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# Lect3 - Mean(ctd Measures of Variation Empirical Rule for...

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Mean ( ctd) Measures of Variation Empirical Rule for Normals Outline Mean ( ctd) Geometric Mean Return Measures of Variation Range Interquartile Range Box-and-whiskers plot Variance, Standard Deviation Empirical Rule for Normals z -scores 1 / 32 Xinghua Zheng Quantitative Data Description: II

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Mean ( ctd) Measures of Variation Empirical Rule for Normals Geometric Mean Return Rate of Return In finance, a quantity of big interest is the so-called rate of return (ROR) , or return on investment (ROI); or sometimes just return I The ratio of money gained or lost on an investment relative to the amount of money invested: ROR = final value - initial value initial value I The higher the ROR, the better. 2 / 32 Xinghua Zheng Quantitative Data Description: II
Mean ( ctd) Measures of Variation Empirical Rule for Normals Geometric Mean Return Example 1, Returns An investment of \$100,000 declined to \$50,000 at the end of Year 1 and rebounded to \$100,000 at the end of Year 2. What is the ROR for each year? 1. For the first year, ROR R 1 is R 1 = 50 , 000 - 100 , 000 100 , 000 = - 0 . 5 = - 50 % 2. For the second year, R 2 is R 2 = 100 , 000 - 50 , 000 50 , 000 = 1 = 100 % The overall two-year return is , since it started and ended at the same level. 3 / 32 Xinghua Zheng Quantitative Data Description: II

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Mean ( ctd) Measures of Variation Empirical Rule for Normals Geometric Mean Return Arithmetic and Geometric Mean ROR I The (arithmetic) mean ROR over Year 1 and Year 2 is - 50 %+ 100 % 2 = 25 % . Misleading! Geometric mean ROR : the constant return that yields the same wealth at the end of the whole investment period as do the actual return: ¯ R G = (( 1 + R 1 ) × . . . ( 1 + R n )) 1 / n - 1 ; where R i is the ROR in time period i . I For Example 1, the geometric mean ROR over the two years is ¯ R G = (( 1 + R 1 ) × ( 1 + R 2 )) 1 / 2 - 1 = (( 1 - 0 . 5 ) × ( 1 + 1 )) 1 / 2 - 1 = 0 . Correctly expresses the fact that the value of the investment is unchanged after two years. 4 / 32 Xinghua Zheng Quantitative Data Description: II
Mean ( ctd) Measures of Variation Empirical Rule for Normals Range Interquartile Range Box-and-whiskers plot Variance, S Example 2, Picking a Stock Suppose you want to choose between two stocks, A and B, and to hold it for one month. During the last 5 years, their respective monthly RORs are distributed like Monthly ROR (in %) of Stock A -8 -6 -4 -2 0 2 4 6 0.00 0.05 0.10 0.15 Monthly ROR (in %) of Stock B -6 -4 -2 0 2 4 6 8 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 I Which stock would you pick? 5 / 32 Xinghua Zheng Quantitative Data Description: II

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Mean ( ctd) Measures of Variation Empirical Rule for Normals Range Interquartile Range Box-and-whiskers plot Variance, S Example 2, Picking a Stock ( ctd) Monthly ROR (in %) of Stock A -8 -6 -4 -2 0 2 4 6 0.00 0.05 0.10 0.15 Monthly ROR (in %) of Stock B -6 -4 -2 0 2 4 6 8 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 I RORs of Stock A and B are around - 1% and 1% resp. I RORs of A and B spreadout in similar magnitude I The magnitude of spreadout is a measure of risk I Choose Stock B over Stock A since it has higher expected ROR but similar risk 6 / 32 Xinghua Zheng Quantitative Data Description: II
Mean ( ctd) Measures of Variation Empirical Rule for Normals Range Interquartile Range Box-and-whiskers plot Variance, S Example 2, Picking a Stock ( ctd) Now suppose there’s another stock, C. During the last 5

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