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Chapter 6
AN OVERVIEW
The purpose of Chapter 6 is to present an analysis of risk and return early enough in the
text for these concepts to be used throughout the book. Return and risk are the key elements of
investment decisions--in effect, everything else revolves around these two factors. It makes
sense, therefore, to analyze and discuss these concepts in detail.
Chapter 6 focuses only on understanding and measuring realized returns and wealth.
This allows students to concentrate on this one issue in a comprehensive manner. All of the
equations for calculating the various types of returns needed in a basic Investments course are
included in this chapter. Beginning students are unlikely to use anything beyond what is
contained here with regard to realized returns.
Chapter 6 provides a complement to Chapter 7, which covers expected returns and risk
and the basic calculations of portfolio theory. Thus, in Chapter 6 we analyze and calculate
realized returns, while in Chapter 7 we analyze and calculate expected returns, based on
probability distributions.
This discussion centers on the definition and meaning of return and risk including the
components of return, the sources of risk, and types of risk. The emphasis is on how to both
understand and measure return and risk. Considerable attention is devoted to explaining the
total return (TR), return relative (RR), and cumulative wealth calculations, which are used
throughout this text and are comparable to the definitions used in such prominent sources as the
Ibbotson Classic Yearbook. Numerous examples are presented.
The discussion of returns measures facilitates the presentation of the data on rates of
return and wealth indexes. This data is both important (as benchmarks) and interesting (it can be
the basis of lively class discussion). The data used here were collected and calculated by the
author, and correspond closely with the Ibbotson data.
The use of the geometric mean is fully explored, along with wealth indexes. Although
challenging, this material is important. Calculations include measuring the yield component and
capital gains component of total returns and cumulative wealth separately, measures of inflationadjusted returns, and risk premiums.
Definitions of risk are presented and discussed. While examples include calculating the
standard deviation, the emphasis here is on understanding and using it.
This chapter contains an extensive problem set.
NOTE: The data for returns on major asset classes is unique, based on calculations over many
years by Jack W. Wilson and Charles P. Jones. In particular, the geometric mean, arithmetic
mena and standard deviation for the S&P 500 Index differs from the numbers given in the
Ibbotson Classic Yearbook. The reason for this is that Wilson and Jones have calculated a more
comprehensive set of returns for the years 1926-1956 for the S&P 500 than is typically used.
The Ibbotson Classic Yearbook uses only 90 companies during this time period because that was
the data set available to researchers until Wilson and Jones developed the expanded set of
companies. The 90 companies typically used included mostly large companies, which performed
better than the full set of companies during the Depression and many of the years thereafter.
Therefore, the means presented in the Ibbotson Classic Yearbook are larger than those presented
here. Wilson and Jones have argued that the data for the 90 companies overstates the return for
the S&P 500 during that time period.
CHAPTER OBJECTIVES
To explain the meaning and measurement of both return and risk.
To illustrate the use of such measures as the geometric mean and standard deviation.
To present the well-known data on rates of return for major financial assets for long
periods of time.
To present and illustrate virtually all the calculations needed for a thorough
understanding of return and risk.
MAJOR CHAPTER HEADINGS [Contents]
An Overview
Return
The Two Components of Return
[Yield; Capital gain/loss; total return = the sum of these two; examples]
Measuring Returns
Total Return
[definition; explanation; examples using the S & P 500 Index]
Return Relative
[definition; example]
Cumulative Wealth Index
[definition; example; relation between cumulative wealth and total return]
Taking a Global Perspective
International Returns and Currency Risk
[how currency changes affect investors; calculating currency-adjusted returns; the
dollar and investors]
Summary Statistics for Returns
Arithmetic Mean
Geometric Mean
Arithmetic Mean Versus Geometric Mean
Inflation-Adjusted Returns
[definition; relation between nominal return and real return; the CPI]
Risk
Sources of Risk
[sources include: interest rate; market; inflation; business; financial; liquidity;
exchange rate; country]
Measuring Risk
Variance and Standard Deviation
[definition; formulas; example]
Risk Premiums
[definition; equity risk premium; calculation; the expected risk premium]
Realized Returns And Risks From Investing
Total Returns and Standard Deviations for the Major Financial Assets
[linkage between arithmetic and geometric mean; data on rates of return for major
asset classes]
Cumulative Wealth Indexes
[cumulative wealth graph showing major financial assets; inflation-adjusted
cumulative wealth; the yield and price change components of cumulative wealth;
compounding and discounting]
The Components of Cumulative Wealth and the Geometric Mean
[the cumulative price change; the cumulative dividend yield; how the components go
together]
Compounding and Discounting
POINTS TO NOTE ABOUT CHAPTER 6
Exhibits, Figures and Tables
Exhibit 6-1 should be reviewed carefully with students as examples of how to calculate
total returns and return relatives for three different securities. This is an important calculation for
the entire course, and students should be very comfortable with doing such calculations.
Table 6-1 shows annual S&P 500 data--prices and dividends--from the beginning of 1926
through the latest year possible. Calculated total returns for each year are presented. This table
provides a good source of data for discussions throughout the text involving market returns as
measured by the S&P 500 Index. These data were calculated and compiled by Jack Wilson and
Charles Jones, and have been used in several articles.
Table 6-2 illustrates the impact of currency movements on an investor’s returns.
Tables 6-3 and 6-4 involve the calculation and interpretation of the arithmetic and
geometric means using TRs. Instructors should stress the meaning of the geometric mean.
Figure 6-1 shows the spread in returns for the major financial assets covered in Table 6-6.
As we would expect, both stock categories have wider spreads than do bonds, and small stocks
have a wider dispersion than does the S&P 500.
Table 6-5 (for historical data) shows calculations for the standard deviation and can be
handled by students on their own or emphasized by instructors to the extent thought necessary.
Table 6-6 is an important table on rates of return and should be used in class discussion.
This table is important for numerous reasons: investors need to know the historical return series
for benchmark purposes, it illustrates the nature of the return--risk tradeoff, and it allows you to
talk about the variability in returns over time by analyzing the arithmetic and geometric means as
well as the standard deviations presented in the table. In addition, other points can be developed,
such as the “small” stock effect, and so forth.
NOTE: This table corresponds quite well with the comparable table from the Ibbotson Classic
Yearbook. The major difference is the use of more stocks for the years 1926-1956 as compared
to the 90 stocks which Ibbotson uses.
Figure 6-2 shows cumulative wealth indices for the major financial assets since the
beginning of 1926. This is a good source of class discussion because students find this
interesting—how much $1 can compound to over time. Instructors may wish to emphasize how
these values are calculated (which is covered in the chapter).
ANSWERS TO END-OF-CHAPTER QUESTIONS
6-1.
Historical returns are realized returns, such as those reported by Ibbotson Associates.
Expected returns are returns expected to occur in the future. They are the most likely
returns for the future, although they may not actually be realized because of risk.
6-2.
A Total Return can be calculated for any asset for any holding period. Both monthly and
annual TRs are often calculated, but any desired period of time can be used.
6-3.
Total return for any security consists of an income (yield) component and a capital gain
(or loss) component.
• The yield component relates dividend or interest payments to the price of the security.
•
The capital gain (loss) component measures the gain or loss in price since the security
was purchased.
While either component can be zero for a given security over a specified time period,
only the capital change component can be negative.
6-4.
TR, another name for holding period return, is a decimal or percentage return, such as
+.10 or -15%. The term “holding period return” is sometimes used instead of TR.
Return relative adds 1.0 to the TR in order that all returns can be stated on the basis of
1.0 (which represents no gain or loss), thereby avoiding negative numbers so that the
geometric mean can be calculated.
6-5.
The geometric mean is a better measure of the change in wealth over more than a single
period. Over multiple periods the geometric mean indicates the compound rate of return,
or the rate at which an invested dollar grows, and takes into account the variability in the
returns.
The geometric mean is always less than the arithmetic mean because it allows for the
compounding effect--the earning of interest on interest.
6-6.
The arithmetic mean should be used when describing the average rate of return without
considering compounding. It is the best estimate of the rate of return for a single period.
Thus, in estimating the rate of return for common stocks for next year, we use the
arithmetic mean and not the geometric mean. The reason is that because of variability in
the returns, we will have to earn, on average, the arithmetic rate in order to achieve a
compound rate of growth which is given by the smaller geometric mean.
6-7.
See Equation 6-11. Knowing the arithmetic mean and the standard deviation for a
series, the geometric mean can be approximated.
6-8.
An equity risk premium is the difference between stocks and a risk-free rate (proxied by
the return on Treasury bills). It represents the additional compensation, on average, for
taking the risk of equities rather than buying Treasury bills.
6-9.
As Table 6-6 shows, the risk (standard deviation) of large common stocks was about two
and one-half times that of government and corporate bonds. Therefore, common stocks
are clearly more risky than bonds, as they should be since larger returns would be
expected to be accompanied by larger risks over long periods of time.
6-10. Market risk is the variability in returns due to fluctuations in the overall market. It
includes a wide range of factors exogenous to securities themselves.
Business risk is the risk of doing business in a particular industry or environment.
Interest rate risk and inflation risk are clearly directed related. Interest rates and inflation
generally rise and fall together.
6-11. Systematic risk: market risk, interest rate risk, inflation risk, exchange rate risk, and
country risk.
Nonsystematic risk: business risk, financial risk, and liquidity risk.
6-12. Country risk is the same thing as political risk. It refers to the political and economic
stability and viability of a country’s economy. The United States can be used as a
benchmark with which to judge other countries on a relative basis.
Canada would be considered to have relatively low country risk. Mexico seems to be on
the upswing economically, but certainly has its risk in the form of nationalized
industries, overpopulation, drug cartels and other issues. Mexico also experienced a
dramatic devaluation of the peso.
6-13. The return on the Japanese investment is now worth less in dollars. Therefore, the
investor’s return will be less after the currency adjustment.
EXAMPLE: Assume an American investor in the Japanese market has a 30% gain in
one year but the Yen declines in value relative to the dollar by 10%. The percentage of
the original investment after the currency risk is accounted for is (0.9)(130%) = 117%.
Therefore, the investor’s return is 17%, not 30%. In effect, the investor loses 10% on the
original wealth plus another 10% on the 30% gain, or a total of 13 percentage points of
the before-currency-adjustment return of 130% of investment.
6-14. Risk is the chance that the actual outcome from an investment will differ from the
expected outcome. Risk is often associated with the dispersion in the likely outcomes.
Dispersion refers to variability, and the standard deviation is a statistical measure of
variability or dispersion.
Standard deviation measures risk in an absolute sense.
6-15. A wealth index measures the cumulative effect of returns over time, typically on the
basis of $1 invested. It measures the level rather than changes in wealth.
The geometric mean is the nth root of the cumulative wealth index. Alternatively, adding
1.0 to the decimal value of the geometric mean and raising this number to the nth power
produces the cumulative wealth index.
6-16. No. Although the long-run average (where long-run is on the order of 75 or 80 or more
years) has consistently been around 10 percent, deviations from that average do occur,
even for periods of 10 years or so. The decade 2000-2009 saw a low average return on
stocks because of the market decline in 2000-2002 and the financial crisis of 2008.
6-17. Dividing 1.0 + the geometric mean return for common stocks by 1.0 + the geometric
mean for inflation for a given period, and subtracting out the 1.0, produce the inflationadjusted rate of return.
6-18. The two components of the cumulative wealth index are the yield (income) component
and the price change (capital gain or loss) component. Multiplying these two
components together produces cumulative wealth. Knowing one of these components,
the other can be calculated by dividing the known component into the cumulative wealth
index number.
6-19. No. These relationships are not linear, nor is there any reason why they should be. The
risk on common stocks relative to bonds has been more than twice as great.
6-20. Yes. Cumulative wealth can be stated on either a nominal or real (inflation-adjusted)
basis.
CFA
6-21. The following risk exposures should be reported as part of an Enterprise Risk
Management System for Ford Motor Company:

Market risks
o Currency risk, because expenditures and receipts denominated in nondomestic
currencies create exposure to changes in exchange rates.
o Interest rate risk, because the values of securities that Ford has invested in are
subject to changes in interest rates. Also, Ford has borrowings and loans, which
could be affected by interest rate changes.
o Commodity risk, because Ford has exposure in various commodities and finished
products.
o Credit risk, because of financing provided to customers who have purchased
Ford’s vehicles on credit.
o Liquidity risk, because of the possibility that Ford’s funding sources may be
reduced or become unavailable and Ford may then have to sell its securities at a
short notice with a significant concession in price.
o Settlement risk, because of Ford’s investments in fixed-income instruments and
derivative contracts, some of which effect settlement through the execution of
bilateral agreements and involve the possibility of default by the counterparty
o Political risk, because Ford has operations in several countries. This exposes it to
political risk. For example, the adoption of a restrictive policy by a non-U.S.
government regarding payment of dividends by a subsidiary in that country to our
parent company could adversely affect Ford.
CFA
6-22. (a) We calculate the sample mean by finding the sum of the 10 values in the table and
then dividing by 10. Thus the arithmetic mean return R =
10
∑ Ri
R=
t=1
-------------------
10
(46.21-6.1+8.04+22.87+45.90+20.32+41.20-9.53-17.75-43.06)/10=108.02/10=10.802 or
10.80 percent.
(b). The geometric mean requires that all the numbers be greater than or equal to 0. To
ensure that the returns satisfy this requirement, after converting the returns to
decimal form we add 1 to each return. We use the equation for the geometric mean
return, Rg.
10
Rg = [Π (1 + Ri)]1/10 - 1
t=1
Which can also be written as
Rg =
- 1
To find the geometric mean in this example we take the following five steps:
i. Divide each figure in the table by 100 to put the returns into decimal representation.
ii. Add 1 to each return to obtain the terms 1 +Rt
Return
46.21%
-6.18%
8.04%
22.87%
45.90%
20.32%
41.20%
-9.53%
-17.75%
-43.06%
Return in Decimal Form
0.4621
-0.0618
0.0804
0.2287
0.4590
0.2032
0.4120
-0.0953
-0.1775
-0.4306
1 + Return
1.4621
0.9382
1.0804
1.2287
1.4590
1.2032
1.4120
0.9047
0.8225
0.5694
iii. Multiply together all the numbers in the third column to get 1.9124
iv. Take the 10th root of 1.9124 to get
=1.0670. On most calculators, we evaluate
x
using the y key. Enter 1.9124 with the yx key. Next enter 1/10=0.10. Then press the
= key to get 1.0670.
v. Subtract 1 to get 0.0670, or 6.70 percent a year. The geometric mean return is 6.70 percent.
This result means that the compound annual rate of growth of the MSCI Germany Index was 6.7
percent annually during the 1993-2002 period. Note that this value is much less than the
arithmetic mean of 10.80 percent that we calculated in the solution to (a).
CFA
6-23.
A. The time-weighted rate of return for the investment manager is:
Rtwr = (1+0.0125)(1+0.0347)(1 + [-0.0236])(1+0.0189)
(1+[-0.0267])(1+0.0257) – 1
= 0.0405 or 4.05%
B. Adding the subperiod rates of return gives 0.0125 + 0.0347 + (-0.0236) + 0.0189 + (-0.0267)
+ 0.0257 = 0.0415 or 4.15 percent
Characteristically, the additive calculation gives a higher return number (4.15 percent)
than the time-weighted calculation (4.05 percent). In general, the time-weighted rate of
return is a better indicator of long-term performance because it takes account of the effects of
compounding.
ANSWERS TO END-OF-CHAPTER PROBLEMS
6-1.
Calculating Total Returns (TRs) for these assets:
(a) TRps = (Dt + (PE - PB)) / PB
where Dt = the preferred dividend
PE = ending price or sale price
PB = beginning price or purchase price
TR = (5 + -8) / 70
= -4.29%
(b) TRw = (Ct + PC) / PB
where Ct is any cash payments paid (there are none for a warrant)
PC = price change during the period
TR = (0 + 3)/10
= 30% for the three month period
(c) TRb = (It + PC) / PB
= (240* + 90) / 830
= 39.76% for the two year period.
*interest received is $120 per year (12% of $1000) for two years.
Calculating Return Relatives (RRs) for these examples:
(a) a TR of -4.29% is equal to a RR of .9571 or (1.0+ [-.0429])
(b) a TR of 30% is equal to a RR of 1.3
(c) a TR of 39.76% is equal to a RR of 1.3976
6-2.
Returns for the S&P 500 for 2001-2004 are -11.85%, -22.10%, 28.37%, and 10.75%.
For 2005 the return was 4.83%.
The arithmetic mean for 2001-2004 was 1.29%; the geometric mean, -0.60%.
The arithmetic mean for 2001-2005 was 2%; the geometric mean, 0.46%.
6-3.
$1 (.9686)(1.3)(1.0743)(1.0994)(1.0129)(1.3711)(1.2268)(1.3310)
(1.2836) (1.2087) = 5.2324= the cumulative wealth index for this period.
(5.2324)1/10 = 1.18
1.18-1= .18 = 18%
6-4.
The geometric mean for the S&P 500 was 10.05% for 1926-2007, a period of 82 years.
Raise 1.1005 to the 82nd power to obtain $2,572.67.
The value of the trust would be $100,000 X 2,572.67 = $257,267,000.
6-5.
The geometric mean for small stocks was 12.47%.
Raise 1.1247 to the 82nd power to obtain $15,311.19
The value of the trust would be $100,000 X 15,311.19 = $1,531,190,000.
6-6.
The geometric mean for Treasury bonds was 5.22%.
Raise 1.0522 to the 82nd power to obtain $64.8725
The value of the trust would be $100,000 X 64.8725 = $6,487,246.37.
6-7.
The geometric mean for Treasury bills was 3.71%
Raise 1.0371 to the 82nd power to obtain 19.8286.
The value of the trust would be $100,000 X 19.8286 = $ 1,982,860.
6-8.
The geometric mean for corporates is 5.89%. 1.0589 raised to the 82nd power =
109.1706. Therefore, cumulative wealth per dollar invested over this period was
$109.17.
6-9.
Given a cumulative wealth of $19.90 for Treasury bills for 1926-2007, the geometric
mean would be 3.71%. This is determined by taking the 82nd root of 19.90 and
subtracting the 1.0.
6-10. The nominal cumulative wealth for small common stocks for 1926-2007 is given as
$14,315. Raise the inflation rate (as a decimal) + 1.0, which is 1.0405, to the 82nd power to
obtain 25.9329.
Inflation-adjusted cumulative wealth for small common stocks = $14,315 / 25.9329 =
$552.0015.
6-11. Divide 72 by 4 to obtain 18 years.
6-12. Divide ending value by beginning value to obtain 13.71, and take the 82nd root, then
subtract 1.0 to obtain
3.24% (rounded).
6-13. (1.04)82 = 24.93 = cumulative wealth index for the yield component
$2,572.67 (Figure 6-2) is the cumulative wealth index value for stocks, year-end 2007.
$2,572.67 / 24.93 = 103.19 = cumulative wealth index value for the capital gain or price
change component.
6-14. Raise 1.1247 to the 100th power to obtain $126,960.76. This would be the cumulative
wealth per dollar invested.
6-15. The two ways to calculate inflation-adjusted returns are:
1. 1.0522 / 1.0405 = 1.0112; (1.0112)82 = 2.5016
2. (1.0522)82 = 64.8725; (1.0405)82 = 25.9329; 64.8725 / 25.9329= 2.5016
6-16. The 6 TRs for 1926-1931 are, in order starting with 1926: 8.2%, 32.76%, 38.14%,
-10.18%, -26.48%, and -43.49%. Converting to RRs, we have: 1.082, 1.3276, 1.3814,
.8982, .7352, and .5651. Multiply these 6 RRs together to obtain .7405. Take the 6th root
and subtract the 1. 0 to obtain .9512. Subtract .9512 from 1.0 to convert to a decimal
geometric mean of -0.0488, or -4.88%.
Of course, if we already know that the ending wealth index (cumulative wealth index) for
the six years 1926-1931 is 0.7405, we can calculate the geometric mean by taking the
sixth root of this wealth index and subtracting out the 1.0 to obtain the same answer as
above, -0.0488, or -4.88%.
6-17. Any set of TRs that are identical will produce a geometric mean equal to the arithmetic
mean; for example, 10%, 10% and 10%, or any other set of three identical numbers.
6-18. The calculated results for 1981-1991 are:
Arithmetic Mean
Standard Deviation
Geometric Mean
15.77%
13.15%
15.07%
As we can see, the standard deviation for the shorter period was less than that of the
entire period. This is because of the good years in the 1980s that were more similar than
in a typical 10 or 11 year period. Also, there were only two negative years during this
period, whereas the historical norm for many years was 3 negative years out of 10 (this
did not occur in the 1990s).
6-19. A compound rate of return of 10.4% for 10 years has a cumulative effect of 1.104 raised
to the 10th power, or 2.6896. Therefore, $20,000 would grow to $20,000 X 2.6896 =
$53,792. Thus, the $20,000 portfolio would be the better alternative.
COMPUTATIONAL PROBLEMS
6-1.
First, convert the TRs to Return Relatives: .909, .881, .779, 1.287, and 1.107.
Multiply these RRs together to obtain .8888, the cumulative wealth for the first 5 years.
The cumulative wealth for the 1970s was (1.0588)10 = 1.7707. Divide this result by
.8888 to obtain 1.9922. Take the 5th root of this result to obtain 1.1478. Subtract the 1.0
to obtain .1478 or 14.78%.
Thus, the geometric mean for the last 5 years must be 14.78% if the entire decade is to
equal the performance of the 1970s.
6-2.
Cumulative wealth for the first 5 years is .8888 (from Computational Problem 6-1).
Cumulative wealth for 10 years, given a geometric mean of 10.35,
= (1.1035)10 = 2.6775.
If one of the next 5 years has a loss of 40%, the cumulative wealth for 6 years would be
.8888 ×.6 = .5333.
Therefore, divide 2.6775 by .5333 to obtain 5.0206.
Take the 4th root of 5.0206 to obtain 1.4969; subtract 1.0 to obtain 49.69%.
Therefore, the geometric mean of the remaining 4 years must be 49.69% in order for the
decade to match the 20th Century geometric mean of 10.35%.
6-3.
Knowing these two items, the geometric mean for the total return and the geometric mean
for the dividend yield component, we can calculate the other component of total return.
(a) The other component is the price change component.
(b) A total return index for common stocks of $4,028.97 (calculated as (1.10259)85) and a
yield component index of 29.45, calculated as (1.0406)85, implies an ending wealth
for the price change component of $138.81 (calculated as 4028.97 / 29.45).
6-4.
The linkage between the geometric mean and the arithmetic mean is given, as an
approximation, by Equation 6-12.
(1 + G)2 ≈ (1 + A.M.)2 - (S.D.)2
G
= the geometric mean of a series of asset returns
A. M. = the arithmetic mean of a series of asset returns
S. D. = the standard deviation of the arithmetic series of returns
Thus, if we know the arithmetic mean of a series of asset returns and the standard
deviation of the series, we can approximate the geometric mean for this series. As the
standard deviation of the series increases, holding the arithmetic mean constant, the
geometric mean decreases.
Using the data given
(1 + G)2 ≈ (1.184395)2 - (.379058)2
(1 + G)2 ≈ 1.4028 - .1437
(1 + G)2 ≈ 1.2591
1 + G ≈ 1.1221; G = 12.21%
In this example, the very high standard deviation for this category of stocks results in a
considerably lower geometric mean annual return despite the high arithmetic mean.
Variability matters!