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Risk Return

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Risk Return

1. 1. LECTURE 1 Risk, Return, and Uncertainty: A Review of Principles Useful in Finance
2. 2. The Concept of Return <ul><li>Measurable return </li></ul><ul><li>Expected return </li></ul><ul><li>Return on investment </li></ul>
3. 3. Measurable Return <ul><li>Definition </li></ul><ul><li>Holding period return </li></ul><ul><li>Arithmetic mean return </li></ul><ul><li>Geometric mean return </li></ul><ul><li>Comparison of arithmetic and geometric mean returns </li></ul>
4. 4. Definition <ul><li>A general definition of return is the benefit associated with an investment </li></ul><ul><ul><li>In most cases, return is measurable </li></ul></ul><ul><ul><li>E.g., a \$100 investment at 8%, compounded continuously is worth \$108.33 after one year </li></ul></ul><ul><ul><ul><li>The return is \$8.33, or 8.33% </li></ul></ul></ul>
5. 5. Holding Period Return <ul><li>The calculation of a holding period return is independent of the passage of time </li></ul><ul><ul><li>E.g., you buy a bond for \$950, receive \$80 in interest, and later sell the bond for \$980 </li></ul></ul><ul><ul><ul><li>The return is (\$80 + \$30)/\$950 = 11.58% </li></ul></ul></ul><ul><ul><ul><li>The 11.58% could have been earned over one year or one week </li></ul></ul></ul>
6. 6. Arithmetic Mean Return <ul><li>The arithmetic mean return is the arithmetic average of several holding period returns measured over the same holding period: </li></ul>
7. 7. Arithmetic Mean Return (cont’d) <ul><li>Arithmetic means are a useful proxy for expected returns </li></ul><ul><li>Arithmetic means are not especially useful for describing historical returns </li></ul><ul><ul><li>It is unclear what the number means once it is determined </li></ul></ul>
8. 8. Geometric Mean Return <ul><li>The geometric mean return is the n th root of the product of n values: </li></ul>
9. 9. Arithmetic and Geometric Mean Returns <ul><li>Example </li></ul><ul><li>Assume the following sample of weekly stock returns: </li></ul>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. 10. Arithmetic and Geometric Mean Returns (cont’d) <ul><li>Example (cont’d) </li></ul><ul><li>What is the arithmetic mean return? </li></ul><ul><li>Solution: </li></ul>
11. 11. Arithmetic and Geometric Mean Returns (cont’d) <ul><li>Example (cont’d) </li></ul><ul><li>What is the geometric mean return? </li></ul><ul><li>Solution: </li></ul>
12. 12. Comparison of Arithmetic & Geometric Mean Returns <ul><li>The geometric mean reduces the likelihood of nonsense answers </li></ul><ul><ul><li>Assume a \$100 investment falls by 50% in period 1 and rises by 50% in period 2 </li></ul></ul><ul><ul><li>The investor has \$75 at the end of period 2 </li></ul></ul><ul><ul><ul><li>Arithmetic mean = (-50% + 50%)/2 = 0% </li></ul></ul></ul><ul><ul><ul><li>Geometric mean = (0.50 x 1.50) 1/2 –1 = -13.40% </li></ul></ul></ul>
13. 13. Comparison of Arithmetic & Geometric Mean Returns <ul><li>The geometric mean must be used to determine the rate of return that equates a present value with a series of future values </li></ul><ul><li>The greater the dispersion in a series of numbers, the wider the gap between the arithmetic and geometric mean </li></ul>
14. 14. Expected Return <ul><li>Expected return refers to the future </li></ul><ul><ul><li>In finance, what happened in the past is not as important as what happens in the future </li></ul></ul><ul><ul><li>We can use past information to make estimates about the future </li></ul></ul>
15. 15. Standard Deviation and Variance <ul><li>Standard deviation and variance are the most common measures of total risk </li></ul><ul><li>They measure the dispersion of a set of observations around the mean observation </li></ul>
16. 16. Standard Deviation and Variance (cont’d) <ul><li>General equation for variance: </li></ul><ul><li>If all outcomes are equally likely: </li></ul>
17. 17. Standard Deviation and Variance (cont’d) <ul><li>Equation for standard deviation: </li></ul>
18. 18. Semi-Variance <ul><li>Semi-variance considers the dispersion only on the adverse side </li></ul><ul><ul><li>Ignores all observations greater than the mean </li></ul></ul><ul><ul><li>Calculates variance using only “bad” returns that are less than average </li></ul></ul><ul><ul><li>Since risk means “chance of loss” positive dispersion can distort the variance or standard deviation statistic as a measure of risk </li></ul></ul>
19. 19. Some Statistical Facts of Life <ul><li>Definitions </li></ul><ul><li>Properties of random variables </li></ul><ul><li>Linear regression </li></ul><ul><li>R squared and standard errors </li></ul>
20. 20. Definitions <ul><li>Constants </li></ul><ul><li>Variables </li></ul><ul><li>Populations </li></ul><ul><li>Samples </li></ul><ul><li>Sample statistics </li></ul>
21. 21. Constants <ul><li>A constant is a value that does not change </li></ul><ul><ul><li>E.g., the number of sides of a cube </li></ul></ul><ul><ul><li>E.g., the sum of the interior angles of a triangle </li></ul></ul><ul><li>A constant can be represented by a numeral or by a symbol </li></ul>
22. 22. Variables <ul><li>A variable has no fixed value </li></ul><ul><ul><li>It is useful only when it is considered in the context of other possible values it might assume </li></ul></ul><ul><li>In finance, variables are called random variables </li></ul>
23. 23. Variables (cont’d) <ul><li>Discrete random variables are countable </li></ul><ul><ul><li>E.g., the number of trout you catch </li></ul></ul><ul><li>Continuous random variables are measurable </li></ul><ul><ul><li>E.g., the length of a trout </li></ul></ul>
24. 24. Variables (cont’d) <ul><li>Quantitative variables are measured by real numbers </li></ul><ul><ul><li>E.g., numerical measurement </li></ul></ul><ul><li>Qualitative variables are categorical </li></ul><ul><ul><li>E.g., hair color </li></ul></ul>
25. 25. Variables (cont’d) <ul><li>Independent variables are measured directly </li></ul><ul><ul><li>E.g., the height of a box </li></ul></ul><ul><li>Dependent variables can only be measured once other independent variables are measured </li></ul><ul><ul><li>E.g., the volume of a box (requires length, width, and height) </li></ul></ul>
26. 26. Populations <ul><li>A population is the entire collection of a particular set of random variables </li></ul><ul><li>The nature of a population is described by its distribution </li></ul><ul><ul><li>The median of a distribution is the point where half the observations lie on either side </li></ul></ul><ul><ul><li>The mode is the value in a distribution that occurs most frequently </li></ul></ul>
27. 27. Populations (cont’d) <ul><li>A distribution can have skewness </li></ul><ul><ul><li>There is more dispersion on one side of the distribution </li></ul></ul><ul><ul><li>Positive skewness means the mean is greater than the median </li></ul></ul><ul><ul><ul><li>Stock returns are positively skewed </li></ul></ul></ul><ul><ul><li>Negative skewness means the mean is less than the median </li></ul></ul>
28. 28. Populations (cont’d) Positive Skewness Negative Skewness
29. 29. Populations (cont’d) <ul><li>A binomial distribution contains only two random variables </li></ul><ul><ul><li>E.g., the toss of a coin </li></ul></ul><ul><li>A finite population is one in which each possible outcome is known </li></ul><ul><ul><li>E.g., a card drawn from a deck of cards </li></ul></ul>
30. 30. Populations (cont’d) <ul><li>An infinite population is one where not all observations can be counted </li></ul><ul><ul><li>E.g., the microorganisms in a cubic mile of ocean water </li></ul></ul><ul><li>A univariate population has one variable of interest </li></ul>
31. 31. Populations (cont’d) <ul><li>A bivariate population has two variables of interest </li></ul><ul><ul><li>E.g., weight and size </li></ul></ul><ul><li>A multivariate population has more than two variables of interest </li></ul><ul><ul><li>E.g., weight, size, and color </li></ul></ul>
32. 32. Samples <ul><li>A sample is any subset of a population </li></ul><ul><ul><li>E.g., a sample of past monthly stock returns of a particular stock </li></ul></ul>
33. 33. Sample Statistics <ul><li>Sample statistics are characteristics of samples </li></ul><ul><ul><li>A true population statistic is usually unobservable and must be estimated with a sample statistic </li></ul></ul><ul><ul><ul><li>Expensive </li></ul></ul></ul><ul><ul><ul><li>Statistically unnecessary </li></ul></ul></ul>
34. 34. Properties of Random Variables <ul><li>Example </li></ul><ul><li>Central tendency </li></ul><ul><li>Dispersion </li></ul><ul><li>Logarithms </li></ul><ul><li>Expectations </li></ul><ul><li>Correlation and covariance </li></ul>
35. 35. Example <ul><li>Assume the following monthly stock returns for Stocks A and B: </li></ul>4% 1% 4 5% 4% 3 0% -1% 2 3% 2% 1 Stock B Stock A Month
36. 36. Central Tendency <ul><li>Central tendency is what a random variable looks like, on average </li></ul><ul><li>The usual measure of central tendency is the population’s expected value (the mean) </li></ul><ul><ul><li>The average value of all elements of the population </li></ul></ul>
37. 37. Example (cont’d) <ul><li>The expected returns for Stocks A and B are: </li></ul>
38. 38. Dispersion <ul><li>Investors are interest in the best and the worst in addition to the average </li></ul><ul><li>A common measure of dispersion is the variance or standard deviation </li></ul>
39. 39. Example (cont’d) <ul><li>The variance ad standard deviation for Stock A are: </li></ul>
40. 40. Example (cont’d) <ul><li>The variance ad standard deviation for Stock B are: </li></ul>
41. 41. Logarithms <ul><li>Logarithms reduce the impact of extreme values </li></ul><ul><ul><li>E.g., takeover rumors may cause huge price swings </li></ul></ul><ul><ul><li>A logreturn is the logarithm of a return </li></ul></ul><ul><li>Logarithms make other statistical tools more appropriate </li></ul><ul><ul><li>E.g., linear regression </li></ul></ul>
42. 42. Logarithms (cont’d) <ul><li>Using logreturns on stock return distributions: </li></ul><ul><ul><li>Take the raw returns </li></ul></ul><ul><ul><li>Convert the raw returns to return relatives </li></ul></ul><ul><ul><li>Take the natural logarithm of the return relatives </li></ul></ul>
43. 43. Expectations <ul><li>The expected value of a constant is a constant: </li></ul><ul><li>The expected value of a constant times a random variable is the constant times the expected value of the random variable: </li></ul>
44. 44. Expectations (cont’d) <ul><li>The expected value of a combination of random variables is equal to the sum of the expected value of each element of the combination: </li></ul>
45. 45. Correlations and Covariance <ul><li>Correlation is the degree of association between two variables </li></ul><ul><li>Covariance is the product moment of two random variables about their means </li></ul><ul><li>Correlation and covariance are related and generally measure the same phenomenon </li></ul>
46. 46. Correlations and Covariance (cont’d)
47. 47. Example (cont’d) <ul><li>The covariance and correlation for Stocks A and B are: </li></ul>
48. 48. Correlations and Covariance <ul><li>Correlation ranges from –1.0 to +1.0. </li></ul><ul><ul><li>Two random variables that are perfectly positively correlated have a correlation coefficient of +1.0 </li></ul></ul><ul><ul><li>Two random variables that are perfectly negatively correlated have a correlation coefficient of –1.0 </li></ul></ul>
49. 52. Linear Regression <ul><li>Linear regression is a mathematical technique used to predict the value of one variable from a series of values of other variables </li></ul><ul><ul><li>E.g., predict the return of an individual stock using a stock market index </li></ul></ul><ul><li>Regression finds the equation of a line through the points that gives the best possible fit </li></ul>
50. 53. Linear Regression (cont’d) <ul><li>Example </li></ul><ul><li>Assume the following sample of weekly stock and stock index returns: </li></ul>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
51. 54. Linear Regression (cont’d) <ul><li>Example (cont’d) </li></ul>Intercept = 0 Slope = 0.96 R squared = 0.99
52. 55. R Squared and Standard Errors <ul><li>Application </li></ul><ul><li>R squared </li></ul><ul><li>Standard Errors </li></ul>
53. 56. Application <ul><li>R-squared and the standard error are used to assess the accuracy of calculated statistics </li></ul>
54. 57. R Squared <ul><li>R squared is a measure of how good a fit we get with the regression line </li></ul><ul><ul><li>If every data point lies exactly on the line, R squared is 100% </li></ul></ul><ul><li>R squared is the square of the correlation coefficient between the security returns and the market returns </li></ul><ul><ul><li>It measures the portion of a security’s variability that is due to the market variability </li></ul></ul>
55. 59. Standard Errors <ul><li>The standard error is the standard deviation divided by the square root of the number of observations: </li></ul>
56. 60. Standard Errors (cont’d) <ul><li>The standard error enables us to determine the likelihood that the coefficient is statistically different from zero </li></ul><ul><ul><li>About 68% of the elements of the distribution lie within one standard error of the mean </li></ul></ul><ul><ul><li>About 95% lie within 1.96 standard errors </li></ul></ul><ul><ul><li>About 99% lie within 3.00 standard errors </li></ul></ul>
57. 61. Runs Test <ul><li>A runs test allows the statistical testing of whether a series of price movements occurred by chance. </li></ul><ul><li>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. </li></ul>
58. 62. Notation <ul><li>R = number of runs (3 in this example) </li></ul><ul><li>n 1 = number of observations in the first category. For instance, here we have a total of 14 “ups”, so n 1 =14. </li></ul><ul><li>n 2 = number of observations in the second category. For instance, here we have a total of 3 “downs”, so n 2 =3. </li></ul><ul><li>Note that n 1 and n 2 could be the number of “Heads” and “Tails” in the case of a coin toss. </li></ul>
59. 63. Statistical Test
60. 64. Example <ul><li>Let the number of runs R=23 </li></ul><ul><li>Let the number of ups n 1 =20 </li></ul><ul><li>Let the number of downs n 2 =30 </li></ul>
61. 65. About 2.5% of the area under the normal curve is below a z score of -1.96.
62. 66. Interpretation <ul><li>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. </li></ul><ul><li>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) </li></ul>