# Measuring Returns Using Historical Stock Prices

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```					                               CHAPTER 17
RISK AND DIVERSIFICATION

STUDENT LEARNING 0BJECTIVES

After reading Chapter 17, students should be able to answer the following questions:

1.     What is risk aversion, and why are investors, as a group, risk averse?
2.     What are the general investment implications of risk aversion?
3.     Why is standard deviation a good measure of risk, and how does an investor
compute standard deviations for both individual securities and portfolios?
4.     What is the impact of security correlations impact portfolio risk?
5.     What are the benefits of diversification, and how investors achieve them?
6.     What is the meaning of efficient diversification and modern portfolio theory/

SUGGESTIONS FOR USE AND TEACHING TIPS

The material in Chapter 17 is conceptual and somewhat difficult for students who are
intimidated with math. Chapter 17 develops the concept of risk aversion and the
mathematics of portfolio theory. This chapter is the foundation of modern portfolio
theory (MPT) and Chapters 18, 19, 20, and 21 build on the concepts presented in
this chapter. Needless to say, this chapter is very important and we suggest one or
two full class periods be devoted to the chapter concepts and problems. If time
permits, we suggest discussing the concepts and reconfirming the knowledge by
having students work on a select problems from the End-of-chapter discussion
problems or critical thinking exercises. We found that many students learn from
working out a problem or discussion question rather than just listening to a lecture on
a difficult concept.

The question of how much to cover on risk aversion seems to evoke some
controversy. Regardless of your philosophy, it is important to stress the concept of
risk aversion early on in the development of MPT. Given that they are convinced that
most investors are risk averse, we can establish how investors will make investment
decisions and then develop the beginnings of MPT. Knowing that most investors are
risk averse allows us to make preference choices for them; they prefer higher returns
and lower risk. This lends itself to the definition of mean-variance dominance.

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Next, the mathematics to measure return and risk is introduced. The statistical
terms may be review for most students; however, if time permits a quick review may
be helpful. Of course, given time constraints, an assignment to calculate return and
risk (from the Discussion questions or Critical thinking exercises) would be just as
effective.

The essence of portfolio theory is also covered in Chapter 17. Most students accept
the idea of diversification and this chapter explains why it works. The text chooses to
emphasize the essentials without going into mathematical derivations. However, we
feel it may help promote understanding of diversification if examples are used. We
suggest that you take a problem such as 26 or 27 from the End-of-chapter exercises
and generate the investment opportunity set. Assign each row a different X%
invested in the first stock and (1-X)% invested in the second stock. Have each row
calculate the portfolio ERp and SDp. Display the results as they provide the answers
to the assignment. Graph it and show how the investment opportunity set is
generated with their numbers. Also if they see that the calculation isn’t so difficult,
they are less intimidated by the formulas.

Foremost in this chapter, students should feel comfortable with the concept of
diversification because it leads to the definition of two types of risk: market risk and
diversifiable risk. These concepts are important as it links portfolio theory to CAPM
in Chapter 18.

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LECTURE OUTLINE

I.     What is risk aversion?

A.     Risk aversion.
B.     Risk aversion and expected returns
C.     Relative risk aversion and expected returns.

II.    Measuring Risk and Return: Individual Securities

A.     Measuring returns.

1.       Ex-ante or expected returns.
2.       Ex-post or historical returns.
B.     Measuring risk.

1.       Range
2.      Number of negative outcomes
3.      Standard deviation (or variance)

C.     Calculating standard deviations.

1.      Ex-ante or expected risk.
2.      Ex-post or historical risk.

D.     Security selection

1.      Mean-variance dominance or mean-variance efficient

2.      Coefficient of variation (CV)

III.   Portfolio risk and return

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A.    Portfolio return.

1.     Ex-ante portfolio return, ERp
2.     Ex-post portfolio return, Mp.

B.     Standard deviation of a two-security portfolio.

1.       Covariance (COV(A,B))
2.      Correlation coefficient CORR(A,B)
3.      CORR(A,B) = COV(A,B)/ (SD A)(SD B)
4.      Standard deviation for a two-security portfolio.

C.     Correlation and portfolio standard deviation.
D.     Investment opportunity set for two-security portfolio.
1.     Minimum variance portfolio
E.     Standard Deviation of an N-Security Portfolio.

IV.   Diversification.

A.      Diversification across securities

B.     Two types of portfolio risk

C.     Mathematical effects of diversification

D.     Diversification across time

E.     Efficient diversification

F.     How to find an efficient frontier

V.    Implications for Investors

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INVESTMENT PROFILE BOX – Harry Markowitz

The Investment Profile in Chapter 17 is a short look at the father of Modern Portfolio
Theory, Harry markowitz. It’s just historical look at one of the important founders of
Portfolio Theory.

INVESTMENT INSIGHT BOX – Should Companies Diversify?

The Investment Insight Box in Chapter 17 discusses whether companies should
apply the theory of diversification and acquire other companies in different line of
business. While the extension sounds reasonable the article shows that investors
can diversify by investing in various firms without the corporation acquiring other
firms, especially because acquisitions require paying a premium for the target firm in
order to gain corporate control.

Questions you might want to ask the students are:

   Johnson & Johnson successfully acquired various businesses; can you think
of other firms that fit this category?

   What did firms that could not successfully diversify do? What happened to
their stock prices when they divested? PepsiCo is one example where they
divested Frito-Olay, Taco Bell, Kentucky Fry Chicken, etc.

ANSWERS TO END OF CHAPTER EXERCISES

Mini Cases

Mini- Case 1

Please refer to Instructor data disk for answers. The instructor could take this
opportunity to motivate students to use a spreadsheet such as EXCEL.
The commands in EXCEL are:

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=AVERAGE(first cell:last cell) to calculate mean return.
=STDEV(first cell:last cell) to calculate standard deviation.

Mini- Case 2

Please refer to Instructor data disk for answers. This case is an excellent out of
class assignment to motive students to use spreadsheets and to apply theoretical
concepts from Chapter 17 (particularly, risk aversion, portfolio theory and the
investment opportunity set).

Review Exercises

1. A risk neutral investor’s utility function increases at a constant rate and will gain
the same amount of utility as wealth increases while a risk averse investor’s utility
increases at a decreasing rate. For example, as wealth level increases from 10
to 20, a risk averse investor’s utility increases by 30% but a wealth level increase
from 20 to 30 increases the utility by only 13%. The risk neutral investor’s utility
function would increase by 30% for both wealth level increases.

2. Risk aversion implies that the long-term relationship between risk and return
should be positive.

3. The expected return from investing in eBay stock is:
Expected return = .40(100%) + .60(0%) = 40%
Since a 40% return on eBay investment is (much) greater than investing in T-bill
(4%), even risk averse investors may prefer eBay investment to T-bills.

4.     HPR(TOYS) = (36.625-21.75)/21.75 = 0.6839
HPR(TOOT) = -0.0779
HPR(HP) = 0.152

5.     HPR(Boe) = 0.199
HPR(Tex) = 0.273
HPR(IBM) = 0.474

6.     HPR(STAR) = 0.526
HPR(Duke) = 0.079
HPR(Dis) = -0.016

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7. The actual holding period return represents a historical return while an expected
return represents a future return using predicted stock price for a future period.
The formulas also show the differences:
Actual return = (P t – Pt-1 + DIVt)/Pt-1
Expected return = (P t+1 – Pt + DIVt+1 )/Pt

8. a. Actual return = (\$65 - \$58 + \$1.50)/\$58 = 14.66%
b. Expected return = (\$72 - \$65 + \$2.00)/ \$65 = 13.85%

9.     a. Actual return = .202
b. Expected return = .167

10. Following the procedure outlined in the text:
a. ER (Bud) = 14.5%; ER(Toy) = 22%
b. SD (Bud) = 5.7%; SD (Toy) = 9.5%
c. Portfolio return = 18.25%
d. Correlation coefficient = -0.0054/(.057)(.095) = -0.997
e. Portfolio SD = 1.95%

11.    a.   ER(JNJ) = 7.9% ER(Dis) = 33.7%
b.   SD(JNJ) = 4.89%         SD(Dis) = 10.68%
c.   ERp = (.50)(7.9%) + (.50)(33.7%) = 20.8%
d.   CORR = -.9991 or –1.0
e.   SDp = [.0005978 + .0028569 - .0012938]1/2 = 4.63%

12.  a. M(Pep) = 33.30% M(HP) = 50.92%
b. Using both equations the portfolio mean = 42.11%.
Using Equation 12.6b, ERp = (.50)(33.30%) + (.50)(50.92%) = 42.11%.
c. CORR(Pep, HP) = .55
d. Both equations should give SDp= 19.48%
2       2      2       2                                1/2
SDp = [(.5) (.1373) + (.5) (.2972) + 2(.5)(.5)(.1373)(.2972)(.835)] = 0.1948

13. ERp = (.20)(33.30%) + (.80)(50.92%) = 42.30%
SDp = [(.20)2(.1373)2 + (.8)2(.2972)2 + 2(.20)(.80)(.1373)(.2972)(.835)]1/2 = 0.253

14.    a. M(Toys) = -3.375%       M(Toot) = 10.16%
b. Both equations should give you Mp = .0746.
c. CORR(Toys, Toot) = 0.61

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d. Both equations should give you SDp = .1523

15. Mp = 8.81%              SDp = 16.67%

16.    a.   M(IBM) = 13.44% M(Tex) = 13.61%
b.   Both equations should give 13.52%.
c.   CORR(IBM,Tex) = .079
d.   Both equations should give SD = 11.59%.

17. Mp = 13.56%       portfolio standard deviation = 11.78%

18. ERp = 19%; SDp = 14.94%

19. XA = .6966 or rounded to 70%
a. ERp = 14.5%
b. SDp = 13.57%

20. The results are summarized below.
Portfolio           ERp             SDp           CV
(P1) 40% in A-60% in B     .19             .1494         .789
(P2) 70% in A-30% in B     .145            .1357         .936

Even though P2 has the minimum risk, comparing the coefficient of variation, it has
greater risk per return making P1 more attractive. However, it would depend on the
individual risk preference as to the choice between P1 and P2 if the investor does
not have the option to invest in the riskfree asset.

21.    ERp = 27% SDp = 24.2%

22. The minimum variance portfoli consists of 64% in stock J and 36% in Stock K.
.This portfolio expected return, ERp = 23.6% and portfolio standard deviation of
18.3%.

23. The 30-70% portfolio (prob 22) has a greater return, but also greater risk, so it
would depend on the investor’s risk preference as to the choice between the two
combinations.

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24. The minimun variance portfolio has 74% invested in Stock A and 26% in Stock
E.. It has a portfolio expected return of 14.1% and a standard deviation of
5.2%.

25. Recall the following data:      Stock         ER     SD
A             .12    .06
E             .20    .10

XA = 0.16/.0256 = .625 and X E = (1 – XA) = (1 - .625) = .375
ERp = 15% and SDp = 0.03%

26.      a. Estimated minimum variance portfolio has 70% in Stock X and 30% in
Stock Y.
b. The actual proportions calculated using Equation 12.12 are 67.7% in
Stock X and 32.3% in Stock Y.

27.
X in L     1-X     ERp     SDp
1.0        0.0     .13     .1500
0.9        0.1     .138    .1409
0.8        0.2     .146    .1367
0.7        0.3     .154    .1379
0.6        0.4     .162    .1442
0.5        0.5     .170    .1551
0.4        0.6     .178    .1697
0.3        0.7     .186    .1871
0.2        0.8     .194    .2066
0.1        0.9     .202    .2277
0.0        1.0     .210    .2500

a. The minimum variance proportion is approximately 80% in Stock L and 20% in
Stock M.
b. Using Equation 12.12, the minimum variance proportion is 77% in Stock L and
23% in Stock M.

28. The correlation coefficient is the only variable in the SDp formula that can be
negative and thereby reduce the overall portfolio risk.

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29. a.        Based on portfolio theory, the stock combination with the lower
correlation will provide the greatest potential to diversify. X and Z have more
diversification potential than X and Y because the correlation between X and
Z (-.35) is lower than the correlation between X and Z (+.35).
b.      Using the two-asset portfolio worksheet in the I-Wizard workbook, you
should be able to generate the investment opportunity set between X and Z,
and X and Y. The numbers should equal the ones in the table below.
c.      You should observe that no combinations of X and Y dominate any
combinations of X and Z – in other words combinations of X and Y never have
a higher return and/or lower standard deviation than combinations of X and Z.

(part of problem 29)
%                (1-X)   ER(XY)     SD(XY)      ER(XZ)   SD(XZ)
invested
in Stock X

1.0             0.0      .050       .0800       .050      .0800
0.9             0.1       .057      .0785       .057      .0682
0.8             0.2       .064      .0796       .064      .0604
0.7             0.3       .071      .0832       .071      .0583
0.6             0.4       .078      .0890       .078      .0624
0.5             0.5       .085      .0966       .085      .0716
0.4             0.6       .092      .1055       .092      .0843
0.3             0.7       .099      .1156       .099      .0992
0.2             0.8       .106      .1265       .106      .1154
0.1             0.9       .113      .1380       .113      .1324
0.0             1.0       .120      .1500       .120      .1500

30. Relevant data are shown in the table below. Graphs can be easily costructed
from these data.
Portfolio Standard Deviation
XA invested in Portfolio             A and B           A and C         A and D
Stock A         Return
1.0             10%               5.00%             5.00%           5.00%
0.9             11                5.50              5.07            3.63

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0.8               12                6.00              5.29              2.37
0.7               13                6.50              5.63              1.53
0.6               14                7.00              6.08              1.84
0.5               15                7.50              6.61              2.96
0.4               16                8.00              7.21              4.29
0.3               17                8.50              7.86              5.69
0.2               18                9.00              8.54              7.11
0.1               19                9.50              9.26              8.55
0.0               20                10.00             10.00             10.00

Portfolio AD offers the greatest diversification benefits due to the lowest (negative)
correlation coefficient. Given that all the portfolio combinations, AB, AC, and AD
have the same ERp, the one with the lowest correlation offers the best risk/return
opportunity.

31.    Using the Two-Security Portfolio worksheet in the I-Wizard workbook, you can
graph the investment opportunity sets for each portfolio JK and JL.
The graph will show that at approximately 90% in J & 10% in K and at 80% in
J & 20% in L, the ER(JK)=ER(JL) and SD(JK)=SD(JL) (.1228 vs .1259).
Beyond that JL curves in and is dominated by JK at every other investment
points. So JK dominates JL entirely and there are no points where JL
dominates JK.

32.    Market risk is the systematic risk that is inherent in the market “system” and is
the security’s risk contribution to the market portfolio. Market risk is also
called systematic risk or nondiversifiable risk. Firm-specific risk is risk that
can be diversified because it is risk related to the company only. Firm-
specific risk is also called diversifiable risk or nonsystematic risk.

33.     Disagree because the correlation coefficient is the important variable to
consider when combining securities into portfolios. Rather than just
examining ER and SD, the correlation coerricient determines the degree of
diversification benefit when securities are combined. For example, both
portfolios, AB and AC, have ERp equal to 17.5%, but the SDp for AC equals
0.47 while SDp for AB is much greater (.122). Since the ERp is the same
and AC has a lower risk than AB for an equally weighted portfolio, AC is
preferred by risk averse investors.

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34.    Disagree. Even if D has a lower SD risk, its correlation with F is higher that
the correlation between F and E. So it’s most likely that the portfolio risk is
lower for EF combinations than for DF combinations. We show that the
portfolio return equals .1920 for both portfolios DF and FE, and as expected
the portfolio risk is lower for EF (.18) compared to DF (.22).

Critical Thinking Exercises

1. Please refer to Instructor and Student data disk for answers. This exercise is a
good opportunity for students to create equations to perform the returns and in part
d, the portfolio ER and SD. If the results are somewhat different for part d, it could be
due to rounding.

2. This problem is very much like mini-case 2 where the student must calculate the
stock mean return and standard deviation. By creating ten portfolios with the two
stocks, the student will create the investment opportunity set for the two stocks. Have
them graph it on the same return-standard deviation graph from part b. Discuss how
they chose the two stocks and whether it made a difference to diversify with the two
stocks as compared to the ten individual stocks graphed in part b.

This exercise may bring out the trade-off between efficient (high return-low SD)
individual stocks and correlation between stocks. It’s possible that two stocks may
have very high returns and low SDs (risk), but very highly correlated. So combining
the two, may not be portfolio efficient. However, a stock with low return and high risk
may be a bad choice individually; however, if its correlation with another stock is low,
it may be portfolio efficient.

You should encourage your students to use the Two-Asset Portfolio worksheet in the
I-Wizard workbook. It will relieve them of some of the more tedious calculations. If
you wish to go even further, ask students to find “optimal” portfolios –those portfolios
with the highest return, for a given level of risk, or the lowest risk for a given return.
The Optimal Portfolio worksheet in the I-Wizard workbook will find the optimal
portfolio for up to a five stock portfolio combination.

EXTRA PROBLEMS

1. Today is year t, last year was t-1, and next year will be t+1. The prices and
disbursed cash dividends for a stock for these years are listed below.

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Year           Price         Dividends
t-1            \$50            \$1.25
t            \$55            \$1.50
t+1            \$67            \$1.75
a. Calculate the actual holding period return.
b. Calculate the expected return.
c. Discuss the difference between the two.

ANS: a. HPR = 0.13         b. ER = 0.25 c. Difference is that HPR is past earned
return and ER is an expected or exante return.

2. Today is year t, last year was t-1, and next year will be t+1. The prices and
disbursed cash dividends for a stock for these years are listed below.

Year           Price         Dividends
t-1            \$100           \$2.25
t            \$115           \$2.50
t+1            \$130           \$2.75
a. Calculate the actual holding period return.
b. Calculate the expected return.
c. Discuss the difference between the two.

ANS: a. HPR = 0.175         b. ER = 0.154       c. Difference is that HPR is past or
expost return and ER is an expected or exante return.

3. The following probabilities are given for each state of the economy and the
respective stock returns:

Returns
State of Economy Probability Cisco Systems                      Merck
Good                  60%                    +75%              +30%
Normal                30%                    +10%              +35%
Poor                  10%                    -20%              +40%
a. What is the expected return for each state of the economy for each stock?
b. Which stock do you think will have greater risk? Explain.
c. What is the standard deviation for each stock?

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d. Do you think these two stocks move in the same or opposite direction? Explain.
Now calculate the correlation coefficient.
e. What is the portfolio expected return from investing 50 percent in each stock?
f. What is the portfolio standard deviation for an equally weighted portfolio of the two
stocks?

ANS: a. Cisco ER = 46% Merck ER = 32.5%
b. Cisco is riskier because of the spread between returns is greater.
c. Cisco SD = 36.5%         Merck ER = 3.35%
d. The two stocks seem to move somewhat in the opposite direction though
not exactly.
CORR = -.012
e. 50% in each stock results a portfolio ERp = 39.25%
f. 50% in each stock results in a portfolio SDp = 18.3%

4. Five years of returns are given for Stocks C and D.

Year          Stock C        Stock D
1991          +40%           -15%
1992          +30            - 5
1993          +20              0
1994          +10            +10
1995          -10            +20

a. What is the actual mean return and standdard deviation for each stock?
b. Suppose you invested 80 percent in Stock C and 20 percent in Stock D.
Calculate the mean return for the portfolio using Equation 12.3 and again using
equation 12.6b.
c. Calculate the correlation coefficient between Stocks C and D.
d. If you invested 80 percent in Stock C and 20 percent in Stock D, calculate the
portfolio standard deviation using Equation 12.5 and again using Equation 12.11b.
e. Suppose you decided to invest only 30 percent in Stock C and 70 percent in D.
Using the data from problem 2, what are the mean portfolio return and the standard
deviation.
ANS: Stock C: Mc = .18 SDc = .192           Stock D: Md = -.07 SDd = .273
b. Mp = .13 for both equations.
c. CORR = -.748
d. SDp = .119 for both equations.

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e. Mp = .005 and SDp = .153

5. What is the minimum variance portfolio for problem 4? What are its mean
portfolio return and standard deviation?

ANS: Minimum variance portfolio for CD is approximately 68% in C and 32% in D.
Mp = (.68)(.18) + (.32)(-.07) = .10
SDp = .087 with 68% in Stock C and 32% in Stock D.

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