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Chapter 1
A Brief History of Risk and Return
Slides
1.1. Chapter 1
1.2. A Brief History of Risk and Return
1.3. Learning Objectives
1.4. Example I: Who Wants To Be A Millionaire?
1.5. Example II: Who Wants To Be A Millionaire?
1.6. A Brief History of Risk and Return
1.7. Dollar Returns
1.8. Percent Returns
1.9. Example 1.1: Concannon Plastics Calculating Total Dollar and Total
Percent Returns
1.10. Annualizing Returns, I
1.11. Annualizing Returns, II
1.12. A $1 Investment in Different Types of Portfolios, 1926—2015
1.13. Financial Market History, 1801-2015
1.14. The Historical Record: Total Returns on Large-Company Stocks
1.15. The Historical Record: Total Returns on Small-Company Stocks
1.16. The Historical Record: Total Returns on Long-term U.S. Bonds
1.17. The Historical Record: Total Returns on U.S. T-bills
1.18. The Historical Record: Inflation
1.19. Historical Average Returns
1.20. Average Annual Returns for Five Portfolios and Inflation, 1926—2015
1.21. 2008: The Bear Growled and Investors Howled
1.22. World Stock Market Capitalization
1.23. Average Annual Returns and Risk Premiums for Five Portfolios, 1926—2015
1.24. Average Returns: The First Lesson
1.25. Why Does a Risk Premium Exist?
1.26. Return Variability Review and Concepts
1.27. Frequency Distribution of Returns on Common Stocks, 1926—2015
1.28. Return Variability: The Statistical Tools
1.29. Example: Calculating Historical Variance and Standard Deviation
1.30. Historical Returns, Standard Deviations, and Frequency Distributions: 1926
—2015
1.31. The Normal Distribution and Large Company Stock Returns
1.32. Good Times for the Dow Jones Index
1.33. Bad Times for the Dow Jones Index
1.34. Arithmetic Averages versus Geometric Averages
1.35. Example: Calculating a Geometric Average Return
1.36. Geometric versus Arithmetic Averages, 1926—2015
1.37. Arithmetic Averages versus Geometric Averages
1.38. Dollar-Weighted Average Returns, I
1.39. Dollar-Weighted Average Returns, II
1.40. Dollar-Weighted Average Returns and IRR
1.41. Risk and Return
1.42. Historical Risk and Return Trade-Off
1.43. A Look Ahead
1.44. Useful Internet Sites
1.45. Chapter Review, I
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McGraw-Hill
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A Brief History of Risk and Return 1-2
1.46. Chapter Review, II
Chapter Organization
1.1 Returns
A. Dollar Returns
B. Percentage Returns
C. A Note on Annualizing Returns
1.2 The Historical Record
A. A First Look
B. A Longer Range Look
C. A Closer Look
D. 2008: The Bear Growled and Investors Howled
1.3 Average Returns: The First Lesson
A. Calculating Average Returns
B. Average Returns: The Historical Record
C. Risk Premiums
D. The First Lesson
1.4 Return Variability: The Second Lesson
A. Frequency Distributions and Variability
B. The Historical Variance and Standard Deviation
C. The Historical Record
D. Normal Distribution
E. The Second Lesson
1.5 More on Average Returns
A. Arithmetic versus Geometric Averages
B. Calculating Geometric Average Returns
C. Arithmetic Average Return or Geometric Average Return?
D. Dollar-Weighted Average Returns
1.6 Risk and Return
A. The Risk-Return Trade-off
B. A Look Ahead
1.7 Summary and Conclusions
Selected Web Sites
finance.yahoo.com (reference for authors’ favorite financial web site)
www.globalfinancialdata.com (reference for historical financial market data
—not free)
www.mhhe.com/jmd8e (the website for this text)
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McGraw-Hill
Education.
A Brief History of Risk and Return 1-3
Annotated Chapter Outline
1.1 Returns
This chapter uses financial market history to provide information about risk and
return. In general, two key observations emerge:
There is a reward for bearing risk and, on average, the reward has been
considerable.
Greater rewards are accompanied by greater risks.
The important point is that risk and return are always linked together.
A. Dollar Returns
Total dollar return: the return on an investment measured in dollars that
accounts for all cash flows and capital gains or losses.
When you buy an asset, your gain or loss is called the return on your investment.
This return is made up of two components:
The cash you receive while you own the asset (interest or dividends), and
The change in value of the asset, the capital gain or loss.
The total dollar return is the sum of the cash received and the capital gain or loss
on the investment. Whether you sell the stock or not, this is a real gain because
you had the opportunity to sell the stock at any time.
B. Percentage Returns
Total percent returns: the return on an investment measured as a
percentage of the original investment that accounts for all cash flows and
capital gains or losses
When you calculate percent returns, your return doesn't depend on how much
you invested. Percent returns tell you how much you receive for every dollar
invested. There are two components of the return:
Dividend yield, the current dividend divided by the beginning price
Capital gains yield, the change in price divided by the beginning price
C. A Note on Annualizing Returns
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A Brief History of Risk and Return 1-4
To compare investments, we need to “annualize” the returns, which we refer to
as the Effective Annual Return (or EAR).
1 + EAR = (1 + holding period return)m
Where m is the number of holding periods in a year.
1.2 The Historical Record
The year-to-year historical rates of return on five important categories of
investments are analyzed in this section. These categories are:
Large-company stocks, which is based on the Standard & Poor's 500 index
(S&P 500).
Small-company stocks, where "small" corresponds to the smallest 20% of the
companies listed on the major U.S. exchanges, as measured by the market
value of outstanding stock.
Long-term corporate bonds, which is a portfolio of high-quality bonds with 20
years to maturity.
Long-term U.S. government bonds, which is a portfolio of U.S. government
bonds with 20 years to maturity.
U.S. Treasury bills (T-bills) with a three-month life.
The annual percentage changes in the Consumer Price Index (CPI) are also
calculated as a comparison to consumer goods price inflation.
A. A First Look
When we examine the returns on these categories of investments from 1926
through 2015, we see that the small-company investment grew from $1 to
$24,113, the larger common stock portfolio to $4,955, the long-term government
bonds to $121, and T-bills to $22. Inflation caused the price of an average
consumer good to grow from $1 to $13 over the 90 years. An obvious question
resulting from examining this graph would be, "Why would anyone invest in
anything other than small-company stocks?" The answer lies in the higher
volatility of the small- company stocks. This topic will be discussed later in the
chapter.
B. A Longer Range Look
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McGraw-Hill
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A Brief History of Risk and Return 1-5
When we look at a longer term, back to 1801, we see that the return from
investing in stocks is much higher than investing in bonds or gold. Over this 215-
year period, one dollar invested in stocks grew to an astounding $21.9 million,
whereas bonds only returned $39,134, and gold (until the past few years) has
simply kept up with inflation. The moral is, "Start investing early."
C. A Closer Look
As you examine the bar graphs you can observe that the return on stocks,
especially small-company stocks, was much more variable than bonds or T-bills.
The returns on T-bills were much more predictable than stocks. Although the
largest one-year return was 143% for small-company stocks and 53% for large-
company stocks, the largest T-bill return was only 14.6%. The largest historical
return for long-term government bonds was 47.14%, which occurred in 1982.
D. 2008: The Bear Growled and Investors Howled
The S&P 500 index plunged -37 percent in 2008, which is behind only 1931 at
-44 percent. Moreover, there were 18 days during 2008 on which the value of the
S&P changed by more than 5 percent. From 1956 to 2007 there were only 17
such days.
1.3 Average Returns: The First Lesson
This section provides simple measures to accurately summarize and describe all
of these numbers, starting with calculating average returns.
A. Calculating Average Returns
The simplest way to calculate average returns is to add up the annual returns
and divide by the number of years. This will provide the historical average. So,
the average return for the large-company stocks over the 90 years is 11.9%.
B. Average Returns: The Historical Record
Table 1.2 also shows that small-company stocks had an average return of 17.5%,
government bonds returned 6.5% on average, and T-bills only returned 3.6%.
Note that the return on T-bills is just slightly more than the inflation rate of 3.0%.
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A Brief History of Risk and Return 1-6
C. Risk Premiums
Risk-free rate: the rate of return on a riskless investment.
Risk premium: the extra return on a risky asset over the risk-free rate.
The rate of return on T-bills is essentially risk free because there is no risk of
default. So we will use T-bills as a proxy for the risk-free rate, our investing
benchmark. If we consider T-bills as risk-free investing and investing in stocks as
risky investing, the difference between these two returns would be the risk
premium for investing in stocks. This is the additional return we receive for
investing in the risky asset, or the reward for bearing risk.
The U.S. Equity Risk Premium: Historical and International
Perspectives: Earlier periods suggest a lower risk premium than in recent
periods, while international risk premiums also tend to be slightly lower.
Based on evidence and expectations, 7 percent seems to be a reasonable
estimate for the risk premium.
D. The First Lesson
When we calculate the risk premium for large-company stocks (stock return
minus the T-bill return) we get 8.3% and for government bonds 2.6%. Of course,
the risk premium for T-bills is zero. So we see that risky assets, on average, earn
a risk premium, or "there is a reward for bearing risk." The next question is, "Why
is there a difference in the risk premiums?" This is addressed in the next section
and relates to the variability in returns.
1.4 Return Variability: The Second Lesson
A. Frequency Distributions and Variability
Variance: a common measure of volatility.
Standard deviation: the square root of the variance.
Variance and standard deviation provide a measure of return volatility or how
much the actual return differs from this average in a typical year. This is the same
variance and standard deviation discussed in statistics courses.
B. The Historical Variance and Standard Deviation
The variance measures the average squared difference between the actual
returns and the average return. The larger this number, the more the actual
returns differ from the average return. Note how the stocks have a much larger
standard deviation than the bonds and were therefore more volatile.
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A Brief History of Risk and Return 1-7
Lecture Tip: Although all students should have been exposed to calculating
variance and standard deviation, many still have difficulty with it. Another method
to illustrate how to calculate the variance is to structure it in the form of a table
where each step is a separate column. This is illustrated below using the data in
the text.
For 1926-1930, the average return for large-company stocks (as represented by
the S&P 500) = (11.14 + 37.13 + 43.31 – 8.91 – 25.26) / 5 = 57.41 / 5 = 11.48%
RNRN - RA(RN - RA)2
11.14% 11.14 -11.48 = -.34 (-.34)2=.12
37.13% 37.13 -11.48= 25.65 (25.65)2=657.82
43.31% 43.31 -11.48= 31.83 (31.83)2=1,013.02
-8.91% -8.91 -11.48 = -20.39 (-20.39)2=415.83
-25.26% -25.26 -11.48 = -36.74 (-36.74)2=1,349.97
Sum of squares = 3,436.77
(RN - RA)2 / (N-1) = 3,436.77/(5-1) = 859.19 Variance
Square root = 29.31% Standard
deviation
Lecture Tip: Note the difference between using N-1 and N as the divisor when
calculating variance and standard deviation. You use N when you have the entire
population, as opposed to N-1 when you have a sample of the population.
Lecture Tip: After calculating variance and standard deviation, ask what units
are attached to each. The students will most likely have to puzzle on this. Of
course, variance is percent squared, whereas standard deviation is percent. This
provides a starting point for the discussion on how to interpret the resulting value
for standard deviation.
Lecture Tip: You may want to point out that this example calculates variance
and standard deviation using historical data. When expected futures values are
used, there is another method that must employ probabilities. This method will be
discussed in a later chapter.
C. The Historical Record
The standard deviation for the large-company stock portfolio is more than six
times the standard deviation for the T-bill portfolio. Also notice that the distribution
is approximately normal. This allows us to use the fact that plus or minus one
standard deviation from the mean return gives us the range of returns that would
result 2/3 of the time. If we take plus or minus two standard deviations from the
mean, there is a 95% probability that our investment will be within this range of
returns.
D. Normal Distribution
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A Brief History of Risk and Return 1-8
Like most statistical concepts, students will struggle remembering the concept of
a normal distribution. In our experience, this is mostly because they are unsure of
their understanding—not that they have not “seen” the material before.
For many different random events in nature, a particular frequency distribution,
the normal distribution (or bell curve) is useful for describing the probability of
ending up in a given range. For example, the idea behind “grading on a curve”
comes from the fact that exam scores often resemble a bell curve.
Figure 1.10 illustrates a normal distribution and its distinctive bell shape. As you
can see, this distribution has a much cleaner appearance than the actual return
distributions illustrated in Figure 1.8. Even so, like the normal distribution, the
actual distributions do appear to be at least roughly mound shaped and
symmetric. When this is true, the normal distribution is often a very good
approximation.
Also, you will have to remind students that the distributions in Figure 1.9 are
based on only 90 yearly observations, while Figure 1.10 is, in principle, based on
an infinite number. So, if we had been able to observe returns for, say, 1,000
years, we might have filled in a lot of the irregularities and ended up with a much
smoother picture. For our purposes, it is enough to observe that the returns are
at least roughly normally distributed.
The usefulness of the normal distribution stems from the fact that it is completely
described by the average and the standard deviation. If you have these two
numbers, then there is nothing else to know. For example, with a normal
distribution, the probability that we end up within one standard deviation of the
average is about 2/3. The probability that we end up within two standard
deviations is about 95 percent. Finally, the probability of being more than three
standard deviations away from the average is less than 1 percent.
E. The Second Lesson
Observing that there is variability in returns from year-to-year, we see that there
is a significant chance of a large change in value in the returns. So the second
lesson is: The greater the potential reward, the greater the risk.
1.5 More on Average Returns
A. Arithmetic versus Geometric Averages
The geometric average return answers the question: “What was your average
compound return per year over a particular period?”
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A Brief History of Risk and Return 1-9
The arithmetic average return answers the question: “What was your return in an
average year over a particular period?”
B. Calculating Geometric Average Returns
Let us use data from the example above to calculate an arithmetic average and a
geometric average:
RN1+RN1+RN,1 x 1+RN,2 x
…
11.14% 1.1114 1.1114
37.13% 1.3713 1.5241
43.31% 1.4331 2.1841
-8.91% .9109 1.9895
-25.26% .7474 1.4870
Sum: 57.41 Raised to 1/5th
Power: 1.0826
Arithmetic Geometric
Average: 11.48% Average: 8.26%
C. Arithmetic Average Return or Geometric Average Return?
Two points are worth stressing:
First, generally, when one sees a discussion of “average returns,” the return in
question is an arithmetic return.
Second, there is a nettlesome problem concerning forecasting future returns
using estimates of arithmetic and geometric returns. The problem is: arithmetic
average returns are probably too high for longer periods, and geometric average
returns are probably too low for shorter periods. Fortunately, Blume’s formula
provides a way to weight arithmetic and geometric averages for a T-year average
return forecast using arithmetic and geometric averages which have been
calculated for an N-year period (T cannot exceed N).
Blume’s formula is:
AverageArithmetic
1N
TN
AverageGeometric
1N
1T
R(T)
As is readily apparent from this formula, as T (the length of time of the forecast)
increases, the geometric average receives a higher weight relative to the
arithmetic average. That is, if N = T, the arithmetic average receives no weight,
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A Brief History of Risk and Return 1-10
and the resulting forecast stems entirely from the geometric average. If T = 1,
then the geometric average receives a zero weight. In this case, the resulting
forecast comes only from the arithmetic average.
D. Dollar-Weighted Average Returns
If an investor adds money to or subtracts money from an account, his actual
return will likely be different than either the arithmetic or geometric average. The
dollar weighted return (or internal rate of return, IRR) captures the impact of cash
flows, giving the average compound rate of return earned per year.
1.6 Risk and Return
A. The Risk-Return Trade-off
If we are unwilling to take on any risk, but we are willing to forego the use of our
money for a while, then we can earn the risk-free rate. We can think of this as the
time value of money. If we are willing to bear risk, then we can expect to earn a
risk premium, on average. We can think of these two factors as the "wait"
component and the "worry" component.
Notice that the risk premium is not guaranteed; it is "on average." Risky
investments by their very nature of being risky do not always pay more than risk-
free investments. Also, only those risks that are unavoidable are compensated by
the risk premium. There is no reward for bearing avoidable risk.
B. A Look Ahead
The remainder of the text focuses on financial assets only: stocks, bonds, options
and futures. Remember that to understand the potential reward from an
investment, you must understand the risk involved.
1.7 Summary and Conclusions
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