Statistical Measures of Asset Returns
Sample Mean (Arithmetic Mean)
\[ \bar{X} = \frac{\sum_{i=1}^{n} X_i}{n} \tag{1} \]
Where:
- \(X_i\): value of observation \(i\)
- \(\bar{X}\): sample mean
- \(n\): number of observations in the sample
View Markdown Source
## Sample Mean (Arithmetic Mean)
$$
\bar{X} = \frac{\sum_{i=1}^{n} X_i}{n} \tag{1}
$$
Where:
* $X_i$: value of observation $i$
* $\bar{X}$: sample mean
* $n$: number of observations in the sampleRange
\[ \text{Range = Maximum value − Minimum value} \tag{2} \]
Where:
- \(\text{Maximum value}\): largest observation in the dataset
- \(\text{Minimum value}\): smallest observation in the dataset
View Markdown Source
## Range
$$
\text{Range = Maximum value − Minimum value} \tag{2}
$$
Where:
* $\text{Maximum value}$: largest observation in the dataset
* $\text{Minimum value}$: smallest observation in the datasetMean Absolute Deviation (MAD)
\[ \text{MAD} = \frac{\sum_{i=1}^{n} |X_i - \bar{X}|}{n} \tag{3} \]
Where:
- \(X_i\): value of observation \(i\)
- \(\bar{X}\): sample mean
- \(n\): number of observations in the sample
- \(| ... |\): indicate the absolute value of what is contained within these bars
View Markdown Source
## Mean Absolute Deviation (MAD)
$$
\text{MAD} = \frac{\sum_{i=1}^{n} |X_i - \bar{X}|}{n} \tag{3}
$$
Where:
* $X_i$: value of observation $i$
* $\bar{X}$: sample mean
* $n$: number of observations in the sample
* $| ... |$: indicate the absolute value of what is contained within these barsSample Variance
\[ s^2 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n - 1} \tag{4} \]
Where:
- \(s^2\): sample variance
- \(X_i\): value of observation \(i\)
- \(\bar{X}\): sample mean
- \(n\): number of observations
View Markdown Source
## Sample Variance
$$
s^2 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n - 1} \tag{4}
$$
Where:
* $s^2$: sample variance
* $X_i$: value of observation $i$
* $\bar{X}$: sample mean
* $n$: number of observationsSample Standard Deviation
\[ s = \sqrt{\frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n - 1}} \tag{5} \]
Where:
- \(s\): sample standard deviation
- \(X_i\): value of observation \(i\)
- \(\bar{X}\): sample mean
- \(n\): number of observations
View Markdown Source
## Sample Standard Deviation
$$
s = \sqrt{\frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n - 1}} \tag{5}
$$
Where:
* $s$: sample standard deviation
* $X_i$: value of observation $i$
* $\bar{X}$: sample mean
* $n$: number of observationsSample Target Semideviation Formula
\[ S_{\text{target}} = \sqrt{ \sum\nolimits_{\text{for all} X_i \le B}^{n} \frac{(X_i - B)^2}{n - 1} } \tag{6} \]
Where:
- \(S_{\text{target}}\): target semideviation
- \(X_i\): value of observation \(i\)
- \(B\): target return
- \(n\): total number of observations
View Markdown Source
## Sample Target Semideviation Formula
$$
S_{\text{target}}
=
\sqrt{
\sum\nolimits_{\text{for all} X_i \le B}^{n}
\frac{(X_i - B)^2}{n - 1}
} \tag{6}
$$
Where:
* $S_{\text{target}}$: target semideviation
* $X_i$: value of observation $i$
* $B$: target return
* $n$: total number of observationsCoefficient of Variation (CV)
\[ CV = \frac{s}{\bar{X}} \tag{7} \]
Where:
- \(CV\): coefficient of variation
- \(s\): sample standard deviation
- \(\bar{X}\): sample mean
View Markdown Source
## Coefficient of Variation (CV)
$$
CV = \frac{s}{\bar{X}} \tag{7}
$$
Where:
* $CV$: coefficient of variation
* $s$: sample standard deviation
* $\bar{X}$: sample meanSample Skewness (Approximation)
\[ \text{Skewness} \approx \left(\frac{1}{n}\right)\frac{\sum_{i=1}^{n} (X_i - \bar{X})^3}{s^3} \tag{8} \]
Where:
- \(\text{Skewness}\): is computed as
- the average cubed deviation from the mean,
- standardized by dividing by the standard deviation cubed
- to make the measure free of scale.
- \(X_i\): value of observation \(i\)
- \(\bar{X}\): sample mean
- \(s\): sample standard deviation
- \(n\): number of observations (100 or more)
View Markdown Source
## Sample Skewness (Approximation)
$$
\text{Skewness} \approx \left(\frac{1}{n}\right)\frac{\sum_{i=1}^{n} (X_i - \bar{X})^3}{s^3} \tag{8}
$$
Where:
* $\text{Skewness}$: is computed as
* the average cubed deviation from the mean,
* standardized by dividing by the standard deviation cubed
* to make the measure free of scale.
* $X_i$: value of observation $i$
* $\bar{X}$: sample mean
* $s$: sample standard deviation
* $n$: number of observations (100 or more)Sample Excess Kurtosis (Approximation)
\[ K_E \approx \left(\frac{1}{n}\right)\frac{\sum_{i=1}^{n} (X_i - \bar{X})^4}{s^4} - 3 \tag{9} \]
Where:
- \(K_E\): sample excess kurtosis
- \(X_i\): value of observation \(i\)
- \(\bar{X}\): sample mean
- \(s\): sample standard deviation
- \(n\): number of observations (100 or more)
View Markdown Source
## Sample Excess Kurtosis (Approximation)
$$
K_E \approx \left(\frac{1}{n}\right)\frac{\sum_{i=1}^{n} (X_i - \bar{X})^4}{s^4} - 3 \tag{9}
$$
Where:
* $K_E$: sample excess kurtosis
* $X_i$: value of observation $i$
* $\bar{X}$: sample mean
* $s$: sample standard deviation
* $n$: number of observations (100 or more)Sample Covariance
\[ s_{XY} = \frac{\sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})}{n - 1} \tag{10} \]
Where:
- \(s_{XY}\): sample covariance between \(X\) and \(Y\)
- \(X_i\): observation \(i\) of variable \(X\)
- \(Y_i\): observation \(i\) of variable \(Y\)
- \(\bar{X}\): sample mean of \(X\)
- \(\bar{Y}\): sample mean of \(Y\)
- \(n-1\): ensures that the sample covariance is an unbiased estimate of population covariance.
- \(n\): number of paired observations
View Markdown Source
## Sample Covariance
$$
s_{XY} = \frac{\sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})}{n - 1} \tag{10}
$$
Where:
* $s_{XY}$: sample covariance between $X$ and $Y$
* $X_i$: observation $i$ of variable $X$
* $Y_i$: observation $i$ of variable $Y$
* $\bar{X}$: sample mean of $X$
* $\bar{Y}$: sample mean of $Y$
* $n-1$: ensures that the sample covariance is an unbiased estimate of population covariance.
* $n$: number of paired observationsSample Correlation Coefficient
\[ r_{XY} = \frac{s_{XY}}{s_X s_Y} \tag{11} \]
Where:
- \(r_{XY}\): sample correlation coefficient
- \(s_{XY}\): sample covariance between \(X\) and \(Y\)
- \(s_X\): sample standard deviation of \(X\)
- \(s_Y\): sample standard deviation of \(Y\)
View Markdown Source
## Sample Correlation Coefficient
$$
r_{XY} = \frac{s_{XY}}{s_X s_Y} \tag{11}
$$
Where:
* $r_{XY}$: sample correlation coefficient
* $s_{XY}$: sample covariance between $X$ and $Y$
* $s_X$: sample standard deviation of $X$
* $s_Y$: sample standard deviation of $Y$