Statistical Concepts and Market Returns
Statistics : What is it ?
Measures of Central Tendency
Measures of central tendency compute the middle point or the average of a data set. This middle point or average then represents the typical value to the expected in the data set. Measures of central tendency are mean, median and mode. We will examine each one of them individually beginning with mean;
Arithmetic mean is the sum of the observation values in the data set divided by the number of observations in the data set. There are two types of arithmetic means; population mean and sample mean.
Population mean is the sum of the observations values in the population divided by the number of observations in the population. A population mean is unique, meaning a population can have only one mean.
Sample mean is the sum of the observations values in the sample divided by the number of observations in the sample. Sample mean is also unique. Sample mean is used to draw inferences about the population mean. Sample mean is the best estimate of the true mean of the sample.
Arithmetic mean is the most popular measure of central tendency. It has the following properties;
- A data set can have only one arithmetic mean
- Arithmetic mean considers all the values in the data set.
- The sum of the deviations of each observation from the mean is always zero.
Drawback of the arithmetic mean is that inclusion of a few unusually large or small values in the data set can disproportionately alter the mean. For example, if the values in the data set are 1,2,3,4,5 and 50, the mean turns out to be 10.83 i.e. approximately 11, which is not at all representative of majority of the values in the set.
Computation of the weighted mean recognizes the fact that inclusion of a few unusually large or small values in the data set can disproportionately alter the mean. Weighted mean is calculated by multiplying each observation value with its weight and summing them all up. The formula is given as;
W1 X1 + W2X2 +………….+WnXn
Weights are the proportionate weight of a value in the set. For example, if a portfolio has 50% stocks, 40% bonds, 10% cash and the return on stocks is 12%, on bonds is 8% and on cash is 2%, the weighted average return of the portfolio can be computed as,
0.50 *0.12 + 0.40*0.08 + 0.10 *0.02 = 0.09 = 9%.
Even though the return on cash is a small value compared to the returns on the other two asset classes, the weighted mean is representative of the majority of the return values in the portfolio.
Median is the middle value when the observation values in a data set are arranged in ascending or descending order. Half the values lie below the median and the other half above it. For example, if a data set has values 3,1,4,2 and 5, then we can find the median by first arranging the values in ascending or descending order as 1,2,3,4,5. Now we know the Median is 3.
Again in a set having even number of observations the median is the average or arithmetic mean of the middle two values. For example, if we add one more observation to our set so that it becomes 1,2,3,4,5,6, then the median is 3+4/2 = 3.50.
Median is a more useful measure of central tendency than the arithmetic mean in the event the mean is distorted due to the presence of a few unusually large or small values in the data set.
Mode is the value that occurs most frequently in a dataset. A data set can have more than one modes or no mode at all. If it has one mode, it is called unimodal. If it has 2 or 3 modes, it is called bimodal or trimodal.
For example, in a data set having values 3,4,5,3,6,7,3, the mode is 3.
Sharpe ratio is given as ;
A leptokurtic distribution tends to have more values that either clustered around the mean or are far away from the mean. Therefore leptokurtic distributions are highly peaked in the centre with fatter tails compared to a normal distribution and mesokurtic distribution.
Platykurtic and Mesokurtic Distribution
In quantitative analysis risk managers tend to focus on the tails because that’s where the risk is. Therefore, a distribution with positive excess kurtosis and negative skew is an indication of high risk.