Reading 07: Statistical Concepts and Market Returns
Basic terms
Descriptive vs. Inferential Statistics
Population vs. Sample
Measurement scales
N O I R
Descriptive Statistics: describes data set
Inferential statistics: uses samples to make forecasts about the population.
Population: includes all members of a specified group
Sample: Subset of population, are used to draw inferences about the population
Nominal scale: Only NAME makes sense
Ordinal Scale: ORDER also makes sense
Interval Scale: INTERVALS also makes sense
Ratio Scale: RATIOS makes senses. Has absolute value
Distribution
Parameter vs. Sample statistics
Parameter: describe a characteristic of a population.
Sample Statistic: describe a characteristic of a sample.
Frequency Distribution
Relative vs. Cumulative Frequencies
Relative Frequency: % of total observation in each interval.
Cumulative Frequency: shows % of observations that is less than the upper bound of each interval
Histogram
Polygon: A line connects all midpoints (middle value of each interval ; frequency of such inteval)
Measure of Central Tendency
Arithmetic mean
Geometric mean
Weighted mean
Harmonic mean
Median: Midpoint of a dataset when arranged from largest to smallest
Mode: Value that occurs the most frequently in a dataset
Quantile
Definition: a quantile is where a sample is divided into equal-sized, adjacent, subgroups
Example
Quartiles: distribution is divided into quarters
Quintiles: Distribution is divided into fifths
Deciles: Distribution is divided into tenths
Percentiles: Distribution is divided into hundredths
Dispersion
Measure of risk
Range: Max Value - Min Value
Mean Absolute Deviation (MAD)
Variance
Definition: Mean of squared deviation from the Arithmetic Mean
Population Variance:
σ2=∑Ni=1(Xi−¯X)N
Sample Variance:
\(S^{2}=\frac{\sum_{i=1}^{n}\left (X_{i}-\overline{X} \right )}{n-1} \)
Standard Deviation: is the positive square root of Variance
Chebyshev's inequality
Definition: Proportion of the observation within k standard deviation (SD) of the mean is \( \geq 1-\frac{1}{k^{2}} \) for all k >1
Specifically
\( \pm \)1.25 SD: 36% of observations lie within
\( \pm \)1.5 SD: 56% of observations lie within
\( \pm \)2 SD: 75% of observations lie within
\( \pm \)3 SD: 89% of observations lie within
\( \pm \)4 SD: 94% of observations lie within
Sharpe Ratio
Coefficient of Variation: \( CV=\frac{s}{\overline{X}} \)
Sharp Ratio: measures excess return per unit of risk
\( Ratio_{Sharpe}=\frac{\overline{r}_{p}-\overline{r}_{f}}{\sigma _{p}} \)
Skewness & Kurtosis
Skewness
Definition: Describes the degree to which a distribution is not symmetric about its mean. If Absolute value of skew > 0.5, it is considered significantly different from 0
Right Skew: Positive skewness.
Mean > Median > Mode
Left Skew: Negative skewness
Mean < Median < Mode
Kurtosis
Definition: Measure the peakedness of a distribution & the probability of extreme outcomes (Thickness of tails)
Kurtosis = 3 a.k.a Mesokurtic: Normal Distribution
Kurtosis >3 a.k.a Leptokurtic (Núi Trẻ)
Kurtosis < 3 a.k.a Platykurtic (Núi Già)
How to create Frequency Distribution?
Step 1: Define the intervals.
Step 2: Tally the observations.
Step 3: Count the observations.
Horizontal axis shows intevals
Vertical axis shows frequency
POSITION of the observation: \(L=\frac{\left(n+1\right)\times y}{100} \)
Formula: \(Sample\ skewness\ =\ \frac{\sum_{i=1}^{N}\frac{{(X_i-\bar{X})}^3}{N}}{s^3}\)
Sample kurtosis formula: \(Sample\ kurtosis\ =\ \frac{\sum_{i=1}^{N}\frac{\left(X_i-\bar{X}\right)^4}{N}}{s^4} \)
LINEAR INTERPOLATION
is a tabular presentation that summarizes data by assigning it to specified intervals
Population mean: \( \mu =\frac{\sum_{i=1}^{n}X_{i}}{N} \)
Sample mean: \( \overline{X}=\frac{\sum_{i=1}^{n}X_{i}}{n} \)
Good for forecasting future single period return
\( \overline{X}_{W}=\sum_{i=1}^{n}W_{i}\times X_{i} \)
weights each value according to its influence.
Used to find compound growth rate:
\( G=\sqrt[n]{X_{1}\times X_{2}\times ... \times X_{n}} \)
Good for forecasting compound return over multiple period.
Used to find average purchase price
\( \overline{X}_{H}=\frac{N}{\sum_{i=1}^{N}\frac{1}{X_{i}}} \)
Total Money paid divided by Total of shares bought
The greater the difference between number, the more Arithmetic > Geometric > Harmonic
\(X^{th}\ Observation\ Value\ +\ Location\ Value\ behind\ decimal\ point\ \times[{{(X+1)}^{th}\ Observation\ Value\ -X}^{th}\ Observation\ Value]\)
When the desired location is between 2 observation X1 and X2, we need to find the exact value of observation which reflect that location.
Lower CV is better, less risk per unit of return
Higher likelihood of extreme value as compared to a normal distribution.
However, large fluctuations are more likely
For Risk-seeking investor (higher chance of extreme gain)
negative excess kurtosis
lower likelihood of extreme value as compared to a normal distribution.
For Risk-averse investors (less chance of extreme loss)
is the average of the absolute value of deviations from Mean
\(MAD=\frac{\sum_{i=1}^{n}\left | X_{i}-\overline{X} \right |}{n} \)
Higher MAD means greater risk