January 2, 2020

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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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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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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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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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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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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