5. Measures of Central Tendency

1. The Arithmetic Mean

2. The Median

3. The Mode

4. Other Concepts of Mean

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5 1 The Arithmetic Mean


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5.1 The Arithmetic Mean

Arithmetic Mean: sum of the observation values

divided by the number of observations

Population Mean

Sample Mean

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= Xi / N

X = Xi / N

5 2 The Median


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5.2 The Median

Middle item of a set of items that has been sorted into ascending

or descending order

For 2, 5, 7, 11, 14 Median = 7

For 3, 9, 10, 20 Median = (9+10) /2 = 9.5

Less affected by extreme values than the mean

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5 3 The Mode


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5.3 The Mode

Most frequently occurring value in a distribution.

2, 4, 5, 5, 7, 8, 8, 8, 10, 12 Mode = 8

Data sets can have more than one mode (bimodal, trimodal, etc)

A data set might not have any mode

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5 4 Other Concepts of Mean


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5.4 Other Concepts of Mean

Weighted Mean

Geometric Mean

Harmonic Mean

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Weighted Mean


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Weighted Mean

Different observations are given different

proportional influence on the mean

Consider the following portfolio:

Stock A = $40 million

Stock B = $60 million

Stock C = $100 million

If returns were 5% on A, 7% on B and 9% on C,

what was the portfolio return?

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Practice Question


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Practice Question

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A portfolio manager wishes to compute the weighted mean of a portfolio that h

following asset allocation:

Local Equities: 25%

International Equities: 13%Bonds: 27%

Mortgage: 18%

Gold: 17%

The returns on the above mentioned assets on December 31, 2012, were 5.4%, 8.9%, -

-7%, 11%, respectively. What is the weighted mean for the portfolio?

The Geometric Mean


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The Geometric Mean

Geometric Mean = (X1X2X3Xn)1/n

The geometric mean is frequently used to average rates

of change over time or to compute growth rate of avariable

Example: An investment account had returns of 20%,

20%, and -40% over the last three years. What is the

arithmetic mean return? What is the geometric mean

return?

G = [(1+R1)(1+R2).(1+Rn)]n - 1

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Practice Question


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Practice Question

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Judith buys a share at $45 on January 1, 2011. The price of the share is $54 on January 1, 2

January 1, 2013. What is the geometric mean annual return? Assume that no dividends we

The Harmonic Mean


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The Harmonic Mean

The harmonic mean is a special type of

weighted mean in which an observations

weight is inversely proportional to its

magnitude.

You purchase $2,000 of your company stock

every month. The purchase price over the last

three months has been $20, $24, and $30. On

average how much did you pay per share.

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XH = n / (1

Practice Question


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Practice Question

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Sam is an investor. He purchases $1000 of stock every quarter and the share price over the

quarters has been $10, $12, and $15. What is the average purchase price?

Comparison of AM GM and HM


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Comparison of AM, GM and HM

If the returns are constant over time: AM = GM = HM

If returns are variable: AM > GM > HM

The greater the variability of returns over time, the more thearithmetic mean will exceed the geometric mean

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6 Other Measures of Location: Quan


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6. Other Measures of Location: Quan

Quartiles: distribution is divided into quarters

Quintiles: distribution is divided into fifths

Deciles: distribution is divided into tenths

Percentile: distribution is divided into hundredths (percents)

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Say you have the following return data on 20 mutual funds (data in ascending order)

1 2 3 4 5 6 7 8 9 1

1.25 1.70 1.75 1.85 1.98 1.99 2.05 2.40 2.49 2

11 12 13 14 15 16 17 18 19 2

2.90 3.00 3.24 3.75 3.90 4.00 4.20 4.30 4.40 4

Fund Number

Return in %

Fund Number

Return in %

Location


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Location

At a given percentile, y, with n data points sorted in ascending order, thelocation of the observation is given by:

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Ly

= (n + 1) (y/100)

1 2 3 4 5 6 7 8 9 10

1.25 1.70 1.75 1.85 1.98 1.99 2.05 2.40 2.49 2.60

11 12 13 14 15 16 17 18 19 202.90 3.00 3.24 3.75 3.90 4.00 4.20 4.30 4.40 4.50

L10 = (20 + 1) (10/100) = 2.1

For small samples the location is approximate.

It becomes more precise as the sample size increases.

Practice Question


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Practice Question

Consider the data set:

47 35 37 32 40 39 36 34 35 31 44

Using Ly = (n + 1) (y/100)

Find the 75th percentile point

Find the 1st quartile pointFind the 5th decile point

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Practice Question - Answer


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Practice Question Answer

First arrange the data in ascending order:

31, 32, 34, 35, 35, 36, 37, 39, 40, 44, 47

Location of the 75th percentile is the

L75 = (11 + 1) (75/100) = 9th value. i.e. P75=40

Location of the 1st quartile is the

L25 = (11 + 1) (25/100) = 3rd

value. i.e. P25=34

Location of the 5th decile is the

L50 = (11 + 1) (50/100) = 6th value. i.e. P50=36

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Linear Interpolation


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Linear Interpolation

Linear Interpolation is used when Ly is not a whole number/integer,and Ly lies between two closest numbers.

Find the 6th

decile of the data set in the previous example L60 = (11 + 1) (60/100) = 7.2

Use linear interpolation, which estimates an unknown value on the basis of two

known values that surround it.

31, 32, 34, 35, 35, 36, 37, 39, 40, 44, 47

In the above case, the 7th

value is 37 and the 8th

value is 39. The 6th

decile is: P60= 37.4 which is 0.2 times the linear distance between 37 and 39

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7. Measures of Dispersion


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7. Measures of Dispersion

The Range

The Mean Absolute Deviation

Population Variance and Population Standard Deviation

Sample Variance and Sample Standard Deviation

Semivariance, Semideviation and Related Concepts (not a LO

Chebyshevs Inequality

Coefficeint of Variation

Sharpe Ratio

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The Range


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e a ge

Range: difference between the maximum and minimum

in a data set

Range = Max value Min value

Annual returns data: 10%, -5%, 10%, 25%. What is the r

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Answer: Range = 30%

The Mean Absolute Deviation


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Mean Absolute Deviation (MAD): average of the absolute valu

deviations from the mean

Annual returns data: 10%, -5%, 10%, 25%. What is the MAD?

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Answer: MAD = 7.5

Practice Question


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QThe number of pages read by a group of students per day has been tabula

presented as follows:

Which of the following is most likely to be the mean absolute deviation (M

A. 36.5

B. 55.6

C. 75.6

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Patrick 100 Fiona 175

Ginni 65 James 50

Melisa 27 Margaret 20

Tina 120 Samuel 75

Thomas 34 Hailey 90

Solution


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Answer: A

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Population Variance and Standard Dev


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p

Population variance is the mean of the squared deviations from the mea

population variance is computed using all members of a population.

Annual returns data: 10%, -5%, 10%, 25%. What is the population varianc

Population standard deviation is defined as the positive square root of th

variance

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Practice Question


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The dividend yield for five hypothetical companies is given below.

The population variance is most likely to be:

A. 36.89

B. 45.20

C. 56.49

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Paknama 10.50%

Genie Ltd. 16.25%

Mirinda Corp. 27.00%

Tina Travels Ltd. 12.00%

Thomas Press Ltd. 7.80%

Solution


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Practice Question


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Little Wonder 10.5%

Genesis Ltd. 16.25%

Moral Corp. 9.81%

Travis Ltd. 12.0%

The return on equity for four hypothetical companies is given below. W

the population standard deviation?

Solution


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Sample Variance and Standard Devia


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Sample variance applies when we are dealing with a subset, or sample, o

population

Annual returns data: 10%, -5%, 10%, 25%. What is the sample variance?

Sample standard deviation is defined as the positive square root of the sa

variance

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Practice Problem


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Paknama 10.50%

Genie Ltd. 16.25%

Mirinda Corp. 27.00%

Tina Travels Ltd. 12.00%

Thomas Press Ltd. 7.80%

The dividend yield for five hypothetical companies from a list of ten c

given below. What is the sample variance?

Solution


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Using the Calculator


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Annual returns data: 10%, -5%, 10%, 25%. What is the population and sample standard dev

Keystrokes Explanation

[2nd] [DATA] Enter data entry mode

[2nd] [CLR WRK] Clear data registers10 [ENTER]

[] [] 5+/- [ENTER]

[] [] 10 [ENTER]

[] [] 25 [ENTER]

[2nd] [STAT] [ENTER] Puts calculator into stats mode.

[2nd] [SET] Press repeatedly till you see

[] Number of data points

[] Mean

[] Sample standard deviation

[] Population standard deviation



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Chebyshev's inequality states that for any set of observations, the proport

observations within k standard deviations of the mean is at least: 1 (1/k2

Chebyshevs inequality can be used to measure maximum amount of dispe

regardless of the shape of the distribution

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Practice Problem


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In accordance with the Chebyshevs inequality, 88.89% of any distribution will lie withi

standard deviations of the mean?

Coefficient of Variation


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Coefficient of variation expresses how much dispersion exists relative to

a distribution and allows for direct comparison of dispersion across differe

It is used in investment analysis to compare relative risks.

Example: Investment A has a mean return of 7% and a std dev of 0.05. Inv

has a mean return of 12% and a std dev of 0.07. Which is riskier?

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Practice Problem


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Asset Class Arithmetic mean return (%) Standard deviation of return (%)

Bond A 16.4% 4.9%

Bond B 12.6% 3.5%

Bond C 14.8% 4.2%

The table below provides data for three bonds. Which bond has the low

per unit of return?

Answer: Bond B

The Sharpe Ratio


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The Sharpe ratio tells us whether a portfolio's returns a

smart investment decisions or a result of excess risk.

Sharpe Ratio = Rp - Rf

p

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Practice Problem


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The table below provides data for three portfolios. Given that the mean a

free rate is 10.5%, which portfolio is mostlikely to have the highest Sharpe r

Portfolio Arithmetic mean return (%) Variance of (%)

Portfolio A 16.4% 4.9%

Portfolio B 12.6% 3.5%

Portfolio C 14.8% 4.2%

8. Symmetry and Skewness in Return Dist


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Skewness, or skew, refers to the extent to which adistribution is not symmetrical.

Normal distribution has skew = 0

A positively skewed distribution is characterized bymany outliers in the upper region, or right tail.

Mean > Median > Mode

A negatively skewed distribution has adisproportionately large amount of outliers that fallwithin its lower (left) tail.

Mode > Median > Mean

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Practice ProblemWhich of the following distribution is most likely characterized by frequ


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Which of the following distribution is most likely characterized by frequ

losses and a few extreme gains?

A. Normal distribution

B. Negatively skewed

C. Positively skewed

Answer: C

A positively skewed distribution has frequent small losses and a few extreme gains.

skewed distribution has frequent small gains and a few extreme losses. A normal d

symmetrical.

Practice Problem


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Which of the following is mostlikely to be true for a negatively skewed dist

A. Mean < median < mode

B. Mode < median < mean

C. Median < mean < mode

Answer: A

For a negatively skewed distribution, the mean is less than the median, which is less than

mode.

Practice Problem


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Which of the following is most likely to be a positively skewed distribution?

A. A distribution skewed to the right

B. A distribution skewed to the left

C. A distribution skewed upward

Answer: A

A positively skewed distribution is skewed to the right whereas a negatively skewed distri

skewed to the left.

9. Kurtosis in Return Distributions


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Measure the degree to which a distribution is more or less p

than a normal distribution.

Kurtosis Subtypes:

Mesokurtic K = 3 (Normal)

Leptokurtic K > 3

Platykurtic K < 3

Leptokurtic is more peaked with fatter tails (more extreme outliers)

Platykurtic is less peaked

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Practice ProblemWhich of the following is most likely to be a measure of when a distributio


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Which of the following is ost likely to be a measure of when a distributio

or less peaked compared to a normal distribution?

A. Skewness

B. Coefficient of variation

C. Kurtosis

Answer: C

Kurtosis is the statistical measure that tells to when a distribution is more or less peaked t

normal distribution.

Practice ProblemA di t ib ti id ti l t l di t ib ti i


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A distribution identical to a normal distribution is:

A. leptokurtic

B. mesokurtic

C. platykurtic

Answer: B

A distribution identical to a normal distribution is mesokurtic.

10. Using Geometric and Arithmetic M


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Geometric mean is appropriate for making investment statements ab

performance

Started with $100 two years ago; return of 100% in Y1 and -50% in

was the mean return?

Arithmetic mean is appropriate for making investment statements in f

looking context Starting with $100 now. Two possible returns for upcoming year ar

or -50%. What is the expected return?

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Summary


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Measures of Central Tendency

Measures of Dispersion


Sharpe Ratio

Skewness and Kurtosis

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Conclusion


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Review learning objectives

Examples and practice problems from the

curriculum

Practice questions from other sources

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r07 statistical concepts and market returns

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