1. Risk and Return in Capital Markets
Chapter 11
Risk and Return in Capital Markets

2. Chapter Outline 11.1 A First Look at Risk and Return
11.2 Historical Risks and Returns of Stocks
11.3 The Historical Tradeoff Between Risk and Return
11.4 Common Versus Independent Risk
11.5 Diversification in Stock Portfolios

3. Learning Objectives
Identify which types of securities have historically had the highest returns and which have been the most volatile
Compute the average return and volatility of returns from a set of historical asset prices
Understand the tradeoff between risk and return for large portfolios versus individual stocks
Describe the difference between common and independent risk
Explain how diversified portfolios remove independent risk, leaving common risk as the only risk requiring a risk premium 

4. 11.1 A First Look at Risk and Return
Consider how an investment would have grown if it were invested in each of the following from the end of 1929 until the beginning of 2010:
Standard & Poor’s 500 (S&P 500)
Small Stocks
World Portfolio
Corporate Bonds
Treasury Bills

5. Figure 11. 1 Value of $100 Invested at the End of 1925 in U. S
Figure Value of $100 Invested at the End of 1925 in U.S. Large Stocks (S&P 500), Small Stocks, World Stocks, Corporate Bonds, and Treasury Bills

6. Table Realized Returns, in Percent (%) for Small Stocks, the S&P 500, Corporate Bonds, and Treasury Bills, Year-End 1925–1935

7. 11.2 Historical Risks and Returns of Stocks
Computing Historical Returns
Realized Returns
Individual Investment Realized Returns
The realized return from your investment in the stock from t to t+1 is:
(Eq. 11.1)

8. Example 11.1 Realized Return
Problem:
Microsoft paid a one-time special dividend of $3.08 on November 15, Suppose you bought Microsoft stock for $28.08 on November 1, 2004 and sold it immediately after the dividend was paid for $ What was your realized return from holding the stock?

9. Example 11.1 Realized Return
Solution:
Plan:
We can use Eq 11.1 to calculate the realized return. We need the purchase price ($28.08), the selling price ($27.39), and the dividend ($3.08) and we are ready to proceed.

10. Example 11.1 Realized Return
Execute:
Using Eq. 11.1, the return from Nov 1, 2004 until Nov 15, 2004 is equal to
This 8.51% can be broken down into the dividend yield and the capital gain yield:

11. Example 11.1 Realized Return
Evaluate:
These returns include both the capital gain (or in this case a capital loss) and the return generated from receiving dividends. Both dividends and capital gains contribute to the total realized return—ignoring either one would give a very misleading impression of Microsoft’s performance.

12. Example 11.1a Realized Return
Problem:
Health Management Associates (HMA) paid a one-time special dividend of $10.00 on March 2, Suppose you bought HMA stock for $20.33 on February 15, 2007 and sold it immediately after the dividend was paid for $ What was your realized return from holding the stock?

13. Example 11.1a Realized Return
Solution:
Plan:
We can use Eq 11.1 to calculate the realized return. We need the purchase price ($20.33), the selling price ($10.29), and the dividend ($10.00) and we are ready to proceed.

14. Example 11.1a Realized Return
Execute:
Using Eq. 11.1, the return from February 15, 2007 until March 2, 2007 is equal to
This -0.2% can be broken down into the dividend yield and the capital gain yield:

15. Example 11.1a Realized Return
Evaluate:
These returns include both the capital gain (or in this case a capital loss) and the return generated from receiving dividends. Both dividends and capital gains contribute to the total realized return—ignoring either one would give a very misleading impression of HMA’s performance.

16. Example 11.1b Realized Return
Problem:
Limited Brands paid a one-time special dividend of $3.00 on December 21, Suppose you bought LTD stock for $29.35 on October 18, 2010 and sold it immediately after the dividend was paid for $ What was your realized return from holding the stock?

17. Example 11.1b Realized Return
Solution:
Plan:
We can use Eq 11.1 to calculate the realized return. We need the purchase price ($29.35), the selling price ($30.16), and the dividend ($3.00) and we are ready to proceed.

18. Example 11.1b Realized Return
Execute:
Using Eq. 11.1, the return from October 18, 2010 until December 21, 2010 is equal to
This 12.98% can be broken down into the dividend yield and the capital gain yield:

19. Example 11.1b Realized Return
Evaluate:
These returns include both the capital gain (or in this case a capital loss) and the return generated from receiving dividends. Both dividends and capital gains contribute to the total realized return—ignoring either one would give a very misleading impression of LTD’s performance.

20. 11.2 Historical Risks and Returns of Stocks
Computing Historical Returns
Individual Investment Realized Returns
For quarterly returns (or any four compounding periods that make up an entire year) the annual realized return, Rannual, is found by compounding:
(Eq. 11.2)

21. Example 11.2 Compounding Realized Returns
Problem:
Suppose you purchased Microsoft stock (MSFT) on Nov 1, 2004 and held it for one year, selling on Oct 31, What was your annual realized return?

22. Example 11.2 Compounding Realized Returns
Solution:
Plan:
We need to analyze the cash flows from holding MSFT stock for each quarter. In order to get the cash flows, we must look up MSFT stock price data at the purchase date and selling date, as well as at any dividend dates. From the data we can construct the following table to fill out our cash flow timeline: 

23. Example 11.2 Compounding Realized Returns
Plan (cont’d):
Next, compute the realized return between each set of dates using Eq Then determine the annual realized return similarly to Eq by compounding the returns for all of the periods in the year.

24. Example 11.2 Compounding Realized Returns
Execute:
In Example 11.1, we already computed the realized return for Nov 1, 2004 to Nov 15, 2004 as 8.51%. We continue as in that example, using Eq for each period until we have a series of realized returns. For example, from Nov 15, 2004 to Feb 15, 2005, the realized return is

25. Example 11.2 Compounding Realized Returns
Execute (cont’d):
The table below includes the realized return at each period.

26. Example 11.2 Compounding Realized Returns
Execute (cont’d):
We then determine the one-year return by compounding.

27. Example 11.2 Compounding Realized Returns
Evaluate:
By repeating these steps, we have successfully computed the realized annual returns for an investor holding MSFT stock over this one-year period. From this exercise we can see that returns are risky. MSFT fluctuated up and down over the year and ended-up only slightly (2.75%) at the end.

28. Example 11.2a Compounding Realized Returns
Problem:
Suppose you purchased Health Management Associate’s stock (HMA) on March 16, 2006 and held it for one year, selling on March 15, What was your realized return?

29. Example 11.2a Compounding Realized Returns
Solution:
Plan:
We need to analyze the cash flows from holding HMA stock for each period. In order to get the cash flows, we must look up HMA stock price data at the start and end of both years, as well as at any dividend dates. From the data we can construct the following table to fill out our cash flow timeline: 

30. Example 11.2a Compounding Realized Returns
Plan (cont’d):
Next, compute the realized return between each set of dates using Eq Then determine the annual realized return similarly to Eq by compounding the returns for all of the periods in the year.
Date
Price
Dividend
16-Mar-06
21.15
10-May-06
20.70
0.06
9-Aug-06
20.62
8-Nov-06
19.39
15-Feb-07
20.33
2-Mar-07
10.29
10.00
15-Mar-07
11.07

31. Example 11.2a Compounding Realized Returns
Execute:
In Example 11.1a, we already computed the realized return for February 15, 2007 to March 2, 2007 as -.2%. We continue as in that example, using Eq for each period until we have a series of realized returns. For example, from August 9, 2006 to November 8, 2006, the realized return is

32. Example 11.2a Compounding Realized Returns
Execute (cont’d):
The table below includes the realized return at each period.
Date
Price
Dividend
Return
16-Mar-06
21.15
10-May-06
20.70
0.06
-1.84%
9-Aug-06
20.62
-0.10%
8-Nov-06
19.39
-5.67%
15-Feb-07
20.33
4.85%
2-Mar-07
10.29
10.00
-0.20%
15-Mar-07
11.07
7.58%

33. Example 11.2a Compounding Realized Returns
Execute (cont’d):
We then determine the one-year return by compounding.

34. Example 11.2a Compounding Realized Returns
Evaluate:
By repeating these steps, we have successfully computed the realized annual returns for an investor holding HMA stock over this one-year period. From this exercise we can see that returns are risky. HMA fluctuated up and down over the year and yielded a return of only 4.11% at the end.

35. 11.2 Historical Risks and Returns of Stocks
Average Annual Returns
Average Annual Return of a Security
(Eq. 11.3)

36. Figure 11. 2 The Distribution of Annual Returns for U. S
Figure The Distribution of Annual Returns for U.S. Large Company Stocks (S&P 500), Small Stocks, Corporate Bonds, and Treasury Bills, 1926–2009

37. Figure 11. 3 Average Annual Returns in the U. S
Figure Average Annual Returns in the U.S. for Small Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury Bills, 1926–2009

38. 11.2 Historical Risks and Returns of Stocks
The Variance and Volatility of Returns:
Variance
Standard Deviation
(Eq. 11.4)
(Eq. 11.5)

39. Example 11.3 Computing Historical Volatility
Problem:
Using the data from Table 11.1, what is the standard deviation of the S&P 500’s returns for the years ?

40. Example 11.3 Computing Historical Volatility
Solution:
Plan:
First, compute the average return using Eq because it is an input to the variance equation. Next, compute the variance using Eq and then take its square root to determine the standard deviation as shown in Eq

41. Example 11.3 Computing Historical Volatility
Execute:
In the previous section we already computed the average annual return of the S&P 500 during this period as 3.1%, so we have all of the necessary inputs for the variance calculation: 
Applying Eq. 11.4, we have:

42. Example 11.3 Computing Historical Volatility
Execute (cont'd):
Alternatively, we can break the calculation of this equation out as follows:
Summing the squared differences in the last row, we get
Finally, dividing by (5-1=4) gives us 0.233/4 =0.058
The standard deviation is therefore:

43. Example 11.3 Computing Historical Volatility
Evaluate:
Our best estimate of the expected return for the S&P 500 is its average return, 3.1%, but it is risky, with a standard deviation of 24.1%.

44. Example 11.3a Computing Historical Volatility
Problem:
Using the data from Table 11.1, what is the standard deviation of small stocks’ returns for the years ?

45. Example 11.3a Computing Historical Volatility
Solution:
Plan:
First, compute the average return using Eq because it is an input to the variance equation. Next, compute the variance using Eq and then take its square root to determine the standard deviation as shown in Eq
Year
2005
2006
2007
2008
2009
Small Stocks’ Return
5.69%
16.17%
-5.22%
-36.72%
28.09%

46. Example 11.3a Computing Historical Volatility
Execute:
Using Eq. 11.3, the average annual return for small stocks during this period is:
We now have all of the necessary inputs for the variance calculation: 
Applying Eq. 11.4, we have:

47. Example 11.3a Computing Historical Volatility
Execute (cont'd):
Alternatively, we can break the calculation of this equation out as follows:
Summing the squared differences in the last row, we get
Finally, dividing by (5-1=4) gives us /4 =0.0615
The standard deviation is therefore:

48. Example 11.3a Computing Historical Volatility
Evaluate:
Our best estimate of the expected return for small stocks is its average return, 1.71%, and they are risky, with a standard deviation of 24.80%.

49. Figure 11. 4 Volatility (Standard Deviation) of U. S
Figure Volatility (Standard Deviation) of U.S. Small Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury Bills, 1926–2009

50. 11.2 Historical Risks and Returns of Stocks
The Normal Distribution
95% Prediction Interval
About two-thirds of all possible outcomes fall within one standard deviation above or below the average
(Eq. 11.6)

51. Figure 11.5 Normal Distribution

52. Example 11.4 Confidence Intervals
Problem:
In Example 11.3 we found the average return for the S&P 500 from to be 3.1% with a standard deviation of 24.1%. What is a 95% confidence interval for 2010’s return?

53. Example 11.4 Confidence Intervals
Solution:
Plan:
We can use Eq to compute the confidence interval.

54. Example 11.4 Confidence Intervals
Execute:
Using Eq. 11.6, we have:
Average ± (2  standard deviation) = 3.1% – (2  24.1%) to 3.1% + (2  24.1% ) = –45.1% to 51.3%.

55. Example 11.4 Confidence Intervals
Evaluate:
Even though the average return from 2005 to 2009 was 3.1%, the S&P 500 was volatile, so if we want to be 95% confident of 2010’s return, the best we can say is that it will lie between –45.1% and +51.3%.

56. Example 11.4a Confidence Intervals
Problem:
The average return for small stocks from was 1.71% with a standard deviation of 24.8%. What is a 95% confidence interval for 2010’s return?

57. Example 11.4a Confidence Intervals
Solution:
Plan:
We can use Eq to calculate the confidence interval.

58. Example 11.4a Confidence Intervals
Execute:
Using Eq. 11.6, we have:

59. Example 11.4a Confidence Intervals
Evaluate:
Even though the average return from was 1.71%, small stocks were volatile, so if we want to be 95% confident of 2010’s return, the best we can say is that it will lie between % and %.

60. Example 11.4b Confidence Intervals
Problem:
The average return for corporate bonds from was 6.49% with a standard deviation of 7.04%. What is a 95% confidence interval for 2010’s return?

61. Example 11.4b Confidence Intervals
Solution:
Plan:
We can use Eq to calculate the confidence interval.

62. Example 11.4b Confidence Intervals
Execute:
Using Eq. 11.6, we have:

63. Example 11.4b Confidence Intervals
Evaluate:
Even though the average return from was 6.49%, corporate bonds were volatile, so if we want to be 95% confident of 2010’s return, the best we can say is that it will lie between -7.59% and %.

64. Table 11.2 Summary of Tools for Working with Historical Returns

65. 11.3 Historical Tradeoff between Risk and Return
The Returns of Large Portfolios
Investments with higher volatility, as measured by standard deviation, tend to have higher average returns

66. Figure 11.6 The Historical Tradeoff Between Risk and Return in Large Portfolios, 1926–2010

67. 11.3 Historical Tradeoff between Risk and Return
The Returns of Individual Stocks
Larger stocks have lower volatility overall
Even the largest stocks are typically more volatile than a portfolio of large stocks
The standard deviation of an individual security doesn’t explain the size of its average return
All individual stocks have lower returns and/or higher risk than the portfolios in Figure 11.6

68. 11.4 Common Versus Independent Risk
Types of Risk
Common Risk
Independent Risk
Diversification

69. Table 11.3 Summary of Types of Risk

70. Example 11.5 Diversification
Problem:
You are playing a very simple gambling game with your friend: a $1 bet based on a coin flip. That is, you each bet $1 and flip a coin: heads you win your friend’s $1, tails you lose and your friend takes your dollar. How is your risk different if you play this game 100 times in a row versus just betting $100 (instead of $1) on a single coin flip?

71. Example 11.5 Diversification
Solution:
Plan:
The risk of losing one coin flip is independent of the risk of losing the next one: each time you have a 50% chance of losing, and one coin flip does not affect any other coin flip. We can compute the expected outcome of any flip as a weighted average by weighting your possible winnings (+$1) by 50% and your possible losses (-$1) by 50%. We can then compute the probability of losing all $100 under either scenario.

72. Example 11.5 Diversification
Execute:
If you play the game 100 times, you should lose about 50 times and win 50 times, so your expected outcome is 50  (+$1) + 50  (-$1) = $0. You should break-even. Even if you don’t win exactly half of the time, the probability that you would lose all 100 coin flips (and thus lose $100) is exceedingly small (in fact, it is , which is far less than even %). If it happens, you should take a very careful look at the coin!

73. Example 11.5 Diversification
Execute (cont’d):
If instead, you make a single $100 bet on the outcome of one coin flip, you have a 50% chance of winning $100 and a 50% chance of losing $100, so your expected outcome will be the same: break-even. However, there is a 50% chance you will lose $100, so your risk is far greater than it would be for 100 one dollar bets.

74. Example 11.5 Diversification
Evaluate:
In each case, you put $100 at risk, but by spreading-out that risk across 100 different bets, you have diversified much of your risk away compared to placing a single $100 bet.

75. Example 11.5a Diversification
Problem:
You are playing a very simple gambling game with your friend: a $1 bet based on a roll of two six-sided dice. That is, you each bet $1 and roll the dice: if the outcome is even you win your friend’s $1, if the outcome is odd you lose and your friend takes your dollar. How is your risk different if you play this game 100 times in a row versus just betting $100 (instead of $1) on a roll of the dice?

76. Example 11.5a Diversification
Solution:
Plan:
The risk of losing one dice roll is independent of the risk of losing the next one: each time you have a 50% chance of losing, and one roll of the dice does not affect any other roll. We can compute the expected outcome of any roll as a weighted average by weighting your possible winnings (+$1) by 50% and your possible losses (-$1) by 50%. We can then compute the probability of losing all $100 under either scenario.

77. Example 11.5a Diversification
Execute:
If you play the game 100 times, you should lose about 50% of the time and win 50% of the time, so your expected outcome is 50  (+$1) + 50  (-$1) = $0. You should break-even. Even if you don’t win exactly half of the time, the probability that you would lose all 100 dice rolls (and thus lose $100) is exceedingly small (in fact, it is , which is far less than even %). If it happens, you should take a very careful look at the dice!

78. Example 11.5a Diversification
Execute (cont’d):
If instead, you make a single $100 bet on the outcome of one roll of the dice, you have a 50% chance of winning $100 and a 50% chance of losing $100, so your expected outcome will be the same: break-even. However, there is a 50% chance you will lose $100, so your risk is far greater than it would be for 100 one dollar bets.

79. Example 11.5a Diversification
Evaluate:
In each case, you put $100 at risk, but by spreading-out that risk across 100 different bets, you have diversified much of your risk away compared to placing a single $100 bet.

80. 11.5 Diversification in Stock Portfolios
Unsystematic Versus Systematic Risk
Stock prices are impacted by two types of news:
Company or Industry-Specific News
Market-Wide News
Unsystematic Risk
Systematic Risk

81. Figure 11.7 Volatility of Portfolios of Type S and U Stocks

82. Figure 11.8 The Effect of Diversification on Portfolio Volatility

83. 11.5 Diversification in Stock Portfolios
Diversifiable Risk and the Risk Premium
The risk premium for diversifiable risk is zero
Investors are not compensated for holding unsystematic risk

84. Table 11.4 Systematic Risk Versus Unsystematic Risk

85. 11.5 Diversification in Stock Portfolios
The Importance of Systematic Risk
The risk premium of a security is determined by its systematic risk and does not depend on its diversifiable risk

86. 11.5 Diversification in Stock Portfolios
The Importance of Systematic Risk
There is no relationship between volatility and average returns for individual securities

87. Table 11.5 The Expected Return of Type S and Type U Firms, Assuming the Risk-Free Rate Is 5%

88. Chapter 11 Quiz
Historically, which types of investments have had the highest average returns and which have been the most volatile from year to year? Is there a relation?
For what purpose do we use the average and standard deviation of historical stock returns?
What is the relation between risk and return for large portfolios? How are individual stocks different?
What is the difference between common and independent risk?
Does systematic or unsystematic risk require a risk premium? Why?