January 2, 2020

Chapter 11

Diversification and Risky Asset Allocation

Concept Questions

1. Based on market history, the average annual standard deviation of return for a single, randomly

2. If the returns on two stocks are highly correlated, they have a strong tendency to move up and down

6. The common answer might be that over time volatility cancels out; however, this is incorrect and is

7. An investment with high volatility could actually reduce the risk of the overall portfolio if its

8. The importance of the minimum variance portfolio is that it determines the lower bound of the

10. If two assets have zero correlation and the same standard deviation, then evaluating the general

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Education.

Solutions to Questions and Problems

NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple

steps. Due to space and readability constraints, when these intermediate steps are included in this

solutions manual, rounding may appear to have occurred. However, the final answer for each problem is

found without rounding during any step in the problem.

Core Questions

1. .25(–.08) + .5(.13) + .25(.23) = 10.25%

4.

Calculating Expected Returns

Roll Ross

(1)

State of

Economy

(2)

Probability of

State of

Economy

(3)

Return if

State

Occurs

(4)

Product

(2) × (3)

(5)

Return if

State

Occurs

(6)

Product

(2) × (5)

Bust .40 –10% –.0400 21% .0840

5.

(1)

State

of Economy

(2)

Probability of

State of

Economy

(3)

Return Deviation

from Expected

Return

(4)

Squared

Return

Deviation

(5)

Product

(2) × (4)

Roll

Bust .40 –.2280 .0520 .0208

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Education.

6.

Expected Portfolio Return

(1)

State of

Economy

(2)

Probability of

State of

Economy

(3)

Portfolio Return if State Occurs

(4)

Product

(2) × (3)

7.

Calculating Portfolio Variance

(1)

State of

Economy

(2)

Probability of

State of Economy

(3)

Portfolio Return

if State Occurs

(4)

Squared Deviation from

Expected Return

(5)

Product

(2) × (4)

8. E[RA] = .3(.04) + .4(.09) + .3(.12) = 8.40%

9. a. boom: E[Rp] = .25(.18) + .50(.48) + .25(.33) = .3675

good: E[Rp] = .25(.11) + .50(.18) + .25(.15) = .1550

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Education.

10. Notice that we have historical information here, so we calculate the sample average and sample

standard deviation (using n – 1) just like we did in Chapter 1. Notice also that the portfolio has less

risk than either asset.

Annual Returns on Stocks A and B

Year Stock A Stock B Portfolio AB

2012 11% 21% 17.00%

Intermediate Questions

11. Boom: .35(15%) + .45(18%) + .20(20%) = 17.35%

12. E(RP) = .50(.14) + .50(.10) = 12.00%

σP

2

13.

σP

2

= .502(.422)+ .502(.312) + 2(.50)(.50)(.42)(.31)(1.0) = .13323; σP = 36.50%

σP

2

As the correlation becomes smaller, the standard deviation of the portfolio decreases. In the extreme,

14. w3 Doors =

= .33709; wDown = (1 – .33709) = .66291

σP

2

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15.

Risk and Return with Stocks and Bonds

Portfolio Weights Expected Standard

Stocks Bonds Return Deviation

1.00 0.00 12.00% 21.00%

16. wD =

17. E(RP) = .6345(.13) + .3655(.16) = 14.10%

σP

2

18. wK =

.182 - . 28 ×. 18 ×. 40

.282+. 182 - 2 ×. 28 ×. 18 ×. 40

= .1737; wL = (1 – .1737) = .8263

19. wBruin =

.572 - . 42 ×. 57 ×. 25

. 422+. 572 - 2 ×.57 ×. 42×. 25

= .6946; wWildcat = (1 – .6946) = .3054

20. E(R) = .45(12%) + .25(16%) + .30(13%) = 13.30%

σP

2

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21. wJ =

.192 - . 54 ×. 19 ×. 50

.542+. 192 - 2 ×.54 ×.19 ×.50

= –.0675; wS = (1 – (–.0675)) = 1.0675

σP

2

Even though it is possible to mathematically calculate the standard deviation and expected return of

a portfolio with a negative weight, an explicit assumption is that no asset can have a negative weight.

The reason this portfolio has a negative weight in one asset is the relatively high correlation between

is:

σmin

σmax

> . In this case,

22. Look at

σP

2

:

σP

2

2×σB

, which is precisely the expression for the

variance on a two–asset portfolio when the correlation is +1.

23. Look at

σP

2

:

σP

2

2+ 2 × x A× xB×σA×σB×(-1)

, which is precisely the expression for

24. From the previous question, with a correlation of –1:

Set this to equal zero and solve for x to get:

This is the weight on the first asset.

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25. Let stand for the correlation, then:

σP

2

=

xA

2×σA

2+ xB

Take the derivative with respect to x and set equal to zero:

Solve for x to get the expression in the text.

CFA Exam Review by Kaplan Schweser

1. b

Simply increasing return may not be appropriate if the risk level increases more than the return.

2. a

3. b

4. c

5. c

Since the beta of Beta Naught is zero, its correlation with any of the other funds is zero. Thus, the

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