Download Statistical Concepts and Market Returns

Welcome To
Quantitative
Methods in
Investment
Management
BUSINESS STATISTICS
 Session 1: Statistical Concepts and Variation
of Data
 Session 2: Probability Theory and Probability
Distribution
 Session 3: Statistical Estimation and
Statistical Hypothesis Testing
 Session 4: Liner Regression and Correlation
Statistical Concepts and
Variation of Data
Statistics
 Descriptive statistics describe the properties
of a large data set
 Inferential statistics uses a sample from a
population to make probabilistic statements
about the characteristics of a population
 A population is a complete set of outcomes
 A sample is a subset drawn from a population
Measurement Scales
 Nominal - only names make sense
e.g. robin, parrot, seagull
 Ordinal - order makes sense
e.g. large-cap, mid-cap, small-cap
 Intervals - intervals make sense
o
o
o
e.g. 40 F is 10 greater than 30 F
 Ratio - ratios make sense (absolute zero)
e.g. $200 is twice as much as $100
Statistics Terms
 A parameter describes a characteristic of a
population
 A sample statistic describes a characteristic of a
sample (drawn from a population)
 A relative frequency distribution shows the
percentage of a distribution’s outcomes in each
interval
 A cumulative frequency distribution shows the
percentage of observations less than the upper bound
of each interval
A Histogram
A Frequency Polygon
Measures of Central Tendency:
Population and Sample Means
 Population and sample means have
different symbols but are both arithmetic
means
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
Actually, the compound rate of return is the
geometric mean of the price relatives, minus 1
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
Weighted Mean
A mean in which different observations are
given different proportional influence on the
mean
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!
Median
 Midpoint of a data set, half above and half below
 With an odd number of observations
2, 5, 7, 11, 14
Median = 7
 With an even number of observations, median is the
average of the 2 middle observations
3, 9, 10, 20
Median = (9 + 10)/2 = 9.5
Less affected by extreme values than the mean
Mode
Value occurring most frequently in a data set
2, 4, 5, 5, 7, 8, 8, 8, 10, 12
Mode = 8
Data sets can have more than 1 mode (bimodal,
trimodal, etc.)
Quantiles
 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
For data with 17 observations, the 70th percentile is at
observation (17+1) × 0.70 = 12.6
For ordered observations, this is six-tenths of the way
from observation 12 to observation 13
Range and MAD
Annual returns data: 15%, –5%, 12%, 22%
Range (the difference between the largest and smallest
value in a data set) = 22% – (–5%) = 27%
Mean Absolute Deviation (MAD): average of the
absolute values of deviations from the mean.
Mean = (15 – 5 + 12 + 22)/4 = 11%
MAD = (|15 – 11| + |–5 – 11| + |12 – 11| + |22 – 11|)/4 =
32/4 = 8%
Population Variance and Std.
Deviation
Variance is the average
of the squared deviations
from the mean
Standard deviation is
the square root of
variance
Population Variance
2
(σ )
Example:
Returns on 4 stocks: 15%, –5%, 12%, 22%
Population Mean (µ)= 11%
Population Standard Deviation
Variance: (σ 2)= 98.5
 Standard deviation is in the same units as the
observations (percent returns in our example)
2
(s )
Sample Variance
and
Sample Standard Deviation (s)
Key difference between calculation of σ2 and s2
is that the sum of the squared deviations for s2
is divided by n – 1 instead of n
Skewness
 Skew measures the degree to which a distribution
lacks symmetry
 A symmetrical distribution has skew = 0
Positive Skew = Right Skew
 Positive skew has outliers in the right tail
 Skew absolute values > 0.5 are significant
 Mean is most affected by outliers
‘Pull’ on right tail
to get
positive/right
skew
Negative Skew = Left Skew
 Negative skew has outliers in the left tail
 Again, mean is most affected by outliers
Kurtosis
 Measures the degree to which a distribution is
more or less peaked than a normal distribution
 Leptokurtic (kurtosis > 3) is more peaked with
fatter tails (more extreme outliers)
Kurtosis
 Kurtosis for a normal distribution is 3.0
 Excess Kurtosis is Kurtosis minus 3
 Excess Kurtosis is zero for a normal
distribution
 Excess kurtosis greater than 1.0 in absolute
value is considered significant