READING 3 STATISTICAL MEASURES OF ASSET RETURNS Flashcards
Identify the center, or average, of a dataset.
Measures of central tendency
The sum of the observation values divided by the number of observations.
Arithmetic mean
The sum of all the values in a sample of a population, ΣX, divided by the number of observations in the sample, n.
Sample mean
Midpoint of a dataset.
Median
The median is important because the arithmetic mean can be affected by?
Outliers
The value that occurs most frequently in a dataset.
Mode
When a distribution has one value that appears most frequently, it is said to be?
Unimodal
When a dataset has two or three values that occur most frequently, it is said to be
Bimodal or Trimodal
For continuous data, such as investment returns, we typically do not identify a single outcome as the mode. Instead?
We divide the relevant range of outcomes into intervals, and we identify the modal interval as the one into which the largest number of observations fall
A researcher may decide that outliers should be excluded from a measure of central tendency. What is the technique he should use?
Trimmed mean
For dealing with outliers, instead of discarding the highest and lowest observations, we substitute a value for them. what is this method called?
Winsorized mean
General term for a value at or below which a stated proportion of the data in a distribution lies.
Quantile
The distribution is divided into quarters.
Quartile
The distribution is divided into fifths.
Quintile
The distribution is divided into tenths.
Decile
The distribution is divided into hundredths (percentages).
Percentile
The difference between the third quartile and the first quartile (25th percentile) is known as the?
Interquartile range
The variability around the central tendency.
Dispersion
Relatively simple measure of variability, but when used with other measures, it provides useful information. The distance between the largest and the smallest value in the dataset
The range
Average of the absolute values of the deviations of individual observations from the arithmetic mean
MAD
The measure of dispersion that applies when we are evaluating a sample of n observations from a population.
Sample variance
This systematic underestimation causes the sample variance to be a biased estimator of the population variance.
Using n instead of n - 1 in the denominator
A major problem with using variance is the difficulty of?
Interpreting it. How does one interpret squared percentages, squared dollars, or squared yen? This problem is mitigated through the use of the standard deviation.
Square root of the sample variance
Standard deviation
For instance, suppose you are comparing the annual returns distribution for retail stocks with a mean of 8% and an annual returns distribution for a real estate portfolio with a mean of 16%.
A direct comparison between the dispersion of the two distributions is not meaningful because of the relatively large difference in their means.
To make a meaningful comparison, what measure should you use?
Relative dispersion which is commonly measured with the coefficient of variation (CV)
The amount of variability in a distribution around a reference point or benchmark.
Relative dispersion
Measures the amount of dispersion in a distribution relative to the distribution’s mean.
Coefficient of variation
You have just been presented with a report that indicates that the mean monthly return on T-bills is 0.25% with a standard deviation of 0.36%, and the mean monthly return for the S&P 500 is 1.09% with a standard deviation of 7.30%.
Your manager has asked you to compute the CV for these two investments and to interpret your results.
T-Bills = 0.00036/0.00025 = 1.44
S&P = 0.0730/0.0109 = 6.69
These results indicate that there is less dispersion (risk) per unit of monthly return for T-bills than for the S&P 500 (1.44 vs. 6.70)
A lower CV is better
When we use variance or standard deviation as risk measures, we calculate risk based on outcomes both above and below the mean.
In some situations, it may be more appropriate to consider only outcomes less than the mean (or some other specific value) in calculating a risk measure. In this case, what are you measuring?
Downside risk
One measure of downside risk is?
Target downside deviation, which is also known as target semi deviation.
What refers to the extent to which a distribution is not symmetrical?
Skewness
Observations extraordinarily far from the mean, either above or below
Outliers
Characterized by outliers greater than the mean (in the upper region, or right tail). It is said to be skewed right because of its relatively long upper (right) tail.
A positively skewed distribution
Has a disproportionately large number of outliers less than the mean that fall within its lower (left) tail. It is said to be skewed left because of its long lower tail.
A negatively skewed distribution
For a symmetrical distribution, the mean, median, and mode are?
Equal
Effect of Skewness on Mean, Median, and Mode for a positive skew?
Mean > Median > Mode
Effect of Skewness on Mean, Median, and Mode for a negative skew?
Mean < Median < Mode
Sum of the cubed deviations from the mean divided by the cubed standard deviation and by the number of observations.
Sample skewness
When a distribution is right skewed, sample skewness is?
Positive
When a distribution is left skewed, sample skewness is?
Negative
Measure of the degree to which a distribution is more or less peaked than a normal distribution.
Kurtosis
Describes a distribution that is more peaked than a normal distribution.
Leptokurtic
Refers to a distribution that is less peaked, or flatter than a normal one.
Platykurtic
A distribution is that has the same kurtosis as a normal distribution.
Mesokurtic
A distribution that has either more or less kurtosis than the normal distribution is said to exhibit?
Excess kurtosis
The computed kurtosis for all normal distributions is?
Three
Statisticians, however, sometimes report excess kurtosis, which is defined as?
Kurtosis minus three
Thus, a normal distribution has excess kurtosis equal to?
Zero
A leptokurtic distribution has excess kurtosis greater than?
Zero
Platykurtic distributions will have excess kurtosis less than?
Zero
Approximated using deviations raised to the fourth power
Sample kurtosis
A method for displaying the relationship between two variables.
Scatterplot
Measure of how two variables move together
Covariance
A standardized measure of the linear relationship between two variables is called?
Correlation coefficient or simply correlation
The properties of the correlation of two random variables, X and Y.
If ρXY = 1.0?
The random variables have perfect positive correlation.
The properties of the correlation of two random variables, X and Y.
If ρXY = -1.0?
The random variables have perfect negative correlation.
This means that a movement in one random variable result in an exact opposite proportional movement in the other relative to its mean.
The properties of the correlation of two random variables, X and Y.
If ρXY = 0?
There is no linear relationship between the variables
The variance of returns on Stock A is 0.0028, the variance of returns on Stock B is 0.0124, and their covariance of returns is 0.0058.
Calculate and interpret the correlation of the returns for Stocks A and B
- Square root Stock A for standard deviation, Thus Stock A equals 0.0529
- Square root Stock B for standard deviation, Thus Stock B equals 0.1114
- Multiply both standard deviations
- Correlation equals covariance divided by both standard deviations multiplied equals 0.9842
The fact that this value is close to +1 indicates that the linear relationship is not only positive, but also is very strong
Correlation that is either the result of chance or present due to changes in both variables over time that is caused by their association with a third variable.
Spurious correlation
True or false
Mean is less affected by outliers more than the Median
False
Median is less affected by outliers more than the Mean
