Chapter Six
Why Diversification is a Good Idea
KEY POINTS
This chapter augments chapter six and shows logically (rather than mathematically) why
diversification is a good idea. Portfolio risk depends on both the risk of the individual
components and their interactions. Portfolio programming seeks to find the least risky
way of achieving a particular level of expected return.
Dominance is a central idea introduced here. Investors do not voluntarily accept
additional risk unless they expect to be rewarded for doing so. The work of Harry
Markowitz is a featured attraction of any portfolio course, and it is covered in this
chapter. The development of the efficient frontier is a central idea in portfolio theory.
The Evans and Archer research on naive diversification is arguably one of the most
important discoveries in portfolio construction. This should be a memorable point that all
students take away from the course.
The chapter concludes with a discussion of the capital asset pricing model, the estimation
of beta via a scattergram, and a brief discussion of the arbitrage pricing theory and its
prospects.
TEACHING CONSIDERATIONS
Students must fully understand the idea of dominance after finishing this chapter.
Dominance underlies most of finance theory and it should become second nature to those
anticipating a career in finance.
It is useful to use the random number generator on Lotus 1-2-3 or Microsoft Excel to
show the effects of naive diversification. Prepare five columns of random numbers, and
then form two, three, four, and five security equally weighted portfolios. Only bad luck
will keep the portfolio variance from declining as the number of securities increases. Do
this before class and put the results on a set of overheads. You also can prepare a figure
like Figure 6-9 from the data. Everyone should fully understand the implications of the
Evans and Archer research and be able to explain why it really isn't necessary for
everyone to hold the market portfolio.
It is also a good idea to stress that securities may have a negative expected return and be
properly priced, as shown in Figure 6-7. The risk-reduction benefits of negative betas are
analogous to the purchase of fire insurance on a house or collision insurance on a car. If
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Chapter Six
Why Diversification is a Good Idea
the value of a house goes down because of a fire, the value of having the insurance policy
goes up. A homeowner does not feel bad at the end of the year if there was no fire and
the insurance policy was unused. The same idea holds true with financial assets.
ANSWERS TO QUESTIONS
1. Diversification is a good idea because, properly done, it enables the investor to reduce
risk while not sacrificing any expected return. Expected utility is a negative function
of risk and a positive function of expected return, so a rational, risk averse investor
will always diversify if possible.
2. Your ego can make you reluctant to admit a mistake or cause you to assume a greater
degree of risk than you really should.
3. A significant success with a first investment can cause the investor to get cocky and
lose perspective on the nature of risk. Too much success early makes it all seem easy.
4. Risk that can be diversified away is unnecessary, and the investor is not rewarded for
bearing it. The market prices systematic risk only.
5. Risk from Security A, from Security B, and from the interaction of Securities A and
B.
6. The variance of a well-diversified portfolio equals the portfolio's beta squared times
the variance of the market.
7. These securities could be used to lock in a certain return with no risk. Whether this is
advantageous or not depends on the alternatives.
8. They should have the same level of systematic risk.
9. A gadget costs 34 cents or four for a dollar. No one should ever buy three; the
purchase of four dominates the purchase of three.
10. It shows that it is not necessary to invest in the market portfolio. Very good risk
reduction can be achieved with a relatively small portfolio.
11. Selecting securities whose returns are poorly correlated reduces portfolio risk.
Securities in the same industry are likely to be highly correlated.
12. Portfolio theory holds that only systematic risk is rewarded, and beta measures
systematic risk.
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Chapter Six
Why Diversification is a Good Idea
13. There may be multiple security portfolios with a lower variance and the same
expected return as the single security with the lowest variance. If this is the case, the
single security is dominated and does not lie on the efficient frontier.
14. There is no way to get a higher expected return. Despite its risk, no other security
dominates this one.
15. Risk reduction benefits usually accrue anytime a security is added because securities
are generally less than perfectly correlated. If, however, the securities have widely
different standard deviations, this may not be the case.
16. Most large institutional investors are precluded from placing more than a certain
percentage of their money into any single security. The lower this percentage, the
more securities that must be purchased.
17. It increases.
18. The efficient frontier extends from the riskfree lending rate to the market portfolio,
following the curve until it reaches the tangent point to the line extending from the
borrowing rate to the efficient frontier for risky assets. The efficient frontier then
follows the extension of the borrowing rate line.
19. Naive diversification is unable to guarantee that expected return will be at or above a
specified level unless the security universe is carefully selected to include only
securities with an expected return of the desired level or greater.
20. They reduce the volume of computer output and show the relative merits of the
portfolio components available.
21. Because the standard deviation of a constant is zero and division by zero is not
allowed, the correlation between a random variable and a constant is also undefined.
22. The expected return of the market periodically changes as the general level of
optimism in the marketplace changes, and as the merits of investment alternatives
change (for psychological or other reasons).
23. The CML is measured against variance, while the SML is measured against beta.
24. Beta sometimes changes as the firm's characteristics change (such as its degree of
financial leverage) or the risk of its future earnings changes.
25. The market model uses past information to estimate beta, while the CAPM is a
theoretical statement about future expected returns.
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Chapter Six
Why Diversification is a Good Idea
26. The statement is true. The actual beta is unobservable; empirical evidence is used to
estimate it.
27. Availability of data and the ultimate use of the beta. There is some evidence that
daily betas contain too much noise and should be avoided.
28. Normative theories that are developed without a positive underpinning amount to ad
hoc theorizing that have very little to do with reality. Theories should be used to test
hypotheses, and hypotheses should originate from observation of actual events,
directly or indirectly.
29. This is a subjective consideration that cannot be globally answered.
30. The CAPM uses a single “beta” statistic, while the APT uses several.
31. The problem with the logic in this statement is that the objective is not to simply
maximize the expected return because risk must be taken into account. The optimal
combination of expected return and risk would be a more appropriate objective.
Borrowing more money does increase the financial risk of a business, which explains
why beta increases. Higher risk, however, increases the expected return the
shareholders require. Investors would best be served by an optimal combination of
the two.
32. If the risk-free rate increases, the optimal point (the point of tangency between the
CML and the efficient frontier) would move up the efficient frontier. Since stock B
has the higher standard deviation (as well as the higher expected return), a higher
percentage of stock B would be contained in the optimal portfolio. Thus, an increase
in the risk-free rate, while holding the percentage of the risk-free asset constant,
would require rebalancing the portfolio to include a slightly higher percentage of
stock B and a corresponding lower percentage of stock A.
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Chapter Six
Why Diversification is a Good Idea
ANSWERS TO PROBLEMS
1.
~
~
~
a. E ( Rp )  xA E ( RA )  xB E ( RB )
= .5(.12) + .5(.13) = .125 = 12.5%
 p  x A  A  xB  B
b.
= .5(1.10) + .5(1.20) = 1.15
2.
A 
~ ~
cov( A, M )
 M2
~ ~
cov( A, M )   A M2
= 1.10 x .0002 = 2.2 x 10 –4
3.
 2p  xA2 A2  xB2 B2  2 xA xB  AB A B
= (.5)2(.021)2 + (.5)2(.029)2 + 2(.5)(.5)(.6)(.021)(.029) =
= (1.10 x 10-4) + (2.10 x 10-4) + (1.83 x 10-4) = 5.03 x 10-4
4.
5.
1.83x10 4
= 36.4%
5.03x10  4
 2p  xA2 A2  xB2 B2  xC2 C2  xA xB  AB A B  xA xC  AC A C  xB xC BC B C
min  2p
subject to
xA + xB + xC = 1.0
xA, xB, xC > 0 (optional)
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Chapter Six
6.
Why Diversification is a Good Idea
~
~
~
E ( Rp )  xA E ( RA )  xB E ( RB )
= .5(8%) + .5(13%) = 10.5%
 2p  xA2 A2  xB2 B2  2 xA xB  AB A B
= 0  x B B2  0
= (.5)2(.029)2= 2.10 x 10-4
7. Student response.
8. The minimum variance combination is 61.02% Stock C and 38.98% Stock E.
9. The minimum variance combination is 36.19% Stock A and 63.98% Stock D.
10. From Chapter 5:
 B2   A B  AB
Min variance xA = 2
 A   B2  2 A B  AB
A, B > 0 if AB > 0
For xB = 0, xA = 1.0
If this is true,
 A2   B2  2 A B  AB   B2   A B  AB
 A2  2 A B  AB   A B  AB
 A2   A B  AB  0
 A2   A B  AB
 A   B  AB which is possible
Therefore the statement is true.
11. Student response.
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Chapter Six
Why Diversification is a Good Idea
12. (1) Form a portfolio of 50% SEC1 and 50% SEC2. Its beta is 1.10 and its expected
return is 11%.

~
(2) Buy SEC3   1.10, E ( R )  13%

(3) Sell short the SEC1/SEC2 portfolio
This leaves you with a beta of zero and an expected return of 2%.
13.  p  (.26).5  .5099
xxz = (.20).5 = .4472
.4472
 .8771
.5099
 < .8771; therefore, yes
14. Student response.
15. Student response.
16. Regression output:
Constant
Std Err of Y Est
R Squared
No. of Observations
Degrees of Freedom
X Coefficient
Std Err of Coef.
~
~
-0.00298
0.007314
0.950234
10
8
0.763740
0.061794
17. E ( R )  R f  E RM  R f

= .08 + 1.23[.14 - .08] = 15.38%

~
18. 14%  .07  1.00 E ( RM )  .07
 E ( R~ )  14%

~
E (R )  .07  2.00[.14  .07]  21%
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Chapter Six
19. a. Regression output:
Constant
-0.00540
Std Err of Y Est
0.010474
R Squared
0.933100
No. of Observations
5
Degrees of Freedom
3
X Coefficient
0.783422
Std Err of Coef.
0.121110
b. Regression output:
Constant
0.000095
Std Err of Y Est
0.003554
R Squared
0.988584
No. of Observations
5
Degrees of Freedom
3
X Coefficient
0.722929
Std Err of Coef.
0.044850
20. Student response.
21.  = 0.9833 +/- 2(.3354)
.3125 <  < 1.6541
The beta estimate is not very reliable.
22. 1.22 +/- 2(.04)
1.14 <  <1.30
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Why Diversification is a Good Idea