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  • 1. Chapter 6 - Risk and Rates of Return  2005, Pearson Prentice Hall
  • 2. Chapter 6: Objectives <ul><li>Inflation and rates of return </li></ul><ul><li>How to measure risk </li></ul><ul><li>(variance, standard deviation, beta) </li></ul><ul><li>How to reduce risk </li></ul><ul><li>(diversification) </li></ul><ul><li>How to price risk </li></ul><ul><li>(security market line, Capital Asset Pricing Model) </li></ul>
  • 3. Inflation, Rates of Return, and the Fisher Effect Interest Rates
  • 4. Interest Rates Conceptually :
  • 5. Interest Rates Conceptually : Nominal risk-free Interest Rate k rf
  • 6. Interest Rates Conceptually : Nominal risk-free Interest Rate k rf =
  • 7. Interest Rates Conceptually : Nominal risk-free Interest Rate k rf = Real risk-free Interest Rate k*
  • 8. Interest Rates Conceptually : Nominal risk-free Interest Rate k rf = Real risk-free Interest Rate k* +
  • 9. Interest Rates Conceptually : Nominal risk-free Interest Rate k rf = Real risk-free Interest Rate k* + Inflation- risk premium IRP
  • 10. Interest Rates Conceptually : Nominal risk-free Interest Rate k rf = Real risk-free Interest Rate k* + Inflation- risk premium IRP Mathematically :
  • 11. Interest Rates Conceptually : Nominal risk-free Interest Rate k rf = Real risk-free Interest Rate k* + Inflation- risk premium IRP Mathematically : (1 + k rf ) = (1 + k*) (1 + IRP)
  • 12. Interest Rates Conceptually : Nominal risk-free Interest Rate k rf = Real risk-free Interest Rate k* + Inflation- risk premium IRP Mathematically : (1 + k rf ) = (1 + k*) (1 + IRP) This is known as the “ Fisher Effect ”
  • 13. <ul><li>Suppose the real rate is 3%, and the nominal rate is 8%. What is the inflation rate premium? </li></ul><ul><li>(1 + k rf ) = (1 + k*) (1 + IRP) </li></ul><ul><li>(1.08) = (1.03) (1 + IRP) </li></ul><ul><li>(1 + IRP) = (1.0485), so </li></ul><ul><li>IRP = 4.85% </li></ul>Interest Rates
  • 14. Term Structure of Interest Rates <ul><li>The pattern of rates of return for debt securities that differ only in the length of time to maturity. </li></ul>
  • 15. Term Structure of Interest Rates <ul><li>The pattern of rates of return for debt securities that differ only in the length of time to maturity. </li></ul>yield to maturity time to maturity (years)
  • 16. Term Structure of Interest Rates <ul><li>The pattern of rates of return for debt securities that differ only in the length of time to maturity. </li></ul>yield to maturity time to maturity (years)
  • 17. Term Structure of Interest Rates <ul><li>The yield curve may be downward sloping or “inverted” if rates are expected to fall. </li></ul>yield to maturity time to maturity (years)
  • 18. Term Structure of Interest Rates <ul><li>The yield curve may be downward sloping or “inverted” if rates are expected to fall. </li></ul>yield to maturity time to maturity (years)
  • 19. For a Treasury security, what is the required rate of return?
  • 20. For a Treasury security, what is the required rate of return? Required rate of return =
  • 21. For a Treasury security, what is the required rate of return? <ul><li>Since Treasuries are essentially free of default risk , the rate of return on a Treasury security is considered the “ risk-free ” rate of return. </li></ul>Required rate of return = Risk-free rate of return
  • 22. For a corporate stock or bond , what is the required rate of return?
  • 23. For a corporate stock or bond , what is the required rate of return? Required rate of return =
  • 24. For a corporate stock or bond , what is the required rate of return? Required rate of return = Risk-free rate of return
  • 25. For a corporate stock or bond , what is the required rate of return? <ul><li>How large of a risk premium should we require to buy a corporate security? </li></ul>Required rate of return = + Risk-free rate of return Risk premium
  • 26. Returns <ul><li>Expected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc. </li></ul><ul><li>Required Return - the return that an investor requires on an asset given its risk and market interest rates. </li></ul>
  • 27. Expected Return <ul><li>State of Probability Return </li></ul><ul><li>Economy (P) Orl. Utility Orl. Tech </li></ul><ul><li>Recession .20 4% -10% </li></ul><ul><li>Normal .50 10% 14% </li></ul><ul><li>Boom .30 14% 30% </li></ul><ul><li>For each firm, the expected return on the stock is just a weighted average : </li></ul>
  • 28. <ul><li>State of Probability Return </li></ul><ul><li>Economy (P) Orl. Utility Orl. Tech </li></ul><ul><li>Recession .20 4% -10% </li></ul><ul><li>Normal .50 10% 14% </li></ul><ul><li>Boom .30 14% 30% </li></ul><ul><li>For each firm, the expected return on the stock is just a weighted average : </li></ul><ul><li>k = P(k 1 )*k 1 + P(k 2 )*k 2 + ...+ P(k n )*kn </li></ul>Expected Return
  • 29. Expected Return <ul><li>State of Probability Return </li></ul><ul><li>Economy (P) Orl. Utility Orl. Tech </li></ul><ul><li>Recession .20 4% -10% </li></ul><ul><li>Normal .50 10% 14% </li></ul><ul><li>Boom .30 14% 30% </li></ul><ul><li>k = P(k 1 )*k 1 + P(k 2 )*k 2 + ...+ P(k n )*kn </li></ul><ul><li>k (OU) = .2 (4%) + .5 (10%) + .3 (14%) = 10% </li></ul>
  • 30. Expected Return <ul><li>State of Probability Return </li></ul><ul><li>Economy (P) Orl. Utility Orl. Tech </li></ul><ul><li>Recession .20 4% -10% </li></ul><ul><li>Normal .50 10% 14% </li></ul><ul><li>Boom .30 14% 30% </li></ul><ul><li>k = P(k 1 )*k 1 + P(k 2 )*k 2 + ...+ P(k n )*kn </li></ul><ul><li>k (OI) = .2 (-10%)+ .5 (14%) + .3 (30%) = 14% </li></ul>
  • 31. <ul><li>Based only on your expected return calculations, which stock would you prefer? </li></ul>
  • 32. RISK? Have you considered
  • 33. What is Risk? <ul><li>The possibility that an actual return will differ from our expected return. </li></ul><ul><li>Uncertainty in the distribution of possible outcomes. </li></ul>
  • 34. What is Risk? <ul><li>Uncertainty in the distribution of possible outcomes. </li></ul>
  • 35. What is Risk? <ul><li>Uncertainty in the distribution of possible outcomes. </li></ul>Company A return
  • 36. What is Risk? <ul><li>Uncertainty in the distribution of possible outcomes. </li></ul>return Company B Company A return
  • 37. How do We Measure Risk? <ul><li>To get a general idea of a stock’s price variability, we could look at the stock’s price range over the past year. </li></ul>52 weeks Yld Vol Net Hi Lo Sym Div % PE 100s Hi Lo Close Chg 134 80 IBM .52 .5 21 143402 98 95 95 49 -3 115 40 MSFT … 29 558918 55 52 51 94 -4 75
  • 38. How do We Measure Risk? <ul><li>A more scientific approach is to examine the stock’s standard deviation of returns. </li></ul><ul><li>Standard deviation is a measure of the dispersion of possible outcomes . </li></ul><ul><li>The greater the standard deviation, the greater the uncertainty, and, therefore, the greater the risk. </li></ul>
  • 39. Standard Deviation <ul><li>= (k i - k) 2 P(k i ) </li></ul> n i =1 
  • 40. <ul><li>Orlando Utility, Inc. </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 41. <ul><li>Orlando Utility, Inc. </li></ul><ul><li>( 4% - 10%) 2 (.2) = 7.2 </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 42. <ul><li>Orlando Utility, Inc. </li></ul><ul><li>( 4% - 10%) 2 (.2) = 7.2 </li></ul><ul><li>(10% - 10%) 2 (.5) = 0 </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 43. <ul><li>Orlando Utility, Inc. </li></ul><ul><li>( 4% - 10%) 2 (.2) = 7.2 </li></ul><ul><li>(10% - 10%) 2 (.5) = 0 </li></ul><ul><li>(14% - 10%) 2 (.3) = 4.8 </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 44. <ul><li>Orlando Utility, Inc. </li></ul><ul><li>( 4% - 10%) 2 (.2) = 7.2 </li></ul><ul><li>(10% - 10%) 2 (.5) = 0 </li></ul><ul><li>(14% - 10%) 2 (.3) = 4.8 </li></ul><ul><li>Variance = 12 </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 45. <ul><li>Orlando Utility, Inc. </li></ul><ul><li>( 4% - 10%) 2 (.2) = 7.2 </li></ul><ul><li>(10% - 10%) 2 (.5) = 0 </li></ul><ul><li>(14% - 10%) 2 (.3) = 4.8 </li></ul><ul><li>Variance = 12 </li></ul><ul><li>Stand. dev. = 12 = </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 46. <ul><li>Orlando Utility, Inc. </li></ul><ul><li>( 4% - 10%) 2 (.2) = 7.2 </li></ul><ul><li>(10% - 10%) 2 (.5) = 0 </li></ul><ul><li>(14% - 10%) 2 (.3) = 4.8 </li></ul><ul><li>Variance = 12 </li></ul><ul><li>Stand. dev. = 12 = 3.46% </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 47. <ul><li>Orlando Technology, Inc. </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 48. <ul><li>Orlando Technology, Inc. </li></ul><ul><li>(-10% - 14%) 2 (.2) = 115.2 </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 49. <ul><li>Orlando Technology, Inc. </li></ul><ul><li>(-10% - 14%) 2 (.2) = 115.2 </li></ul><ul><li>(14% - 14%) 2 (.5) = 0 </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 50. <ul><li>Orlando Technology, Inc. </li></ul><ul><li>(-10% - 14%) 2 (.2) = 115.2 </li></ul><ul><li>(14% - 14%) 2 (.5) = 0 </li></ul><ul><li>(30% - 14%) 2 (.3) = 76.8 </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 51. <ul><li>Orlando Technology, Inc. </li></ul><ul><li>(-10% - 14%) 2 (.2) = 115.2 </li></ul><ul><li>(14% - 14%) 2 (.5) = 0 </li></ul><ul><li>(30% - 14%) 2 (.3) = 76.8 </li></ul><ul><li>Variance = 192 </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 52. <ul><li>Orlando Technology, Inc. </li></ul><ul><li>(-10% - 14%) 2 (.2) = 115.2 </li></ul><ul><li>(14% - 14%) 2 (.5) = 0 </li></ul><ul><li>(30% - 14%) 2 (.3) = 76.8 </li></ul><ul><li>Variance = 192 </li></ul><ul><li>Stand. dev. = 192 = </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 53. <ul><li>Orlando Technology, Inc. </li></ul><ul><li>(-10% - 14%) 2 (.2) = 115.2 </li></ul><ul><li>(14% - 14%) 2 (.5) = 0 </li></ul><ul><li>(30% - 14%) 2 (.3) = 76.8 </li></ul><ul><li>Variance = 192 </li></ul><ul><li>Stand. dev. = 192 = 13.86% </li></ul>= (k i - k) 2 P(k i )  n i =1 
  • 54. <ul><li>Which stock would you prefer? </li></ul><ul><li>How would you decide? </li></ul>
  • 55. <ul><li>Which stock would you prefer? </li></ul><ul><li>How would you decide? </li></ul>
  • 56. <ul><li>Orlando Orlando </li></ul><ul><li> Utility Technology </li></ul><ul><li>Expected Return 10% 14% </li></ul><ul><li>Standard Deviation 3.46% 13.86% </li></ul>Summary
  • 57. <ul><li>It depends on your tolerance for risk! </li></ul><ul><li>Remember, there’s a tradeoff between risk and return. </li></ul>
  • 58. <ul><li>It depends on your tolerance for risk! </li></ul><ul><li>Remember, there’s a tradeoff between risk and return. </li></ul>Return Risk
  • 59. <ul><li>It depends on your tolerance for risk! </li></ul><ul><li>Remember, there’s a tradeoff between risk and return. </li></ul>Return Risk
  • 60. Portfolios <ul><li>Combining several securities in a portfolio can actually reduce overall risk . </li></ul><ul><li>How does this work? </li></ul>
  • 61. Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated). rate of return time
  • 62. Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated). rate of return time k A
  • 63. Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated). rate of return time k A k B
  • 64. What has happened to the variability of returns for the portfolio? rate of return time k A k B
  • 65. What has happened to the variability of returns for the portfolio? rate of return time k p k A k B
  • 66. Diversification <ul><li>Investing in more than one security to reduce risk . </li></ul><ul><li>If two stocks are perfectly positively correlated , diversification has no effect on risk. </li></ul><ul><li>If two stocks are perfectly negatively correlated , the portfolio is perfectly diversified. </li></ul>
  • 67. <ul><li>If you owned a share of every stock traded on the NYSE and NASDAQ, would you be diversified? </li></ul><ul><li>YES! </li></ul><ul><li>Would you have eliminated all of your risk? </li></ul><ul><li>NO! Common stock portfolios still have risk. </li></ul>
  • 68. Some risk can be diversified away and some cannot. <ul><li>Market risk ( systematic risk) is nondiversifiable. This type of risk cannot be diversified away. </li></ul><ul><li>Company-unique risk (unsystematic risk) is diversifiable . This type of risk can be reduced through diversification. </li></ul>
  • 69. Market Risk <ul><li>Unexpected changes in interest rates. </li></ul><ul><li>Unexpected changes in cash flows due to tax rate changes, foreign competition, and the overall business cycle. </li></ul>
  • 70. Company-unique Risk <ul><li>A company’s labor force goes on strike. </li></ul><ul><li>A company’s top management dies in a plane crash. </li></ul><ul><li>A huge oil tank bursts and floods a company’s production area. </li></ul>
  • 71. <ul><li>As you add stocks to your portfolio, company-unique risk is reduced. </li></ul>
  • 72. <ul><li>As you add stocks to your portfolio, company-unique risk is reduced. </li></ul>portfolio risk number of stocks
  • 73. <ul><li>As you add stocks to your portfolio, company-unique risk is reduced. </li></ul>portfolio risk number of stocks Market risk
  • 74. <ul><li>As you add stocks to your portfolio, company-unique risk is reduced. </li></ul>portfolio risk number of stocks Market risk company- unique risk
  • 75. Do some firms have more market risk than others? <ul><li>Yes . For example: </li></ul><ul><li>Interest rate changes affect all firms, but which would be more affected: </li></ul><ul><li>a) Retail food chain </li></ul><ul><li>b) Commercial bank </li></ul>
  • 76. <ul><li>Yes . For example: </li></ul><ul><li>Interest rate changes affect all firms, but which would be more affected: </li></ul><ul><li>a) Retail food chain </li></ul><ul><li>b) Commercial bank </li></ul>Do some firms have more market risk than others?
  • 77. <ul><li>Note </li></ul><ul><li>As we know, the market compensates investors for accepting risk - but only for market risk . Company-unique risk can and should be diversified away. </li></ul><ul><li>So - we need to be able to measure market risk. </li></ul>
  • 78. This is why we have Beta. <ul><li>Beta: a measure of market risk. </li></ul><ul><li>Specifically, beta is a measure of how an individual stock’s returns vary with market returns. </li></ul><ul><li>It’s a measure of the “sensitivity” of an individual stock’s returns to changes in the market. </li></ul>
  • 79. <ul><li>A firm that has a beta = 1 has average market risk . The stock is no more or less volatile than the market. </li></ul><ul><li>A firm with a beta > 1 is more volatile than the market. </li></ul>The market’s beta is 1
  • 80. <ul><li>A firm that has a beta = 1 has average market risk . The stock is no more or less volatile than the market. </li></ul><ul><li>A firm with a beta > 1 is more volatile than the market. </li></ul><ul><ul><li>(ex: technology firms) </li></ul></ul>The market’s beta is 1
  • 81. <ul><li>A firm that has a beta = 1 has average market risk . The stock is no more or less volatile than the market. </li></ul><ul><li>A firm with a beta > 1 is more volatile than the market. </li></ul><ul><ul><li>(ex: technology firms) </li></ul></ul><ul><li>A firm with a beta < 1 is less volatile than the market. </li></ul>The market’s beta is 1
  • 82. <ul><li>A firm that has a beta = 1 has average market risk . The stock is no more or less volatile than the market. </li></ul><ul><li>A firm with a beta > 1 is more volatile than the market. </li></ul><ul><ul><li>(ex: technology firms) </li></ul></ul><ul><li>A firm with a beta < 1 is less volatile than the market. </li></ul><ul><ul><li>(ex: utilities) </li></ul></ul>The market’s beta is 1
  • 83. Calculating Beta
  • 84. Calculating Beta -5 -15 5 10 15 -15 -10 -10 -5 5 10 15 XYZ Co. returns S&P 500 returns
  • 85. Calculating Beta -5 -15 5 10 15 -15 -10 -10 -5 5 10 15 XYZ Co. returns S&P 500 returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
  • 86. Calculating Beta -5 -15 5 10 15 -15 -10 -10 -5 5 10 15 XYZ Co. returns S&P 500 returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
  • 87. Calculating Beta -5 -15 5 10 15 -15 -10 -10 -5 5 10 15 XYZ Co. returns S&P 500 returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beta = slope = 1.20
  • 88. Summary: <ul><li>We know how to measure risk, using standard deviation for overall risk and beta for market risk. </li></ul><ul><li>We know how to reduce overall risk to only market risk through diversification . </li></ul><ul><li>We need to know how to price risk so we will know how much extra return we should require for accepting extra risk. </li></ul>
  • 89. What is the Required Rate of Return? <ul><li>The return on an investment required by an investor given market interest rates and the investment’s risk . </li></ul>
  • 90. Required rate of return =
  • 91. Required rate of return = + Risk-free rate of return
  • 92. Required rate of return = + Risk-free rate of return Risk premium
  • 93. market risk Required rate of return = + Risk-free rate of return Risk premium
  • 94. market risk company- unique risk Required rate of return = + Risk-free rate of return Risk premium
  • 95. market risk company- unique risk can be diversified away Required rate of return = + Risk-free rate of return Risk premium
  • 96. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>Beta Let’s try to graph this relationship!
  • 97. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Risk-free rate of return (6%) Beta 12% 1
  • 98. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Risk-free rate of return (6%) Beta 12% 1 security market line (SML)
  • 99. <ul><li>This linear relationship between risk and required return is known as the Capital Asset Pricing Model (CAPM). </li></ul>
  • 100. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Risk-free rate of return (6%) Beta 12% 1 SML 0
  • 101. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Risk-free rate of return (6%) Beta 12% 1 SML 0 Is there a riskless (zero beta) security?
  • 102. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>Beta . 12% 1 SML 0 Is there a riskless (zero beta) security? Treasury securities are as close to riskless as possible. Risk-free rate of return (6%)
  • 103. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Beta 12% 1 SML Where does the S&P 500 fall on the SML? Risk-free rate of return (6%) 0
  • 104. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Beta 12% 1 SML Where does the S&P 500 fall on the SML? The S&P 500 is a good approximation for the market Risk-free rate of return (6%) 0
  • 105. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Beta 12% 1 SML Utility Stocks Risk-free rate of return (6%) 0
  • 106. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Beta 12% 1 SML High-tech stocks Risk-free rate of return (6%) 0
  • 107. The CAPM equation:
  • 108. <ul><li>k j = k rf + j (k m - k rf ) </li></ul>The CAPM equation: 
  • 109. <ul><li>k j = k rf + j (k m - k rf ) </li></ul><ul><li>where: </li></ul><ul><li>k j = the required return on security j, </li></ul><ul><li>k rf = the risk-free rate of interest, </li></ul><ul><li>j = the beta of security j, and </li></ul><ul><li>k m = the return on the market index. </li></ul>The CAPM equation:  
  • 110. Example: <ul><li>Suppose the Treasury bond rate is 6% , the average return on the S&P 500 index is 12% , and Walt Disney has a beta of 1.2 . </li></ul><ul><li>According to the CAPM , what should be the required rate of return on Disney stock? </li></ul>
  • 111. k j = k rf + (k m - k rf ) <ul><li>k j = .06 + 1.2 (.12 - .06) </li></ul><ul><li>k j = .132 = 13.2% </li></ul><ul><li>According to the CAPM, Disney stock should be priced to give a 13.2% return. </li></ul>
  • 112. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Beta 12% 1 SML 0 Risk-free rate of return (6%)
  • 113. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Beta 12% 1 SML 0 Theoretically, every security should lie on the SML Risk-free rate of return (6%)
  • 114. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Beta 12% 1 SML 0 Theoretically, every security should lie on the SML If every stock is on the SML, investors are being fully compensated for risk. Risk-free rate of return (6%)
  • 115. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Beta 12% 1 SML 0 If a security is above the SML, it is underpriced. Risk-free rate of return (6%)
  • 116. <ul><li>Required </li></ul><ul><li>rate of </li></ul><ul><li>return </li></ul>. Beta 12% 1 SML 0 If a security is above the SML, it is underpriced. If a security is below the SML, it is overpriced. Risk-free rate of return (6%)
  • 117. Simple Return Calculations
  • 118. Simple Return Calculations t t+1 $50 $60
  • 119. Simple Return Calculations = = 20% P t+1 - P t 60 - 50 P t 50 t t+1 $50 $60
  • 120. Simple Return Calculations P t+1 60 P t 50 - 1 = -1 = 20% = = 20% P t+1 - P t 60 - 50 P t 50 t t+1 $50 $60
  • 121. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 Feb $63.80 Mar $59.00 Apr $62.00 May $64.50 Jun $69.00 Jul $69.00 Aug $75.00 Sep $82.50 Oct $73.00 Nov $80.00 Dec $86.00
  • 122. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 Mar $59.00 Apr $62.00 May $64.50 Jun $69.00 Jul $69.00 Aug $75.00 Sep $82.50 Oct $73.00 Nov $80.00 Dec $86.00
  • 123. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 0.100 Mar $59.00 Apr $62.00 May $64.50 Jun $69.00 Jul $69.00 Aug $75.00 Sep $82.50 Oct $73.00 Nov $80.00 Dec $86.00
  • 124. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 0.100 Mar $59.00 -0.075 Apr $62.00 May $64.50 Jun $69.00 Jul $69.00 Aug $75.00 Sep $82.50 Oct $73.00 Nov $80.00 Dec $86.00
  • 125. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 0.100 Mar $59.00 -0.075 Apr $62.00 0.051 May $64.50 Jun $69.00 Jul $69.00 Aug $75.00 Sep $82.50 Oct $73.00 Nov $80.00 Dec $86.00
  • 126. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 0.100 Mar $59.00 -0.075 Apr $62.00 0.051 May $64.50 0.040 Jun $69.00 Jul $69.00 Aug $75.00 Sep $82.50 Oct $73.00 Nov $80.00 Dec $86.00
  • 127. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 0.100 Mar $59.00 -0.075 Apr $62.00 0.051 May $64.50 0.040 Jun $69.00 0.070 Jul $69.00 Aug $75.00 Sep $82.50 Oct $73.00 Nov $80.00 Dec $86.00
  • 128. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 0.100 Mar $59.00 -0.075 Apr $62.00 0.051 May $64.50 0.040 Jun $69.00 0.070 Jul $69.00 0.000 Aug $75.00 Sep $82.50 Oct $73.00 Nov $80.00 Dec $86.00
  • 129. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 0.100 Mar $59.00 -0.075 Apr $62.00 0.051 May $64.50 0.040 Jun $69.00 0.070 Jul $69.00 0.000 Aug $75.00 0.087 Sep $82.50 Oct $73.00 Nov $80.00 Dec $86.00
  • 130. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 0.100 Mar $59.00 -0.075 Apr $62.00 0.051 May $64.50 0.040 Jun $69.00 0.070 Jul $69.00 0.000 Aug $75.00 0.087 Sep $82.50 0.100 Oct $73.00 Nov $80.00 Dec $86.00
  • 131. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 0.100 Mar $59.00 -0.075 Apr $62.00 0.051 May $64.50 0.040 Jun $69.00 0.070 Jul $69.00 0.000 Aug $75.00 0.087 Sep $82.50 0.100 Oct $73.00 -0.115 Nov $80.00 Dec $86.00
  • 132. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 0.100 Mar $59.00 -0.075 Apr $62.00 0.051 May $64.50 0.040 Jun $69.00 0.070 Jul $69.00 0.000 Aug $75.00 0.087 Sep $82.50 0.100 Oct $73.00 -0.115 Nov $80.00 0.096 Dec $86.00
  • 133. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 Feb $63.80 0.100 Mar $59.00 -0.075 Apr $62.00 0.051 May $64.50 0.040 Jun $69.00 0.070 Jul $69.00 0.000 Aug $75.00 0.087 Sep $82.50 0.100 Oct $73.00 -0.115 Nov $80.00 0.096 Dec $86.00 0.075
  • 134. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 0.049 Feb $63.80 0.100 0.049 Mar $59.00 -0.075 0.049 Apr $62.00 0.051 0.049 May $64.50 0.040 0.049 Jun $69.00 0.070 0.049 Jul $69.00 0.000 0.049 Aug $75.00 0.087 0.049 Sep $82.50 0.100 0.049 Oct $73.00 -0.115 0.049 Nov $80.00 0.096 0.049 Dec $86.00 0.075 0.049
  • 135. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 0.049 0.012321 Feb $63.80 0.100 0.049 0.002601 Mar $59.00 -0.075 0.049 0.015376 Apr $62.00 0.051 0.049 0.000004 May $64.50 0.040 0.049 0.000081 Jun $69.00 0.070 0.049 0.000441 Jul $69.00 0.000 0.049 0.002401 Aug $75.00 0.087 0.049 0.001444 Sep $82.50 0.100 0.049 0.002601 Oct $73.00 -0.115 0.049 0.028960 Nov $80.00 0.096 0.049 0.002090 Dec $86.00 0.075 0.049 0.000676
  • 136. (a) (b) monthly expected month price return return (a - b) 2 Dec $50.00 Jan $58.00 0.160 0.049 0.012321 Feb $63.80 0.100 0.049 0.002601 Mar $59.00 -0.075 0.049 0.015376 Apr $62.00 0.051 0.049 0.000004 May $64.50 0.040 0.049 0.000081 Jun $69.00 0.070 0.049 0.000441 Jul $69.00 0.000 0.049 0.002401 Aug $75.00 0.087 0.049 0.001444 Sep $82.50 0.100 0.049 0.002601 Oct $73.00 -0.115 0.049 0.028960 Nov $80.00 0.096 0.049 0.002090 Dec $86.00 0.075 0.049 0.000676 0.0781 St. Dev: sum, divide by (n-1), and take sq root:
  • 137. Calculator solution using HP 10B: <ul><li>Enter monthly return on 10B calculator, followed by sigma key (top right corner). </li></ul><ul><li>Shift 7 gives you the expected return. </li></ul><ul><li>Shift 8 gives you the standard deviation. </li></ul>

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