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From Chapter 7 of Financia l M anag ement: Principle s and Appl ications, Twelf th Edition. Sheridan Titman, Arthur J. Keown, and John D. Martin .
Copyright © 2014 by Pearson Education, Inc. All rights reserved.
209
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Realized and Expected
Objective 1. Calculate realized and expected rates
of return and risk.
Rates of Return and Risk
A Brief History of Financial ____
Objective 2. Describe the historical pattern of financial
market returns.
Market Returns
Geometric vs. Arithmetic _____ Objective 3. Compute geometric
(or compound)
and arithmetic average rates of return.
Average Rates of Return
What Determines Stock ---- Objective 4. Explain the effic ient market hypothesis
and why it is important to stock prices.
Prices?
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Princi p es C2 and C4 Applied
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To develop an understandi ng of why different investments earn
different returns, we will focus much of our attention on
C Principle 2: There Is a Risk-Return Tradeoff. Specifically ,
although investing in higher-risk investments does not always
result in higher realized rates of return (that's why they call it
risk), higher-risk investments are expected to realize higher re­
turns , on average. So, as we review historical rates of retu rn
realized on securities with different risks, we will be looking to
see whether or not the riskier investments are indeed rewarded
with higher returns. In addition, C Principle 4: Market Prices
Reflect Information will help us understand the wisdom of mar­
kets and how investor purchases and sales of a security drive its
price to fully reflect all the relevant information about the secu­
rities future cash flow s.
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Suppose that in January of 1926, your great-gr andfather set up a trust for you, his unknown heir, with an invest­
ment of $100. If your great-grandfather was very conservative and invested the trust f und in long-term bonds
issued by the U.S. government, the trust f und woul d have grown at a rate of 5.7 percent f rom January 1926
through December 2011 (a period of 86 years) to be worth about $11,761. This is quite a nice windfall that will
help with your books and spending money at college. If, however, the money had been invested in a portfolio of
large U.S. stocks, it would have grown at a rate of 9.8 percent and would be worth about $310,314, enough to pay
for all of your tuition, and then some. But if your great-grandfather was willing to really gamble with your future
and invested the money in the common stock of a portfolio of the smallest publicly traded f irms, the investment
would have grown at a compound annual rate of return of 11.9 percent and would now be worth more than $1.58
million, enough to pay for your tuition and living expenses and then buy
a pretty nice house and car for your graduation present.
In this chapter, we answ er three important questions that he lp
us understand why eac h investment just described y ielded very dif­
ferent retur ns. First, how do we measur e the risk and return for an
individual investment? Second, w hat is the history of f inanc ial mar­
ket returns on var ious classes of f inancial assets, including domestic
and international debt and equity securities as well as real estate
A
and commodities? Finally , w hat returns should investors expect f rom
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investing in risky f inanc ial assets? In answering these questions, our
introduction to the history of f inanc ial market returns w ill f oc us pri­
marily on securities suc h as bonds and shares of common stoc k, as
most corporations use these securities to f inance their investments.
As we disc uss late r, the expe cted rates of return on these secur ities
prov ide the basis for determining the rate of return that f irms require
when they invest in new plant and equ ipment, sales outlets, and the
deve lop ment of new products.
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211
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A n Int ro d uct io n t o R isk and Ret u r n
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Regardless of Your Major . . .
Statistics permeate almost all areas of busi­
ness. Because financial markets provide rich
sources of data, it is no surprise that the tools
used by statistic ians are so widely used in financ e. In this chapter, we use the basic tools of
descriptive statistics, such as the mean and measures of dispersion, to analyze the riskiness of
potential i nvestments. These tools, which are essential for the study of finance, are widely used
in all business disci pli nes as well as in the social .sciences. A good understanding of statist ics is
ext remely useful, regardless of your major.
Realized and Expected Rates
of Return and Risk
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We begin our discussion of risk and return by defining some key terms that are critical to de­
veloping an understanding of the risk and return inherent in risky investments. We will focus
our examples on the risk and return encountered when investing in various types of securities in the financial markets but the methods we use to measure risk and return are equally
applicable to any type of risky investment, such as the introduction of a new product line.
Specifically , we provide a detailed definition of both realized and expected rates of return.
In addition , we begi n our analy sis of risk by showing how to calculate the variance and the
standard deviation of historical, or realized, rates of return.
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Calculating the Realized Return f rom an Investment
If you bou ght a share of stock and sold it one year later, the return you would earn on your
stock investment would equal the ending price of the share (plus any cash distributions such as
dividends) minu s the beginnin g price of the share. This gain or loss on an investment is called
a cash return, which is summarized in Equation (1) as follows:
Cash
Ending
Cash Distribution
Beginning
=
+
(1)
Return
Price
(Dividend )
Price
-
.
Consider what you would have earned by investing in one share of Dick 's Sporting Goods
(DKS) stock at the end of May 2008 and then selling that share one year later at the beginning
of June 2009. Substituting into Equation (1), you ,-vould calculate the cash return as follows:
Cash _ Ending + Cash Distribu tion _ Beginning = $ l . + O.OO _
23 . 15 = -$5.
7 80
Return
Price
(Dividend )
35
Price
In this instance, you would have realized a loss of $5.35 on your investment, because the
firm's stock price dropped over the year from $23.15 down to $17.80 and the firm did not
make any cash distributions to its stockholders.
The method we have just used to compute the return on Dick 's Sporting Goods stock
provide s the gain or loss we experienced during a period. We call this the cash return for
the period .
In addition to calculating a cash return, we can calculate the rate of return as a percen tage. As a general rule, we summarize the return on an investment in terms of a percen tage
return, because we can compare these percen tage rates of return across different investmen ts.
The rate of return (sometime s referred to as a holding period return) is simply the cash ·
return divided by the beginning stock price, as defined in Equation (2) :
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A n I nt ro d uct i o n t o R isk a nd Ret ur n
Ending
Rate of
Cash Return
Return
Beginning Price
+
Price
Cash Distribution
-
Beginning
(Dividend )
Price
(2)
Beginning
Price
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Table 1 contains begin ning prices, dividends (cash distributions) , and ending prices span­ ning
a one-year holding period for five public firms. We u se this data to compute the realized rates
of return for a one-year period of time begin nin g on October 8, 2008, and ending with October
9, 2009. To illustrate, we calculate the rate of return earned from the investment in Dick 's
Sporting Goods stock as the ratio of the cash return (found in Column D of Table 1) to your
investmen t in the stock at the beginning of the period (found in Column A). For this
investment, your rate of return is a whopping 45 o/o = $7.37/15.32. Even though Dick's paid
no cash dividends, its stock price rose from $15.32 at the beginning of the period to $22.69,
or by $7.37 over the year you would have earned a 45 percen t rate of return on the stock if
you had bought and sold on these dates.
Notice that all the realized rates of return found in Table 1 are positive except for Walmart
(WMT), which experienced a negative rate of return. Does this mean that if we purchase shares of
Walmart stock today we should expect to realize a negative rate of return over the next year? The
answer is an emphatic no. The fact that Walmart' s stock earned a negative rate of return in the past
is evidence that investing in stock is risky. So, the fact that we realized a negative rate of return
does not mean we should expect negative rates of return in the future. Future returns are risky and
they may be negative or they may be positive; however, Principle 2: There Is a Risk-Return
Tradeoff tells us that we will expect to receive higher returns for assuming more risk (even though
there is no guarantee we will get what we expect).
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Calculating the Expected Return f rom an Investment
We call the gain or loss we actually experienced on a stock during a period the realized rate of
return for that period. However, the risk-return tradeoff that investors face is not based on re­
alized rates of return ; it is in stead based on what the investor expect s to earn on an investment
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Table 1
Measuring an Investor's Realized Rate of Return from Investing in Common Stock
- r•••
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'"'"!:'"""""'V-,,,_--
.. ..
. ...
,. .c '
R
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.,,...,
f SE
ill. .
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• etum
-
Company
A
B
C
D = C + B -A
-------------+---------------+------·------+--------1------Dick 's Sporting Goods (DKS)
$15.32
$22.69
$0.00
$ 7.37
16.38
Duke Energy (DUK)
15.82
------------+
--Emerson Electric (EMR)
32.73
37.75
1.16
$ 0.60 .
E =DIA
45.0%
l .8o/o
--- -=
- ---+-------,
1.32
$ 6.34
l 9.4o/o
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5/ .74
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---'---
67.86
--
.
-·------- ---'---
17.5%
---- ---
.
--
-9.1%
---- - ---·
- --
Legend:
We formalize the return calculations found in Columns D and E using Equations (1) and (2) :
Column 0 (Cash or Dollar Return)
Cash = Ending + Cash Distribution _ Beginning
,
Return
Price
(Dividend)
Price
= PEnd
+ Div - Paeginning
(1)
Column E (Rate of Retum)
Rate of
Cash Return
PEnd + Div - Paeginning
Return, r
Beginning Price
Paeginning
(2)
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213
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A n I nt r oduct io n t o R is k and Ret u r n
in the future. We can think of the rate of return that will ultimately be realized from making
a risky investment in term s of a range of possible return outcomes , much like the distribution
of grades for a class at the end of the term. The expected rate of return is the weighted aver­
age of the possible returns , where the weights are determined by the probability that it occurs.
To illustrate the calculation of an expected rate of return, consider an investment of
$10,000 in shares of common stock that you plan to sell at the end of one year. To simplify
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the computations we will assume that the stock will not pay any dividends during the year, so
that your total cash return comes from the difference betw een the beginning-of -year and end­
of-year prices of the shares of stock, which will depend on the state of the overall economy. In
Table 2 we see that there is a 20 percent probability that the economy will be in recession at
year 's end and that the v alue of your $10,000 investment will be worth only $9,000, prov iding
you with a loss on your investment of $1,000 (a -10 percent rate of return). Similarly , there
is a 30 percent probabilit y the economy will experience moderate growth, in which case you
will realize a $1,200 gain and a 12 percent rate of return on your investment by year 's end.
Finally, there is a 50 percent chance that the economy will experience strong growth , in which
case your investment will realize a 22 percent gain.
Column G of Table 2 contains the products of the probability of each state of the econ­
omy (recession, moderate growth, or strong growth) found in Column B and the rate of return
earned if that state occurs (Column F). By adding up these probability -weighted rates of return
for the three states of the economy , we calculate an expected rate of return for the investment
of 12.6 percent.
Equation (3) summarizes the calculation in Column G of Table 2, where there are
n possible outcomes.
Expected Rate
of Return
Rate of
Probability
Return 1 X of Return 1
Rate of
+
Probability
Return 2 X of Return 2
Rate of
+ ·· · +
Probability
Return n X of Return n
(3)
[ E( r ) J
We can use Equation (3) to calculate the expected rate of return for the investment in Table
2, where there are three possible outcomes, as follows :
E( r ) = ( -lO o/o
X .2)
+
( 12o/o X .3)
+
( 22o/o X .5) = 12.6 o/o
Measuring Risk
In the example we ju st examined, we exp ect to realize a 12.6 percent return on our investment;
however, the return could be as little as -10 percent or as high as 22 percent. There are two
methods financial analy sts can use to quantify the variability of an investment 's returns. The
Table 2
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--- . -
..
.
Column A
Column B
Column C
Column D
Column
E =C -D
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Column
F=E+ D
Column
G=BXF
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Recession
20%
$ 9,000
$10,000
$(1,000)
-10% = - $1,000 + $10,000
-2.0%
Moderate growth
30%
11,200
10,000
1,2.00
12% = $1,200 + $10,000
3.6%
--------+--------------!----------+------------+---·------Strong growth
Sum
50%
100%
12,200
10,000
2,200
22% = $2,200 + $10,000
11%
12.6%
"The probabilities assigned to the three possible economic conditions have to be determined subj ectively, which requires management to have a thorough
understanding of both the investment cash flows and the general econom¥
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2 14
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A n I nt r o duct io n t o R is k and Ret u r n
first is the variance of the investment returns and the second is the standard deviation, which
is the square root of the variance. Recall that the variance is the average squared difference
between the individual realized returns and the expected return. To better understand this we
will examine both the variance and the standard deviation of an investment 's rate of return.
Calculating the Variance and Standard Deviation of the Rate of Return
on an Investment
Let 's compare two possible investment alternatives:
1. U.S. Treasury Bill. A short-term (maturity of one year or less) debt obligation of the U.S.
government. The particular Treasury bill that we consider matures in one year and prom­
ises to pay an annual return of 5 percen t. This security has a risk-f ree rate of return,
which means that
if we purch ase and hold this security for one year , we can be confident
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of receiving no more and no less than a 5 percent return. The term risk -f ree security
specifically ref ers to a security for which there is no risk of def ault on the promised
payments.
2. Common Stock of the Ace Publishing Company. A risky investment in the common
stock of a company we will call Ace Publishing Company .
The probability distribution of an investment 's returns contains all the possible rates of
return from the investment that might occur, along with the associated probabilities for each
outcome. Figure 1 contains a probability distribution of the possible rates of return that we
might realize on these two investments. The prob ability distribution for a risk-free investment
in Treasury bills is illustrated as a single spike at a 5 percent rate of return. This spike indicates
that if you purchase a Treasury bill , there is a 100 percen t chance that you will earn a 5 percen t
annual rate of return . The probability distribution for the common stock investment, however,
includes returns as low as -10 percent and as high as 40 percen t. Thus, the common stock
investment is risky , whereas the Treasury bill is not.
Probability Distribution of Returns for a Treasury Bill and the Common Stock of the Ace Publishing Company
A probabil ity distribution provides a tool for describing the possible outcomes or rates of return from an investment and the associated
probabilities for each possible outcome. Technically, the following probabil ity distribution is a discrete distribution because there are only five
possible returns that the Ace Publi shing Company stock can earn. The Treasury bil l investment offers only one possible rate of return (5%)
because this investment i s risk-free .
1.0 -
,"!!& Treasury
.WAI bill
0.4 0.35 I--
0.3
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0.2 5
0.15
Publishing
1 chance in 10 (10%)
Co.
-10%
2 chances in 10 {20o/o )
5%
4 chances in 10 (40%)
15%
>--
(
>--
2 chances in 10 (20%)
25%
1 chance in 10 (10%)
40%
.
>--
0.1 >-0.05 f--,,
O '-------10%
5%
15%
2 5%
40 %
Possible returns
>> END FIGURE 1
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A n Int ro duct i o n t o R is k and Ret urn
Using Equation (3), we calculate the expected rate of return for the stock investmen t as
follows:
E (r)
= ( .10) ( -lO o/o )
+
( .20 ) ( 5°/o )
+
( .40) ( 15o/o )
+
( .20 ) ( 25 o/o )
+
-
( .10) (40 o/o ) = 15 °/o
Thus, the common stock investment in Ace Publishing Company gives us an expected rate
of return of 15 percent. As we saw earlier, the Treasury bill investment offers an expected
rate of return of only 5 percen t. Does this mean that the common stock is a better investment
than the Treasury bill because it offers a higher expected rate of return ? The answer is no, be­
cause the two investments have very different risks. The common stock might earn a negative
10 percent rate of return or a positive 40 percent, whereas the Treasury bill offers only one
positive rate of 5 percent.
One way to measure _the risk of an investment is to calculate the variance of the possible
rates of return, which is the average of the squared deviations from the expected rate of re­
turn. Specifically , the formula for the return variance of an investment with n possible future
returns can be calculated using Equation (4) as follows:
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Variance in
Rates of Return =
Rate of
Return 1
Expected Rate
of Return
( T1)
E (r)
( CT 2 )
+
Rate of
Expected Rate
Return 2 - of Return
( r2)
+ ... +
2
E (r)
Expected Rate
Rate of
Return 3 - of Return
( rn)
E( r )
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2
Probability
x of Return 1
( Pb1 )
-
Probability
X of Return 2
( Pb2 )
2
Probability
X of Return n
( Pbn)
(4)
Note that the variance is measured using squared deviations of each possible return from the mean
or expected return. Thus, the variance is a measure of the average ''squared'' deviation around the
mean. For this reason it is customary to measure risk as the square root of the variance which,
as we learned in our statistics class, is called the standard deviation.
For Ace Publishing Company 's common stock, we calculate the variance and standard
deviation using the following five-step procedure:
Step 1. Calculate the expected rate of return using Equation (3). This was calculated previ­
ously to be 15 percen t .
Step 2. Subtract the expected rate of return of 15 percen t from each of the possible rates of
return and square the difference.
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Step 3. Multiply the squared differences calculated in Step 2 by the probability that those
outcomes will occur .
Step 4. Sum all the values calculated in Step 3 together . The sum is the variance of the
distribution of possible rates of return. Note that the variance is actually the aver­
ag e squa red difference between the pos sibl e rates of retu rn and the expe cted rate of
return.
Step 5. Take the square root of the variance calculated in Step 4 to calculate the standard de­
viation of the distribution of possible rates of return. Note that the standard deviation
(unlike the varian ce) is measured in rates of return.
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Table 3 illustrates the application of this procedu re, which results in an estimated
standard deviation for the common stock investment of 12.85 percen t . This standard de­
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viation compares to the 0 percent standard deviation of a risk-free Treasury bill invest­
ment. The investmen t in Ace Publishing Company carries high er risk than inv esting in the
Treasury bill beca use it can potentially result in a return of 40 percen t or possibly a loss
of 10 percen t . The standard deviation measure captures this difference in the risks of the
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two investmen ts.
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216
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A n Int r oduct io n t o R is k and R etur n
Measuring the Variance and Standard Deviation of an Investment in Ac e
Publishing's Common Stock
-------------·
-·-·
·
·
--------·
--
Computing the variance and standard deviation in the rate of return earned from a stock investment
can be carried out using the following five-step process :
Step 1. Calculate the expected rate of return,
Step 2. Subtract the expected rate of return from each of the possi ble rates of return and square
the difference,
Step 3. Multiply the squared differences calculated in Step 2 by the probability that those
outcomes will occur.
Step 4. Sum all the values calculated in Step 3 together to calculate the variance of the possible
rates of return ,
Step 5. Take the square root of the variance calculated i n Step 4 to calculate the standard
deviation of the distribution of possible rates of return.
----------""'l ----_ -,..------,.,,,_.-.......----..----- """.""--.....
. Rate of · ·· Cbance or , :
Stateof
the Wf>t1tt f
O•MW'k'f. >\ ...;c.; ·· ·· •·•,· ·.•··.,· · ·
.
''
'
..
Return ,_ .' . Probabilltf, ·.· '
,,,, ,;,,·, , ·
a
b
. ·;.;
· ··
. .,, .· .·. ·. ,
'
..........,
'
• ·•
•
· '
!
-
-0.10
0.10
-0.01
3
0.05
0.20
0.01
4
-·-0.15
•• •·•··• ,' '
''
'
e = [b - E(r)]2
f=exc
0.0625
0.00625
0.0100
0.00200
I
I
1
. Sielf,i - .
;;;fi ' ' ,;_, ..., • •
d=bXc
c
I
·'
. '.
•
'!
-
--
--
-
.... ......_ __
0.40
0.06
0.0000 .
0.05
0.0100
I
·-
0.00000
'
'
'
4
0.25
0.20
0.00200
:------+-------------.--------------+--------''
'
5
0.40
---------'-----'--'
0.10
0.04
-- --· -----
Step 1: Expected Return (E(r)) = ---­
0.0625
----
0.00625
---- - - ----
0.15
Step 4:Variance = ---------------------------c
Step 5: Standard Deviation = -----------------
0.0165
0.1285
Alternatively , we can formalize the five-step procedure above for the calculation of the
standard deviation as follows:
-
Standard
. .
=
Dev1at1on, <7
y'
.
Var1ance
•
(5)
(J'
=
r; =
E( r) =
-
standard deviation
possible return i
the expected return
r-- --·--· ------------------·- ------------Pb; = the probability of return i
·-
Using Equation (5) to calculate the standard deviation we find:
(J'
=
( ( -. 10 - .15) 2 x .10) + ( ( .05 - .15) 2
+ ( ( . 15 - .15)2 x .40) + ( ( .25 - .15) 2
+ ( ( .40 - .15) 2 x .10 )
x
x
1 2
1
.20) .20 )
.0165 = .1285 or 12.85°/o
Now, let's suppose that you are considering putting all of your wealth in either the Ace Publish­
ing Company or in a quick-oil-change franchise. The quick-oil-change franchise provides a high
expected rate of return of 24 percent, but the standard deviation is estimated to be 18 percent.
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21-7
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A n Int r oduct ion t o R isk and R et ur n
Which investment would you prefer? The oil-change franchise has a higher expected rate
of return, but it also has more risk, as is evidenced by its larger stand ard deviation . So your
choice will be determined by your attitude toward risk. You might select the publish ing
company , whereas another investor might choose the oil-change investment, and neither
would be wrong . You would each simply be expressing your tastes and preferences about
risk and return.
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Evaluating an Investment 's Return and Risk
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Clarion I nvestment Advisors is evaluating the distribution of returns for a new stock investment and has come up with five
possible rates of return for the coming year. Their associated probabilities are as follows:
1 chance in 10 (10%)
2 chances in 10 (20o/o )
4 chances in 10 (40%)
2 chances in 10 (20%)
1 chance in 10 (10%)
-20%
0%
15%
30%
50%
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a. What expected rate of return might they expect to realize from the investment?
b. What is the risk of the investment as measured using the standard deviation of possible future rates of return?
STEP 1: Picture the problem
The distribution of possible rates of return for the investment, along with the probabilities of each, can be depicted
in a probability distribution as follows:
45 %
40 %
35%
3 0%
'
2 5%
e 20%
-
15%
-
-
'
I'
\
10%
5%
1
0%
-2 0 %
0%
15%
Rates of return
I
'
30%
'
50%
The probabilities of each of the potential rates of return are read off the vertical axis and the returns are found on
the horizontal axis .
STEP 2: Decide on a solution strategy
We use the expected value of the rate of return to measure Clarion's expected return from the investment and
the standard deviation to evaluate its risk. We can use Equations (3) and (5) for these tasks.
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218
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A n Int roduct ion t o R is k and Ret urn
STEP 3: Solve
Calculating the Expected Return.
We use Equation (3) to calculate the expected rate of return for the investment as follows :
E( r) = r 1 Pb 1 + r2Pb2 + · · · + rnPbn
E( r)
(3)
= (-20% x . 10) + (0% x .20) + ( 15% x .40) + (30% x .20 ) + (50% x . 10) = 15%
Calculating the Standard Deviation.
Next, we calculate the standard deviation using Equation (5) as follows:
u = V( [ r 1
u
=
-
E( r ) ] 2Pb 1 ) + ( [ r2 - E(r) ] 2Pb2 ) + · · · + ( [ rn - E( r ) ] 2 Pbn )
v( [ -.20 - . 15 ] 2 . 10)
(5)
+ ( [ .oo - . 15 J 2 .20J + ( [ . 15 - . 15 J 2 .4o) + ( [ .3o - . 15 J 2 .20 ) + ( [ .5o - . 15 J 2 . 10)
•
u =
.0335 = . 183 or 18.3%
STEP 4: Analyze
The expected rate of return for the investment is 15 percent ; however, because there is a 10 percent chance that
the actual return may be 50 percent and a 10 percent chance that the actual return may be -20 percent , it i s
obvious that this is a risky investment . In this example, the standard deviation, which is a measure of the average
or expected dispersion of the investment returns, is equal to 18.3 percent. Because the distribution of returns
is described in terms of five discrete return possibilities, we can make probability statements about the possible
outcomes from the i nvestment suc h as the following: There is a 1O percent probability of a realized rate of return
of 50 percent, and a 20 percent probability of a return of 30 percent, and so forth.
STEP 5: Check yourself
·._ J'
Compute the expected return and standard deviation for an investment with the same rates of return as in the
previous example but with probabilities for each possible return equal to .2, .2, .3, . 2, and . 1 .
ANSWER: Expected return = 11 .5 percent and standard deviation = 21 . 10 percent.
Your Turn: For more practice, do related Study Problems 1 and 6 at the end of this chapter.
>> END Checkpoint1
Tools of Financial Analysis -Measuring Investment Returns
Name of Tool
I Formula
.
.
.
.,
-·
l What Jt Jells,Y.ou
8\
-
. .
,
Cash (Dol lar)
Retitrn
•
..
.
Ending
-
.
Ra te of
Retu rn r =
-"-
...
-' .
+
.
Ra te of
.
Beg inning
Price, Pseg inning
PEnd + Div - Peeg i11ning
Cash ( Dol lar) Retu rn
Expe cted Rate
..
-
( Dividend, Di v )
Beg inning P rice
.
.
Cash Distribution
Price, PEn d
'
-
.
Pee.gi1znin g
•
.
.. •
.
.
·-···---···'- " - --
MeasuFes the return from investin g in a
security in dollars
• The higher the cash return, the greater the
return earned by the investment (measured
in dollars).
• Measures the return from investing in a
•
•
security as a percent Q.f the dollars invested
A higher rate of return means a greater
return earned by the investment (measured
as a percent of the initial investment).
The probability weighted average rate of
'
Retu rn 1 will occur (Pb1) ,
.Retu rn 1( r1 )
,
Rate of
P robability that
x
,Retu rn 2( r2 )
·•
..
.
+
•
of Retu rn, E( r)
-
-
P robability that
x
•
.
•
$. .
.
•
'
+
.
investor.
Retu rn 2 will occur (Pb2 )
,
return anticipated for an investment
The higher the expected rate of return,
the greater its impact on the wealth of the
'
Rate of
,Retu rn 3( r3 )
x
P robability that
Return 3 will occur ( Pb 3 )
•
•
•
2 19
'
•
A n Int roduct io n to R is k and Ret ur n
x
Re turn
2
Ra te of
+ ...
x
Retu rn
+
l Occur ( Pb2 )
Rate of
Retu rn n( r,,)
of Retu rn, E( r)
Retu rn n will Occur (Pbn)
Before you move on to 2
1 . If you invested $100 one year ago that is worth $1 10 today, what rate of return did you earn on your investment?
2. What is the expected rate of return, and how is it different than the realized rate of return?
3. What is the variance in the rate of return of an investment?
4. Why is variance used to measure risk?
A Brief History of Financial Market
Returns
Now that we have learned how to measure the risk and return of an investment, we can use
these measuremen t tools to analyze how securities have performed in the past . This is use­
ful when an investor wants to assess whether or not to invest in a security . Let 's look at the
historical returns earned on a wide variety of domestic and international investments. As we
might expect from c::I Principle 2: There Is a Risk-Return Tradeoff, investors have histori­
cally earned higher rates of return on riskier investments.
Note , however , that having a higher expected rate of return simply means that you expect
to realize a higher rate of return, not that you will alw ays receive a higher return . In fact, the
very definition of risk suggests that there will be times when you are not rewarded for assum­
ing more risk. Think back to Table 1 where we looked at the realized rates of return for five
different companies' stock for the year that ended in June 2009. In all five example compa­
nies, the realized rates of return were negative, suggesting that, at least for this time period ,
risk w as not rewarded. This is what we mean by risk you face the prospect of not realizing
your expected return !
U.S. Financial Markets: Domestic Investment Returns
In the introduction to this chapter we talked about a $100 investmen t made by a benev olent
great-grandf ather that grew over a period of 84 years. In this example, we saw how different
investment options with different levels of risk can result in very different returns. Let's take
220
'
..
A n Int ro d uct i o n t o R isk and Ret urn
a look at how different investments have perf ormed. Figure 2 shows the historical returns
earned on four types of investments over the period 1926-2009:
• Small stocks. Shares of the smallest 20 percent of all companies whose stock is traded
on the public exchanges. (Firm size is measured using the market capitalization of the
company 's equity, which is equal to the share price multiplied by the number of shares
outstanding.)
• Large stocks. The Standard & Poor 's (S&P) 500 stock index, which is a portf olio that
consists mainly of large company stocks such as Walmart (WMT), Intel (INTC), and
Microsof t (MSFT).
• Government bonds. 20-year bond s issued by the federal government. These bond s are
typically considered to be free of the risk of default or non-pay ment because the govern­
ment is the most credit-worth y borrower in the country.
• Treasury bills. Short-term securities issued by the federal government that have maturi­
ties of one year or less.
Principle 2: There Is a Risk-Return Tradeof f tells us that higher-risk investments
should expect to receive higher rates of return. Let's see what would have happened if your
great-grandf ather invested $1 in each of these investment alternatives.
The graph in Figure 2 shows the value of a $1 investment made in each of these asset
categories in 1926 and held until the end of 2011. Large and small stocks have provided the
Figure 2
Historical Rates of Return for U.S. Financial Securities: 1926-2011
The following graph provides historical insight into the performance characteristics of various asset classes
over an 35:year period of time. This graph illustrates the hypothetical growth of inflation and a $1 invest­
ment in four traditional asset classes over the time period January 1, 1926, through December 31 , 2011.
$16,808
$10,000
>---
Compound annual return
- Small stocks
•
1,000 ""--
- Large stocks
- Government bonds
- Treasury bills
- Inflation
11.9%
$3,334
9.8%
5.7 %
3.6%
3.0%
$124
100 r-
$2 1
-
..... ..-
10
-
$13
1
0.10
1926
1936
- 1946
1956
1966
1976
-
Legend:
•
1986
1996
2006
\
Small stocks in this example are represented by the fifth capitalization quintile of stocks on the NYSE for 1926-1981
and the performance of the Dimensional Fund Advisors, Inc. (DFA) U.S. Micro Cap Portfolio thereafter. Large stocks
are represented by the Standard & Poor's 90 index from 1926 through February 1957 and the S&P 500® index
thereafter, which is an unmanaged group of securit i es and considered to be representative of the U.S. stock market
in general. Government bonds are represented by the 20-year U.S. government bond, Treasury bills by the 30-day
U.S. Treasury bill , and inflation by the Consumer Price Index. Underlying data is from the Stocks, Bonds, Bills, and
Inflation® (SBBfID) Yearbook , by Roger G. Ibbotson and Rex Si nquefield, updated annually. An i nvestment cannot be
made directly in an i ndex .
Source: © 2012 Morningstar. All rights reserved. Used with permission.
>> END FIGURE 2
221
'
A n Int ro d uct i o n t o R is k and R eturn
highest returns and largest increase in wealth over the past 86 years. Fixed-income invest­
ments provided only a fraction of the growth provided by stocks. However, the higher returns
achieved by stocks are associated with much greater risk, which can be identif ied by the fluc­
tuation of the graph lines. Moreov er, in the following table we see that the standard deviations
of the annual rates of return for the four investment alternatives are highest for small company
stocks and lowest for the risk-free Treasury bills.
-·-· ---,--- ----. .,..,,,.
·..
,-.
.....................
Governm ttt .··•
Bonds
,., ·. • Treasury Bills
.........--+
Compound annual return
Standard deviation
l.9o/o
•
32.8%
9.8%
5 .7o/o
20.5o/o
9.6%
-
3.6%
i
3.1%
Lessons Learned
A review of the historical returns in the U.S. financial markets reveals two important lessons :
• Lesson #1. The riskier investmen ts have historically realized higher returns. The riski­
est investment class is comprised of the stocks of the smallest set of firms followed by
the stocks of large companies, then corporate bonds, long-term U.S. government bond s,
and finally Treasur y bills. The difference betw een the returns of the riskier stock in­
vestments and the less risky investments in govern ment securities is called the equity
risk premium. For example, referring to the previous compound annual return table,
the premiu m of large company common stocks over long-term government bond s aver­
ages 9.8% - 5.4% = 4.4% . A similar comparison to short-term Treasury bills reveals an
average risk premiu m of 9.8 % - 3.7% = 6.1o/o . The risk premiu ms for small company
stocks are even higher because the average returns earned by the smaller and riskier firms
are higher.
• Lesson #2. The historical returns of the higher-risk investment classes have higher
standard deviations. Small stocks had a standard deviation of 32.8 percent, whereas the
standard deviation of Treasury bill returns was only 3.1 percent. Note that these stan­
dard deviations are computed from the annual rates of return realized over the entire
period from 1926 to 2011, such that there is some variation even in the Treasury bill rate
over time.
U.S. Stocks versus Other Categories of Investments
Figure 3 illustrates the growth in the value of $1 invested in 1980 until the end of 2009 for
five different asset classes:
1. U.S. stocks.The common shares of companies headqu artered in the United States whose
shares are traded in the U.S. stock market.
2. Real estate.Ownership of real property such as office build ings, land , and apartmen ts as
well as mortgages or loans used to finance the purchase of real estate. Real estate invest­
ment trusts (REITs) are financial institutions that raise money from investors and either
purchase real estate or mortgages on real estate.
3. International stocks. The common shares of companies headquartered o tside of the
United States.
4. Commodities. Basic resources such as iron ore, crude oil, coal, ethanol, salt, sugar, cof ­
fee beans, soybeans, aluminum, copper, rice, wheat, gold, silver, and platinum.
5. Gold. This particular commodity has historically been u sed as a store of value by many
investors. Its value tends to rise with inflation such that investors often purchase gold
as a means of preservin g the v alue of their savings during times of rising prices or
inflation .
222
>
'
A n Int rod uct io n t o R isk and Ret urn
Stocks, Bonds, Commodities, and Real Estate
This image illustrates the hypothetical growth of a $1 i nvestment in domestic stocks, i nternational
stocks, commodities such as copper, REITs (real estate investment trusts that invest in commer­
cial real estate and real estate mortgage loans) , and commodities over the time period January 1 ,
1980, to December 31 , 201 1 .
Compound annual return
$40
- REITs
- Stocks
12.1%
11.1%
- Bonds
10.2o/o
$39.01
$28.67
$22 .57
- Commodities 7.1%
- Treasury bills 5.1%
- Inflation
3 .4%
$9.05
10
!....----- $4.98
$2 .94
*
'
' '
1
0.60
1980
1985
1990
1995
2000
2 005
2 010
Legend:
Stocks in this example are represented by the Standard & Poor' s 500® , which is an unmanaged group of
securities and considered to be representative of the U.S. stock market in general . Bonds i n this example
are represented by the 20-year U.S. government bond, Treasury bills by the 30-day U.S. Treasury bill , and
inflation by the Consumer Price Index. Commodities are represented by the Morningstar Long-Only Commod­
ity Index and REITs by the FTSE NAREIT All Equity REITs Index® . An investment cannot be made directly in
an i ndex.
Source: © 2012 Morningstar. Al l rights reserved. Used with permission .
>> END FIGUR E 3
Global Financial Markets: International Investing
Figure 4 compares the historical returns from investing in U.S. stocks and bond s to the re­
turns on international stocks and bonds. These annual ranges of return s provide an indication
of the historical risk experienced by investments in various global markets. This fluctua­
tion in rates of return earned over a period of time is called the investment's volatili ty. We
measure investment return volatility using the standard deviation as we discussed earlier.
For example, an investment in Pacif ic stocks generated annual rates of return as high as
I
107.5 percen t or as low as -36.2 percent. In contrast, U.S . stocks had the narrowest range
of returns, which implies that U.S. stocks experienced less volatility than an investment in
other regions of the world.
Figure 5 compares the average rates of return earned from investing in developed coun­
tries , such as the U .S., Europe, and some parts of Asia, to the returns from investin g in the
equities of companies located in emerging markets. An emerging market is one located in
an economy with low-to-middle per capita income. These countries constitu te roughly 80
percent of the world's population and represent about a fifth of the world 's economies.
223
,
•
A n Int r o duct i o n t o R is k a nd R e t urn
· --------------------------------Historical Rates of Return in Global Markets: 1970-2011
This figure reports the ranges of annual returns for domestic and international composites, as well
as the Europe and Pacific regional composites, over the period 1970 through 201 1.
Annual ranges of returns
12 5%
107.5%
100%
75 %
-
t--
79.8%
•
69.9%
50%
t--
37.6%
2 5%
9.6%
0%
10.1%
9.3 %
Comp0Ul)d
annual r-eturn
-25%
-36.2 %
-43 .l %
' -46.1%
-50%
United States
International
Europ e
Pacif ic
Legend:
U.S. stocks in this example are represented by the Standard & Poor's 500® index, which is an unmanaged
group of securities and considered to be representative of the U.S. stock market in general. International stocks
are represented by the Morgan Stanley Capital International Europe, Australasia, and Far East (EAFE®) Index,
European stocks by the Morgan Stanley Capital International Europe Index, and Pacific stocks by the Morgan
Stanley Capital International Pacific Index. An investment cannot be made directly in an index. The data as­
sumes reinvestment of income and does not account for taxes or transaction costs .
Source: © 2012 Morningstar. All ri ghts reserved. Used with permission.
>> E ND FIG UR E 4
-
China and India are perh aps the best know n and largest of the emerging market economies.
A developed country is sometimes i·eferred to as an industrialized country , where the term is
used to identify those countries such as the U.S., Great Britain, France, and so forth that have
highly sophistic ated and well-developed economies. The average rates of return from invest­
ing in developed countries were generally lower than those earned in the emerging market
group. However, the most apparent difference in the tw o relates to risk as reflected in the
range of annual rates of return . The top of the bar chart indicates the maximu m rate of return
realized over the period covered by the chart and the bottom reflects the minimu m, so the span
of the bar reflects the variability of past rates of return. Note that the emerging market rates of
return were much more volatile over the period 1988-2011.
If investing in the stock of companies from emerging markets is so much more risky than
investing in domestic equities or equities of companies from developed countries, why do
it? The a11swer may well come from a consideration of the risk-reduction benefits that come
about when you invest in both types of securities.
---/-
•
-,
224
•
A n Int ro duct io n t o R isk and Ret urn
-- -Investing in Emerging Markets: 1988-2011
The following graph il lustrates the range of returns as wel l as the compound annual return of
selected developed and emerging countries. Although both sets experienced growth, emerging
markets experienced a much greater upside and often deeper downside.
Developed markets
150%
Emerging markets
*compound annual return
125
100
75
;s---
•
50
'i.7%
_f °-""""',1,.---
0
-2 5
Austr alia
U.S.
U.K.
Japan
Mexico
Korea
Taiwan
Chile
Legend:
Equities for the U K , Australia, Japan, Taiwan, and Mexico are represented by the Morgan Stanley Capital I nter­
national country indexes. Equities for Korea and Chile are represented by the Morgan Stanley Capital International
Emerging Market country indexes. United States equities are represented by the Standard & Poor's 500®, which
is an unmanaged group of securities and considered to be representative of the U.S. stock market in general. An
investment cannot be made directly in an index. Keep in mind that the countries il lustrated do not represent invest­
ment advice. The developed countri es i ll ustrated are a common range of investment options. Emerging- market
countries were chosen based on avai lability of historical data; those with the longest stream of data were selected.
*Compounded annual return
-
Source : © 2012 Morningstar All rights reserved. Used with permission .
>> END FIGURE 5
You have just finished savi ng up for a "once- in-a- lifetime"
vacation. Three weeks before you plan to leave, you lose your
job. How would you handle this situation? Specifically, choose
the one response from the fol lowing ist that best describes what
you would do
Take a much more modest vacation .
Go as scheduled, reasoning that you need the time to pre­
pare for a job search.
Extend your vacation because this might be your last chance
.
-
'
--
.
An important factor affecting i ndividuals' deci sions as to how to
As you might guess, the alternative that you select suggests
something about your personal toleranc e for risk. you want to
learn more about your risk tolerance, take a look at the Rutgers'
-
-
'Risk Tolerance Quiz Source: Grable, J. E., & Lytton, R. H. (1 999). Financial risk tolerance revisited: The development of a risk assessment instrument . Financial Services
Review, 8, 163-181 . Ruth Lytton and John Grabl e, Investment Risk Tolerance Quiz, www .rce . rutgers .edu/money/riskquiz/
225
•
A n Int ro duct i o n t o R is k and R et ur n
Before you move on to 3 •
Concept Check 2
1. How well does the risk-return princi pl e hold up in light of historical rates of return? Explain.
2. What is the equity risk premium, and how is it measured?
•
3. Does the historical evidence suggest that i nvesting in emerging markets is more or less risky than investi ng in developed
markets?
Geometric Versus Arithmetic Ave rage
Rates of Return
When evaluating the possibility of investing in a security or financial asset such as those dis­
cussed in the previous section , investors generally begin by looking at how that investment
perf ormed in the past. This often entails looking at how the investment has perf ormed over
many years. It is common to summarize the past returns as a yearly average. For example, if
you held a stock for two years that realized a rate of return of 10 percent in the first year and
20 percent in the second year, you might simply add the two rates together and divide by two
to get an average rate of 15 percent. This is a simple arithmetic average return. However, as
we will describe, the actual return you realized from holding the stock for two years is some­
what less than 15 percen t per year . To describe the actual two-y ear return you would need to
know the geometric or compound average return.
Let 's look at an example. Suppose you invest $100 in a particular stock. Af ter one year,
y our investment rises to $150. But unfortunately, in the second year it falls to $75. What was
the average return on this investment? In this first year , the stock realized a rate of return
of 50 percent and in the second year, it realized a rate of return of -50 percent. If we took
the simple average of these two rates, we get 0 percent, indicating that the average yearly
investment return over the two-year period is 0 percent. However , this does not mean that you
earned a 0 percent rate of return, because you began with $100 and ended two years later with
only $75 ! In actuality, over the two-year investment period , the $100 investment lost the
equivalent of -13.4 percent.
In the above example, the 0 percent rate is ref erred to as the arithmetic average rate of
return, whereas the -13.4 percent rate is ref erred to as the geometric or compound average
rate. The arithmetic average is the simple average we have already learned to calculate in this
chapter. The geometric average is different because it takes compounding into account . For
example, a 50 percent increase in value from $100 is $50, but a 50 percent decrease in value
from $150 is $75. The geometric average rate of return answers the question, ''What was the
growth rate of your investment ?'' whereas the arithmetic average rate of return answers the
question, ''What was the average of the yearly rates of return ?''
•
Computing the Geometric or Compound
Average Rate of Return
The geometric average rate of return for a multiyear investment spanning n years is calculated
as follows:
Geometric
Average Return
+
1
Rate of Return
for Year 1, Trear J
x
1
+ Rate of Return
for Year 2,
Trea r 2
x ... x
1
+
Rate of Return
for Year n, Trear n
1/ n_
- 1
(6)
Note that we multiply together 1plus the annual rate of return for each of the n years, and then
take the nth root of the prod uct to get the geometric average of (1 + annual rate of return) , and
then subtract 1 to get the geometric average rate of retu1·n.
To illustrate the calculation of the geometric average rate of return , consider the return
earned by the $100 investment that grew in value by 50 percent to $150 in Year 1 and dropped
by 50 percent to $75 in. Year 2. The arithmetic average rate of return. is 0 percent.
226
•
•
A n Int r od uct i on t o R is k a nd Ret ur n
We can calculate the geometric annual rate of return for this investment using Equation (6)
as follows:
•
Geometric
= [ (l
Average Return
+
rYear 1
)X
(l +
r Year 2 ) ] l/Z -
1
.
=
[(1 + .50)
X
(1
+ (-.50)] l/2
-
1 = .866025 - 1 = -13.40 o/o
So, over the two-year investment period, the $100 investment lost the equivalent of -13.40
percent per year.
We could also solve for the geometric mean or compou nd rate of return using a financial
calculator, taking the initial investment and final value and solving for i:
Enter
2 .
- 100
l/Y
0
75
•
Solve for
Using either approach we find the geometric mean or compound average rate of return
to be -13.4 percen t.
Choosing the Right ''Average''
Which average should we be using ? The answer is that they both are importan t and, dependin g
on what you are trying to measure, correct. The following grid provides some guidance as to
which average is appropriate and when:
-
What annual rate of return can we
expect for next y·ear?
'i The arithmetic average rate of return calculated
I
using annual rates of return.
What annual rate of return can we
The geometric, or compound, average rate of return
_e_xe_c_t _o_v_
er_a_ m_u_It - ar ho
_r_iz
_o
_n_?__ _c_a_lc_u_la_t_ed o r a similar past period .
j
•
It's important to note that arithmetic average rates of return are only appropriate for think­
ing about future period s that are equal in duration to the period over which the historical
returns were calculated . For example, if we want to evalu ate the expected rate of return for
a peri od of one year and our data corresponds to quarters, we would w ant to convert these
quarterly returns to annual returns using a geometric average, and then use the arithmetic
mean of these annual rates of return (not four times the qu arterly rate of return , as some
might assume) .
•
-
Computing the Arithmetic and Geometric Average
Rates of Return
Five years ago Mary's grandmother gave her $10 , 000 worth of stoc k in the shares of a publicly traded company
founded by Mary's grandf ather . Mary is now considering whether she should continue to hold the shares, or perhaps
· sell some of them . Her f irst step in analyzing the investment is to eval uate the rate of return she has earned over the
pa fiw
ars .
/
The following table contains the beginning value of Mary's stock five years ago as well as the values at the end of
each year up until today (the end of Year 5) :
(2 CONTI N UED >> ON NEXT PAG E)
•
-
227
,
'
-
A n Int ro d uct i o n t o R isk and R et ur n
0
1
2
3
4
5
$10,000.00
11,000.00
12,650.00
10,752.50
12,903.00
14,193.30
---- ----· -·------- -
10.0%
15.0%
-15.0%
20.0%
10.0%
--
What rate of return did Mary earn on her investment in the stoc k given to her by her grandmother?
STEP 1: Picture the problem
•
The value of Mary's stock investment over the past five years looks l i ke the following:
$16,000.00
$14,000.00
12,903 .00
$12 ,000.00
CJ)
Q)
.5
B
CJ)
10,752 .50
$10,000.0 0
$8,000.00
-
CJ)
$6,000.0 0
Q)
..a
$4,000.00
•
$2,000.00
$0
1
0
2
3
Year
4
5
6
.
STEP 2: Decide on a solution strategy
-
Our first thought might be to just calcul ate an average of the five annual rates of return earned by the stock
investment . However, this arithmeti c average fai l s to capture the effect of compound i nterest . Thus , to esti­
mate the compound annual rate of return we calculate the geometri c mean using Equat ion (6) or a f i nancial
calculator .
STEP 3: Solve
Calculate the Arithmetic Average Rate of Return for the Stock Investment.
The arithmetic average annual rate of return is calculated by summi ng the annual rates of return over
the past f ive years and dividing the sum by 5 . Thus the arithmetic average annual rate of return equals
8 .00 percent .
Note that the sum of the annual rates of return is equal to 40 percent and when we divide by five years we
get an arithmetic average rate of return of 8.00 percent. Thus, based on the past performance of the stock, Mary
should expect that it would earn 8 percent next year.
-
Calculate the Geometric Average Rate of Return for the Stock Investment.
We calculate the geometric average rate of return using Equation (6) :
Geometric
= [(1
Average Return
+
. 1)(1
+
. 15)(1
+ (-. 15))( 1 +
.20) (1
+
. 10)]1/5 - 1
= (1 .41 93) 115 - 1 = .0725 or 7.25%
I
-·•
228
'
A n I nt ro duct io n t o R is k and R et ur n
Alternatively, using a financial calcul ator and solving for i we get:
,
Enter
- 10,000
5
0
14,193.30
l/Y
Solve for
7.25
STEP 4: Analyze
The arithmetic average rate of return Mary has earned on her stock i nvestment is 8 percent , whereas the geo­
metric, or compound, average is 7 .25 percent. The reason for the lower geometric , or compound, rate of return
is that it incorporates consideration for compoundi ng of interest; it takes a lower rate of i nterest with annual com­
pounding to get a particular future value. The important thing to recognize here is that both of these averages
are useful and meaningful, but they answer two very different questions. The arithmetic mean return of 8 percent
answers the question, What rate of return should Mary expe ct to earn from the stock investment over the next
year; assuming all else remains the same as in the past? However, if the question is What rate of return should
Mary expect over a five-year period? (during which the effect of compounding must be taken into account) , the
answer is 7.25 percent, or the geometric average.
t
STEP 5: Check yourself
Mary has decided to keep the stock given to her by her grandmother. However, she now wants to consider the
prospect of selli ng another gift made to her fi ve years ago by her other grandmother . What are the arithmetic and
geometric average rates of return for the following stock investment?
·, •
I
0
1
2
3
4
5
$10,000.00
8,500.00
9,775.00
12,2 18.75
15,884.38
14,295.94
-15.0%
15.0o/o
25.0%
30.0%
-10.0%
'·,.•.·'
ANSWER: 9 percent and 7.4 1 percent.
Your Turn:For more practice, do related Study Problem 8 at the end of this chapter.
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Tools of Financial Analysis -Geometric Mean Rate of Return
;" r.1e o Iool - . .. t.fo1Jnlll
---.
··=
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• Measures the compou nd rate of re­
Geometric
Ave rag e
Retu rn
>> END Checkpoint 2
1+
Rate of Return
for Year 1, rrear I
Rate of Retu rn
-·or Year 1, rrear n
x 1
+ Rate of Return
for Year 1, rrear 2
- 1
x ... x
turn earned from an investment using
multiple annual rates of return
• The higher the estimated rate of return,
the higher is the value of the invest­
ment at the end of the holding period
in n years.
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How is a simple arithmetic average computed? For example, what is the arithmetic average of the following annual rates
of return: 10 percent,
percent, and 5 percent?
annual rates of return : 1O percent,
/
percent, and 5 percent?
Why is the geometric average different from the arithmetic average?
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A n Int r o duct io n to R is k and R et ur n
What Determines Stock Prices?
-
Our review of financial market history tells us that stock and bond returns are subject to sub­
stantial fluctu ations. As an investor, how should you use this info1111ation to form your portfo­
lio? Should you invest all of your retirement account in stocks, becau se historically stocks have
perf ormed very well? Or, should you be timing the market, buying stocks when the returns
look good and buying bonds when the stock market is looking rather weak ? Note that this is ex­
actly the question the great-grandf ather faced in the example we used to introduce this chapter.
To answer these questions, we must first understa nd what causes stock prices to move
from month to month. In short, stock prices tend to go up when there is good news about
fu ture prof its, and they go down when there is bad news about fu ture prof its. This, in part,
explain s the favorable returns of stocks in the United States over the past 80 years , and it also
explains the very bad returns of 2008 through early 2009. Althou gh the country certainly has
gone through some challenging times, for the most part the last century was quite good for
American bu sinesses and, as a result, stock prices did quite well.
One might be ten1pted to use this logic and invest more in stocks when the economy is
doing well and less in stocks when the economy is doing poorly . Indeed, one might think that
it is possible to do even better by picking the individu al stocks of companies whose prof its are
likely to increase. For example, one might w ant to buy oil stocks when oil prices are increas­
ing and at the same time sell a i rline stocks, as the prof its of these firms will be hurt by the
increased cost of jet fuel.
Unfortunately , according to the efficient markets hypothe sis, a strategy of shifting one's
portfolio in response to public information, such as changes in oil prices, will not result in
higher expected returns. This is becau se in an efficient market, stock prices are forward look­
ing and reflect all av ailable public inf ormation about future prof itability. Strategies that are
based on such information can generate higher expected returns only if they expose the in­
vestor to higher risk. This theory underlies much of the study of financial markets and is the
foundation for the rest of this chapter.
•
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The Efficient Markets Hypothes is
-
The concept that all trading opportunities are fairly priced is referred to as the efficient markets
hypothesis (EMH), which is the basis of 1:i:J Principle 4: Market Prices Reflect Information.
The efficient markets hypothesis states that securities prices accurately reflect future expected
cash flows and are based on all info1·1nation available to investors.
An efficient market is a market in which all the available information is f ully incorpo­
rated into securities prices, and the returns investors will earn on their investments cannot be
predicted . Taking this concept a step further, we can distinguish between weak-form efficient
markets, semi-strong-f orm efficient markets, and strong-f orm efficient markets, depend­
ing on the degree of efficiency :
1. The weak-form efficie nt market hypot hesis asserts that all past security market inf orma­
tion is fully reflected in securities prices. This means that all price and volume inf orma­
tion is already reflected in a security 's price.
2. The semi-st rong -form efficien t market hypothe sis asserts that all publicly available inf or­
mation is fully reflected in securities prices. This is a stronger statement becau se it isn't
limited to price and volume inf ormation, but includes all public information. Thu s, the
firm 's financial statements ; new s and announcement s about the economy , industry, or
company; analysts' estimates on future earnings; or any other publicly available inf orma­
tion is already reflected in the security 's price. As a result, taking an investments class
w on't be of any v alue to you in picki ng a winner.
3. The strong -fo rm eff icient ma rket hypothe sis asserts that all information , regardless of
whether this inf ormation is public or priv ate, is fully reflected in securities prices. This
form of the efficient market hypothesis encompasses both the weak-f orm and semi­
strong-form efficient market hypotheses. It asserts that there isn't any inf ormation that
isn't already embedded into the prices of all securities. In other words, even insider
inf ormation that is, material information that isn 't available to any other investor is
of no use.
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A n I nt ro d uct i o n t o R is k and R et ur n
Do We Expect Financial Markets
to Be Perfectly Efficient?
A famous quote from Milton Friedman says that ''there is no such thing as a free lunch." In other
words, everything that has benefits also has costs. The efficient markets hypothesis can be viewed
as a special case of Milton Friedman 's notion of ''no free lunch."The basic idea is that if someone is
offering free lunches, the demand for those lunches will explode, and will be impossible to satisfy.
Similarly, if there were a simple trading strategy that made money without subjecting in­
vestors to risk, then every investor would want to invest with that strategy. However, this is
clearly impossible, because for every stock that is bought, there must be someone selling. In
other words, the stock market can offer you a free lunch (in this case, an underpriced stock) only
when other investors exist who are willing to provide millions of free lunches to both you and
all the other investors who would be very pleased to buy underpriced stocks and sell overpriced
stocks. Individuals generally like to think that when they buy and sell stock they are trading with
an impersonal ''market." In reality, when you buy or sell a stock, in most cases you are trad­
ing with professional investors representing institutions such as Goldman Sachs, Fidelity, and
Merrill Lynch. What this means is that when you buy a stock because you think it is under­
priced, you are likely to be buying it from someone who thinks the same stock is overpriced !
This argument suggests that one should not expect to find prof itable investment strategies
based on publicly available information. In other words, markets should be at least weak-f orm
and semi-stron g-form efficient. If there did exist simple profitable strategies, then they would
attract the attention of investors who, by implementing those strategies, would compete away
their prof its. For example, suppose that it became known that the stocks of well-managed
firms tended to realize higher rates of return . This would encourage investors to increase their
holdings of well-managed companies, thereby increasing the stock prices of these firms to the
point where their stocks would be no better or worse long-term investments than the stocks of
poorly managed firms.
What about investment strategies that require private inf ormation, or that are complicated
and require quite a bit of work to figure out? If the market were so efficient that investment
strategies, no matter how complex, earned no prof its, then no one would bother to take the
time and effort to understand the intricacies of security pricing. Indeed, it is hard to imagine
how security markets could be efficient if no one put in the time and effort to study them . For
this reason, we would not expect financial markets to be strong-form efficient. We expect the
market will partially, but not perf ectly, reflect inf ormation that is priv ately collected.
To understand this concept, let's think about how biotech stock prices are likely to re­
spond when a promising new drug receives Food and Drug Administration (FDA) approv al.
If almost all market investors ignored information about drug approvals, the market might re­
spond very little. This would allow those investors who collected and interpreted information
about new dru gs to be able to exploit the inf ormation to earn significant trading prof its. How­
ever, if those prof its are very high, then we might expect more investors to become interested
in collecting inf ormation of this type, which would in turn make the market more efficiently
incorporate this type of inf ormation into market prices. However, if there were absolutely no
prof its to be made from collecting this type of inf ormation, then the incentive to collect the
information would be eliminated . For this reason , we expect markets to be just ineff icient
enough to provide some investors with an opportunity to recoup their costs of obtaining inf or­
mation, but not so inefficient that there is easy mon ey to be made in the stock market.
•
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The Behavioral View
Milton Friedman 's ''no free lunch'' view of markets assumes that investors, as a group, are
pretty rational. This was the view taken by most economists until very recently . Financial
economists have started to study the implications of the fact that individuals are not strictly
rational. This new approach to the study of finance has gained a strong following and even
resulted in a Nobel Prize for Princeton psy chologist Daniel Kahneman in 2002.
If we believe that investors do not rationally process information, then market prices may
not accurately reflect even public information. As an example, economists have suggested that
overconf ident investors tend to underreact when a company 's management announces earnings
or makes other statements that are relevant to the value of the firm's stock. This is because inves­
tors have too much .confidence in their own views of the company 's true value and tend to place
231
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A n I nt roduct io n t o R i s k and Ret ur n
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# l .. Value stocks
outperf orm growth
stocks
Value stocks, which are stocks with tang ible assets that g enerate
citrrent earnings, ha:iAe tended to outpeiform g rowth stocks, which
are stocks with low current earnings that are expected to g r0w in the
f uture. More specifically, stocks with low price-to"earnirrgs ratios, low
price-to-cash-flow ratios, and low price-to-book-value ratios tend to
outperform the market.
#2. Momentum in
stock returns
Stocks t:fu.at have performed well in the past 6 to 12 months tend to
continue to outperf orm other stocks.
•
#3. Over- and
underreaction
to corporate
announcements
The market has tended to underreact to many corpor ate events. For
example, stock prices react favorably on dates when firms announce
favorable earnings news, which is exactly what we would expect in
an efficient market. However, on the day.s after favorable earnings
news, stock .returns continue to be positive on average. This is known
as post-earnings announcement drif t. Similarly, there is evidence of
some degree of predictability in stock returns following other major
annouacements, s.uch as the issuance of stock or bonds.
too little weight on new information provided by management. As a result, this new information,
even though it is publicly and freely available, is not completely reflected in stock prices.
Market Efficiency: What Does the Evide nce Show ?
The extent to which financial markets are efficient is an important question with broad impli­
cations. As a result, this question has generated thousands of empirical studies. Although this
is a topic that has generated considerable debate and disagreemen t, our interpretation of the
matter is that, historically, there has been some evidence of inefficiencies in the financial mar­
kets. Most of the evidence of market inefficiency can be summarized by three observations
found in Table 4. Note that evidence that the equity market is inefficient is tantamou nt to say­
ing that investors can earn returns greater than their investment 's risks would warrant by
engaging in a trading strategy designed to take advantage of the mispricing.
We should stress that although the evidence relating to the return patterns described in
Table 4 is quite strong for studies that examine returns prior to 2000, more recent evidence
suggests that strategies that exploit these patterns have been quite risky and have not been suc­
cessful after 2000. Indeed, the quantitative hedge funds that exploited those patterns lost consid­
erable amounts of money during the 2007 to 2009 financial crisis period . What do we learn from
the initial success and demise of strategies using these patterns? The first lesson is that there
may be information that predicts returns that are not well known. However, when the informa­
tion becomes widely known, which was the case after the publication of academic research that
documented these return patterns, we expect institution al investors to trade aggressively on
the patterns, and thereby eliminate the inefficiencies. This suggests that, looking forward, one
should probably assume that the financial markets are pretty efficient, at least in the semi-strong
form. In particular, we do not expect the simple momentum and value strategies that worked so
well prior to 2000 to work well going forward . However, we cannot rule out the possibility that
one of our clever readers will develop an innovative and successful strategy.
•
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Concept Check 4
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1. What is an "eff icient market"?
2. What are the three categories of information that are commonly used to categorize tests of the efficient market
hypothesis?
3. How do behavioral biases affect the efficiency of market prices?
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Applying the Principles of Finance to This Chapter
Principle 2: There Is a Risk-Return Tradeoff In examining
historical rates of return realized on securities with differ ent risks, we see that
Principle 2 does indeed hold true riskier investments are indeed rewarded
with higher expected returns. However, it should be pointed out that although
investors expect to receive higher returns for assuming more risk, there is no
guarantee that they will get what they expect.
Principle 4 Market Prices Reflect Information This helps us
understand the wisdom of markets and how investor purchases and sales of a
security drive its price to reflect everything that is known about that security's
risk and expected return and provides the basis for the efficient markets
hypothesis.
..
tJmmar1es
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Calculate realized and expected rates of return and risk.
SUMMARY : We refer to the actual rate of return earned on an investment as the realiz ed rate of
;·eturn. This can be expressed as a percen tage or as a cash amount gained or lost on the
'
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investment.
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But because investment return s are uncertain, w e mu st speak in terms of expected
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expe cted rate of retu rn is the rate we anticipate earning on an investmen t and is the rate
Ireturns. The
relied on
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when evaluating a particular investment opportunity . We can calculate the expected rate of return
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u sing Equation (3):
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Rate of
Expected Rate
of Return-
Probability
Rate of
+
Return 1 X of Return 1
Probability
Return 2 X of Return 2
[ E ( r) J
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+ ...
( Pb2 )
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Rate of
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+
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Probability
(3)
Return n X of Return n
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The risk of an individu al asset can be measu red by the dispersion in possible return outcomes from
an investment in that asset. We measure dispersion u sing the variance, which is calculated using
Equation (4):
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Rate of
Variance in
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Return 1 -
Rates of Return =
( 0'2 )
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Rate of
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+
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of Return
Expected Rate
Return 2 -
2
of Return
X of Return 2
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Expected Rate
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+ ... +
'
X of Return 1
Probability
E( r )
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Probability
(Pb1)
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2
E (r)
( r1)
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Expected R ate
Return n -
I
of Return
2
Probability
X of Return n
E( r )
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(4)
( Pbn )
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Risk is also measured u sing the square root of the variance or the standa rd d eviation. The latter
provides the same indication of investment risk but is stated in terms of percent returns, so it is
sometimes preferred becau se of its easier interpretatio n.
KEY TERMS
Cash return The monetary increase (decrease)
in the value of an investment measured over a
particular span of time.
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possible rates of return, where each pos sible re­
turn is weighted by the probability that it might
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occur.
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earned by investing for a specific period of time,
such as one year or one month.
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Expected rate of return The average of all
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Probability distribution For an investmen t's
rate of return, a description of all possible rates of
return from the investment along with the associ­
ated probabilities for each outcome.
Rate of return See Holding period return .
Risk-free rate of return The rate of return
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Holding period return The rate of return
•
earned
by
investin
g in a
security
that
always
pays the promised rate of return (without risk) .
233
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. .
A n Introduct ion to Ris k and Retur n
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of return. As such, the variance is a measure of
the average squared difference in possible and
expected rates of return.
Standard deviation The square root of the
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vari ance.
Variance The average of the squared difference
in possible rates of return and the expected rate
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Cash
'
Ending
Return
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+
Price
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Cash Distribution
(Dividend )
Ending
Price
Rate of
I
Price
Beginning
(Dividend )
Price
-
Cash Return
(2)
-
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(1)
Cash Distribution
+
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Beginning
Return
Beginning
Price
--
Beginning Price
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Expected Rate
Rate of
of Return
Probability
Rate of
+
Return 1 X of Return 1
[ E ( r) J
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+ · ·· +
I
Return 2 X of Return 2
( r2)
( Pb1)
Rate of
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Probability
Probability
•
(3)
Return n X of Return n
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Variance in
Rate of
Expected Rate
Return 1
of Return
('1)
E( r )
2
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Concept Check I 1
Rate of
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---- ----·
-------
+
1. If you invested $100 one year
ago that is worth $1 1O today,
Expected Rate
Return 2 -
of Return
( r2 )
what rate of return did you earn
on your investment'?
I
Rates of Return =
( u2)
+ ... +
2. What is the expected rate of
Probability
X of Return 1
( Pb 1 )
2
Probability
X of Return 2
E( r )
Rate of
Expected Rate
Return n - of Return
(4)
( Pb2 )
2
Probability
X of Return n
return, and how is it different
than the realized rate of return?
3. What is the vanance i n the rate
Standard
of return of an i nvestment'?
Deviation, u
- VVariance or
4. Why is variance used to
,
measure risk?
--·
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(5)
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Describe the historical pattern of financial market returns.
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Concept Check I 2
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SUMMARY: Perhaps the most important observ ation we can make about the historical returns of
different types of investmen ts is that the average rates of return earned on more risky investments
have been higher than the average rates of return earned on investments that have less risk. Specifi­
cally, equity securities have. earned higher returns than debt securities, corporate debt securities
have earned higher returns than government debt securities , and long-term debt securities have
earned higher return s than short-term debt securities.
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and how is it measured?
1 . How well does the risk-return
principle hold up i n light of
historical rates of return?
Explai n.
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I 2. What is the equity nsk premium,
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3. Does the historical evidence suggest that
i nvesting 1 n emerging markets is more
or l ess risky than investing i n developed
markets?
234
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Developed country Sometimes
represent about a fifth of the world 's economies .
China and India are perhaps the
best known and largest of the
emerging-market economies.
referred to as an industrialized
country, where the term is u sed to
identify those countries such as the
United States, Great Britain, France,
and so forth that have highly
sophisticated and well-developed
•
economies.
Equity risk premium The
difference betw een returns of the
riskier stock investments and the
less risky investments in
government securities .
Emerging market One located in
an economy with low -to-middle percapita
income. These countries constitute
roughly 80 percent of the world 's
population and
Volatility Another term for the
fluctuation in returns.
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A n Intr oduction to R is k and R etu r n
Compute geometric (or compound) and arithmetic average rates of return.
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SUMMARY: When analyzing how a particular investment has perf ormed in the past, we typically
begin by calculating the rates of return earned over several years . These annual rates of return are
then averaged to calculate the arithmetic average in an effort to understand how the investment
has perf ormed in comparison with other investments. The geometric mean is the preferred type of
average for use when analyzing compou nd average rates of return, becau se it provides the rate at
which the investmen t's value has grown .
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Concept Check I 3
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How is a simple arithmetic
average computed? For
example, what is the arithmetic
average of the following annual
rates of return: 1O percent,
KEY TERMS
Arithmetic average return
The sum of the set of returns divided by their
number.
·
How i s a geometric average
rate of return computed? For
example, what is the geometric
average of the following annual
rates of return: 1O percent,
•
Geometric or compound average re­
turns The rate of return earned on an invest­
ment that incorporates consideration for the
effects of compou nd interest .
KEY EQUATIONS
Geometric
Average Return
3. Why is the geometric average
1+
Rate of Return
for Year 1
x
Rate of Return
average?
x ... x
1+
I
1
+
Rate of Return
for Year 2
l/n
1
(6)
for Year n
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Explain the efficient market hypothesis and why it is important to stock
pr•ices.
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SUM MARY: The concept of efficient markets describes the extent to which information is incor­
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porated into security prices. In an efficient market, security prices reflect all available
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at all times; and, becau se of this, it is impossible for an investor to consistently
inf ormation I'
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earn high rates of
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return withou t taking substantial risk.
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Market efficiency is a relative concept. We do not expect financial markets to reflect
100 percent of the available information, but we also do not expect to see very many easy prof it
opportunities. In general , we expect financial markets to be weak -form eff icient, which means
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that information about past prices and volumes of trading are fully reflected in current prices. For
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the most part we also expect financial markets to be semi -st rong -fo rm efficient, which means that
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market prices fully reflect all publicly available information (that is, information from the firm's
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publicly released financial statements , inf ormation revealed in the financial press, and so forth). Fi­
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nally , to a lesser extent, finance markets are st rong -form efficie nt, meaning that prices fully
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reflect I'
privately held inf ormation that has not been released to the general public.
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Concept Check
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KEY TERMS
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1 . What is an "efficient market"?
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2. What are the three categories
of information that are
commonly used to categorize
tests of the efficient market
hypothesis?
•
3. How do behavioral biases
affect the efficiency of market
prices?
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Efficient market A market in which prices
Strong-form efficient market
quickly respond to the annou ncement of new
information .
A market in which even private information is
fully and quickly reflected in market prices.
Efficient markets hypothesis (EMH) This
Weak-form efficient market
hypothesis states that securities prices accurately
reflect fu ture expected cash flows and are based
on all information available to investors .
A market in which current prices quickly and
accurately reflect information that can be derived
from patterns in past security prices and trading
volumes.
Semi-strong-form efficient market A mar­
ket in which all publicly available information is
quickly and accurately reflected in prices.
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235
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A n Int roduct ion to R isk and Return
a common stock dividend . What rate of return would you have earned on your in­
vestment had you purchased the shares on December 24, 2007?
2.
(Calculating rates of return) The S&P stock index represen ts a portf olio com­
prised of 500 large publicly traded companies. On December 24, 2007, the index
had a valu e of 1,410 and on December 23, 2008, the index was approximately 890.
If the average dividend paid on the stocks in the index is approximately 4 percen t of
the value of the index at the beginning of the year, what is the rate of return earned
on the S&P index ? What is your assessment of the relative riskiness of the Google
investmen t (analyzed in the previous problem) compared to investing in the S&P
index?
3.
(Calculating rates of return) The common stock of Placo Enterprises had a market
price of $12 on the day you purchased it just one year ago. Durin g the past year the
stock had paid a $1 dividend <1:nd closed at a price of $14. What rate of return did you
earn on your investment in Placo 's stock?
4.
(Calculating rates of return) Blaxo Balloons manuf actures and distributes birth day
balloons. At the beginning of the year Blaxo' s common stock was selling for
$20 but by year end it was only $18. If the firm paid a total cash dividend of
$2 during the year , what rate of return would you have earned if you had purchased
the stock exactly one year ago? What would your rate of return have been if the firm
had paid no cash dividend ? .
5.
(Computing rates of return) From the following price data, compute the annual
rates of return for Asman and Salinas.
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1
2
3
4
$10
12
11
13
--
--
·-..
$30
28
32
35
-·-·-··----·-·"
--·- ·-·· --
How wou ld you interpret the meaning of the annu al rates of return?
6.
(Related to Checkpoint 1 ) (Expected rate of return and risk)
B. J. Gautney Enterprises is evaluating a security. One-year Treasury bills are
currently paying 2.9 percen t. Calculate the following investmen t 's expected return
and its standard deviation. Should Gautney invest in this security ?
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Return
.40
.15
7.
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-3%
2%
4%
- --·'-··---·· ·------
6%
(Expected rate of return and risk) Syntex , Inc. is considering an investment in one
of two common stocks. Given the information that follows, which investment is bet­
ter, based on risk (as measured by the standa1·d deviation) and return?
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Common
A
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.30
.40
.30
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237
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Return
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(Related to Checkpoint 2 ) (Calculating the geometric and arithmetic average rate
of return) Caswell Enterpri ses had the following end-of-year stock prices over the
last five years and paid no cash dividends:
-
Geometric vs.Arithmetic Average Rates of
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$10
15
12
1
2
3
4
5
-
9
10
•
a. Calculate the annual rate returns for each year from this information.
b. What is the arithmetic average rate of return earned by investing in Caswell 's
stock over this period ?
c. What is the geometric average rate of return earned by investing in Caswell's
stock over this period ?
d. Considering that the begin ning and ending stock prices for the five-year period
are the same, which type of average rate of return best describes the average
annual rate of return earned over the period (the arithmetic or geometric) ?
9.
•
(Calculating the geometric and arithmetic average rate of return) The common
stock of the Brangu s Cattle Company had the following end-of -year stock prices
over the last five year s and paid no cash dividends:
B ra14gus "aHie"11"1tmpa. qv .
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2
3
4
5
•
•
$15
10
12
1
••
23
25
a. Calculate the annual rate of return for each year from this inf ormation .
b. What is the arithmetic average rate of return earned by investing in the compan y 's
stock over this period ?
c. What is the geometric average rate of return earned by investing in the company 's
stock over this period ?
d. Which type of average rate of return best describes the average annual rate of
return earned over the period (the arithmetic or geometric) ? Why ?
10.
•
•
•
(Comprehensive problem) Use the following end-of -year price data to answer the
following questions for the Barris and Carson Companies.
ime
$10
12
8
15
1
2
-
3
4
..Carson
Barris
--
$20
28
-
32
27
--
a. Compute the annual rates of return for each time period and for both firms.
b. Calculate both the arithmetic and geometric mean rates of return for the entire
three-year period using your annual rates of return from part a. Note: You may
assume that neith·er fir111pays any dividends.
c. Compute a three-year rate of return spanning the entire period (i.e., using the
beginning price for Period 1 and ending price for Period 4).
d. Because the rate of return calculated in part c is a three-year rate of return, convert
it to an annual rate of return by using the following equation:
3
1+
238
'
Three-Year
Rate of Return
-
1
+ Annu al Rate
•
of Return
-
--
'
'
A n Int roduct ion t o Risk and Ret u.r n
e. How is the annual rate of return calculated in part d related to the geometric rate
of return ? When you are evaluating the perf ormance of an investment that has
been held for several years, what type of average rate of return should you use
(arithmetic or geometric)? Why ?
Mini...Case
After graduating from college last spring with a major in ac­
counting and finance, Jim Hale took a jo b as an analyst trainee
for an investment company in Chicago. His first few weeks
w ere filled with a series of rotations through out the firm' s·vari­
ous operating units , but this week he was assigned to one of the
firm's trader s as an analyst. On his first day Jim's boss called
him in and told him that he w anted to do some rudimentary
analysis of the investment returns of a semiconductor manu ­
facturer called Advanced Micro Devices, Inc. (Ticker: AMD) .
Specifically , Jim w as given the following month-end clos­
ing prices for the company spanning the November 1, 2011,
through No vember 1, 2012:
Questions
·. Date
l-Nov-11
l-Dec-11
3-Jan-12
l-Feb-12
l-Mar-12
2-Apr-12
1-May-12
5.69
5.4
6.71
7.35
8.02
7.36
6.08
l -Jun-12
2-Jul-12
l-Aug- 12
4-Sep-12
l-Oct-12
l-Nov-12
3.72
3.37
2.05
1.88
He was then instructed by his boss to complete the follow­
ing tasks usin g the AMD price data (note that AMD paid no
dividend during 2008).
---------------------------------------------------------------------------------------------
1. Compute the monthly realized rates of return earned by
AMD for the entire year.
2. Calculate the average monthly rate of return for AMD usin g
5. Now calculate the annual rate of return using the geo­
metric average monthly rate of return using the following
relationship:
both the arithmetic and geometric averages.
Compound Annual
Rate of Return
3. Calculate the year-end price for AMD, computing the com­
pound value of the beginning-of-year price of $5.69 per
share for 12 months at the monthly geometric average rate
of return calculated earlier:
•
End-of-Year _ Begin ning-of -Year
-
Stock Price
1
+
Stock Price
12
Geometric Average
Monthly Rate of Return
4. Compute the annual rate of return for AMD using the
beginning stock price for the period and the ending price
(i.e., $5.69 and $1.88).
•
re I ,_ S ____ _____ _ _ .,_
1+
Geometric Average
Monthly Rate of Return
12
-1
6. If you were given annual rate of return data for AMD or
any other company 's stock and you were asked to estimate
the average annual rate of return an investor would have
earned over the sample period by holding the stock, would
you use an arithmetic or geometric average of the historical
rates of return ? Explain your response as if you were talk­
ing to a client who has had no formal training in finance or
investments.
_
_ ___ ---------- -·---- ·----.. ·----- ---·-··-·· ··----- ·-·· ·-- --
Credits are listed in order of appea rance.
Pixelbliss/Fotolia; Runzelkorn/Shutterstock
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