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Day 1Quantitative Methods for Investment

Managementby Binam Ghimire

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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

Basic ConceptStatistics

Descriptive statisticsInferential statisticsPopulation

ParameterSample

Statistics

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Basic ConceptVariable Measurement Scale

Variable ScaleNominalOrdinalIntervalRatio

Guides what type of test we need to perform

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Less Informative

More Informative

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

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Measures of location - Averages

Meaning & CalculationMean: Arithmetic, Weighted and GeometricModeMedian

Formula Activity: Football Game

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Weighted Mean as Portfolio Return

Weighted Mean is useful to find return of a portfolioReturn of Portfolio is basically (W1xR1) + (W2xR2) + (W3xR3) … (WnxRn)

where W is weight and R is return

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Weighted Mean as a Portfolio Return

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Example:Actual Portfolio

Return Weight Cash 5% × 0.10 =

0.5% Bonds 7% ×0.35 = 2.45%

Stocks 12% × 0.55 = 6.6% Σ =

9.55%Same method works for expected portfolio returns!

Geometric Mean

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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

1)]R)...x(1R)x(1R[(1R 1/nn21Geom

Geometric Mean: Example

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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 %1)]R)...x(1R)x(1R[(1R 1/n

n21Geom

Measures of location Meaning and Calculation

MaximumMinimumQuantile: Quantile is a method for dividing a

range of numeric values into categoriesQuartile, Percentiles, Deciles

75% of the data points are less than the 3rd quartile

60% of the data points are less than the 6th decile

50% of the data points are less than the 50th percentile

FormulaActivity: Football Game 11

Measures of Dispersion

Meaning and CalculationRangeInter-quartile rangeSemi-interquartile rangeMean Absolute Deviation VarianceStandard Deviation

Formula Activity: Football Game

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Measures of Association

MeaningCo-varianceFormula:

Calculation

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Covariance = sXY = S(X - X)(Y - Y)n

Measures of Association:Covariance

Co-variance has a sign

Covariance = 10

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X values

Y va

lues

X Y12 2014 2416 2818 32

Measures of Association:Covariance

Co-variance has a sign

Covariance = -10

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X Y12 3214 2816 2418 20

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X values

Y va

lues

Measures of Association:Covariance

Co-variance has a sign

Covariance = 6.94

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X values

Y va

lues

X Y12 2015 2518 2814 2216 2619 3015 23

Measures of Association:Covariance

Co-variance has a sign

Covariance = -7.49

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X Y12 3017 2218 1714 2616 2619 2115 23

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X values

Y va

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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.

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Measures of Distributions

Distribution ShapeSkewnessKurtosis

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Measures of Distributions:Skewness

Concept: Skewness characterizes the degree of

asymmetry of a distribution around its meanPositive 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

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Measures of DistributionPositive Skewness

Skewness = 0.45

Tail to the higher values. Mean > Median > Mode Exercise in Excel

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Frequency Distribution

X values

Freq

uenc

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Measures of Distribution :Negative Skewness

Skewness = - 0.45

Tail to the lower. Mean < Median < Mode Exercise in Excel

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0123

4567

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Frequency Distribution

X values

Freq

uenc

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Measures of Distribution :No Skewness

Skewness = 0

Tail to the lower. Mean = Median = Mode (Symmetrical/ Normal)

Exercise in Excel 23

0123456789

Frequency Distribution

X values

Freq

uenc

ies

Measures of DistributionKurtosis

ConceptKurtosis 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

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Measures of Distribution Positive Kurtosis

Kurtosis = 1.68

Positive Kurtosis: Peaked relative to the Normal Exercise in Excel

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02468

1012141618

Frequency Distribution

X values

Freq

uenc

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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

0123456789

Frequency Distribution

X values

Freq

uenc

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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)

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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%.

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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).

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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%.

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CV

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

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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.

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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

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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)

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Stock Price at the beginning of the

year

Price at the end of the year

Cash dividend during the year

X £20 £10 £0 Y £40 £50 £2 Z £100 £105 £4