Cfa Quant Review - Statistics(1)

7/28/2019 Cfa Quant Review - Statistics(1)

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Investment Tools

Statistics

SASF CFA Quant. Review


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2

Statistical Concepts

Population is defined as all members of a specified group.

Sample is a subset of a defined population.

Frequency Distribution: is a tabular display of data summarized into a

relatively small number of intervals.

Frequency distribution is the list of intervals together with the

corresponding measures of frequency for the variable of interest.

A histogram - graphical equivalent of a frequency distribution; it

is a bar chart where continuous data on a random variables

observations have been grouped into intervals.

A frequency polygon is the line graph equivalent of a frequency

distribution; it is a line graph that joins the frequency for each

interval, plotted at the midpoint of that interval.


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Frequency Distribution Table

Raw Data:

24, 26, 24, 21, 27, 27, 30, 41, 32, 38

Class Frequency

15 but < 25 3

25 but < 35 535 but < 45 2


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Frequency Distn Table Steps

1. Determine Range

2. Select Number of Classes

Usually Between 5 & 15 Inclusive

3. Compute Class Intervals (Width)

4. Determine Class Boundaries (Limits)

5. Compute Class Midpoints

6. Count Observations & Assign to Classes


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5

01

2

3

4

5

Histogram

Frequency

RelativeFrequency

Percent

0 15 25 35 45 55

Lower Boundary

BarsTouch

Class Freq.

15 but < 25 325 but < 35 535 but < 45 2

Count


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6

0

1

2

34

5

Frequency Polygon

Midpoint

FictitiousClass

0 10 20 30 40 50 60

Class Freq.

15 but < 25 325 but < 35 535 but < 45 2

Frequency

RelativeFrequency

Percent

Count


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Measures ofCentral Tendency summarize the location on which the data are

centered.

Population Mean: calculated as

where there areN members in the population and each observation isXi i =1, 2,

N.

Sample Mean: calculated aswhere there are n observations in the sample and each observation isXi i =1, 2,

n. It is also the arithmetic mean of the sample observations.

Median: calculated as the middle observation in a group that has been ordered

in either ascending or descending order.

In an odd-numbered group this is the (n+1)/2 position.In an even numbered group it is the average of the values in the n/2 and (n+1)/2

positions.

Mode: is the most frequently occurring value in the distribution. A distribution

may have one, more than one, or no mode.

Measures of Central Tendency

n

iiXnX 1

1

N

i

iX XN 1

1


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Other Definitions for Means

Measures ofcentral tendency summarize the location on which thedata are centered.

Weighted Mean: calculated as

where there are n observations, each observation isXi, and the weight

associated with each observation is wi i =1, 2, n. Ifwi = 1/n, then this is

the sample mean. Ifwi is the probability ofXi occurring then this weighted

mean is the expected value of the random variableX.

Geometric Mean: calculated as

where there are n observations and each observation isXi.

nnXXXG 21

n

i

iiWeighted XwX1


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Measures of Dispersion

Range: is the difference between the maximum and minimum values ina dataset.

Mean Absolute Deviation: is the average of the datas absolutedeviations from the mean.

Population Variance: is the average of the populations squareddeviations from the mean.

The population standard deviation is simply the square root of thepopulation variance.

Sample Variance: is the average of the sample datas squareddeviations from the sample mean.

The sample standard deviation is simply the square root of the samplevariance.

n

i

i XXn

MAD1

1

N

i

iXN 1

22 1

n

i

i XXn

s1

22

1

1


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Useful Measures for Returns

Holding Period Return: is expressed in percent terms, i.e.independent of currency units, and is calculated over a period of time.

Holding Period Return = Rt

Share Price end of time t = Pt

Share Price end of time t-1 = Pt-1

Cash Distributions during period t = Dt

Holding Period Return, Rt, consists of capital gains over the period plus

distributions during the period divided by the beginning price (distribution

yield).

1

1

t

tttt P

DPPR


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Coefficient of variation,CV shows relative dispersion. If X is returns on an asset then CV shows

the amount of risk (measured by sample standard deviations) for every

% of mean return on the asset. The lower an assets CV, the more

attractive it is in risk per unit of return.

Sharpe measure,

SM is a more precise return-risk measure as it takes into account an

investor can earn the risk-free rate, rp, without bearing any risk. Hence aportfolios risk (measured by its standard deviation sp) must be

compared to its return in excess of the risk-free rate . The higher is SM,

the better the return-risk tradeoff on the portfolio for an investor

X

sCV

p

fp rrSM

Measures of Risk vs. Return


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Shape

1. Describes How Data Are Distributed2. Measures of Shape

Kurtosis = How Peaked or Flat

Skew = Symmetry

Positive-SkewedNegative-Skewed Symmetric

Mean = Median = ModeMean Median Mode Mode Median Mean


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Measures of Shape

Frequency distribution that is not symmetric is skewed.

Positively-skewed distribution is characterized by many small losses but a few

extremely large gains. It has a long tail on the right side of the distribution.

Negatively-skewed distribution is characterized by many small gains but a few

extremely large losses. It has a long tail on the left-hand side of the distribution.

Skewness arises as a result of the properties of asset prices and returns. A

share price can never be negativethere is a lower limit on the assets

returns (-100%) but no theoretical limit on its upper limitso an assets

return may be positively-skewed.

i. Symmetrical distribution: Mean = Median = Mode

ii. Positively-skewed distribution: Mean > Median > Mode

iii. Negatively-skewed distribution: Mean < Median < Mode


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Measures of Shape

A frequency distribution that is more or less peaked than a

Normal distribution is said to exhibit kurtosis. If the

distribution is more peaked than a Normal (i.e. exhibits fat

tails) it is leptokurtic. If it is less peaked than a Normal it is

called platykurtic.

Positive excess kurtosis, i.e. a leptokurtic distribution, means that largepositive and negative deviations from the mean have higher

probabilities for occurring than they would under a Normal

distribution.

If an portfolios returns are leptokurtic then its true risk is higher than

the risk suggested by an analysis that assumes returns are Normallydistributed. This is important for Value at Risk (VAR) calculations that

must assume distributions for asset returns in a portfolio.


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Frequencies

19. An analyst gathered the

following data:63.5 96.9 112.3 134.1

66.4 98.3 116.2 138.5

75.6 99.5 116.9 139.8

77.5 100.7 118.3 140.7

84.4 102.0 122.0 143.087.6 105.5 122.2 153.9

89.9 108.4 124.5 155.5

Five classes as follows:1. 60 < x < 80.2. 80 < x < 100

3. 100 < x < 120

4. 120 < x < 140

5. 140 < x < 160

In constructing a frequency

distribution using five classes, if thefirst class is "60 up to 80," the class

frequency of the third class is:

A. 4.

B. 5.

C. 6.D. 8.

Hence there are 8 observations inthe third class.

Note the misleading way thequestion is asked! Always readthe question carefully!!!!!


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Geometric Mean

21. A portfolio of non-dividend-paying stocks earned a geometricmean return of 5 percent betweenJanuary 1, 1995, and December31, 2001. The arithmetic meanreturn for the same period was 6

percent. If the market value of theportfolio at the beginning of 1995was $100,000, the market value ofthe portfolio at the end of 2001was closest to:

A. $135,000.B. $140,710.

C. $142,000.

D. $150,363.

Identify what you are being asked for

Portfolio Ending value P12/31/2001

Given the following:

Portfolio Beginning value = P1/1/1995=$100,000

Geometric mean return = 5%

Arithmetic mean return = 6%

Number of periods = 7

Non-dividend paying stocks in portfolio.

Identify correct approach usegeometric mean return and formula

Pt+7 = (1+r)7 PtPt+7 = (1.05)

7 $100,000 = $140,710


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Other Questions

23. Which of the following statementsabout standard deviation is TRUE?

Standard deviation:

A. is the square of the variance.

B. can be a positive or a negative

number.

C. is denominated in the same

units as the original data.

D. is the arithmetic mean of the

squared deviations from the mean.

25. A stock with a coefficient of variationof 0.50 has a(n):

A. variance equal to half the stock'sexpected return.

B. expected return equal to half the

stock's variance.C. expected return equal to half thestock's standard deviation.

D. standard deviation equal to halfthe stock's expected return.

If

then

21

XsCV

Xs2

1


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Simple Linear Regression


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Y

Y = mX + b

b = Y-intercept

X

Change

in Y

Change in X

m = Slope

Linear Equations & Regression

1. Answer to What Is the Relationship Between the Variables?

2. Regression Equation Used

1 Numerical Dependent (Response) Variable

Variable to be Predicted

1 or More Numerical or Categorical Independent (Explanatory) Variables

3. Used to Test Theories and for Prediction


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Y Xi i i 0 1

Linear Regression Model

Relationship Between Variables Is a LinearFunction

Dependent(Response)Variable

Independent(Explanatory)Variable

Population

Slope

Population

Y-Intercept

Random

Error


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Probabilistic Models

Hypothesize 2 Components involved in explainingbehavior of a variable of interest.

Deterministicbased on relevant theory

Random Errorreflects unknown elements

Example: Want to explain the return on a companysstock.

Theory: Return on Company j is 1.50 Times Return onOverall Stock Market Plus Random Error

Probabilistic Model:Rj = 1.5 RMkt + j

Random Error May Be Due to Company-specific Factors.


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e2

Y

X

e1 e3

e4

Least Squares (LS) Graphically

Y b b X ei i i 0 1

Y b b Xi i 0 1

LS Minimizes e e e e eii

n2

112

22

32

42


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Interpretation of LS Coefficients

1. Slope (b1)Estimated Ychanges by b1 for each 1unit change

inX

Ifb1 = 2, then Company Return (Y) is expected to

increase by 2 for each 1 unit increase in MarketsReturn (X)

2. Y-Intercept (b0)

Average Value ofY whenX= 0 Ifb0 = 2, then Average Company Return (Y) Is

Expected to Be 2 When Market Return (X) Is 0

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Cfa Quant Review - Statistics(1)