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Chapter 3 A Review of Statistical Principles Useful in Finance. Statistical thinking will one day be as necessary for effective citizenship as the ability to read and write. - H.G. Wells. Outline. Introduction The concept of return Some statistical facts of life. Introduction.

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Chapter 3 a review of statistical principles useful in finance l.jpg

Chapter 3A Review of Statistical Principles Useful in Finance


Slide2 l.jpg

Statistical thinking will one day be as necessary for effective citizenship as the ability to read and write.

- H.G. Wells


Outline l.jpg
Outline effective citizenship as the ability to read and write.

  • Introduction

  • The concept of return

  • Some statistical facts of life


Introduction l.jpg
Introduction effective citizenship as the ability to read and write.

  • Statistical principles are useful in:

    • The theory of finance

    • Understanding how portfolios work

    • Why diversifying portfolios is a good idea


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The Concept of Return effective citizenship as the ability to read and write.

  • Measurable return

  • Expected return

  • Return on investment


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Measurable Return effective citizenship as the ability to read and write.

  • Definition

  • Holding period return

  • Arithmetic mean return

  • Geometric mean return

  • Comparison of arithmetic and geometric mean returns


Definition l.jpg
Definition effective citizenship as the ability to read and write.

  • A general definition of return is the benefit associated with an investment

    • In most cases, return is measurable

    • E.g., a $100 investment at 8%, compounded continuously is worth $108.33 after one year

      • The return is $8.33, or 8.33%


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Holding Period Return effective citizenship as the ability to read and write.

  • The calculation of a holding period return is independent of the passage of time

    • E.g., you buy a bond for $950, receive $80 in interest, and later sell the bond for $980

      • The return is ($80 + $30)/$950 = 11.58%

      • The 11.58% could have been earned over one year or one week


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Arithmetic Mean Return effective citizenship as the ability to read and write.

  • The arithmetic mean return is the arithmetic average of several holding period returns measured over the same holding period:


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Arithmetic Mean Return (cont’d) effective citizenship as the ability to read and write.

  • Arithmetic means are a useful proxy for expected returns

  • Arithmetic means are not especially useful for describing historical returns

    • It is unclear what the number means once it is determined


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Geometric Mean Return effective citizenship as the ability to read and write.

  • The geometric mean return is the nth root of the product of n values:


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Arithmetic and effective citizenship as the ability to read and write. Geometric Mean Returns

Example

Assume the following sample of weekly stock returns:


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Arithmetic and Geometric Mean Returns (cont’d) effective citizenship as the ability to read and write.

Example (cont’d)

What is the arithmetic mean return?

Solution:


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Arithmetic and Geometric Mean Returns (cont’d) effective citizenship as the ability to read and write.

Example (cont’d)

What is the geometric mean return?

Solution:


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Comparison of Arithmetic & effective citizenship as the ability to read and write. Geometric Mean Returns

  • The geometric mean reduces the likelihood of nonsense answers

    • Assume a $100 investment falls by 50% in period 1 and rises by 50% in period 2

    • The investor has $75 at the end of period 2

      • Arithmetic mean = (-50% + 50%)/2 = 0%

      • Geometric mean = (0.50 x 1.50)1/2 –1 = -13.40%


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Comparison of Arithmetic & effective citizenship as the ability to read and write. Geometric Mean Returns

  • The geometric mean must be used to determine the rate of return that equates a present value with a series of future values

  • The greater the dispersion in a series of numbers, the wider the gap between the arithmetic and geometric mean


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Expected Return effective citizenship as the ability to read and write.

  • Expected return refers to the future

    • In finance, what happened in the past is not as important as what happens in the future

    • We can use past information to make estimates about the future


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Return on Investment (ROI) effective citizenship as the ability to read and write.

  • Definition

  • Measuring total risk


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Definition effective citizenship as the ability to read and write.

  • Return on investment (ROI) is a term that must be clearly defined

    • Return on assets (ROA)

    • Return on equity (ROE)

      • ROE is a leveraged version of ROA


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Measuring Total Risk effective citizenship as the ability to read and write.

  • Standard deviation and variance

  • Semi-variance


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Standard Deviation and Variance effective citizenship as the ability to read and write.

  • Standard deviation and variance are the most common measures of total risk

  • They measure the dispersion of a set of observations around the mean observation


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Standard Deviation and Variance (cont’d) effective citizenship as the ability to read and write.

  • General equation for variance:

  • If all outcomes are equally likely:


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Standard Deviation and Variance (cont’d) effective citizenship as the ability to read and write.

  • Equation for standard deviation:


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Semi-Variance effective citizenship as the ability to read and write.

  • Semi-variance considers the dispersion only on the adverse side

    • Ignores all observations greater than the mean

    • Calculates variance using only “bad” returns that are less than average

    • Since risk means “chance of loss” positive dispersion can distort the variance or standard deviation statistic as a measure of risk


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Some Statistical Facts of Life effective citizenship as the ability to read and write.

  • Definitions

  • Properties of random variables

  • Linear regression

  • R squared and standard errors


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Definitions effective citizenship as the ability to read and write.

  • Constants

  • Variables

  • Populations

  • Samples

  • Sample statistics


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Constants effective citizenship as the ability to read and write.

  • A constant is a value that does not change

    • E.g., the number of sides of a cube

    • E.g., the sum of the interior angles of a triangle

  • A constant can be represented by a numeral or by a symbol


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Variables effective citizenship as the ability to read and write.

  • A variable has no fixed value

    • It is useful only when it is considered in the context of other possible values it might assume

  • In finance, variables are called random variables

    • Designated by a tilde

      • E.g.,


Variables cont d l.jpg
Variables (cont’d) effective citizenship as the ability to read and write.

  • Discrete random variables are countable

    • E.g., the number of trout you catch

  • Continuous random variables are measurable

    • E.g., the length of a trout


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Variables (cont’d) effective citizenship as the ability to read and write.

  • Quantitative variables are measured by real numbers

    • E.g., numerical measurement

  • Qualitative variables are categorical

    • E.g., hair color


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Variables (cont’d) effective citizenship as the ability to read and write.

  • Independent variables are measured directly

    • E.g., the height of a box

  • Dependent variables can only be measured once other independent variables are measured

    • E.g., the volume of a box (requires length, width, and height)


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Populations effective citizenship as the ability to read and write.

  • A population is the entire collection of a particular set of random variables

  • The nature of a population is described by its distribution

    • The median of a distribution is the point where half the observations lie on either side

    • The mode is the value in a distribution that occurs most frequently


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Populations (cont’d) effective citizenship as the ability to read and write.

  • A distribution can have skewness

    • There is more dispersion on one side of the distribution

    • Positive skewness means the mean is greater than the median

      • Stock returns are positively skewed

    • Negative skewness means the mean is less than the median


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Populations (cont’d) effective citizenship as the ability to read and write.

Positive Skewness

Negative Skewness


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Populations (cont’d) effective citizenship as the ability to read and write.

  • A binomial distribution contains only two random variables

    • E.g., the toss of a die

  • A finite population is one in which each possible outcome is known

    • E.g., a card drawn from a deck of cards


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Populations (cont’d) effective citizenship as the ability to read and write.

  • An infinite population is one where not all observations can be counted

    • E.g., the microorganisms in a cubic mile of ocean water

  • A univariate population has one variable of interest


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Populations (cont’d) effective citizenship as the ability to read and write.

  • A bivariate population has two variables of interest

    • E.g., weight and size

  • A multivariate population has more than two variables of interest

    • E.g., weight, size, and color


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Samples effective citizenship as the ability to read and write.

  • A sample is any subset of a population

    • E.g., a sample of past monthly stock returns of a particular stock


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Sample Statistics effective citizenship as the ability to read and write.

  • Sample statistics are characteristics of samples

    • A true population statistic is usually unobservable and must be estimated with a sample statistic

      • Expensive

      • Statistically unnecessary


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Properties of effective citizenship as the ability to read and write. Random Variables

  • Example

  • Central tendency

  • Dispersion

  • Logarithms

  • Expectations

  • Correlation and covariance


Example l.jpg
Example effective citizenship as the ability to read and write.

Assume the following monthly stock returns for Stocks A and B:


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Central Tendency effective citizenship as the ability to read and write.

  • Central tendency is what a random variable looks like, on average

  • The usual measure of central tendency is the population’s expected value (the mean)

    • The average value of all elements of the population


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Example (cont’d) effective citizenship as the ability to read and write.

The expected returns for Stocks A and B are:


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Dispersion effective citizenship as the ability to read and write.

  • Investors are interest in the best and the worst in addition to the average

  • A common measure of dispersion is the variance or standard deviation


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Example (cont’d) effective citizenship as the ability to read and write.

The variance ad standard deviationfor Stock A are:


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Example (cont’d) effective citizenship as the ability to read and write.

The variance ad standard deviationfor Stock B are:


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Logarithms effective citizenship as the ability to read and write.

  • Logarithms reduce the impact of extreme values

    • E.g., takeover rumors may cause huge price swings

    • A logreturn is the logarithm of a return

  • Logarithms make other statistical tools more appropriate

    • E.g., linear regression


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Logarithms (cont’d) effective citizenship as the ability to read and write.

  • Using logreturns on stock return distributions:

    • Take the raw returns

    • Convert the raw returns to return relatives

    • Take the natural logarithm of the return relatives


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Expectations effective citizenship as the ability to read and write.

  • The expected value of a constant is a constant:

  • The expected value of a constant times a random variable is the constant times the expected value of the random variable:


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Expectations (cont’d) effective citizenship as the ability to read and write.

  • The expected value of a combination of random variables is equal to the sum of the expected value of each element of the combination:


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Correlations and Covariance effective citizenship as the ability to read and write.

  • Correlation is the degree of association between two variables

  • Covariance is the product moment of two random variables about their means

  • Correlation and covariance are related and generally measure the same phenomenon


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Correlations and Covariance (cont’d) effective citizenship as the ability to read and write.


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Example (cont’d) effective citizenship as the ability to read and write.

The covariance and correlation for Stocks A and B are:


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Correlations and Covariance effective citizenship as the ability to read and write.

  • Correlation ranges from –1.0 to +1.0.

    • Two random variables that are perfectly positively correlated have a correlation coefficient of +1.0

    • Two random variables that are perfectly negatively correlated have a correlation coefficient of –1.0


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Linear Regression effective citizenship as the ability to read and write.

  • Linear regression is a mathematical technique used to predict the value of one variable from a series of values of other variables

    • E.g., predict the return of an individual stock using a stock market index

  • Regression finds the equation of a line through the points that gives the best possible fit


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Linear Regression (cont’d) effective citizenship as the ability to read and write.

Example

Assume the following sample of weekly stock and stock index returns:


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Linear Regression (cont’d) effective citizenship as the ability to read and write.

Example (cont’d)

Intercept = 0

Slope = 0.96

R squared = 0.99


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R Squared and effective citizenship as the ability to read and write. Standard Errors

  • Application

  • R squared

  • Standard Errors


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Application effective citizenship as the ability to read and write.

  • R-squared and the standard error are used to assess the accuracy of calculated statistics


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R Squared effective citizenship as the ability to read and write.

  • R squared is a measure of how good a fit we get with the regression line

    • If every data point lies exactly on the line, R squared is 100%

  • R squared is the square of the correlation coefficient between the security returns and the market returns

    • It measures the portion of a security’s variability that is due to the market variability


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Standard Errors effective citizenship as the ability to read and write.

  • The standard error is the standard deviation divided by the square root of the number of observations:


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Standard Errors (cont’d) effective citizenship as the ability to read and write.

  • The standard error enables us to determine the likelihood that the coefficient is statistically different from zero

    • About 68% of the elements of the distribution lie within one standard error of the mean

    • About 95% lie within 1.96 standard errors

    • About 99% lie within 3.00 standard errors