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Chapter 07 - Introduction to Risk and Return
CHAPTER 7
Introduction to Risk and Return
The values shown in the solutions may be rounded for display purposes. However, the answers were
derived using a spreadsheet without any intermediate rounding.
Answers to Problem Sets
1.
Expected payoff = (.10 × $500) + (.50 × $100) + (.40 × $0)
= $100
Rates of return:
($500 – 100) / $100 = 400%
($100 – 100) / $100 = 0%
($0 – 100) / $100 = –100%
Expected rate of return = (.10 × 400%) + (.50 × 0%) + (.40 × –100%)
Expected rate of return = 0%
Variance = .10(400% – 0)2 + .50(0% – 0)2 + .40(–100% – 0)2
Variance = 20,000
Standard deviation = 20,000.5
Standard deviation = 141.42%
Est. Time: 01 – 05
2.
a.
Average nominal return = (.172 + .010 + .161 + .331 + .127) / 5
Average nominal return = .1602, or 16.02%
Variance = [(.172 – .1602)2 + (.010 – .1602)2 + (.161 – .1602)2 + (.331 –
.1602)2 + (.127 – .1602)2] / 5
Variance = .010595
Standard deviation = .010595.5
Standard deviation = .1029, or 10.29%
b.
Average real return = {[(1.172 / 1.015) – 1] + [(1.010 / 1.030) – 1] + [(1.161
/ 1.017) – 1] + [(1.331 / 1.015) – 1] + [(1.127 /
1.008) – 1]} / 5
Average real return = .1412, or 14.12%
Est. Time: 01 – 05
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Chapter 07 - Introduction to Risk and Return
3.
Ms. Sauros:
Average return = [.249 + (–.009) + .186 + .421 + .152] / 5
Average return = .1998, or 19.98%
Variance = [(.249 – .1998)2 + (–.009 – .1998)2 + (.186 – .1998)2 + (.421 –
.1998)2 + (.152 – .1998)2] / 5
Variance = .019485
Standard deviation = .0194
Standard deviation = .1396, or 13.96%
S&P 500:
Average return = (.172 + .010 + .161 + .331 +.127) / 5
Average return = .1602, or 16.02%
Variance = [(.172 – .1602)2 + (.010 – .1602)2 + (.161 – .1602)2 + (.331 –
.1602)2 + (.127 – .1602)2
Variance = .010595
Standard deviation = .010595.5
Standard deviation = .1029, or 10.29%
Est. Time: 01 – 05
4.
a.
False. Investors prefer diversified portfolios because diversification
reduces variability and therefore reduces risk. However, the diversification
of an individual company does not necessarily make it less risky.
b.
True. Stocks must be less than perfectly positively correlated in order to
obtain diversification benefits.
c.
False. The risk eliminated by diversification is called specific risk, or the
risk surrounding an individual company or industry. Market risk will still
exist in a fully diversified portfolio.
d.
False. It is true that the greatest benefit to diversification occurs when
stocks are uncorrelated. However, most stocks tend to move in the same
direction. There are still benefits to diversification any time stocks are less
than perfectly positively correlated.
e.
False. The contribution to portfolio risk depends on the relationship of the
stock to the market as a whole.
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Chapter 07 - Introduction to Risk and Return
f.
True. Market risk is the risk that relates to a diversified portfolio.
g.
True. The only risk inherent in a diversified portfolio is market risk.
h.
False. An undiversified portfolio with a beta of 2 is twice as risky as the
market portfolio. Part of that risk will be market risk and part will be
specific risk.
Est. Time: 01 – 05
5.
Perfect negative correlation does the most to reduce risks because the stocks
always move in opposite directions. When one goes up, the other goes down,
and vice versa.
Est. Time: 01 – 05
6.
Est. Time: 01 – 05
7.
a.
σp = 1.3 × 20% = 26%
b.
σp = 0 × 20% = 0%
c.
βp = 15% / 20% = .75
d.
βp = Less than 1
βp = 20% / 20% = 1, This would be the beta if the portfolio were
diversified. Since the portfolio is non-diversified, the beta must be less
than 1 because part of the portfolio’s risk is specific, or unique, risk.
Est. Time: 01 – 05
8.
βp = {(5 × 1.2) + [(10 – 5) × 1.4)]} / 10
βp = 1.3
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Chapter 07 - Introduction to Risk and Return
Beta measures systematic risk which cannot be eliminated by diversification.
Est. Time: 01 – 05
9.
The beta of each stock is given by the slope of the line, or the rise divided by the
run. The run is the range of the market returns while the rise is the range of the
stock returns.
BetaA = (0 – 20) / (–10 – 10) = 1
BetaB = (–20 – 20) / (–10 – 10) = 2
BetaC = (–30 – 0) / (–10 – 10) = 1.5
BetaD = (15 – 15) / (–10 – 10) = 0
BetaE = (10 – (–10) / (–10 – 10) = –1
Est. Time: 01 – 05
10.
a.
r = [(1 + R) / (1 + h)] – 1
r1929 = {[1 + (–.145)] / (1 + .002)} – 1 = –.1467, or –14.67%
r1930 = {[1 + (–.283)] / [1 + (–.060)]} – 1 = –.2372, or –23.72%
r1931 = {[1 + (–.439)] / [1 + (–.095)]} – 1 = –.3801, or –38.01%
r1932 = {[1 + (–.099)] / [1 + (–.103)]} – 1 = .0045 or .45%
r1933 = [(1 + .573) / (1 + .005)] – 1 = .5652, or 56.52%
b.
Average real return = [–.1433 + (–.2372) + (–.3801) + .0045 + .5652] / 5
Average real return = –.0382, or –3.82%
c.
Risk premium1929 = –.145 – .048 = –.1930, or –19.30%
Risk premium1930 = –.283 – .024 = –.3070, or –30.70%
Risk premium1931 = –.439 – .011 = –.4500, or –45.00%
Risk premium1932 = –.099 – .010 = –.1090, or –10.90%
Risk premium1933 = .573 – .003 = .5700, or 57.00%
d.
Average risk premium = [–.1930 + (–.3070) + (–.4500) + (–.1090) + .5700]
/5
Average risk premium = –0978, or –9.78%
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Chapter 07 - Introduction to Risk and Return
e.
σRisk premium = {[–.1930 – (–.0978)]2 + [–.3070 – (–.0978)]2 + [–.4500 – (–
.0978)]2 + [–.1090 – (–.0978)]2 + [.5700 – (–.0978)]2]} / 5
σRisk premium = .1246, or 12.46%
Est. Time: 06 – 10
11.
a.
A long-term United States government bond is always considered to be
safe in terms of the dollars received. However, the price of the bond
fluctuates as interest rates change, and the rate at which coupon
payments received can be invested also changes as interest rates
change. And, of course, the payments are all in nominal dollars, so
inflation risk must also be considered.
b.
It is true that stocks offer higher long-run rates of return than do bonds, but
it is also true that stocks have a higher standard deviation of return. So,
which investment is preferable depends on the amount of risk one is
willing to tolerate. This is a complicated issue and depends on numerous
factors, one of which is the investment time horizon. If the investor has a
short time horizon, then stocks are generally not preferred.
c.
Unfortunately, 10 years is not generally considered a sufficient amount of
time for estimating average rates of return. Thus, using either a 5- or 10year average is likely to be misleading.
Est. Time: 06 – 10
12.
The risk to Hippique shareholders depends on the market risk, or beta, of the
investment in the black stallion. The information given in the problem suggests
that the horse has very high unique risk, but we have no information regarding
the horse’s market risk. So, the best estimate is that this horse has a market risk
about equal to that of other racehorses, and thus this investment is not a
particularly risky one for Hippique shareholders.
Est. Time: 01 – 05
13.
In the context of a well-diversified portfolio, the only risk characteristic of a single
security that matters is the security’s contribution to the overall portfolio risk. This
contribution is measured by beta. Lonesome Gulch is the safer investment for a
diversified investor because its beta of .10 is lower than the beta of Amalgamated
Copper of .66. For a diversified investor, the standard deviations are irrelevant.
Est. Time: 01 – 05
14.
a.
σP2 = .602 × .102 + .402 × .202 + 2(.60 × .40 × 1 × .10 × .20)
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Chapter 07 - Introduction to Risk and Return
σP2 = .0196
b.
σP2 = .602 × .102 + .402 × .202 + 2(.60 × .40 × .50 × .10 × .20)
σP2 = .0148
c.
σP2 = .602 × .102 + .402 × .202 + 2(.60 × .40 × 0 × .10 × .20)
σP2= .0100
Est. Time: 06 – 10
15.
a.
Refer to Figure 7.13 in the text. With 100 securities, the box is 100 by 100.
The variance terms are the diagonal terms, and thus there are 100
variance terms. The rest are the covariance terms. Because the box has
(100 × 100) terms altogether, the number of covariance terms is:
Number of covariance terms = 1002 – 100 = 9,900
Half of these terms (i.e., 4,950) are different.
b.
Once again, it is easiest to think of this in terms of Figure 7.13. With 50
stocks, all with the same standard deviation (.30), the same weight in the
portfolio (.02), and all pairs having the same correlation coefficient (.40),
the portfolio variance is:
σ2 = 50(.02)2(.30)2 + [(50)2 – 50](.02)2(.40)(.30)2 = .03708
σ = .193, or 19.3%
c.
For a fully diversified portfolio, portfolio variance equals the average
covariance:
σ2 = (.30)(.30)(.40) = .036
σ = .190, or 19.0%
Est. Time: 06 – 10
16.
a.
Refer to Figure 7.13 in the text. For each different portfolio, the relative
weight of each share is (1 / number of shares (n) in the portfolio), the
standard deviation of each share is .40, and the correlation between pairs
is .30. Thus, for each portfolio, the diagonal terms are the same, and the
off-diagonal terms are the same. There are n diagonal terms and (n2 – n)
off-diagonal terms. In general, we have:
Variance = n(1 / n)2(.4)2 + (n2 – n)(1 / n)2(.3)(.4)(.4)
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Chapter 07 - Introduction to Risk and Return
For one share:
Variance = 1(1)2(.4)2 + 0 = .160000
For two shares:
Variance = 2(.5)2(.4)2 + 2(.5)2(.3)(.4)(.4) = .104000
The results are summarized in the second and third columns of the
table in part (c) below.
b.
The underlying market risk that cannot be diversified away is the second
term in the formula for variance above:
Underlying market risk = (n2 – n)(1 / n)2(.3)(.4)(.4)
As n increases, [(n2 – n)(1 / n)2] = [(n – 1) / n] becomes close to 1, so that
the underlying market risk is: [(.3)(.4)(.4)] = .048.
c.
This is the same as Part (a), except that all of the off-diagonal terms are
now equal to zero. The results are summarized in the fourth and fifth
columns of the table below.
(Part a)
No. of
Shares
1
2
3
4
5
6
7
8
9
10
Variance
.160000
.104000
.085333
.076000
.070400
.066667
.064000
.062000
.060444
.059200
(Part a)
Standard
Deviation
.400
.322
.292
.276
.265
.258
.253
.249
.246
.243
(Part c)
Variance
.160000
.080000
.053333
.040000
.032000
.026667
.022857
.020000
.017778
.016000
(Part c)
Standard
Deviation
.400
.283
.231
.200
.179
.163
.151
.141
.133
.126
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McGraw-Hill Education.
Chapter 07 - Introduction to Risk and Return
Graphs for Part (a):
Portfolio Standard Deviation
Portfolio Variance
0 .5
Standard Deviation
0 .2
Variance
0 .1 5
0 .1
0 .0 5
0
0 .4
0 .3
0 .2
0 .1
0
0
2
4
6
8
10
12
0
2
N u m b e r o f S e c u ritie s
4
6
8
10
12
N u m b e r o f S e c u ritie s
Graphs for Part (c):
Portfolio Standard Deviation
Portfolio Variance
0 .5
Standard Deviation
0 .2
Variance
0 .1 5
0 .1
0 .0 5
0
0 .4
0 .3
0 .2
0 .1
0
0
2
4
6
8
N u m b e r o f S e c u ritie s
10
12
0
2
4
6
8
10
12
N u m b e r o f S e c u ritie s
Est. Time: 16 – 20
17.
The table below uses the format of Figure 7.13 in the text in order to calculate the
portfolio variance. The portfolio variance is the sum of all the entries in the
matrix. Portfolio variance is .03178.
BHP
BP
Fiat
Heineken
Korea Electric
Nestlé
Sony
Tata Motors
BHP
.0006126
.0003781
.0005062
.0000893
.0002841
-.0000090
.0002636
.0006050
BP
.0003781
.0013231
.0007832
.0002051
.0003290
.0000529
.0008359
.0005157
Korea
Fiat Heineken
Electric
.0005062 .0000893 .0002841
.0007832 .0002051 .0003290
.0028971 .0002063 .0003558
.0002063 .0005085 .0001334
.0003558 .0001334 .0012102
-.0000653 .0001203 .0000042
.0013274 .0004677 .0003120
.0008420 .0002866 .0002211
Nestlé
-.0000090
.0000529
-.0000653
.0001203
.0000042
.0001470
.0001563
.0000474
Sony
.0002636
.0008359
.0013274
.0004677
.0003120
.0001563
.0031416
.0005206
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Tata
.0006050
.0005157
.0008420
.0002866
.0002211
.0000474
.0005206
.0023900
Chapter 07 - Introduction to Risk and Return
Est. Time: 21 – 25
18.
“Safest” means lowest risk; in a portfolio context, this means lowest variance of
return. Half of the portfolio is invested in British Petroleum stock (BP), and half of
the portfolio must be invested in one of the other securities listed. Thus, we
calculate the portfolio variance for seven different portfolios to see which is the
lowest (see table below). The safest attainable portfolio is comprised of British
Petroleum stock (BP) and Nestlé.
Company with BP:
Variance
Variance
BHP
= .52 × .19802 + .52 × .29102 + 2 × .5 × .5 × .42 × .1980 × .2910 =
.04307
Fiat Chrysler
= .52 × .43062 + .52 × .29102 + 2 × .5 × .5 × .40 × .4306 × .2910 =
.09259
Heineken
= .52 × .18042 + .52 × .29102 + 2 × .5 × .5 × .25 × .1804 × .2910 =
.03587
Korea Electric
= .52 × .27832 + .52 × .29102 + 2 × .5 × .5 × .26 × .2783 × .2910 =
.05106
Nestlé
= .52 × .09702 + .52 × .29102 + 2 × .5 × .5 × .12 × .0970 × .2910 =
.02522
Sony
= .52 × .44842 + .52 × .29102 + 2 × .5 × .5 × .41 × .4484 × .2910 =
.09819
Tata Motors
= .52 × .39112 + .52 × .29102 + 2 × .5 × .5 × .29 × 39.11 × .2910 =
.07591
Est. Time: 11 – 15
19.
a-1.
Change in stock’s rate of return = .05 × –.25 = –.0125, or –1.25%
a-2.
Change in stock’s rate of return = –.05 × –.25 = .0125, or 1.25%
b.
“Safest” implies lowest risk. Assuming the well-diversified portfolio is
invested in typical securities, the portfolio beta is approximately one. The
largest reduction in beta is achieved by investing the $20,000 in a stock
with the negative beta.
Est. Time: 06 – 10
20.
E(r)P = .12 = .1wa + .15(1 – wa); wa = .60; (1 – wa) = .40
σp = (.602 × .202 + .402 × .402 + 2 × .60 × .40 × .50 × .20 × .40).5
σp = .2433, or 24.33%
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Chapter 07 - Introduction to Risk and Return
Est. Time: 06 – 10
21.
a.
In general:
σP = (x1212 + x2222 + 2x1x21212).5
Thus:
P = (.52 × .35802 + .52 × .21002 + 2 × .5 × .5 × .30 × .3580 × .2100).5
P = .2331, or 23.31%
b.
We can think of this in terms of Figure 7.13 in the text, with three
securities. One of these securities, T-bills, has zero risk and, hence, zero
standard deviation. Thus:
P = [(1/3)2 × .35802 + (1/3)2 × .21002 + 2 × (1/3) × (1/3) × .30 × .3580 ×
.2100].5
P = .1554, or 15.54%
Another way to think of this portfolio is that it is comprised of one-third
T-Bills and two-thirds a portfolio which is half Bank of America and half
Starbucks. Because the risk of T-bills is zero, the portfolio standard
deviation is two-thirds of the standard deviation computed in Part (a)
above:
P = (2/3) × 23.31% = 15.54%
c.
With 50% margin, the investor invests twice as much money in the
portfolio as he had to begin with. Thus, the risk is twice that found in Part
(a) when the investor is investing only his own money:
P = 2  23.31% = 46.62%
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McGraw-Hill Education.
Chapter 07 - Introduction to Risk and Return
d.
With 100 stocks, the portfolio is well diversified, and hence the portfolio
standard deviation depends almost entirely on the average covariance of
the securities in the portfolio (measured by beta) and on the standard
deviation of the market portfolio. Thus, for a portfolio made up of 100
stocks each with Bank of America’s beta of 1.57, the portfolio standard
deviation is approximately:
P = 1.57  23.0% = 36.11%
For stocks like Starbucks, the approximate standard deviation is:
P = .83  23.0% = 19.09%
Est. Time: 11 – 15
22.
For a two-security portfolio, the formula for portfolio risk is:
σP2= x1212 + x2222 + 2x1x21212
If security one is Treasury bills and security two is the market portfolio, then 1 is
zero, and 2 is 20%. Therefore:
σP2 = x2222 = x22 × .202
σP = .20x2
Portfolio expected return = x1(.06) + x2(.06 + .085)
Portfolio expected return = .06x1 + .145x2
Portfolio
x1
x2
1
2
3
4
5
6
1.0
.8
.6
.4
.2
.0
.0
.2
.4
.6
.8
1.0
Expected
Return
.060
.077
.094
.111
.128
.145
Standard
Deviation
.000
.040
.080
.120
.160
.200
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McGraw-Hill Education.
Chapter 07 - Introduction to Risk and Return
Portfolio Risk & Return
.16
Expected Return
.14
.12
.10
.08
.06
.04
.02
.00
.00
.05
.10
.15
.20
.25
Standard Deviation
Est. Time: 11 – 15
23.
The matrix below displays the variance for each of the eight stocks along the
diagonal and each of the covariances in the off-diagonal cells:
Variances/Covariances BHP
BP
Fiat
Heineken
Korea
Nestlè
Sony
Tata
σs,mkt
BHP
0.0392040 0.0241996 0.0323983 0.0057151 0.0181841 -0.0005762 0.0168688
0.0387189
0.0218391
BP
0.0241996 0.0846810 0.0501218 0.0131241 0.0210562 0.0033872 0.0534986
0.0330049
0.0353842
Fiat
0.0323983 0.0501218 0.1854164 0.0132056 0.0227688 -0.0041768 0.0849557
0.0538905
0.0548225
Heineken
0.0057151 0.0131241 0.0132056 0.0325442 0.0085349 0.0076995 0.0299298
0.0183442
0.0161372
Korea
0.0181841 0.0210562 0.0227688 0.0085349 0.0774509 0.0002700 0.0199664
0.0141496
0.0227976
Nestlè
-0.0005762 0.0033872 -0.0041768 0.0076995 0.0002700 0.0094090 0.0100038
0.0030349
0.0036314
Sony
0.0168688 0.0534986 0.0849557 0.0299298 0.0199664 0.0100038 0.2010626
0.0333202
0.0562007
Tata
0.0387189 0.0330049 0.0538905 0.0183442 0.0141496 0.0030349 0.0333202
0.1529592
0.0434278
The covariance of BP with the market portfolio (σBP, Market) is the mean of the eight
respective covariances between BP and each of the eight stocks in the portfolio.
(The covariance of BP with itself is the variance of BP.) Therefore, σBP, Market is
equal to the average of the eight covariances in the second row or, equivalently,
the average of the eight covariances in the second column. Beta for BP is equal
to the covariance divided by the market variance, which we calculated at 0.03178
in problem 17. The covariances and betas are displayed in the table below:
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McGraw-Hill Education.
Chapter 07 - Introduction to Risk and Return
BHP
BP
Fiat
Heineken
Korea
Nestlè
Sony
Tata
Covariance
0.0218391
0.0353842
0.0548225
0.0161372
0.0227976
0.0036314
0.0562007
0.0434278
Market
Variance
0.0317801
0.0317801
0.0317801
0.0317801
0.0317801
0.0317801
0.0317801
0.0317801
Beta
0.6871943
1.1134082
1.7250607
0.5077763
0.7173556
0.1142674
1.7684269
1.3665106
Est. Time: 21 – 25
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