Chapter 11
Diversification and Risky Asset
Allocation
Slides
11-1. Chapter 11
11-2. Diversification and Risky Asset Allocation
11-3. Learning Objectives
11-4. Diversification
11-5. Diversification and Asset Allocation
11-6. Expected Returns, I.
11-7. Expected Return, II.
11-8. Expected Return, III.
11-9. Calculating Expected Returns
11-10. Expected Risk Premium
11-11. Calculating the Variance of
Expected Returns
11-12. Example: Calculating Expected Returns and Variances: Equal State
Probabilities
11-13. Expected Returns and Variances,
Starcents and Jpod
11-14. Calculating Expected Returns Unequal Probabilities
11-15. Portfolios
11-16. Portfolios: Expected Returns
11-17. Example: Calculating Portfolio Expected Returns
11-18. Variance of Portfolio Expected Returns
11-19. Example: Calculating Variance of Portfolio Expected Returns
11-20. Example II: Calculating Variance of Portfolio Expected Returns
11-21. Diversification and Risk, I.
11-22. Diversification and Risk, II.
11-23. The Fallacy of Time Diversification, I.
11-24. The Fallacy of Time Diversification, II.
11-25. The Fallacy of Time Diversification, III.
11-26. The Fallacy of Time Diversification, IV.
11-27. The Very Definition of Risk—a Wider Range of Possible Outcomes from
Holding Equity
11-28. So, Should Younger Investors Put a High Percent of Their Money into
Equity?
11-29. Why Diversification Works, I.
11-30. Why Diversification Works, II.
11-31. Why Diversification Works, III.
11-32. Why Diversification Works, IV.
11-33. Why Diversification Works, V.
11-34. Calculating Portfolio Risk
11-35. The Importance of Asset Allocation, Part 1.
11-36. Correlation and Diversification, I.
11-37. Correlation and Diversification, II.
11-38. Correlation and Diversification, III.
11-39. More on Correlation and the Risk-Return Trade-Off (The Next Slide is an
Excel Example)
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Diversification and Asset Allocation 11-2
11-40. Example: Correlation and the Risk-Return Trade-Off, Two Risky Assets
11-41. The Importance of Asset Allocation, Part 2.
11-42. Risk and Return with Multiple Assets, I.
11-43. Risk and Return with Multiple Assets, II.
11-44. The Markowitz Efficient Frontier
11-45. Useful Internet Sites
11-46. Chapter Review, I.
11-47. Chapter Review, II.
Chapter Organization
11.1 Expected Returns and Variances
A. Expected Returns
B. Calculating the Variance of Expected Returns
11.2 Portfolios
A. Portfolio Weights
B. Portfolio Expected Returns
C. Portfolio Variance of Expected Returns
11.3 Diversification and Portfolio Risk
A. The Effect of Diversification: Another Lesson from Market
History
B. The Principle of Diversification
C. The Fallacy of Time Diversification
11.4 Correlation and Diversification
A. Why Diversification Works
B. Calculating Portfolio Risk
C. The Importance of Asset Allocation, Part 1
D. More on Correlation and the Risk-Return Trade-Off
11.5 The Markowitz Efficient Frontier
A. The Importance of Asset Allocation, Part 2
11.6 Summary and Conclusions
Selected Web Sites
www.thestock411.com (to find expected returns)
www.investopedia.com (for more on risk measures)
www.teachmefinance.com (also contains more on risk measure)
www.morningstar.com (measure diversification using “instant x-ray”)
www.moneychimp.com (review modern portfolio theory)
www.efficientfrontier.com (check out the online journal)
www.wolframalpha.com (work the web box: to find efficient portfolio)
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McGraw-Hill
Education.
Diversification and Asset Allocation 11-3
Annotated Chapter Outline
11.1 Expected Returns and Variances
A. Expected Returns
Expected Return: the weighted-average of possible future returns on a
risky asset.
The expected return on a security is equal to the sum of the possible returns
multiplied by their probabilities, or the weighted average of all the possible
returns of the security. The risk premium is the difference between this expected
return and the risk-free weight.
n
1i
ii
RpRE
B. Calculating the Variance of Expected Returns
To find the variance of expected returns, first determine the squared deviations
from the expected return. Then multiply each possible squared deviation by its
probability. Finally, sum these up, and the result is the variance.
n
1s
2
iss
2
ii
REREpσRVAR
Lecture Tip: It should not be surprising that students have problems with
expected return and variance, even thought statistics usually is a prerequisite for
business finance, and business finance is usually a prerequisite for investments.
The half-life of statistical prowess is quite short. Therefore, it is vital to point out
that the expected returns and variances calculated in this chapter are projected
future returns, as compared to Chapter 1 where the returns and variances are
based on historical returns. If this difference is not emphasized to students, it is
sometimes difficult for them to remember the correct application. When
presenting examples of return and variance, it is usually better to show examples
using unequal probabilities (such as example 11.2 in the text). This may require a
few more steps, but it makes the concept less confusing for students in the long
run. Otherwise, students sometimes mistakenly think they can always take the
simple average of returns to obtain the expected return.
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McGraw-Hill
Education.
Diversification and Asset Allocation 11-4
11.2 Portfolios
A Portfolio is a group of assets, such as stocks and bonds, held by an investor.
Portfolios are often described by portfolio weights.
A. Portfolio Weights
A Portfolio Weight is the proportion, or fraction, of the total value of the portfolio
invested in a particular asset. However, portfolio weights are often expressed as
percentages. For example, if Molly has three-fifths of her wealth invested in
domestic equities, one-fifth in foreign bonds, one-tenth in foreign equities, and
one-tenth in cash, her portfolio weights are .60, .20, .10, and .10, or 60%, 20%,
10%, and 10%, respectively.
It is important to remember that in calculating a portfolio return or standard
deviation, use the fractions, not the percentages.
B. Portfolio Expected Returns
The portfolio expected return is the weighted average combination of the
expected returns of the assets in the portfolio.
n
1i
iiP REwRE
C. Portfolio Variance of Expected Returns
The portfolio standard deviation (variance) is not the weighted average of the
individual security’s standard deviations (variance). The portfolio variance of
expected returns is calculated as follows:
n
1s
2
pss
2
pp REREpσRVAR
11.3 Diversification and Portfolio Risk
A. The Effect of Diversification: Another Lesson from Market History
Table 11.7 shows how the standard deviation of a portfolio of securities declines
as more securities are added to the portfolio. Notice that the risk decreases
rapidly as the first ten stocks are added to the portfolio, but the diversification
effects taper off when about 20 or more securities are added to the portfolio.
Notice that this is based on an equally weighted portfolio of randomly selected
NYSE securities. Figure 11.1 graphs this relationship and dramatically shows the
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McGraw-Hill
Education.
Diversification and Asset Allocation 11-5
rapid decrease in risk and the tapering off after investing in twenty or more
securities.
Remember, though, this assumes randomly selected securities. If you have 30
stocks all from the same industry, you have not eliminated as much risk.
B. The Principle of Diversification
Principle of Diversification: Spreading an investment across a number
of assets will eliminate some, but not all, of the risk.
Figure 11.1 shows how diversifying across many securities eliminates some of
the risk of individual securities. This figure also gives the alternative names for
diversifiable risk (unsystematic and company-specific) and non-diversifiable risk
(systematic and market). The two important points are:
Forming portfolios can eliminate some of the risk associated with
individual assets.
There is a minimum level of risk that cannot be eliminated by simply
diversifying.
C. The Fallacy of Time Diversification
Many people, including some investment professionals, assume that a longer
time horizon makes equity less risky since volatility “should average out.”
However, deviation increases at the square root of time, so more time actually
equals more risk.
This does not mean a person shouldn’t have more equity when they are young. It
does mean, though, that the reason is different. Specifically, a big loss at a young
age can be offset by change in work habits and other factors, which is difficult for
older workers to do.
11.4 Correlation and Diversification
A. Why Diversification Works
Correlation: The tendency of the returns on two assets to move together.
Correlation measures the tendency of two stocks’ returns to move together. It is
written Corr(RA, RB) or A,B. Facts about correlation:
-1.0 +1.0
Perfect positive correlation: +1.0 and gives no risk reduction
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McGraw-Hill
Education.
Diversification and Asset Allocation 11-6
Perfect negative correlation: -1.0 and gives complete risk reduction
Correlation between -1.0 and +1.0 gives some, but not full, risk reduction
B. Calculating Portfolio Risk
Investment Opportunity Set: Collection of possible risk-return
combinations available from portfolios of individual assets.
Efficient Portfolio: A portfolio that offers the highest return for its level of
risk.
For a portfolio of two assets, the portfolio variance is:
For a portfolio of three assets, the portfolio variance is:
Note that six terms appear in this equation. There is a term involving the squared
weight and the variance of the return for each of the three assets (A, B, and C)
as well as a cross-term for each pair of assets. The cross-term involves pairs of
weights, pairs of standard deviations of returns for each asset, and the
correlation between the returns of the asset pair.
If you had a portfolio of six assets, you would have an equation with 21 terms. If
you had a portfolio of 50 assets, the equation for the variance of this portfolio
would have 1,275 terms!
The equation illustrates that purchasing a security with a high volatility does not
necessarily mean the portfolio will become more risky. The opposite could
actually happen if the correlation of the risky security to the current portfolio was
low enough.
C. The Importance of Asset Allocation, Part 1
To illustrate why correlation and asset allocation are important, practical, real-
world considerations, suppose that as a very conservative, risk-averse investor
you decide to invest all of your money in a bond mutual fund. Based on your
analysis, you think this fund has an expected return of 6 percent with a standard
deviation of 10 percent per year. A stock fund is available, however, with an
expected return of 12 percent, but the standard deviation of 15 percent is too
high for your taste. Also, the correlation between the returns on the two funds is
about .10.
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McGraw-Hill
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σP
2=xA
2σA
2+xB
2σB
2+2xAxBσAσBCorr (RARB)
σP
2=xA
2σA
2+xB
2σB
2+xC
2σC
2+2x AxBσAσBCorr(RARB)
+2xAxCσAσCCorr (RARC)+ 2xBxCσBσCCorr(RBRC)
Diversification and Asset Allocation 11-7
Is the decision to invest 100 percent in the bond fund a wise one, even for a very
risk-averse investor? The answer is no; in fact, it is a bad decision for any
investor. To see why, Table 11.11 shows expected returns and standard
deviations available from different combinations of the two mutual funds. In
constructing the table, we begin with 100 percent in the stock fund and work our
way down to 100 percent in the bond fund by reducing the percentage in the
stock fund in increments of .05.
Beginning on the first row in Table 11.11, we have 100 percent in the stock fund,
so our expected return is 12 percent, and our standard deviation is 15 percent.
As we begin to move out of the stock fund and into the bond fund, we are not
surprised to see both the expected return and the standard deviation decline.
However, the standard deviation falls only so far and then begins to rise again. In
other words, beyond a point, adding more of the lower risk bond fund actually
increases your risk!
Figure 11.5 plots the various combinations of expected returns and standard
deviations. Note that the returns plot on a smooth curve (in fact, for the
geometrically inclined, it’s a hyperbola—if we were plotting returns versus
variance, the curve would be a parabola).
Now we see clearly why a 100 percent bonds strategy is a poor one. With a 10
percent standard deviation, the bond fund offers an expected return of 6 percent.
However, Table 11.11 shows us that a combination of about 60 percent stocks
and 40 percent bonds has almost the same standard deviation, but a return of
about 9.6 percent. Comparing 9.6 percent to 6 percent, we see that this portfolio
has a return that is fully 60 percent greater (6% × 1.6 = 9.6%) with approximately
the same risk. Our conclusion? Asset allocation matters.
D. More on Correlation and the Risk-Return Trade-Off
Figure 11.5 shows how the shape of the investment opportunity set is a
hyperbola. The shape of this curve changes as the correlation changes. Figure
11.6 shows how the shape of this curve varies. The extremes occur at Corr =
+1.0 and Corr = -1.0. At a correlation of +1.0 the portfolios lie on a straight line
between the two stocks. At a correlation of -1.0 the portfolios lie on two straight
lines; one connecting Stock A to a return with zero standard deviation (Y axis),
and another connecting Stock B to the same point with zero standard deviation
(Y axis). This shows how two securities with a correlation of -1.0 can be
combined to give a portfolio with zero risk. To calculate the weight of asset A in
the minimum variance portfolio:
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xA
¿=σB
2−σAσBCorr (RARB)
σA
2+σB
2−2σAσBCorr (RARB)
Diversification and Asset Allocation 11-8
11.5 The Markowitz Efficient Frontier
Markowitz Efficient Frontier: The set of portfolios with the maximum
return for a given standard deviation.
A. The Importance of Asset Allocation, Part 2
We can illustrate the importance of asset allocation using only three assets by
using mutual funds. A mutual fund that holds a broadly diversified portfolio of
securities counts as only one asset. So, with three mutual funds that hold
diversified portfolios, we can construct a diversified portfolio with three assets.
Figure 11.7 shows the result of combining U.S. stocks, U.S. bonds, and foreign
stocks in to one portfolio. The result is the Markowitz efficient frontier, which
represents the set of portfolios with the maximum return for a given standard
deviation. This figure shows how asset allocation and diversification matter.
The Markowitz analysis is not usually extended to a large number of assets
because of the data requirements. The inputs into the analysis include:
Expected returns on all assets;
Standard deviations on all assets; and
Correlations between every pair of assets.
If we do an analysis with 2,000 securities, this would encompass almost 2 million
unique pairs of assets!
11.6 Summary and Conclusions
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McGraw-Hill
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