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# 4 - lecture 3 Descriptive Statistics for Quantitative Data...

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lecture 3, Descriptive Statistics for Quantitative Data II 1 / 32 Mean ( ctd) Geometric Mean Return Measures of Variation Range Interquartile Range Box-and-whiskers plot Variance, Standard Deviation Empirical Rule for Normals z -scores lecture 3, Descriptive Statistics for Quantitative Data II Outline 1 Mean ( ctd) Geometric Mean Return 2 Measures of Variation Range Interquartile Range Box-and-whiskers plot Variance, Standard Deviation 3 Empirical Rule for Normals z -scores

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lecture 3, Descriptive Statistics for Quantitative Data II 2 / 32 Mean ( ctd) Geometric Mean Return Measures of Variation Range Interquartile Range Box-and-whiskers plot Variance, Standard Deviation Empirical Rule for Normals z -scores lecture 3, Descriptive Statistics for Quantitative Data II Mean ( ctd) 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 The ratio of money gained or lost on an investment relative to the amount of money invested: ROR = final value - initial value initial value The higher the ROR, the better.
lecture 3, Descriptive Statistics for Quantitative Data II 3 / 32 Mean ( ctd) Geometric Mean Return Measures of Variation Range Interquartile Range Box-and-whiskers plot Variance, Standard Deviation Empirical Rule for Normals z -scores lecture 3, Descriptive Statistics for Quantitative Data II Mean ( ctd) Geometric Mean Return Example 1, ROR 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? For the first year, ROR R 1 is R 1 = 50 , 000 - 100 , 000 100 , 000 = - 0 . 5 = - 50 % 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.

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lecture 3, Descriptive Statistics for Quantitative Data II 4 / 32 Mean ( ctd) Geometric Mean Return Measures of Variation Range Interquartile Range Box-and-whiskers plot Variance, Standard Deviation Empirical Rule for Normals z -scores lecture 3, Descriptive Statistics for Quantitative Data II Mean ( ctd) Geometric Mean Return Arithmetic / Geometric Mean ROR 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 . 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.
lecture 3, Descriptive Statistics for Quantitative Data II 5 / 32 Mean ( ctd) Geometric Mean Return Measures of Variation Range Interquartile Range Box-and-whiskers plot Variance, Standard Deviation Empirical Rule for Normals z -scores lecture 3, Descriptive Statistics for Quantitative Data II Measures of Variation Example 2, Picking a Stock Suppose you want to choose between two stocks, A and B, and to hold it for one month.

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