1. 1 Day 1 Quantitative Methods for Investment Management by Binam Ghimire

2. 2 Statistical Concepts and market returns and Probability Concepts  Identify measures of central tendency and measures of Dispersion  Understand that measures of central tendency give an indication of the expected return of an investment and measures of dispersion measure riskiness of an investment  Use of Excel on the topic Objective

3. Basic Concept Statistics  Descriptive statistics  Inferential statistics  Population  Parameter  Sample  Statistics 3

4. Basic Concept Variable Measurement Scale  Variable Scale  Nominal  Ordinal  Interval  Ratio  Guides what type of test we need to perform 4 Less Informative More Informative

5. Descriptive Statistics: Histogram and Frequency Polygons  Histogram: Grouped data. The area of each rectangle is proportion to the frequency  Frequency Polygon: a line graph drawn by joining all the midpoints of the top of the bars of a histogram  Activity: Excel – Histogram and Frequency Polygon 5

6. Measures of location - Averages  Meaning & Calculation  Mean: Arithmetic, Weighted and Geometric  Mode  Median  Formula  Activity: Football Game 6

7. Weighted Mean as Portfolio Return  Weighted Mean is useful to find return of a portfolio  Return of Portfolio is basically  (W1xR1) + (W2xR2) + (W3xR3) … (WnxRn) where W is weight and R is return 7

8. Weighted Mean as a Portfolio Return 8  Example: ActualPortfolio ReturnWeight Cash5% ×0.10= 0.5% Bonds7% ×0.35= 2.45% Stocks12% ×0.55 = 6.6% Σ = 9.55% Same method works for expected portfolio returns!

9. Geometric Mean 9  Geometric mean is used to calculate compound growth rates  If the returns are constant over time, geometric mean equals arithmetic mean  The greater the variability of returns over time, the more the arithmetic mean will exceed the geometric mean Actually, the compound rate of return is the geometric mean of the price relatives, minus 1

10. Geometric Mean: Example 10 An investment account had returns of 15.0%, –9.0%, and 13.0% over each of three years Calculate the time-weighted annual rate of return = 5.75 %

11. Measures of location  Meaning and Calculation  Maximum  Minimum  Quantile: Quantile is a method for dividing a range of numeric values into categories  Quartile, Percentiles, Deciles  75% of the data points are less than the 3 rd quartile  60% of the data points are less than the 6 th decile  50% of the data points are less than the 50 th percentile  Formula  Activity: Football Game 11

12. Measures of Dispersion  Meaning and Calculation  Range  Inter-quartile range  Semi-interquartile range  Mean Absolute Deviation  Variance  Standard Deviation  Formula  Activity: Football Game 12

13. Measures of Association  Meaning  Co-variance  Formula:  Calculation 13

14. Measures of Association: Covariance  Co-variance has a sign  Covariance = 10 14

15. Measures of Association: Covariance  Co-variance has a sign  Covariance = -10 15

16. Measures of Association: Covariance  Co-variance has a sign  Covariance = 6.94 16

17. Measures of Association: Covariance  Co-variance has a sign  Covariance = -7.49 17

18. Measures of Association: Covariance in Investment Management  For example, if two stock prices tend to rise and fall at the same time, these stocks would not deliver the best diversified earnings. 18

19. Measures of Distributions  Distribution Shape  Skewness  Kurtosis 19

20. Measures of Distributions: Skewness  Concept:  Skewness characterizes the degree of asymmetry of a distribution around its mean  Positive skewness indicates a distribution with an asymmetric tail extending toward more positive values  Negative skewness indicates a distribution with an asymmetric tail extending toward more negative values  No Skewness: symmetrical 20

21. Measures of Distribution Positive Skewness  Skewness = 0.45  Tail to the higher values. Mean > Median > Mode  Exercise in Excel 21

22. Measures of Distribution : Negative Skewness  Skewness = - 0.45  Tail to the lower. Mean < Median < Mode  Exercise in Excel 22

23. Measures of Distribution : No Skewness  Skewness = 0  Tail to the lower. Mean = Median = Mode (Symmetrical/ Normal)  Exercise in Excel 23

24. Measures of Distribution Kurtosis  Concept  Kurtosis characterizes the relative peakedness or flatness of a distribution compared with the normal distribution  Positive kurtosis indicates a relatively peaked distribution  Negative kurtosis indicates a relatively flat distribution  No or zero Kurtosis = normal distribution 24

25. Measures of Distribution Positive Kurtosis  Kurtosis = 1.68  Positive Kurtosis: Peaked relative to the Normal  Exercise in Excel 25

26. Measures of Distribution Negative Kurtosis  Kurtosis = - 0.34  Negative Kurtosis: Flat relative to the Normal  Zero Kurtosis: Peak similar to Normal Distribution  Exercise in Excel 26

27. Kurtosis: Other names  A distribution with a high peak is called leptokurtic (Kurtosis > 0), a flat-topped curve is called platykurtic (Kurtosis < 0), and the normal distribution is called mesokurtic (Kurtosis = 0) 27

28. Semivariance  Semivariance is calculated by only including those observations that fall below the mean on the calculation.  Sometimes described as “downside risk” with respect to investments.  Useful for skewed distributions, as it provides additional information that the variance does not.  Target semivariance is similar but based on observations below a certain value, e.g values below a return of 5%. 28

29. Coefficient of Variance (CV)  Coefficient of Variance (CV) = standard deviation mean  In investments for example; CV measures the risk (variability) per unit of expected return (mean). 29

30. CV  Example: Suppose you wish to calculate the CV for two investments, the monthly return on British T-Bills and the monthly return for the S&P 500, where: mean monthly return on T-Bills is 0.25% with SD of 0.36%, and the mean monthly return for the S&P 500 is 1.09%, with a SD of 7.30%. 30

31. CV  CV (T-Bills) = 0.36/0.25 = 1.44  CV (S&P 500) = 7.30/1.09 = 6.70 31

32. CV  Interpretation: CV is the variation per unit of return, indicating that these results indicate that there is less dispersion (risk) per unit of monthly returns for T-Bills than there is for the S&P 500, i.e. 1.44 vs 6.70. 32

33. We now should know the followings  Concept, Formula and Calculation  Mean  Median  Quartiles  Percentile  Range  Interquartile and semi-interquartile Range  Mean Deviation  Variance, Semi Variance  Standard Deviation  Covariance, Coefficient of Variance  Use of Excel for the above and Skewness and Kurtosis 33

34. Can we solve the following?  An investor holds a portfolio consisting of one share of each of the following stocks:  For the 1-year holding period, the portfolio return is closest to: a) 6.88% b) 9.13% c) 13.13% and, d) 19.38%  Now practice Examples Day 1 (Some questions require knowledge from other chapters) 34