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Chapter 3
Statistical Concepts
1 of 59
Simple Probability distribution

Return calculation for a 1 period investment

P  CF PEND  PBEG  CF Pend  CF
r


 1.0
PBEG

PBEG
PBEG
Example: Today’s price is $40. We expect the price to go
to $44 and receive $1.80 in dividends. What is the return?
44  1.80
r
 1.0  14.5%
40
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Adjustments are necessary to annualize
the return.

 P  CF 

HPR   end
 PBEG

N = years of investment


2 years N = 2
1
N
 1.0
6 months N = .5
What if it took 2 years to get the same result?
1/ 2
 44  1.80 
HPR  

40


 1.0  7.00%
Beginning Value
Ending Value
Investment Cash Flows
Investment Time (Yrs)
$40.00
$44.00
$1.80
2.000
Return
HPR (annualized return)
14.50%
7.00%
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HPR
The Normal Distribution
Absence Symmetry, standard deviation as a measure of
risk is inadequate.
If normally distributed returns are combined into a
portfolio, the portfolio will be normally distributed.
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Descriptive Statistics


In order to describe what a distribution looks like we need to
measure certain parameters
Population

1) Mean = expected return = weighted average
• P = probability


r = return
E(r )   pi * ri 
2) variance = potential deviation from the mean or average

Sample




2   ri  E(r ) * pi   pi * ri2  E(r )2
1) Mean
2
r

r
i
n

2) Variance

r  r


2
2
i
n 1
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Normal and Skewed
(mean = 6% SD = 17%)
Normal distribution have a skew of 0.
Positive skew – standard deviation overestimates risk
Negative – underestimates risk
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Normal and Fat Tails Distributions
(mean = .1 SD =.2)
Kurtosis of a normal distribution is 0.
Kurtosis greater than 0 (fat tails), standard
deviation will underestimate risk
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Joint Probability


Joint probability statistics
Covariance

Population
COVab   pi * ra  E(ra ) * rb  E(rb )

Sample
COVab


r


ai

 ra * rbi  rb
n 1

Fig 3-3 pg 37 Pos. covariance
• Note the data points are
concentrated in Quadrants I & III
Fig 3.4 pg 38
neg. Covariance
• Data points in Quadrants II & I
Fig 3.5 pg 38
zero covariance
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Correlation
Perfect
positive


Correlation
Perfect positive correlation


Fig 3.8 and 3.9 pg 41
Fig 3.10 pg 42
Imperfect correlations



Perfect
negative
Perfect negative correlation


ab
COVab

a * b
(Fig 3.11 and 3.12 pg 43)
There is some error in this
technique (hence the imperfect)
(Use regression coefficient b =
correlation) = slope of the line
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Correlation

standardizes the COV


covariance can have an unlimited magnitude
It produces a line of best fit that minimizes the sum of the
squared deviations
Correlation has a range of values
 -1.0
0.0
+1.0
perfect negative
zero
perfect positive


Can rewrite COV
COVab  ab * a * b

Coefficient of determination


Describes the amount of variation in one invest that can be
associated with or explained by another
how much does one explain about the movement of
another
CD  
2
ab
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How does the CD work?

Correlation between Asset A and B = 1.0;
CD = 1.0 (100%)


If you can predict the returns of asset A; you can
exactly predict the returns in Asset B
Correlation between Asset C an D = .70;
CD = .49 (49%)

If you can predict the returns of asset C; you can
predict 49% of the returns in Asset D
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Stock and Market Portfolios

Market portfolio contains every risky asset in
the investment world


Each asset is held in its proportion to all the
others in the market


(stocks, bonds, coins, real estate, etc.)
Value weighted index
To create your own index you would buy .01%
of the total market value of each and every risky
asset. You would own the same amount of
control in each company, but each asset would
have a different value
12 of 59
Characteristic Line





Fig 3.13 pg 45
plot the return of the asset at a
certain time against the return of
the market at the same time
can create a line that best
describes the relationship (Line of
best fit) using regression statistic
Shows the return you would expect
to get in the asset given a
particular return in the market
index
The line is not a perfect fit, it does
not show the exact return you will
get. There are errors made in the
estimation.
13 of 59
Variables for Characteristic Line

Beta factor




indicator of the degree to which the stock responds to changes in the
return in the market
slope of the characteristic line
Note the similarity between the formulas for Beta and the correlation
coefficient
COVj,m
• Only the denominator
j 
 b coefficient
2
m
Alpha

This term represents the return on the stock if the market had a
return of zero


A j  rj   j * rm  cons tan t

Residual


since we will seldom if ever see perfect relationships, there will
always be some “error” in the representation of the line of best fit
It is a line of “best” fit not “perfect” fit
 j,t  rj,t  A j   j * rmt   rj,t  E(rj,t )
14 of 59
How much error?


Variance measures the deviation from the mean or expected return
Residual variance measures the deviation from the characteristic
line
2, j 

n2
As the residual variance approaches zero, the correlation between
the returns of the stock and the returns of the market approaches
either +1 or -1,


2

 j,t
also then the coefficient of determination would approach 1.0
the closer to +/- 1 the better the explanation of the data, so you
could use this as a prediction of the stocks returns, given an
estimate of the markets returns

There would be less error in the prediction, if the residual variance was
very low
15 of 59
Examples
data pts
1
2
3
4
5
6
Mean
Stnd Deviation
Coef. of Var
Return Asset A Returns Asset B Mkt returns Returns
residuals A(mkt)residuals B(mkt)
5.00%
-1.00%
4.00%
0.000011
0.000018
8.00%
3.00%
6.00%
0.000075
0.000012
9.00%
8.00%
8.00%
0.000000
0.000001
10.00%
12.00%
10.00%
0.000054
0.000078
12.00%
19.00%
12.00%
0.000028
0.000180
15.00%
22.00%
14.00%
0.000044
0.000018
9.833%
10.500%
9.000%
0.0002
0.0003
3.430%
8.961%
3.742%
0.3488
0.8534
0.4157 Correlation Matrix
A with mkt
B with Mkt
Alpha
0.0173
(0.1097)
Beta
0.9000
2.3857
Residual variance
0.0053%
0.0077%
Portfolio Statistics
weights
1
2
3
4
5
6
50.00%
2.50%
4.00%
4.50%
5.00%
6.00%
7.50%
Correlation Matrix
A
B
1.000
0.973
0.982
A
B
C
20.00%
-0.20%
0.60%
1.60%
2.40%
3.80%
4.40%
C
1.000
0.996
1.000
30.00% sum returns
sum squares
1.20%
3.50%
0.0039
1.80%
6.40%
0.0011
2.40%
8.50%
0.0001
3.00%
10.40%
0.0000
3.60%
13.40%
0.0014
4.20%
16.10%
0.0041
Mean
9.717%
Std Dev
3.390%
Coef of Var
2.87
Historical Data returns
16 of 59