7/28/2019 Cfa Quant Review - Statistics(1)
1/26
Investment Tools
Statistics
SASF CFA Quant. Review
7/28/2019 Cfa Quant Review - Statistics(1)
2/26
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.
7/28/2019 Cfa Quant Review - Statistics(1)
3/26
3
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
7/28/2019 Cfa Quant Review - Statistics(1)
4/26
4
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
7/28/2019 Cfa Quant Review - Statistics(1)
5/26
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
7/28/2019 Cfa Quant Review - Statistics(1)
6/26
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
7/28/2019 Cfa Quant Review - Statistics(1)
7/26
7/28/2019 Cfa Quant Review - Statistics(1)
8/268
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
7/28/2019 Cfa Quant Review - Statistics(1)
9/269
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
7/28/2019 Cfa Quant Review - Statistics(1)
10/2610
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
7/28/2019 Cfa Quant Review - Statistics(1)
11/2611
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
7/28/2019 Cfa Quant Review - Statistics(1)
12/2612
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
7/28/2019 Cfa Quant Review - Statistics(1)
13/26
13
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
7/28/2019 Cfa Quant Review - Statistics(1)
14/26
14
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
7/28/2019 Cfa Quant Review - Statistics(1)
15/26
15
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.
7/28/2019 Cfa Quant Review - Statistics(1)
16/26
16
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!!!!!
7/28/2019 Cfa Quant Review - Statistics(1)
17/26
17
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
7/28/2019 Cfa Quant Review - Statistics(1)
18/26
19
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
7/28/2019 Cfa Quant Review - Statistics(1)
19/26
Simple Linear Regression
7/28/2019 Cfa Quant Review - Statistics(1)
20/26
21
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
7/28/2019 Cfa Quant Review - Statistics(1)
21/26
22
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
7/28/2019 Cfa Quant Review - Statistics(1)
22/26
23
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.
7/28/2019 Cfa Quant Review - Statistics(1)
23/26
7/28/2019 Cfa Quant Review - Statistics(1)
24/26
7/28/2019 Cfa Quant Review - Statistics(1)
25/26
26
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
7/28/2019 Cfa Quant Review - Statistics(1)
26/26
27
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
Top Related