W E B
A P P E N D I X
CHAPTER
1A
A Review of Statistics and the Security
Market Line
Although many of these concepts will be familiar to you from earlier statistics or introductory corporate finance courses, a quick review of this appendix will be helpful as you work your way through the
remainder of the chapters.The first section reviews variance and standard deviation, section two deals
with covariance, and the last section reviews some of the security market line fundamentals.
1.1
COMPUTING VARIANCE AND STANDARD DEVIATION
Variance and standard deviation are measures of how actual values differ from the expected values
(arithmetic mean) for a given series of values. In this case, we want to measure how rates of return differ
from the arithmetic mean value of a series. There are other measures of dispersion, but variance and
standard deviation are the best known because they are used in statistics and probability theory.Variance is defined as:
n
Variance (S2 ) ⴝ a (Probability ) (Possible Return ⴚ Expected Return ) 2
iⴝ1
n
ⴝ a (Pi )[Ri ⴚ E (Ri )] 2
iⴝ1
Consider the following example, as discussed in Chapter 1:
Probability of Possible
Return (Pi)
Possible Return (Ri)
Pi Ri
0.15
0.15
0.70
0.20
⫺0.20
0.10
0.03
⫺0.03
0.07
⌺ = 0.07
This gives an expected return [E(Ri)] of 7%. The dispersion of this distribution as measured by
variance is:
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Probability (Pi)
Return (Ri)
Ri ⴚ E(Ri)
[Ri ⴚ E(Ri)]2
Pi[Ri ⴚ E(Ri)]2
0.15
0.15
0.70
0.20
⫺0.20
0.10
0.13
⫺0.27
0.03
0.0169
0.0729
0.0009
0.002535
0.010935
0.000630
⌺ = 0.014100
WEB APPENDIX 1A
1
2
WEB APPENDIX 1A
The variance (␴ 2) is equal to 0.0141.The standard deviation is equal to the square root of the variance:
n
Standard Deviation (S2 ) ⴝ
Da
iⴝ1
Pi[Ri ⴚ E (Ri )] 2
Consequently, the standard deviation for the preceding example would be:
Si ⴝ 10.0141 ⴝ 0.11874, or 11.874%
Therefore, you could describe this distribution as having an expected value of 7% and a standard deviation of 11.87%.
In many instances, you might want to calculate the variance or standard deviation for a historical
series in order to evaluate an investment’s performance.Assume that you are given the following information on annual rates of return (HPY) for common stocks listed on the TSX:
Year
Annual Rate of Return
2010
2011
2012
2013
2014
0.07
0.11
⫺0.04
0.12
0.06
In this case, we are evaluating actual returns; therefore, we assume equal probabilities.The expected
value (in this case the mean value, R) of the series is the sum of the individual observations in the series
divided by the number of observations, or 0.04 (0.20/5).The variances and standard deviations are:
Year
Ri
Ri ⴚ R
(Ri ⴚ R)2
2010
2011
2012
0.07
0.11
⫺0.04
0.03
0.07
⫺0.08
0.0009
0.0049
0.0064
2013
2014
0.12
⫺0.06
0.08
⫺0.10
0.0064
0.0110
⌺ ⫽ 0.0286
␴2 ⫽ 0.0286/5
⫽ 0.00572
␴ ⫽ 10.00572
⫽ 0.0756
We can interpret the performance of TSX common stocks during this period of time by saying that
the average return was 4% and the standard deviation of annual returns was 7.56%.
1.1.1
Coefficient of Variation
In some instances, you might want to compare the dispersion of two different series.The variance and
standard deviation are absolute measures of dispersion, which means they can be influenced by the magnitude of the original numbers.To compare series with greatly different values, you need a relative measure of dispersion.The coefficient of variation is a measure of relative dispersion:
Coefficient of Variation (CV) ⴝ
Standard Deviation of Returns
Expected Rate of Return
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WEB APPENDIX 1A
3
A larger value indicates greater dispersion relative to the arithmetic mean of the series. For the previous example, the CV would be:
CV1 ⴝ
0.0756
ⴝ 1.89
0.0400
It is possible to compare this value to a similar figure having a markedly different distribution. For
example, assume you wanted to compare this investment to another investment that had an average
return of 10% and a standard deviation of 9%. The standard deviations alone tell you that the second
series has greater dispersion (9% versus 7.56%) and might be considered to have higher risk. In fact, the
relative dispersion for this second investment is much less.
CV1 ⴝ
0.0756
ⴝ 1.89
0.0400
CV2 ⴝ
0.0900
ⴝ 0.90
0.1000
Considering the relative dispersion and the total distribution, most investors would probably prefer
the second investment.
1.2
COVARIANCE
Because most students have been exposed to the concepts of covariance and correlation, the following
discussion is set forth in intuitive terms with examples. A detailed, rigorous treatment is contained in
DeFusco, McLeavey, Pinto, and Runkle (2004).
Covariance is an absolute measure of the extent to which two sets of numbers move together
over time, that is, how often they move up or down together. In this regard, move together means they
are generally above their means or below their means at the same time. Covariance between i and j is
defined as
COVij ⴝ
a (i ⴚ ៮i ) (j ⴚ ៮j )
n
If we define (i ⫺ ៮i ) as i⬘ and ( j ⫺ ៮j ) as j⬘, then
aij
œ œ
COVij ⴝ
n
If both numbers are consistently above or below their individual means at the same time, their products will be positive, and the average will be a large positive value. However, if the i value is below its
mean when the j value is above its mean or vice versa, their products will be large negative values,
giving negative covariance.
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4
WEB APPENDIX 1A
Calculation of Covariance
Exhibit 1A.1
Observation
i
j
i ⴚ ៮i
j ⴚ ៮j
i⬘j⬘
1
3
8
⫺4
⫺4
16
2
6
10
⫺1
⫺2
2
3
8
14
⫹1
⫹2
2
4
5
12
⫺2
0
0
5
9
13
⫹2
⫹1
2
6
11
15
⫹4
⫹3
12
⌺
42
72
7
12
Mean
COVij ⫽
34
34
⫽ ⫹5.67
6
Exhibit 1A.1 should make this clear. In this example, the two series generally moved together, so
they showed positive covariance. As noted, this is an absolute measure of their relationship and, therefore, can range from ⫹⬁ to ⫺⬁. Note that the covariance of a variable with itself is its variance.
1.2.1
Correlation
To obtain a relative measure of a given relationship, we use the correlation coefficient (rij), which
is a measure of the relationship:
COVij
rij ⴝ
SiSj
Recall earlier that standard deviation was:
2
a (i ⴚ ៮i )
Si ⴝ
T
N
If the two series move completely together, then the covariance would equal ␴i␴j and
COVij
SiSj
ⴝ 1.0
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WEB APPENDIX 1A
5
The correlation coefficient would equal unity in this case, and we would say the two series are perfectly correlated. Because we know that
COVij
rij ⴝ
SiSj
we also know that COVij = rij ␴i␴j.This relationship may be useful when computing the standard deviation of a portfolio, because in many instances the relationship between two securities is stated in terms
of the correlation coefficient rather than the covariance.
Continuing with the example from Exhibit 1A.1, the standard deviations are computed in
Exhibit 1A.2, as is the correlation between i and j. As shown, the two standard deviations are rather
large and similar but not exactly the same. Finally, when the positive covariance is normalized by the
product of the two standard deviations, the results indicate a correlation coefficient of 0.898, which
is obviously quite large and close to 1.00. This implies that these two series are highly related.
Exhibit 1A.2
Calculation of Correlation Coefficient
Observation
i ⴚ ៮i
1
2
a
( i ⴚ ៮i ) 2
j ⴚ ៮j
( j ⴚ ៮j )2
⫺4
16
⫺4
16
⫺1
1
⫺2
4
3
⫹4
1
⫹2
4
4
⫺2
4
0
0
5
⫹2
4
⫹1
1
6
⫹4
16
⫹3
9
a
42
s 2j ⫽ 42/6 ⫽ 7.00
s 2j ⫽ 34/6 ⫽ 5.67
s j ⫽ 17.00 ⫽ 2.65
rij ⫽ COVij/si sj ⫽
1.3
34
sj ⫽ 15.67 ⫽ 2.38
5.67
5.67
⫽
⫽ 0.898
12.652 12.382
6.31
SECURITY MARKET LINE
The security market line (SML) reflects the combination of risk and return available on various
investments. Investors would select investments that are consistent with their risk preferences; some
would consider only low-risk investments, whereas others welcome high-risk investments. Exhibit 1A.3
graphs the expected relationship between risk and return.
As noted in Chapter 1, beginning with an initial SML, three changes can occur. First, changes in
the perceived risk of individual investments can result in changed positions on the SML. Second, if
investor attitudes toward risk change, so will the slope of the SML. Lastly, the SML can experience a
parallel shift due to a change in the real risk-free rate (RRFR) or the expected rate of inflation—that
is, a change in the nominal risk-free rate (NRFR).These three possibilities are discussed below.
1.3.1
Movements along the SML
Investors place investment alternatives somewhere along the SML based on their perceptions of the
risk of the investment. Obviously, if an investment’s risk changes due to a change in one of its risk
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6
WEB APPENDIX 1A
Exhibit 1A.3
Relationship between Risk and Return
Expected Return
Low
Risk
Average
Risk
Security Market
Line
(SML)
High
Risk
The slope indicates the
required return per unit
of risk
NRFR
Risk
(business risk, etc., or systematic risk)
Exhibit 1A.4
Changes in the Required Return Due to Movements along the SML
Expected
Return
SML
NRFR
Movements along the
curve that reflect
changes in the risk
of the asset
Risk
sources (e.g., business risk), it will move along the SML. For example, if a firm increases its financial
risk by issuing debt, investors will perceive its common stock as riskier and the stock will move up
the SML to a higher risk position. Investors will then require a higher return. Any change in an asset
that affects its fundamental risk factors or its market risk (i.e., its beta) will cause the asset to move
along the SML, as shown in Exhibit 1A.4. Note that the SML does not change, only the position of
specific assets on the SML.
1.3.2
Changes in the Slope of the SML
The slope of the SML indicates the return per unit of risk required by all investors.Assuming a straight
line, it is possible to select any point on the SML and compute a risk premium (RP) for an asset through
the equation:
1A.1
RPi ⴝ E (Ri ) ⴚ NRFR
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WEB APPENDIX 1A
7
where:
RPi ⫽ risk premium for asset i
E(Ri) ⫽ the expected return for asset i
NRFR ⫽ the nominal return on a risk-free asset
If a point on the SML is identified as the portfolio that contains all the risky assets in the market
(referred to as the market portfolio), it is possible to compute a market RP as follows:
RPm ⴝ E (Rm ) ⴚ NRFR
1A.2
where:
RPm ⫽ the risk premium on the market portfolio
E(Rm) ⫽ the expected return on the market portfolio
NRFR ⫽ the nominal return on a risk-free asset
This market RP is not constant because the slope of the SML changes over time. Although we do not
understand completely what causes these changes in the slope, we do know that there are changes in
the yield differences between assets with different levels of risk even though the inherent risk differences are relatively constant.
These differences in yields, or yield spreads, change over time. For example, if the yield on a
portfolio of AA-rated bonds is 7.50% and the yield on a portfolio of BBB-rated bonds is 9.00%, the
yield spread is 1.50%.This 1.50% is referred to as a credit risk premium because the BBB-rated bond
is considered to have higher credit risk or a higher probability of default.We can see substantial changes
in the yield spreads on AA-rated bonds and BBB-rated bonds in Exhibit 1A.5.
Yield Spreads on Canadian Corporate Bonds
Exhibit 1A.5
Basis points
500
500
450
AA
A
BBB
Investment-grade Index
400
350
450
400
350
300
300
250
250
200
200
150
150
100
100
50
50
0
0
1990
1995
2000
2005
1989–96: Scotia Capital corporate bond yields minus mid-term
GoC yields
1997–2008: Merrill Lynch option-adjusted spreads
1989–91: monthly averages; 1992–2008: end-of-month values
Sources: Bank of Canada, Financial System Review, June 2008, http://www.bankofcanada.ca/en/fsr/2008/fsr_0608.pdf
(Chart 4, page 12). Reprinted by permission.
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8
WEB APPENDIX 1A
Although the underlying risk factors for the portfolio of bonds in the AA-rated and the BBB-rated
bond indices would probably not change dramatically over time, it is clear from the time-series plot in
Exhibit 1A.5 that the difference in yield spreads has experienced changes of more than 100 basis points
(1%) in a short period of time (e.g., see the yield spread increase between 1992 and 1993 and the dramatic declines in 2006). Such a significant change in the yield spread during a period where there is
no major variation in the fundamental risk characteristics of BBB bonds relative to AA bonds would
imply a change in the market RP. Specifically, although the intrinsic financial risk characteristics of the
bonds remain relatively constant, investors have changed the yield spreads they demand to accept this
relatively constant difference in financial risk.
This change in the RP implies a change in the slope of the SML, as presented in Exhibit 1A.6.The
exhibit assumes an increase in the market risk premium, which means an increase in the slope of the
market line. Such a change will affect the required return for all risky assets. Irrespective of where an
investment was on the original SML, its required return will increase, although its individual risk characteristics remain unchanged.
Change in Market Risk Premium
Exhibit 1A.6
Expected Return
New SML
Original SML
Rm′
•
Rm
•
NRFR
•
Risk
1.3.3
Changes in Capital Market Conditions or Expected Inflation
Exhibit 1A.7 shows what happens to the SML when there are changes in one of the following factors:
(1) expected real growth in the economy, (2) capital market conditions, or (3) the expected rate of inflation. For example, an increase in expected real growth, temporary tightness in the capital market, or an
increase in the expected rate of inflation will cause the SML to experience a parallel shift upward as
shown in Exhibit 1A.7.The parallel shift occurs because changes in expected real growth or in capital
market conditions or a change in the expected rate of inflation affect the economy’s NRFR that
impacts all investments, no matter what their levels of risk are.
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WEB APPENDIX 1A
Exhibit 1A.7
9
Capital Market Conditions, Expected Inflation, and the Security Market Line
Expected Return
New SML
Original SML
NRFR´
NRFR
Risk
Key Terms
correlation coefficient, p. 4
covariance, p. 3
security market line (SML), p. 5
yield spreads, p. 7
Problems
1. Your return expectations for the common stock of Gray
Disc Company during the next year are:
GRAY DISC CO.
Possible Rate of Return
Probability
⫺0.10
0.00
0.10
0.25
0.25
0.15
0.35
0.25
a. Compute the expected return [E(Ri)] on this investment,
the variance of this return (␴ 2), and its standard deviation (␴).
b. Under what conditions can the standard deviation be
used to measure the relative risk of two investments?
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c. Under what conditions must the coefficient of variation be used to measure the relative risk of two investments?
2. Your return expectations for the stock of Cornerbrook
Computer Company during the next year are:
CORNERBROOK COMPUTER CO.
Possible Rate of Return
Probability
⫺0.60
⫺0.30
⫺0.10
0.20
0.40
0.80
0.15
0.10
0.05
0.40
0.20
0.10
10
WEB APPENDIX 1A
a. Compute the expected return [E(Ri)] on this stock,
the variance (␴2) of this return, and its standard
deviation (␴).
b. On the basis of expected return [E(Ri)] alone, discuss
whether Gray Disc or Cornerbrook Computer is
preferable.
c. On the basis of standard deviation (␴) alone, discuss
whether Gray Disc or Cornerbrook Computer is
preferable.
d. Compute the coefficients of variation (CVs) for Gray
Disc and Cornerbrook Computer and discuss which
stock return series has the greater relative dispersion.
3. The following are annual returns for Canadian T-bills
and U.K. common stocks.
Year
Canadian T-Bills
U.K. Common Stock
2010
2011
2012
2013
2014
0.063
0.081
0.076
0.090
0.085
0.150
0.043
0.374
0.192
0.106
a. Compute the arithmetic mean return and standard
deviation of return for the two series.
b. Discuss these two alternative investments in terms of
their arithmetic average returns, their absolute risk, and
their relative risk.
c. Compute the geometric mean return for each of these
investments. Compare the return and geometric mean
return for each investment and discuss this difference
between mean returns as related to the standard deviation of each series.
4. As a new analyst, you have calculated the following
annual returns for the stocks of both Granum Corporation and Leader Industries.
Year
Granum’s Rate
of Return
Leader’s Rate
of Return
2009
2010
2011
2012
2013
5
12
⫺11
10
12
5
15
5
7
⫺10
Your manager suggests that because these companies
produce similar products, you should continue your
analysis by computing their covariance. Show all
calculations.
5. You decide to go an extra step by calculating the coefficient of correlation using the data provided in Problem
4. Prepare a table showing your calculations and explain
how to interpret the results.Would the combination of
the common stock of Granum and Leader be good for
diversification?
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