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measures of dispersion

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```									Measures of Dispersion

These measure the dispersion or variability of the data, and therefore indicate whether the
mean is a reliable average.

1. Range :

The range is the difference between the highest and the lowest value.

2. Mean Absolute Deviation

This is the mean of the absolute value of the deviations from the mean.

(8)

where

X = mean

Xi = ith observation

|Xi - X| = absolute value of the difference between Xi and X, it is always zero or a positive
number

n = number of observations

Example 9: Mean absolute deviation

The following observations are recorded: 2, 5, 6, 8, 9.

The range is 9 - 2 = 7

The MAD =[ | (2 - 6) | + | (5 - 6) | + | (6 - 6) | + | (8 - 6) | + | (9 - 6) | ]/5 = 2

3. Variance

Variance is calculated using the squares of the deviations from the mean, so whether a
difference from the mean is positive or negative by the time it is squared it is positive. This
avoids the problem of positive and negative deviations canceling out. Variance and standard
deviation are the most common risk measures used in investment.

Variance of a population is calculated as:

(9)
where

σ2 = variance of a population

μ = population mean

Xi = ith observation

N = number of observations in the population

4. Standard Deviation

This is the square root of the variance, and it is therefore in the same units as the original
data. Standard deviation of a population is:

(10)

where

σ = Standard deviation of a population

μ = population mean

Xi = ith observation

N = number of observations in the population

Example 10: Variance and standard deviation

The ages of all the children in a violin lesson are 9,11,12,13, and 15.

First calculate μ = 60/5 = 12

The population variance is calculated as:

σ2 = [(9 - 12)2 + (11 - 12)2 + (12 - 12)2 + (13 - 12)2 + (15 - 12)2]/5 = 4

The standard deviation = σ = 2

The larger the standard deviation or variance of a distribution is, the wider the dispersion of
the observations away from the mean.

When the data is for a sample, rather than the whole population, then the sample variance is
given by the equation below:

(11)
where

S2 = variance of a sample

X = sample mean

Xi = ith observation

n = number of observations in the sample

The denominator in Equation 11 is now (n - 1), using n would tend to underestimate the
variance if we are using a sample, rather than the whole, of the population.

As before, the standard deviation of the sample is the square root of the sample variance.

It can be argued that most investors are only concerned with downside risk, or the risk of
returns falling below the mean. This is measured by semivariance which is the average
squared deviation below the mean. The formula for semivariance is given in Equation 12, and
semideviation can be calculated by taking the square root of the semivariance.

(12)

where

Ssemi2 = semivariance of a sample

X = sample mean

Xi = ith observation

n* = number of observations that are smaller than the mean

Example 11: Semivariance

The returns from a portfolio over six months are -10%,-6%, + 5%, +6%, +8% and +9%.

The sample mean is (-10% - 6% + 5% + 6% +8% + 9%)/6 = 2%.

Using Equation 11 variance of the sample is

To calculate semivariance, note that there are only 2 returns below the mean. Using Equation
12, semivariance is
This is significantly higher than the variance since the lower readings are well below the
means showing relatively high downside risk.

Target semivariance refers to when there is a target return and the target semivariance is
calculated by taking the sum of the squared deviations of observations below the target,
divided by the number of such observations minus one.

using Chebyshev's inequality.

Chebyshev's Inequality

For both discrete and continuous data distributions Chebyshev's Inequality states that:

For any set of observations at least 1-1/k2 percent of readings fall within k standard
deviations of the mean, when k >1.

Coefficient of Variation

A measure is needed to compare the dispersions of two or more distributions. The coefficient
of variation is the standard deviation divided by the mean of a distribution, often expressed as
a percentage. The higher the coefficient of variation, the higher the dispersion and vice versa.

(13)

where

CV = coefficient of variation, as a percentage

s = standard deviation

X = sample mean

Example 12: Coefficient of variation

Data is collected on (i) the monthly performance of a mutual fund, the sample mean is 2%
and the standard deviation is 0.5% (ii) the monthly performance of a pension fund, the
sample mean is 1.2% and the standard deviation is 0.4%.

The coefficients of variation are calculated using Equation 8-13

1. CV(mutual fund)= (0.5%/2%) × 100 = 25%
2. CV(pension Fund) = (0.4%/1.2%) × 100 = 33.3%

This means that there is more dispersion relative to the mean in the distribution of the returns
of the pension fund than the mutual fund.

An alternative measure is the Sharpe ratio, which is used to measure excess returns for each
unit of risk taken.

(14)
where

rp = mean return of a portfolio p

rf = mean return from the risk free asset

SP = standard deviation of a portfolio p

Example 14: Sharpe ratio

A portfolio is providing a mean return of 12% per annum compared with a return of 5% per
annum from the risk-free asset, the standard deviation of the portfolio returns is 10%.

Sharpe ratio = (12% - 5%)/10% = 0.7

This indicates the portfolio earned a 0.7% return for each unit of risk; a high Sharpe ratio is
more attractive than a low one.

Symmetric and Skewed Distributions

A symmetric or normal distribution has the following characteristics:

1. The mean and median are equal.
2. The mean and variance completely describe the distribution.
3. 68.3% of observations lie between (mean ± 1 standard deviation)

95.5% of observations lie between (mean ± 2 standard deviations)

99.7% of observations lie between (mean ± 3 standard deviations)

A nonsymmetrical distribution is skewed
In a positively skewed distribution the mean will be higher than the median, which will be
higher than the mode; the opposite will be the case for a negatively skewed distribution.

If a distribution of portfolio returns is positively skewed it indicates that poor returns occur
frequently but losses are small, whereas very high returns occur less frequently but are more
extreme.

Skewness is calculated using the cubes of the deviations, thereby keeping the 'direction' of the
deviations i.e. whether the observations are above or below the mean. When interpreting
investment returns positive skewness is considered attractive since it indicates that there is a
greater probability of very high returns.
Kurtosis

Kurtosis measures whether a distribution is more peaked (leptokurtic) or less peaked
(platykurtic) than a normal distribution.

Excess positive kurtosis, or leptokurtosis, would indicate that a distribution has fatter tails
than a normal distribution. This is very important since, if a distribution of stock returns
exhibits excess kurtosis (a sample excess kurtosis of 1.0 or larger would be considered
unusually large) the probability of very good or very bad returns is more likely than if it were
a normal distribution. If measures such as Value at Risk (VaR) are used to estimate the
probability of a specified loss occurring, and the distribution has been assumed to be normal,
then the chance of a major loss will have been underestimated.

Semi-Logarithmic Scale

When looking at graphs to assess historic returns, semi-logarithmic scales are often the most
appropriate. A semi-logarithmic scale is logarithmic on the vertical axis and arithmetic on the
horizontal axis. On a logarithmic scale equal distances represent equal percentage
movements.

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