Risk Return

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  • 1. LECTURE 1 Risk, Return, and Uncertainty: A Review of Principles Useful in Finance
  • 2. The Concept of Return
    • Measurable return
    • Expected return
    • Return on investment
  • 3. Measurable Return
    • Definition
    • Holding period return
    • Arithmetic mean return
    • Geometric mean return
    • Comparison of arithmetic and geometric mean returns
  • 4. Definition
    • 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%
  • 5. Holding Period Return
    • 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
  • 6. Arithmetic Mean Return
    • The arithmetic mean return is the arithmetic average of several holding period returns measured over the same holding period:
  • 7. Arithmetic Mean Return (cont’d)
    • 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
  • 8. Geometric Mean Return
    • The geometric mean return is the n th root of the product of n values:
  • 9. Arithmetic and Geometric Mean Returns
    • Example
    • Assume the following sample of weekly stock returns:
    1.000 0.0000 4 1.0021 0.0021 3 0.9955 -0.0045 2 1.0084 0.0084 1 Return Relative Return Week
  • 10. Arithmetic and Geometric Mean Returns (cont’d)
    • Example (cont’d)
    • What is the arithmetic mean return?
    • Solution:
  • 11. Arithmetic and Geometric Mean Returns (cont’d)
    • Example (cont’d)
    • What is the geometric mean return?
    • Solution:
  • 12. Comparison of Arithmetic & 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%
  • 13. Comparison of Arithmetic & 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
  • 14. Expected Return
    • 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
  • 15. Standard Deviation and Variance
    • 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
  • 16. Standard Deviation and Variance (cont’d)
    • General equation for variance:
    • If all outcomes are equally likely:
  • 17. Standard Deviation and Variance (cont’d)
    • Equation for standard deviation:
  • 18. Semi-Variance
    • 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
  • 19. Some Statistical Facts of Life
    • Definitions
    • Properties of random variables
    • Linear regression
    • R squared and standard errors
  • 20. Definitions
    • Constants
    • Variables
    • Populations
    • Samples
    • Sample statistics
  • 21. Constants
    • 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
  • 22. Variables
    • 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
  • 23. Variables (cont’d)
    • 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
  • 24. Variables (cont’d)
    • Quantitative variables are measured by real numbers
      • E.g., numerical measurement
    • Qualitative variables are categorical
      • E.g., hair color
  • 25. Variables (cont’d)
    • 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)
  • 26. Populations
    • 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
  • 27. Populations (cont’d)
    • 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
  • 28. Populations (cont’d) Positive Skewness Negative Skewness
  • 29. Populations (cont’d)
    • A binomial distribution contains only two random variables
      • E.g., the toss of a coin
    • A finite population is one in which each possible outcome is known
      • E.g., a card drawn from a deck of cards
  • 30. Populations (cont’d)
    • 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
  • 31. Populations (cont’d)
    • 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
  • 32. Samples
    • A sample is any subset of a population
      • E.g., a sample of past monthly stock returns of a particular stock
  • 33. Sample Statistics
    • Sample statistics are characteristics of samples
      • A true population statistic is usually unobservable and must be estimated with a sample statistic
        • Expensive
        • Statistically unnecessary
  • 34. Properties of Random Variables
    • Example
    • Central tendency
    • Dispersion
    • Logarithms
    • Expectations
    • Correlation and covariance
  • 35. Example
    • Assume the following monthly stock returns for Stocks A and B:
    4% 1% 4 5% 4% 3 0% -1% 2 3% 2% 1 Stock B Stock A Month
  • 36. Central Tendency
    • 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
  • 37. Example (cont’d)
    • The expected returns for Stocks A and B are:
  • 38. Dispersion
    • 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
  • 39. Example (cont’d)
    • The variance ad standard deviation for Stock A are:
  • 40. Example (cont’d)
    • The variance ad standard deviation for Stock B are:
  • 41. Logarithms
    • 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
  • 42. Logarithms (cont’d)
    • 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
  • 43. Expectations
    • 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:
  • 44. Expectations (cont’d)
    • The expected value of a combination of random variables is equal to the sum of the expected value of each element of the combination:
  • 45. Correlations and Covariance
    • 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
  • 46. Correlations and Covariance (cont’d)
  • 47. Example (cont’d)
    • The covariance and correlation for Stocks A and B are:
  • 48. Correlations and Covariance
    • 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
  • 49.  
  • 50.  
  • 51.  
  • 52. Linear Regression
    • 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
  • 53. Linear Regression (cont’d)
    • Example
    • Assume the following sample of weekly stock and stock index returns:
    0.0005 0.0000 4 0.0019 0.0021 3 -0.0048 -0.0045 2 0.0088 0.0084 1 Index Return Stock Return Week
  • 54. Linear Regression (cont’d)
    • Example (cont’d)
    Intercept = 0 Slope = 0.96 R squared = 0.99
  • 55. R Squared and Standard Errors
    • Application
    • R squared
    • Standard Errors
  • 56. Application
    • R-squared and the standard error are used to assess the accuracy of calculated statistics
  • 57. R Squared
    • 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
  • 58.  
  • 59. Standard Errors
    • The standard error is the standard deviation divided by the square root of the number of observations:
  • 60. Standard Errors (cont’d)
    • 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
  • 61. Runs Test
    • A runs test allows the statistical testing of whether a series of price movements occurred by chance.
    • A run is defined as an uninterrupted sequence of the same observation. Ex : if the stock price increases 10 times in a row, then decreases 3 times, and then increases 4 times, we then say that we have three runs.
  • 62. Notation
    • R = number of runs (3 in this example)
    • n 1 = number of observations in the first category. For instance, here we have a total of 14 “ups”, so n 1 =14.
    • n 2 = number of observations in the second category. For instance, here we have a total of 3 “downs”, so n 2 =3.
    • Note that n 1 and n 2 could be the number of “Heads” and “Tails” in the case of a coin toss.
  • 63. Statistical Test
  • 64. Example
    • Let the number of runs R=23
    • Let the number of ups n 1 =20
    • Let the number of downs n 2 =30
  • 65. About 2.5% of the area under the normal curve is below a z score of -1.96.
  • 66. Interpretation
    • Since our z-score is not in the lower tail (nor is it in the upper tail), the runs we have witnessed are purely the product of chance.
    • If, on the other hand, we had obtained a z-score in the upper (2.5%) or lower (2.5%) tail, we would then be 95% certain that this specific occurrence of runs didn’t happen by chance. (Or that we just witnessed an extremely rare event)