1. Risk and Return in Capital Markets: Historical Perspectives
Chapter 11
Risk and Return in Capital Markets: Historical Perspectives

2. Chapter Outline 1. A First Look at Risk and Return
2. Historical Risks and Returns of Stocks
3. The Historical Tradeoff Between Risk and Return
4. Common Versus Independent Risk
Diversification in Stock Portfolios
Expected Return and Risk
Arithmetic average versus Geometric average
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 

3. Bull Market, Bear Markets, Risk and Return
Talking about return
Internet stock boom and bust
DOW fell 16.8% in 2002
NASDAG plunged 31.5% in 2002
Stock inflow of $12 billion in 2002
Bond inflow of $128 billion in 2002
Good in 2003
Inflation is negligible
Low interest rate
Low unemployment at 6% or less
Bad
Uncertainty in War with Iraq
Source: Google Finance

4. Recent History: Yield Spreads

5. Returns on Some “Extreme” Days

6. The Market of 1987 – 2009, “Bumps Along the Way”
Period
% Decline in S&P 500
Aug. 25, 1987 – Oct. 19, 1987
-33.2%
Oct. 21, 1987 – Oct. 26, 1987
-11.9%
Nov. 2, 1987 – Dec. 4, 1987
-12.4%
Oct. 9, 1989 – Jan. 30, 1990
-10.2%
July 16, 1990 – Oct. 11, 1990
-19.9%
Feb. 18, 1997 – Apr. 11, 1997
-9.6%
Jul. 17, 1998 – Oct. 8, 1998
-19.2%
Jul. 19, 1999 – Oct. 18, 1999
-12.1%
Aug. 28, 2000 – Sep. 30, 2002
-46.2%
May. 16, 2008 – Mar. 6, 2009
-52.1%
Talking about risk
Adapted from the T. Rowe Price Report, Fall 1997 issue and historical information from Yahoo! Investor
DOW fell 16.8% in 2002
NASDAG plunged 31.5% in 2002
Stock inflow of $12 billion in 2002
Bond inflow of $128 billion in 2002
Good in 2003
Inflation is negligible
Low interest rate
Low unemployment at 6% or less
Bad
Uncertainty in War with Iraq

7. Historical Returns, Standard Deviations, and Frequency Distributions: 1926—2009

8. Geometric versus Arithmetic Averages, 1926-2006, 1926-2012
Source: Jordan, Miller, and Dolvin (2012) textbook

9. 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

10. 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

11. Computing historical return and risk

12. 11.2 Historical Risks and Returns of Stocks
Computing Historical Returns
Individual Investment Realized Returns
The realized return from your investment in the stock from t to t+1 is:
(Eq. 11.1)

13. Example 11.1 Realized Return
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?
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:

14. 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)

15. 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?

16. Example 11.2 Compounding Realized Returns
Solution:
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.
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.
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

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

18. Example: GM Stock

19. Example: FORD Stock Problem:
What were the realized annual returns for Ford stock in 1999 and in 2008?
Date
Price ($)
Dividend ($)
Return
12/31/1998
58.69
12/31/2007
6.73
1/31/1999
61.44
0.26
5.13%
3/31/2008
5.72
-15.01%
4/30/1999
63.94
4.49%
6/30/2008
4.81
-15.91%
7/31/1999
48.5
-23.74%
9/30/2008
5.2
8.11%
10/31/1999
54.88
0.29
13.75%
12/21/2008
2.29
-55.96%
12/31/1999
53.31
-2.86%

20. Example (cont’d) Solution
Note that, since Ford did not pay dividends during 2008, the return can also be computed as:

21. Realized Return for the S&P 500, GM, and Treasury Bills, 1999–2008

22. 11.2 Historical Risks and Returns of Stocks
Average Annual Returns
Example: S&P 500 Annual Returns
(Eq. 11.3)
𝑅 = ( … +26.5) / 5 = 3.1%

23. 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, 1926–2012

24. 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, 1926–2012,

25. 11.2 Historical Risks and Returns of Stocks
The Variance and Volatility of Returns:
Variance
Standard Deviation
(Eq. 11.4)
(Eq. 11.5)

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

27. Example 11.3 Computing Historical Volatility
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:
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%.

28. 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

29. 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)

30. Frequency Distribution of Returns on Common Stocks, 1926—2009

31. Price Changes and Normal Distribution
Coca Cola - Daily % change
Proportion of Days
Daily % Change

32. The Normal Distribution and Large Company Stock Returns

33. 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?
Solution:
Average ± (2  standard deviation)
= 3.1% – (2  24.1%) to 3.1% + (2  24.1% ) = –45.1% to 51.3%.
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%.

34. 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?
Solution:
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 %.

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

36. Historical tradeoff between risk and return

37. 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
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

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

39. Systematic risk, unsystematic risk, and diversification

40. 11.4 Common Versus Independent Risk
Types of Risk
Common Risk
Independent Risk
An Example: Consider two types of home insurance an insurance company might offer: theft insurance and earthquake insurance

41. 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?
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.

42. 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!
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.
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.

43. 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?
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.

44. 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!
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.
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.

45. Diversification using Historical Returns: ExxonMobil (XOM) and Panera Bread (PNRA)

46. Why Diversification Works,
Covariance:
A measure of how the returns for two securities moves together.
Correlation Coefficient
Correlation is a standardized covariance scaled by the product of two standard deviations, so that it can take a value between -1 and +1. The correlation coefficient is denoted by Corr(RA, RB) or simply, A,B.

47. Figure 11.8 The Effect of Diversification on Portfolio Volatility

48. 11.5 Diversification in Stock Portfolios
Unsystematic vs. Systematic Risk
Stock prices are impacted by two types of news:
Company or Industry-Specific News, or unsystematic risk, diversifiable risk
Market-Wide News, or systematic risk, non-diversifiable risk
Let’s re-define risk more precisely.
Stand-alone Risk (or total risk)
= Market Risk + Diversifiable Risk

49. Systematic Risk
Risk factors that affect a large number of assets such as market-wide news
Also known as non-diversifiable risk or market risk
Includes such things as changes in GDP, inflation, interest rates, etc.

50. Unsystematic Risk
Risk factors that affect a limited number of assets such as firm-specific news
Also known as unique risk, diversifiable risk, idiosyncratic risk, and firm-specific risk
Includes such things as labor strikes, part shortages, etc.

51. Pop Quiz: Systematic Risk or Unsystematic Risk?
The government announces that inflation unexpectedly jumped by 2 percent last month.
Systematic Risk
One of Big Widget’s major suppliers goes bankruptcy.
Unsystematic Risk
The head of accounting department of Big Widget announces that the company’s current ratio has been severely deteriorating.
Congress approves changes to the tax code that will increase the top marginal corporate tax rate.

52. Diversification and Risk

53. 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
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
Table Systematic Risk Versus Unsystematic Risk

54. 11.5 Diversification in Stock Portfolios
The Importance of Systematic Risk
While volatility or standard deviation might be a reasonable measure of risk for a large portfolio, it is not appropriate for an individual security.
There is no relationship between volatility and average returns for individual securities.
Consequently, to estimate a security’s expected return we need to find a measure of a security’s systematic risk.

55. An Intuitive Example
Suppose type S firms are only affected by the systematic risk.
Holding a large portfolio of type S stocks will not diversify the risk.
Suppose type U firms are only affected by the unsystematic risks.
Holding a large portfolio of type U firms will diversify the risk because these risks are firm-specific.
Of course, actual firms are not similar to type S or U firms. Typical firms are affected by both systematic, market-wide risks, as well as unsystematic risks.
When firms carry both types of risk, only the unsystematic risk will be eliminated by diversification when we combine many firms into a portfolio
The volatility will therefore decline until only the systematic risk, which affect all firms, remain.

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

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

58. Expected return and risk: Probability-weighted return

59. Common Measures of Risk and Return
Probability Distributions
When an investment is risky, there are different returns it may earn. Each possible return has some likelihood of occurring. This information is summarized with a probability distribution, which assigns a probability, PR , that each possible return, R , will occur.
Assume BFI stock currently trades for $100 per share. In one year, there is a 25% chance the share price will be $140, a 50% chance it will be $110, and a 25% chance it will be $80.

60. Probability Distribution of Returns for BFI

61. Probability Distribution of Returns for BFI

62. Expected Return Expected (Mean) Return
Calculated as a weighted average of the possible returns, where the weights correspond to the probabilities.

63. Variance and Standard Deviation
The expected squared deviation from the mean
Standard Deviation
The square root of the variance
Both are measures of the risk of a probability distribution

64. Variance and Standard Deviation (cont'd)
For BFI, the variance and standard deviation are:
In finance, the standard deviation of a return is also referred to as its volatility. The standard deviation is easier to interpret because it is in the same units as the returns themselves.

65. Another Example

66. Probability Distributions for BFI and AMC Returns

67. Alternative Example Problem Probability Return .25 8% .55 10% .20 12%
TXU stock is has the following probability distribution:
What are its expected return and standard deviation?
Probability
Return
.25 8%
.55
10%
.20
12%

68. Alternative Example (cont’d)
Solution
Expected Return
E[R] = (.25)(.08) + (.55)(.10) + (.20)(.12)
E[R] = = = 9.9%
Standard Deviation
SD(R) = [(.25)(.08 – .099)2 + (.55)(.10 – .099) (.20)(.12 – .099)2]1/2
SD(R) = [ ]1/2
SD(R) = /2 = = 1.338%

69. Arithmetic average vs. Geometric average

70. Arithmetic Avg. vs. Geometric Avg.
Ex.) Buy a stock for $100. 1st year: falls to $50 2nd year: rises to $100 Historical Avg. = 𝑌𝑒𝑎𝑟𝑙𝑦 𝑅𝑒𝑡. 𝑅 = − = 25% ???
- 50%
+ 100%
0% or 25%?
Geometric Avg. Return
What was your avg. compound return per year?
Arithmetic Avg. Return
What was your returns in an average (≈ typical) year?

71. Arithmetic Avg. vs. Geometric Avg.
Arithmetic Avg. = 𝑅𝑖 𝑁 =25% Geometric Avg. = [(1+R1)(1+R2)…(1+RN)]1/N -1 = [(1-.5)(1+1)] 1/2 - 1 = [(.5)(2)]1/2 -1 = [1]1/2 – 1 = 0%

72. Arithmetic Averages versus Geometric Averages
The arithmetic average return answers the question: “What was your return in an average year over a particular period?”
The geometric average return answers the question: “What was your average compound return per year over a particular period?”
When should you use the arithmetic average and when should you use the geometric average?
First, we need to learn how to calculate a geometric average.

73. Example: Calculating a Geometric Average Return

74. Arithmetic Averages versus Geometric Averages
The arithmetic average tells you what you earned in a typical year.
The geometric average tells you what you actually earned per year on average, compounded annually.
When we talk about average returns, we generally are talking about arithmetic average returns.
For the purpose of forecasting future returns:
The arithmetic average is probably "too high" for long forecasts.
The geometric average is probably "too low" for short forecasts.

75. Geometric versus Arithmetic Averages, 1926-2006, 1926-2012
Source: Jordan, Miller, and Dolvin (2012) textbook

76. 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?