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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 MAD = mean absolute deviation 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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