Corporate Finance - Banks and Markets
Download
Report
Transcript Corporate Finance - Banks and Markets
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
17
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.
18
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
24
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)
27
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%.
28
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).
29
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%.
30
CV
CV (T-Bills) = 0.36/0.25 = 1.44
CV (S&P 500) = 7.30/1.09 = 6.70
31
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.
32
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)
34