Day 1
Quantitative Methods for Investment
Management
by Binam Ghimire
1
Objective
Statistical Concepts and market returns and
Probability Concepts
Identify measures of central tendency and measures of
Dispersion
Understand that measures of central tendency give an
indication of the expected return of an investment and
measures of dispersion measure riskiness of an
investment
Use of Excel on the topic
2
Basic Concept
Statistics
Descriptive statistics
Inferential statistics
Population
Parameter
Sample
Statistics
3
Basic Concept
Variable Measurement Scale
Variable Scale
Nominal
Ordinal
Interval
Ratio
Less Informative
More Informative
Guides what type of test we need to perform
4
Descriptive Statistics:
Histogram and Frequency Polygons
 Histogram: Grouped data. The area of each rectangle is
proportion to the frequency
 Frequency Polygon: a line graph drawn by joining all the
midpoints of the top of the bars of a histogram
 Activity: Excel – Histogram and Frequency Polygon
5
Measures of location - Averages
 Meaning & Calculation
Mean: Arithmetic, Weighted and Geometric
Mode
Median
 Formula
 Activity: Football Game
6
Weighted Mean as Portfolio Return
 Weighted Mean is useful to find return of a portfolio
Return of Portfolio is basically
(W1xR1) + (W2xR2) + (W3xR3) … (WnxRn) where
W is weight and R is return
7
Weighted Mean as a Portfolio Return
Example:
Actual
Return
Cash
5%
×
Bonds
7%
×
Stocks 12% ×

Portfolio
Weight
0.10 = 0.5%
0.35 = 2.45%
0.55 = 6.6%
Σ = 9.55%
Same method works for expected portfolio
returns!
8
Geometric Mean
Geometric mean is used to calculate compound
growth rates
If the returns are constant over time, geometric mean
equals arithmetic mean
The greater the variability of returns over time, the more
the arithmetic mean will exceed the geometric mean
R Geom  [(1  R 1 )x(1  R 2 )...x(1  R n )]1/n  1
Actually, the compound rate of return is the geometric mean
of the price relatives, minus 1
9
Geometric Mean: Example
An investment account had returns of 15.0%, –9.0%, and
13.0% over each of three years
Calculate the time-weighted annual rate of return
R Geom  [(1  R 1 )x(1  R 2 )...x(1  R n )]1/n  1
= 5.75 %
10
Measures of location
 Meaning and Calculation
Maximum
Minimum
Quantile: Quantile is a method for dividing a range of
numeric values into categories
Quartile, Percentiles, Deciles
 75% of the data points are less than the 3rd quartile
 60% of the data points are less than the 6th decile
 50% of the data points are less than the 50th
percentile
Formula
 Activity: Football Game
11
Measures of Dispersion
 Meaning and Calculation
Range
Inter-quartile range
Semi-interquartile range
Mean Absolute Deviation
Variance
Standard Deviation
 Formula
 Activity: Football Game
12
Measures of Association
 Meaning
Co-variance
Formula:
_
Covariance = sXY =
_
S (X - X)(Y - Y)
n
 Calculation
13
Measures of Association:
Covariance
 Co-variance has a sign
Y
20
24
28
32
34
30
Y values
X
12
14
16
18
26
22
18
14
 Covariance = 10
10
10
12
14
16
18
20
X values
14
Measures of Association:
Covariance
 Co-variance has a sign
34
Y
32
28
24
20
30
Y values
X
12
14
16
18
26
22
18
14
10
 Covariance = -10
10
12
14
16
18
20
X values
15
Measures of Association:
Covariance
 Co-variance has a sign
Y
20
25
28
22
26
30
23
34
30
Y values
X
12
15
18
14
16
19
15
26
22
18
14
10
10
12
14
16
18
20
X values
 Covariance = 6.94
16
Measures of Association:
Covariance
 Co-variance has a sign
Y
30
22
17
26
26
21
23
34
30
Y values
X
12
17
18
14
16
19
15
26
22
18
14
10
10
12
14
16
18
20
X values
 Covariance = -7.49
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Measures of Association:
Covariance in Investment Management
 For example, if two stock prices tend to rise and fall at
the same time, these stocks would not deliver the best
diversified earnings.
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Measures of Distributions
 Distribution Shape
Skewness
Kurtosis
19
Measures of Distributions:
Skewness
 Concept:
Skewness characterizes the degree of asymmetry of
a distribution around its mean
Positive skewness indicates a distribution with an
asymmetric tail extending toward more positive
values
Negative skewness indicates a distribution with an
asymmetric tail extending toward more negative
values
No Skewness: symmetrical
20
Measures of Distribution
Positive Skewness
 Skewness = 0.45
Frequency Distribution
9
8
Frequencies
7
6
5
4
3
2
1
0
X values
 Tail to the higher values. Mean > Median > Mode
 Exercise in Excel
21
Measures of Distribution :
Negative Skewness
 Skewness = - 0.45
Frequency Distribution
9
8
Frequencies
7
6
5
4
3
2
1
0
X values
 Tail to the lower. Mean < Median < Mode
 Exercise in Excel
22
Measures of Distribution :
No Skewness
 Skewness = 0
Frequency Distribution
9
8
Frequencies
7
6
5
4
3
2
1
0
X values
 Tail to the lower. Mean = Median = Mode (Symmetrical/
Normal)
23
 Exercise in Excel
Measures of Distribution
Kurtosis
 Concept
Kurtosis characterizes the relative peakedness or
flatness of a distribution compared with the normal
distribution
Positive kurtosis indicates a relatively peaked
distribution
Negative kurtosis indicates a relatively flat
distribution
No or zero Kurtosis = normal distribution
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Measures of Distribution
Positive Kurtosis
 Kurtosis = 1.68
Frequency Distribution
18
16
Frequencies
14
12
10
8
6
4
2
0
X values
 Positive Kurtosis: Peaked relative to the Normal
 Exercise in Excel
25
Measures of Distribution
Negative Kurtosis
 Kurtosis = - 0.34
Frequency Distribution
9
8
Frequencies
7
6
5
4
3
2
1
0
X values
 Negative Kurtosis: Flat relative to the Normal
 Zero Kurtosis: Peak similar to Normal Distribution
 Exercise in Excel
26
Kurtosis:
Other names
 A distribution with a high peak is called leptokurtic
(Kurtosis > 0), a flat-topped curve is called platykurtic
(Kurtosis < 0), and the normal distribution is
called mesokurtic (Kurtosis = 0)
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Semivariance
 Semivariance is calculated by only including those
observations that fall below the mean on the calculation.
 Sometimes described as “downside risk” with respect to
investments.
 Useful for skewed distributions, as it provides additional
information that the variance does not.
 Target semivariance is similar but based on observations
below a certain value, e.g values below a return of 5%.
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Coefficient of Variance (CV)
 Coefficient of Variance (CV)
= standard deviation
mean
 In investments for example; CV measures the risk
(variability) per unit of expected return (mean).
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CV
 Example: Suppose you wish to calculate the CV for two
investments, the monthly return on British T-Bills and
the monthly return for the S&P 500, where: mean
monthly return on T-Bills is 0.25% with SD of 0.36%,
and the mean monthly return for the S&P 500 is 1.09%,
with a SD of 7.30%.
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CV
 CV (T-Bills) = 0.36/0.25 = 1.44
 CV (S&P 500) = 7.30/1.09 = 6.70
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CV
 Interpretation: CV is the variation per unit of return,
indicating that these results indicate that there is less
dispersion (risk) per unit of monthly returns for T-Bills
than there is for the S&P 500, i.e. 1.44 vs 6.70.
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We now should know the followings
 Concept, Formula and Calculation
 Mean
 Median
 Quartiles
 Percentile
 Range
 Interquartile and semi-interquartile Range
 Mean Deviation
 Variance, Semi Variance
 Standard Deviation
 Covariance, Coefficient of Variance
 Use of Excel for the above and Skewness and Kurtosis
33
Can we solve the following?
 An investor holds a portfolio consisting of one share of
each of the following stocks:
Stock
X
Y
Z
Price at the
beginning of the
year
£20
£40
£100
Price at the end
of the year
Cash dividend
during the year
£10
£50
£105
£0
£2
£4
 For the 1-year holding period, the portfolio return is
closest to:
a) 6.88% b) 9.13% c) 13.13% and, d) 19.38%
 Now practice Examples Day 1 (Some questions require
knowledge from other chapters)
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