STATISTICS Types of Data Data can be presented in many ways. The following four data descriptions are in order of least to most in-depth. [Tip: remember with the acronym “NOIR”] -
Normal; Ordinal; Interval; Ratio;
discrete data placed in no particular order. the data is placed in order. ordinal data which is also grouped into equally sized groups. interval data which also has a natural zero point.
Means Below are four common ways of taking a mean average… - Arithmetic: (single period returns in future period) - Geometric: (future compound returns over multiple periods) - Weighted: - Harmonic: Variances Three common variance formulae… - Mean Arithmetic Deviation: - Variance: - Sample Variance: Other Key Formulae Some other key statistical formulae we’ll need to know… - Chebyshev’s Inequality: - Coefficient of Variation: (sigma, σ, is equal to the standard deviation) - Sharpe Ratio: (higher value => lower risk => better investment)
Kurtosis Remember, the kurtosis of a normal distribution is 3. - Positive Kurtosis (clustered x-axis => higher peak)… “Leptokurtic” - Negative Kurtosis (dispersed x-axis => lower peak)… “Platykurtic” The Median, Quartiles and Interpolation The median is the 50th percentile and a special case of the below formula where y = 50. -
Quartile:
Example: to find the third quintile value (i.e. value that comes 60% along the values when placed in order) of 13 items… (n+1)(y)/100 = ((13)+1)(60)/100 = 8.4th item. Money Weighted Return (MWR) Steps for calculation… 1. The cash flows are netted for each period, 2. MWR = IRR (Internal Rate of Return). Time Weighted Return (TWR) Steps for calculation… 1. Find the IRR for each period, 2. Take the compounded mean (geometric mean) of the results. Example: if in year 1 the IRR is found to be +31.0% and then in year 2 -1.8% then... TWR = [((1+(31%)) x (1+(-1.8%)) ^ 0.5 ]-1 = 13.4% Larger Threshold Rate Roy’s Safety First Criterion < Sharpe Ratio