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# 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)